---



HHH. a TaK»re ocooeHHOCTen a
ropHTMOB B
qHCJIeHHn nOSBOJIHeT nPH
pememlB oceCHMMeTpH'\IHbIX sa.Ilaq HCnOJIbSOBaTb OqeBll.l\H
e aHaJIorH
H TaKHM oopaSOM Hs6eraTb rpOMOS.IlKHX BbIKJIaJ\OK.
npaKTllqeCKH Bce npHBe.IleHH
e B paOOTe OO
He nOCTpoeHHH
HJIJIlOCTPHpYJOTCH KOHKpeTHbiMIJ npHMepaMllpaCqeTOB. CaMa TeXHHKa
npOBe.IleHllH paCqeTOB .IlOBOnbHO npOCTa, n03TOMY OCHOBHoe BHllMaHHe
Y.IleneHO aHMHSY peSYJIbTaTOB. HCKJIJO'leHHeM HBJIHJOTCH JIJIlllb npHMe-
pbI nOCJIe,l\Hen rJIaBbi. KOTop
e nOKaS
BaJOT peaJIHsyeMocTb npe.l\J10>KeH-
Horo MeTOJla.


BBE.LJ;EHIIE



 1. IIocTaHoBRa I'paHH'IHbIX 3a;a;a'l TeopHH yrrpyroCTH
TeopHH ynpymcTH K8K C8MOCTOHTeJIbHbIM pas.n:eJI MeXaHHKH cnJIOlll-
HbIX Cpe.n: CJIORUIJIaCb B nepBOH nOJIOBHHe XIX BeKa B pesyJIbTaTe YCH-
JIHH BbI.ll:alOll.{HXCH yqeHbIX, cpe.n:11 KOTOpbIX B nepBYIO oqepe.n:b CJle.n:yeT
Ha3BaTb K. n. HaBbe, A. JI. KOIIIl1, C. TIyaccoHa, r. JIaMe H
5. D. E. KJIaneHpoHR. EJIaro.n:apH HX p860TaM S8.n:aqa onpe.n:eJIeHHH
BHYTpeHHHx YCHJIHM H nepeMell.{eHHH B TBep.n:OM TeJIe npH ero .n:
op-
MHpOBaHHH COrJIaCHO saKoHY rYK8 CBe.n:eHa K rpaHHqHOH sa.n:aqe M8-
reMaTHqeCKOH <):JHSHKH. TIpH 3TOM HsyqeHHe COCTOHHHH ynpyrom TeJI8.
Haxo.n:Hll.{emCH no.n: .n:eHcTBHeM BHellIHHX CHJI H onpe.n:eJIeHHbIM o6pasoM
3aKpenJIeHHOrO, OCHOBHB8eTCH H8 HCCJIe.n:OBaHHH peIIIeHHH HeKoTopoH
CHCTeMbI ypaBHeHHH B qaCTHblX npoHsBo.n:HbIX. Heo6xo.n:HM:ble .n:JIH KOHK-
peTHSaIJ.HH pellIeHHH rpaHHqHble YCJIOBHH OTpaR<alOT xapaKTep BsaHMO-
.n:eiicTBHH paccMaTpHBaeMoro TeJIa c oKpYR<aIOIl.{HMH TeJIaMH H <):JOPMY-
JIHPYIOTCsr 06blqHO B BH.n:e CooTHOllIeHI1M, sa.n:alOIIJ,HX Ha rp8HI1IJ.e CMe-
ll.{eHHH, Hanpsr1KeHHH HJIH HX KOM6HHaIJ.mL
HcXO.n:HYIO CHCTeMY .n:H<):J<):JepeHIJ.HaJIbHbIX ypaBHeHHM M01KHO c<):Jop-
MYJIl1pOBaTb OTHOCHTeJIbHO BeKTopa CMeIIJ,eHHM HJIH TeHsopa Hanpsr-
j{{eHHM. B S8BHCl1MOCTH OT 3Toro paSJIHTlalOT nOCTaHOBKY sa.n:aqH TeopHH
ynpymcTH B CMell.{eHHHX HJIH HanpH1KeHHHx.
Ey.n:eM HCXO.n:HTb HS nOCTaHOBKH rpaHHqHbIX sa.n:aq TeopHH
ynpyrocTH B CMell.{eHHHX. BeKTop ynpyrHx cMern.eHHH U onpe.n:eJIsreTcsr HS
BeKTopHoro ypaBHeHHH paBHoBecHH (.n:BHR<eHHH) JIaMe [1241:


m -..1. ( . P a
u
2 m _ 2 grad dlv U - rot rot U = 0 . G at'J. ! ,


(1.1)


r.n:e m - qHCJIO TIyaccoH8, G - MO.n:YJIb c.n:BHra, p - nJIOTHOCTb
MaTepH8JIa, f - BpeMH. B 3TOM ypaBHeHHH 06'beMHble CI1JIbI onYIIJ,eHbI,
o.n:HaKO yqeCTb HX BJIHHHHe npH HeCBHsaHHblX nOJIHX JIerKO.
3HaH BeKTop ynpymx CMell.{eHHH B TeJIe, MOJE;HO onpe.n:eJIHTb Bce
BeJIHqHHbl, xapaKTepHsYlOll.{He em HanpH1KeHHo-.n:e<):JOpMHpoBaHHoe co-
CTOHHHe. TIpH3TOM y.n:06HO HCnOJIbSOBaTb npe.n:cTaBJIeHHe BeKTopa
CMell.{eHHM qepes em npoeKIJ.I1H u i (i = I, 2, 3) Ha TpH e.n:HHHqHbIX opTa
e i (i = I, 2, 3), onpe.n:eJIHIOIIJ.He opTOrOHaJIbHYIO KpHBOJIHHeMHYIO CH-
cTeMY Koop.n:HJ-IaT ql' q2' q3:
U = e1u 1 + e2u2 + e:.JU 3 ' (1.2)
7




0603HaqHM qepe3 Hi (i = 1, 2, 3) cooTBeTCTBYIOIlI.He napaMeTpbI
.naMe Koop.n.HHaTHOH CHCTeMbI. Tor.n.a ,n:JIH onpe,n:eJIeHIHI KOMnOHeHTOB
TeH30pa .n.eepopMaILHH qepe3 nepeMeIll.eHH5I HcnOJIb3yeM CJle,n:YIOIlI.He epop-
MY JIbI:


8 = 10 1 + 10 2 + 10 3 .
flapaMeTpbI, Bxo,n:HIlI.He B BblpameHHH (1.3) H (1.4), HMelOT COOTBeT-
CTBeHHO CJIe,n:YIOIlI.He 3HaqeHHH B ,n:eKapToBoH, ILHJIHH,n:plIIlecKoH H cepe-
pHqeCKOH CHCTeMax Koop,n:HHaT:
ql = X, q2 = y, q3 = Z, HI = H 2 = H3 = 1;
ql=Z, q2=r, q3=8, H 1 = 1, H 2 = 1, H3=r;
ql=f, q2=t1, q3=qJ, H 1 = 1, H 2 =r, H 3 =rsmit.


(1.5)
(1.6)
( 1.7)


r,n:e n - OpT BHelllHeH HOpMaJIH K rpaHHIlHOH nOBepXHOCTH S TeJIa,
Fn - BeKTOp BHelllHHX HarpY30K Ha ?OBepXHOCTb S. YCJIOBHe (1.2)
onpe.n.eJIHeT BeKTOp ynpyrHx CMeIll.eHHH C TOqHOCTblO .11.0 nepeMeIll.eHHH
ynpyroro TeJIa KaK meCTKoro ueJIoro.
CqHTaH BeKTop
A (qt, q2' q3) = e1A 1 + e 2 A 2 + e 3 A 3 (1.9)
H CKaJIHp cD 3a.n.aHHbIMH epYHKILH5IMH Koop,n:HHaT, Bxo.n.HIlI.He B ypaB-
HeHHH (1.1) H rpaHHqHble YCJIOBHH (1.8) .n.HepepepeHILHaJIbHble onepaTopbI
MomeM npe,n:cTaBHTb B CJIe,n:YIOIUeM BHlle fII8]:
1 a<D
(grad cD)q; = Hi Oqi (i = 1, 2, 3),
1 { a 0 a }
div A = H _ a (H2H3AI) + ---a- (H 1 H 1 A 2 ) +
 (H 1 H 2 A 3 ) ,
HIH2 3 q] q2 qs
(rot A)q, -- 1I2 1H 3 r a
2 (H3 A 3) -- d
s (H 2 A 2 )] , (1.10)
(rot A)q, = H:H 1 [ a
3 (HIAI) - O
l (H3 A 3) I '
(rot A)q, = H:H 2 [ a
l (H 2 A 2 )- a
s (H1A 1 )].
B CJIyqae BTOpOH rpaHHqHOH 3a.n.aqH BeKTOp ynpyrHx CMeIll.eHHH
o,n:H03HaqHO onpe,n:eJJHeTCH YCJIOBHeM paBeHcTBa ero Ha nOBepxHocTH
TeJIa HeKoTopoMY 3a,n:aHHoMy BeKTOpy Un:
U Is = Un" (1. 11)
OCHOBHaH CMelllaHHaH rpaHH4HaH 3a,n:aqa COOTBeTcTByeT TOMY
CJIyqalO, Kor,n:a Ha qaCTH SJ rpaHHqHOH nOBepXHOCTH TeJIa S 3a.n.aH!,I
YCHJIllH, a Ha OCTaJIbHOH qaCTH rpaHHILbI S2 - BeKT
p CMeIll.eHHH.
1-13 B03MomHbiX cpopMYJIHpOBOK rpaHHqHbIX YCJIOBHH YKameM Ha
YCJIOBHH rJIa,n:Koro KOHTaKTa, B03HHKalOIlI.He npH ,n:eepopMHpoBaHHH
ynpyrora TeJIa a6COJIIOTHO TBep,n:bIM rJJa,n:KHM TeJIOM [62]. flpH 3TOM Ha
nOBepxHocTH KOHTaKTa 3a,n:aIOTCH HOpMaJIbHble COCTaBJIHIO
He BeKTopa
CMeIll.eHHH H KaCaTeJIbHble COCTaBJIHIOIlI.He BeKTopa YCHJIHH. PH,n: ,n:py-
rHX nOCTaHOBOK rpaHHqH
X 3a,n:aq TeopHH ynpyrocTH H 060cHoBaHHe
HX KoppeKTHocTH YKa3aHbI B pa60Te [176]. B KHHre OCHOBHoe BHHMaHH

6y,n:eT y,n:eJIeHO paccMOTpeHHIO npaKTH4ecKH HaH60JIee BamHbJX nepBoH
H OCHOBHOH cMelllaHHoH rpaHH4HbJX 3a,n:aq.
rpaHH4H
e 3a,n:aqH TeopHH ynpyrocTH HBJIHIOTCH O,n:HHMH H3 HaH-
60JIee Tpy,n:HbIX B MaTeMaTHqeCKOM OTHOllleHHH rpaHH4HbJMH 3a.n.a4aMH
MaTeMaTHqeCKOH epH3HKH H OTHOCHTCH K THny HeHMaHoBcKHX BeKTOpHbIX
rpaHH4HbIX 3a,n:a4 [132]. PellleHHe HX conpHmeHo co 3HaqHTeJIbHbIMH
MaTeMaTHqeCKHMH Tpy,n:HOCTHMH. 3,n:ecb B03HHKalOT ,n:Be TeCHO CBH3aH-
HbIe Mem,n:y C060H np06JIeMbI - H3yqeHHe xapaKTepa epYHKILHH, y,n:OB-
JIeTBOpHIOIlI.HX OCHOBHbIM ,n:HepepepeHILHaJIbHbJM ypaBHeHHHM, H onpe-
.n.eJIeHHe npOH3BOJIbHbIX nOCTOHHHbIX H epYHKILHH, BXO,n:5JIlI.HX B Ha60pbl
qaCTHbJX pellleHHH, no 3a,n:aHHbIM rpaHH4HbIM YCJIOBHHM.
9


I


10 -
 ( OUI +..!!L aH I +
 aH 1 )
1 - HI aql H 2 aq2 H3 aq3 '
10 _
 ( aU2 +
 aH 2 +
 aH2 ) ,
2 - H 2 aq2 HI oqJ Hs Oq3
10 3 =
 ( aus +..!!L aHs + --.!2 aH S ) , (1.3)
H3 aqa HI oql H 2 oq2
_ 210 _
 aU 2 +
 aUI _
 iJH L _
 aH 2 ,
1'12 - 12 - H] aql H 2 aq, H]H 2 0q2 H]H 2 aqi
_ 210 _
 aus +
 aUI _
 ijH 1 _
 aHa
"V13 - 13 - HI oq] H3 iJq:J H]Hs oqs H]Hs oq] ,
I' - 210 -
 aU 3 +
 iJu 2 _
 iJH s _
 oH 2
23 - 23 - H 2 aq2 H3 oq3 H2H3 oq? H2H3 iJq3 .
HOpMaJIbHble U i H KaCaTeJIbHble 1: i i = Tii KOMnOHeHTbI TeH30pa
HanpHmeHHH B npOH3BOJIbHOH opTOraHaJIbHOH CHCTeMe Koop.n.HHaT BbI-
pamalOTCH 4epe3 ,n:eepopMauHH no 3aKOHY rYKa:


1 f)
--w- u; = E i + m - 2 '


1
G Tii = "Vii


(i, j = I, 2, 3), (1.4)


I-!3 Bcex B03MomHbIX pellleHHH ypaBHeHHH (1.1) ,n:JJH aHa.TJH3a Ha-
npmKeHHocTH KOHKpeTHoro TeJIa He06xo,n:HMO oTo6paTb Te, KOTopble
y,n:oBJIeTBOpHIOT rpaHHqHbIM YCJIOBHHM Ha era nOBepxHocTH. rpaHH4-
Hble YCJIOBHH, ,n:onYCKalOIll.He o,n:H03HaqHOe HJIH C TOqHOCTblO ,n:o CMeIll.e-
HHH ynpyroro TeJIa B ILe.TJOM onpe,n:eJIeHHe BeKTopa cMeIUeHHH U H3 COBO-
KynHoCTH pellleHHil ypaBHeHHH (1.1), MorYT 6bJTb pa3.TJH4HbIMH. flpem,n:e
Bcera, CJIe,n:yeT BbJ,n:eJIHTb TPH OCHOBHble rpaHH4Hble 3a,n:aqH - nepBYlO,
BTOPYIO H CMelllaIIHYIO [133].
B CJIY4ae nepBoH rpaHHqHOH 3a,n:a4H Ha nOBepxHocTH TeJIa 3a,n:aeTCH
BeKTop BHelllHHX YCHJIHH. Tor.n.a .n.JIH BeKTopa ynpYfHx CMeIll.eHHH no-
JIY4aeM YCJIOBHe
2G [n


 + (n . grad) u + i- n X rot u] = F no (1.8)
8




_ LITO KaCaeTCfI llCCJIe.n:OBaHllfl CTPYKTypbl perneHllSl ypaBHemIii
paBHOBeCllfl II .n:Bll>KeHHfI ynpyroro TeJIa, TO K HaCT05IIIJ.eMY BpeMeHH
B pa3.n:eJIe TeOpllll ynpyrocTll, nOCBSlIIJ.eHHOM ll3ytteHl-UO Tal< Ha3bIBae-
MbIX 06II{llX perneHHH, .n:OCTllrHYT 3HattllTeJIbHbIH nporpecc. DOJIytte-
HbI npe.n:CTaBJIeHHSI BeKTopa ynpyrnx CMeII{eHHH ttepe3 rapMOHlltte-
CKlle <PYHKUllll B 3a.n:attaX CTaTllKll II BOJIHOBbIe <PYHKUHll B 3a.n:attaX
.n:llHaMHKll. 3.Tll pe3YJIbTa!bI .n:alOT B03MO>KHOCTb CTpOllTb Ha60pbI ttaCT-
HbIX perneHllll ypaBHeHllll paBHOBeCllfl llJIll .n:Bll>KeHllSl B KOOp.n:I-IH8THbIX
CllCTeMax, .n:onYCKalOII{llX pa3.n:eJIeHlle nepeMeHHblX B TpeXMepHbIX ypaB-
HeHllSlX JIanJIaca llJIll reJIbMrOJIbLJ:a. -
B CJIyttae CTaTllttecRllX 3a.n:att Teopllll ynpyrocTll Hall60JIee rnllpOKoe
pacnpOCTpaHeHlle nOJIyttllJIO npe.n:CTaBJIeHHe DanKOBlltta - HeH6epa
[139, 243]:


JIeTBOpeHHfI rpaHHttHbIX YCJIOBllH KOM6llHaUllH ttaCTHbIX perneHHH ypaB-
HeHllSl JIaMe pernaeTcSI MeTo.n:OM c06CTBeHHbIX BeKTopHbIX <PYHKUllH
[179, 216].
_ rpaHHttHbIe 3a.n:attH Teopllll ynpyrocTll SlBJISlIOTCSI BeKTopHbIMll.
11mI' HllX nOJIHYIO cHcTeMY c06cTBeHHbIx BeKTopHbIX <PYHKUllH Ha no-
BepxHocTll TeJIa MO>KHO nOCTpOllTb He. e.n:llHCTBeHHbIM 06pa30M.TaK,
HanpHMep, nOJIHYIO CllCTeMY c06CTBeHHbIX BeKTopHblX <PYHKUllH MO>KHO
nOJIYttllTb, BbIpa>KaSl Bce KOMnOHeHTbI BeKTopOB CMeII{eHllH llJIll HanpSl-
>KeHllH c06CTBeHHbIMH <PYHKUllSlMll COOTBeTCTBYIOII{eH CKaJmpHOH 3a-
A1'ittll. BbIpa>KeHllSl YKa3aHHbIX BeKTopOB coxpaHSllOT He06xo.n:llMbIe
CBOHCTBa opToroHaJIbHOCTH H nOJIHOTbI npH llcnOJIb30BaHHH HapSl.n:y C
co6CTEeHHbIMll <PYHKUH5IMll CKaJISlpHOH 3a.n:attH HeKOTopbIX KOM6HHaUHH
HX nepBbIX npOll3Bo.n:HbIX.
BbIpa>KeHll51 AJISI BeRTOpa ynpyrnx CMeII{eHHH nOCTpOllM TaKllM
06pa30M, ttT06bI ynpoCTHTb 3aAatty BbI60pa npOll3BOJIbHbIX nOCT05lHHbIX
H <PYHKUllH no 3a.n:aHHbIM rpaHHttHbIM YCJIOBHSlM. Hall60JIee nOCJIe.n:o-
BaTeJIbHO TaKofl no.n:xo.n: OCYIII.eCTBJIeH B nOCJIe.n:Hee BpeMfI npll pa3-
pa60TKe MeTo.n:a c06CTBeHHbIX BeKTopHbIX <PYHKUHH B rpaHllttHbIX 3aAa-
ttax Teopllll ynpyroCTll. I1cnOJIb30BaHlle .:noro MeTO.n:a .n:aeT B03MO>K-
HOCTb nOJIYttllTb perneHlle npOCTpaHCTEeHHbIX 3a.n:att TeopllH ynpyrocTll
.n:JI5I TeJI, orpaHlltteHHbIX Koop.n:llHaTHbIMll nOBepXHOCT5IMH o.n:Horo ce-
MeHCTBa, eCJIll ll3BeCTHO perneHHe COOTBeTCTBYlOIII.eH 3a.n:attll Teopllll no-
TeHUllaJia.
MeTOA c06CTBeHHbIX BeKTopHbIX <PYHKUllH 5IBJI5IeTC5I o606III.eHveM
MeTo.n:a co6CTBeHHbIX <PYHKUI1H B CKa,,1S1pHbIX 3a.n:attaX Ha CJIyttaH BeK-
TOpHbIX nOJIeH. CYIII.HOCTb :lTOro MeTo.n:a MO>KHO oxapaKTepH30BaTb
CJJe.n:YlOIII.HM 06pa30M. I1cKoMoe BeKTopHoe nOJIe .n:eJIHTCSI Ha .n:Be ttac-
TH - npo.n:OJIbHYlO (rpaAHeHT CKaJISlpHOrO nOTeHUllaJIa) H nonepettHYlO
(pOTOp BeKTopHoro nOTeHUHaJIa). DpO.n:OJIbHbIH KOMnOHeHT BeKTopHoro
nOJI5I npe.n:CTaBJI5IeTCSI c06CTBeHHbIMH <PYHKUHSlMll CKaJISlpHOH 3a.n:attll.
DonepettHbIe KOMnOHeHTbI npe.n:CTaBJI5IIOTC5I HeKOTopOH KOM6HHaUlleH
nepBbIX npOH3BOAHblX c06CTBeHHbIX <PYHKUHH CK8JI51pHOH aa.n:attll,
ROTopbIe TaK>Ke 06pa3YIOT nOJIHYlO CllCTeMY Ha nOBepXHOCTH TeJIa.
TaRoH nOA»OA .n:aeT HaH60JIee npOCTbIe COOTHOllleHllSl AJISI onpf.n:
JIeHHSI
npOll3BOJIbHbIX nOCT05lHHbl'X II <PYHKUllH. co.n:ep>KaIII.llXC5I B 06III.eM pe-
llleHHH, no 3a.n:aHHbIM rpaHHttHbIM YCJIOBHSlM. DOJIytteHHbIe pe3YJIbTa-
Tbl 5IBJI5IIOTCSI HCXO.n:HbIMH npll paCCMOTpeHHH rpaHHttHbIX 3a.n:att AJI5I TCJI
KOHettHbIX pa3MepOB.


m
u= B-- 4 (m-I) grad(r. B+Bo), (1.12)
r.n:e Boo II B - COOTBeTCTBeHHO rapMOHHtteCKHH CKaJI5Ip H rapMOHll-
tteCKllH B.,eKTop. B 3a.n:attaX .n:HHaMllKH npOCToe BbIpa,KeHHe BeKTopa
CMeIII.eHHH nOJIyttaeTC5I Henocpe.n:CTBeHHO Ha OCHOBe TeopeMbI reJIbM-
rOJIbua [1
2J a B03MO>K
OCTH npe.n:CTaBJIeHll51 BeKTopHoro nOJISI lJepe3
CKaJISlpHblli H BeKTopHblli nOTeHUHaJIbI B BH.n:e
u = grad <D + rot A, (1.13)
divA=O.
11JISI CKaJISlpHoro <D H BeKTopHoro A nOTeHUllaJIOB Ha OCHOBe (1.1)
nOJIyttalOTcH cJIe.n:YIOIIJ.He ypaBHeHHH:
V 2 <D = -.L d2(J) V2A _ -.L iJ2A ( 1.l 4)
a 2 dt 2 ' - b'l iJt 2
r.n:e a = V 2 : =
 g - CKopOCTb BOJIH pacrnHpeHH5I. b = V
 -
CKOpOCTb BOJIH c.n:Blfra.
LlpyrnM <popMaM 06II{llx pellleHHfl ypaBHeHllH TeopHlf ynpyrocTll
nOCB5IIII.eHa 06IJlllpH8H JIllTepaTypa [165, 168, 182, 2341. 3HattHTeJIbHOe
l/HCJIO pa60T nOCB5IIII.eHO ll3ytteHlllO CB5I3H Me>K.llY HHMH [38, 114, 2341
H BonpocaM llX nOJIHOTbI [141, 159, 169, 2091. He 6y.n:eM OCTaHaBJIHBaTb-
CH Ha aHaJIll3e YKa3aHHbIx pa60T, nOCKOJIbKY 3.n:eCb pacCMaTpllBalOTCH
rp
HllttHbIe 3a.n:attll B Tex I{OOpAHHaTHbIX CHCTeMaX, r.n:e Ha60pbl <PYHK-
UHll,
BJIHIOIII.H
C5I perneHHeM ypaBHeHHH (1.1), Y>Ke nOCTpOeHbI. Oc-
HOBHOll 3a.n:attell 6y.n:eT HBJI5ITbC5I BbI60p HY>KJJbIX KOM6HHaUHH 9TllX
<PYHKUHH II y.n:oBJIeTBOpeHHe C IlX nOMOIIJ.blO rpaHHttHbIM YCJIOBHSlM.
PerneHHe 3TllX BonpOCOB He CBfl3aHO C <pOpMOH npe.n:CTaSJIeHllH 06mero
pellleHll51 ypaBHeHllH (1.1).
Dpll paccMoTpeHHll rpaHllttHbIX 33.n:aQ Teopllll YIJPyroCTH llJIR TeJI,
rpaHllttH351 nOBepxHocTb S KOTOpbIX cOBrta.n:aeT C KoopAHHaTHoH no-
BepxHocTblO TpllopTOrOHaJIbHOH CllCTeMbI KoopAlmaT, AonycKalOlUeH pa3-
.n:eJIeHHe nepeMeHHblx B BeKTopHoM ypaBHemm JIanJIaca llJIH reJIbM-
rOJIbua, Bonpoc 0 nOCTpoeHHH HaH60JIee npHcnoco6JIeHHbiX J1.JISI y.n:OB-
10



 2. rpaHU'IHbIe aaAa'I11 AJIJI TeJI ROHe'IHbIX paSMepOB
Ynpyrne TeJIa, rpaHllttHbIe 3a.n:attI1 .n:JIH KOTOpbIX 6y.n:YT paccMaT-
pHBaTbcSI B KHllre, 06'be.n:HHHlOTCSI B O.n:HH KJIaCC nOH5ITlleM TeJIa Ko-
HetlHbIX pa3MepoB. XapaKTepHoH oc06eHHOCTblO reoMeTpllll ynpyroro
TeJIa KOHettHbIX pa3MepoB 5IBJI5IeTCSI TO, ttTO rpaHHttHa51 nOBepXHOCTb
era MO>KeT 6b1Tb COCTaBJIeHa H3 ttacTefI Koop.n:HHaTHbIX nOBepXHOCTefl
pa3JIlllJHbIX ceMeHCTB (pllC. 1). TaKa51' Tp3KToBKa nOH5ITll51 TeJIa KOHelJ-
HbIX pa3MepoB npHAaeT eMY HeCKOJIbKO YCJIOBHbIH xapaKTep, nOCKOJIbKY


11




OTHOCHT K TeJIaM KOHeqHbIX pa3MepOB nOJIy6ecKOHeqHbIH l.{HJIHH,l(p
H HCKJIlOqaeT 3aMKHYTYlO C<pepy. O,l(HaKO C TOqKH 3peHH5I cnoc060B
y,l(OBJIeTBOpeHH5I rpaHHqHbIX YCJIOBHH Me)f(,l(y KOHeqHbIM H nOJIy6ec-
KOHeqH
M l.{HJIHH,l(paMH HeT HHKaKHX pa3JIHqHH H rpaHHqH
e 3a,l(aqH
,l(JI5I HHX, HeCOMHeHHO, ,l(OJI)f(HbI 6bITb OTHeCeH
 K O,l(HOMY KJIaccy rpa-
HHqHbIX 3a,l(aq TeOpHH ynpyroCTH.
Pe3YJIbTaTbI, nOJIyqeHHble B TeOpHH ynpyroCTH npH paCCMOTpeHHH
rpaHHqHbIX 3a,l(aq ,l(JI5I 06JIaCTeH, rpaHHqHa5I nOBepXHOCTb KOTOpbIX
06pa30BaHa KOOp,l(HHaTHbIMH nOBepX-
HOCT5IMH O,l(HOrO CeMeHCTBa TPH-
opToroHaJIbHOH CHCTeMbI KOOp,l(H-
HaT, 5IBJIHlOTCH HCXO,l(HbIMH npH pac-
CMOTpeHHH rpaHHqHbIX 3a,l(aq ,l(JIH
TeJI KOHeqHbIX pa3MepOB. B HaCTOH-
Il.{ee BpeMH nOJIyqHJIH umpOKoe pac-
npOCTpaHeHHe JJ:Ba nO,l(xoJJ:a K HC-
nOJIb30BaHHlO 3THX pe3YJIbTaTOB.
O,l(HH H3 HHX, KOTOpbIH Bnoc-
JIe,l(CTBHI1 B pa60Tax A. 11. JIYPbe
PRC. L [1241 nOJIyqHJI Ha3BaHHe MeTOJJ:a
O,l(HOpOJJ:HbIX perneHlIH, BnepBble
6bIJI npHMeHeH B pa60Tax n. A. llIH<p<pa [257] H B. A. CTeKJIOBa [166J.
3TOT MeTO,l( MO)f(HO paCCMaTpl1BaTb KaK 0606Il.{eHHe KJIaCCHqeCKOrO Me-
TO,l(a co6cTBeHHblx <PYHKl.{HH Ha rpaHHqHble 3a,l(aqH TeOp1111 ynpyroCTI1.
npl1 TaKOM 0606Il.{eHHI1 B03HHKaeT BeKTopHaH rpaHHqHaH 3a,l(aI.J:a 0
nOCTpOeHI1H c06CTBeHHblx <PYHKl.{HH Ha qaCTH rpaHHl.{bI TeJIa, KOTopaH
npI1Ha,l(JIe)f(I1T KaKoH-HH6Y,l(b O,l(HOH KOOp,l(HHaTHoH nOBepXHOCTI1, npH
HYJIeBbIX HanpH)f(eHI15!X HJII1 CMeIl.{eHHHX Ha OCTaJIbHOH qaCTH rpaHI1l.{bI.
TaKa5! rpaHH'lHaH. 3a,l(aqa OKa3bIBaeTCH cYIl.{eCTBeHHo CJIO)f(Hee aHaJIO-
rl1qHOH CKaJIHpHOH 3a,l(aql1. B03HI1KalOIl.{He npH perneHHH 3TOH 3aJJ:aqH
BonpoCbI xoporno pa3pa60TaHbI B TeopHH llITypMa - JIHYBHJIJI5!.
11cCJIe,l(OBaHHH nOKa3bIBalOT, qTO 6a3HcHble <PYHKl.{HH O,l(HOP0,l(HbIX
perneHHH HBJI5IlOTC5I c06cTBeHHbIMH 3JIeMeHTaMI1 HeKoTopbIX HecaMOCO-
np5I)f(eHHbIX KpaeBblx 3a,l(aq. BCJIeJJ:cTBHe 3TOro HapyrnaeTC5I YCJIOBl1e
B3aHMHoH opTOrOHaJIbHOCTH 6a3HcHbIX <PYHKl.{HH, a CaIVTH 3TH <PYHK-
l.{HI1 H COOTBeTCTBYlOIl.{He HM c06CTBeHHble 3HaqeHH5I HBJI5IlOTCH KOMn-
JIeKCHbIMH. B CB5I3H C 3THM OCHOBHbIMI1 BonpocaMH, B03HHKalOIl.{HMH
npl1 HCnOJIb30BaHHI1 OJJ:HOpO,l(HbIX perneHI1H, 5IBJIHlOTCH BonpOCbI nOJIHO-
TbI co6CTBeHHbIX 3JIeMeHTOB H xapaKTep CXO,l(IIMOCTH pa3JIO)f(eHIIH no
HI1M B 3aBHCliMOCTH OT CBOHCTB pa3JIaraeMbIX <PYHKl.{HH. np06JIeMbI
6a3lIca OJJ:HOpO,l(HbIX perneHHH B nOCJIe,l(Hlie roJJ:bI 6bIJIH paCCMOTpeHbI
pH,l(OM aBTOpOB. HaH60JIee nOJIHble pe3YJIbTaTbI B 3TOH 06JIaCTII nOJIY-
qeHbI B HCCJIeJJ:oBaHI1HX 11. 11. BOpOBHqa II OTpa)f(eflbI B pa60Tax [57.
59].
npaKTlIqeCKOe npHMeHeHHe MeTO.ll.a O,l(HOP0,l(HbIX perneHHH npH
pernemm KOHKpeTHblx 3a,l(aq COnpH)f(eHO C npeO,l(OJIeHlIeM 3HaqHTeJIb-
HbIX BblqHCJII1TeJIbHbIX TPY,l(HOCTeH. n03TOMY, HeCMOTp5! Ha TO qTO
,l(JIfI pfI,l(a ynpymx TeJI O,l(HOpO,l(Hble perneHHH H3BeCTHbI ,l(aBHO (3,l(eCb,
12


"pOMe l.{HTlipOBaHHblx BbHlIe pa60T n. A. lllH<p<pa H B. A. CTeKJIOBa
MO)f(HO YKa3aTb Ha HCCJIe.ll.oBaHIiH 11. D:yraJIJIa [2051, JI. <l>aHJIOHa [212]
H n <1>. nanKoBH4a (1401), npaKTHqeCKOe npl1MeHeHHe 3TH perneHHH
nOJIyqHJII1 JIHrnb B nOCJIe,l(Hee Bpel',15I 6JIarO,l(apH pa60TaM B K npo-
KonOBa. nOJIyqeHHble HM pe3YJIbTaTbI OCBeIl.{eHbI B MOHorpa<pI1H
A. 11. JIYPbe [1241 H B 0630pHOM ,l(OKJIa,l(e B. K npOKonoBa [149J.
3<p<peKTHBHoCTb perneHHH, nOCTpoeHHbIX B. K. npOKonoBbIM, 11 B03-
MO)f(HOCTb HCnOJIb30BaTb 3BM npH CJIO)f(HbIX paCqeTaX nOCJIY)f(HJIIi
CTHMYJIOM K ,l(aJIbHeHrneMY pacrnHpeHHlO Kpyra 3a,l(aq, nOJIyqHBrnHX pe-
rneHHe C nOMOIl.{blO MeTO,l(a O,l(HOpO,l(HbIX perneHHH [33, 137, 138].
.MeTo,l( O,l(HOP0,l(HbIX perneHHH CTaJI MOIl.{HbIM cpeJJ:cTBoM aCIIMnTOTH-
qeCKOrO aHaJIH3a npH6JIH)f(eHHbIX TeopHH nJIaCTHH H o60JIoqeK, a TaK)f(e
OCHOBOH ,l(JIH nocTpoeHHH HOBbIX TeopHH, n03BOJIHlOIl.{HX nOCJIe,l(OBa-
TeJIbHO npoBecTH yqeT BeJIHqHH O,l(HOrO nOpH,l(Ka MaJIOCTH. CYIl.{e-
CTBeHHble pe3YJIbTaTbI 6bIJIH nOJIyqeHbI B pa60Tax 11. 11. BOpOBHqa H era
yqeHHKOB [12, 14, 29].
3aKaHqHBaH xapaKTepHcTHKY MeTOJJ:a O,l(HOP0,l(HbIX perneHHH, YKa-
)f(eM, qTO OTMeqeHHble Bblrne pa60TbI HH B KoeH Mepe He HCqepnbIBalOT
Bcero 06'beMa HCCJIeJJ:OBaHHH, BbmOJIHeHHJ.!X B paMKax 3Toro MeTOJJ:a.
3,l(eCb YKa3aHbI pa6OTbI, 3HaKOMCTBO C KOTOpbIMH, no HarneMY MHeHHlO,
n03BOJIHeT COCTaBHTb ,l(OCTaTOqHO nOJIHOe npe,l(CTaBJIeHHe 0 CYIl.{HOCTH
MeTO,l(a H B03HHKalOIl.{HX npH ero HCnOJIb30BaHHI1 np06JIeM.
D:pyroH no,l(xo,l( K HCCJIe,l(OBaHHlO rpaHHqHbIX 3a,l(aq TeopHH
ynpyrocTH ,l(JIH TeJI KOHeqHbIX pa3MepoB OCHOBbIBaeTCfI Ha pa3BHTHI1
II,l(eH, BblcKa3aHHoH r. JIaMe B JIeKl.{HHX no TeopHH ynpyrocTH [232].
B 3TIIX JIeKUHHX r. JIaMe npe,l(npHHHJI nOnbITKY ,l(aTb TOqHOe perneHHe
nepBoH OCHOBHOH rpaHHqHOH 3a,l(aqH TeopHH ynpyrocTH ,l(JIH napaJIJIe-
JIenlIne,l(a. KOHKpeTHo paCCMaTpl1BaJIaCb 3a,l(aqa 0 cHMMeTpHqHOM OT-
HOCHTeJIbHO Tpex nJIOCKOCTeH cHMMeTpHH HanpH)f(eHHOM COCTOHHHH
napaJIJIeJIenHne,l(a, 06YCJIOBJIeHHOM HOpMaJIbH
MH HarpY3KaMH Ha
rpaHHX.
K TOMY BpeMeHH r. JIaMe H 5. n. E. KJIaneHpOH nOJIyqHJIIi perneHHe
3aJJ:aqH 0 nepHO,l(HqeCKH ,l(e<popMHpoBaHHoM ynpyroM CJIoe. 11cnOJIb-
30BaHHe O,l(HOrO TaKoro perneHHH, OqeBH,l(HO, ,l(aeT B03MO)f(HOCTb nOJI-
HOCTblO Y,l(OBJIeTBOpHTb rpaHHqHbIM YCJIOBHHM Ha ,l(BYX npOTHBono-
JIO)f(HbIX rpaHHX napaJIJIeJIenHne,l(a, npHqeM CO,l(ep)f(aIl.{IIeCH B HeM
npOH3BOJIbHble nOCTOHHHble (K03<P<PHl.{HeHTbI JJ:BOHHoro pH,l(a <l>YPbe)
onpe,l(eJIHlOTCH H3 npOCTbIX aJIre6paHqeCKHX COOTHorneHHH. CJIe,l(YlO-
Il.{HH rnar r. flaMe npe,l(CTaBJIHeTCH BnOJIHe eCTeCTBeHHbIM - eCJIH
Heo6xo,l(HMO Y,l(OBJIeTBOpHTb rpaHHqHbIM YCJIOBHHM Ha Tpex napax
rpaHeH, TO HY)f(HO HCnOJIb30BaTb TPH COOTBeTCTBYlOIl.{HX perneHHH
JJ:JIH CJIOH. D:pymMH CJIOBaMH, npHcTynaH K perneHHlO 3a,l(aqH 0 napaJI-
JIeJIenHne,l(e, r. JIaMe HaqaJI C nOCTpoeHHH o6Il.{ero perneHHH Kpae-
BOH 3aJJ:aqH, T. e. TaKoro perneHHH ypaBHeHHH paBHOBeCH5!, caMa

)OpMa KOTOpOro ,l(aBaJIa 6bI YBepeHHocTb B B03MO)f(HOCTH nOJIyqHTb
nOJIHOe y,l(oBJIeTBOpeHHe rpaHHqHbIM YCJIOBHHM.
AJIre6paTH3al.{HH <PYHKl.{HOHaJIbHbIX ypaBHeHHH, Bblpa)f(alOIl.{Hx rpa-
I1H'lIlble YCJIOBHH Ha rpaHHx napaJIJIeJIenHne,l(a, npHBeJIa K 6eCKOHeq-


13




HblM ClfCTeMaM JIlfHeHHbIX ypaBHeHllH OTHOClfTeJIbHO K03<P<PlfUlleHTOB
pH.n.OB, co.n.ep:>Ka
llXCH B Ka:>K.n.OM pemeHllll .n.JIH CJIOH. MeTO.n.bl llCCJIe.n.O-
BaHllH If pemeHlfH TaKllX CllCTeM He 6bIJIll pa3BlfTbl, If ll.n.eH r. naMe,
3a llCKJIIOIleHlfeM 3nll30.n.llIleCKllX pa60T [235, 249], .n.OJIroe BpeMH He
Haxo.n.llJIa npllMeHeHlfH. HaJIlfIllle CTpOHHOH Teoplfll 6eCKOHeIlHblX CH-
CTeM II nOHBJIeHHe 3JIeKTpOHHblX MamHH C03.n.aJIH B HaIlaJIe 50-x ro.n.OB
npe.n.nOCblJIKll .n.JIH B03pO:>K.n.eHHH H.n.eH r. naMe, KOTOpaH Tenepb .n.
JIa
B03MO:>KHOCTb nOJIYIlHTb KOJIHIleCTBeHHble oueHKll lf3YIlaeMblX nOJIeH H
3HaIll1TeJIbHO pacmHpHTb Kpyr pemeHHblX 3a.n.aIl TeopHH ynpyrocTH,
HMeIO
llX mHpOKYIO 06JIaCTb nplfMeHeHllH. vb nepBblX pa60T 3TOro
nepHo.n.a CJIen:yeT yKa3aTb pa60Tbl E. n. A6paMHHa [1 1 If X. CaHTO [2551.
YKa3dHHYIO pa60TY E. n. A6paMHHa CJIen:yeT Bbl.!J.eJIHTb oc060.
PaccMaTpHBaH 3a.n.aIlY OU ocecllMMeTpHIlHoM HanpH:>KeHHOM COCTOHHHH
UHJIHH.n.pa KOHeIlHOH .n.JIllHbl, aBTOp BnepBble .n.OKa3aJI perYJIHpHOCTb
6ecKoHeIlHblx CllCTeM, nopO:>K.n.aeMbIx rpaHHIlHblMlf YCJIOBHHMH. 3TOT
pe3YJIbTaT HBJIHeTCH Ba:>KHbIM 3TanOM He TOJIbKO B CMblCJIe 060CHOBaHllH
MeTO.n.a r. naMe, HO H C TOIlKll 3peHHH nOCTpoeHlfH .paKTHIleCKoro pe-
meHllH 3a.n.aIlH, nOCKOJIbKY .n.OKa3aTeJIbCTBO perYJIHpHOCTlf 6eCKOHeIl-
HblX CHCTeM .n.aeT B03MO:>KHOCTb YKa3aTb aJIrOpllTM OTblCKaHHH HX
npH6JIH:>KeHHOrO pemeHHH.
B HaCTOH
ee BpeMH HCnOJIb30BaHHe no.n.xo.n.a, no cy
eCTBY COB-
na.n.aIO
ero C no.n.xo.n.OM r. naMe, npH pemeHlfH rpaHHIlHblX 3a.n.aIl .n.JIH
TeJI KOHeIlHblX pa3MepOB nOJIYIlHJIO mHpOKoe pacnpOCTpaHeHHe. npe.n.-
CTaBJIeHlfe 06 06
eM 06'beMe BbInOJIHeHHblX llCCJIe.n.oBaHHH MO:>KHO co-
CTaBllTb Ha OCHOBe 0630pa [51. AHaJIH3 ny6JIlfKaUHH, OTHOCH
HXCH K
paccMaTpHBaeMblM B .n.aHHoH pa60Te 3a.n.aIlaM, 6y.n.eT npoBe.n.eH B co-
oTBeTcTBYIO
HX rJIaBax.
3aKaHIlHBaH xapaKTeplfcTlfKY MeTO.n.a r. naMe, YKa:>KeM, IlTO Bbl-
nOJIHeHHble B nOCJIe.n.HHe ro.n.bl pa60Tbl nOKa3bIBaIOT 60JIbmyIO ero nJIO-
.n.OTBOpHOCTb, a HaCTOH
YIO pa60TY MO:>KHO paCCM::lTpHBaTb KaK pe-
3YJIbTaT pa3BllTlfH H YTOIlHeHHH If.n.eH r. naMe Ha OCHOBe COBpeMeH-
HblX MeTO.n.OB pemeHlfH rpaHllIlHbIX 3a.n.aIl TeopHH ynpyroCTlf.
He06xo.n.HMOCTb YTOIlHeHHH no.n.xo.n.a r. naMe K nOCTpoeHHIO 06
ero
pemeHHH rpaHHIlHoH 3a.n.aIllf .n.JIH TeJIa KOHeIlHbIX pa3MepoB CJIe.n.yeT
ll3 paccMoTpeHHH CJIe.n.YIO
ero npllMepa.
nYCTb He06xo.n.HMO H3YIlHTb HanpH:>KeHHOe COCTOHHHe ynpyroro
TeJIa (CM. pHC. 1), rpaHllIlHaH nOBepxHocTb KOTOpOro 06pa30BaHa
IlaCTHMlf Koop.n.HHaTHblx nOBepxHocTeH
 = C 1 H '11 = C 2 . l1cxo.n.HblM
nYHKToM paccMoTpeHHH HBJIHeTCH npe.n.nOJIO:>KeHHe 0 B03MO:>KHOCTll
nOCTpoHTb 06
ee pemeHlfe rpaHllIlHblX 3a.n.a'l .n.JIH ynpymx TeJI,
orpaHHIleHHblx nOBepxHocTHMH
 = C 1 H '11 = C 2 COOTBeTCTBeHHO. npll
.paKTHIlecKoM nOCTpoeHHH 3THX pemeHllH, KOTopoe He06xo.n.lfMO Bbl-
nOJIHHTb .n.JIH nOJIYIleHHH 06
ero pemeHHH rpaHHIlHoH 3a.n.a'llf .n.JIH TeJI

KOHe'lHbIX pa3MepoB (CM. pHC. 1), cpa3Y :>Ke CTaJIKHBaeMCH C HeKOTopOH
Heonoe.n.eJIeHHOCTbIO.
LleJIo B TOM, IlTO KaK .n.JIH nepBoro, TaK If .n.JIH BToporo ll3 YKa3aH-
HblX ynpymx TeJI <popMa o6
ero pemeHHH rpaHH'lHOH 3a.n.aIllf 3aBllCHT
OT XapaKTepa rpaHlfIlHblX YCJIOBHH Ha BceH nOijepXHOCTlf
 = C 1 lfJIH
14


'11 = C 2 . COXpaHH5I OCHOBHoe CBOHCTBO .n.ocTaBJIHTb HY:>KHYIO CTeneHb
.pYHKUllOHaJIbHOrO npOH3BOJIa .n.JIH y.n.oBJIeTBOpeHllH rpaHllIlHbIX
YCJIOBHH Ha IlaCTll nOBepxHocTll, nOJIYIlHM pa3JIHIlHble BbIpa:>KeHllH npll
3a.l1.aHHH Ha OCTaJIbHOH IlaCTlf HYJIeBbIX rpaHHIlHbIX YCJIOBHH llJIlf YCJIO-
BllH, OTBe'laIO
lfX B HeKoTOpoM cMblCJIe nepHo.n.HIlecKoMY np0.n.OJI:>KeHllIO
3allaHHblX Ha BCIO nOBepxHocTb.
D.JIH paccMoTpeHHoro r. flaMe CJIYIlaH .n.e<popMllpOBaHHH napaJIJIe-
JIenllne.n.a 3TO 06CTOHTeJIbCTBO nplfBo.n.lfT K B03MO:>KHOCTlf llCnOJIb30-
BaTb peweHHe .n.JIH CJIOH He TOJIbKO B .popMe .n.BOHHbIX pH.n.OB, HO II B
.pop Me .n.BOHHbIX lfHTerpaJIOB <PYPbe. nOJIY'leHHOe nplf 3TOM pemeHlle
TaKllie 06J1a.n.aJIO 6bl HYlliHOH CTeneHbIO nOJIHOTbl .n.JIH y.n.oBJIeTBopeHHH
Bcex rpaHlfIlHblX YCJIOBllH.
B03MO:>KHOCTb .n.BYx no.n.Xo.n.oB K nocTpoeHHIO 06
ero pemeHllH
rpaHH'lHOH 3a.n.a'lll .n.JIH KOHKpeTHoro ynpyroro TeJIa peaJIH30BaHa B
nepBoH r JIaBe pa60Tbl nplf paccMaTpHBaHllH 3a.n.aIlll .n.JIH npHMoyroJIb-
HHKa. Ha 3TOM npHMepe J1.ocTaToIlHO IleTKO Bll.n.Hbl pa3JIHIlHH OCHOBHbIX
ypaBHeI-IHH, KOTopble He06xo.n.lfMO pa3pemHTb npH aHaJIH3e HanpHllieH-
HOCTlf ynpyroro TeJIa.
ECTeCTBeHHblM cnoc06oM npe.n.onpe.n.eJIeHHH onHcaHHOH Heo.n.HO-
3HaIlHOCTll HBJIHeTCH peKoMeH.n.aUHH BbI6HpaTb pemeHlle B TOH .popMe,
KOTOpaH n03BOJIHeT ynpocTlfTb 3a.n.aIlY y.n.oBJIeTBOpeHHH rpaHllIlHbIM
YCJIOBHHM. 3THM nOJIO:>KeHHeM PYKOBo.n.CTByeMcH npH paccMoTpeHllll
KOHKpeTHblX rpaHllIlHblX 3a.n.aIl.
npllBe.n.eHHble Bblme paccYlli.n.eHHH 0 nocTpoeHlfll o6
ero pemeHllH
rpaHH'lHOH 3a.n.aIlll HOCHT .popMaJIbHblH xapaKTep. B03MO:>KHOCTb .paK-
TlfIlecKoro pemeHlfH rpaHHIlHoH 3a.n.aIlll C nOMO
bIO nOJIYIleHHoro
onlfcaHHblM cnoco60M BblpallieHlfH .n.JIH BeKTopa cMe
eHHH MOllieT 6bITb
060cHoBaHa pa3JIlfIlHblMll cnoc06aMH. C O.n.HOH CTOpOHbl, TaKHM 060-
CHOBaHHeM CJIY:>KHT .n.OKa3aTeJIbCTBO pa3pemHMocTll 6ecKoHeIlHblx CllCTeM
II .n.pyrHx cooTHomeHllH, K KOTOpblM nplfBo.n.HT rpaHlfIlHble YCJIOBllH.
C .n.pyroH CTOpOHbl, TaKoe 060cHoBaHHe MO:>KeT 6a3lfpoBaTbcH Ha 06
llX
TeopeMax 0 CXO.n.HMOCTll aJIrOpHTMa HeHMaHa - lllBapua [1061 B 3a-
.n.aIlax TeopllH ynpyrocTH. LloKa3aTeJIbCTBO TeopeM 0 CXO.n.HMOCTH aJI-
ropllTMa HeHMaHa - UIBapua .n.JIH .n.BYx OCHOBHblX rpaHHIlHblX 3a.n.aIl
TeoplfH ynpyrOCTlf .n.aHO B pa60Tax C. n. C060JIeBa [1631. E. H. HllKOJIb-
CKOro [1351 If 11. Ea6YIllKlf [1931.
CBH3b Mellin:y H3JIOllieHHblM no.n.xo.n.OM II aJIropllTMoM HeHMaHa-
lllBapIl.a MOllieT 6blTb YCTaHOBJIeHa B BH.n.e TeopeM, aHaJIOrHIlHblX Teope-
MaM lf3 TeoplfH nOTeHIl.HaJIa [1061. l1cxo.n.H H3 TOro IlTO paccMaTpHBae-
Mble ynpyrlfe TeJIa HBJIHIOTCH nepeceIleHHeM 06JIaCTeH, He lfMeID
llX
06
lfX Y'laCTKOB rpaHHIl.bl, 3a HCKJIIOIleHHeM KOHeIlHoro IlllCJIa TOIleR
(nJIOCKHH CJIYIlaH) HJIH JIHHHH (npOCTpaHCTBeHHbLH CJIYIlaH), MOlliHO
nOKa3aTb, IlTO npHMeHeHHe aJIropHTMa HeHMaHa - lllBapIl.a 3KBH-
BaJIeHTHO npHMeHeHHIO MeTO.n.a nOCJIe.n.OBareJIbHbIX nplf6JIH:>KeHlfH npH
pemeHlfH 6eCKOHe'lHbIX ClfCTeM, HHTerpaJIbHblX ypaBHeHHH H CHCTeM
CMemaHHoro THna. nOCKOJIbKY nOIlTlf BO Bcex paCCMaTpHBaeMblX HlfJKe
3aaallax npOBo.n.lfTCH .n.eTaJIbHOe lfCCJIe.n.OBaHHe YKa3aHHblX COOTHome-
HHH, Ha .n.OKa3aTeJIbCTBe TaKHX TeopeM oCTaHaBJIHBaTbCH He 6Y.lleM.


u




npHBe.n:eHHoe H3JIOmeHHe CYIlI,eCTBa MeTo.n:a, HCIlOJIb3yeMoro B
pa60Te, opHeHTHpOBaHO Ha CTaTHIIeCKHe 3a.n:aIIH TeopHH YIlPyroCTH.
Te .n:OIlOJIHHTeJIbHble TPy.n:HOCTH, KOTopble B03HHKalOT IlPH HCIlOJIb-
30BaHHH ero .n:JISI aHaJIH3a YCTaHOBHBIlIHXCSI KOJIe6aHHH TeJI KOHeIIHblX
pa3MepOB, OIlHcaHbl B rJI. 6 pa60TbI.
BbIIIIe .n:aHa xapaKTepHcTHKa .n:BYX IlO.n:XO.n:OB K H3YIIeHHIO HaIlpH-
meHHoro cocrOSlHHSI TeJI KOHeIIHblX pa3MepoB - MeTo.n:a O.n:HOpO.n:HbIX
peIlIeHHH H MeTO.n:a JIaMe. 06a YKa3aHHblx MeTO.n:a SlBJIHIOTCH Ilp06JIeMHO
opHeHTHpoBaHHbIMH, T. e. opHeHTHpoBaHHblMH Ha oIlpe.n:eJIeHHblH
KJIaCC 3a.n:aII. EcrecTBeHHo, IITO Ilporpecc B peIlIeHHH YKa3aHHoro KJIaC-
ca 3a.n:aII TeCHO CBSl3aH TaKme C pa3pa60TKoH 60JIee 06W:HX MeTO.n:OB pe-
IlIeHlm rpaHHIIHblX 3a.n:aII TeopHH YIlPyroCTH. 3.n:ecb CJIe.n:yeT YKa3aTb
Ha .n:Ba IlepcIleKTHBHblX HaIlpaBJIeHHSI, IlOJIYIIHBIlIHX IlIHpoKoe pac-
IlpocrpaHeHHe.
OCHOBOH IlepBoro H3 HHX HBJISIeTCH IlOJIYIIeHHoe B IlOCJIe.n:Hee BpeMSI
Ilpe.n:cTaBJIeHHe 06U1,ero peIlIeHHSI ypaBHeHHH paBHoBecHSI JIaMe IIepe3
aHaJIHTHIIeCKHe <PYHKIJ.HH [181. LlJISI IlpaKTHIIecKH BamHoro CJIYIIaSi
ocecHMMeTpHIIHoH 3a.n:aIIH YIlpyrocTH CBSl3b Mem.n:y ocecHMMeTpHIIHblMH
H Il10CKHMH IlOJISlMH H3YIIaJIaCb B pa60Tax [16, 19, 1451. PeIlIeHHe pSl.n:a
KOHKpeTHblx 3a.n:aII B paMKax ::noro MeTo.n:a co.n:epmHTcSI B Ily6JIHKa-
IJ.HSlX [20, 21, 107] H .n:p.
Tpy.n:HOCTH, J{oTopble Ilpe.n:cTOH'I Ilpeo.n:OJIeTb IlPH .n:aHHOM aHaJIH-
THIIecKOM Ilo.n:xo.n:e K rpaHHIIHblM 3a.n:aIIaM .n:JIH TeJI KOHeIIHblX pa3-
MepoB, COCTOHT B Haxom.n:eHHH SlBHoro BH.n:a 0606Il1,eHHblX aHaJIHTHIIe-
CKHX <PYHKIJ.HH .n:JIH HeKaHOHHIIeCKHX 06JIacreH MepH.n:HoHaJIbHOrO
ceIIeHHH YIlpyroro TeJIa.
EOJIbiliOe 3HaIIeHHe IlPH IlocrpoeHHH peIlIeHHH 06IlIHpHOro Kpyra
3a.n:aII TeopHH YIlpyroCTH HMelOT MeTo.n:bl, OCHOBaHHble Ha 06Il1,HX Ba-
pHaIJ.HOHHbIX TeopeMax. B03MomHbl pa3JIHIIHble IlYTH HCIlOJIb30BaHHSI
BapHaIJ.HOHHbIX CBOHCTB IlOJIeH HaIlpSlmeHHH H .n:e<popMaIJ.HH [120, 130,
1361. B03MomHocTH BapHaIJ.HoHHbIX MeTo.n:OB peIlIeHHH rpaHHIIHblX 3a-
.n:aII 3HaIIHTeJIbHO pacIlIHpSlIOTCH B CBSl3H C pa3BHTblMH B IlOCJIe.n:Hee Bpe-
MSI cIloc06aMH aHaJIHTHIIeCKoro OIlHcaHHH CJIOmHblX rpaHHIIHblX IlO-
BepxHocTeH [152, 1531. BamHble pe3YJIbTaTbl peIlIeHHH C IlOMOIll,blO
TeopeMbl KaCTHJIbSiHO HaH60JIee CJIOmHblX 3a.n:aII .n:JIH TeJI KOHeIIHblX
pa3MepoB (3a.n:aIIa .n:JIH IlapaJIJIeJIeIlHIle.n:a) IlpHHa.n:JIemaT M. M. cI>HJIO-
HeHKo-EopO.n:HIIY [183, 1841. npe.n:JIOmeHHblH HM cIloco6 IlocrpoeHHSI
Koop.n:HHaTHblx <PYHKIJ.HH Il03BOJIHJI IlOJIYIIHTb 3<p<peKTHBHoe peIlIeHHe
IlJIOCKOH 3a.n:aIIH .n:JISI IlpSiMoyroJIbHHKa [971. He.n:ocTaTKoM BapHaIJ.Hol;'-
HblX IlO.n:XO.n:OB K rpaHHIIHblM 3a.n:aIIaM TeopHH YIlpyroCTH .n:JISI TeJI KO-
HeIIHblX pa3MepoB SlBJISleTCSI Tpy.n:HOCTb OIJ.eHKH KaIIeCTBa IlOJIYIIeHHoro
peIlIeHHSl, OC06eHHO IlPH H3YIIeHHH JIOKa.1bHbIX 3<p<peKToB B HCCJIe.n:ye-
MblX IlOJISIX.
JII060H MeTO.n: peIlIeHHSI MaTeMaTHIIeCKH C<popMyJmpOBaHHblx 3a.n:aII
B <pH3HIIecKHx TeopHSIx .n:OJImeH 06ecIleIIHBaTb IlOJIYIIeHHe pe3yJIbTaTa,
KOTOpblH 6bl KOJIHIIeCTBeHHblM H KaIIeCTBeHHblM 06pa30M OTpamaJI
Bce cYIll,eCTBeHHble IIepTbl H3YIIaeMoro SlBJIeHHH. I1MeHHo B 3TOM CJIY-
IJae MomHO roBOpHTb 0 ;<pH3HIIecKH TOIJHOM peIlIeHHH 3a.n:aQH.


B OCIlOBe HCIlOJIb30BaHHoro MeTo.n:a IlocrpoeHHSI 06Il1,ero peIlIeHHH
1'I'II1111 1 1111>IX 3a.n:aII JIemHT OIIeHb IlpocTaSi H.n:eSl, BblCKa3aHHaSi r. JIaMe
I'II
C' 1\ l"t'pe.n:HHe IlpOWJIOro BeKa. .:ha H.n:eSl IlOJIYIIHJIa 3HaIIHTeJIbHOe
pI! IJIIITlle Ja IlOCJIe.n:HHe ro.n:bl H B HaCTOSlIll,ee BpeMSI Ha ee OCHOBe c<pop-
Mllpllll:J.11l"H .n:OBOJIbHO 06W:HH MeTO.n: peweHHSI rpaHHIIHblX 3a.n:aII TeopHH
Y"PVronH .n:JISI TeJI KOHeIIHblX pa3MepoB. PeweHHe, IlOJIYIIeHHoe B
II/IMI(:JX iToro MeTo.n:a, Il03BOJISIeT paccIIHTbIBaTb Ha IlOJIYIIeHHe <pH3H-
'11'1'1<11 TOQHOro peweHHSI cooTBeTcTBYIOw:eH rpaHHIIHoH 3a.n:aIIH. O.n:HaKo
ell/II< l'II'ICCKaSi peaJIH3a LHSI MeTo.n:a B 60JIbIlIHHCTBe cOBpeMeHHblx Ily6-
JIIII(aIlIIH TaKOBa, IITO 3aJIOmeHHble B 06Il1,eM peweHHH B03MomHOCTH He
l/l'IIO.III>:lYIOTCSI H TeM caMblM IlOTeHIJ.HaJIbHO MOIll,HblH MeTO.n:, IlO CYIll,eCT-
"Y, ypaBHHBaeTCSI C MHOrOIIHCJIeHHblMH IlpH6JIHmeHHblMH Ilo.n:xo.n:aMH.
I IOJl rnepm.n:eHHeM TaKoro IlOJIOmeHHSI MomeT CJIymHTb TO 06CTOSlTeJIbCT-
1\0, '1'1'0 B JIHTepaType IlpaKTHIIeCKH OTCYTCTBYIOT pa60Tbl, CBSl3aHHble C
1I.IY'ICIIHeM KaIIeCTBeHHblX oc06eHHOCTeH IlOJISI HaIlpSlmeHHH B OKpeCT-
IIOrTlI yr JIOBblX TOIIeK B paMKax YKa3aHHoro MeTo.n:a.
Ol'HOBHaSi IJ.eJIb Ilpe.n:IlpHHSlTOro aBTopOM HCCJIe.n:OBaHHSI 3aKJIIO-
"laI'n'S! B TOM, IIT06bl .n:OCTHIIb IlPH paCCMOTpeHHH TOIIHblX aHaJIHTH-
'IC'C'I(l1 x peweHHH 3a.n:aII .n:JISI TeJI KOHeIIHblX pa3MepOB TOH me creIleHH
111JIt'pIlICHHOCTH, KOTOpaSi xapaKTepHa IlPH COOTBeTcTBylOW:HX 3a.n:aIIax
JVUI «oeCKOHeIIHbIX» YIlpyrHx TeJI. .:ha IJ.eJIb MomeT 6bITb .n:OCTHrHYTa
JIIIIIII> B TOM CJIYIIae, eCJIH IlOJIYIIaeMble 06Il1,He <pOpMyJIbl IlpaBHJIbHO
orpamalOT KaIIeCTBeHHble oc06eHHOCTH H3YIIaeMblX IlOJIeH H .n:OIlYCKaIOT
IIpOBCJ
CHHe BblIIHCJIeHHH C JII060H <pH3HIIeCKH OIlpaB.n:aHHOH TOIIHOCTblO.
B CBSl3H C IlOCJIe.n:HHM Tpe60BaHHeM B KHHre y.n:eJIeHO 60JIbiliOe
BIIIIMallHe IlOJIYIIeHHIO IIHCJIeHHblX pe3YJIbTaTOB .n:JISI Kam.n:OH H3 pac-
CMII rpHl3aeMblX 3a.n:aII. HaH.n:eHHble KOJIHIIeCTBeHHble oueHKH He TOJIbKO
1I,1ImOCTpHpYIOT 3<p<peKTHBHOCTb pa3BHToro Ilo.n:xo.n:a, HO H CIloc06-
C'TIIYIOT Bblpa60TKe 60JIee IlOJIHblX Ilpe.n:cTaBJIeHHH 06 oc06eHHocTSIX
Jlt'eIJopMHpOBaHHSI YIlPYrHX TeJI KOHeIIHblX pa3MepoB.
npoIJ.ecc IlocrpoeHHSI peweHHH rpaHHIIHblX 3a.n:aII .n:JISI TeJI KOHeII-
II 101 X pa3MepoB, H3JIOmeHHblH B KHHre, Ilpe.n:IlOJIaraeT HaJIHIIHe peIlIeHHH
I'pallHIIHblX 3a.n:aII .n:JISI cooTBeTCTBYIOIlI,HX 6ecKoHeIIHblx 06JIaCTeH.
B CBS!3H C 3THM WHpOKO HCIlOJIb3YlOTCSI HaKOIlJIeHHble K HaCTOSlIll,eMY
BI>l'MeHH peIlIeHHH IlpocrpaHcTBeHHblx 3a.n:aII TeopHH YIlpyroCTH, 0606-
IIlt'lIlIble B MOHorpa<pHSIx A. JISiBa [125], C. n. THMOIlIeHKO [171],
!\ 11. JIypbe [1241, 51. C. Y<pJISlH.n:a [1811. nO.n:06HaSi 6H6JIHOrpa<pHSI
110 IIpocrpaHcTBeHHblM 3a.n:aIIaM co.n:epmHTCSI. B YKa3aHHoH Bblille MO-
lIorplH
HH A. 11. JIypbe, a TaKme .n:OIlOJIHHeTCSI 0630paMH [5, 57, 127,
2:J4, 241, 2631. AHaJIH3 HCCJIe.n:oBaHHH, OTHOCSlIlI,HXCSI K paccMaTpHBae-
MI>IM 13 KHHre 3a.n:aIIaM, 6y.n:eT .n:aH B cooTBeTcTBYIOW:HX rJIaBax.
0611
He peIlIeHHSI Bcex paccMaTpHBaeMblx B KHHre 3a.n:aII Ilpe.n:cTaB-
JUIIOTCH c06cTBeHHblMH <PYHKIJ.HSlMH, B03HHKalO[IJ.HMH IlPH peIlIeHHH rpa-
1I1I'IIIIoIX Ja.n:aII .n:JISI ypaBHeHHSI JIaIlJIaca. B CBSl3H C 3THM IlOCTOSlHHO
IICIIOJlhJYIOTCSI pe3YJIbTaTbl TeopHH IlOTeHIJ.HaJIa, H3JIOmeHHble B Ily6-
J/IIKallHHX (64, 69, 101, 117, 118, 132, 1601. a TaKme JIHTepaTypa
CnpllBoIIHoro xapaKTepa, B KOTOpon paccMaTpHBalOTcSI CBoHcTBa CIle-
IllldJIblIhlX <PYHKIJ.HH [45, 46, 47, 48, 54, 68, 218].


16


2 7
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£7


Ii
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Ii
II





 3. OCHOBHbIe CBe.u;eHlIH H3 TeopHII 6eCKOHe'lHhIX
c
cTeMRoHnoBllqa
npHMcHcmlc CBSI3aHHoro C pa3BHTHcM YKa3aHHoH BblWC H.lI.eH
JIaMc MeTO.lI.a pcrncHHSI rpaHHqHblX 3a.ll.aq TCOpHH ynpyrocTH .l{JHJ TCJI
KOHCqHblX pa3McpoB npHBo.ll.HT K pcrncHHIO 6CCKOHCqHbiX CHCTeM, CH-
CTCM HHTcrpaJlbHbIX ypaBHcHHH H CMcrnaHHoro THna, CBSI3bIBalO!l.lHX
3HaqeHHSI <PYHKl.lHH .lI.HCKpcTHoro H HcnpcpbIBHoro aprYMeHToB.
B npol.lCCCC pcrnCHHSI KpacBoH 3a.ll.aqH .lI.JlSI ynpyroro TCJla onpC.l{C_1CHHe
HeH3BCCTHbiX B YKa3aHHbiX ypaBHcHHSIX SIBJlSICTCSI JlHrnb npOMC)KYTOq-
HblM 3TanOM. HaH.lI.cHHblC 3HaqCHHSI HCH3BCCTHbiX SIBJlSIIOTCSI K03<P<P H l.lHCH-
TaMH 6CCKOHCqHbiX pSI.lI.OB HJlH nJlOTHOCTSIMH HHTcrpaJlbHblX Bblpa)KCHHH,
KOTOpbIMH npC.l{CTaBJlSIIOTCSI xapaKTcpHcTHKH HanpH)KeHHO-.lI.c<popl\'mpo-
BaHHoro COCTOSIHHSI ynpyroro TCJla. ECTCCTBCHHO, qTO KaqeCTBO no-
JlyqaeMblX Ol.leHOK .lI.JlSI cMern.cHHH H HanpSI)KCHHH Cyrn.CCTBCHHO 3a-
BHCHT OT CKOpOCTH CXO.ll.HMOCTH YKa3aHHIJIX pSI.l{OB H Hec06cTBCHHbiX
HHTCrpaJlOB, KOTOpblC npH BblqHCJlCHHSIX KaKHM-JlH60 06p330M 06Pbl'
BalOTCSI .
3T0 06CTOHTeJlbCTBO .lI.CJlaCT qpC3BblqaHHO Ba)KHbIM HC TO_lbKO H3Y-
'leHHC pa3pcrnHMoCTH 6CCKOHCqHbiX CHCTeM H pO.ll.CTBCHHblX HM COOT-
HorncHHH, HO H HCCJIC.lI.OBaHHe aCHMnTOTHqeCKHX CBOHCTB HCH3BeCTHbiX
B 3THX CHCTCMax. 3HaHHc 3THX CBOHCTB n03BOJlSICT 3HaqHTCJIbHO nOBbl-
CHTb 3<P<PCKTHBHOCTb BblqHCJICHHH nYTeM HCnOJIb30BaHHH MCTO.l{OB Y.'Iyq-
WCHHSI CXO.ll.I1MOCTH pSI.lI.OB H BbISICHHTb KaqCCTBCHHble Oc06eHHOCTH
nOBC.l{CHHH xapaKTepHcTHK HanpSI)KCHHO-.lI.C<pOpMHpoBaHHoro COCTOHHHH
BO,ilH3H yr JIOBblX TOl{CK..
TCOpHSI 6CCKOHeqHbIX CHCTCM, B KOTOpOH nOMI1MO BonpOCOB cY-
!l.lCCTBOBaHI1H pcrnCHHH 60JIbrnOC BHIiMaHHC Y.ll.CJIeHO H3yqCHHIO aCHMIl-
TOTHqCCKHX CBOHCTB HCH3BCCTHblX, pa38HTa E. M. KOHJIOBHqCM rl1l,
112J. YKa3aHHbic pa60Tbi nOCBHrn.CHbl HCCJlC.lI.08aHHIO napHblx 6ecKo-
HCqHblX CIiCTCM JIHHCHHblX aJIrC6paHqCCKHX ypaBHcHHH. I1cxo.l{HOH MC-
XaHHqCCKOH 33,.lI.aqCH, npHBJlCKrnCH BHHMaHHC aBTOpa K HCCJIe.l{OBaHHIO
TaKHX CIiCTCM, 6blJla H3BCCTHaSI 3a.ll.aqa 06 H3rH6c )KeCTKO-3a!l.lCI\'WCHHOH
npHMoyrOJIbHOH nJlaCTHHKH r 113]. Eha 3a.ll.al1a H pacCMaTpHBaCMbie
J3 pa60Te rpauHH'lHbIC 3a.ll.aqH .lI.JISI ynpyrHx TCJI KOHCqHblX pa3MepOB B
3HaqIiTeJlbHOlf Mcpe pO.ll.CTBeHHbl, qTO npC.lI.cTaBJISICT B03MO)KHOCTb rnH-
pOKoro HCnOJIb30BaHHH PC3YJIbTaTOB E. M. KOH.l0BHqa npH HX pcrne-
IIHH.
B pa60Tax E. M. KOHJIOBHqa HCCJle.ll.yeTcSI cJle.l{yl{)rn.aH napHaH 6cc-
I<OHe'lHaSI CHCTCMa ypaBHcHI1H C HCH3BCCTHblMH x'l Ii Y n (n = 1. 2. ...):


1'11I'Tl'Mbl H ec cB060.ll.HbiX qJICHOB npHHHMal{)TCSI 06blqHbiC OrpaHHqCHHSI,
IIIIJHII{)!l.lHCCH 0606rn.eHHcM H3BCCTHbiX YCJIOBHH pcryJIHpHOCTW


00


00



 akn = 1 - rpl (n),
k=l



 akn = I -. '1)11 (n),
k=l


(3.2)




K
<K.
(f'I (n) "" , 'lJ'1 (n)
OrnOCHTCJIbHO <PYHKl.lHH % (n) H '1)11 (n) .l{OnYCKaCTCH B03MO)KHOCTb I1X
lIC'ipa!l.leHHH B HYJIb npH CTpCMSIrn.CMCSI K 6CCKOHCqHOCTI1 3HaqCHHH apry-
Ml'IITa. T aKHM 06pa30M, CHCTCMa (3.1) HC SIBJIHCTCSI BnOJIHe pcry JIHp-
lIoii ECJII1 .lI.JISI CBOOO.ll.HblX qJICHOB CHCTCMbl (3.1) BbillOJIHCHO YCJIOBHe
(:1.:3) npH HCKOTOpOM onpc.lI.cJ1CHHOM K, TO OHa .lI.OnYCKaCT OqCBH.lI.HYIO
I:IMCHY HCH3BCCTHbIX, nepCBO.ll.Hrn.ylO ec B CHCTCMY C nOJIO)KHTCJlbHblMH
l"lIo60.l{HbIMH qJICH3MH, H nOTOMY BnpC.lI.b MO)KHO CqHTaTb Fn >- 0 II
(III >- O. npHHHTblC YCJIOBI1H (3.2) H (3.3) 06ccnCqHBalOT cyrn.ccTBoBa-
II Ill' OrpaHHqCHHOrO rJIaBHOrO pcrnCHHH CHCTCMbl (3.1), HC npcBocxO-
JlSIIIlero no a6COJIIOTHOH BCJIl1qHHC qHCJla K. 3TO perncHHc MO)KCT 6bITb
lIai'l.l{CHO MCTO.lI.OM PC.lI.YKl.lHH HJIH MCTO.l{OM nOCJIC.lI.OBaTCJIbHblX npH6JIH-
iI\CIIHH. nCpBblM Ba)KHbIM PC3YJlbTaTOM E. M. KOHJIOBl1qa HBHJIOCb .lI.O-
l\:naTCJIbCTBO TOro, qTO npH nOJIO)KHTCJIbHbIX CB060.l{HbIX qJICHaX rJIaB-
lIoe pcrncHHc CHCTeMbl (3.1) TaK)KC nOJIO)KHTCJIbHO, T. c. .lI.JIH Bcex HC-
IUBCCTHblX CnpaBC.lI.JII1Bbl HcpaBcHcTBa x n >- 0, Y n >- 0 (n = I, 2, ...).
."ho 06CTOHTCJIbCTBO nOCJIY)KHJIO OCHOBOH nocTpoeHHSI TCOpHH, n03BOJISIIO-
I1ll'H BbIHCHHTb aCHMnTOTHqCCKHC CBoHcTBa HcH3BCCTHbiX B 6eCKO-
IIC'lHbIX CHCTCMax H qro HC MCHCC Ba)KHO, .lI.aTb cnoc06 HX pcrncHHH,
II03BOJISII{)!l.lHH HaHT
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I1anepc.l{ 3a.l{aHHOH TOqHOCTbl{) 3a KOHCqHOC qHCJIO rnarOB.
npC)K.l{C Bccro, 113 <paKTa nOJIO)KHTCJIbHOCTH rJIaBHOrO pcrnCHHH
l'HCTCMbl (3.1) BblTCKaCT B03MO)KHOCTb nocTpoeHHH HCKOropblX Bbl-
pa)KCHHH, Ha3BaHHbiX JIlIMHTaHTaMH, C nOMorn.blO KOTOpbIX MO)KHO OIlC-
I1HTb nOBC.lI.CIIHe HCH3BCCTHbiX C BblCOKHMI1 HOMcpaMH, CCJIH H3BC-
CTlibI 3HaqCHHSI ncpBblx HCH3BCCTHbIX.
nYCTb B CHCTCMC (3.1) H3BCCTHbi 3HaqCHHH j 11 r HCH3BCCTHbiX Yn If
xn' CocTaBHM cJIe.ll.YIOrn.Hc Bblpa)KCHHH:
i
Fn +
 aknYk
k=1
i
<PI (n) +
 a kn
k=l


(3.3)


v =
"


00


(3.1)


w=
m


i'
Qm +
 r:J.kmXk
k=l
i'
1jJ1 (m) +
 a km
R=l


(3.4)


00


X n =
 a1inYk + Fit>
k=1


r.ll.e akn, akn. F n'
aIm C'lHTaIOTCSI


Yn -,-
 a1inXk + Q,.,
k=J
Qn - .lI.aHHblC BeLUCCTBeHHblC qMC,na, npHqCM akn H
nOJIO)KHTCJIbHbIMH. OTHOCHTeJIbHG Ko34xjJHIlHeHTOB


KOTopble 6Y.ll.CM paccMaTpHBaTb KaK <PYHKl.lHH n H m npH CJIC.lI.YIO-
IIllIX 3IIa'lCHHHX aprYMcHToB: n = j' + I, j' + 2, ..., m = j + I,


18



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19


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j + 2, ... TIpe)JJIOJIO)!{HM, 'ITO y,n:aJIOCb HaHTH ,n:Ba 'IIICJIa h H H TaKHe,
'ITO h - HaHMeHbwee 3Ha'IeHHe BeJIII'IHH V n H W m Ha BceM HHTepBaJIe
H3MeHeHHH n H m. a H - COOTBeTCTBeHHO HaH60JIbWee HX 3HaIleHHe.
Tor,n:a ,n:JIH Bcex HeH3BeCTHblX x n lIPH n > j' H Y m lIPH m > j clIpaBe,n:-
JIIiBbI oueHKH
h -< x n -< H (n > j'); h -< Y m -< H (m > j). (3.5)
3TO YTBep)!{,n:eHHe COCTaBJIHeT CMbICJI OCHOBHOH TeopeMbI E. M. KOHJIO-
BHIla, KOTopaH 06HapY)!{HBaeT CJIe,n:YIOIlI.HH Ba)!{HbIH <paKT: 3HaIleHHHMH
nepBbIX HeH3BeCTHblX B r JIaBHOM peweHHH YCTaHaBJIHBalOTCH HeKOTO-
pble rpaHHIJ:bI ,n:JIH 3HaqeHHH Bcex OCTaJIbHbIX HeH3BeCTHblX. TaK KaK
3TH rpaHHIJ:bI OlIpe,n:eJIHIOTCH 3HaIleHHIIMH Bblpa)!{eHHH V H W TO
lIOCJIe,n:HHe Ha3bIBalOTCH JIHMHTaHTaMH. n m'
OCHOBHoe CBOHCTBO JIHMHTaHT BO MHorHX lIpaKTHIleCKHX CJIYIlaHX
lI03BOJIHeT lIOJIYIlHTb Ba)!{Hble KaIleCTBeHHble OIJ:eHKH peweHHH rpa-
HHIlHbIX 3a,n:aq, npHBo,n:HIlI.He K CHCTeMaM THIIa (3.1). TaK, 3aIlaCTYlO
y,n:aeTCH 6e3 peweHHH CHCTeMbI YCTaHOBHTb, IlTO h =1= 0, T. e. HeH3Be-
CTHble B CHCTeMe He CTpeMHTCH K HYJIIO. 31'0 06CTOHTeJIbCTBO BMeCTe C <paK-
TOM nOJIO)!{HTeJIbHOCTH peweHHH CHCTeMbI (3.1) ,n:aeT B03MO)!{HOCTb OIJ:e-
HHTb xapaKTep CXO,n:HMOCTH pH,n:OB, npe,n:cTaBJIHIOIlI.HX KOMlIOHeHTbI
BeKTopa CMeIll.eHHH H TeH30pa HalIpH)!{eHHH Ha lIoBepxHocTH YlIpyroro
TeJIa.
B pa60Te paccMaTpHBalOTcH 3a,n:aIlH TeopHH YlIPyroCTH ,n:JIH TeJI
Ko
eIlHblx pa3MepoB XapaKTepHOH oc06eHHoCTbIO reoMeTpHH rpaHHIl-
HOH nOBeI:xHocTH 9THuX TeJI HBJIHeTCH HaJIHIlHe yrJIOBbIX TOIleK (yrJIo-
BbIX JII1HHH). EOJIblllOH HHTepec npe,n:cTaBJIHeT BbIHCHeHHe oc06eHHocTeH
nOBe,n:eHHH HanpH)!{eHHOrO COCTOHHHH B6JIH3H TaKHX pe3KHx H3MeHeHI1H
<POpMbI rpaHHIJ:bI. 31'01' BOlIpOC B paMKax pa3BHBaeMoro MeTo,n:a MO)!{HO
pellllITb JIHlllb Ha OCHOBe 60JIee rJIy60KHx 3HaHHH 0 lIOBe,n:eHHH HeH3-
BeCTHblX B 6ecKoHeIlHblx CHCTeMax.
1l0lIOJIHHTeJIbHble CBe,n:eHHH 0 HeH3BeCTHblX B CHCTeMe (3.1) He
y,n:aeTCH lIOJIYIlHTb, He ,n:eJIaH HOBbIX npe,n:IIOJIO)!{eHHH OTHOCHTeJIbHO
CHCTeMbI (3.1), T. e. He HaJIO)!{HB KaKHX-JIH60 orpaHHIleHHH Ha ee KO-
3<pqmUI1eHTbI H cB060,n:Hble IlJIeHbI, KpoMe (3.2) H ( 3.3). D. M. KOH-
JIOBI1IlY y,n:aJIOCb lIOJIYIlHTb Ba)!{HOe CBOHCTBO peweHHH CHCTeMbI (3.1),
KOTopoe OH Ha3BaJI 3aKOHOM aCHMIITOTHIleCKHX Bblpa)!{eHHH. CYIll.eCTBO
9TOro 3aKOHa Bblpa)!{aeTCH paBeHCTBoM
lim x n = Jim Y n = Ao. (3.6)
n-+co n-+oo


2) MO:n{HO YKa3aTb TaKoe lIOJIO)!{HTeJIbHOe IlHCJIO tJ, lIPH KOTOpOM
IlI'paneHCTBa


P
fJJl (n) +
 akn >- 8,
k=1
i[p
1}J I (n) +
 u'kn> 8
k=1


(3.8)


OH YCTaHOBHJI, qTO C pOCTOM HOMepa HeH3BeCTHble CTpeMHTcH K
06Il1.eMY ,n:JIH HHX OTJIHqHOMY 01' HYJIH lIpe,n:eJIY.
1l0Ka3aTeJIbCTBo. cnpaBe.ll.JIHBOCTH COOTHoweHHH (3.6) 6a3HpyeTcH
Ha CJIe,n:YIOIlI.Hx CBOHCTBax K03<pqmIJ:HeHTOB CHCTeMbI (3.1):
1) MO)!{HO HaHTH ,n:Ba TaKHX lIOJIO)!{HTeJIbHbIX IlHCJIa 1 H L, lIPH KO-
TOpbIX ,n:JIH Bcex k, MeHbWIIX n, cnpaBe,n:JIHBbI HepaBeHcTBa
1
 Ukn L 1 akn
"'" 'PI (n) -<, -< 'ljJl (n) -< L (3.7)
npH JII06bIX 3HaIleHI1HX n;


('lIp.lBe,n:JIHBbI,n:JIH n = p + I, P + 2, ..., 2p npH JII060M p.
3aKoH aCHMIITOTHIleCKI1X Bblpa)!{eHHH ,n:aeT B03MO)!{HOCTb nOJIHOCTblO
P('IIIHTb BOlIpOC 0 nOBe,n:eHHH H3YIlaeMblx lIOJIeH B6JII13I1 yrJIOBbIX TO-
'I('K. TaK, ,n:JIH lIepBoH OCHOBHOH rpaHHIlHoH 3a,n:aIlH y,n:aeTcH lIOKa3aTb,
'11'0 Bblpa)!{eHHH .lI.JIH HalIpH)!{eHHH B yrJIOBbIX TOIlKax He co,n:ep)!{aT
oc06eHHOCTeH, a caMH HalIpH)!{eHHH HMelOT 3,n:eCb BlIOJIHe OlIpe,n:eJIeHHble
IWlleIlHble 3HaIleHHH.
3aKOH aCHMIIToTHIleCKllx Bblpa)!{eHHH H OTMeqeHHble Bblwe CBOHCTBa
IIIMHTaHT HBJIHIOTCH OCHOBOH HOBora clIoco6a peweHllH 6eCKOHeIlHblx
cilCTeM, KOTOpbIH 3HaIlHTeJIbHO COKpaIll.aeT 06'beM He06xo,n:HMbIX BbI-
'1IICJleHI1H H CYIlI.eCTBeHHO lIOBblwaeT KaIleCTBO lIOJIYIleHHoro pe3YJIb-
T.lTa. TIpllMepbI lIpHMOro HClIOJIb30BaHI1H 3TI1X B03MO)!{HOCTeH ,n:aHbI B
IIqmoH II BTOpOH rJIaBaX pa60TbI. I1ClIOJIb30BaHHe CBOHCTB JIHMllTaHT
I
a.llo B03MO)!{HOCTb B paCCMOTpeHHbIX 3a,n:aIlax lIOJIYIlHTb Ha OCHOBe pe-
III(,IIHH KOHeIlHOH CHCTeMbI Bce HeH3BeCTHble H3 6eCKOHeQHOH CllCTeMbI
(' O'leHb BbICOKOH TOIlHOCTblO.
TIOMllMO 60JIbWOrO 3HaqeHlfH, KOTopoe OlIllcaHHble pe3YJIbTaTbI llMe-
111'1 IIPH peweHHH OT,n:eJIbHbIX rpaHllIlHbIX 3a,n:aIl, CJIe,n:yeT YKa3aTb Ha
II x Ba)!{HOCTb C TOIlKH 3peHI1H H3YIleHHH CBOHCTB HalIpH)!{eHHOra CO-
ITOHHHH B6JIH3H yrJIOBbIX TOIleK B TeJIaX KOHeIlHbIX pa3MepOB. 3HaHHe
JTIIX CBO, ikTB lIOJIYIleHHoe lIPH paCCMOTpeHHH lIpOCTeHwllX 3a,n:aIl,
l',;IY)!{HT OCHOBOH lIOCTpoeHHH 3<tJ<peKTllBHoro MeTo,n:a H3YIleHHH HalIpH-
il\l'IIIIOCTH B 60JIee CJIO)!{HbIX CJIYIlaHX, Kor,n:a B03HHKalOIlI.lle lIpll y,n:OB-
.1l'TBOpeHHH rpaHHIlHbIM YCJIOBHHM COOTHoweHHH ropa3.l1.0 CJIO)!{Hee
('IICTeMbI (3.1). llJIH 3TH X CJIYIlaeB B pa60Te lIpe,n:JIO)!{eH ClIoco6 pelIIeHllH
OCIIOBHbIX COOTHoweHHH, 6a3HpYIOIlI.HXCH Ha YCTaHOBJIeHHbIX B OT.lI.eJIb-
IIblX 3a.ll.aQaX CBH3HX Me)!{.lI.Y xapaKTepOM HalIpH)!{eHHOrO COCTOHHHH B
yrJIOBbIX TOIlKax II lIOBe,n:eHHeM pH,n:OB H HHTerpaJIOB, OlIHCbIBalOIIJ:llX

 ro COCTOHHHe.
TIpH paCCMOTpeHHH rpaHllIlHbIX 3a,n:aIl .lI.JIH ylIpymx TeJI KOHeIlHbIX
pa.!MepOB lIOMllMO CHCTeM THIIa (3.1) BCTpeQalOTCH TaK)!{e po,n:CTBeHHble
11M CHCTeMbI HHTerpaJIbHbIX ypaBHeHHH:


x (t) = S Y (s) a (t, s) ds + b (t).
o


IX>
Y (t) = 5 x (s) u, (t, s) ds + B (t), (3.9)
o
/I nlK:»<e CHCTeMbI CMewaHHoro THlIa, CBH3bIBalOIlI.He <PYHKIJ:llH ,n:llcKpeT-
1101 0 H nelIpepbIBHoro aprYMeHTa:


20


21




co


x (t) =
 a k (t) Y" + b (t),
k=1


co


paeeY>KJ1.emUI B. M. KOSJJIOBHqa 0 CHCTeMax (3.9) H (3.10). LlJISJ xapaK-
TI'IJlICTHKH cy
ecTBa 3THX YCJIOBHll npHMeHHM HeKoTopbIe BbIKJIaJ1.KH
I{ l'IICTeMe (3.10).
PaccMoTpHM CHCTeMY (3.10), J1.JISJ K03tPqJHlI,HeHTOB KOTOpOll BbInOJI-
111'111,1 YCJIOBHSJ (3.12). CqHTaeM, qTO HaJ1.JIe>Ka
eH 3aMeHoH HCKOMbIX
nt'JlllqHH 06ecneqeHa nOJIO>KHTeJIbHOCTb ee CBo6o.D.HbIX qJIeHOB. Dpl1
raKHX npe.D.nOJIO>KeHHSJx CHCTeMa (3.10) HMeeT nOJIOLKHTeJIbHOe rJIaBHOe
pelUeHHe, KOTopoe MOLKeT 6bITb Hall.D.eHO MeTO.D.OM nOCJIe.D.OBaTeJIbHbIX
IIpIl6JIH>KeHHll.
Dpe.D.CTaBHM COOTHOllleHHSJ CHCTeMbI (3.10) B BH.D.e


Y n = \ an (s) X (s) ds + !)n-
ii
DpeJ1.JIO>KeHHe 0 nOJIO>KHTeJIbHOCTH SJ.D.ep HHTerpaJIbHbIX ypaBHeImH
H K03tPtPHlI,HeHTOB 6eCKOHeqHOH CHCTeMbI nOqTH He HapywaeT 06
HOCTH
npH paCCMOTpeHHH KpaeBbIX 3a.D.aq, .D.aeT B03MOLKHOCTb .D.OKa3aTb cy
e-
CTBOBaHHe OrpaHHqeHHbIX rJIaBHbIX peweHHH, eCJIH .D.JIS1 CooTHoweHHll
(3.9) H (3.10) BbInOJIHSJlOTCH YCJIOBHSJ THna YCJIOBHll perYJIS1pHOCTI1 .D.JIH
6eCKOHeqHbIX CflCTeM.
LlJIH CHCTeMbI HHTerpaJIbHbIX ypaBHeHHll (3.9) 3TH YCJIOBHSJ I1MelOT
BH.D.


co


I a (t, s) ds = I - qJ (t),
o


qJ (t) > O.
I b (t) I c.. K
qJ (t) ...." ,


(3.10)


co


co N
x (t) =
 a k (t) Yk +
 a k (t) Yk + b (t),
k=N+J k=J
T
Yn = .\ an (s) X (s) ds + ,\ an (s)x (s) ds +
n'
T 0


(3.13)


S a (t. s) ds = 1 - 'IjJ (t),
o


YCJIOBHH perYJIS1pHOCTH 3TOll CHCTeMbI .D.alOT CJIC.D.YlOIl1.He paBeHcTBa:


'IjJ(t) > 0,

 c.. K
'IjJ (t) ...." .


(3.11 )


o N
1 =
 a k (t) -f
 a k (t) + qJ (t),
k=N+1 k=1
T
1 = .\ an (s) ds + ,\ an (s) ds + 'IjJ (n).
T 0


(3.14)


AHaJIOmqHO 3anHCbIBalOTCSJ YCJIOBHH perYJISJpHOCTH .D.JISJ COOTHO-
llIeHHH (3.10):


3.D.eCb H B (3.11) K - HeKOTopaH nOJIOLKHTeJIbHaH nOCTOHHHaH. ECJIH
3HaqeHHe 3TOll nocToHHHOll onpe.D.eJIeHO, TO OqeBH.D.HOH 3aMeHOll HC-
KOMbIX BeJIHqHH CBo6o.D.HbIe qJIeHbI B COOTHOllIeHHHX (3.9) H (3.10) MorYT
6bITb C.D.eJIaHbI nOJIO>KHTeJIbHbIMH. CJle.D.yeT nO.D.qepKHYTb, qTO Oll,eHKH
B (3.11) H (3.12) OTHollleHHll CBo60.D.HbIX qJIeHOB K cOOTBeTcTBYlO
HM
q,YHKlI,HHM, xapaKTepH3YlO
HM perYJIHpHOCTb, .D.OJI>KHbI 6bITb paBHo-
MepHbIMH, T. e. .D.OJI>KHbI BbInOJlHHTbCH .D.JIH Bcex 3HaqeHHll nepeMeHHoll t
H napaMeTpa n.
CHcTeMbI COOTHOllIeHHH (3.9) H (3.10) nOpO>K.D.alOTCH rpaHHqHbIMH
3a.D.aqaMH .D.JIH «noJIyorpaHHqeHHbIx» ynpymx TeJl (noJlynoJIoca, nOJIY-
6ecKoHeqHbIll lI,HJIHH.D.P, nOJIynpocTpaHcTBO C lI,HJlHH.D.pHqeCKOll nOJIOC-
TblO H .D.p.). <PH3HqeCKaH pO.D.CTBeHHocTb HX C 3a.D.aqaMH .D.JIS1 aHaJIO-
rHqHbIX KOHeqHbIX ynpyrHx TeJI HBJIHeTCH OCHOBaHHeM .D.JIH TOro, QTo6bI
O>KH.D.aTb CnpaBe.D.JIHBOCTH 3aKOHa aCHMnTOTHQeCKHX BbIpa>KeHHH .D.JIH
HeH3BeCTHblX B (3.9) H (3.10). Heo6xo.D.HMbIe .D.JIH 3Toro cooTBeTCTBYlO-

He tPopMaJIbHbIe YCJIOBHH, HaJIaraeMbIe Ha H.D.pa HHTerpaJIbHbIX ypaB-
HeHIIi! H K03cpcpHlI,HeHTbI pS1.D.a B (3.10), JIerKO nOJIYQHTb, nOBTOpHH



 a k (t) = I - qJ (t).
k=J


qJ(t) > 0,
l!iQl c.. K
<p (t) "" ,


00
S an (s) ds = 1 - 'IjJ (n),
o


YMHO>Ka5I paBeHcTBa (3.14) Ha HeKOTopoe npOH3BOJIbHOe QIlCJIO M
11 BbIqHTaH I1X H3 cooTBeTcTBylO
Hx ypaBHeHHH CHCTeMbI (3.13), Haxo.
.D.HM


'IjJ(n) > O.
I
nl / K
'IjJ(n)
 .


(3. 12)


> J N
,-,
x(tI-M=
 ak(t)(Yk- M ) + ,b(t) + kjak(t)Yk-
.
N-!-I I k=J


- M [qJ (t) + k
 a k (t)]} , (3.15)
Yn - M = T an (s) [x (s) - M] ds + {
n + f an (s) X (s) ds-
T 0


-M [
(n>+ ! a n (,) d,]I.


DOJIYQeHHa5I CHCTeMa ypaBHeHHH (3.15), paccMaTpHBaeMaH KaK
CHCTeMa .D.JIH onpe.D.eJIeHHH Yn - M npH n > N H X (t) - M npH t >
> T, OQeBH.D.HO, Y.D.OBJIeTBopSJeT Hco6xo.D.HMbIM YCJIOBHHM perYJIHpHoCTII.
ECJIII ee cBo6o.D.HbIe QJleHbI 6Y.D.YT nOJIO)KHTeJIbHbI, TO H rJIaBHOe pellIeHHC
6Y.D. eT nOJIO>KHTeJIbHO. OTClO.D.a, BbI6paB B KaQeCTBC M MeHbllIee 113 .D.BYX


22


23




q'HCeJI,


Y,D.OBJIeTBOpHIOIIJ.HX HepaBeHCTBaM
N
b (t) + .
 4k (t) Vk
k=1
N
cp(t) +
 Qk (t)
k=l


M<:


(t > T),


T

n + S r:L n (8) X (s) ds
M<: 0


(3.16)


npHcTyn8H K HCCJIe,D.OBaHHlo CBoikTB JIHMHT8HT (3.4) npu YBe-
JIHlIeHHH j Hi', D. M. KOHJIOI3HlI OpHHHJI j = j', T. e. Hsy'I8JI CBoikTB8
HeHSBeCTHbIX B 6eCKOHelIHOH CHCTeMe H8 OCHOl3e ope,D.nOJIOLKeHHH 00
O,D. HO l3peMeHHOM YBeJIHlIeHHH KOJIHlIeCTB8. nepBbIX HeHsBecTHbIX X n H
Un> npHBJIeKaeMbIX ,D.JUI oueHKH SHalIeHHH HeH3BeCTHbiX C BbICOKHMM
HOMep8MH. TIOJIYlIeHHble HMpeSYJIbT8TbI COXpaHSIIOT CMJIY npH npOHS-
BOJIbHOH .cml3H MeLK)J.Y j H j' BH)J.a j = yi', r,D.e 'V - npOH3BOJIbHOe KO-
HelIHoe u.eJIOe lIHCJIO. AH8JIOrHlIH8SI CBSISb Me:m,D.Y N M T ,D.OJI:mHa ObITb
YCT8HOBJIeHa npH HSYlIeHHH CBOHCTB Bb!p8LKeHHH (3.18). IVUI: npOCTQTbI
6Y)J.eM C'IHT8Tb B ,D.aJIbHeHllleM T = N.
He npHBJIeKaSI HHKRKHX )J.onOJIHHTeJlbHbIX ope,D.OOJIOLKeHHH OTHO-
CHTeJIbHO Ko3<jxpHlmeHToB CHCTeMbI (3.10), H8 OCHOBe aHaJIHSa BbIpa-
:meHHH (3.18) )J.JISI JIHMHTaHT JIerKO YCT3.HOBHTb, lITO C YBeJIHlIeHHeM T
H NaepXHHe H HHLKHHe rpaHHl!bI JIHMHT8HT 06p8SYIOT COOTBeTCTBeHHO
y6bIB8I01II.YIO H BospaCTaI01II.YIO nOCJIe,D.OBaTeJIbHOCTH nOJIOLKHTeJIbHblX
lIHCeJI. TIOCKOJIbKY rJIaBHOe pellleliHe C.\ICTeMbi (3.10) He MOLKeT npeBSOH-
TH BeJIHlIHHY K B (3.12), TO OTCIO,D.a CJIe,D.yeT cYIIJ.ecTBOBaHHe BepxHero
M H HHLKHero m npe,D.eJIOB
THX nOCJIe,D.OB8TeJIbHOCTeH. .lloKaSaTeJIbCT-
130 TOTO, lITO m = M, H eCTb )J.OKaSaTeJIbCTBOM cop8I3e)J.JIHBOCTH saKOH8
aCHMnTOTHlIeCKHX abIpaLKeHHH )J.JISI HeHSBeCTHblX B (3.10).
PaccMOTpHM BbipaLKeHHSI )J.JISI JIHMHT8HT (3.18) npM OpHHSITOM Ha-
MH YCJIOBHH T = N:


T
'IjJ (
) + S r:L n (s) ds
o


(n > N).


CJIe,D.yeT


x(t»M (t>1); Yn>M (n>N). (317)
BepxHHe Ol!eHKH ,D.JISI HeHSBeCTHbIX B CHCTeMe (3.15) nOJIYlIHM, eCJIH
BbloepeM M TaE:, IITOObI ee CBooo)J.HbIe
IJIeHbI ObIJIH OTpIIUaTeJIbBbIMH.
DO,D.bITOLKHBaSI Ope.lJ.b!)J.Y1II.lle paccYLK.D.eHHSI, 3aKJIIOlJ;feM. lITO SHaHEe
3Ha'IeHHH HeCKOJIbKHx nepBblx HeH3BeCTHbIX Un (n - 1, 2. .... N)
H 3HalIeHHH X (t) (0 <: t <: T) n03BOJISIeT OOJIYIIHTb BepXHHe H HII:mHIIe
oueHKII )J.JIR OCTaJIbHbIX HeH3BeCTHblX Un II SHalIeHIIH <PYHKl!IIH X (t)
npII t> T. TIpll 3TOM HeOOXO,D.HMO naHTH HaHMeHbIllYIO HIILKHIOIO II HaH-
OOJIblllYIO BepXHlOIO rpaHHl!bl CJIe)J.YIOIIJ.IIX BblpaLKeHHH, HaSbIB8eMbIX
JIllMllTaHT8MH:


(3 18)


W (t) =


N

n + S an (s) X (s) ds
o
N
'1]) (n) + S r:Ln (s)ds
o
N
b (t) +
 ak (t) Uk
k=I
N
q (t) +
 Qk (t)
k=1


(n > N),


W (t) =


T
an + S r:L n (s) X (s) ds
V n = 0 T
'IjJ (
) + S r:L n (S) ds
o
N
b (t) + .L Qk (t) Uk
k=l
N
CjJ. (t) +
 Qk (t)
k=1


v=
n


(n >N),


(3.19)


(t> N).


(1) T).


BbioepeM HeKOTopoe 3HaqeHlIe N. HaopHMep N o - TIYCTb Mo.
 TOII-
HaR BepXHSISI rpaHlU
8 3HaqeHHH V n II W (t) Opll N = No ,D.JISI n > No,
t> No. .llJISI 3HalIeHHH n H t nOJIOLKllM
Un = Mo --l)n (n > No); X It) = Mo - Y (t) (t > Nil). (3.20)
r,D.e, OqeBHJ(HO, Bce IJ Ii H Y (t) 60JIbllle HYJIH.
3aMeTIIM, (no ,D.JISI ,D.OCTaTOIIHO OOJIhlllHX No pa3HocTb Mo - M
MOLKeT OI,ITb CKQJIb yro,D.HO MaJIOH.
TIYCTI, Teneph N I - npOH3BOJIbHOe 'IHCJIO, opeBoCxoJ(H1II.ee No,
II M I - TOlIH8SI BepxHSISI rpaHHI
a 3HaqeHIIH V n II W (t) npH n > N}
H t > N 1. TIpHMeM, ,D.JISI onpe)J.eJIeHHOCTH, LJTO BepXHHSI rpaHHl!a ,D.a.eTCSI
JllfMHTaHTOM V n npH HeKOTOpOM n = n l > N 1 , He HCKJIIOlIaR CJIyqaSI
"l = 00.


HeOOXO)J.HMbIe ,D.JISI cnp8Be,D.JIHBOCTR 38KOHa aCHMnTOTll'leCKHX BbI-
pa:meHHH OTHOCHTeJIbHO HeH3BecTHbIx B CHCTeMe (3.1) yCJIOERSI (3.7)
II (3.8) YCTaHOBJIeBb! D. M. KOSIJIOBlIIJeM nYTeMTIIJ.8.TeJIbHoro aH8.JIIU
a
CBOHCTB JIHMHTaHT (3.4). 3TH CBOHCTBa .n;8HbI B pSI,D.e TeopeJ\l, nOBTope

HHe )J.OK83aTeJIbCTB KOTOpbrx B nOJIHOM OO"beMe npHMeHHTe.1bHO K Bbl

paLKeHMHM (3.18) BPR)J. JIH l!eJIeCOOOp8SHO. OCTaBoBllMcR Ha paccMoT-
peHIIH JIRlllb Tex MOMeHTOS B 06IlI,eH TeopIIIf JI.HMlfTaHT, B
{QTOpbIX no-
Tpe6oBaJIOCb BBe,D.eHlfe )J.onOJIHlITeJIbHbIX npe,D.nOJIo:meHHH (3.7) 11 (3.8)
o K03<p<p H l!HelrTax CHCTeMI,I (3.1), If YKClLKeM HX 130SMOLKHYIO tPOPMYJIH-
pOBKY npIIMeHHTeJIbHO .K COO1'HOllleHHHM (3.1O).
24


25


III




C YIJeTOM COOTHDlIleHHH (3.20) npeJlCT8BHM Bblp8>KeHVle llJHI V n B
CJIellYfOIlI.eM BHJle:
No N, N,
(3n +- S an (5) x (s) ds +- Mo S an (s) ds \ an (s) l' (5) ds
o No No
No N, N,
'lj1 (n) +- S an (s) ds +- .\ IXn (s) ds 'ljJ (n) +- 5 IXn (s) ds
o
 0
'-IHc.JIlne.JIb H 3HaMeHaTeJIb nepBoH llpo6H B 910M BbIpa>KeHHH o6pa-
30BaHbI C.JIO>KeHHeM 'IHCJIHTeJIeH H 3HaMeHaTeJIeH JlBYX Jlpo6eH, H3 KO-
TOpbIX nepBaH ne MO>KeT 6bITb 60JlblIle Mo, 8 BTOpaH paBHa Mo H, CJJe-
JlOBaTe.JIbHO, nepBoe c.JIaraeMOe B (3.21) He MO>KeT npen30HTll Mo. Ho no
npHHHToMY npeJlno.JIO>KeHHfO npH n = n I BbIpa)KeHHe Jl.JIH V" JlOJJ)KHO
paBHHTbC51 MI' KOTopoe llJJ51 JlOCTaTOIJHO OOJIblIlHX No CKOJIb yro]tHo
Ma.JIO 01JIH'IaeTCH OT M. OTCfOJla c.JIeJlyeT, <iTO ]tJIH n = n I llpo6b
Nt
.\' an (s) l' (s) ds
1) (n) = No


OTCIOJla CJIeJlYeT OueHKa


Nt
5
 (n.) an, (s) l' (s) ds
N.


(3.24)


v=
n


(3.21)


N <.8,
K 2 +- \

 (n.) an, (s) ds
11


nle 8 - CKO,JIb yrOJlHO MaJIOe IJHCJIO.
PaCCM8TpHBaH Bblpa>KeHHe Jl,JI51 '11 (n) npH npOH3BO,JIbHOM n Ha OC-
HOBe Oll.eHKH (3.23), HaXOJlHM


N,
\ an, (s)
 (n.) l' (s) ds
i.!.


L2
'11 (n) <: [2" .


L2
N, <: [2"8 2 .
K 2 + \ an, (s)
 (n.) ds
o


3TO HepaBeHCTBO YK83bIB8eT, 'ITO CnpaBeJlJJHBOCTb Oll.eHOK (3.23),
rJle cjJYHKlI.HH
 (n) He onpe]teJI5IeTCH HenOCpeJlCTBeHHO tJepe3 !jJYHKlI.HH.
xapaKTepH3YfOlilHe perYJIHpHOCTb CHCTeMbI, BJIeIJeT 3a c060H MaJJOCTb
BToporo CJJaraeMoro B (3.21) npH JJf06bIX n > N l'
nepeHAeM Tenepb K nepBOH Jlpo6H B Bblpa>KeHHH (3.21). PaccMoTpHM
ee 'IHCJJHTeJ1b H 3HaMeHaTeJIb, pa3Jle.'1HB HX npeJlBaplITeJIbHO Ha OJlHY H
N,
TY llie BeJIH'IHHY .\ a (s) as. TorJla 'IHCJJHTeJIb npe.n.CTaBHTcH B BHJle
No


(3.22)


N,
'ljJ (n) +- \ an (s) ds
.0
JlOJIlliH8 6bITb CKO,JIb yrOJlHO MaJJOH npH llOCTaTOQHO 60,JIblIlOM No He-
31BHCHMO OT 3Ha'IeHHH N I .
npeJlbIJlYlilHe paCCYlliJleHH51 6bIJIH, no CYlIlecTBY, nOBTopeHHeM H3-
JIOllieHlH1 pa60Tbl E. M. KOHJIOBHIJa [1111. OTCTynJIeHHe B Jla,JIbHeHlIleM
OT nOJ1XOlla E. M. K05lJIOBHIJa JlHKTyeTcH Heo6xoJlHMOCTbfO paccM8T-
pHBaTb npH pelIleHHH 3aJla'J TeopHH ynpyrocTH CHCTeMbI, JlJIH KOTO-
pbIX TeopHH E. M. KOHJJOBi1'Ja EO Bcex ee JleTaJI5IX HenpHrOJlHa.
lI,JIH JlOKa3aTeJIbCTBa TOro, 'ITO HHTerpaJI '11 (n) B (3.22) HBJI5IeTC5I
Ma.'IbIM He TOJIbKO npH n = nIt HO H JlJIH JJf06oro n > N I , B TeopHH
E. M. KOHJIOBHIJa BBOJlHTCH npeJlnOJIOllieHHe 06 OJlHHaKOBOM xapaKTepe
y6bIB8HHH K03!jJ!jJHuHeHTOB CHCTeMbI H !jJYHKUHH, xapaKTepH3YfOlIlHX
perYJIHpHOCTb COOTHOlIleHHH (3.7). B HeKoTopbIX CJIYIJaHX (CM.
 5)
3TO npeJlnOJIOllieHHe He BbIno.mmeTCH. OllHaKO eC.'1H H8HlleTCH T8Ka51
<fJYHKIlH5I S (n), 'ITO 'IjJ (n) S (n) < K 2 H
n
 (n) < K I npH n = I, 2,
..., a JlJI51 Bcex n > s HMeeM
1 <: a" (s)
 (n) <: L (n > S),
TO MOlliHO nOK83aTb, 'ITO H3 !jJaKT8 MaJIOCTH fIHTerpaJI8 '11 (n) npH He-
KOTOpOM n = n I CJJellyeT ero MaJIOCTb JlJI51 Bcex n > N l'
lIeHcTBHTeJIbHO, npH n = n I , C O)l.HOH CTOpOHbI, HMeeM '11 (nl) < B,
a C JlpyroH,-


No
(3n +- I" IXn (S) x (s) ds
()
N,
S an (s) d&
No


Mo+


(3.25)


11.'151 BTOpOrO CJIaraeMOrO B nOJIY'IeHHOM BblpallieHHH C YIJeTOM HepaBeH-
CTBa X (s) <: K HaXOJlHM OIleHKY


(3.23)


N.
1; (n) (3n +- 5 1; (n) an (s) x (s) ds
o


K 1 + KLN o
<: 1 (N 1 - No) ,


N,
5 1; (n) an (s) ds
N.
H3 KOTOpOH CJJeJlyeT, tJTO HaJlJIelliaIll.HM BbI60pOM N 1 BTopoe CJJaraeMOe
B (3.25) MOllieT 6bITb CJleJIaHO CKOJIb yrOJlHO MaJIbIM npH JIf060M
Il > N I . TaKHM 06pa30M, tJHCJIHTeJIb B nepBOM CJJaraeMOM BblpallieHHH
(3 21) MOllieT 6bITb CJleJIaH CKOJIb yroJlHO MaJIO OTJIH'IafOIlI.HMCH OT Mo.
AHaJIOrH'IHbJe paCCYlliJleHH51 npH paCCMOTpeHHH 3HaMeHaTeJIH
lIepnoro CJIaraeMoro B (3.21) n03BOJI51fOT 3aK.lfOlJHTb, tJTO ero BeJJH'IHHa


N,
5
 (n.) an, (s) 'I' (s) ds
i!1(n I ) >- No N,
K 2 +- .r 1; (n I ) ant (s) ds
o


26


27


I


II
I


u


)"1
1111




CnellHaJIbHbIM BbI6opOM N 1 M02KeT 6bITb C
eJIaHa CKOJIb yro
HO 6JIH3KOH
K e
HHHlle.
CYMMHPYH nOJIyqeHHble OlleHKH BXO
HIllHX B Bblpa>KeHHe (3.21)

JIH V n BeJIHqHH, 3aKJIlOqaeM, qTO npH CYIlleCTBOBaHHH <PYHKllHH
 (n),
06eCneqHBaIOIlleH BblIIOJIHeHHe HepaBeHCTBa (3.23), He TOJIbKO HaH60.'lb

llIee 3HaqeHHe V n npH n = nl> HO H Bce 3HaqeHHH V n
JIH n > N 1
CKOJIb yro
Ho MaJIO OTJIHqalOTCH OT TOqHOH BepXHeH rpaHHllbI JIHMHTaHT
M npH
OCTaTOqHO 60JIbllII1X No H N 1 .
nO
06Hble npHBe
eHHbIM BbIllIe BbIKJIa
KH MorYT 6bITb nOBTopeHbI
OTHOCHTeJIbHO JIHMHTaHTa W (i),
alOIllero TOqHYlO Hl1mHlOlO rpaHHllY

JIH HeH3BeCTHbIX, B npe
nOJIOmeHHH 0 cyIlJ,ecTBoBaHHH TaKoH <PYHKllHH
q (i), qTO


npe
nO.IIomHM
aJIee, qTO Ha HHTepBaJIe No
 t
 2N o npH npo-
H3BOJIbHOM No cnpaBe
JIHBO HepaBeHcTBo
N.
'P (t) +
 Uk (t) > (j (No
 i <: 2N o ), (3.30)
k=1
r
e 8 - HeKoTOpaH nOCTOHHHaH. Tor
a H3 HepaBeHcTBa (3.29) CJIe
yeT
X (i) < M + £- (M -m) 8 (N o <: t
 2N o ). (3.31)


b (t) q (t) <: Ql' 'P (t) q (t) <: Q2 (0 <: i < 00);
1 <: Uk (i) q (i) <: L (t> k).
npH 3TOM 06IllHH BbIBO
, XapaKTepH3YlOIllHH CBoHcTBa BblpameHHH
(3.19), 6y
eT 3aKJIlOqaTbCH B TOM, qTO He TOJIbKO TOqHble BepXH5IH H
HHmHHH rpaHHllbI JIHMHTaHTOB V n H W (i) cTpeMHTcH npH N -+ 00
K onpe
eJIeHHbIM npe
eJIaM, HO H caMH 3TH JIHMHTaHTbI CTpeMHTCH K TeM
me npe
eJIaM npH JI106bIX n H i, npeBocxo
HIllHX N. Tenepb y2Ke He-
TPY
HO
oKa3aTb, qTO M = m.
npe
nOJIOmHM, qTO m =F M. nYCTb K npe
eJIY M cTpeMHTCH JIHMfi-
TaHT V no a K npe
eJIy m - JIHMI1TaHT W (t). Tor
a
JIH CKOJIb yro
HO
MaJIOro 8 MomHO YKa3aTb TaKoe No, qTo6w
M +8> V n >M-8 (n>N o ),
m + 8> W (i) > m - 8 (t> No).
113 OCHOBHOrO cBoiicTBa JIHMHTaHT CJIe
yeT
M + 8>Un>m-8 (n>N o ),
M + 8 > x (i) > m - € (i > No).
npe)lCTaBHM COOTHOllIeHHe CHCTeMbI (3.10)
JIH onpe
eJIeHHH x (i)
qepe3 Uk


(3 26)


HepaBeHCTBO (3.31) nOKa3bIBaeT, qTO npe
nOJIOmeHIIe 0 HeCOB-
na
eHHII BepXHeH M II HHmHeH m rpaHIIll JIHMIITaHT B CJIyqae CYIlleCT-
BOBaHIIH OTJIHqHOH OT HYJIH nOCTOHHHOH 8 B (3.31) npIfBO
IIT K TOMY,
qTO HeIf3BeCTHaH <PYHKllIIH x (i) .B CHCTeMe (3.10) OTJIIIqaeTCH OT M Ha
KOHeqHYlO BeJIHqIIHY
JIH Bcex No <: i <: 2N o npII JI1060M No.
OCHOBbIBaHCb Ha 3TOM <paKTe, paCCMOTpIIM 3HaqeHIIe JIHMHTaHTa
V n npH N = 2N o . I1cxo
H 113 HepaBeHCTBa (3.28) H (3.31), MomeM HaHTH
(M + 1') [ 'II' (n) + t an (s) ds . / +IM+E - (M-m) !J] 21" an (s) ds
V 0

n < 2N.
'II' (n) + S a. n (s) ds
o


(3.27)


= M + 8 - (M - m) 8 2N.
IjJ (n) + S an (s) ds
o


2N.
S an (S) ds
N.


(3.32)


(3.28)


lI.JIH BXO
HIlleH B 3TO HepaBeHCTBO
p06If, co
epmaIlleH HHTerpaJIbI
OT an (s), HMeeM Ha OCHOBe npe
nOJIOmeHHH 0 nOBe
eHHH <PYHKllHII
'IjJ (n) If COOTHOllIeHIIH (3.23) )lJIH an (s) CJIe
YlOIllYlO olleHKy:


II


N. 00
x (i) = b (i) +
 Uk (i) Uk +
 Uk (i) Uk'
k=I k=N.+I


2N.
S an (s) ds
N.
2N. >
'II' (n) + S an (S) ds
()
C yqeTOM nOJIOmHTeJIbHOCTH V n H3 HepaBeHCTB (3.32) II (3.33)
nOJIyqaeM HepaBeHCTBO


[No [
K 2 + 2LN o > Kz +2L


(3.33)


HCXO
H H3 HepaBeHCTB (3.27),
JIH nepBbIX
BYX CJIaraeMblX B 3TOM
BblpameHHH HaXO
IfM OlleHKY
N. [ N. ]
b (i) + k
I Uk (t) Uk < (m + 8) 'P (i) + k
 Uk (i)


(t>NI)'


[
V n <M+8-(M-m)8 K 2 +2L


(3.34)


I1cnOJIb3YH ee npH paCCMOTpeHHH BblpameHI1H
JIH X (t), C yqeTOM He-
paBeHCTB (3.28) HaXO
IIM, qTO
JIH Bcex i > No
X (i) < M + 8 - (M - m) [ 'P (t) + k
 Uk (i) J . (3.29)
28


CJIe
OBaTeJIbHO,
JIH npOII3BOJIbHOrO CKOJIb yro
Ho 60JIbllIOro No
311a4eHHe V n npH N = 2N o OTJIIIqaeTCH OT M Ha KOHeqHYlO BeJIHqHHY
[
(M - m) 8 K 2 + 2L '
29




a 31'0 npOTHBOpelJHT YCJIOBHlO, lJTO V n CTpeMHTCH npH N --+ 00 K npe-
11.eJIY M. f1poTHBOpelJHe, nOpO}K)l,aeMoe npe11.nOJIOLKeHHeM 0 TOM, lJTO
M =1= m, H 11.OKa3blBaeT 3aKOH aCHMnTOTHlJeCKHX BbIpaLKeHHH.
f1pe11.nOJIOLKeHHe 0 TOM, lJTO, nOMHMO HepaBeHcTB (3.23), BXOllHIllHe
B BbIpaLKeHHH 11.JIH VII tPYHKUHH Y11.OBJIeTBOpHlOT HepaBeHcTBY
N
'IjJ (n) + I an (s) ds> e (n = N + 1, N + 2, ..., 2N), (3.35)
o


M;HI O11.HHM H3 HepaBeHCTB (3.12) CPYHKUHH 'IjJ (n) = 0. YlJHTbIBaH,
'I ro <P (t) =1= 0, nYTeM HCKJIlOlJeHHH He1l3BeCTHbIX Yli B nepBoM ypaBHe-
IIHH MOLKHO CBeCTH YKa3aHHYlO CHCTeMY K HHTerpaJIbHoMY ypaBHeHHlO
(1)pe11.rOJIbMa BToporo p011.a, OTHOCHTeJIbHO KOTOpOro H3BeCTHO, lJTO
l"OOTBeTCTBYlOIllee Ol1.HOp011.Hoe ypaBHeHHe HMeeT TOJIbKO HYJIeBOe pe-
meHHe. f10JIYlJeHHOe TaKHM 06pa30M HHTerpaJIbHOe ypaBHeHHe npHHa-
(JIeLKHT K THI1Y ypaBHeHHH C 0606IlleHHbIM H11.pOM f1HHKepJIe - rypca
[173], nOCTpoeHHe perneHHH KOTOpOro nOJIHOCTblO 3KBHBaJIeHTHO perne-
l11ilO HeKoTopoH 6eCKOHelJHOri CHCTeMbI. Eha CHCTelVIa OKa3bIBaeTCH pery-
JIHpHOH, O11.HaKO yCJIOBHH CYIlleCTBOBaHHH OrpaHHlJeHHOrO perneHH5I
IIpHB011.HT K tPH3HlJeCKH He061,51CHHMbIM Tpe60BaHH5IM K nOpH11.KY y6b1-
naHHH BeJIHlJHH
 n H XapaKTepy rpaHHlJHbIX YCJIOBHH Ha cooTBeTcTBYlO-
IUHX nOBepXHOCTHX ynpyroro TeJIa.
CooTHorneHHH THna (3.10) H (3.9) B03HHKalOT npH perneHHH rpaHHlJ-
IIbIX 3a11.alJ 11.JIH ynpyrHx TeJI, rpaHHlJHa51 nOBepxHOCTb KOTOpbIX co-
}(ep)j{HT YlJaCTKH 6eCKOHelJHOH npOTmr.:eHHoCTH (noJIynoJIoca, nOJIY-
IIpocTpaHCTBo C nOJIOCTblO H 11.p.). HeH3BecTHbIe X (t) H Y (t) HrpalOT
IIPH 3TOM pOJIb nJIOTHOCTeH B cooTBeTcTBYlOIllHX HHTerpaJIbHbIX npe11.-
CTaBJIeHHHX KOMnOHeHTOB BeKTOpa nepeMelileHHH H TeH30pa Hanp5lLKe-
IIHH, nOJIYlJaeMblX Ha OCHOBe pa3JIHlJHbIX HHTerpaJIbHbIX npe06pa30-
naHHH. ECTecTBeHHoe Tpe60BaHHe 06palileHHH B HYJIb Ha 6eCKOHelJHOCTH
YKa3aHHbIX xapaKTepHCTHK HanpHLKeHHO-11.etPopMHpoBaHHoro COCTOH-
IIH5I ynpyroro TeJIa npHB011.HT K TOMY, lJTO tPH3HlJeCKHH CMbICJI HMelOT
.IHIl1b Te perneHHH CHCTeM (3.9) H (3.10), B KOTOpbIX tPYHKUHH X (t) H
Y (t) HMelOT onpe11.eJIeHHbIH xapaKTep rIOBe11.eHHH B OKpeCTHocTH TOlJKH
t O.
CBH3b MeLK11.Y xapaKTepoM y6b1BaHHH tPYHKUHH Ha 6eCKOHelJHOCTH
Ii 3HalJeHHHMii B HYJIe nJIOTHOCTH B npe11.CTaBJIeHHH ee HHTerpaJIaMH
<1>ypbe H XaHKe.n51 MO)KHO HaHTii B MOHorpatPHH [I 721-
f1YCTb OTHOCIiTeJIbHO Heli3BeCTHOH tPYHKUHIi X (t) B (3.10) 1i3 rpa-
IIHlJHbIX YCJIOBIiH Ha 6eCKOHelJHOCTH 1i3BeCTHO, lJTO OHa 11.OJI)j{Ha HMeTb B
TOlJKe t = 0 HYJIb HeKoToporo nOp5l11.Ka. 06bIlJHO TaKoe Tpe60BaHHe
lIe npOTHBOpelJliT xapaKTepy nOBe11.eHIiH K03<p<pIiUHeHTOB a k (t) H tPYHK-
IlHH b (t). Tor11.a 11.OJILKHO cYlilecTBoBaTb HeKoTopoe KOHelJHOe lJliCJIO
To TaKoe, lJT06bI HHTerp.aJI


I
II


n03BOJIHeT npHHTH K TaKOMY LKe BbIB011.Y npli paCCMOTpeHHH JIHMHTaHTa
W (t), CTpeMHlilerocH, no npe11.nOJIOLKeHHlO, K npe11.eJIY m.
f1poBe11.eHHbIH aH3J1H3 YCJIOBHH CnpaBe11.JIHBOCTH 3aKOHa aCHMnTO-
THlJeCKHX BbIpaLKeHHH 11.JI5I HeH3BeCTHbIX B CHCTeMe (3.10) n03BOJIHJl
YCTaHoBHTb, lJTO nepBoe YCJIOBHe E. M. K05lJIOBHlJa (3.7) MOLKeT 6bITb
3aMeHeHO YCJIOBH5IMH (3.23) H (3.26). He06xo11.HMOCTb 11.pyrorl, OTJIIllJ-
Hoil 01' (3.7), tPOPMYJIHpOBKH 3Toro YCJIOBHH B03HHKaeT npH paccMoT-
peHHH KOHKpeTHbIx I'paHHlJHbIX 3a11.alJ (CM.
 5). 41'0 KacaeTCH BToporo
YCJIOBHH (3.8), TO npHHHTbIe BbIrne HepaBeHcTBa (3.30) H (3.35) HBJIHlOT-
CH JIHrnb JJ.pyroH tPOpMOH era 3anHCH npHMeHHTeJIbHO K BH11.Y COOT-
HorneHHH (3.10). f10 nOB011.Y 3TOro YCJIOBHH CJIellyeT T3K)j{e 33MeTiiTb,
lJTO B TOM CJIYlJae, KOI'lla <PI (n) H 'ljJJ (n) B (3.8) y6hIBalOT C pOCTOM n K3K
I/n, HepaBeHCTBa (3.8) HBJIHlOTCH OlJeBH11.HbIM CJIellCTBHeM HepaBeHcTB3
(3.7). 31'0 33MelJ3HHe OCTaeTCH cnpaBe11.JIHBbIM TaK)Ke OTHOCHTeJlbHO
HepaBeHcTB (3.30) H (3.35), eCJIH
 (n) B (3.23) H q (t) B (3.26) paCTYT C
pOCTOM apI'YMeHToB COOTBeTCTBeHHO KaK n H t.
CtPOPMYJIHPQBaHHbIH BbIrne HTor nepecMoTpa YCJIOBHH, HaJIaraeMbIX
Ha K03<p<pHUHeHTbI 6eCKOHelJHbIX CHCTeM npH H3YlJeHHH aCHMnTOTH-
lJeCKHX CBOHCTB HeH3BeCTHbIX, n03BOJIHeT npe011.OJIeTb HeKoTopbIe
tPopMaJIbHbIe 3aTpY11.HeHHH npH perneHHH tPH3HlJeCKH p011.CTBeHHbIx
3a11.alJ, npHB011.HlilHXCH K CHCTeMaM C pa3JIHlJHbIMH xapaKTepHcTHKaMH
perYJIHpHoCTH. f1pH 3TOM, eCTeCTBeHHO, npe11.noJIaraJIOCb, lJTO xapaK-
TepH3Y lO lil He perYJIHpHOCTH $YHKUHH <P (t),
1 (n) H
1 (t) B COOTHorne-
HHHX (3.11) H (3.12) OTJIHlJHbI 01' HY JIH H B HHX MO)j{HO YKa3aTb KOHelJHOe
3HalJeHHe nocToHHHOH K.
3HalJHTeJIbHbIe TPY11.HOCTH B03HHKalOT B Tex CJIYlJaHX, KOI'11.a rpa-
HHlJHaH 3a11.aqa npHBo11.HTCH K COOTHorneHHHM THna (3.9) HJIH (3.10) C
BblpOLK11.eHHO H perYJIHpHOCTblO, T. e. KOr11.a, HanpHMep, 'IjJ (n) == 0
HJIH B (3.11) <p (t) = 'IjJ (t) = O. <1>opMaJIbHbIH aHaJIH3 cooTBeTcTBYlOIllHX
CHCTeM npH 3TOM CTaHOBHTCH 3aTpY11.HHTeJIbHbIM, a BO BTOpOM H3 YKa-
3aHHbIX CJIYlJaeB, KOr11.a O11.Hopo11.Ha51 CHCTeMa (3.9) HMeeT HeTpHBHaJIbHOe
perneHHe, H BOBce HeB03MO)j{eH. f1pe011.OJIeTb BblTeKalOlilHe H3 YKa3aH-
HbIX 06CTOHTeJIbCTB TPY11.HOCTH MOLKHO Ha OCHOBe aHaJIH3a TOll pOJIH,
KOTOpylO HrpalOT HeH3BeCTHbIe B CHCTeMax (3.9) H (3.10) B npe11.CTaBJIe-
HHH 06lilero perneHHH rpaHHlJHOll 3a11.alJH. CYlilecTBo TaKoro n011.xoJJ.a
npOHJIJIlOCTpHpyeM n011.p06HbIM paccMoTpeHHeM CHCTeMbI (3.10), HMeH
B BH11.Y, lJTO Bce CKa3aHHoe npH 3TOM MO)j{eT 6bITb nOBTopeHO H npH aHa-
JIH3e CHCTeMbI (3.9).
Pc:CCMOTpHM cHcTeMY CooTHoIIleHHH (3.10), (';lJHTaH, lJTO onpe).l,
lHe-
30:


TO
I = S an (S) X (s) ds
o


(3.66)


OCTaBaJICH KOHetIHbIM 11.JIH npOH3BOJIbHOro n, a caMa tPYHKIlHH X (8)
ObIJIa MOHOTOHHOH npH 0 -< 8
 To.
CtPOPMYJIHpOBaB Tpe60BaHHH K perneHHlO cHcTeMbI (3.10), BbITe-
KalOLUHe H3 tPH3HlJeCKOll CYlilHOCTH rpamIlJHOH 3a11.alJH, MO>KeM nepe
TH
K paccMoTpeHHlO 11.BYX OCHOBHbIX BonpocoB: CYlileCTByeT JIH pellieHHe
CHCTeMbI (3.10), 06JIa11.alOIllee HY)KHbIMH cBoHCTBaMH, H eC,lm 11.a, TO KaK
ero HaHTlf. f1pH 3TOM CJIe11.yeT HMeTb B BH11.Y, lJTO OTHOCHTeJIbHO K03<iJ-
CIJHUHeHToB CHC1'eMbI (3.10) ClJHTaeM BbIDOJIHeHHblMH Bce He06xo11.HMbIe
:lJIH cYIllecTBoBaHHH OrpaHHlJeHHOrO perneHHSI YCJIOBHH B (3.12), QT_
1I0CHlilHeCH K _ CPYHKUiIfH <p (t).


3.t


I..




To
1jJ (n) = S an (s) ds.
o


(3.39)


IllY II KIUUI 1jJ (n) onpe.n:e.neHa, ro K HeH B nOJIHOH Mepe npHJIO:>KeH!>I
YI'JIOBH5I cnpaBe.n:JIHBOCTH 3aKOHa aCHMnTOTH1J:eCKHX Bblpa:>KeHHH.
I I Ol:JIe.n:CTBH5I BblpO:>K.n:eHH5I perYJI5IpHOCTH CKa3bIBaJOTC5I npH 3TOM JIHilib
1\ TOM, 1J:TO .n:JI5I CHCTeMbI THna (3.10) He y.n:aeTC5I nOCTpOHTb BbIpa:>KeHH5I
JIIIMHTaHT. 3THM HCKJIJ01J:aeTC5I B03MO:>KHOCTb HcnOJIb30BaHH5I npH HX
pcweHHH MeTo.n:a cTyneHeH [1111. O.n:HaKo, KaK 6y.n:eT nOKa3aHO npH
pcweHHH KOHKpeTHblx 3Ma1J:, 3HaHHe aCHMnTOTH1J:eCKHX CBOHCTB- He-
H3BeCTHblX 13 CHCTeMax C BblpO:>K.n:eHHoH perYJI5IpHoCTbJO n03BOJI5IeT
3Ha1J:HTe.nbHO YJIY1J:IlIHTb MeTO.n: pe.n:YKIJ.HH H nOJIY1J:HTb Ba:>KHble .n:aHHble
o Ka1J:eCTBeHHblX oc06eHHOCT5IX Hanp5l:>KeHHOrO COCTOSlHHSI 13 oKpecTHoc-
TH yrJIOBbIX T01J:eK.
3aKaH1J:HBa51 KpaTKHH 0630p TeopHH 6eCKOHe1J:HbIX CHCTeM, CJIe.n:yeT
OCTaHOBHTbCSI Ha oueHKe 3Ha1J:eHH5I ee BWBO.n:OB C T01J:KH 3peHH5I HX
nOJIe3HOCTH npH peIlIeHHH rpaHH1J:HbIX 3a.n:a1J: TeopHH ynpyrocTH. 3.n:ecb
pe1J:b 6y.n:eT H.n:TH 0 6eCKOHe1J:HbIX CHCTeMax JIHHeHHblx aJIre6paH1J:eCKHX
ypaBHeHHH, HO Bce c.n:e.naHHble 3aMe1J:aHH5I 13 nOJIHOM 06'beMe OTHOC5ITC5I
H K po.n:cTBeHHbIM HM CHCTeMaM mna (3.9) H (3.10).
B npe.n:bI.n:y
eM naparpa<pe 6bIJI H3JIO:>KeH MeTO.n: nOCTpOeHH5I 06-

ero peIlIeHH5I KpaeBblx 3a.n:a1J: .n:JI5I ynpyrHx Te.n KOHe1J:HbIX pa3MepoB.
I10.n:1J:HHHB 3'1'0 peIlIeHHe yCJIOBH5IM Ha rpaHHue .n:JISI onpe.n:eJIeHH5I CO-
.n:ep:>K
HXCSI 13 HeM nOCT05lHHbIX, nOJIY1J:HM 6eCKOHe1J:Hble CHCTeMbI
JIHHeHHbIX aJIre6paH1J:eCKHX ypaBHeHHH. ECJIH .n:JI5I HHX y.n:aeTC5I YC-
TaHOBHTb perYJI5IpHOCTb, TO 3'1'0, KaK npaBHJIO, 03Ha1J:aeT, 1J:TO:
1) peIlIeHHe rpaHH1J:HOH 3a.n:a1J:H 13 npHH5ITOH <POp Me cyruecTByeT,
2) 3Ha1J:eHH5I HeH3BeCTHblX MorYT 6bITb HaH.n:eHbI MeTo.n:OM nOCJIe-
.n.OBaTe.nbHbIX npH6JIH:>KeHHH HJIH MeTo.n:OM pe.n:YKU HH .
I1pH 3TOM 6e3 OTBeTa OCTaJOTC5I .n:Ba Ba:>KHbIX C T01J:KH 3peHH5I <p aK -
TH1J:eCKOrO peIlIeHH5I rpaHH1J:HOH 3a.n:a1J:H Bonpoca. OCTaeTC5I He5lCHbIM,
KaKoe KOJIH1J:eCTBO HeH3BeCTHblX .n:OJI:>KHO 6bITb y.n.ep:>KaHO 13 pe.n:YUH-
pOBaHHoH CHCTeMe .n.JI5I y.n:oBJIeTBOpHTeJIbHOrO onHCaHH5I Hanp5l:>KeHHH H
cMe
eHHH 13 ynpyroM TeJle. KpoMe TOro, .n:OKa3aTeJlbCTBO pa3peIlIH-
MOCTH CHCTeMbI e
e He .n:aeT B03MO:>KHOCTH H3Y1J:HTb Ka1J:eCTBeHHble OCO-
6eHHOCTH HanpSl:>KeHHoro COCT05lHH5I B6JIH3H yrJloBbIX T01J:eK.
ECJlH npH OTBeTe Ha nepBbIH Bonpoc
e MO:>KHO peKOMeH.n:OBaTb
13 Ka1J:eCTBe KpHTepH5I .n:Jl5I BbI60pa nop5l.n:Ka KOHe1J:HOH CHcreMbI Tpe-
6yeMYJO CTeneHb T01J:HOCTH y.n.OBJleTBopeHH5I rpaHH1J:HbIX YCJlOBHH
(3TOT KpHTepHH 130 MHorHX CJIY1J:a5lX npaKTH1J:eCKH HepeaJlH3yeM, CM.

 8), TO OTBeTHTb Ha BTOpOH Bonpoc Bo06rue HeB03MO:>KHO 6e3 3HaHHfI
xapaKrepa nOBe.n:eHH5I HeH3BeCTHWX Ha 6eCKOHe1J:HOCTH.
OTMe1J:eHHble 06CT05lTeJlbCTBa n03BOJI5IJOT oueHHTb TO qOJlbilioe 3Ha-
1J:eHHe, KOTopoe HMeeT 3aKOH aCHMnTOTH1J:eCKHX Bblpa:>KeHHH npH
H3Y1J:eHHH Hanp5l:>KeHHo-.n.e<pOpMHpoBaHHoro COCT05lHH5I ynpyrHx TeJl KO-
He1J:HbIX pa3MepOB. ECJlH cnpaBe.n:JlHBOCTb 3Toro 3aKOHa .n:Jl5I HeH3BeCT-
HbIX 13 6eCKOHe1J:HOH CHCTeMe, nOpO:>K.n:eHHOH paCCMaTpHBaeMOH rpaHl11J:-
HOH sa.n:a1J:eH, YCTaHOBJleHa, TO n05lBJl5IeTC5I BOSMO:>KHOCTb:
1) .n.aTb nOJlHbIH Ka1J:eCTBeHHbIH aHaJlHS Hanp5l:>KeHHOrO COCT05lHH5I
ynpyroro TeJla B6J1H3H yrJIOBbIX T01J:eK,
2) Ha OCHOBe aHaJlHSa CXO.n.l1MOCTH p5l.n.OB YKa3aTb nopSl.n. OK pe.n.y-


Bonpoc 0 TOM, KaK xapaKTep nOBe.n:eHl1S1 x (t) B OKpeCTHOCTH HYJISI
CKa3bIBaeTC5I Ha 3Ha1J:eHH5IX HeH3BeCTHbIX Yn (n = 1,2, ...) H X (t) (t >
> To), peWaeTC5I npl1 paCCMOTpeHHH CJIe.n.YJO
eH CHCTeMbI COOTHOIlIeHHH:


X(t) -
 a k (t) Yk + b (t)
k=1


(t > To),


00 To
Y n = S an (s) x (s) ds + S an (s) X (s) ds +
n
To 0


(3.37)
(n = 1, 2, .. .).


3Ta CHCTeMa, 01J:eBH.n.HO, 51BJlSleTC5I perYJI5IpHOH H, CJle.n:OBaTeJlbHO,
06J1a.n:aeT OrpaHH1J:eHHbIM r JlaBHbIM peIlIeHHeM, eCJll1 .n:Jl5I Bcex n 1, 2,
... cy
ecTByeT TaKoe K =i= 0, npH KOTOpOM cnpaBe.n:JlHBO HepaBeHCTBO
I J
 an (s) x (s) ds T I3n I
111 (n) < K, (3.38)


r.n.e


LITO KaCaeTC5I nepBoro CJlaraeMoro 13 1J:l1CJlHTeJle Bblpa:>KeHH5I (3.38),
TO, HCXO.n:51 H3 CBOHCTB X (S) npH 0 < S < To, l1MeeM
I J an (s) x (s) ds I
1jJ (n) < I X (To) I. (3.40)
TaKHM 06pa30M, Bonpoc 0 cyrueCTBOBaHHH peIlIeHH5I CHCTeMbI THna
(3.10) C BblpO:>K.n:eHHoH perYJl5lpHOCTbJO (fan (s) ds = 1) B03HHKaJOruHX
npH pacCMoTpeHHH rpaHH1J:HbIX 3a.n:a1J: C 06bI1J:HbIMH 13 TeopHH ynpyrocTH
YCJlOBH5IMH Ha 6eCKOHe1J:HOCTH peiliaeTC5I 13 paMKax TeopHH perYJlSlpHbIX
CHCTeM, eCJlH 'ljJ (n) onpe.n:eJlHTb COOTHOIlIeHHeM (3.39). I1pH 3TOM
HaJlH1J:He paBHoMepHoH oueHKH OTHOIlIeHH5I I
n I /'IjJ (n) < K YKa3bIBa-
eT Ha TO, 1J:TO cy
ecTByeT OrpaHH1J:eHHOe peweHHe CHCTeMbI (3.37).
PaccMaTpHBa51 nepBoe ypaBHeHHe 13 (3.10) npH 0 < t < To 13 HaH60Jlee
He6J1aronpH5ITHOM cJlyqae Yk= K, 06bI1J:HO JlerKO y6e:>K.n:aeMC5I, liTO npo-
THBopellHH Me:>K.n:y Tpe6yeMbIMH CBOHCTBaMH X (t) 13 OKpeCTHOCTH HYJl51
H nOBe.n:eHHeM peIlIeHHfI CHCTeMbI (3.37) He B03HHKaeT. OTbICKaTb npH-
6J1H:>KeHHOe peIlIeHHe CHCTeMbI (3.10) MO:>KHO C nOMOIlIbJO MeTo.n:a pe.n.YK-
UHH.
B npe.n:bI.n:y
eM H3JIO:>KeHHH, OTHOC5I
eMC5I 13 OCHOBHOM K CJIY1J:aJO
CHCTeMbI C HeBblpO:>K.n:eHHoH perYJl5lpHOCTbJO, 60JlblliOe BHHMaHHe 6bIJlO
y.n:eJleHo <POPMYJlHpoBKe YCJlOBHH, npH KOTOpbIX MO:>KHO nOJlYllHTb 60-
JIee nOJlHYJO HH<p0pMauHJO 06 aCHMnTOTHlIeCKHX CBoHcTBax HeH3BeCT-
HbIX. I1oCKOJlbKY 13 CHCTeMe (3.37) C BblpO:>K.n:eHHoH perYJl5lpHoCTbJO
32


3 7-2464


33




UHpOBaHHOH KOItelJHOH CHCTeMbI, He06xo.1I;IIMbIH ,nJls:J: .n:OCTH)f(eHHSI onpe-
.n:eJleHHOH TOlJflOCTH BbIilOJlHeHHSI rpaHHlJHbIX YCJlOBHil,
3) TIOCTpOHTb aJlropHTM perneHHH 6eCKOHelJHOH 'CHCTeMbI,3Ha-
lJWreJlbHO npeBocxo.n:SIlli.HH no
<P<PeKTHBIi:OCTH Meto.n: npocToH pe.n.YK-
U,HH.
DpHMepbI HCnOJIb3'OB8HHfI YKa3aHHblx B03MO}KHocTeH npH perneHHH
pSI.n:a KOHKpeTHbIx rpaHHlJ:HblX a8.n:alJ co.n:ep>KaTCH B nOCJle.n:YIOlli.HX rJl
-
Bax. HeKoTopbI'e Ha paCC'MaTpHBaeMbIX a,a.n.alJ npHBO.B.}'jTClI HenOCpe:D;CT
BeHHO K OTHOlI1elHfSIM (3.1), (-:3..9) H (3.10). llpyme, pOl{CTBeHHbIe HM no
<pHaHlJeCKOH 'CYIllHOC'fH, llopojRi/;alO'f 60Jlee 'CJIOX<HbI€ CHCTeMbI COOTHO-
rneHHH. HenOCpe.lKTBeHHoe npHMeHeHHe peaYJlbTaTOB D. M. KOSIJlOBHqa
H H);: 0606IlleHHH K TaKHM CR'cTeMaM HeBOaMO}KHO. OCHOBO'H 3$$eKTHB-
HOro HX pellIeiuHI SIBJISIIOTCSI YCTaHOBJI'eHHble B OTHOCHTeJIbHO npocTbIX
3a.n.alJaX CBSI3H MeJK.n.y <paKToM Ha.nHlJHSI yrJIOBl:ilx TOlJeK H xap8KTepoM
nOBe.n;eHHSI K03<P<pHu,lfeH'fOB Pll,t(OB H nJlOTHOCTeH IfHTerpaJIbHbIX npe.n;-
CTaBJI€HHH B o6iJiJ.eM peIlIeHHH rpaHHlJHO'H aa.n:-alJ'H.


r.!£aea 1
II.JIOCKlIE rp AI-IllqHbIE 3AAA qII A.JIll TEJI
KOHEqHblX P A3MEPOB


fpaHI1lJHble 3a.n.alJH TeopHH ynpyrocTH .n.JISI npSIMoyroJIbHHKa Ii'
nOJIynOJIOCbI npI1Ha.n.JIe>KaT K HaI160JIee npocTbIM 3a.n:alJaM 113 paCCMaT-
pHBaeMoro B pa60Te KJIaCCa rpaHHlJHbIX aa.n.alJ. 3aBHCHMOCTb OCHOBHbIX
xapaKTepHCTHK HanpmKeHHo-.n:eqJOpMHpOBaHHoro COCTOSIHHSI JIHrnb OT
.n:BYX KOOp.n:HHaT, npOCTOTa <PYHKU,HH, lJepea KOTopble BbIpa}KaIOTCSI
06IllHe perneHHSl,- Bce 31'0 C03.n.aeT npe.n.nOCbIJIKH .n.JIH Toro, lJTo6bI npH
HX paCCMOTpeHHH nOJIYlJI1Tb npe.n:CTaBJIeHHe 0 xapaKTepe Tpy.n.HOCTeH,
B03HHKaIOIllHX npH H3YlJeHHH HanpSI}KeHHOrO COCTOSIHHSI TeJI KOHelJHbIX
pa3MepOB. HeCOMHeHHO, o6Illee 3HalJeHHe HMelDT TaK}Ke pa3BHTble np"
paCCMOTpeHHH nJIOCKHX 3a.n:aQ cnoc06bI npeo.n.OJIeHHH 3THX Tpy.n.HOCTeH.
llJISI perneHHSl nJIOCKHX 3a.ZI.alJ TeapHH ynpyrocTH COBpeMeHHBJI
HaYKa pacnOJIaraeT 3<p<peKTHBHbIM MeTo.n.OM, 6a3HpYIOIllHMCH Ha B03-
MO}KHOCTH npe.n:CTaBHTb H3YlJaeMOe nJIOCKOe nOJIe HanpSl}KeHHH (CMe-
IlleHHH) lJepe3 aHaJIHTHlJeCKHe <PYHKUHH. B CBS!aH C 3TI1M paCCMaT-
pHBaeMble 3.n.eCb 3a.n.alJH, B03MO}KHO, H He HMelOT caMOCTOHTeJIbHOrO
3HalJeHHSI, XOTSl CJIe.n.yeT YKa3aTb, lJTO nOCTpoeHHe perneHHSI OCHOBHbIX
rpaHHlJHbIX 3a.n.alJ .n.JISl npSlMoyrOJIbHHKa H nOJIynOJIOCbI B paMKax
MeTo.n:OB TeopHH <PYHKUHH KOMnJIeKcHoro nepeMeHHoro He SIBJISIeTCH
TpHBHaJIbHOH 3a.n.alJeH. Tpy.n:HOCTH, KOTopble B03HHKaIDT npH perneHHH
nJIOCKHX 3a.n.alJ .n.JISI npSIMoyroJIbHHKa H nOJIynOJIOCbI C HCnOJIb30BaHHeM
TaKoro MOIllHoro MaTeMaTHlJeCKoro annapaTa, KaKoH AOCTaBJISIeT Teo-
pHSI aHaJIHTHlJeCKHX <PYHKU,HH, YKa3bIBaIOT Ha TO, lJTO HaJIHlJHe yrJIOBbIX
TOlJeK B ynpyroM TeJIe CYIlleCTBeHHO YCJIO}KHSIeT KapTHHY HanpSI}KeHHO-
ro COCTOSIHHSI. 3TH Tpy.n:HOCTH 06YCJIOBJIeHbI oco6eHHocTSIMH H3YQaeMO-
ro nOJISI HanpSI}KeHHH.
DepBOH pa60TOH, B KOTOpOH npe.n.npHHSITa nonbITKa H3YliHTb Ha-
npSI}KeHHOe COCTOSIHHe npSIMoyrOJIbHHKa npH 3a.n.aHHbIX Ha rpaHHue
HanpSl}KeHHSIX, SIBJISIeTCSI pa60Ta 3. MaTbe [2351. DpH nocTpoeHHH pe-
rneHHSI 3. MaTbe HcnOJIb30BaJI MeTO.n., aHaJIOmlJHbIH no.n.xo.n.y f. JIaMe
K 3a.n:alJe 0 napaJIJleJIenI1ne.n:e. VIM 6bIJIH no.nYlJeHbI 6eCKOHelJHble CH-
CTeMbI ypaBHeHHH H HaH.n.eHO HX npH6JIH}KeHHOe perneHHe C HCnOJIb30-
B8HHeM MeTo.n:OB pe.n.YKUHH H nOCJIe.n.OBaTeJIbHbIX npH6JIH>KeHHH. C TOlJKH
3peHHSI COBpeMeHHoro CTaH.n:apTa nOJIHOTbI HCCJIe.n.OBaHHSI B TaKoro po.n.a
aa.n:aqax He.n.OCTaTKOM npo.n.eJIaHHOH 3. MaTbe pa60TbI SIBJISIeTCSI 01'-
CYTCTBHe .n:OKa3aTeJIbCTBa perYJI5IpHOCTH 6eCKOHelJHbIX CHCTeM.
BnOCJIe.n.cTBHH paccMoTpeHHIO 3a.n:alJ .n.JISI npSIMoyroJIbHHKa nOCBS!-


I


3*


35


\1




I
'I


m.aJIHCb paoOTbI [195, 210, 254 H ,D;p.]. B 3THX pa60Tax rpaHHqHbIe
YCJIOBHH Sa,D;aqH y,D;oBJIeTBOpHJIIfCb TOJIbKO Ha JI.BYX npOHIBOnOJImKHbIX
cTopoHax npHMoyroJIbHHKa. Ha ,D;BYX ,D;pymx cTOpoHax rpaHHqHbIe
YCJIOBHH y,D;OBJIeTBOpHJIHCb BeCbMa npHOJIHLKeHHo.
CTpeMJIeHHe YTOqHHTb xapaKTep HanpH>KeHHoro COCTOHHHH Sa
ct:zeT oOJIee nOJIHOrO Y,D;OBJIeTBOpeHHH rpaHHqHbIX YCJIOBHH Ha Bcex
cTopoHax npHMoyrOJIbHHKa OOYCJIOBHJIO nOHBJIeHHe
eJIoro pH,D;a pa-
OOT, B KOTOpbIX ObIJIH npe,D;JIO>KeHbl paSJIHqHbIe cnOCOObI nOCTpoeHHH
nOJIHblX perneHHH KpaeBOH sa,D;aqH. O,D;HH HS TaKHX cnOCOOOB CBHsaH C
IfcnOJIbSOBaHHeM MeTO,D;a O,D;HOpO,D;HbIX perneHHH [122, 140, 148]. 00-
JIyqeHHbIe C nOMOlI1.blO MeTO,D;a O,D;HOpO,D;HbIX perneHHH BbIpa>KeHHH ,D;JIH
cMem.eHHH H HanpRLKeHHH OOJIa,D;aIOT ,D;OCTaTOqHbIM npOHSBOJIOM ,D;JIH
nOJIHOrO y,D;oBJIeTBOpeHHH rpaHHqHbIX YCJIOBHH. O,D;HaKo SHaqHTeJIbHbIe
BblqHCJIHTeJIbHbIe TPY,D;HOCTH OOYCJIOBJIllBalOT nocTaHoBKY CMHrqeHHbIX
rpaHHqHbIX YCJIOBHH. PaSJIHqHbIe BapHaHTbI HX OOYCJIOBJIHBalOTCH <PH-
3Ht:zeCKHMH cooopaLKeHHHMH.
3aTPy,nHeHHH, BOSHHKalOm.He npH nonbITKe HenOCpe,D;CTBeHHo onH-
caTb HanpH>KeHHOe COCTOHHHe npHMoyroJIbHHKa C nOMOlI1.blO CllCTeM
<PYHK
HH, nopO>K,D;aeMbIX pO,D;CTBeHHbIMH Sa,D;aqaMH TeOpJIJI nOTeH
Ha-
JIa, nplfBenH K paspaooTKe pH,D;a nO,D;XO,D;OB, OCHOBY KOTOpbIX COCTaB-
mIIOT cne
JIaJIbHO CKOHCTpYllpoBaHHbIe CHCTeMbI <PYHK
HH. 11s paOOT
3TOro l!anpaBJIeHHH CJIe,D;yeT OTMeTHTb npeLK,D;e Bcero [70, 71 I. 3<P<peK-
THBHOCTb pasBHToro aBTopaMIf YKasaHHbIx paOOT nO,D;XO,D;a OOYCJIOB-
JIHBaeTCH TeM, t:zTO HM Y,D;aJIOCb CBeCTll OHrapMOHHqeCKYIO npOOJIeMY K
rapMOHll'leCKOH.
.D.PyrylO, nOJIHYIO B npHMoyroJIbHHKe CHCTeMY <PYHKII.HH npe,D;JIO-
lKHJI M. M. cI>llJIOHeHKO-BopOJI.llq [185]. 11cKoMbIe xapaKTepHCTHKH Ha-
npHLKeHHO-,D;e<popMllpoBaHHoro COCTOHHHH npe,D;CTaBJIHIOTCH B BH,D;e ,D;BYX
COCTaBJIHIOlI1.llX. OepBaH llS HHX Y,D;OBJIeTBOpHeT ypaBHeHHHM paBHOBe-
CHH H rpaHHt:zHbIM YCJIOBHHM 3a,D;aqH. BTopaR, KOTopaH CTpOHTCH C
nOMOlI1.blO npe,D;nOJIOLKeHHH CHCTeMbI <PYHK
llH, TaK>Ke Y.D:OBJIeTBOpHeT
ypaBHeHllHM paBHOBeCHH H O,D;HOpO,D;HbIM rpaHHqHbIM YCJIOBHHM, coxpa-
HHH npH 3TOM onpe,D;eJIeHHYIO CTeneHb npOHSBOJIa. Pacnop5DKaHcb 3THM
npOllSBOJIOM, MOLKHO ,D;OOHTbCH Toro, qTO OOlI1.He BbIpa>KeHHH ,D;JIH Ha-
npHLKeHHlloY,D;YT Y,D;OBJIeTBOpHTb YCJIOBHHM COBMeCTHOCTH ,D;e<popMa
HH,
KOTopbIe BbIpaLKalOTCH BapHa
HoHHoH TeopeMoH KaCTHJIbHHO [119].
OOJIHOe perneHHe Sa,D;aqH ,D;JIH npHMoyrOJIbHHKa C HCnOJIbSOBaHHeM onH-
caHHoro MeTO,D;a CO,D;ep>KHTCH B paooTe [971.
11cCJIe,D;OBaHHIO nJIOCKOH: Sa,D;aqll TeopHH ynpyrocTH ,D;JIH npHM&
. yroJIbHHKa B paMKax HCnOJIbsyeMoro B ,D;aHHOH paoOTe MeTO,D;a nOCBR'
m.eHO pH,D; nYOJIHKa
HH: B. JI. AopaMHHa Hero yqeHHKOB [2, 4, 8].
Ha 3Tane nocTpoeHHH 06Il(Hx BbIpaLKeHHH ,D;JIH <PYHK
HH HanpH>.KeHHH
:>Tll paooTbI B sHat:zHTeJIbHOH Mepe CBH3aHbI C H,D;eHMH, BbICKasaHHbIMH
B onHcaHHbIX BbIrne paooTax r. JIaMe [232] H 3. MaTbe [2351. O,D;HaKO
fiX aBTopbI rrpO,D;BHHYJIHCb SHaqllTeJIbHO ,D;aJIbrne KaK B CMbICJIe OOOCHO-
-BaHHH MeTO,D;a, TaK H C TOqKH 3peHHH <paKTHqeCKOrO pellIeHHH KOHKpeT.
HbIX rpaHHqHbIX sa,D;aq. I1MeHHo B pesYJIbTaTe paOOT, npOBeJI.eHHbIX
-B apMHHCKOH: llIKone MexaHHKOB, C<POpMHpOBaJICH JI.OCTaTOqHO


I
!
I
I,
I
I'
I


36


I


060CHOBaHHbIH MeTO,D; pellIeHllH rpaHHqHbIX sa,D;aq )I.JIH ynpyrHX TeJI KO-
HeqHbIX paSMepOB.
3HaqHTeJlbHbIM npeHMym.eCTBOM 3Toro MeTO,D;a RBJIHeTCH TO, qTO
npll nOCTpOeHlIM Oom.HX pellIeHHH HcnOJIbSYIOTCH xoporno HSY'leHHbIe
B sa,D;a-.:J:ax TeopHH nOTeHII.HaJIa CHCTeMbI <PYHK
llH. YLKe O,D;HO 3TO
OOCTOHTeJIbCTBO OOYCJIOBJII1BaeT BOSMOLKHOCTb paCCMOTpeHHH Ha era
OCHOBe llIHpOKOro Kpyra rpaHH'lHbIX 3a,D;aq, 3TH BOSMOLKHOCTll pea-
JIllSOBaHbI, B qaCTHOCTll, npH perneHHH paSJIHqHbIX CMellIaHHl>IX sa,D;atJi
,D;JIH npHMoyrOJIbHHKa [24, 50, 63].
B paMKax MeTO,D;a, HCnOJIbSOBaHHoro B.
HTHpoBaHHbIx BbIllIe pa-
oOTax, BbIIIOJIHeH elI1.e pH,D; HCCJIe,D;OBaHHH paSJIHqHbIMH al3TopaMH
[174,224,273 H ,D;p.]. XapaKTepHoH ocooeHHoCTblO 3THX paOOT HBJIHeTCH
TO, qTO B HHX He CTaBHTCH Bonpoc 00 ocooeHHOCTHX HanpHLKeHHoro
COCTOHHIfH, nOpOLK,D;aeMbIX HaJIHqHeM OCOObIX TOqeK rpaHH
I (yrJIOBbIX
TOqeK). B CBHSH C 3THM nOJIyqeHHbIe KOJIUqeCTBeHHbIe o
eHKH HanpH-
>KeHHOrO COCTOHHHH He Bcer,D;a TOqHbI.
OOOCHOBaHHe MeTO,D;a, pasBHToro B YKasaHHbIX BbIllIe paOOTax
B. JI. AopaMHHa, 6asHpyeTcH Ha ,D;OKaSaTeJIbCTBe perYJIHpHOCTH oeCKO-
HeqHbIX CHCTeM. ECTeCTBeHHbIM CJIe,D;YIOlI1.HM 3TanOM B ero paSBHTHH
HBJIHeTCH pellIeHHe Ba>KHOrO Bonpoca 0 Kat:zeCTBeHHbIX ocooeHHOCTHX
HanpHLKeHHoro COCTOHHHH BOJIHSH yrJIOBbIX TOqeK rpaHH
. .D.JIH 3TOro
HeOOXO,D;HMO He TOJIbKO SHaTb, qTO pellIeHHe oeCKOHeqHOH CHCTeMbI
cym.eCTByeT, HO H ,D;OBOJIbHO TOqHO SHaTb XapaKTep nOBe,D;eHlUI HeHS-
BeCTHbIX C pOCTOM HOMepa O,D;HH HS BOSMOLKHbIX cnOCOOOB OTBeTa Ha
3TOT Bonpoc HSJIO>KeH HH>Ke Ha OCHOBe paooT [72, 86, 87]. CJIe,D;yeT no,D;-
t:zepKHYTb, qTO YCTaHOBJIeHHaH B 3THX paoOTax CBHSb Me)K,l(y <paKToM
HaJIHqllH yrJIOBbIX TOqeK 11 xapaKTepoM nOBe,D;eHHH K03<p<pI1
HeHTOB
oeCKOHeqHbIX pH,D;OB B 06lI1.eM pellIeHIfI1 ,D;aeT BOSMOLKHOCTb He TOJIbKO
,D;OCTllqb nOJIHOH onpe,D;eJIeHHOCTI1 B OnI1CaHI1H KaqeCTBeHHOH KapTHHbI
HanpmI{eHHOrO COCTOHHI1H, HO 11 SHaqI1TeJIbHO YCOBepllIeHCTBOBaTb aJI-
ropHTMbI qHCJIOBOH oopaOOTKH OOlI1.I1X <POPMYJI.
B 3TOH: rJIaBe peqb oY,D;eT H,D;TH JII1llIb 0 nepBOH OCHOBHOH rpaHHqHOH
sa,D;aqe. CnOCOObI HsyqeHI1H ocooeHHOCTeH nOJIH HanpHLKeHI1H B TOM
CJIyqae, KOr,D;a yrJIOBaH TOqKa rpaHI1
bI HBJI5IeTCH O,D;HOBpeMeHHO H
TOqKOH CMeHbI xapaKTepa KpaeBbIX YCJIOBHH. HSJIOLKeHbI BO BTOpOH
r JIaBe.
Pa3BHTbIe npII Hsyt:zeHIIII Hanp5lLKeHHOro COCTOHHII5I np5lMoyrOJIb-
HIIKa MeTO,D;bI pellIeHHH rpaHIIqHbIX sa,D;aq nOqTII oes IISMCHeHHH MorYT
ObITb HCnOJIbSOBaHbI npII pellIeHIIH sa,D;aq ,D;JI5I nOJIynOJIocbI. qosop
OOJIbrnoro KOJIIIqeCTBa nYOJIIIKa
IIH, B KOTOpbIX paCCMaTpHBalOTCH
paSJIIIt:zHbIe Sa,D;a'lH ,D;JIH nOJIynOJIOCbI, CO,D;epLKIITCH B paooTax [60, 200,
269]. He OCTaHaBJIHBaHCb Ha aHaJIHSe OT,D;eJIbHbIX paOOT, YKaLKeM, 'ITO
lIS OOJIbllIOrO HX qHCJIa, nO>KaJIYH:, JIHllIb B ,D;BYX paOOTax paSBIIBalOTCH
nO,D;XO,D;bI, ,D;alOm.He BOSMOLKHOCTb yqeCTbKaqeCTBeHHbIe ocooeHHOCTII
HanpHLKeHHoro COCTOHHIIH [60, 1941. B ,D;pyrnx aBTOpbI JIIIOO BOBce He
OCTaHaBJIIIBalOTCH Ha Bonpocax KOJIIIqeCTBeHH0I1 o
eHKII HanpHLKeHHoro
COCTOHHIIH 1267], JIIIOO nplfMeHHIOT nO,D;XO,D;bI, saBe)l.OMO CBHSaHHble c
cym.ecTBeHHbIMII norpellIHocTHMII [221].
37




II
I



 4. IIocTpoeHHe o6Iqero pemeHna OCHOBHblX
rp3HH'IHblX 33Aa'I AJIH npHMoyrOJIbHHK3
fpaHHIlHbIe san.a'IH nJIOCKOH TeopMH ynpyrocTH Henocpen.CTBeHHo
B03HMKaroT npH H3yqeHHH n.BYx CJIY'IaeB paBHoBecHH ynpyroro TeJIa:
B cJIyqae nJIOCKOn n.e<popMauHM H B cJIyqae n.e<popMauHH nJIOCKOH
nJIaCTHHKH CHJIaMH, ,l(enCTByrolll.H-
MH B ee nJIOCKOCTH. KOHKpeTHbIe
BbIKJIa,l(KH B paoOTe Be,l(YTC5I npH-
MeHHTeJIbHO K cJIyqaro nJIOCKOn ,l(e-
<popMaUMM, O,l(HaKO nOJIY'IeHHbIe npH
3TOM pe3YJIbTaTbI JIerKO nepeHo-
C5ITC5I Ha cJIyqan OOOOlll.eHHoro rrJIOC-
KOro Hanp5I/KeHHOrO COCTOHHH5I [1331.
PaccMaTpHPaH OCHOBHbIe. rpaHHq-
HbIe 3a,l(aqn ,l(JI5I np5IMoyrOJIbHHKa
(pMC. 2), nCXO,l(HM H3 OqeBH,l(HOn
B03MO/KHOCTH pa30HemI5I 06ll1.ero
nOJI5I CMelll.eHHn Ha qaCTH, oOJIa,l(arorqHe onpe,l(eJIeHHbIMH cBoncTBaMH
cHMMeTPHH OTHOCHTeJIbHO },{oop,l(HHaTHbIx ocen. TaKoe pa30HeHHe rro-
3BOJI5IeT H3ae/KaTb H3JIHlllHeii rpOM03,l(KOCTH B H3JIO/KeHHH cYlll.eCT-
Ba nO,l(XO,l(a K nocTpoeHHro oalll.ero pellleHH5I rpaHn'IHOn 3a,ua'IM, KD-
TopbIn npHMeHMM BHaqaJIe K cJIyqaro CHMMeTpnqHoro OTHOCHTeJIbHO
ooeHx KOOp,l(HHaT ocen nOJI5I CMelll.eHHn.
Boo,l(5I oe3pa3MepHbIe KOOp,l(HHaTbI COOTHOllleHlUlMH x = x'fa, y =
= y'la, TJ = bla, paccMoTpHM nepBYro OCH08HYlO rpaHH'IHYro 3a,l(3qy
,l(Jl5I npSlMoyrQJIhHHKa. B 3TOM cJIY'Iae COCT05IHHe yrrpyroro TeJIa oxa-
paKTepH3yeM BeK7OpoM CMeIJ:J.eHlin u = i u (x, y) + jw (x, y), KOTOpbIn
,l(OJI]KeH yn.oBJIeTBopHTb ypaBHeHHlO JIaMe (1.1). B&r'IHCJIeHHbIe no HeMY
fia OCROBe 3aKOHa fYKa Hanp5I/KeHH5I HMeIOT Ha rpaHliue 3a,l(aHHble
3Ha'IeHH5I:
t I
2G Q", (::I:: 1, y) = /1 (y), 2G TXII (::t: I. y) =::1:: CPl (y),
I I
2G Oy (x,
 '!J) = 12 (x), 2G 1: xy (x, + 11) = + CP2 (X).


fI'


npe,AOO.J1Q{.11;UM., \11'0
a,l(mpp\He I?Heuume B03,l(eHCTB.H5I <PYHJ\l\HH y,l(OB.-
JIeTBOp5IroT CJIe,l(YlOllI.HM YCJIO:&H5IM:
11 (y} = fl (- y), 12, (x) = [2 (-x),
CPl (
) = - eel (- y), CP2 (x) = - cp
d- x).
B COOTBeTCTBHH C H3JIO/KeHHbIM B
 2 MeTO,l(OM nOCTpOeHHJI 06w.ero
pellleHH5I rpaHH'IHbIX 3a,l(a'I ,l(JI5I TeJI KOHeqHbIX pa3MepoB KOMnOHeHTbI
MCKOMOro BeKTop
 cMeru.eHHii npe){CTaBHM B Bli){e
u = U 1 + U 2 , W = WI + W 2 . (4.3)
nepBbIe CJIaraeMbIe B paaeHcTBax (4.3) c
.nyeT BbIopa:b ,l(OCTaTO'IHO
npOH3BOJIbHO ,lI.JI5I y,l(OBJIeTBOpeHH5I rpaHMqHbIX YCJIOBMH B HanpsnKe-
HHSlX Ha nOBepXHOCT5IX Y = ::I::'l! npH -I -<
 -< I. .
370 TpeooBaHHe He onpe,l(eJI5IeT 0,lI.H03HaqHQ B.H,l( BbIpa/KeHHH n.JI5I
Ul H WI H, BOOOlll.e roBOp5I, ,l(OnYCKaeT oeC,!HCJIeHHOe MHO/KeCTBO pe
e-
HHH. TaK, BbIOHpa5I B KaqeCTBe U I H WI H3
ecTHoe 06ll1.ee pellleHife rpa-
HH'IHoii 3a,l(a'IH ,l(JI5I CHMMeTpHqHO n.e<POpMHpOBaHHoii oeCKOHeqHO n.JIHH-
HOH nOJIOCbl, nOJIyqaeM
U 1 =

 I [A(s) ch sy + C (8) 3r;n,;;;; 4 ch sy + C (8) ysh sY] sin sxds,
o


(4.2)


I


fpaHH'IHbIe 3a,lI.at!U ,l(JI5I np5IMoyroJIhHMKa MorYT paccMaTpHB8TbC5I
KaK qaCTHbIii cJIY'Iaii oOJIee oOlll.en 3an.a'IH 0 .netPOpMHpoBaHMM na-
paJIJIeJIenMne,l(a. TIpH nOCTpoeHMH oOlll.ero pellleHH5I B COOTBeTCTBHM C
MeTO,lI.OM JIaMe rpaHH'IHbIe 3an.a'IM ,l(JI5I napaJIJIeJIerrHrre,l(a JIerKO CBD-
.n5ITC5I K oeCKOHe'IHbIM CHCTeMaM JIHHeiiHbIx ypaBHeHHll [32, 53, 2681.
LIJI5I 9TMX CHCTeM BO MHOnIX CJIY'Ia5IX YCTaHOBJIeHa perYJIIIpHOCTb
[26, 156 H ,l(P.J, qro n03BOJIIIeT nOJIyqHTb ,l(OCTaTO'IHO Ha,l(e/KHbIe n.aHHbIe
o HanpSl/KeHHOCTH BO BHYTpeHHHx TO'IKaX OOJIaCTH.
fcc.Tje,l(oBaHMe X3-
paKTepa Hanp5I/KeHHOrO COCT05IHH5I BOJIH3H rpaHH'IHbIX nOBepxHocTeii.
H ocooeHHo BOJIH3M BeplllHH napaJIJIeJIenHrre,l(a, Tpe6YeT 3HaHM5I aCHMn-
TOTM'IeCKHX cBoHcTB HeH3BeCTHbIX B oeCKOHeqHbIX CHCTeMax, KOTopble
Jl.0 CHX rrop He H3yqeHbI.


00


a


-<::I


i.


)('


WI =
 .\ [A (s) sh sy +C (s) ych sy] cossxds. (4.4)
o
3,lI.eCb m - qHCJIO nyaccoHa; A (s), C (s) - npOM3BOJIbHbIe <PYHKUHM.
BbIqHCJI5ISI C nOMOlll.bro 3aKOHa fyKa 3HaqeHHH Hanp5I/KeHHii, COOT-
£jeT
l'BYlOllI.He CMeru.eHH5IM H.4), HaXO,l(tfM
;0 (J
l) =

 J[A(S)SChSY+C(S) 3m;;:2 chsy+
+C (s) sy SA $y lcos
ds.
00
_
_ g:
) =:;;; .
 .\[A (s}sch
 + C (s) m ,n2- (,:hsy+
0.
+c (s) sy sh sy] cos sx-ds, (4.5)
00
do T

 = -
 .n It (
J

q
II + C (s) 2 Tn :;; 1 &h sJ{ +
o
+ C(s) sych SY] sin sxds.
C"pYj(!'l'I}f\ I1tlT
rp?JI9ij
YPb
 I10
I!QJH.JroT YTBep
.!J.I
Tb, '11'9 C nQ-
Pt'IOIA\>'1Q I!J>mfUI
tm"ii (4,fi) Mq?f{HO YJJ:q:
.1J
1't:JOJnnb PPQH&BQ
bHl>JJd rpa-
IJIiI,JHWl'4 n\II(C)E\
tfM "
 nQJJepxfIoqfl y = '¥} nD" I x I -< I. .ll
UI on.HO-

HijHijQrQ gJ1peAt;!J1
tll1ft (;p.qt\P
'!ll\ti
c
 .
 (4.&) IIpOij3f!P.1!bFji>I
 <pytI
-
1m" C
fic!)l.Y
T &&A.rn

ijall
HHH Ha!lP
U

mHn
fW gl'P
Jia I x I -< L





PEc.2.


(4. I}


38




B qaCTHOCTII, Harrp5DKeHIIH Ha :JTOM qaCTII rpaHIII.{bI rrOJIOCbI MOrYT 6bITb
,l(JIH Orrpe,l(eJIeHHOCTII rrpIIHHTbI paBHbIMII HYJIIO.
CCPOPMYJIIIpOBaHHOe BbIllle Tpe60BaHIIe OTHOCIITeJIbHO nepBbIX CJIa-
raeMbIX B (4.3) M01KeT 6bITb TaK1Ke Y,l(OBJIeTBOpeHO, eCJIII B KaqeCTBe
U I II WI IICnOJIb30BaTb o6r.qee pellleHIIe,l(JIH rrepIIO,l(IIqeCKII ,l(ecpOpMIIpOBaH-
HOM rrOJIOCbr:


U I = -
 ( A ch n<t y + C 3m - 4
 ch
 y +
n=1 n T n m nJT T


h nn ) . nn
+Cnys r Y smyx,
WI = AoY +
 (An sh n; y + Cnych n; Y) COs n; x.
n=1 I
3,l(eCb Ao, An> C n - rrpOII3BOJIbHble rrOCTOHHHble.
CooTBeTcTBYIOr.qHe CMer.qeHIIHM {4.6) HanpH1KeHIIH BblqIICJIHIOTCH no
cpopMYJIaM


(4.6)


I (I)
20 ax -


- ,; (A
ch nn + c 3m-2
...::.J, 11 T T Y n m
'1=1
nn nn \ nn
+ CnryshTy)cosTx,
00
A '1 ( A nn h nn m-2
0+;;:1 nr c r Y + C n m


nn
chry+


Ao
m-2


h nn
C r Y +


I (I)
20 a y


m-l
m-2


C nn h nn ) . nn
+ nr Ys rY cosT x,
2
 .r
 = - ,
(An n; sh n.;-r Y + C n 2 m;;: I sh
Tt y +
C nn h nn ) . nn
+ n-r' yc -r' y smTx.
CO,l(ep1Kar.qIIeCH B 3TIIX Bblpa1KeHIIHX rrpOII3BOJIbHble nOCTOHRHble Ao.
An> C I1 rr03BOJIHIOT Y,l(OBJIeTBOpIITb JII06ble rpaHIIqHble YCJIOBHH .n;JIH
a y II .xy Ha IIHTepBaJIe - T <: x <: T, II, c.J:le,l(oBaTeJIbHO, perneHIIH
ypaBHeHIIM JIaMe B BII,l(e (4.6) MorYT 6bITb IIcnO.J:lb30BaHbl B KaqeCTBe
nepBblx CJIaraeMblX B (4.3)- npII T> 1. ECJIIi npIIHHTb T> 1, TO He-
OPXO,l(IIMble ,l(JIH o.n;H03HaqHOro orrpe,l(eJIeHHH nocroHHHbIX An II C n 3Ha-
qeHIIH HarrpH1KeHIIH a y II .xy Ha IIHTepBaJIe I <: I x I <: T MorYT 6bITb
3a.n;aHbI npOII3BOJIbHO.
C006pa1KeHIIH, npIIBe,l(IllIIe K nocrpoeHIIIO rrepBblX CJIaraeMbIX B
(4.3) B BII,l(e (4.4) II (4.6), MorYT 6bITb nOJIHOCTbIO nOBTopeHbI rro OT-
HOllleHIIIO KO BTOpblM CJIaraeMbIM, KOTopble HBJIHIOTCH OTBeTCTBeHHbIMII
3a y.n;oBJIeTBOpeHIIe rpaHJiqHbIX YCJIOBIIH Ha nOBepxHoc'rHX X = + I
rrpE
'I') <: y <: '1'). DpII npOII3BOJIbHOM cIIMMeTpIIqHOM npO,l(OJI1KeHIIII
rpaHIIqHbIX YCJIOBIIH BHe OTpe3Ka I y I <: 'I') lIMeeM {;JIe,l(YIOr.qIIe Bblpa1Ke-J


(4.7)


Il(}


l


HIIH ,l(JIH u z , W z :


co
2 \ '
U z = - [B (s) sh sx + D (s) X ch sx) cos syds,
n.
o


(4.8)


11


W z = -
 S [B (s) ch sx + D (s) 3m
 4 ch sx +
o
+ D(s)x sh sx]sin syds.
O.n;H03HaqHOe onpe,l(eJIeHIIe rrpOH3BOJIbHbIX CPYHKI.{IIH B (s) II D (s) B
3THX Bblpa1KeHIIHX B03M01KHO JIHlllb rrpII 3a,l(aHIIII onpe,l(eJIeHHblX rpaHIIq
HblX YCJIOBIIM ,l(JIH a x II .>;y BHe oTpe3Ka I y I <: '1'),
B cJIyqae rrepIIOAIIqeCKoro npo,l(OJI1KeHIIH rpaHIIqHbIX YCJIOBHH Ha
rroBepxHocTHX x = + I C rrepIIo,l(oM p BTopaH qaC'rb o6r.qero perneHIIH
,l(JIH rrpHMoyroJIbHIIKa npII06peTaeT BH,l(
'1 ( nn D nn ) nn
U z = Box-": Bnsh -x+ lIxch- X cos- y,
n=1 P P P
co
" (B h nn . + D 3m - 4 P h nn +
W =
 C -X -C -X
Z n=l n P n m nn p
nn ) . nn
+Dnxsh-x sm-y,
f) p
m-l Bo-
 ( Bn
Ch nn x+Dn m-2 ch
x+
m - 2 n=1 ' p P m p
+D nn. nn ) nn
n-XSh-X cos-y,
p p P
I (2) Bo
 ( nn nn 3m-2 nn
20 . a y = m-2 +
. B -ch-x+D --ch-x+
n
 n p p n m p


I (2)
20 ax


(4.9)


nn nn ) nn
+Dn-xsh-x COS-y,
p P P
1 (2)
 (B nn h nn + 2D m - 1 h nn +
20 .xy = "'-i n - S - X n S - X
11=1 P P m p
nn nn ) nIl
+ D -xch-x sm-y.
n p p p
3,l(eCb TaK1Ke ,l(JIH orrpe,l(eJIeHIIH nOCTOHHHbIX Bn II DI1 Heo6xo,l(HMO
3a,l(aHIIe 3H3qeHHH HanpH1KeHHH ax II .xy Ha HHTepBaJIe 'I') <: y <: p.
TaKHM o6pa30M. o6e COCTaBJIHIOr.qIIe BeKTopa nepeMer.qeHIIM B (4,3)
rrocTpoeHbI. CJIe.n;yeT nO,l(qepKHYTb, qTO nOCTpoeHIIe COCTaBJIHIOr.qnx
UI> WI> U 2 II W 2 npOBO,l(IITCH cOBepllleHHo He3aBIICIIMO. B CBH3II C 3THM
BblpameHIIH ,l(JIH U II W B (4.3) MorYT 6bITb o6pa30BaHbI pa3JIHqHbIMII
cnoco63MII, T. e. BbI6paB, HanpIIMep, ,l(JIH U I II WI npe.n;CTaBJIeHlIe B BII,l(e
(4.4), ,l(JIH u 2 II W z M01KHO IICnOJIb30B3Tb JII060e II3 Bblp21KeHIIH (4.8) IIJIII
(4.9). Ba1KHO oTMeTIITb, qTO IICnOJIb30BaHIIe Ka1K.n;oM II3 .n;onycTHMbIX


11/1


III
II I
II
/ 1,
II
I
ii,
II
III
I
I


41


!,


I,ll




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CTaB,;ls!IQll{HX BeKTopa nepeMeIll.eHHH B (4.3) HMelOT BfI.ll



 ( nn h mt ) . nn
WI = n
 An ch l' y-I- eny s ----r Y Sin ----r x,
00
,-, I nn D h nn ) . nn
U 2 = ..:...J Bnch-x-I- '1 xs -x Slll-Y,
'=1' p P P
W 2 = Box -I-
 ( B sh.!!:!!:.. X -I- D -L 3m - 4 sh nn X -I-
=1 n p n nn m p


(4.10)


nn ) nn
+ Dnxch - X cas -Yo
p p
BbIpa>KeHHS! 1(JI5I CMeIll.eHHH C .llpyrnMH THnaMH cHMMeTpHll MorYT CbITb
nOJIYlIeHbI H3 pemeHHH (4.6), (4.9) II (4.10) nYTeM CHCTeMaTllllecKoro
npHMeHeHH5I CJIe.llYIOIlI.HX 3aMeH ch ( )
 sh (), sh ()
 ch ( ),
cos ( )
 -sin ( ), sin ( )
 cos ( ).
Ha 3TOM 3aIWH'lHM paccMoTpeHHe Bonpoca 0 nOCTpoeHHll 06Il1.HX
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yroJIbHOH 06JIaCTH. <f>opMaJIbHbIll aHaJIH3 3TBX peweHHH nOKa3blBaeT,
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peWeHH5I .llJISI 3TOro CJIYlIa51 51BJUleTC5I pemeHHe B pH
ax npH P = 1}
H T = 1. BbIpa}KeHH5I .llJUI KOMnOHeHT BeKTOpa CMeIll.eHHH B 3TOM cny-
lIae npllMYT Bll.ll


00


u = Box-
 (Bnshanx -I- Dnxchanx)cosany-
n=1


- n
 (A n ch
nY + C n 3:;: 4 ch
nY + eny sh
nY) sin f3nx,


w=AoY+
(A"shf3ng+Cnychf3ny)coS
nx+ (5.1)
n=1


U 1 = AoY +
 ( An sh n r Tl Y + en
 3m - 1 sh lt r rf y-I-
n==1 nJl m


" (8 D 3m - 4 D ) '
-I-..:...J n ch a..x T n ch anx -I- nX sh a"x SID rJ.nY
n=l rnt:X. n


nn ) nll
-I- C nY ch ----r Y cos ----r X,


tan =
n , f3n = n:n} .


4









+ D n m
 2 ch a,
 + D nanX sh a,,x) cos anY- n
 (AnBn ch r I1Y +
C .. 3m-2 )
+ n m Ch
l1Y + C
nY sh
IlY: cos
nX'


C nOMor.u;blQ I3bIpa.memIi:r (5.1) H (5.2) MorYT ObITb TRK.me peIlIeHb/
HeKOTopbJe Cnel);Ha.IJbHbI€ CMeIlIaHHbI€ rpaHH l lHbIe 3a.na l IH HaHOOJIb-
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6hITh nOJIYII€HO oea 6eCKOHe
HbIX CHC1'€M. K TaKtrM 3a)],aQaM OTHOCHTCH
rpaHUIIHb/e 3a.nallll C YCJIOI3HHMU CKpJIb3Hr.u;eH 3a.ne.IJKH no .nBYM npoTH-
BOnOJIo.mHbIM CTopOHaM, T. €. Kor.na 3a.naIOTCH HOpMaJIhHble CMer.u;eHIfH H
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KOTOpbIH, KCTaTlI, 13 3TOM cJIyqae npHBO)],HT K BbIpa.meHHIO cMer.u;eHHH B
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aprYMeHT 13 nOJIb3Y npHHS!Toro cnoc06a npo.nOJl.meHHH rpaHHqHbIX YC-
JIOBHM Ha BCIO nOBepxHocTh 6ecKoHellHoH nOJlOCbI npH nocTPoeHHII CM€-
r.u;emlH 13 BH.ne (5.1).
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tIbte YCJIOBHSI (4.1) npHxo.nHM K CJI€.nYIOIl1;eM CtrcTeMe Q>YHKIUIOHaJIbHbIX
ypaBHeHHH;
m-I 1 A
 ( . . m-2 h
m-2 Bo+ m-2 0- n::; ,BnanchCtiJ+Dn m C a n +
+ D"a n sh an) cosany - n
 (- l)n (An
n ch
IlY +
+ C n 3m;;2 ch
nY + Cn
"Y sh
')Y)= fdY),
n
 (B"a n sh an + D,,2 m;; 1 sh a" + D'Jan ch all) sin anY = CjJi (y), (5.3)


06'be.nHHSIH COOTHOIlIemfH (4.7) H (4.9), Haxo.nHM cooTl3eTC1'J3YIO-
r.u;lIe CMe
eHHHM (5.1) 3HalleHHH KOMnOHeHT TeH30pa HanpSIJReHHll:

 _ m-I B 1 A _ " (B h i
20 (iN - m _ 2 fJ + m _ 2 0 "'-J. "rx" C anx
n=l.


2
 U y = m I 2 Bo + :
 Ao +
 (B"a n ch anx +
/1=1
+ Dn 3m;; 2 ch allx + DJlanX sh allx) cas anY +
 (A,,
n ch
"Y +
,,=1
+ m m 2 C" ch Bl1y+Cn
nY sh
nY ) C05'
"x. (5.2)
2 1 0 TNy = .i (Bnrxn sh anx + D.,2 m . 1 sh anx +
11=1 m
\ 00
+ D fjanx ch anx) sin anY- ,,
 (An
n sh
"Y +
+ C n2 m m. 1 sh
"Y + C .,
nY ch
"Y) sin
nx.
I1cnOJIb30BaHHe I3bIpa.meHHH (5.1) U (5.2) npll peilieHuH Tpex
OCHOBHbIX rpaHHIIHhIX 3a.naII Teopull ynpyrocTH npHBo.nHT K 6eCKOHetI-
HbIM eHCTeMaM JIHH€HHbIX ypaBHeHHH OTUOCHTeJlbHO co.nep.mar.u;HXCH B
HHX npOH3BOJlbHbIX nOCTOSIUHb/X. DOCKOJIbKY Henocpe.ncTB€HHOe He

nOJIb30BaHHe 6ecKoHellHbIx elieTeM npH nOJIyqeHHH KOJIHlleCTl3eHHbIX
xapaKTepHcTHK H3YllaeMbIX nOJIeM HeB03Mo.mHO, TO ycnex 13 p€IlIeHHH
YKa3aHHbIx rpaHHIIHb/X3a.nall e13H3aH e pa3pa60TKoH 3<p<peKTHI3HbIX Me-
1'0)],013 HX yeeqemuI. TpH OCHOBHbI€ rpaHH
HbI€ 3a.nallH nopo)K)].alOT pa3-
JII1I1HbIe THnbI 6eCK0HeQHbIX CliCTeM C pa3JIHtIHhIM xapaKTepoM nOBe.ne-
HIlS! HeIl3BeCTHbIX B HIlX. ECTeCTB€HHO, liTO 3<p<peKTHBHOCTb MeTo.na
yeelleHllH 6eeKOHellHbIX ClIC1'eM 130 MHOroM onpelXeJIHeTCSI ero cnocoo-
HOCTblO YlleCTb 3TH pa3jJtrllllH, 3a KOTOpblMtr CKpblBaIOTCSI KaqeeTB{)H-
Hble oc06eHHOCTIl II3YllaeMblx nOJIeH.
B Hae1'OHr.u;eM naparpa<pe no.nP06HO OeTaHOBHMCH Ha nepBOH oe-
HOBHOM rpaHIlIIHOM 3a)],aIIe H 1I0Ka.meM, liTO nOJIYI/€HHb/€ 3.necb oec-
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plm E. M. KmWOBHlla. Cnoc06 peIlIeHHH 6eCKOHeqHhIX cHcTeM, B03HHKa-
IOr.u;HX npH paccMoTpeHIlH CMeillatIHbIX Ba.na4. 13 cJ1Yllae eOl3na.neHHH
TOIIeK pa3)],eJIa rpaHHI/Hb/X YCjJOBHH C yrJIOBb/MH TOIIKaMH 06JIaCTIl H3-
.lJO:>KeH npH peIlIeHHH rpaHHllHoH 3a)],allIl JJ.JIH /.J.HJIUH.npa KOHellHoM .nJIl1-
RbI (
 11).


m
2 Bo+ : ; Ao+ _
 (- l)n (Bnanehrxnx +
+ Dn 3m;; 2 ch a"x + Dna"x sh aflx) + 1
 (An
n ch
n1) +
m-2 h . h ) '
+ C n m e
lIfJ + C rI Bn1')S
n1') eosanx= '2 (x),
-
 (Afl
n sh
nfJ + C 1I 2 m-l sh
,,1) + CII
n1) ch
n1') ) sin
nx = <P2(-'I
11=1 m
(-1J<Y<fJ, -1
x<1).
.D.J1H .naJlbHeHIlIHx npeo6pa30BatIHH CHCTeMbI (5.3) npeJJ.cTaBHI\11
Q>YHK/.J.HH, Ba.naIOr.u;He BHellIHIOIO tIarpY3KY Ha C1'OpOHa.X npHMoyrojJb-
Hl1Ka., pH.naMH <Pyphe;
00
<Pi (yJ =t
 <PIn sin anY.
 (x) =
 <jJ2n sin
"x;
,,=] fl=1


00


00


00


(5.4)


Ii (y) = liO +
 frn cosa"y,
n=1


,
 (x) =/20 +
 fZlJ cos
nx.
n=l


44


45




Tor.na BTOpOe H qeTBepTOe ypaBHeHH5J B CHCTeMe (5.3) npHBO,QHT K
cJIe.ll.YlOmeH CB5J3H MelK,QY HCKOMbIMH BeJIHqHHaMH:
anB" = -Dn!2 m-I + anctha n ) + h lJ!ln ,
. m I s an


(5 5)


TaKHM 06pa30M, CHCTeMa <j:JYHKLU'!oHaJIbHbIX ypaBHeHHIT (5.3) 3a-
MeHeHa 3KBHBaJIeHTHOH eH nOqTH BCIO,QY CHCTeMOH anre6paHqeCKHX
COOTHOUleHHH (5.5), (5.7) - (5.10). CHCTeMa .lI.BYX ypaBHeHHH (5.7) H
(5 9) onpe,QeJI5JeT 3HaqeHlHI nOCTosmAbIX Ao H Bf)' Onpe,QeJIHB BeJIHqHHbI
Au H Bo H nOJIOlKHB ocraJIbHbIe nOCTOHHHbIe B (5.1) paBHbIMH HY JIIO,
nOJIyqHM, no cymeCTBY, peUleHHe rpaHHqHOH 3a.n.aqH ,QJI5J npHMoyrOJIb-
HHKa npH HHTerpaJIbHOM Y,QoBJIeTBOpeHHH rpamIqfIbIX YCJIOBHH. Do-
nblTKa YTOqHHTb peUleHHe paccMoTpeHHeM caMoypaBHoBeUleHHbIX no
KalK,QoH CTopOHe HarpY30K IipnBo,QHT K He06xo.n.lIMOCTIi peUlaTb nap-
HYIO 6eCKOHeqHYlO CHCTeMY ypaBHeHHH (5.8) H (5.10)
YpaBHeHH5J (5.7) H (5.9), onpe.n:eJIHlOmHe 9JIeMeHTapHbIe peUleHH5J,
HMeIOT HeCKOJIbKO HeY,Q06HYIO <j:JOPMY B CBH3H C HaJIHqHeM B HHX CJIa-
raeMbIX, npe,QCTaBJIeHHbIX 6eCKOHeqHbIMH pH,QaMH. 3TH BeJIHqHHbI, o.n.-
HaKO, JIerKO MorYT 6bITb BbIpa}KeHbI qepe3 HHTerpaJIbHbIe xapaKTepHCTH-
KH KaCaTeJIbHbIX HanpHlKeHHH, .n.eHCTBYIOmHX no cropoHaM npHMO.
yroJIbHHKa. PaCCMOTpHM, HanpHMep, p5J,Q, BXO,QHmHH B ypaBHeHHe (5.7).
Hcxo.n.5J H3 onpe.n.eJIeHHH K09(pqJlllmeHTOB <I>YPbe H pa3JIOmeHH5J


( m-l ) IP2n
p"An = -Cn 2 -m- + l:\n'Y)cth(3,,1') - shr
n11
DpHBe.n;eHHe nepBoro H TpeTbero paBeHcTB B CHCTeMe (5.3) K aJI-
re6paHQeCKHM COOTHOUleHH5JM ocymeCTBJI5JeTCH Ha OCHOBe CJIe,Q\}(I-
UlHX npe.n.cTaBJIeHHH .n.JIH rHnep60JIHQeCKHX
YHKUHH:
00
h _ sh ah ,
 2a (- I)n sh ah
C ax - ah T
 ( 2 COS YnX,
n=1 h Y n + a 2 )


00
h _ chah shah +
 2a(-I)n c hah +
xs ax---

 2 COSYnX
a a n=1 Y n + a 2
00 n 2

 2(-1) {Yn-(2)shah
+
 2 COSYn X ,
n=1 h (Yn + ( 2 )2
nn
3.neCb Yn = -;z H 9TH paBeHCTBa cnpaBe.n.JIHBbI npH JII060M a ,QJI5J Bcex
x, HaXO,QHIl.{HXC5J B HHTepBaJIe I-h, h). HCnOJIb30BaHHe COOTHOUleHHH
(5.4) - (5.6) n03BOJIHeT 3aMeHHTb nepBoe ypaBHeHHe B CHCTeMe (5.3)
CJIe.n.YIOIl.{HMH aJIrropaH'leCKHMH ypaBHeHHHMlf:


co


m-I B 1 A
 k Cf>2k
m - 2 0 + m _ 2 0 -t
 (- 1)
 = fIo;
k=1 vkf)


- 2
Dn ( Cha n +
 ) =
(-I)k+nCk 4I1k a n S hBkf) _
sh an
=I f) W
 + a
l'


co


-
 (- l)k+nfP2k ;Bk 2 + /In + fPln dh an
k=1 11 W k + an)
(n = I, 2, .. .).
AJIre6paH3aUH5J TpeTberO
YHKUHOHaJIbHOrO ypaBHeHH5J
(5 3) .lI.aeT TaKHe C'OOTHOUleHH5J:
1 B m - 1 A
 (- I)k .
m -I 0 + m - 2 0 +
 fPlk ----r;;- = '20'
k=1
- 2
C ( ch f3n1J + Bnf) ) =
 (_ l)k+nD 4akl1nshak +
n . sh /11111
 k ( a2 + A2 ) 2
..-I k fJn


+
 (- l)k+nfPlk 2 2ak 2 - fP2n cth (3n 1 1- hn
k=J (a k + /1,,)
(n = 1, 2, ...).
46


(5.6)


00


2 (-I ) "
-- Sin R X
f3k Pk




k=1


(-I <X< 1),


(5.11)


X=-


Haxo.n.I1M


(5.7)


co 1 00
:I 1 k Cf>2k 1

 2 (- Il .
(-- ) - =
 fP.) (X,
, . smp xdx-
k=1 {'\k11 fl 0 - ,;::j Bk .; . k ...:...
1
!=

 5 X'P.e (x) ax.
11
o


(5.12)


(5.8)


AHaJIOmQHO MomeT 6bITb nOJIYQeHO BbrpalKeHHe ,QJHI CYMMbI p5J,Qa, BXO-
.n5Jmero B ypaBHeHHe (5.9). B HTore CHCTem'a ,QJISI onpe,QeJIeHH5J nocro-
5JHHbIX Af) H Bo npH06peTaeT BH.n.


B CHCTeMe


J
:
 Bo + m
 2 Ao = 110 + * I X({J2 (x) dx.
11
I m-I r
tn- 2 Bf) + m-2 Ao = 120 + J YfPl (Y) dy.
o


(5.13)


(5.9)


(5 10)


Dpe>K,Qe "leM npHCTynHTb K HCCJIeJI.OBaHHIO 6eCKOHe'lHOH CHCTeMbI,
06pa30BaHHoH COOTHOIlleHHfIMM (5.8) H (5.10), BBe.n.eM HOBble HeMS-
BecTHbIe COOTHO
Hl!5J
X .
 (
 1 , ) k'l) a sh a
k - k k k.


Ii I
Yk = (- 1) C k - r:\k sh f3k't}.
'()


(5.14)


'47




Tenepb nO.ll.JIe)!{a
aH pell1eHHlO CHCTeMa npH06peTaeT BH.lI.
00 2
X = 1
 4a n \jJn
n t" (an)
 y" (13;; + a
)2 + t" (an) ,


Yn =
 X
Tj2t"n\f3n TJ) t:: k
LI.JIH COKpa
eHl1H 3anHCl1 BBe.ll.eM


4 f3 n 2
+ (lJn
(a;; + f3
)2 TJ2t"n (f3n1])
0603HaqeHHH


(5 15)


/). (x) =
 ( cth X + - h
 ),
x s x


00


qr n= /In (- It + fPln (- It cth an -
 (- ItfP2k 2f3k
k=l TJ (f);; + a
) ,


(5.16)
- fP2n (- It cth
n'r) - t2n (- It.


00
<D -
 ( _ l ) k 2ak
n - ...:J fPlk ( 2 + R2 )
k=l a k Pn
I1cCJIe.ll.OBaHHe 6eCKoHeqHOH CHCTeMbI (5.15) 6Y.lI.eM npOBO.ll.HTb
B paMKax H3JIO)!{eHHOH B npe.ll.bI.lI.y
eH rJIaBe TeopHH 5. M. KOHJIOBHqa.
npl1 3TOM nOCTOHHHO 6Y.lI.eM HMeTb B BH.lI.Y He06xo.ll.HMOCTb YBH3aTb
CBOHCTBa HeH3BeCTHbIX B 6eCKOHeqHOH CHCTeMe C oc06eHHOCTHMH no-
Be.ll.eHHH XapaKTepHCTHK HanpH)!{eHHo-llecpopMHpoBaHHoro COCTOHHHH
ynpyroro TeJIa.
HaqHeM C .lI.OKa3aTeJIbCTBa perYJIHpHOCTH CHCTeMbI (5.15). I1CXO.ll.H
H3 pa3JIO)!{eHHH (5.6), HaXO.ll.HM'



 4a
 2
Sl (n) =
 2 2 - /). (an) -
,
k=I (f3 k + a n )2 an


(5.17)


413 2
S2 (n) =
 n 1 1 2/). ( A 'I'l ) _ 2
_ ( 2 + R2 ) 2 = Pn'.
.
k




B CBH3H C 3THMH paBeHCTBaMH .lI.JIH CYMMbI K03cpcpHIlHeHTOB B CTpOKaX
6eCKOHeqHOH CHCTeMbI HMeeM Ou:eHKH


1
t" (an) Sl (n) = 1 - an < 1,


an =


2
a
t" (an)


(5.18)


1
TJ2t" (1 3 nTJ) S2 (n) = I - b n < 1,


2
b n =
f3
TJ2t" (f3nTJ)


PaBeHCTBO e.ll.HHHu:e CYMMbI K03cpcpHU:HeHTOB .lI.OCTHraeTCH JIHll1b .lI.JIH
6eCKOHeQHoro HOMepa CTpOKH. CnpaBe.ll.JIHBOCTb COOTHOll1eHHH (5.18)
H .lI.OKa3bIBaeT perYJIHpHOCTb 6eCKOHeQHOH CHCTeMbI (5.15).
llaJIbHeHll1ee HCCJIe.ll.OBaHHe K03cpcpHU:HeHTOB B CHCTeMe (5.15) npo-
Be.ll.eM C u:eJIblO YCTaHOBJIeHHH CnpaBe.ll.JIHBOCTH 3aKOHa aCHMnTOTHQe-
CKHX BbIpa)!{eHHH .lI.JIH CO.ll.ep)!{a
HXCH B HeH HeH3BeCTHbIX. DepBoe He-
06XO.ll.HMoe .lI.JIH 3TOro YCJIOBHe, BbIpa)!{alO
eeCH HepaBeHCTBaMH

8


(3.7), B .lI.aHHOM CJIYQae npH06peTaeT BH.lI.
2a

I < 2 2 < L,
(f3 k + a n )2
2f3

1<. 2 2 < L, k < n, n = 1,2, ...
(f3 n + ak)2
.nemo npOBepHTb, QTO 3TH HepaBeHCTBa 6Y.lI.YT CnpaBe.ll.JIHBbI .lI.JIH
k < n npH npOH3BOJIbHOM n, eCJIH npHHHTb L = 2, a B KaQeCTBe I
2 21']4
BbI6paTb MeHbll1ee H3 .lI.BYX QHCeJI II = (TJ2 + 1)2 ' 1 2 = (TJ2 + 1)2
BTopoe He06xo.ll.HMOe YCJIOBHe 6Y.lI.eT BbIllOJIHeHO, eCJIH B COOT-
BeTCTBHH C (3:8) YKa)!{eM TaKoe nOJIO)!{HTeJIbHOe QHCJIO e, npH KOTOpOM
HepaBeHcTBa


(5.19)


2 1
 p 4a

+ 2 >- e,
a
t" (an) /1 (an) 11=1 ([3;; + a n )2
p 4 f3 2 n
2 + I
 >- e (5.20)
f3
TJ2t,,(f3nTJ) 1]2/1 (f3n 1]) k
 (f3
 + a;;)2
6Y.lI.YT CnpaBell.JIHBbI .lI.JIH BCeX n = p + 1, P + 2, ..., 2p npH JIlO60M p.
PaCCMOTpHM nepBoe HepaBeHCTBO B (5.20). Ha OCHOBaHHH ou:eHOK
(5.19)


.
'I


a
t" 2(a n ) ( 1 +
 (f3
 ::
)2 ) >- a

:an)
YQI1TbIBaH .lI.Hana30H H3MeHeHHH n H xapaKTep nOBe.ll.eHHH
TeJIH B npaBOH QaCTH HepaBeHCTBa (5.21), nOJIYQHM ou:eHKY
2pl ::;;" iTJ 1
2 ) 9"J1 :Tt
ant" (an cth - +
11


(5.22)


(5.21)


3HaMeHa-


:Tt


:Tt
TJ sh 2 -
Tj


AHaJIOrHQHO .lI.JIH JIeBOH QaCTH BTOpOro HepaBeHCTBa B (5.20) HaXO.ll.HM
2 (
 4f3
 ) ::;;" 1 1 (5.23)
f3
TJ2t" (f3n 1 ]) I +
 (f3
 + a
)2 9" cth:TtTJ +
 :rt1] .
sh 2 :Tt1]


II


TaKHM 06pa30M, npHXO.ll.HM K 3aKJIlOQeHHlO, QTO H BTopoe YCJIOBHe,:
He06xo.ll.HMOe .lI.JIH cnpaBe.ll.JIHBOCTH 3aKOHa aCHMnTOTHQeCKHX BbIpa)!{eHHH
B CHCTeMe (5.15), 6Y.lI.eT BbmOJIHeHO, eCJIH B KaQeCTBe e npHHHTb MeHb-
ll1ee H3 .lI.BYX QHCeJI, CTOH
HX B npaBbIX QaCTHX HepaBeHCTB (5.22)
H (5.23). 9THM 3aKaHQHBaeTCH .lI.OKa3aTeJIbCTBO B03MO)!{HOCTH cy
ecT-
BOBaHHH Y HeH3BeCTHbIX B CHCTeMe (5.15) 06
ero .lI.JIH HHX OTJIHQHOro
OT HYJIH npe.ll.eJIa npH CTpeMJIeHI1H K 6eCKOHeQHOCTH HX HOMepOB.



 7-2
64



9




OKOHqaTeJIbHOe 3aKJIIOqeHHe no 3TOMY Bonpocy MOLKHO nOJIyqHTb
nOCJIe aHaJIH3a CB060,n:HbIX qJIeHOB 6eCKOHeqHOH CHCTeMbI.
B COOTBeTCTBHH C (3.3) 3aKOH aCHMnTOTHqeCKHX BbIpaLKeHHH 6y,n:eT
cnpaBe,n:JIHB, eCJIH B03MOLKHO YKa3aTb TaKoe nOJIOLKHTeJIbHOe qHCJIO K,
npH KOTOpOM HepaBeHcTBa


+1'Ifnla
<K,


1 :l
21<Dnl
n <K


(5.24)



')Tlf BbIpaLKeHH5I nOJIyqeHbI C HCnOJIb30BaHHeM CBoHcTBa HenpepbIBHoc-
'I'll (PYHKU,HH (jJ1 (y) H (jJ2 (x), H3 KOToporo B CB5I311 C HX aHTHCHMMeT-
(JlleH, a TaKLKe B npe,n:JIOLKeHHH KYCOqHOH HenpepbIBHocTH BTOpbIX npo-
JI:I130,n:HbIX OT CPYHKU,HH (jJ1 (y) H (jJ2 (x) BbITeKaeT, qTO (jJ1 (0) = (jJ2 (0) =
() CJIaraeMbIe, HMeIOmI1e nOp5I,n:OK n- 3 , B BbIpaLKeHH5IX ,n:m: \pIn
II 'Pz" y,n:OBJIeTBOp5IIOT Tpe60BaHH5IM HepaBeHCTB (5.24), H B ,n:aJIbHeHrneM
II X MOLKHO He paCCMaTpl1BaTb.
00,n:CTaBJI5I5I 3HaqeHH5I K03cpcpHU,HeHTOB <pYPbe (jJIn H (jJZ" H3 (5.25)
II COOTBeTCTBYIOmHe CJIaraeMbIe B 'If n' HMeeM
" nn
 I ) k 2
k
'PIn (- I) cth 1] -
 (- (jJ2k 'I'} (

 + a
)
=
cth nn [(jJ2 (1) - (jJ1 ('I'J)] + 0 (n- 2 ).
nIL 1]
\"aJIOrl1qHO ,n:JI5I COOTBeTCTBYIOmHX CJIaraeMbIX B <D" Haxo,n:HM


(5.26)


6y,n:YT cnpaBe,n:JIHBbI ,n:JI5I Bcex n = I, 2, ... 9THMH HepaBeHcTBaMH
YCTaHaBJIHBaeTC5I onpe,n:eJIeHHbIH nOp5I,n:oK y6bIBaHH5I ,n:JI5I 'ljJn H <D". T. e.,
no cYlIJ,ecTBY, ,n:JI5I K03cpcpI1U,HeHToB <PYPbe CPYHKU,HH, 3a,n:aIOmHX BHern-
HHe HarpY3KH. 3,n:ecb CJIe,n:yeT YKa3aTb, qTO eCJIH YCJIOBH5I (5.24) He
BbillOJIH5IIOTC5I, TO B 06meM CJIyqae HeB03MOLKHO pernHTb Bonpoc 0
cymecTBoBaHHH perneHH5I perYJI5IpHOH 6eCKOHeqHOH CHCTeMbI, KaKOH
5IBJI5IeTC5I CHCTeMa (5.15).
PaccMoTpHM 60JIee no,n:p06HO BbIpaLKeHHe ,n:JI5I 'ljJn" B 06meM cJIyqae
rpaHHqHbIX YCJIOBl{H OHO 06pa30BaHO TpeM5I CJIaraeMbIMH, KaLK,n:oe H3
KOTOpbIX B TOM HJIH ,n:pyroM qaCTHOM CJIyqae MOLKeT 6bITb paBHO HYJIIO.
B CB5I3H C 3THM Tpe60BaHHeM HepaBeHcTBaM (5.24) ,n:OJILKHO y,n:oBJIe-
TBOp5ITb KaLK,n:oe H3 CJIaraeMbIX B OT,n:eJIbHOCTH
00 OTHOrneHI1IO K CJIaraeMoMy h", 5IBJI5IIOmeMYC5I K03cpcpl1U,HeHToM
p5I,n:a <PYPbe ,n:JI5I HOpMaJIbHOrO Hanp5ILKeHH5I, YCJIOBl1e (5.24) 3KBHBa-
JIeHTHO Tpe60BaHI1IO KYCOqHOH HenpepbIBHOCTI1 CPYHKU,HI1 t; (y) Ha HH-
TepBaJIe -'I'J < y < 'I'J [182]. 9TO Tpe60BaHI1e He 5IBJI5IeTC5I cymecTBeH-
HbIM OrpaHl1qeHl1eM. ECJIH B HCXO,n:HOH rpaHl1qHOH 3a,n:aqe HenpepbIB-
HOCTH HarpY30K HeT, TO MOLKHO JIerKO nOCTpOHTb perneHl1e, CB5I3aHHOe C
pa3pbIBaMH (perneHl1e ,n:JI5I 6eCKOHeqHOH nOJIOCbI), a ,n:JI5I era YTOQHeHI15I
YLKe He06xo,n:I1MO peIIlaTb rpaHI1QHyIO 3a,n:aQy C HenpepbIBHbIMI1 YCJIO-
BI15IMH.
HeCKOJIbKO CJIOLKHee 06CTOHT ,n:eJIO C KaCaTeJIbHbII\UI Hanp5ILKeHH5IMH.
Op5IMOe no,n:QI1HeHHe CJIaraeMbIx OT KaCaTeJIbHbIX HanpHLKeHHH B
BbIpaLKeHI1H ,n:JI5I 'ljJn YCJIOBII5IM (5.24) nplIBo,n:HT He TOJIbKO K Tpe60Ba-
HHIO HenpepbIBHOCTH CPYHKU,HH, 3a,n:aIOmHX KaCaTeJIbHbIe Hanp5ILKeHH5I,
HO II K 065I3aTeJIbHOMY 06pameHHIO HX B HY JIb B yr JIOBbIX TOQKaX 06-
JIaCTH. OCJIa6HTb 3TO Tpe60BaHHe MO)i{HO JIHIIlb OTKa3aBIIlHCb OT pa3-
,n:eJIbHOra paccMOTpeHH5I COCTaBJI5IIOmHX, B03HHKaIOmI1X OT KaCaTeJIb-
HbIX Hanp5ILKeHI1H B 'ljJn"
PaCCMOTpI1M 60JIee no,n:p06HO BbIpaLKeHH5I ,n:JI5I K03cpcpHU,HeHToB PH-
,n:OB <PYPbe CPYHKU,HH, 3a,n:aIOmI1X KaCaTeJIbHbIe Hanp5ILKeHI1H, npe,n:no-
.TIaraH 3TH CPYHKU,I1H HenpepblBHbIMH Ha COOTBeTCTBYIOmHX I1HTepBaJIaX.
I1CXo,n:5I 113 onpe.'J;eJIeHI1H K03:pcpHU,HeHTOB p5I,n:OB <PYPbe, HMeeM


""'" ( l ) k(f)lk 2ak - (jJ2n (- I)" cth
n'I'J =
t; - 't' (a
+

)
=
 cth n:n;'I'J [(jJ2 (1) - (jJd'I'J)) + 0 (n- 2 ).
nn
Opv. BblBo,n:e cOOTHoIIleHHH (5.26) H (5.27) HCnOJIb30BaHbI 3HaQeHH5I
cJIe,n:YIOmHX 6eCKOHeQHbIX CyMM:
1 1
=-ctha n --,
an'l'} a
'I'}


(5.27)



 2
k=1 11 (

 + a
)


(5.28)


11 n
2 r ( - nn 2(- I) 0 -6
(jJIn = - J (jJ1 y) Sin - ydy = - (jJ1 ('I']) + (n ),
'I'} 0


I
f 2 (- I)n 3
qJ2n = 2 . (jJ2 (X) sin n:n;xdx = - (jJ2 (1) + 0 (n- ).
o nn


(5.25)


""'" 2 1 1

 =-cth
n'I'J--""2'
k=I 'I'} (a
 +

)
n 'I'}
n
PaCCMaTpl1Ba5I BbIpaLKeHI15I (5.26) H (5.27), 3aKJIIOQaeM, QTO CHCTeMa
(5.15) HMeeT OrpaHHQeHHOe perneHHe, ,n:JI5I KOToporo cnpaBe,n:JIHB 3aKOH
aCI1MnTOTHQeCKHX BbIpaLKeHHH, eCJIH KaCaTeJIbHbIe Hanp5ILKeHH5I He-
npepbIBHbI Ha BceH nOBepxHocTH np5IMoyrOJIbHHKa, BKJIIOQa5I H yr JIOBbIe
rOQKH, r,n:e ,n:OJILKHO BbmOJIH5ITbC5I YCJIOBHe napHocTH KaCaTeJIbHbIX Ha-
IIpHLKeHHH, T. e. (jJl (I) = (jJl ('I'J). u
3aKoH aCHMnTOTHQeCKHX BbIpaLKeHHH npe,n:onpe,n:eJI5IeT cJIe,n:YIOm HH
xapaKTep nOBe,n:eH115I HeH3BeCTHbIX B 6eCKOHeQHOH CHCTeMe (5.15):
Hrn X n = Hrn Yn = ao. (5.29)
n
CXJ fJ-+OO
BbITeKaIOmHe H3 ::noro paBeHcTBa CBoHcTBa 6eCK u OHeQHbIX p5I
OB (5.1)
II (5.2), npe,n:CTaBJI5IIOm11X BeJII1QHHbI Hanp5ILKeHHH H CMemeHHH B np5I-
MoyrOJIbHHKe, 6y,n:YT paccMoTpeHbI B CJIe,n:YIO
eM naparpacpe. 3,n:ecb
LKe MbI OCTaHOBHMC5I Ha aHaJIH3e B03MOLKHOCTeH H3JIaraeMOro MeTo,n:a
npH HCCJIe,n:OBaHI1II rpaHI1QHWX 3a,n:aQ, B KOTOpWX YCJIOBl1e napHocTH
KaCaTeJIbHbIX Hanp5ILKeHHH B yrJIOBOH TOQKe He BbillOJIH5IeTC5I, T. e.
'P2 (1) =1= (jJ. ('1']).


50


<1*


51




I1cnOJIb30BaHHe npe.n.CTaBJIeHlHI BeKTOpa ynpyrux CMern.eHHH B
BH.n.e (5.1), KaK CJIe.n.yeT H3 npe.n.bI.n.yrn.ero paCCMOTpeHH5I, He n03BOJIHeT
nOJIyqIlTb peIlIeHHe rpaHIlqHOH 3a.n.aqj1 C pa3pblBHblMIl 3HaqeHH5II\HI
KaCaTeJIbHblX HanpmKeHHH. B yrJIOBOH TOqKe. B CB5I3H C 3THM nonbl-
TaeMC5I nOJIyqIlTb peIlIeHHe Ha OCHOBe npe.n.CTaBJIeHHH BeKTopa ynpy-
ruX CMern.eHHH B BIl.n.e (4.6) H (4.9) npM HCnOJIb30BaHHfI OTJIMqHOrO
OT npHHHTOro B (5.1) np0.n.OJI1KeHflH rpaHHqHhlX YCJIOBIlH Ha nOBepx-
HOCTHX cooTBeTcTBYIOrn.flX 6eCKOHeqHhlX nOJlOC. Pa3JIflqHfl B npflHHMa-
eMblX . cnoco6ax np0.n.OJI1KeHflH rpaHIlqHblX YCJIOBHH npe.n.CTaBJIeHhl Ha
pflC. 3. Ha pIlC. 3, a nOKa3aH xapaKTep npOJI.OJI1KeHHH rpaHIlqHblX yc-
g g


2n-1
3.n.ech an = 2
rJ
Jl3BOJIbHble nOCT051HHhle
OOCTpoeHHe Bhlpa1KefJflH (5.30) npoBe.n.eHo C ueJIblO H3yqeHlJH
B03M01KHOCTH npHMeHHeMoro MeTo.n.a npfl aHaJIH3e rpaHHqHbIX 3a.n.aq
.n.JIH TeJI KOHeqHhIX pa3MepoB C pa3pbIBHhlMIl 3HaqeHHHMH KaCaTeJIb-
HhIX HanpH1KeHHH B yrJIOBOIl TOqKe. B CBH3H C .9THM npH 4JoPMYJIflpOBKe.
rpaHIlqHblX YCJIOBHH, H3 KOTOphIX CJIe.n.yeT onpe.n.eJIHTh co.n.ep1Karn.Heca
B (5.30) npOH3BOJIbHbIe nOCTOHHHble, M01KHO CqHTaTb OTJIflqHbIMH OT
HYJIHKaCaTeJIhHhU' HanpH1KeHHH JIIlIlIb Ha O.n.HOH nape rpaHeH, nOJIaraH
Bce OCTaJIbHbIe Hanp5!1KeHHH paBHbIMIl HyJI 10 , T. e. paccMaTpIlBaTb CJIe-
.n.YIOrn.YIO rpaHHqHYIO 3a.n.aqy:


31,



,,= 2n
2
 31, An' En' C" H Dn - npo-


I
20 u x ( + 1. Y) = 0,
t
20 uy(x, + 1']) = 0,


I
20 .Xl! ( + I, y) == :f:: CPt (y),
I
20 .Xl! (X, ::I: 1']) =--= 0.


(5.31)


0 X 0 X
a tf
PHc.3.


CJIe.n.yeT no.n.qepKHYTh, qTO ueJIH, npeCJIeJI.yeMble npH nOCTaHOBKe
.9TOH rpaHflqHOH 3a.n.aqIl, He HaKJIa.n.bIBaIOT KaKflX-JIH60 OrpaHHqeHHH
Ha BH.n. 4JYHKUHH CPt (y). Tpe6yeTcH JIHIlIb, qT06b1 CPt (1']) =i= 0. 3To
06CTOHTeJIhCTBO nOCJIY1KHT B .n.aJIbHeHIlIeM OCHOBOH npeo.n.OJIeHIlH He-
KOTOphIX 3aTpy.n.HeHHH, CBH3aHHbIX C OrpaHMQeHIl5!MH, XapaKTepHbIMH
.n.JIH TeopHH peryJIHpHbIX 6eCKOHeQHbIX CHCTeM.
rpaHI.J1lHbIe YCJIOBHH .n.JIH HOpMaJIbHbIX HanpH1KeHHH B (5.31) npH-
BO.n.HT K cJIeJI.YIOrn.eH CBH3Il Me1K.n.y K034J4JHuHeHTaMIl 6eCKOHeQHhIX
pH.n. OB :


I
I


JIOBHH .n.JIH KaCaTeJIbHbIX HanpH1KeHHH, OTBeQaIOIlI.HH peIlIeHHIO ypaB-
HeHHH .naMe B BH.n.e (5.1). J],pyroH B03MO.lliHbIH cnoco6 HX npo.n.oJI.llie-
HHH .n.aH Ha pIlC. 3, 6. Ha 3TIlX pHcYHKax .n.aHhI cnOC06bJ np0.n.OJI.llieHHH
3a.n.aHHbIX Ha cTopoHax npHMoyroJIbHHKa (-1
 x <: 1, Y = + 1'])
rpaHHQHblX YCJIOBIlH Ha CTOpOHbI 6eCKOHeQHOH nOJIOCbI Y = + 1'] (- 00
00 < x < 00). AHaJIoruQHO CTpOHTCH rpaHHQHble YCJIOBHH Ha CTopoHax
nOJIOCbIX = + 1 (-00 < y < 00).
OTBeQalOrn.ee npe)J,cTaBJIeHHoMY Ha pHC. 3, 6 cnoco6y 3a.n.aHHH
rpaHHQHbIX YCJIOBIlH peIlIeHIle ypaBHeHHH .naMe nOJIYQaeM H3 06rn.HX
npe.n.CTaBJIeHHH BeKTopa CMern.eHHH B BH.n.e (4.6) H (4.9). OpH :lTOM
B HIlX CJIe.n.yeT nOJI01KHTh p = 2, T = 21'] H OTJIIlQHbIMH OT HYJIH CQIl-
TaTb npOIl3BOJIhHble nOCTOHHHble JIHIlIb C HeQeTHbIMH HH.n.eKCaMH. OocJIe
nepeHYMepaUHl-I npOH3BOJIbHbIX nOCTo'HHHbIX peIlIeHHe npe.n.cTaBHM B
BH.n.e


( m-2
Bnan = - D n . m


+ a" th aJ .
. .
+
,,1'] th I3n1']) .


(5.32)


( ' m-2
Anl3n=-C" m
I1cnOJIb30BaHHe pagJIO.llieHHH


,
I,


u = -
 (Bn sh anx + Dllx ch a"x) cos any-
n=I


h t
 co 21' (- 1 t cb 'I'h .
 t
s
,=- SIllu
r h ( ,,2 + fh n
n=l J nl
2n-l
H nOJIYQaeMoro Il3 Hero lU14J4JepeHuHpOBaHHeM no '\' npfl l)n = -cg;- 31
n:aeT B03MO.lliHOCTh HaIlTIl H3 YCJIOBHH .n.JIH KaCaTeJIhHbIX HanpH.llieHflH
B (5.31) cJIe.n.ylOrn.ylO 6eCKOHeQHYlO cHcTeMY JIflHeHHbIx aJIre6paHQe-
CKIlX ypaBHeHIlH:


(5.33)


I,;
II
II'



 ( 3m-4 \ .
-,,
 An ch l3"y + C" mfln ch I3nY + CnY sh f3"Y J SIll f3n x ,


,1 I 3m -4 . )
tV =."'-J B" ch a"x + Dn ch anx + D'IX sh anx . sin anY +
n=1 \ man


x"
 ( th an + h
n ) =
 y"
an c an .=1


4a

2 + (- l)"CPln,
(f\
 + a n )2
4f1

(flk +a
)2 '


(5.34)


co


00


+
 (An sh f3"Y + C flY ch I3"Y) cos I3nx.
n=1


(5.30)


Yn fI: (th
nTJ + c

:tJ ) = tt x"
53


52




r
e


n 060HX ypaBHeHHHX CHCTeMbI (5.34) K03cpcpHIJ,HeHTbI Be
YT ce6
 O
H-
lIaKOBO, TO
OCTaTOqHO npOBeCTH npOBepKY 3TH X COOTHOIIleHHH
JIH
o.n.Hoii rpynnbI K03cpq:JHIJ,HeHTOB.
OepBOe YCJIOBHe CymecTBoBaHHH npe.n.eJIbHoro paBeHcTBa (5.36)
6y.n.eT BbIIIOJIHeHO, eCJIH cnpaBe.n.JIHBa cJIe.n.yIOmaH OIJ,eHKa (
(n) =
= n):


X k = (-l)kDkakchak' Yk = (- l)kCk
chPk'l1. (5.35)
1]
L{JIH CYMM K03cpq:JHIJ,HeHTOB npH Yk H x k B 3TH X paBeHCTBax Ha OCHOBe
pa3JIO)KeHHH (5.33), 06JIa.n.aIOmHX paBHoMepHoH CXO.n.HMOCTbIO, HaXO
HM
cJIe.n.YIOmHe 3HaqeHHH:

 2 4a
 2 =
(than- h
n ) ,
k=1 \
k + a n )2 an c an


4a
n
1<. 2 2 <.L

1 (an) (an +
k)2


(k<n, n = I, 2, .. .), (5.37)



 4


tI (f3
 + ak)2



 (th Pn'l1- C
n
1]) .


npH nOJIO)KHTeJIbHbIX He3aBHCHmHX 01' n nOCTOHHHbIX I H L. 3.n.eCb
A ( a ) - th a + a /ch 2 a YqHTblBaH HHTepBaJI H3MeHeHHH k. JIerKO
ill n - n n no
HaHTH


3TH 3HaqeHHH CYMM nOKa3bIBaIOT, qTO CHCTeMa (5.34) HBJIHeTCH
perYJIHpHoH. O
HaKo cpYHKIJ,HH, onpe.n.eJIHIOmHe CKOpOCTb CTpeMJIeHHH
CYMMbI K03cpqJHIJ,HeHTOB K e.n.HHHIJ,e, B 3TOM cJIyqae HMeIOT 3KCnOHeHIJ,H-
aJIbHbIH xapaKTep y6blBaHHH. 31'0 06CTOHTeJIbCTBO YKa3bIBaeT Ha TO,
qTO npH npOH3BOJIbHbIX CB060.n.HbIX qJIeHaX !PIn, HMeIOmHX, KaK K03cp-
cpHIJ,HeHTbI pH.n.a <DYPbe, CTeneHHOll XapaKTep y6bIBaHH5I, .n.JI5I CHCTeMbI
(5.34) HeJIb351 BbIIIOJIHHTb YCJIOBHe (3.3), .n.OCTaTOqHOe .n.JIH cymecTBo-
BaHHH KOHeqHOrO peIIleHH5I. B CBH3H C 3THM 6y.n.eM CqHTaTb B CHCTeMe
(5.34) OTJIHqHbIMH 01' HYJIH JIHIIlb !Pu. a Bce OCTaJIbHble K03<PcpHIJ,HeHTbI
<DYPbe cpYHKIJ,HH !PI (y) 6y.n.eM CqHTaTb paBHbIMH HYJIIO. TaKoe npe.n.no-
JIO)KeHHe, cpopMaJIbHO npe.n.cTaBJI5IIOmeeCH cymecTBeHHbIM, cpaKTHqeCKH
HHKaK He OrpaHHqHBaeT B03MO)KHOCTell pellleHHH rpaHHqHOll 3a.n.aqH
JIH
np5lMoyrOJIbHHKa C pa3pbIBHbIMH YCJIOBHHMH lI.JI5I KaCaTeJIbHbIX Ha-
np5l)KeHHll B yrJIOBbIX TOqKax.
L{ellcTBHTeJIbHO, HaJIHqHe perueHHH rpaHHqHOii 3a
aqH npH !Pu =t= 0
B (5.34) n03BOJIHeT OqeBH.n.HbIM 06pa30M YCTpaHIITb npOH3BOJIbHblH
KOHeqHbIH pa3pbIB KaCaTeJIbHbIX HanpH)KeHHH B yrJIOBOH TOqKe H,
TaKHM 06pa30M, CBeCTH 3a.n.aqy K pa3peIIlHMoH C nOMOmbIO Bblpa)KeHHH
.'l.JIH cMemeHHH B BH.n.e (5.1). CaM cpaKT cymecTBoBaHH5I B03MO)KHOCTH
BbI
eJIHTb C nOMOmbIO OTBeqaIOmero !Pu peIIleHHH TY qaCTb 06mero
nOJIH CMemeHHH H HanpH)KeHHH, KOTOpa51 CBH3aHa c pa3pbIBOM B Ka-
CaTeJIbHbIX HanpH)KeHH5IX, YKa3bIBaeT Ha TO, qTO, OrpaHHqHBaHCb JIHIIlb
nepBOH rapMoHHKOll B % (y), MbI HHqero He TepHeM H C TOqKH 3peHH5I
KaqeCTBeHHoro aHaJIH3a oc06eHHOCTeH Hanp5l)KeHHoro COCTOHHHH.
nop0)K.n.aeMbIX HapYIIleHHeM Tpe60BaHH5I napHOCTH KaCaTeJIbHbIX Ha-
I1p5l)KeHHH B yrJIOBOH TOqKe.
PerYJIHpHaH 6eCKOHeqHaH CHCTeMa (5.34) npH yCJIOBHH !Pu =1= 0,
!PIn = 0 (n = 2, 3, ...) HMeeT OrpaHHqeHHOe peIIleHHe. OOKa)KeM, qTO
nOBe.n.eHHe 3TOro peIIleHH5I Ha oeCKOHeqHOCTH onpe.n.eJIHeTC5I 3aKOHOM
aCHMnTOTHqeCKHX Bblpa)KeHHH, 1'. e.
\im x n = lim Yn = b v =1= O. (5.36)
n-+c<> n-+co


4 4a
n 41]
11 <. 2 2 <. (5.38)
11: (I + 1]2)2
; (an)
1 (an) (an + f3 k ) 2 11: 1 1 (an)
3.n.ecb /1
 (an),
1 (an) - COOTBeTCTBeHHO HaHOOJIbIIlee H HaHMeHblllee
3HaqeHHe BeJIHqHHbI /1 1 (an) .n.JIH BCeX n I, 2, ... 41'0 KacaeTCH HaH-
OOJIbIIlero 3HaqeHHH Bblpa)KeHHH /1 1 (an)' TO yKa)KeM, qTO cpYHKIJ,H5I
/1 1 (x) HMeeT a6COJIIOTHbIH MaKCHMYM B TOqKe X c Xv, r.n.e Xv HBJIHeTCH
KOpHeM ypaBHeHH5I X th X ,1. B CBH3H C 3THM .n.JIH BeJIHqHHbI I MO)KHO
YKa3aTb CJIe.n.YIOmee 3HaqeHHe:
I -= 11: (I + 1]

2
1 (x o ) (5.39)
HaHMeHbIIlee 3HaqeHHe BeJIHqHHbI /1 1 (an) 3aBHCHT 01' COOTHOIIleHHH
CTOpOH npHMoyroJIbHHKa H YMeHbIIlaeTCH C pOCTOM '11. Ba)KHbIM HBJIHeT-
CH TO 06CT05lTeJIbCTBO, qTO npH JII060M KOHeqHOM 'I'] MO)KHO YKa3aTb BnOJI-
He onpe.n.eJIeHHOe 3HaqeHHe BeJIHqHHbI 51 (an) H, CJIe.n.OBaTeJIbHO, no-
CTOHHHOH L B (5.37).
OpH .n.aHHOM xapaKTepe K03cpcpHIJ,HeHTOB CHCTeMbI (5.34) BTOpO

yCJIOBHe (3.8) cnpaBe.n.JIHBOCTH 3aKOHa aCHMnTOTHQeCKHX Bblpa)KeHHH
.n.JIH HeH3BeCTHbIX 51BJI5IeTC5I npocTbIM CJIe.n.CTBHeM OIJ,eHOK (5.37).
TaKHM 06pa30M, cnpaBe.n.JIHBOCTb npe.n.eJIbHoro paBeHcTBa u (5.36)

JIH HeH3BeCTHblX B CHCTeMe (5.34) YCTaHOBJIeHa. TaKHM )Ke CBOHCTBOM
06JIa.n.aIOT HeH3BeCTHble B CHCTeMe (5.15) npH yCJIOBHH HenpepbIBHocTH
KaCaTeJIbHbIX HanpH)KeHHH B yrJIOBbIX TOqKaX 06JIaCTH.
TIpe.n.eJIbHbIMH paBeHcTBaMH (5.29) H (5.36) YCTaHaBJIHBaeTCH on-
pe.n.eJIeHHbIH xapaKTep nOBe.n.eHHH K03cpq:JHIJ,HeHTOB p5l.n.OB B 06mHX
peIIleHHHX (5.1) H (5.30). BbITeKaIOmHe H3 3TOro oc06eHHOCTH nOBe.n.eHH5I
pH.n. OB
JIH cMemeHHll H HanpH)KeHHH Ha CTopOHax np5lMoyrOJIbHHKa,
a TaK*e cooTBeTcTByIOmHe Tpe60BaHH5I K aJIrOpHTMaM npH6JIH)KeHHOrO
peIIleHHH 6eCKOHeqHbIX CHCTeM paccMaTpHBaIOTcH B CJIe.n.YIOmeM napa-
rpacpe. 3
ecb )Ke KOpOTKO OCTaHOBHMCH Ha Bonpocax e
HHCTBeHHOCTH
peIIleHHH CHCTeM (5.15) H (5.34).
BonpocbI e.n.HHCTBeHHOCTH peIIleHH5I perYJI5IpHbIX 6eCKOHeqHbIX
CHCTeM, K KOTOpbIM npHMeHHMa TeOpH5I 5. M. KOHJIOBHqa, 51BHJIHCb
npe
MeToM O)KHBJIeHHOH .n.HCKyCCHH [I 12, 1151. ECTecTBeHHo, qTO
55


L{oCTaTOqHble YCJIOBH5I .n.JI5I cnpaBe.n.JIHBocTH 3TOro paBeHcTBa Bblpa-
)KaIOTCH B
aHHOM cJIyqae COOTHOIIleHHHMH (3.23) H (3.8). OOCKOJIbKY
54




OCTpOTa 3TOro BOnpOCa B 3HaqHTeJIbHOH Mepe CHH*aeTC5I npH paCCMOT-
peHHH CHCTeM, nopO*.n;aeMbIX rpaHHqHbIMlI 3a.n;aqaMlI, .n;JI5I KOTOpbIX
.n;OKa3aHa e.n;lIHCTBeHHOCTb perneHH5I, KaK no nOKa3aHO B CJIyqae pac-
CMaTplIBaeMbIX 3a.n;aq TeoplIlI ynpyroCTlI. O.n;HaKO nOCKOJIbKY peqb lI.n;eT
o nOCTpoeHHH aJIrOplITMOB perneHlI5I 6eCKOHeqHbIX ClICTeM, n03BOJI5IIO-

HX HRXO.ll.lITb onpe.n;eJIeHHOe IlX perneHlIe, TO *eJIaTeJIbHO HMeTb
YBepeHHOCTb, qTO lIMeHHO nOJIyqaeMOe perneHlIe OTBeqaeT paCCMaTplI-
BaeMOH rpaHlIqHOH 3a.n;aqe. TaKa51 YBepeHH0CTb eCTb, eCJIlI .n;OKa3aHa
e.ll.lIHCTBeHHOCTb perneHlIH 6eCKOHeqHOH ClICTeMbI.
34JcpeKTlIBHbIM cnoco60M .n;OKa3aTeJIbCTBa e.n;lIHCTBeHHOCTH perneHH5I
CHCTeM paCCMaTplIBaeMoro KJIaCCa 51BJI5IeTC5I TeopeMa, YCTaHOBJIeHHa51
n. c. EOH.lI.apeHKO [391. TIplIMeHlITeJIbHO K ClICTeMaM TlIna (5.15) CYTb
3TOH TeopeMbI 3aKJIIOqaeTC5I B CJIe.n;YIO
eM. ECJIH nplI HeKoTopoH 3RMeHe
HelI3BeCTHblX


X n = x
Hn> Yn = y;'P n ;
lim Hn = ex>, lim Pn = ex>
tZ-t-()Q n-+txJ


(5.40)


B CHCTeMe OTHOCHTeJIbHO x n H Yn nOJIyqeHHa51 CHCTeMa OTHOCHTe.nbHO
* .
Xn H Yn perYJI5IpHa H lIMeeT OrpaHHqeHHOe perneHlIe, TO perneHlIe lIC-
XO.ll.HOH CHCTeMbI e.n;lIHCTBeHHoe.
BBe.n;eM B CHCTeMe (5.15) HOBbIe HeH3BeCTHble COOTHOrneHlI5I
X n = x: (
 ) , Yn = y
n'Jt. {5.41)
. fj
LI.JIH onpe.ll.eJIeHHH HOBbIX HeH3BeCTHbIX X
 II Y
 nOJIyqHM ypaBHeml>I


. 4a n
. I3k 'I' n
X n = tJ. (an)
 Yk ( 1'12 + n2 ) 2 + antJ. (an) ,
k=1 Pk
n


.
Yn=


413n
 ak
fj2tJ. (I3nfj) .... x; 2 2
k=1 (ak + I3n)2


(5.42)


+ IDn
'1 2 I3 n tJ. ((:In fj)


LI.JI5I HCCJIe.n;OBaHHH perYJIHpHOCTH CHCTeMbI Hc06xo.n;lIMO OueHHTb
CYMMY K03cpcpHUHeHTOB B Ka*.n;OH CTpOKe. T aK, .n;JIH K03cpcp H UlIeHTOB
nepBOH rpynnbI ypaBHeHHH B (5.42) lIMeeM


00 00

 [:\k. <: 2- fj4
 S -2..


 {l3k + a
)2 2 [(m1)
 + 13
j2 + 1t 0 (x2 + a
)2 . (5.43)
AnnpOKCHMaulI5I C H36bITKOM nOH CYMMbI HHTerpaJIOM ocy
eCTBJIHeTC5I
Ha OCHOBe aHaJIH3a xapaKTepa nOBe.n;eHH5I <PYHKUHH x/(x2 + a 2 )2. BbI-
qHCJIHH lIHTerpaJI B (5.43), Haxo.n;HM CJIe.n;yIO
YIO OU
HKY .n;JI5I CYM-
MbI K03cpcpHuHeHToB B nepBoM ypaBHeHlIlI CHCTeMbI (5.42):


4a n

t:. (an) t':


I3k <: 2- .
(13
 + a
)2 1t


(5.44)


56


II


I


AHaJIOrHqHaH oueHKa cnpaBe.n;JIHBa .n;JIH K03cpcpHuHeHTOB BToporo ypaB-
HeHlIH CHCTeMbI (5.42).
TaKHM o6pa30M, nOJIyqeHHaH H3 (5.15) npeo6pa30BaHHeM (5.41)
6eCKOHeqHaH CHCTeMa OKa3aJIaCb BnOJIHe perYJI5IpHoH. OrpaHHqeHHOCTb
ee CB06o.ll.HbIX qJIeHOB OqeBH.n;Ha lI3 oueHOK, npl1Be.n;eHHblx npH pac-
cMoTpeHlIH CHCTeMbI (5.15). B CBH3H C 3THM 3aKJIIOqaeM, qTO ClICTeMa
(5.42) lIMeeT e.n;HHCTBeHHoe OrpaHlIqeHHOe perneHHe H, CJIe.n;oBaTeJIbHO,
e.n;lIHCTBeHHoe perneHHe lIMeeT Hcxo.n;HaH CHCTeMa (5.15).
TIpe.n;bl.n;y
ee paccMOTpeHHe 6eCKOHeqHOH CHCTeMbI (5.15) n03-
BOJIlIJIO YCTaHOBHTb CnpaBe.ll.JIHBOCTb 3aKOHa aClIMnTOTHqeCKlIX Bblpa-
*eHlIH .n;JIH ee r JIaBHoro perneHHH. TIoCJIe.n;YIOIUHH aHaJIH3 nOH CHCTeMbI
nOKa3aJI, qTO 3TO perneHlIe 51BJI5IeTC5I e.n;HHCTBeHHbIM. OTcIO.n;a CJIe.n;yeT
3aKJIIOqHTb, qTO BbITeKaIOIUl1e lI3 3aKOHa aCHMnTOTHqeCKHX Bblpa*eHlIH
oc06eHHocTH nOBe.n;eHHH K03cpcpHuHeHTOB p5l.ll.0B oTpa*aIOT xapaKTep-
Hble CBoHcTBa H3yqaeMbIX <pH3HQeCKHX nOJIeH. HCXO.n;H H3 3TOro 06CTOH-
TeJIbCTBa, npH aHaJIH3e B paMKax H3JIo*eHHoro no.n;XO.ll.a rpaHHQHbIX
3a.n;aQ Bnpe.n;b Mo*eM paccMaTpHBaTb JIHrnb rJIaBHble perneHH5I nopo-
*.n;aeMbIX HMH 6eCKOHeQHblX ClICTeM. TIo.n;TBep*.lI.eHHeM npaBHJIbHOCTH
TaKoro BbI60pa perneHHH Ka*.n;bIH pa3 Mo*eT CJIY*HTb TaK)Ke npoBepKa
y.n;oBJIeTBOpHeMocTH rpaHHQHbIX YCJIOBHH.
OueHHBaH <paKT BnOJIHe perYJI5IpHOCTH CHCTeMbl (5.42), KOTOPYIO
MO)KHO HCnOJIb30BaTb BMecTO CHCTeMbI (5.15) npH perneHHH nepBoH
OCHOBHOH rpaHHQHOH 3a.n;aQH .n;JI5I np5lMoyroJIbHHKa, CJIe.ll.yeT CJI,eJIaTb
CJIe.n;YIOIUee 3aMeQaHHe. C TOQKH 3peHH5I TeopHH 6eCKOHeQHbIX CHCTeM
BonpocbI, B03HHKaIO
lIe npH HaXO)K.lI.eHHH npH6JIlI}HeHHblx 3HaQeHHH
HelI3BeCTHblX lI3 BnOJIHe perYJIHpHOH ClICTe-MbI, 3HaQHTeJIbHO npo
e,
QeM B cJIYQae perYJIHpHbIX CHCTeM. OnpeJl.f'JIeHHbIMlI npeHMy
ecTBaMH
nepe.n; clIcTeMoH (5.15) 06JIa.n;aeT H B HarneM CJIYQae ClICTeMa (5.42).
TIpoBe.ll.eHHbIH aHaJIlI3 nOCJIe.n;HeH nOKa3bIBaeT, QTO e.n;HHCTBeHHoe ee
perneHlIe cy
ecTByeT H nplI OTHa3e OT Tpe60BaHH5I napHocTlI KaCaTeJIb-
HbIX HanpH)KeHlIH B yrJIOBOH TOQKe. O.ll.HaKO cy
ecTBeHHbIM He.ll.OCTaT-
KOM CHCTeMbl (5.42) 51BJIHeTCH TO, QTO B 06
eM cJIYQae He y.n;aeTC5I YCTa-
HOBI1Tb XapaKTep nOBe.n;eHI151 HeH3BeCTHblX B HeH C pOCTOM HX HOMepa.
3TO 06CTOHTeJIbCTBO HCKJIIOQaeT B03MO)KHOCTb nOJIYQHTb Ha OCHOBe
paCCMOTpeHI1H 6eCKOHeQHbIX pH.n;OB npe.n;CTaBJIeHHe 0 KaQeCTBeHHbIX
oc06eHHOCT5IX H3YQaeMbIX nOJIeH CMe
eHHH H Hanp5l)KeHHH B CB5I3H C
oc06eHHOCTHMH reOMeTpHH ynpyrl1x TeJI, a TaK)Ke cnoc060B HX Harpy-
*eHH5I. KpoMe Taro, 3HaHHe npe.n;eJIbHbIX CBOHCTB Hel13BeCTHbIX B pe-
rYJIHpHOH CHCTeMe n03BOJI5IeT C03.n;aTb aJIrOpl1TMbI I1X OTbICKaHH5I,
He YCTynafO
He no 3cpcpeKTHBHOCTH MeTO.n;aM perneHI1H BnOJIHe perYJI5Ip-
HbIX CHCTeM. TaKHM 06pa30M, npe.n;nOQTHTeJIbHOH C TOQKH 3peHI1H rpa-
HHQHOH 3a.n;aQI1 51BJI5IeTCH Ta CHCTeMa, KOTopaH n03BOJIHeT HaH60JIee
npOCTO nOJIYQHTb a.n;eKBaTHble KOJIHQeCTBeHHble 11 KaQeCTBeHHble onH-
CaHH5I H3YQaeMOrO HBJIeHHH.


57




S 6. CBoHcTBa 06
HX perneHHii nepBoii OCHOBHOii
rpaHH'IHOii aa.a;a'IH .a;JIH npHMoyrOJIhHHK3


TblO. 3HaqHTeJIbHa CJIoxmee a6CTaHT ,l(eJIa npH nanbITKe BblqHCJIHTb
HanpmKeHH5I Ha CTOpaHax np5IMayraJIbHHKa.
PaccMaTpHM na,ZJ:pa6Hee BbITeKalOw.ee H3 (6.1) Bblpa.tKeHHe ,ZJ:JI5I U x
npH x = + 1:


B a6eHx paccMaTpeHHblx BbIll1e 3a,ZJ:aqaX, na,ZJ:'lHH5I5I a6w.He pell1eHH5I
rpaHH'lHbIM YCJIaBH5IM, MbI npHll1JIH K napHbIM 6eCKaHe'lHbIM CHCTeMaM
JIHHeHH
X aJIre6paH'leCKHX ypaBHeHHH. 3Ha'lHTeJIbH
e YCHJIHH 6
JIH
3aTpa'leHbI Ha I13Y'leHHe npe,ZJ:eJIbHbIX cBaHcTB HeH3BeCTHbIX, H ,ZJ:JI5I a6e-
HX 6eCKaHe'lHbIX CHCTeM (5.15) II (5.34) 6bIJIa YCTaHaBJIeH8 cnpaBe,ZJ:JIH-
BaCTb 3aKaHa aCHMnTOTHqeCKHX Bblpa.tKeHHH. 3TaT pe3YJIbT8T ,ZJ:aeT Ba3-
Ma.tKHaCTb H3yqHTb xapaKTep cxa,ZJ:HMaCTH 6eCKOHe'lHbIX p5I,ZJ:aB B a6w.Hx
pell1eHHHX, YCTaHaBHTb aca6eHHaCTH HanpmKeHHa-,ZJ:e<p a pMHpaB8HHara
CaCTa5IHII5I B yrJIaB
X Ta'lKaX H C<PapMYJIHpaBaTb Tpe6aBaHH5I K aJI-
ropHTMY pell1eHH5I 6eCKaHe'lHbIX CHCTeM C Ta'lKH 3peHH5I Ta'lHaCTH Ka-
JIH'leCTBeHHblX aueHaK ,ZJ:JIH HanpH.tKeHHH II cMew.eHHH.
PaccMaTpHM BHa'laJIe a6w.ee pell1eHHe rpaHHQHaH 3a,ZJ:aQH ,l(JI5I np5I-
MayroJIbHHKa B <papMe (5.1). BBa,ZJ:5I ,ZJ:JI5I caKpaw.eHH5I 3anHCH a603Ha-
nn
QeHHH an = -,
n = n:n II Y'lHTbIBa5I CaaTHall1eHHH (5.5) H (5.14),
'11
npe,ZJ:cTaBJI5IeM KaMnaHeHTbI BeKTapa cMew.eHHH B BII,ZJ:e


1
2{;-u x (x= + l)


B m - 1 Ao +
 ( 1 ) n ch f3nY
o m - 2 + m - 2
 fjJ2n - sh r
nfJ -



 th --L
 (- I)n xn ( h + an )
-
 ffJ..C anCasfY.nY I """ - h . c an - h . cas any-
an s an s an {
.=1 n=1


- I B


 (ch
nY - BllfJ cth BII T ] ch BnY + BnY sh BnY). (6.2}
n=1 n nfJ



 !l'2n ch \:InY.

u = Box +
 f:\nsh(:\nfJ SIn Bnx-



!fIn sh anx
an sh an


ca
 ally-


06paTHMC5I BHa'laJle K p5I,ZJ:aM, ca,ZJ:ep)f{aw.HM Ka3cpcpHuHeHTbI <I>ypbe 'PIn
H fP211 CPYHKUHH, 3a,ZJ:alOw.HX KaCaTeJIbHble HarpY3KII Ha naBepXHaCT5IX
np}jMayroJIbHHKa. ECJlH KaCaTeJIbHble Hanp5I.tKeHH5I He a6paw.alOTC5I Ii
HYJlb B yrJIaBaH TO'lKe, TO. rJIaBHble 'laCTH 3THX Ka3cpcpHUHeHTaB ,ZJ:alOTC5I
Bblpa.tKeHH5IMII (5.25). Onpe,ZJ:eJl5IeMbIH 3THMH Bblpa)f{eHH5IMH x8paKTep
y6bIBaHH5I BeJIH'lHH fPln H (['2n YKa3bIBaeT Ha TO., LITO. caaTBeTCTBYlOw.He
CJlaraeMble B (6.2) a6J1a,ZJ:alOT a'leHb Me,ZJ:JIeHHaH cxa,ZJ:HMaCTblO. O,ZJ:H8Ka
3HaHIIe rJlaBHbIX 'laCTeH Ka3cpcpHuHeHTOB fPln H ({!2n na3BaJl5IeT HcnaJlb-
.30BaTb npH cYMMHpaBaHHH pH,ZJ:aB 1I3BeCTHble MeTa,ZJ:bI YJlY'lll1eHH5I cxa-
,ZJ:HMaCTH. B ,ZJ:aHHaM cJIY'lae 3HaQHTeJIbHae y JIYQll1eHHe cxa,ZJ:HMaCTH
p5I,ZJ:aB C fPln H fP2n B (6.2) ,ZJ:aCTHraeTC5I HCnaJIb3aBaHHeM paBeHCTB



 (- I)n xn (
- '"'-
 xchanx-cthanshanx- 2
11=1 an s an \


m-l )
sh anx cas anY -
man


-I
n=1


t s
;

1] ( mml1n 2 ch
nY - fJ cth Bn fJ ch B"y + Y sh BnY) sin Bn x ,


J
 1 -(3 (tj-y) 1 I [ 1 -n('I")-YJ ]
1=
Te n =--;:rn -e ,
,=1 n


(6.1)


J 2 = I ( - I)" cas anY = -
 In (2 cas
 Y) .
,,=1 an n \
'I
3Ha'leHH5I CYM,\I p5l,ZJ:aB B (6.3) YKa3bIBalOT, 'lTO nap03Hb Ka.tK,ZJ:ae
CJlaraeMae C fPln II fP 2n B (6.2) HMeeT JlarapHcpMH'leCKYlO aca6eHHaCTb npH
na,ZJ:xa,ZJ:e K yrJlaBaH TO'lKe. O,ZJ:HaKa npHH5ITOe HaMH npe,ZJ:naJla)f{eHHe a
HenpepbIBHacTH KaCaTeJlbHbIX Hanp5I)f{eHHH B yrJIaBaH Ta'lKe (fPl (fJ) -
= fP2 (1» npHBa,l(HT K TOMY, QTa 3TH aca6eHHaCTH B3alIMHa narall1alOT-
C5I. LI.eHcTBIITeJIbHa, paCCMaTpHBa5I rJlaBHble QaCTII caaTBeTCTBYlOW.HX
p5I,l(aB B (6.2), IIMeeM


(6.3)


w=AoY+ I
n=1


<1'111 ch anx.

h SIn allY - '"'-
an s an
11-1


!l'2n sh (:\nY
(:\ h::l. cas I:\n x +
n S nfJ


""
+
 (-I)n xn ( m - 2 h th h h ) .
'"'- C anx - C all C anx + x S a.,x Sill artY +


sh
 m
 I


+I
n=1


(- l)n YnfJ (y ch B"Y -fJ cth Bn fJ sh BnY - 2 :-;n 1 sh BI1Y ) cas Bn X '
(:\" sh (:\,,'11 .


Bblpa.tKalOIlI.He KaMnaHeHTbI BeKTapa cMew.eHHH P5I,ZJ:bI a'leBII,ZJ:Ha 0.6-
JIa,ZJ:alOT paBHaMepHaH cxa,ZJ:IIMaCTblO B Jl106aH BHYTp
HHeH TO'lKe 0.6-
JIaCTII II npH HaJlH'lIIH 3HaQeHHH Ka3cpcplIuHeHTaB x n H Yn na3BaJl5IlOT
npaBa,ZJ:HTb 3cpcpeKTHBHae BbI'lHCJIeHIIe. AHaJlH3 Bblpa.tKeHHH ,ZJ:Jl5I KaM-
naHeHTOB BeKTapa CMeIll.eHHH C YQeTaM cBaHCTBa Ka3cpcpHuHeHTaB
x n II Y n (5.29) naKa3bIBaeT, 'lTO Bxa,ZJ:5IIlI.He B HHX p5I,ZJ:bI Ha rpaHHue 0.6-
JIaCTH TaK.tKe a6J1a,ZJ:alOT ,ZJ:aCTaTa'lHa 6bICTpaH paBHaMepHaH CXa,l(HMac-
58


00
I ([;2n (- 1)"
: t
 - :I fPI II cth all cas a"y =
n=1 n=1
2 1 1 - e-n('I")-Y)
= - 2fP2 (I) J 1 + - fPdfJ) J 2 + ° (I) = - fP2 (1) In +
'11 n 2cos 2
 Y
+0(1). (6.4)


59




3.n;eCb 3HaKOM 0 (1) 0603Ha1leHa HeKOTOpafl KOHellHafl npH no.n;xo.n;e K
yrJIOBOH T01lKe Q:JYHKUHfI y. J1orapHQ:JM OTHOllJeHHfI B nOCJIe.n;HeM paBeH-
CTBe HMeeT npH y -+ 1) BnOJIHe onpe.n;eJIeHHbIH KOHe1lHbIH npe.n;eJI:
. I - e-:rr(tj-y)
hm In . = In 1). (6.5)
y-tj 2(:os
y
2'IJ
YCTaHOBHB COOTHOllleHHfI (6.3) H (6.5), nOJIY1laeM B03MmKHOCTb 3Q:J-
Q:JeKTHBHo npoBecTH BW1lHCJIeHHe TOH llaCTH HanpflJKeHHH U X ' KOTopafl
onpe.n;eJIfleTCfI pfl.n;aMH C (jJ1" 11 (jJ2n B (6.2).
06paTHMcfI Tenepb K paccMoTpeHHIO Tex pH.n;OB B (6.2), K03Q:JQ:JH-
UHeHTbl KOTOpblX onpe.n;eJIfllOTCfI 113 6eCKOHe1lHOH CI1CTeMbl (5.15).
BCJIe.n;cTBHe cnpaBe.n;JII1BOCTH .n;JIfI x" 11 y" 3aKOHa aCHMnTOTH1leCKHX
BbIpaJKeHHH (5.29), xapaKTep CXO.n;HMOCTH 3THX pfl.n;OB TaKoH JKe, KaK H
TOJIbKO IlTO paCCl'I'IOTpeHHblx pH.n;OB C (jJ1" H q:2n' Cy
ecTBeHHoe OCJIOJKHe-
Hl1e Tenepb, o.n;HaKO, B03HI1KaeT BCJIe.n;CTBlie Toro, IlTO 3HalJeHHe rJIaB-
HblX 1laCTeH K03<p<pHUl1eHTOB x" H Yn' T. e. nocToflHHOH Go B (5.29),
HeH3BeCTHO. ECTecTBeHHo, MOJKHO nOnblTaTbCfI npoBecTH BbIIlHCJIeHHfI
6e3 YJIYllllJeHHfI CXO.n;HMOCTH pfl.n;OB C x" H y", onpe.n;eJIfIfI 3Ha1leHHf! KO-
HellHOrO 1lI1CJIa 3THX BeJIH1lHH H3 CHCTeM, nOJIY1leHHbIX pe.n;YKUHeH CH-
CTeMbI (5.15). 1-13 npHBe.n;eHHblx oueHOK xapaKTepa CXO.n;HMOCTH CJIe.n;yeT,
IlTO TIPH TaKOM no.n;xo.n;e .n;JIfI nOJIY1leHI1f1 .n;ocTaT01lHOH T01lHOCTH y.n;OB-
JIeTBOpeHI15I rpaHHIlHblX YCJIOBI1H nOTpe6yeTcfI pemaTb CI1CTeMbI ypaB-
HeHHH, co.n;epJKa
He HeCKOJIbKO COTeH ypaBHeHI1H. TaKaH nepcneKTHBa
.n;eJIaeT npaKTHlleCKH HepeaJIH3yeMbIM 113JIOmeHHblH BblllJe no.n;xo.n; K pe-
llJeHl110 rpaHHIlHOH 3a.n;a1lH KaK 6y.n;eT nOKa3aHO .n;aJI, bllJe MeTO.n; pe-
.n;YKUIiH Bo06
e HCKJII01laeT B03MOJKHOCTb Bbllll1CJIeHHfI HanpHmeHI1H B
yr JIOBOH T01lKe B CI1JIy oc06eHHocTeH CXO.n;I1MOCTI1 pfl.n;OB.
J13JIOJKeHHafl B
 3 TeOpl1f1 6eCKOHe1lHbIX CI1CTeM O. M. KOflJIOBI11la
.n;aeT B BI1.n;e JII1MI1TaHT cpe.n;cTBa .n;JIH oueHKH 3HalleHI1f1 BeJII11lI1Hbl
Go B (5.29). 3HaHI1e 3TOH BeJII11lI1Hbl n03BOJIHeT 3Ha1ll1TeJIbHO yopoc-
TI1Tb 3a.n;a1lY cyMMl1pOBaHl1f! pfl.n;OB B (6.2) 3a ClleT HCnOJIb30BaHI15I cBep-
HYTblX pfl.n;OB B (6.3). CTeneHb ynpo
eHI1f1 BbI1lI1CJIeHI1H 3aBHCHT,
OQeBI1.n;HO, 01' TOro, HaCKOJIbKO 6blCTpO HeH3BeCTHble B (5.15) cTpeMflTcfI
K CBOHM npe.n;eJIbHblM 3HalleHI1f1M. OTBeT Ha 31'01' Bonpoc 6y.n;eT .n;aH B
CJIe.n;YIO
eM naparpaQ:Je. 3.n;ecb YKameM e
e Ha O.n;HO BaJKHOe CJIe.n;cTBHe
H3 3aKOHa aCI1MnTOTl1lleCKI1X BblpameHI1H.
B3f1Tble nop03Hb CJIaraeMbIe C x" H y" B (6.2), KaK 11 paccMoTpeHHble
paHee pfl,lJ;b1 C (jJ1" H (jJ2", 06JIa.n;a1OT npH no.n;xo.n;e K yrJIOBOH TOIlKe JIO-
rapl1Q:JMH1leCKI1MI1 oc06eHHOCTflMI1. <DaKT COBna.n;eHI1f1 npe.n;eJIbHblX 3Ha-
1leI-II1M Hel13BeCTHblX x" H y" yKa3blBaeT Ha TO, 1lTO OTMe1leHHbIe oc06eH-
HOCTH B3aHMHO nOrallJalOTCfI H HanpflJKeHI1f1 HMelOT B yrJIOBOM T01lKe
BnOJIHe onpe.n;eJIeHHbIe KOHe1lHbIe 3Ha1leHI1f1. XapaKTep npe.n;eJIbHOrO
nepexo.n;a npl1 3TOM TaKoH JKe, KaK H B COOTHOllJeHltflX (6.4) 11 (6.5).
BblBO.n; 0 KOHe1lHOCTI1 B yrJIOBOH T01lKe OTHOCI1TCfI K HanpHJKeHI1f1M
npl1 x = + 1, y -+ 1). 06pa
aflcb K BblpaJKeHI1f1M .n;JIfI .n;pyrl1x Hanpfl-
meHI1H, JIerKO y6e.n;I1TbcfI, 1lTO KOHe1lHble 3Ha1leHHH I1MelOT Bce HOpl\1aJIb-
Hble HanpflJKeHI1f1 He3aBHCI1MO 01' Toro, C KaKOH CTOpOHbl no rpaHI1Ue


IIpf!l\lOyrOJIbHI1Ka ocymeCTBJIHeTCfI no.n;xo.n; K yrJIOBOM T01lKe. Bblpa)Ke-
IIIHI .n;JIfI KaCaTeJIbHblX HanpflJKeHHH H yr JIa nOBopOTa co.n;epJKaT pfl.n;bl,
KaJK.n;bIH 113 KOTOpblX CXO.n;HTCfI, H, CJIe.n;OBaTeJIbHO, HI1KaKI1X oc06eH-
HOCTeH Ha rpaHl1ue npflMoyrOJIbHI1Ka 31'11 xapaKTepl1CTI1KI1 HanpmKeHHO'
JlCQ:J0pMl1pOBaHHoro COCTOflHI1f1 He co.n;epJKaT.
3aKaH1lHBaH aHaJII13 BblpaJKeHHH (6.1) 11 (6.2) B CBfl311 C YCTaHOB
JIeHHbIM xapaKTepoM nOBe.n;eHI1f1 K03Q:JQ:JHlll1eHTOB B HI1X, 3aKJIlOllaeM,
1lTO .n;JIfI cy
ecTBoBaHI1f1 pellJeHHfI nepBoH OCHOBHOH rpaHI11lHOH 3a.n;alll1
.n;JIH np5lMoyrOJIbHI1Ka B Q:J0pMe (6.1) He06xo.n;HMO npe.n;nOJIOJKeHl1e 0
napHOCTI1 KaCaTeJIbHblX Hanp5IJKeHI1H B yrJIOBblX T01lKax. Ilpl1 3TOM
YCJIOBI1I1 HHKaKHX oc06eHHocTeH xapaKTepl1CTI1KI1 HanpflJKeHHo-.n;e-
Q:JOpMHpoBaHHoro COCTOflHI1f1 npflMoyrOJIbHI1Ka He HMeIOT. Ilpe.n;bI.n;y
He
paCCMOTpeHI1f1 nOKa3blBalOT TaKme, IlTO nOJIY1leHl1e KOJIl1lleCTBeHHblX
.n;aHHbIX 0 HanpflmeHHflX Ha OCHOBe MeTO.n;a npOCTOH pe.n;YKUI1I1 CI1CTeMbl
(5.15) COnpHJKeHO C 60JIbllJI1M 061>eMOM BbIIlI1CJIeHI1M. Oc06eHHocTH
HanpflJKeHHo-.n;eQ:JopMl1pOBaHHoro COCTOflHHfI npflMoyrOJIbHHKa, CBH'
JaHHble C HapYllJeHHeM YCJIOBI1H napHocTH KaCaTeJIbHbIX HanpflJKeHI1H B
yrJIOBOH TOIlKe, MOJKHO BbIflBI1Tb npl1 aHaJII13e pellJeHHfI B Q:J0pMe (5.30).
K paccMOTpeHl11O
noro pellJeHHfI MbI Tenepb H nepexo.n;I1M.
3Ha1leHHfI KOMnOHeHT TeH30pa HanpflJKeHI1H H yrJIa nOBopoTa C
YlleTOM COOTHOllJeHHH (5.32) 11 (5.35) B paccMaTpHBaeMoM cJIYllae Bbl-
1lIlCJIflIOTCfI no Q:JopMYJIaM


I .
 (-I)"x"
20 Ux = -
 h (a"x sh a"x - a" th a" ch a"x) cos a"y-
,,=1 a"c a"


-

,,=1


(;-
£
" 'IJ WIlY sh B"y + 2 ch B"y - B" 'Il th B,,1) ch B"y) cos B"x,
"c ,,'IJ


1
 (-I)"x"
20 U =
 'h (a"x sh a"x + 2 ch a"x - a" th a" ch anx) cos a"y+
y "=1 a"c an


=
+ r
 t :tt,,"; mnY sh B"y - B" 'Il th B" 'Il ch B"y) cos B"x, (6.6)


1
20 TX1'


<
 .( - I h )"X" (a"x ch a"x + sh a"x - a" th a" sh anx) sin a"y-
_" c an
,,=1


-

,,=1


(- 1)"y"YJ A h h A
A (B"y ch B"y + sh
"y - ...." 'Il t B,,'Il s ...."y) sin
nx,
!:Inch ....n'IJ


m m 1 '"" ( - 1 ) "x " h
0)=2
 s axsinay+
",=1 a" ch a" " "


60


61


+2


m-l
m




,,=1


(- 1)"Yn'IJ h A .
h B S ...."y SIn I)"x.
!:I"c ,,'IJ




K03<l$IlUlleHTbI X n 11 Y n pSI.lI.OB B (6.6) .lI.OJDKHbI HaXO)).lITbCSI 113 oec

KOHetIHOll CllCTeMbI ypaBHeHllll (5.34). B CBSI311 C 9TllM OCHOBOll .lI.nSI
aHanll3a xapaKrepa CXO.ll.llMOCTll pSI.lI.OB B npllBe.ll.eHHbIX BbIllle BblpIDKe-
HI!SIX .lI.nSI BenlltIllH, onpe.ll.emllOIQlIX HanpSI>KeHHO-.lI.e<j.JOpMllpoBaHHoe
COCTOSIHIIe npSIMoyronbHIIKa, SIBnSIeTCSI paBeHcTBo (5.36).
TOT <PaKT. tITO C YBenHtIeHlleM HOMepa K09<j.J<j.JllIJ.lIeHTbI x n 11 Yn He
CTpeMSITCSI K HynlO, .lI.aeT B03MO}KHOCTb npll xapaKTepllCTllKe CXO)).IlMOC-
TII pSI.lI.OB B (6.6) BblCKa3aTb 1e m:e 3aMetIaHllSI, tITO 11 npll aHanlI3eBblpa-
}KeHHll (6.1) 11 (6.2). KaK H B paccMOTpeHHoM BbIllle cnYlIae, HaHxY.ll.lllell
CX0.ll.IIMOCTblO OTnlltIalOTCSI pSI.lI.bI ))..lISI H8npSIm:eHHll Ha rpaHHIJ.e oonacTH.
OCTaeTcSI B CllJIe TaKm:e 38KJIlOtIeHHe a MaJIOll npaKTHtIeCKOll 9<j.J<PeK-
THBHOCTII MeTO.ll.a npocToH pe)).YKI.1;IlH npHMeHlITeJIbHO K CllCTeMe
(5.34).
OCHOBOH 9<j.J<j.JeKTHBHbIX BbIqHCJIeHllll IIO <j.JopMYJIaM (6.6) Mom:eT
CJIym:HTb 3HaHlle BeJIlltIllHbI npe.ll.eJIbHoro 3HatIeHll51 Ko9<j.J<PllIJ.lleHTOB
bi) B (5.36) 1I npOBe.ll.eHHOe B CBSI311 C 9TllM YJIytIllleHUe CXO)).IlMOCTll co-
OTBeTcT
YlOIQIlX PH.lI.OB. C lI.eJIblO nocTpoeHHSI pSI.lI.OB C YJIYtIllleHHOll cxo-
JJ.llMOCTblO paccM01pllM BH8tIa.IIe Bblpa}KeHlle .lI.JISI U x Ha CTOpOH8X npSI-
MoyrOJIbHllKa Y = :f:: 1].
YCJIOBHe paBeHcTBa HYJIIO ax npll x = + 1, OIleBH)!.HO, Y.lI.OBJIeTBO-
pSIeTcH aBTOMaTHtIeCKll:
I .. .
 (_l}n
20 Ox (y = + 1]) = - 2

 y", 1] cas
nx.
I1cnoJIb3YH 3HaQeHUe CyMMbI pHlla


(6.7)


00

 {_ l)n
Sl = -2 ..,;;,. /3n b o 11 cos
f!x = bo'r].
n=1
y)).ooHoe npll J3bltIllCJIeHHSIX Bblpam:eHlle .lI.JISI Ox npe.ll.CTaBHM J3 BH.lI.e


(6.8)


1 ....
 (-1)n
26 ax (y = + 1]) = b o 11 + 2

 (b o - Yn) 1] cas (3nX. (6.9)
EecKoHelIHbIll p.H.lI. B (6.9) CX0.ll.llTCSI ob1cTpee, tIeM p.H.lI. B (6.7), npll-
qeM CX0.ll.HMOCTb pH.lI.a B (6.9) onpe.n.eJIHeTCH co CKOpOCTblO. C KOTOpOll
Hell3BeCTHb1e B CllCTeMe (5.34) cTpeMHTcSI K CBOllM npe1teJIbHb1M 3HatIe-
HllHM. AHaJIIl311PYSI nOCJ1e.ll.Hee Bblpa}KeHHe .lI.JI.H ox' 3aMeqaeM cJIe.n.ylO-
lqee OOCTOHTeJIbCTJ3o. KaK ObIJIO YKa3aHo Bb1IIIe, li3 Bblpa}KeHll.H 1{JI.H
ax B (6.6) nOCJIe npHMoll nO.ll.CTaHOBKll Bb1TeKaer, tITO
I
20 U x (x = :f:: 1) ===:; 0 (-1] <: Y < 11)- (6.10)
C .n.pyroH CTOpOHbI, li3 paBHoMepHoll cXO.n.llMOCTM pHlI.a B (6.9) Cne.ll.yeT
1
20 o-\' (x = I, Y= 11) '- b o 11, (6. II)
T. e. HOpM8JIbHble HanpHm:eHllH !fMeIOTB yrJIOBOH TOtIKe pa3pbIB.
fi2


lI..lISI BblIlMCJIeHllSI Hanp.Hm:eHHll a y nonYtIaeM cJIe.ll.YlOIQYlO <j.JopMyny:


I


""
2
 oy (x = + 1) = - bo'r) + 2
 (: It (X n - bi) cas a"y. (6.12)
n=1 n
U3 KOTOpOll TaK>Ke cne.ll.yeT CYIIJ.eCTBOBaHlle pa3pbIBa .lI.JIH HanpmKeHllH
a b B yr JIOBOll TOtIKe.
PaccMoTpllM cnocoo YJIyqIIIeHllSI CXO.ll.llMOCT!f pH.lI.OB. npe.ll.cTaB-
JISIlOIQllX KaCaTeJIbHble HanpHm:eHllH B (6.6). npllHllMa.H .lI.JIH onpe.n.eJIeH-
HOCTll x = 1. 3H8qeHllH KacaTenbHbIX HanpSI}KeHllll Ha 9TOll cTOpoHe
npHMoyrOJIbHllKa onpe.ll.eJIHlOTCSI <j.JopMYJIoii
I .. I)
 (- Ijn xn ( . h + at) ) .. +
2 0 'tx/J(X= =..,;;,. h s an.
 smany
7=[ an c an . c an


co


+
 an
h1n1J (
nY ch
nY + sh (3nY -
n'r) th 13n1] sh 13nY). (6.13)
TIpll Bb1IlHCJIeHllll nepBoro CJIaraeMoro B 9TOM BblpIDKeH!fll .lI.JISI Txy
Me.ll.JIeHHO CXOASIIQaHCH tIaCTb pH.lI.a MO}KeT Ob1Tb CBepHYTa C nOMOIll.blO
paBeHCTBa



 (_I)n sin ay = _-2L In tg (
 +
 y) . (6.14)
n=1 at) n n . 4 41J
OTClO.lI.a 3aKJIlOIlaeM, liTO nepBoe CJIaraeMOe B (6.13) oOJIa.ll.aeT IIpll
nO))'XO.ll.e K yrJIoBoii TOtIKe JIOrapH<pMlltIeCKOii oc06eHHocTblO.
BTOpoii pSI.lI., BXO.ll.SIIQllll B Bblpam:eHlIe .lI.JISI T-\,y. oOJIa.n.aeT xopoIll.eii
CX0.ll.MMOCTblO B TOtIKaX, .lI.OCTaTOIlHO Y.lI.aJIeHHbIx 01' yrJIoBoii. TIpll nOk
X0.ll.e K yrJIY CXO.ll.HMOCTb ero 3HatIllTeJIbHO YXY.lI.lllaeTcSI 11, CJIe.ll.OBaTeJIb-
RO, 3.n.eCb TaKm:e CJIe.n.yeT Bbl.ll.eJIllTb Me1tJIeHHO CXO.ll.SIIQYlOCH IlaCTb. KO-
TopaSI oOJIa.ll.aeT ocooeHHocTblO npll y - 1]. Bb1.lJ.eJIeHlle Me.n.JIeHHO cxo-
.lI.HrueiicSI tIaCTll oCYIQeCTBJISIeTCH C MCnOJlb30BaHlleM paBeHcTB


3t.
co .-
T)-I/I

 e- 13n (1)-YI _ e
""'" - 1 . -3t(TJ-y) ·
n=1 -e


(6.15)


00


e-f3,lT)-y) = -..!...In dh
 (11- y).
..,;;,. /3n It 4
n=-1
Y1IMTbIBaSI npe.ll.eJIbHb1e paBeHcTB8 (5.36) .n.JISI X n II y"" Ha OCHOBe co-
oTHollleHHH (6.14) 11 (6.15) HaH1{eM CJIe1tYIOIQYIO Y.lI.OOHYIO np!f J3b1tIHC-
JleHllSIX TXI' <j.JOPMYJIY:


2(TI-/J1
I . .. e 2 boYi I n
20 't xy = - bot'] (11- y) 1- e-n(1j-yl + -n In cth""4 (1] -- y) X
X ctg
 (1 +
 )] + }: ( a
)n sin a"y[x n (th a" + Ch
':xn ) -b o ] +
1;=1
63


II





 1J f [ R ch BnY sh BnY R sh
nY ]
+
lf,;""""lYn FnY chBn1J + Ci1BnTJ -p,,'I]th
,,'I] chBn1J -
- b o [-
" ('I] - y) e -13,,(1)-Y) + e -6,,('I)-1I)]} . (6. 16)
C nOMOrn;blO 3TOro Bblpa./KeHIUI MO./KHO ocyrn;ecTBIITb npOBepKY Y.lI,OBJIe-
TBOpeHII5I rpaHIIqHbIX YCJIOBIIH .lI,JI5I KaCaTeJIbHbIX Hanp5I./KeHIIH B JIlO6oH
TOqKe rpaHIIUbI. BKJIlOqa5I II yrJIOBYlO TOqKY, nOCKOJIbKY


L'eCTBeHHoro II3yqeHII5I nOJIeH cMern;eHIIH II Hanp5I./KeHIIH He TOJIbKO B CJIY-
Lwe np5IMoyrOJIbHIIKa, HO II BO Bcex paCCMaTpIIBaeMbIX rpaHIIqHbIX 3a-
.lI,aLlax .lI,JI5I ynpyrIIx TeJI KOI-IeqHbIX pa3MepOB.


S 7. R'OJIH'IeCTBeHHbIe o.qeHKH HanpmKeHHH
B npHMoyrOJIbHHKe


3t
--('I)-y)
. e 2 I
Irrn ('I] - y) I -3t('I)-Y) = n '
Y-+1) - e
lirn In r cth
 ('I] -- y) ctg
 ( 1 + L )] --= - In '1].
Y-+'I) 4 4 1J
Onpe.ll,eJIeHHbIH HHTepec C TOqKII 3peHII5I II3yqeHII5I nOCJIe.ll,CTBIIH
OTKa3a OT Tpe6oBaHII5I napHOCTII KaCaTeJIbHbIX Hanp5I./KeHIIH B yrJIOBOH
TOqKe npe.ll,CTaBJI5IeT nOBe.ll,eHIIe BeJIIIqIIHbI w, xapaKTepII3YlOrn;eH yrJIbI
nOBopOTa. I1cnoJIb3Y5I COOTHOllieHII5I (6.15) II (6. 14), MO./KeM BbI.ll,eJIHTb
r JIaBHble qaCTII B p5I.lI,ax lIJI5I W, npe.ll,CTaBJI5I5I ee B CJle.ll,YlOIlI.eM BII.lI,e:


(6. 17)


Bonpoc 0 BbI60pe KOHKpeTHoro npIIMepa, IIJIJIlOCTpIIpYlOrn;ero Me-
TOll, 06JIa.ll,alOrn;IIH .lI,OBOJIbHO IlIIIpOKIIMII B03MO./KHOCT5IMII, BCer.ll,a 5IB-
JI5IeTC5I CJIO./KHbIM. Onpe.ll,eJIeHHYlO nOMOIll.b B pellIeHHII 3Toro Bonpoca
OKa3bIBaeT KOHKpeTII3aUII5I ueJIlJ npe.ll,npIIHIIMaeMoro paCqeTa. B pe-
3Y JIbTaTe npOBe.ll,eHHoro BbIllIe KaqeCTBeHHoro aHaJIII3a 06Il1.IIX peilieHIIH
60JIbiliOH IIHTepec npe.ll,CTaBJIHeT oueHKa CKOpOCTII, C KOTOpOH HeII3Be-
CTHble B 6eCKOHeqHbIX CIICTeMaX CTpeM5ITC5I K CBOIIM npe.ll,eJIbHbIM 3Haqe-
HII5IM.
3aKOH aCIIMnTOTIIqeCKIIX Bblpa./KeHIIH, KaK CJIe.ll,yeT II3 paCCMOT-
peHHbIX YCJIOBIIH ero CnpaBe.ll,JIIIBOCTII, 5IBJIHeTC5I CJIe.ll,CTBIIeM HeKO-
TOpbIX OC06bIX CBOHCTB K03rpcpIIuIIeHTOB CIICTeMbI. CaMII 3TH K03rpcpII-
u,IIeHTbI 3aBIIC5IT JIIIllIb OT reOMeTpIIII 06JIaCTII, T. e. OT napaMeTpa
'1]. DOJIO./KIIM 'I] = 1, paccMaTpIIBaH nJIOcKYlO 3a.ll,aqy .lI,JI5I KBa.ll,paTa.
DOCKOJlbKY npII C06JIIO.lI,eHIIH napHocTII KaCaTeJIbHbIX HanpH./KeHIIH B
yrJIe HeT pa3HIIUbI B nOBe.ll,eHIIII npaBblx qaCTeH OT HOpMaJIbHbIX II Kaca-
TeJIbHbIX HarpY30K, TO OrpaHHqIIMC5I paccMoTpeHIIeM npoCTeHllieH Kpa-
eBOH 3 a.ll,aq II:
I A I
20 U x ( + 1, y) = (- 1) cos i:ny,
1 !
20 uy(y, + I) = 20 'tXIj(y, ::!:: I) = 0.
EeCKOHeqHa5I CIICTeMa (5.15) .lI,JIH paccMaTpHBaeMoH reOMeTpHI1 II
HarpY3KH npHo6peTaeT BHlI


'I


00
W = 2 m- 1 "., (- l)n .
2.J (x" t h a" - b o ) SIn anY -
m n=1 cx"


00


__ 2 m - 1 }' --"L ( sh BnY _ b e - l3 n ('I)-Y» ) _
m "-J Bn Yn ch Bn1J n
n=1


1
20 't XII ( + I, y) = U;
(7.1)


2 m - I b '11 [ :n :n ( Y )]
- m 0 Jt In cth 4 ('I] - y) tg 4 1 + 1] .


(6.18)


OTClO.lI,a 3aKJIlOqaeM, qTO BeJIHqHHa W o6JIa.ll,aeT npII nO.ll,XO.ll,e K yrJIOBOH
TOqKe oco6eHHOCTblO. BBO.ll,H MeCTHYIO CHCTeMY KOOp.ll,HHaT r = 'I] - y,
BH.lI,IIM, qTO 3Ta oco6eHHOCTb HMeeT xapaKTep In r npH r _ 0. 3TOT BbI-
BO.ll, OCTaeTCH B CHJIe npII nO.ll,XO.ll,e K yrJIOBOH TOqKe B.lI,OJIb CTOpOHbI Y ,
='1].
TaKIIM o6pa30M, HapYllieHIIe napHOCTII KaCaTeJIbHbIX Hanp5I./KeHIIH
B yrJIOBOH TOqKe npIIBO.ll,II. K n05IBJIeHIIlO pa3p
BOB B HOpMaJIbH
X
Hanp5I./KeHIIHX, a yroJI nOBopOTa IIMeeT B 3TOH TOqKe JIOrapIIrpMHqecKYlO
oco6eHHocTb. 3TOT BbIBO.ll, nOJIHOCTblO corJIacyeTC5I" C pe3YJIbTaTaMII,
ll()JIyqeHHbIMII npII paccMoTpeHIIII 3a.ll,aqII C pa3pbIBHbIMII 3HaqeHHHMH
KaCaTeJIbHbIX Hanp5I./KeHIIH .lI,JI5I qeTBepTbnJIOCKOCTII [198, 252!.
OueHIIBa5I pe3YJIbTaT
 IICCJIe.ll,OBaHII5I cBoikTB 06Il1.IIX peilieHIIH
nepBoH OCHOBHOH rpaHIIqHOH 3a.ll,aqII .lI,.15I npHMoyrOJIbHHKa, OTMeTIIM,
qTO 3TIIM IICCJIe.ll,OBaHHeM YCTaHOBJIeHa cnoc06HOCTf> pa3BlJBaeMOro
.lI,JI5I ee peilieHIIH MeTOlIa YJIaBJIIIBaTb Bce oco6eHHocTII HanpH./KeHHO-
.lI,ecpopMIIpoBaHHoro COCT05IHIIH, B03HIIKalOrn;IIe BCJIe.ll,CTBHe HaJIIIqH5I
yr JIOBbIX TOqeK. 3To o6cTOHTe.lbcTBO OqeHb Ba./KHOe, nOCKOJIbKY IIMeHHO
OHO 5IBJI5IeTCH OCHOBOH nOCTpoeHIIH 3rprpeKTIIBHbIX aJIrOpIITMoB KOJIII-


1
 4n 2 '\jJn
x -
. Y +
n - !J. (n:n) ..... :n 2 (k 2 + n 2 )2 !J. (n:n) ,
k=1


(7.2)


00
1
 4n 2
y,
Xk .
!J. (m ., :n 2 (k 2 -I- n 2 )2 .
=1


31IeCb



 = I !J. ;1:n) ,
!J. (n:n)
0,


n = i,


n =1= i;


(7.3)


IJ. (n:n) =
 (cth n:n + h
:n ) .
n:n s n:n
Bblpa./KeHHe .lI,JI5I 'ljJn /11 (n:n) nOKa3bIBaeT, qTO .lI,JIH HaXO./K.lI,eHII5I Be-
JIIIqIIH, OTBeqalOIlI.IIX KpaeBOH 3a.ll,aqe (7.1), HaH.lI,eHHble II3 CIICTeMbI
(7.2) x n II Yn .lI,OJI./KHbI 6bITb YMHO./KeHbI Ha (_I)' A.


64


5 7-2464


65




llJIH nOJIY'IeHllH Hall60JIee nOJIHOH llH<i>opMall.llll 0 nOBeJ{eHllll He-
H3BeCTHbIX B CllCTeMe (7.2) npllMeHHM npll nOCTpoeHllll ee pemeHll5I
MeToJ{u JIllMllTaHT B CO'IeTaHllll C MeTOJ{OM nOCJIeJ{OBaTeJIbHbIX npll6Jlll-
>KeHllll. TIOCKOJIbKY, eCTeCTBeHHO, HeT CMbICJIa paccMaTpHBaTb KOHe'I-
HY
 cllcTeMY nopHJ{Ka n < i, OTBe'IafOlllYfO (7.2), TO B COOTBeTCTBllll C
(3.4) BbIpa>KeHHH J{JIH JIllMllTaHT B J{aHHOM cJIY'Iae npH06peTafOT BH.Q
p
\''' n 4
L..J Yk (k2 + n2)2
V k=1
n=


AHaJIOrll'IHbIH npoll.ecc nOCJIe.llOBaTeJIbHbIX npll6JIll>KeHHH MO>KHO
-(I) -(I
nOCTpollTb, Ha'IllHaH C HeKoTopbIX 'IHCeJI Xk H Yk) (k = 1, 2, ...
..., p), MeHbllJHX, 'IeM 3Ha'IeHHH cooTBeTcTBYfOlllllX nepBbIx Hell3BeCTHbIX
B CllCTeMe (7.2). TIpeJ{eJIbHbIe 3Ha'IeHHH BeJIH'IHH x
), y
) H hI i) TaK>Ke
06JIaJ{afOT Ma>KOpHpYfOlllHMH CBOHCTB3MH no OTHomeHHfO K Hell3Be-
CTHbIM B CHCTeMe (7.2):
- -
xn> x n ' Y n > y" (n <;, p); xn> h, Yn> h (n > p). (7.8)
B cOBoKynHocTll 3Tll J{Ba npoll.ecca J{afOT B03MO>KHOCTb HaXOJ{HTb
BepxHHe 11 HH>KHHe rpaHl1u.bl J{JIH Bcex Hell3BeCTHbIX B CllCTeMe (7.2).
TIOCKOJIbKY OTHOClneJIbHO Hell3BeCTHbIX B CllCTeMe (7.2) ll3BeCTHO, 'ITO
c pOCTOM HOMepa OHH CTpeMHTcH K HeKoTopoH 06llleH J{JIH HllX nOCTOHH-
HOH, TO HMefOTCH npeJ{nOCbIJIKH K C6JIll>KeHHfO 3Ha'IeHllH h II H B (7.7)
II (7.8) npH YBeJIl1'IeHllll p ll, CJIeJ{OBaTeJIbHO, K HaXO>KJ{eHllfO Bcex He-
ll3BeCTHbIX C J{OCTaTO'IHO BbICOKOH TO'IHOCTbfO.
PeaJIH3aUll51 YKa3allHbIx npou.eccoB nOCJIeJ{OBaTeJIbHbIX npll6JIll-
>KeHllH CBH3aHa C oueHKoH Ha Ka>KJ{OM llJare BepxHllx II Hll>KHllX rpa-
Hllll. JIllMllTaHT (7.4). TIoBbImeHl1e TO'IHOCTll 3TllX Oll.eHOK J{OCTllraeTC5I
nYTeM HeKoTopbIX npe06pa30BaHHH BbIpa>KeHl1H (7.4), IWTopbIeTenepb H
npoJ{eJIaeM. Ha'IHeM C ou.eHKll BepXHl1X rpaHllll. JIllMllTaHT.
Ha Ka>KJ{OM mare nOCJIeJ{OBaTeJIbHbIX npll6JIll>KeHllH npH HaXO>K-
J{eHl1ll 3Ha'IeHllH Hell3BeCTH
X C 1136
TKOM He06xoJ{l1MO HaHTll 'IllCJIO
H (i) (V <i) W (t» ( + 1 + 2 )
= max n , " n = p ,p ,..., OrpaHll'IllBafOlllee
cBepxy 3Ha'IeHll51 Bcex Hell3BeCTHbIX C HOMepal\fll n > p. KaKoBo 6

Hll 6
JIO 'IHCJIO H, nOJIO>KllM
V Ii) H ..:(i) (k 1 2 (7 9
k = i- Uk = , ,..., p). . )
TorJ{a BbIpa>KeHlle J{JIH V n 113 (7.4) npl1BeJ{eM K BHJ{Y
1 p
2 H -
 b
)ak (n)
v<
) = H _ k=l (7.10)
p
1 ,-,
2-1 ..o..J ak(n)
k=1
n 4
rJ{e J{JIH COKpameHllH nOCJIeJ{YfOlllllX 3anllceH npllMeM (Xk (n) = (k2 + n2)2 .
113 (7.10) CJIeJ{yeT, 'ITO M5I TOro 'IT06bI V
) 6
JIO He 60JIbllJe H, J{OCTa-
TO'IHO YCJIOBH5I


1
 n 4
2 \,,2 + n2)2
k=1
p (7.4)

v n 4
k=1 (k 2 + n 2 )2


p
1 " n 4
2 + l,..) (k 2 + n2)2
k=1
TIpll nOCTpOeHl1ll 3THX BbIpa>Ke
llH npllHHTO j = j' = p, a TaK>Ke Yf!-
TeHO, 'ITO B (5.18) an = b n = 2t(nn)2 f':.. (nn). TaK KaK p > i, TO F =
= Qn = O. n
OTHOCllTeJIbHO Hell3BeCTHbIX B CllCTeMe (7.2) ll3BeCTHO 3apaHee, 'ITO
o
 X
 (1n)2 0

 (1n)2
"" n "" 2' "" y" '"
 (n =- 1, 2, .. .). (7.5)
TIpeJ{nOJIO>KllM, 'ITO YJ{aJIOCb Haihll J{Ba p5lJ{a 'IHCeJI xi. H II yj.H (k =
= 1, 2, ..., p) TaKllX, 'ITO x k <;, Xj.IJ, Yk <;, Yk l, J{JIH k = 1, 2, ..., p.
Ou.eHllM .3Ha'IeHl1H JIllMllTaHT (7.4) npll 3aMeHe X II Y Ha 'IllCJIa X(I)
y(I) OIl (I) (1) k k k
II k. eBllJ{HO, npll 3TOM V" > V n , W n :;;:.. W n . TIYCTb Hall60JIb-
wee ll3 3Ha " eHl1ll U 6 u V Ol W (I) H (I)
u" J{po ell "ll n paBHHeTCH . OnpeJ{eJIllM He-
KOTOpbIll npou.ecc nOCJIeJ{OBaTeJIbHbIX npll6JIH>KeHllH COOTHomeHllHMll


W-
,,-


X(i+l) _ 1 I vii) 4n 2 H(i)
 4n 2
n - 1'1 (nn) '=1 k n 2 (k 2 + n2)2 + 1'1 (nn) n 2 (k 2 + n2)2 +
l?=p+1
'\jJ"
+ 1'1 (nn) , (7.6)
.y
+I) = 1
V
) 4n 2 H(i)
 4n 2
1'1 (nn) n 2 (k 2 + 122)2 + 1'1 (nn) n 2 (k2 + n 2 )2
k=1 k=p+1


1
 (i)
2 H >
 6
 (XI? (n). (7.11)
k=]
ECJIH Bce 6};) > 0, TO npaBaH 'IaCTb B 3TOM HepaBeHCTBe - B03paCTafO-
maH <i>YHKUl1H n H ee HaH60JIbmee 3Ha'IeHHe OTBe'IaeT n = 00 TorJ{a
ll3 (7.11) C Y'IeTOM (7.9) CJIeJ{yeT


(n <;, p).
113 perYJIHpHOCTll llcxoJ{HOll 6eCKOHe'IHOH CllCTeMbI CJIeJ{yeT CXOJ{llMOCTb
'" 0 0 P X II) X Y IIJ Y (i)
,,1' r n ouecca, T. e. ,,-+- no n - 'I' H -+- H, npll'IeM
Xn<;,X,.. Yn<Y n (n<;,p); xn<..H, Yn<;,H (n>p). (7.7)
Ba>KHO OTMeTl1Tb, 'ITO 3Tll COOTHOllJeHHH cnpaBeJJ.JIHBbI npH JIf060M p.


H:;;:"


P

l VIi)
...;;.J k
k=1
1
2+ P


(7.12)


66


5*


67




3TOT pe3YJIbTaT M01KeT 6bITb nOJIYIJeH HenOCpe.l{CTBeHHO H3 Bblpa1KeHHH
.l{JUI V n B (7.4).
TIpe.l{nOJI01KHM Tenepb, IJTO Cpe.l{H pH.D.a IJHCeJI 6
1 HMelOTCfI oTpHlJ;a-
TeJIbHble. I1pe.l{CTaBHM HepaBeHCTBo (7.11) B BH.l{e


1
-+- (i)
- n
_ 2 H>-...:::J {jk a k (n) +...:::J {jk a k (n).
k . k


(7.13).


3.l{eCb 3HaKaMH «+» H «-» OTMeIJeHbI CYMMbI, pacnpOCTpaHfllOIlJ;HeCH
COOTBeTCTBeHHO Ha nOJI01KHTeJIbHble H oTpHu.aTeJIbHbIe 6
). I1epBoe
CJIaraeMOe B npaBOH IJaCTH - B03paCTalOIlJ;a51 QJYHKU.H5I n, .l{OCTHralO-
III.a51 HaH60JIbllIera 3HaIJeHH5I npH n = 00. BTopoe CJIaraeMOe - y6bI-
BalOIII.a51 QJY HK U.H5I, H, CJIe.l{OBaTeJIbHO, HepaBeHcTBo (7.13) 6Y.l{eT copa-
Be.l{JIHBO .l{JI5I Bcex n > p, eCJIH B351Tb era 3HaIJeHHe npH n = p + 1.
TaKHM 06pa30M, .l{JIH onpe.l{eJIeHH5I H nOJIYIJHM YCJIOBP e
-=-
+ y
) -'
- Y
)ak {p + I)
H>- k k_ (7.14)
{- + Lt + 1. - ak (p + l)
k
3.l{eCb Lt - 06III.ee IJHCJIO nOJI01KHTeJIbHbIX 6
).
BbIIJHCJIeHHe H no QJOpMy JIe (7.14) HaIJHHaeTC5I C npe.l{nOJI01KeHH5I a
3HaKe cooTBeTcTBYlOIII.HX BeJIHIJHH 6
). ECJIH nOCJIe BbIIJHCJIeHII51 H npe.n;-
nOJI01KeHIIe nO.l{TBep1K.l{aeTC5I, TO HaH.l{eHHa51 BeJIHIJIIHa H MOlKeT 6bITb
npIIH5ITa B KaIJeCTBe BepXHeH rpaHIIlJ,bI JIHMIITaHTa V n' B npOTIIBHOM
CJIYIJae Ha.l{O H3MeHIITb npe.LWOJI01KeHHe II nOBTOpHTb CIJeT.
I1poBe.l{51 aHaJIOrHIJHble paCCY1K.l{eHH5I .l{JI5I HaXO>K.l{eHHH BepXHeH
rpaHHIJ;bI JIHMHTaHTa W no nOJIYIJHM COOTHOllIeHHH
p

 X
)
k=1
1 ·
'"2+ p


H>-


(7.15)



+ X
)
- X
)ak (p -1-- I)
H>- \ k
""2 + Lt +
- ak (p + IJ
k
BbI60p HY1KHOH .l{JI5I BbIIJHCJIeHH5I QJOPMYJIbI oCYIII.ecTBJIfleTC5I Ha OCHOBe
?IJ;eHOK 3fiaKOB BeJIHIJHHbI 1'
) B paBeHcTBe
X li) H (i) (k 1 2
k = + 1'k = , ...., p).
B KaIJeCTBe H(i) .l{JI5I CJIe.n.ylOIII.ero npH6JIIDKeHHH BbI6HpaeTcH 60JIbllIee
H3 .l{BYX 3HaIJeHHH H, HaH.l{eHHbIx npH OIJ;eHKe V n H W,;.
TIepeXO.l{HM Tenepb K OIJ;eHKe HH1KHHX rpaHHIJ; JIHMHTaHT. BbIKJIa.l{KH,
BnOJIHe aHaJIOrHIJHbIe npe.l{bI.l{YIII.HM, C YIJeTOM Taro, IJTO Tenepb HaM Ha.l{O
nOJIYIJHTb OIJ;eHKY CHH3y .l{JI5I BeJIHIJHH V n II W n B (7.4), npHBO.l{HT K co-
OTHOllIeHHHM, nO.l{o6HbIM (7.12), (7.14) H (7.15). TIpH BBe.l{eHHH BeJIH'lHHbI


(7. 16)


68


f


j


h paBeHCTBOM


- . 00 2
Yk = h + ()" (k - 1. ,..., p)
J].JI5I ou.eHKH 3fiaIJeHH5I V n nOJIy'laeM HepaBeHCTBa


(7.17)


p

 yLI)ak (p + I)
k=1
P .
{- +
I ak (p + I)

+ y
)ak (p + I) +
- y
)
h./ k k
"""- 1
""2 + L; +
+ ak (p + l)


h<


(7.18)


npe06pa30BaHH5I BeJIHIJIIHbI W n OCYIII.eCTBJIHeM C nOMOIII.blO paBeHCTBa
x
)=h+1'
) (k= 1.2. .... p). (7.19)
COOTHOllIeHHH J].JIH onpe.l{eJIeHII5I BeJIIIIJHHbI h HMelOT BH.l{. aHaJIOrHIJHbIH
(7.18):


p

 x
)ak (p + I)
k=1
P ,
++
 ak(p+ I)
k=1

+ x
)ak (p + I) +
- x
)
k k
1 .'
'"2 + C;+
+ak(p+ I)
k
B CYMMax L+ H
- cyMMHpoBaHHe pacnpocrpaHeHo COOTBeTC'
BeHHO
Ha Te 3HaIJeHHH k, .l{JI5I KOTOpbIX 6
) > 0, 1'
) > 0 H l)
) < 0, 1'
) < O.
- - ,,(i) (i)
BeJIHIJHHbI L" H Lx paBHbI 06IlJ;eMY IJIICJIY OTpHIJ;aTeJIbHbIX Uk H 1'k CO-
OTBeTCTBeHH
. B KaIJeCTBe BeJIIIIJHHbI h(i) .l{JIH nOCJIe.l{YlOIII.erO npH6JIII-
1KeHH5I BbI6I1paeTC5I MeHbllIee H3 .l{BYX 3HaIJeHIIH h, Onpe.l{eJIHeMbIX B
COOTBeTCTBHH C (7.18) II (7.20).
OocJIe Taro KaK .l{aH cnoc06 BbIIJHCJIeHH5I rpaHHIJ; JIHMHTaHTOB, M01K-
HO npIlcTynaTb K HenOCpe.l{CTBeHHoMY pellIeHHlO CHCTeMbI (7.2). BHaqa-
JIe OCTaHOBIIMC5I Ha paccMoTpeHHH cJIYIJa51 1 = 1 B (7.1).
HaIJHeM C OTbICKaHII5I npH6JIH1KeHHbIX 3HaIJeHHH C H36bITKOM. Co-
rJIaCHO (7.6) nOCJIe.l{OBaTeJIbHble npH6JIH1KeHJ151 6Y.l{YT BbIIJHCJIHTbCH
no (j:JopMYJIaM


h<


(7.20)


h



p J
X(i+1) _ 1
 () k (i) 4n2 + H(i) [ 1 2 +
n - tJ. (nn) ,i..J 31 2 (k 2 + n2)2 - (n31)2 tJ. (nnJ
"=1


'l/'n
+ tJ. (n:rr) ,


(7.21)


69




p
yli+\) =
 \'--' ,Ii) 4n 2 H li ) [ 1 2 J
n 11 (nn) 21 1 k n 2 (k 2 + n 2 )2 + - (nn)211 (nn) ;
yli) = Hli) + (i"' X li) = H ll) + I')
II . II ,k Yk .
PeaJUl3aumo npouecca nOCJIell.OBaTeJIbHbIX npli6mf>KeHliii, Onpell.e-
JI5IeMoro COOTHOllleHli5lMli (7.21), MO}KHO ocyruecTBJI5ITb ll.BYM5I nYT5IMli.
MO}KHO .
 caMoro HaqaJIa Sall.aTbC5I onpell.eJIeHHblM SHaqeHHeM p li saTeM
coxpaHHTb ero HeHSMeHHbIM. Oll.HaKO MO}KHO TaK}Ke, HaqaB C HaliMeHb-
mero 1l.0nYCTliMoro p, nOCTeneHHO YBeJIHqlmaTb ero B COOTBeTCTBlfH C no-
JIyqaeMbIMli peSYJIbTaTaMlf. 311.eCb 6Yll.eT npHMeHeH BTOpOH nYTb, HC-
nOJIbSOBaHHe KOToporo cOKparuaeT 06'beM BbIqHCJIeHHH.
1. f1 e p B 0 e n p H 6 JI Ii >K e H H e.
f10CKOJIbKY B ll.aHHOM cJIyqae 1J.Jl =1= 0, a \jJn = 0 (n == 2, 3, ...),
TO B nepBOM npH6JIH}KeHHH p = 1. I1s COOTHOllleHH5I (7.5) cJIell.YeT,
'ITa ace fIeli3BeC'THbie B CHCTeMe (7.2) npH 1 = 1 SaKJIlOqeHbl B npell.eJIax
1[2 1[2
o <. X n <. 2' 0 <. Yn <. 2 . (7.22)
B CB5ISH C 3THM npH OTblCKaHHH ll.JIH HeIiSBeCTHbIX npli6mnKeHliH cBepxy
B KaqeCTBe nepBoro npli6mi}KeHli51 MO>KeM npHH5ITb
X\I) = 5, y\11 = 5, H(!J = 5.
2. B TOp 0 e n p H 6 JI H }K e H H e.
BbIqliCJIeHli51 no q,opMYJIaM (7.21) Oplf P = 1 ll.alOT X\2) = 4,9596;
Yf) = 1,9015. LlJI5I BeJIHqliHbI H (2 ) liS (7.12) li (7.15) HaX0ll.HM COOTBeT-
CTBeHHO H
) = 1 ,2677; H
) = 3,3064. OTCIOll.3 CJlell.yeT qTO B Ka-
00 '
qeCTBe H ll.JI5I TpeTbero npli6JIli>KeHliH CJIe,ll,yeT npliH5ITb H('l J =
= 3,3064. f1epeXOll.H K TpeTbeMY npli6JIli}KeHlilO, YBeJIHqliM Ha ell.HHli-
uy 3H3qeHlie p, npHHHM851 Tenepb p = 2. f10CKOJIbKV Bce HeHSBeCTHble
C HOMepaMH 60JIblIIe ell.HH H Ubi OKaSblBalOTCH MeHblll
Mli qeM H (2 ) TO
B KaqeCTBe liCXOll.HbIX 1l.8HHbIX Ami TpeTbero npH6JIH}Ke
li51 Bbl6HpaeM
CJIell.YIOIUHe:
x1 2 ) = 4,9596;
X
) = 3,3064;


l)
!1 > O. BblqHCJIeHl15l no q,opMYJIaM (7.14) H (7.15) ll.alOT COOTBeT-
CTBeHHO H
) = 1,9025; Hf}) = 2,3676. 3TH 3HaqeHH5I nOKaSbIB8IOT,
LITO Cll.eJIaHHOe npell.nOJIO>KeHHe OKaSaJIOCb BepHblM, H B KaqeCTBe
JHaqeHH5I H I3J CJIe)J,yeT npHH5ITb 60JIblllee HS ll.BYX HaHll.eHHbIX qHCeJJ,
T. e. H I3 ' = 2,3676.
LlaJIbHeHlllHH xapaKTep BblqHCJIeHliH cOBepllleHHo 5ICeH YBeJIH-
'IliBa51 nOCTeneHHO KOJIHqeCTBO BOBJleKaeMblX B paccMOTpeHHe HeHS-
Ta6JIHlla 1


(7.23)


I - I I - I Y n I * I * I - I -
n X n X n Yn x n Y n X n Yn
1 3,5410 3.5412 1,1692 1,1694 3,5374 1,1657 3,5411 1,1693
2 0,9424 0.9431 1,7340 1,7347 0,9272 1,7188 0,9428 1,7343
3 1,1484 1,1498 1,7088 1,7103 1,1141 1,6746 1,1492 1,7094
i 1,2586 1,2612 1,6395 1,6422 1,1982 1,5791 1,2601 1,6405
5 1,3229 1.3270 1,5861 1,5902 1,2301 1,4933 1,3254 1,5874
6 1,3628 1,3685 1,5489 1,5546 1,2323 1,4184 1,3664 1,5E05
7 1,3886 1,3961 1,5230 1,5305 1,2168 1,3512 1,3836 1,5249
8 1,4058 1.4152 1,5046 1,5140 1,1802 1,2889 1,4123 1,5068
9 1,4176 1,4290 1,4912 1,5026 1,1569 1,2305 1,4257 1,4938
10 1,4258 1,4392 1,4813 1,4946 1,1198 1,1753 1,4355 1,4840
>10 1,4312 1,4928 1,4312 1,4928 0 0 1,4598 1,4598


6}2J = 0;


(7.24)


BeCTHblX 1l.0 ll.eC5ITll, nOCJIe ll.BeHall.UaTOrO npH6JIH>KeHHH nOJIyqaeM
3HaqeHH5I HeHSBeCTHblX C H36bITKOM, YKa3aHHbIe B Ta6JI. 1. 3HaqeHHe
BepxHeH rpaHHUbI JIHMHTaHT, OrpaHHqlmalOIUeH cBepxy Bce HeHS-
BeCTHble C HOMepaMH 60JIblllHMH ll.eC5ITH, paBHo H 1,4928. B npe-
.lI.blll.YIUeM Oll.HHHall.uaTOM npH6JIH}KeHHH HIli) = 1,4932. EOJIbllla51 CTe-
HeHb 6JIHSOCTH BepxHHx rpaHHU JIHMHTaHT B ll.BYX nOCJIe)J,OBaTeJIbHbIX
npH6JIH>KeHH5IX YKasbIBaeT Ha TO, qTO ll.aJIbHeHlllee YMeHbllleHHe H
BOSMO>KHO JIHlllb nOCJIe BKJIlOqeHHH B paCCMOTpeHHe HeH3BeCTHblX C
HOMepaMH 60JIblllHMH ll.eC5ITH.
B Ta6JI. 1 B cTOJI6uax ;'n H !in ll.aHbl SHaqeHH5I HeH3BeCTHblX B 6ec-
KOHeqHOH CHCTeMe (7.2) C He)J,OCTaTKOM. B CTpOKe n > 1 0 ll.aHO SHaqeHHe
HH>KHeH rpaHHUbl JIHMHTaHT, BblqHCJIHeMOH no q,OPMY JIaM (7.18) n (7.20).
OueHHBaH nOJIyqeHHble SHaqeHH5I HeHSBeCTHbIX, OTMeTHM, qTO
npHMeHeHHe MeTOll.a pellleHH5I 6eCKOHeqHOH CH, CTeMbl OCHOBaHHoro Ha
licnOJIb30BaHHH CBoHcTBa JIHMHTaHT, ll.aeT BOSMO}KHOCTb npH paccMoT-
peHHH CHCTeMbI ll.Ball.UaTH ypaBHeHHH HaHTH Bce HeH3BeCTHble C TO'I-
HOCTblO 1l.0 -2%.
f1pe>Kll.e qeM nepexOll.HTb K aHaJIHsy Bblpa}KeHHH ll.JI5I Hanp5l}KeHHH
Ii npoBell.eHHIO BbIqHCJIeHHH no HliM, CJIell.yeT OCTaHOBHTbC5I Ha p5l11.e
BonpocoB, CB5I3aHHblX C oueHKoH B03MO}KHOCTeH ll.PyrHx nOll.XOll.OB K
OTblCKaHlilO pellleHli51 6eCKOHeqHOH CliCTeMbl. ECTeCTBeHHO npH 3TOM
cpaBHliBaTb nOJIyqeHHOe pellleHlie C TeM, KOTopoe 1l.0CTaBJIHeT liCnOJIb-
30BaHlie MeTO)I.a pell.YKUliH no OTHOllleHlilO K CliCTeMe (7.2). B CToJI6uax
71


(2) 2
'\'1 = 1,653 ;
y
2) = 0;


V\2) = 1.9015;
y
2) = 3,3064;
if2) = 3,3064.


6\21 = _ 1.4049;



 TpeTbe npli6JIli}KeHH

BblqliCJleHlUI no q,OPMYJI8M (7.21) C HCnOJIb30B8HHeM 1l.8HHblX (7.24)
)J.81OT cJIellYlOruHe pe3Y JIbT8TbI:
xl;!' = 3,8802;
xfJ = 1,6815;


vl 31 = 1.7696;
y
3) == 2,9275.


(7.25)


f1pH BblqHCJIeHHH rp8HHll )I.JI5I OCT8JIbHblX HeH3BeCTHblX JIOrliqHO
npe)I.nOJIO>KHTb OTHOCHTeJIbHO HI:!J y\3) > 0, y
3J < 0 H 6\3) < 0,
70




x: If Y: Ta6JI. 1 )J.aHbI 3Ha'IeHIf51 HeH3BeCTHbIX, Hall)J.eHHble npIf paCCMOT-
peHHH KOHe'IHOll CIfCTeMbI ypaBHeHIfll, nOJIY'IeHHOll 113 (7.2) Ha OCHOBe
npe)J.nOJIO)KeHH5I X n = 0, Yn = ° npIf n > 10.
CpaBHHBa51 3Ha'IeHH5I HeIf3BeCTHWX, Hall)J.eHHWe Ha OCHOBe MeTO)J.a
pe)J.YKIJ,IfIf If C nOMOI.IJ,bIO OIJ,eHKIf rpaHHIJ, JIHMIfTaHT, npIfXO)J.IfM K 3a-
KJIIO'IeHHIO, 'ITO MeTO)J. pe)J.YKIJ,HIf )J.aeT )J.OCTaTO'lHO TO'IHble 3Ha'IeHH5I He-
CKOJIbKHX nepBbIX HeIf3BeCTHbIX. YBeJIIf'IeHHe nOp5l)J.Ka KOHe'IHOll CIf-
CTeMbI, BbICe'IeHHOll If3 (7.2), eCTeCTBeHHO YBeJIIf'IIfBaeT KOJIH'IeCTBO He-
H3BeCTHbIX, onpe)J.eJI5IeMbIX C )J.OCTaTO'lHO BbICOKOll TO'IHOCTbIO. Pe3YJIb-
TaTbI npoBe)J.eHHoro cpaBHeHH5I npIf6JIIf)KeHHbIX If npaKTH'IeCKIf TO'IHbIX
3Ha'IeHHll HeIf3BeCTHblX B 6eCKOHe'IHOll CIfCTeMe (7.2) nOKa3bIBaIOT, 'ITO
npIfMeHeHIfe MeTO)J.a pe)J.YKIJ,IfIf )J.aeT B03MO)KHOCTb )J.OCTaTO'IHO TO'IHO
onHcaTb Hanp5l)KeHHOe COCT05lHIfe B Tex TO'IKaX 06JIaCTIf, r)J.e CXO)J.HMOCTb
p5l)J.OB )J.JI5I CMeI.IJ,eHIfll H Hanp5l)KeHIfll n03BOJI5IeT OrpaHIf'IIfTbC5I npH
Bbl'IIfCJIeHIf5lX If X HeCKOJIbKHMH nepBbIMIf 'IJIeHaMH.
DocJIe npHBe)J.eHHoro aHaJIIf3a CBOllCTB HeIf3BeCTHblX B 6eCKOHe'IHbIX
CIfCTeMax H YCTaHOBJIeHH5I 3aKOHa aCHMnTOTIf'IeCKIfX Bblpa)KeHIfll )J.JI5I
HIfX 60JIee IJ,eJIeC006pa3HbIM 51BJI5IeTC5I IfCnOJIb30BaHIfe npIf ee pemeHIfH
HeCKOJIbKO YJIY'IllIeHHOro MeTO)J.a pe)J.YKIJ,IfH. OCHOBOll )J.JI5I TaKoro YJIY'I-
llIeHIf51 51BJI5IeTC5I qJaKT CTpeMJIeHIf51 HeIf3BeCTHblX B (7.2) K 06I.IJ,eMY )J.JI5I
HHX If OTJIIf'IHOMY OT HY.JI5I npe)J.eJIY. KaK nOKa3bIBaIOT pe3YJIbTaTbI BbI-
'IIfCJIeHIfll, npoBe)J.eHHblx C nOMOI.IJ,bIO OIJ,eHKIf rpaHIfIJ, JIIfMIfTaHT, He-
If3BeCTHble )J.OCTaTO'IHO 6bICTpO npIf6JIH)KaIOTC5I K npe)J.eJIbHbIM 3Ha'le-
HIf5lM. 3TO 06CT05lTeJIbCTBO )J.aeT B03MO)KHOCTb npIf nocTpoeHIfIf KOHe'I-
HOll CIfCTeMbI, cooTBeTcTBYIOI.IJ,ell 6eCKOHe'IHOll, nOJIaraTb
x n = XN (n
 N); Yn = YM (n
M), (7.26)
T. e. C'IIfTaTb, 'ITO Ha'lIfHa51 C HeKoTopbIX HOMepoB N If M 3Ha'IeHIf51 He-
H3BeCTHblX paBHbI Me)K)J.Y C060ll.
Yce'IeHIfe 6eCKOHe'IHOll CIfCTeMbI Ha OCHOBe paBeHcTB (7.26) npHBO-
)J.IfT K KOHe'IHOll CIfCTeMe. OTJIIf'IaIOI.IJ,eiic51 JIIflllb K03
IfIJ,IfeHTaMH npH
XN If YM OT CIfCTeMbI, nOJIY'IaeMOll no MeTO)J.Y pe)J.YKIJ,HIf. Bbl'IIfCJIeHHe
3TIfX K03
IfIJ,IfeHTOB He npe)J.CTaBJI5IeT TpY)J.a B CB5I3H C HaH)J.eHHbIMIf
TO'lHblMIf 3Ha'IeHIf5lMIf CYMM K03qJqJIfIJ,IfeHTOB 6eCKOHe'IHOll CIfCTeMbI B
Ka)K)J.Oll CTpOKe. PeilleHIfe nOJIY'IeHHOll TaKIfM 06pa30M KOHe'IHOll CH-
CTeMbl Tpe6yeT npaKTIf'IeCKIf TaKoro )Ke 'IIfCJIa OnepaIJ,Ifll, KaK H pellleHIfe
CH, CTeMbI nOJIY'IeHHOll MeTO)J.aMIf pe)J.YKIJ,HIf. O)J.HaKO npIf 3TOM )J.OCTH-
raeTC5I CYI.IJ,eCTBeHHoe YJIY'IllIeHIfe Ka'IeCTBa annpOKCIfMaIJ,IfH pellIeHH5I
COOTBeTCTBYIOI.IJ,ell 6eCKOHe'IHOll LIfCTeMbJ. KOHKpeTHYIO OIJ,eHK) 3qJ-
qJeKTa, )J.OCTIfrael\mro npIf 3aMeHe 6eCKOHe'IHOll CIfCTeMbI KOHe'IHOll Ha
OCHOBe paBeHcTB (7.26), MO)KHO npOIf3BeCTH Ha OCHOBe )J.aHHbIX Ta6JI. 1.
B cToJI6IJ,ax x n If !in )J.aHbI 3Ha'IeHIf51 HeH3BeCTHblX B CIfCTeMe (7.2),
Hall)J.eHHble no peilleHIfIO KOHe'IHOll CIfCTeMbI, IIocTpoeHHoll C IfCnOJIb-
30BaHIfeM CooTHoilleHIfll (7.26) npIf N = M = 10. 11.[151 3Ha'IeHIfll He-
If3BecTHblx_ C Ho
epaMH 60JIbille )J.eC5ITIf npIfH5ITO cpe)J.Hee apIfqJMeTH-
'IeCKOe OT X lO If YIO'
Hall)J.eHHble C nOMOI.IJ,bIO YJIY'IllIeHHOro Ha OCHOBe 3HaHH5I aCIfMnTO-
TH'IeCKIfX CBOllCTB HeH3BeCTHbIX MeTO)J.a pe)J.YKIJ,IfIf 3Ha'IeHIf51 HeIf3-


BeCTHbIX npaKTIf'IeCKIf COBna)J.aIOT C TeMH, KOTopble nOJIY'IeHbI C HC-
nOJIb30BaHIfeM OIJ,eHOK rpaHHIJ, JIHMIfTaHT. Ba)KHO oTMeTIfTb, 'ITO COB-
na)J.eHIfe If MeeT MeCTO He TOJIbKO )J.JI5I nepBbIX, BKJIIO'IeHHbIX B KOHe'IHYIO
CIfCTeMY HeH3BeCTHbIX, HO TaK:>Ke If )J.JI5I HeIf3BeCTHbIX C HOMepaMH
BbIllIe )J.eC5ITH
06paTHMC5I Tenepb K Bbl'IIfCJIeHHIO Hanp5l)KeHIfll, B03HHKaIOI.IJ,IfX
B KBa)J.paTHOll npIf3Me (nJIaCTIfHKe) npIf rpaHIf'IHbIX YCJIOBH5IX (7.1),
r)J.e npIfHHMaJIOCb 1 = 1. DpIf 3TOM Heo6xo)J.IfMO, HCnOJIb3Y5I 3Ha'IeHIf51
K03
HIJ,HeHTOB If3 Ta6JI. 1, npocYMMIfpoBaTb cooTBeTcTBYIOI.IJ,He 6ec-
KOHe'IHble P5l)J.bI.
06I.IJ,Ifll aHaJIIf3 CXO)J.IfMOCTH p5l)J.OB )J.JI5I Hanp5l)KeHHll, npoB
)J.eH-
Hblll B npe)J.bI)J.YI.IJ,eM naparpaqJe, o6HapY)KHJI O'IeHb Me)J.JIeHHYIO If X <:xo-
)J.IfMOCTb Ha rpaHIfIJ,e TeJIa If B TO'IKaX, )J.OCTaTO'IHO 6JIIf3KHX K rpaHIfIJ,e.
nOCKOJIbKY HaM H3BeCTHbi rJIaBHble 'IaCTH K03
IfIJ,HeHTOB X n H Yn B
(6.2), TO, IfCXO)J.51 H3 06bI'lHOro cnoco6a YJIY'IllIeHIf51 CXO)J.HMOCTIf p5l)J.OB,
OCHOBaHHOrO Ha BW)J.eJIeHIfH If npe)J.CTaBJIeHHIf B 3aMKHYTOM BH)J.e HX
Me)J.JIeHHO CXO)J.51I.IJ,HXC5I 'IaCTell, )J.JI5I Bbl'IIfCJIeHIf51 HOpMaJIbHbIX Hanp5l-
)KeHIfll B paCCMaTpIfBaeMOM CJIY'Iae nOJIY'IIfM CJIe)J.YIOI.IJ,Ife qJOPMYJIbl:
N
I.
 (_l)n { ( sh nnx ch nrrx
-(J =- - x nnx--nncthnn--
20 x nn n sh nn sh nn
n=1
c hnnx ) ( 1) 1] -n
o-x) }
- shnn -ao[nn x - - e cosnny-
N

 (- W { ( ch nny th ch nny + sh nn y )
- £..J
 Yn. shnn -nnc nn shnn nny shnn -
n=1


_ a [ 1- nn ( 1 - y)I e-n
(]-I/) } cos nnx +
 In b (x, y )
o 211, () (y,
)
-1' (x, y)/fJ (y, x) -1' (y, x)/fJ (x, y),
N
I .
 (- I)n { ( ch nnx th ch nnx +
20 (JIJ= £..J
 x n shnn -nnc nn shnn
n=1


(7.27)


72


+ nnx s h h nnx ' ) - a o [1- nn (1 - x)] e-111tO-x!1 j cos nny +
s nn,
N
+
 (- W {y ( nn y sh nn y . . _ nn cth nn ch nny _
£..J nn n sh nn sh nn
n=1
- c:hnn
 ) + a o [1 + nn (1- y)] e-n
o-Y)} cosnnx +
+ 2a
 In

;:

 + l' (x, y)/fJ (y, X) + l' (y, X)/fJ (x, y);
fJ (x, y) = 1 + 2 cos nxe -1t(l-y) + e-2
(I-y),
l' (X, !J) = ao (1- X) e- nO - xj [cos ny -t- e-
(I-X)]
73




B 3THX Bblpa1KeHHHX QO - SHa'leHHe 06mero !I.JUI x" H y" npe!I.eJHl B
(5.29), a N - 'lHCJIO HeHSBeCTHblX x" H y", Y!I.ep1KaHHbIX npH perneHHH
6eCKOHe'lHOH CHCTeMbI. B paccMaTpHBaeMoM KOHKpeTHoM cJIY'lae N =
= 10. OpH !fJaKTH'leCKOM npOBe!I.eHHH BbI'lHCJIeHHH npH6JIH1KeHHOe
SHa'leHHe Qo onpe!I.eJIHeTCH KaK cpe!I.Hee apH!fJMeTH'leCKOe BepxHeH H
HH1KHeH rpaHHl.{ JIHMHTaHT, 'lTO, B COOTBeTCTBHH C !I.aHHbIMH Ta6JI. I,
!I.aeT Qo = 1,4620.
OpH nOJIY'leHHH saMKHYTbIX Bblpa1KeHHH !I.JIH Me!I.JIeHHO CXO!I.HmHXCSI
qaCTeH pH!I.OB B (7.27) HCnOJIbSOBaHbI CJIe!I.YlOmHe paBeHcTBa:


co

 ( 1)" e- n :n(l-'I1) cos nn
 = -
 In (1 + 2 cos n
e-1t(l+'I1' +

 nIt 2It
n=1


+ r 2 :n(l-'I1'), (7.28)

 ( l) n -n:n(1-'I1) ,.. (cos It
 + e- 1t (I-'I1') e- n (1 '11)
..:..J - e cos nn>:, = - .
n=1 1 + 2cos It
e-:n(l-'I1' +e- 2 :n(1-'I1)


PaHee B 06mHX 'lepTa X 6bIJI paCCMOTpeH Bonpoc 0 XapaKTepe Ha-
npH1KeHHOrO COCTOHHHH np5lMoyrOJIhHHKa B6JIHSH yr JIOBbIX TO'IeK.
OpH 3TOM 6bIJIO nOKasaHO, 'ITO nOHBJIHlOmHeCH B OT!I.eJIbHbIX CJIarae-
MbIX 06mero perneHHH JIOrapH!fJMH'leCKHe oc06eHHOCTH BsaHMHO nora-
rnalOTCH H npH C06JIlO!I.eHHH YCJIOBHH napHOCTH KaCaTeJIhHbIX HanpSI-
1KeHHH Bce !fJaKTopbI HanpH1KeHHO-!I.e!fJOpMHpOBaHHoro COCTOHHHH ynpy-
roro TeJIa HMelOT B yrJIOBOH TO'lKe BnOJIHe onpe!I.eJIeHH
e KOHe'lH
e
SHa'leHHH. PaCCMaTpHBaH Bhlpa1KeHHH (7.27) B oKpecTHocTH yrJIOBOH
TO'lKH, M01KHO YCT3HOBHTb HeKoTop
e !I.OnOJIHHTeJIbHble cBoikTBa
pH!I.OB !I.JIH HanpH1KeHHH, Ba1KHble C TO'lKH speHHH BbIHCHeHHH OCo-
6eHHOCTeH paSBHBaeMoro MeTO!I.a perneHHH rpaHH'lH
X SaJI.a'l !I.JIH TeJI
KOHe'lHbIX paSMepOB.
HaJIH'IHe JIOrapH!fJMH'leCKHX oc06eHHOCTeH B OT!I.eJIbHbIX COCTaBJIH-
IOmHX 06mero perneHHH OTpa1KeHO B COOTHorneHHHX (7.27) 06mHMH !I.JIH
a x H a J; 'lJIeHaMH, cO!I.ep1KamHMH JIOrapH!fJM !I.P06H. OOMHMO HHX MeJI.-
JIeHHO CXO!I.HmHeCH 'laCTH 6eCKOHe'IHbIX pH!I.OB nOp01K!I.alOT B YKasaH-
HbIX COOTHorneHHHX (7.27) CJIaraeMble, npOnOpl.{HOHaJIbHble (1 - x)
II (1 - y). JIerKo BH!I.eTb, 'ITO !I.JIH HHX CnpaBe!I.JIHBbI paBeHcTBa


!im 11 (I, y) = 0, lim/ 1 (x, I) = _5L;
/i_I . x-I n
!im/2(l,y)=_5L. limf2(.\:,I)=0;
/i-I It x_I
y (x, y) I y (y, x)
fl (x, y) = (f( y, x) ' 2 (x, y) = b (x, y) .


(7.29)


9TH paBeHCTBa YKaSbIBaIOT, npe)KJI.e Bcero, Ha TO, 'lTO !fJOPMYJIbI !I.JIH BbI-
tIHCJIeHHH HOpMaJIhHbIX HanpH1KeHHH 6Y!I.YT HMeTh O!I.HH H TOT 1Ke BH.l{
HesaBHCHMO OT Toro, no KaKOH H3 rpaHeH ocymeCTBJIHeTCH npeJI.eJIbHbIH
nepexoA K yrJIOBOH TO'IKe. DOJIee Toro, paCCMlITpHBaH npe.l{eJIhHbIH ne-


74


J


pexo.!! npH x __ I, Y = aX + (l - a) (0 <: a < (0), HaXO!I.HM


iI[x, ax + (I-a») = - n(a
+ 1) ,


(7.30)


a o a 2



/2[X, ax + (I-a)] = - :t(a2+ 1) .
9TH paBeHcTBa cBH.l{eTeJIbCTByIOT 0 TOM, 'lTO c!fJ0PMYJIHpOBaHHbIH Bblrne
BbIBO.l{ OCTaeTCH B CHJIe npH npe!I.eJIhHOM nepeXO!I.e K yr JIOBOH TO'lKe no
JI1060MY nYTH.
OpOBe!I.eHHOe HCCJIe!I.or.aHHe nOBe!I.eHHH CJIaraeMbIX B (7.27), npo-
nOpl.{HOHaJIbHbIX paccToHHHlO TO'lKH, B KOTOpOH BbI'lHCJIHlOTCH HanpH-
1KeHHH, .AO rpaHH'lHOH nOBepxHocTH, nOSBOJIHeT 3aKJIlO'IHTb, 'ITO pH!I.hI
!I.JIH HanpH1KeHHH He 06JIa!I.alOT paBHOMepHOH CX0.AHMOCTblO B OKpeCT-
HOCTH yrJIOBOH TO'lKH. OTClO!I.a CJIe!I.yeT, 'lTO npH HCnOJIb30BaHHH .AJISI
perneHHH 6eCKOHe'lHOH CHCTeMbI, onpe.AeJIHlOmeH K03rp!fJHl.{HeHTbI 3THX
pHllOB. MeTOlla pe!I.YKUHH HanpH1KeHHH B yrJIOBOH TO'IKe MorYT 6bITb
BbI'lHCJIeHbI JIHrnb C TO'lHOCThlO .AO CJIaraeMOro
, KaKHM 6bI 60JIbrnHM
11
HH 6hIJI nOpH.AOK COOTBeTCTBYlOmeH KOHe'lHOH CHCTeMbI. T aKHM 06pa-
30M, npHXO!I.HM K Ba1KHOMY 3aKJIlO'leHHIO 0 TOM, 'ITO HCnOJIbSOBaHHe
MeTO.Aa pe.AYKl.{HH npH perneHHH 6eCKOHe'IHbIX CHCTeM H .Apymx COOT-
HorneHHH, K KOTOpbIM npHBO!I.HT HSJI01KeHHbIH B nepBOH rJIaBe MeTOl!
tIoCTpoeHHH 06mHx perneHHH rpaHH'lHbIX 33JI.3'l !I.JISI TeJI KOHe'IHhIX
pa3MepOB, CBH3aHO C BHeceHHeM B nOJIY'IaeMble pe3YJIhTaTbI HeYCTpa-
HHMOH norpernHOCTH.
LlJIH TOro 'lT06bI 60JIee nOJIHO oueHHTb xapaKTep 3TOH norpernHOCTH,
paCCMOTpHM 3Ha'leHHe pa3HOCTH HOpMaJIbHbIX HanpH1KNlHH B yrJIOBOH
TO'lKe. I1CXO!I.H H3 Bhlpa1KeHHH (6.1), B KOTOpbIX npHHHMalOTcH BO BHH-
MaHHe JIHrnb pH!I.bI C K03!fJ!fJHl.{HeHTaMH x n H Yno HaXOJI.HM, 'ITO npH YJI.ep-
1KaHHH B 6eCKOHe'IHOH CHCTeMe (5.15) 2N (N + N) ypaBHeHHH, pa3-
HOCTb HanpH1KeHHH ax H a y B yrJIOBOH TO'IKe KBa!I.paTa 6Y!I.eT onpeJI.e-
JIHTbCH no
opMYJIe


N N
I ( ) _ 2 V x
 + 2 '5"' II"
20 ax - a y -
 sh 2 nIt __ shinn .
11=1 ,,=\
C JI.pyroH CTOpOHbI, H3 Bblpa1KeHHH (7.27) nYTeM npe.AeJIbHoro nepeXO!I.a
HaXO!I.HM


(7.31)


N N

u-a -2'

+2)
 2.51.
20 I x ,) -
 sh 2 fl1I .:.oJ sh 2 n1l + n'
,,=1 ,,=\


(7.32)


CJIe!I.OBaTeJIhHO, KaKHM 6bI 60JIhWHM HH 6bIJIO 'lHCJIO N B (7.31), BbI'lHC-
JIeHHaR C nOMOmhlO 3Toro COOTHorneHHH pa3HOCTh HOpMaJIhHbIX HanpH-
1KeHHH 6Y!I.eT OTJIH'laTbCH OT 3a!I.aHHOH B rpaHH'lHbIX YCJIOBHHX Ha Be-
a
JIH'lHHY 2
. 3TO 06CTOHTeJIhCTBO !I.eJIaeT MaJI03!fJ!fJeKTHBHhIM noJI.XO.l{
It
K oueHKe Ka'IeCTBa nOJIY'laeMoro C nOMomhlO MeTO!I.a pe!I.YKWiH npH-
6JIH1KeHHOrO peweHHH Ha OCHOBe npOBepKH Y!I.OBJ1eTBopeHHH rpaHH'lHbIX
YCJIOBHH.


75




H3 OCHOBe Ol(eHOE: xapaKTepa CJIaraeMblX B COOTHOlliemIH (7.32)
MO:mH
 YKa3aTb O,n:HY HHTepeCHYIO C TOqKH 3peHHJI HCCJIe,n:OBaHHH pe-
IDeHHH 6ecKoHeqHblX CHCTeM 06JIaCTb ero npHMeH€HHH. C O,n:HOH cro-
pOHbI, BXO,n:HII1.H€ B (7.32) pH,n:bI 06JIa,n:aIOT HaCTOJIbKO 6bICTpOH CXO,n:H-
MOCTblO, 'ITa JJ:.JIH BblqHCJIeHHH CYMM C TOqHOCTblO ,n:o qeTblpex-nHTH
3HaqaII1.HX l(Hcpp ,n:OCTaroqHO y,n:ep:maTb no qeTblpe-nHTb qJIeHOB pH,n:a.
C ,n:pyroH CTOpOHbI, KaK BH,n:HO H3 ,n:aHHblX Ta6JI. I, nepBble HeH3BeCTHble
B 6eCKOHeqHbiX CHCT€MaX onp€,n:eJIHIOTCH C OqeHb BbICOKOH TOqHOCTbJO.
TIOC){OJIbE:Y pa3HocTb HanpH:meHHH ax H a y B yrJIOBOH TOqKe onpe,n:c.
JIJIeTCJI rpaHHqHblMH yCJIOBHHMH, TO, HCnOJIb3YH OTMeqeHHble CBOHCTBa
TaoJIlIua 2


x I y=O I y=0.5 I y=1
0 - 0,4657 -0,0410 0,6048
-0,3501 0 0,3501
0,5 - 0,6479 - 0,0502 0,8937
- 0,5729 0 0,5729
I - 1,0000 0 1,0017
- 1,0000 0 1,0000


Ta6JIliIta 3
x J y=o I y=0,5 I y=1
0 0,3340 0,1805 - 0.0007
0,1815 0 - 0,1815
0,5 0,1713 0,0610 0,0006
0,1507 0 - 0,1507
- 1,1637 - 0,3715 0,0002
- 0,9543 0 0.9543


x = 1 H a y npH Y = 1 HecYII1.eCTBeHHo OTJIHqalOTCH 01' 3a,n:aHHblX rpa-
HHqHbIMH YCJIOBHHMH (7.1). HaH60JIbllIaSI norpelliHocTb B y,n:oBJIeTBO-
peHHII rpaHHqHbIM YCJIOBHHM HMeeT MeCTO B yrJIOBOH TOqKe OTHOCHTeJIb-
HO HanpH:meHHH ax' r,n:e, o,n:HaKO, OHa He npeBbIlliaeT 0,2% aMnJIIITy.n:bI
BHellIHeH HarpY3KI1.
CpaBHeHHe HanpSI:meHHblx COCTOHHHH KBa,n:paTHoH nJIaCTI1HbI II 6ec-
KOHeqHOH rreplIo,n:lIqeCKI1 ,n:ecpOpMI1pOBaHHoH rrOJIOCbI n03BOJIHeT no-
JIyqHTb HeKoTopble Ol..{eHKH BJIIIHHIISI cB060,n:HbIX nOBepxHocTeH (KoHeq-
HbIX pa3MepoB) Ha HanpSI:meHHoe COCTOSIHI1e ynpyroro TeJIa.
ConOCTaBJIHSI 3HaqeHI1SI a x II (j!J BO BHYTpeHHI1X TOqKaX KBa,n:paTHoH
nJIaCTIIHbI II 6eCKOHeqHOH nOJIOCbl, 3aMeqaeM, '11'0 HaJIlIqlIe cB060,n:HbIX
01' HarrpSI:meHIIH nOBepxHoCTeH Y = ::!:: 1 nplIBo,n:IIT K 3HaQlITeJlbHoMY
YMeHbllIeHlIlO CKOpoCTH 3aTyxaHliSI HanpSI:meHIIH, B03HIIKalOII1.IIX 01'
caMoypaBHoBerneHHblx HarpY30K. TIOJIyqeHHOe p€rneHlIe rpaHlIqHOH
3a,n:aqlI YKa3bIBaeT TaK:me Ha ,n:pyrylO Ba:mHYIO oc06eHHocTb nOBe,n:eHIISI
HarrpSI:meHIIH B ynpymx TeJIaX KOHeqHblX pa3MepoB. CpaBHIIBaSI 3Ha-
'leHIISI HanpSI:meHIIH ax Ha OCII. Y = 0 II Ha rpaHII Y = 1 (Ta6JI. 2),
3aMeqaeM, 'ITO B,n:OJIb cB06o,n:HOH nOBepxHocTII HarrpH:meHIIH a x y6b1Ba-
101"3HaqIITeJIbHO Me,n:JIeHHee. 3TI1M no,n:TBep:m,n:aeTcH cYII1.eCTBOBaHlie nplI-
nOBepxHocTHbIX nOrpaHCJIOeB y cB06o,n:HbIX nOBepxHocTeH ynpymx
TeJI KOHeqHbIX pa3MepOB, CYII1.eCTBOBaHlIe KOTOpblX npe,n:cK333HO
11. l1. BOpOBlIqeM [571.
41'0 KaCaeTCSI HanpH:meHIIH a y B nJIaCTIIHKe, TO IICXO,n:SI 1I3 ,n:aHHbIX
Ta6JI. 3 OTMeTflM 3HaqlITeJIbHOe (nopSI,n:Ka 20%) yBeJIlIqeHHe HX B TOqKe
!J = 0, x = 1 no cpaBHemlIo co 3HaqeHlISIMH a y ,n:JIH 6eCKOHeqHOH
nepHo,n:lIqeCKII ,n:ecpopMHpoBaHHoH nOJIothl.
TIOJIyqeHHble KOJIHqeCTBeHHble Ol..{eHKII 3cpcpeKToB KOHeqHOCTII ynpyro-
ro TeJIa, eCTeCTBeHHO, 1I3MeHSITCf! npH 1I3MeHeHlIH reoMeTpHH nJIaCTIIHbI
II xapaKTepa ee Harpy:meHHH. B CBSI31I C 3THM rrpe,n:cT3BJIHeT HHT€peC
1I3yQlITb HX npe,n:eJIbHble 3HaQeHHH B TOM CJIyqae, Kor,n:a B3aHMHoe BJIH-
HHHe Harpy:meHHbIX nOBepXHOCTen HCKJIlOqaeTCSI. B CJIe,n:ylOII1.eM napa-
rpacpe paCCMaTpHB310TCSI rpaHlIqHble 3a,n:aqlI ,n:JISI yrrpyroH nOJIynOJIOCbI
rrpH pa3JIHqHbIX xapaKTepHCTHKax 1I3MeHSIeMOCTH caMoypaBHoBerneH-
HblX HOpMaJlbHbIX Harpy30K Ha Topue.


pH,n:OB B (7.32) H IIX Ko3cfJcpHI..{HeHTOB x n ' Ym MO:mHO JIerKO Haxo,n:HTh
3HaqeHI1e 0;0 C TOqHOCThlO ,n:o TpeX-qeTblpex 3HaqaII1.HX I..{lIcpp.
TIoCJI€ rrpoBe,n:eHHoH rro,n:roToBHTeJlhHOH pa60TbI cpaKTHq€CKOe BbI-
qlICJIeHlIe HanpSI:meHHH no cpOPMYJIaM (7.27) npe,n:cTaBJIHeT 3JIeMeH-
TapHYIO
a,n:aqy, TIOCKOJIbKY KaqeCTBeHHO xapaKTep nOBe,n:eHHH Ha-
npH:meHHH B ,n:aHHOM CJIY'l3e Harpy:meHHSJ ,n:OCTaTOqHO OqeBII,n:eH, TO
np" aH3JIH3e KOJIHqeCTBeHHblX ,n:aHHbIX ,n:OCTaTOqHO OrpaHlIqlIThCH He-
CKOJlhKIIMII xapaKTepHbIMII CeqeHIISIMI1.
B Ta6JI. 2 npuBe,n:eHbI 3HaqeHUH HanpSI:meHHH a x C TOqHOCTblO ,n:o
nOCTOHHHoro
HO:mHTeJIH -2GA (CM. (7.1». Ka:m,n:oMY 3HaqeHlIIO KO-
Op,n:HHaT B 3TOH Ta6.n.,11l..{e OTBeqalOT ,n:Ba 3HaqeHHSI aJi;' CTOHIl(He BBepxy
qHCJIa OTBeqalOT ,n:eIICTB1iTeJIhHbIM HanpH:meHuHM, BblqHCJIeHHhIM no
cpopMYJIaM (7.27) npH N= 10, KOTOPbI€ B03HHKalOT B KBa,n:paTHoH npH3-
Me (nJlaCTlIHK€) npu rpaHHqHblX YCJlOBHHX (7.1). P3CqeTHble 3HaqeHHH
K03cpcplluHeHToB x n H Yn onpe,n:eJIHJIHCb K3K cpe.n:HHe apHcpMeTHqecKHe
Me:m,n:y BepXHHMJI II HH:mHIIMH oneHKaMH 113 Ta6JI. 1. HmKe ,n:JIH
Ka:m,n:oro 3HaqeHIIH Koop.n:HHaTbl x YKa3aHbI 3HaqeHHH HanpSJ:meHHH
ax, B03HHKalOIl(HX B 6eCKOHeqHO JJ:JIHHHOH nOJIoce -1 <;. x < 1,
-00 < y < 00, nepHo,n:UqeCKH ,n:ecpOpMHpoBaHHoH HOpMaJIhHblMH Ha-
I
rpYSKaMI1 2G ax ( + 1) = -A cos ny. TaKoH:me CMblCJI HM€IOT .n:aHHble
Ta6JI. 3, OTHOCHIl(HeCf! K HanpH}KeHHHM a
LIaHHble Ta6JI. 2 H 3 nOSBOJIHIOT. npe5K.n:e Bcero, Ol..{eHHTb KaqeCTBO
nOJIyqeHHoro perneHHSI C TOqKH 3peHHH TOqHOCTH y,n:oBJIeTBOp€HHH rpa-
HlIqHbIX YCJIOBHH. BblqUCJIeHHble no cpopMYJIaM (7.27) gHaqeHHH ax npH
76



 8. rpaHH'IHhIe 3a)1;a'lH )1;JIH nOJIynOJIOChI
K KJIaccy 3a,n:aq, perneHHe KOTOpblX Mo:meT 6bITb nOJIyqeHO Ha OCHoe
Be npeJJ:CTaBJIeHHH 061l(HX perneHHH rpaHHqHbIX sa,n:aq ,n:JIH npHMD-
yrOJIhHHKa, nOCTpO€HHbIX B
 4, OTHOCHTCH 3a,n:aqH ,n:JIH ynpyroH nOJIY-
nOJIOCbI.
113yqeHHe HanpSI:meHHoro COCTOSIHHH npSIMoyrOJIhHlma OCHOBbIBaeT-
CH Ha perneHHH 6eCKOHeqHbIX CHCTeM JIHHeHHblx ypaBHeHHH THna (5.15)
H (5.34), K03cpcpI1I..{HeHTbI KOTOpbIX HBHO sa.BHCSIT 01' reOMeTpHqeCKoro
napaMeTpa 'I'}. 4>H3HqeCKH HCHO, qTO BO MHomx CJIyqaRX B3aHMHblM
BJIHHHHeM YCJIOBHH Ha npoTHBOnOJIo}KHblX cTopoHax MO:mHO rrpeHe-
6peqh, oco6eHHo B Tex CJIyqaHX, Kor,n:a BHernHHH CaMoypaBHoBerneHHaH
HarpY3Ka 06JIa,n:a€T 3HaqHTeJIhHOH H3MeHHeMOCTblO. B 3THX CJIyqaHX
77




I


pemeHIUI sa,lJ,aqH ,lJ,JUI nOJIynOJIOCbI MOrYT 6bITb HCnOJIbSOBaHbI npH aHa-
JIHS€ HanpSDKeHHoro COCTOHHIUI npHMoyrOJIbHbIX 06JIaCT€H pa3JIH'lHOH
reOMeTpHH.
PaCCMOTpHM BHaqaJIe BonpOC 0 nOCTpOeHHH peWeHHH ypaBH€HHH
naMe, nOSBOJIHlOIllHX H3yqaTb HanpmK€HHOe COCTOHHH€ nOJIynOJIOCbI
(pHC. 4). I1SJIO)KeHHe 6Y,lJ,eM BeeTH B 6espasMepHbIx KOop,lJ,HHaTaX x =
r = x'la, y = y'la, npHqeM no,lJ,p06-
1/ HbIe BbIKJIallKH npOBe,lJ,eM llJIH CJIyqaH
cHMMeTpHqHOrO OTHOCHTeJIbHO OCM
J Ox' HanpH)KeHHOro COCTOHHHH.
KaRoH B CJIyqa€ paccMoTpeHHoH
BbIW€ Sa,lJ,aqH llJIH npHMoyrOJIbHHRa,
BbIPB}!{€HHH KOMnOH€HTOB BeKTO-
pa nepeMeIlleHHH D.JIH nOJIynOJIOCbI
o6pa3YlOTcH HS D.BYX COCTaBJIHlOIllHX.
D€pBa.H HS HHX ,lJ,OJI)KHa 06JIa,lJ,aTb
,lJ,OCTaTOqHOH CTeneHblO npOHSBO-
JIa ,lJ,JIH y,lJ,OBJIeTBOp€HHH rpaHHqHbIX
YCJIOBHH B CMeIlleHHHX MJIM HanpH}!{eHHHX Ha CTopOHaX nOJIynOJIOCbI
y = + 1. DOCKOJIbKY 9THM Tpe60BaHHeM y,lJ,OBJIeTBOpHlOT KaK CMM-
MeTpHqHOe, TaK H aHTHCHMMeTpHqHOe OTHOCHTeJIbHO OCH Oy' o6IllHe pe-
WeHH.H rpaHHqHOH SaD.aqH ,lJ,JIH 6eCKOHeqHOH nOJIOCbIY = + I, TO B
KaqeCTBe HCKOMbIX peW€HHH ypaBHeHHH JIaMe MO}!{HO .npMHHTb BbIpa-
}!{€HHH


!


DapaM€Tp an B (8.2) MO}!{eT ObITb onpe,lJ,eJIeH JIHOO KaK an = nJt.
1
JIH60 KaK an 2" (2n - 1) n. I1CXOD.H HS paccMOTpeHH.H Sa,lJ,3qH ,lJ,JIH
npHMoyrOJIbHHKa, nepBoe SHaqeHHe an cne,lJ,yeT HCnOJIbSOB3Tb B TOM
cnyqae, KOr,lJ,a B yrJIOBbIX TOqKaX BbIllOJIHHlOTCH YCJIOBHH napHOCTM
KaCaTeJIbHbIX HanpH}!{eHHH. DpM 3TOM B nepBOH COCTaBJIHlOIlleH o6Illero
pemeHHH (8.1) BbI6HpaeM BepxHHe SHaqeHHH TpHroHoMeTpHqeCKHX
MHO}!{HTeJIeH. PaccMaTpHBaH HH}!{e KOHKpeTHYlO sa,lJ,aqy C HYJIeBbIMH
SHaqeHHHMH KaCaTeJIbHbIX HanpH}!{eHHH Ha rpaHHIJ.e nOJIynOJIOCbI, or-
paHHqHBaeMCH pemeHHHMH, y,lJ,06HbIMH B CJIyqaHX COOJIlO,lJ,eHHH nap-
HOCTH KaCaTeJIbHbIX HanpH}!{eHHH B yrJIax. DpH 9TOM KOMnOHeHTbI
TeHsopa HanpH}!{eHHH HaXO,lJ,HM B BH,lJ,e
00
I 2 r [A 3m-2
20 ax = - n.J (s)schsy + C(s) m chsy +
o
+ C(s) SYShsy]cossxdS + f (Dn m m 2 -anxD n -
n=I
- anBn) e -anx COS a"y,
00
I 2 r f A . m - 2
20 O'y = n J (s) s ch sy + --,yz- C (s) ch 5y +
o
+ C (s) sysh SY] cossxds +
I (B"a" + a"xD n - (8.3)


o


.x'


t::i
C\J


PHc.4.


00
U 1 = -
 r [ A (5) ch 8Y + C (8) 3m - 4 ch sy +

 d ms


3m-2 D ) -IX x
- n e n cosany.
m


] ( sin 5X )
+ C(s)yshsy ds,
cos sx


(8.1)


00
2
 't xy = -
 S[ A (s) 5 sh sy + 2 m m I C (s) sh sy +
o
+ C (s) sy ch SY] sin sxds - n
1 (B"a" + Dnanx -
m-I ' ) -ax
- 2 -----,n- Dn e . n sin anY.
AHaJIOrHH Me}!{,lJ,y rpaHHqHbIMH Sa,lJ,aqaMH ,lJ,JIH nOJIynOJIOCbI H npH-
MoyrOJIbHHKa B CMbICJIe. CBOHCTB 06IllHX pemeHHH npeD.CTaBJIHeTC.H ec-
TeCTBeHHoH. B CBH3H C 3THM He 6Y,lJ,eM OCTaHaBJIHBaTbCH Ha aHaJIH3e
CBOHCTB pH,lJ,OB H HHTerpaJIOB B (8.1), (8.2) H (8.3) npH npOH3BOJIbHbIX
rpaHHqHbIX YCJIOBHHX, a nepeH,lJ,€M HenOCpe,lJ,cTBeHHO K rpaHHqHOH sa-
,lJ,aqe, KOTOPYIO MO]!{HO paccMaTpHBaTb KaK ,lJ,onOJIHeHHe K HCCJ:le,IJ.OBaH-
HOH BbIme 3B,lJ,aqe D.JIH npHMoyrOJIbHHKa:


2 r ( COSSX )
W1=-J [A(s)shSY+C(s)ych5Y]. . . ds,
11; 0 . - sm sx


r,lJ,e o,lJ,HOBpeMeHHo CJIe,lJ,yeT Bhl6HpBTb JIu60 BepXHee, JIu60 HH}!{Hee 3HB-
qeHl'le TpHroHoMeTpHqeCKHX MHO)KHTeJI€H B 3THX BbIpa}!{eHUHX A (s) H
C (s) - npOH3BOJIbHbIe <PYHKIJ.UH.
BTOPYIO COCTaBJIHIDU1.ylO 06Iu.ero pern€HHH ,lJ,JIH nOJIynOJIOCbI ,lJ,aeT
H3BecTHoe peweHHe rpaHHqHOH 3a,lJ,aqH ,lJ.JI.H nepHO,lJ,HlJeCKH ,lJ,e<popMHpO-
BaHHOH nOJIynJIOCKOCTH:


00
U z =
 (B n + D"x) e-anx cosany,
,,=1


00
W 2 =
 ( Bn + D"x- Dn 3m-4 ) . e-a'?sma"y.
n=1 ma. n
3,Qecb Bn If Dn - npoHSBOJIbHbIe nOCTORHHbIe.


(8.2)


I
2G ax = f (y),


I
20 'tx:
 = 0


npH x = 0;


1 I
20 o'y = 2G'txy = 0


(8.4)


npH 11 = :f:: 1.


78


19




HarPYSKY Ha TOpIte nOJIynOJIOCbI, Sa.n:aBaeMYIO IPYHKI.{Hen I (y), ec-
TeCTBeHHO CqHTaeM caMoypaBHoBellleHHOiL .
YCJIOBHH .n:JIH KaCaTeJIbHbIX HanpHMeHHH B (8.4) .n:aIOT .n:JIH npOH3-
BOJIbHbIX nOCTOHHHbIX H IPYHKItHH B (8.3) CJIe.n:YIOrqHe CBH3H:
Bna n. = 2 m- 1 D
m It'


TO YCJIOBHe cyrqecTBoBaHHH OrpaHHqeHHOrO pellleHHH CHCTeMbI (8.7)
npJ!06peTeT Bli.n:


I


A(s)s=-C(S)(2 m m 1 +scths).
C yqeTOM 3TliX COoTHOllleHHH IPYHKI.{HOHaJIbHble ypaBHeHliH B03HliKa-
IOrqHe npH no.n:CTaHoBKe BblpaMeHHH .n:JIH HOpMRJIbHbIX Hanp
MeHHH HS
(8.3)!; rpaHHqHble YCJIOBHH (8.4), npe06pasYIOTCH Ha OCHOBe COOTHO-
IIIeHHH (5.6) H npe.n:CTaBJIeHHH 3KCnOHeHItHaJIbHbIX IPYHKItI'H HHTerpa-
JIaMH <Pypbe


(8.5)


12a n (_1)11. 111. I .:;:::: l( (8.10)
'\j1. (n) """.
113 BblpaMeHJ!H .n:J!SI 1jJ (n) CJIe.n:yeT BbIllOJIHJ!MOCTb YCJIOBHH (8.10).
eCJIH K03IPIPHItlleHTbI <PYPbe In IPYHKItHH I (y) B (8.4) y6bIBaIOT C pOCTOM
1
n KaK
. Vl3 3Toro Tpe60BaHHH BbITeKaIOT TaKHe Me OrpaHHqeHHH Ha
n
CTeneHb rJIa.n:KOCTli IPYHKItHH. 3a.n:aIOrqeH BHelllHIOIO HarpY3KY. KaK H B
sa.n:aqe .n:JIH npHMoyroJIbHJ!Ka.
DpOBepKY YCJIOBHll crrpaBe.n:JIHBOCTH 3aKOHa aCllMrrTOTHqeCKHX
BbIpaMeHHH .n:JIH HeH3BeCTHbIX B (8.7) B CBHSH C OIteHKaMH .n:JIH K03IP-
IPHItJ!eHToB CHCTeMbI (5.15). OqeBH.n:Ho, .n:OCTaTOqHO ocyrqecTBHTb JIHIIIb
.n:JIH HepBoro ypaBHeHHH. ECJIli B KaqeCTBe
 (n) B (3.23) nOJIOMHTb
 (n) =
= n, TO npHMeHHTeJIbHO K paCCMaTpHBaeMOH CHCTeMe He06xo.n:HMble
YCJIOBHH cyrqecTBoBaHHH 06rqero .n:JIH x (s) H Yn npe.n:eJIbHOrO 3HaqeHHH
(3..24) H (3.36) sarrHCbIBaIOTCH CJIe.n:YIOrqHM o6pasoM:


co
e . --f:X:nx = 2 . r an
I cas sxds,
"0 a
 +S2


co 2
--f:X: x 2 5 an-s2
xe 11. = - . .. cossxds
" (a. 2 + 5 2 }2
on
K CJIe.n:YIOrqen CHCTeMe CMelllaHHorO THna:


(8.6)


4 a 4
l<"---r' 2 11. <;L
" (qn + 5 2 )2


(s < n),


4 5 CO q3
Yn=-n XeS) (2 11 21! ds+2a n (-1)n'n'
o an + 5 )


3.n:eCb 1. L
CTOHHHble.
DepBoe
eCJIH


T

 5
o
H 8 - HeKOTopble npOliSBOJIbHble OTJlHqHble


aT HYJIH no-


q3
11 ds:>8
(a.
 + S2)2


(T
 n <; 2T).


(8.11 )


co
X (s) ( dh s + -,-- h S2 ) =,
 Y 45 3 ( 8.7 )
s S .=;. n (a
 +52)2 '
X (5) = 4s sh sC (s), Yn = (- 1)"+12a.,DII'
. .. Dpll aHaJIH3e perYJIHpHOCTH CHCTeMbI (8.7) HCXO.n:HM liS CJIe.n:YIOIILHX
paBeHCTB:


HepaBeHCTBO 6y.n:eT BbInOJIHeHO .n:JIH npOH3BOJIbHOro n,



 2 45 3 =dhs+
-

11=1 (an +
2}2 sh 2 s s'


(8.8)


4,,2 4
1 = (1 + ,,2)2 ' L =
. (8.12)
I1cxo.n:H H3 HHMHell OIteHKH .n:JIH no.n:bIHTerpaJIbHOro BblpaMeHHH BO BTO-
paM HepaBeHCTBe (8.11), Haxo.n:HM
T T
4 5 . a.
 4 5 a.
 .1
- ds=- ds


" (a 2 + 82)2 ,,2n la2 + 52)2 <J? 2 .
o 11 0 n
TaKHM 06paSOM, eCJIH IPYHKItJ!H I (y) 06JIa.n:aeT KyCOqHO-HenpepbIB-
Hall nepBon npOH3Bo.n:HOH, TO .n:JIH HeH3BeCTHbIX B CHCTeMe (8.7) cnpar;re.n:-
JIHBO CJIe.n:YIOrqee npe.n:eJIbHOe paBeHcTBo:
Hrn Yn = Hm x (s) = a o '
n.....oo s...Oo


(8.13)


co


: 5
o


q3
n dS=l
(a
 + 5 2 )2 '


DepBoe ll
 3THX paBeHCTB YKa3bIBaeT, qTO CHCTeMa (8.7) 06JIa.n:aeT
BbIpOM.n:eHHOH perYJI5IpHOCTbIO, paccMoTpeHHoll B
 3. ECJIH B COOTBeT-
CrBMH C (3.40) BBeCTH IPYHKItHIO
To
4 5 a

1jJ (n) = - -.,-- . - ds,
" o (a;; + s2)2


(8.14)


(8.9)


DpH IPaKTHqeCKOM pellleHHH CliCTeM THna (8.7) YCTaHOBJIeHHOe npe-
.n:eJIbHOe paBeHCTBo (8.14) HBJIHeTCH OCHOBOll .n:JIH cyrqecTBeHHoro YJIyq-
IIIeHMH MeTo.n:a pe.n:YKUlllL B COOTBeTCTBHH C 3THM paBeHCTBOM CHCTeMa
(8.7) npe06pa3yeTcH K KOHeqHOll CHCTeMe Ha OCHOBe rrpe.n:nOJrQMeHHH
Y n = YN (n :> N); x (s) = X (T) (s:> T). (8.15)


80


6 7-2464


81




K8K nOK838JIII peSYJIb1'aT'hl JraCQeT0B, npA p:aCCMOTpeHIHf 38.n.8qH .n.JIJi
npRMoyroJIbHHKll 't8KOe npe.n.nOJIOm:eHHe 06eCneqHB8eT .n.OCTaTOqHO Bb1-
COKYIO TOqIIOCTb onpe.n.eJIeHHR HelI3BeCTHbIX H8 BceM HHT€pBllJIe H3Me-
HeHHR HOMep8 HJIH 8prYMeHT8.
l1.JIR onpe.n.eJIeHHR 3H8qeHHH nepBbIX N HeH3BeCTHbIX Yn II <PYHKUHH
X (s) npII 0 <: s <: T H8 OCHOBe p8BeHCTB (8.15) nOJIYIIaeM CJIe.n.yIOIII.YIO
CIICTeMY COOTHOIIIeHfIH:
T
Yn=
 S
o


v
X CO& sxds + -]-
 . ( ry,,
)n (yn - ao) (1 + anx) e-rY,n x cos any-
11=1


00


ry,,3 [x (5 ) - X (T ))
n . ds + X ( T ) + 2a ( _ l ) n f
(a
 + 5 2 )2 n 11'
X (s) (dh s + sh: 6 ) =


_
 r ...!- e -S(l-y! cos sxds +
2Jt J 5
T
a o (I - y) -T(l-y) (I - y) cos Tx - x sin Tx
+ 2n e x 2 + (I _ y)2
-
 In (1 + 2 cos nye-n:c + e- 2nx ) +
4Jt


(8.16)


1 -m
aox -nx + cos nye
+ -e -nx -2m t
2 1 +2cosnye +e
T

 = --
 {'
(chsy + scthschsy-syshsy) X
20 G y 2n J 5 sh 5
o
N
X cossxds + --.!....
 (_I)n (Yn -ao)(I-anx)e-rL rr cosany-
2 £..J an
n=1
00
_
 . o {'
 e-s(l-yJ co
 sxds-
2n J 5
T


N

 53 (Yn - UN) . ( th ' 5 2 )
= 4;:1 {a
 +62)11 + l1N, C S+ Sh25 - S
(n= 1, 2, ... ,N; 0
 s <. T).


OKOHQ8TeJIbHoe cym:.n.eHHe 0 KOHKpeTHbIX 3HaQeHHRX BeJIHQHH N H
T MOm:HO c.n.eJI8Tb JIHIIIb H8 OCHone Ollp€.Il.eJIeHHblX Tpe60B8HHH K T01J-
HOCTH Y1(OBJIeTBOpeHHR rp8HHQHbIX YCJIOBHH. OJJ:H8KO C C8MOro Ha1J8JI8
lICHO, QTO BeJIHQHH8 N He MO>KeT 6bITb MeHbm€ 'mCJIa 1JJIeHOB PR1(8 <I>y-
pbe, He06xo1(HMbIX 1(JIR' 8nnpOKCHMaWIH BH€lllHeH HarpY3KA C TOQHOCTbIO,
Tpe6yeMoH 01' pemeHHH rp8Hlf1JHOH 3a1(aQHB ueJIOM. HeKOTopble KOC-
BeHHbIe oueHKFf .n.JI1J BbI60pB 'BeJIJftIH'H N H T MO>KHO nOJIYIlHTb T8Km:e
nOCJJe pemeHHR CHCTeMbI (8.16), He 06p8III.8HCb K BbI'mCJIeHHIO H8 rpa-
HHue. nOC'KOJIbKY B COOTBeTCTB'HH C (8.14) npe,lJ,eJJbHb'Fe 3RaQeHHH Yn
H x (s) 1(OJIm:HbI COBn81(8Tb, To nOJIY1JeHH81J nOCJJe pemeHHR KOH€I.fH'-JH
CHCTeMbI (8.16) CYIII.eCTBeHH8H pa3HHU8 3H8'!eHHH YN H X (T) 6Y1(eT
YK83bIB8Tb H8 He06xo1(HMoCTb YBeJIHQeHH1J N H T..
nOcJIe peIIIeHHH CHCTeMbI (8.16) MOm:HO BbIlJHCJIHTb H8npHm:eHHH II
CMeIII.eHHH B nOJIynoJIoce. 38'KOII '8CIIMllTOTHlleCKHX Bbfp8JKeHHH (8.14)
YK83bIB8eT Ha TO, Qro PR1(bI H HHTertyaJIbI, Bblp8>K:aIOIII.He KOMnOHeHTbI
BeKropa CMeIII.eHHH, 06JI81(8IOT 1(OCT8TOQHO 6bICTpOH CX01(HMOCTbIO H
QHCJIeHH8H 06P860TK8 HX He COCT8BJIReT TPY1(8. HeCKOJIbKO CJJom:Hee
06CTOHT 1(eJIO C BbIQHCJ1eHH€M H8npH
eHHH. B03HHI\'8IOIII.H.e 31(eCb TPY1(-
HOCTH, K8K H B p8CCMOTpeHHOH BbIllie 381(8Qe 1(JIR' rrpHMoyroJIbHHK8.
CBR38HbI'C OTCYTCTBHeM p8BIIOMepHOH CX01(HMOCTH pR1(OB H HHTerp8JIOB,
B, CBH3'H \C 'leM 1(OCTOBepHbIe 3Hal.feHH1J HX B OTlJ;eJIb"HbI>X 'I'OlfK8X MorYT
6blTb nOJIYQeHbI JIHmb nYTeM npe1(eJIbHOrO nepeXQ,lJ;8.
BbIp8m:8R B (8.3) npoH3BOJ}bHble nOCT0H'HHbIe H <PYHKUHH Qepe3
tin H X (s) H nOJI8raH X (s) = aD, Yn = aD ,lJ;JIH Bcex s> T H n> N,
COOTBeTC'PBeHHQ nOJIYlfHM GJJt}1(YIOIIJ;He $OPM;YJIbI .&MI BhltliiCJI€HH1J H3-
npH}l{eHHH:


l'
,1 1 S :\: (8) ,
"--'- . ' '6 == - - . ' -J(ch sy - s dh s'ch sy + sy ch sy) X
20
 2n 5sh5
o


1 + cos nye-:nx (8.17)
- a
x e- nx 1 + 2 cos nye-nx + e 2nx
a o (1 - II) -T(1-y) (I - y) cos Tx - x sin Tx
2n e x 2 + (I _ y)2
a -nx + -2:nx )
- -3L In (1 + 2 cos nye e ,
4n

"t =
 r x(s) ( cthS sh sy _ Y ch h S y ) sin sxds +
20 xy 2n J sh 5 s 5
o
N
+
 X
 (- It (Yn - aD) e-rLnX sin any-
2 11=1
aox sin nye- nx . +
-
 1 + 2 cos nye- nx + e- 2mc
ao (I - y) -T()-IjJ (1- y) sin Tx + x cas Tx .
+ 2n e x 2 + (I _ y)2
HeKOTopble TPY1(HOCTH npH n01(ClleTe H8npH}l{eHHH no <popMYJIar.

(8.17) B03HHK8IOT npH BbIQHCJI€HHH HHTerpaJI8


00
I' I -S(l-y! d
J (x, y) = J
e cossx.s,
T


(8.18;


,82


6*


8





T

 0" .. = - --,!.- r x (s) ( dh s + - h
 ) ds +
2G y 231 J s S s
o
+..!.. f --,!.- (Yt! - ao) + ;0 (C - 1 + In
 ) .
2 n=1 an . n ,
flOJIY'IeHHeM BbIpameHHH .n;JHI HanpHmeHHH B yrJIOBOH TO'IK: 3aKaH:
'IHBaeTCH nO.n;rOTOBHTeJIbHaH pa60Ta K npOBe.n;eHHm BM'IHCJIeHHH nOJI:"
CMellJ,eHHH H HanpHmeHHH B nOJIynOJIoce. IJ,eJlec006pa3HOCTb YCHJIHH,
HanpaBJIeHHbIX Ha nOJIY'IeHHe paC'IeTHbIX <POPMY JI B yr JIOBOH TO'IKe,
.n;JIH nJIOCKOH 3a.n;a'IH. onpaB.n;bIBa- T a 6 JI H
 a 4
eTCH CJIe.n;ymllJ,HMH 06cT05lTeJIbCT-
BaMH. flpem.n;e Bcero CJIe.n;yeT
HMeTb B BH.n;y, 'ITO e.n;HHCTBeHHbIM
KpHTepHeM .n;JIH oueHKH Ka'IeCTBa
nOJIY'IeHHOrO no H3JIOmeHHOMY
MeTO.n;y perneHHH tlBJIHeTCH TO'I-
HOCTb y.n;OBJIeTBOpeHHH rpaHH'IHbIX
YCJIOBHH. 3Ha'IHTeJIbHbIe OTKJIO:
HeHHH BbI'IHCJIeHHbIX 3Ha'IeHHH
HOpMaJIbHbIX HanpHmeHHH OT 3a-
.n;aHHbIX rpaHlI'IHbIMH YCJIOBHHMH
B OKpeCTHOCTH yrJIOBOH TO'IKH CHH-
mamT .n;OCTOBepHOCTb nOJIY'IeHHbIX
3Ha'IeHHH CMellJ,eHHH H H
mpH-
meHHH BO BceH 06JIaCTlI. C .n;pyroH CTOpOHbI, B CJIY'Iae nP9CTpaH-
CTBeHHbIX 3a.n;a'I TeopHH ynpyrocTH onpe.n;eJIeHHe HanpHmeHHH B yr-
JIOBbIX TO'IKaX npOCTO He06xo.n;HMO, a pa3BHTbIe npH paCCMOTpeHHH nJIO-
CKOH 3a.n;a'IH cnoco6bI BbI'IHCJIeHHH HaXOllHT npH 3TOM Henocpe.n;CTBeHHoe
npHMeHeHHe. . . .
I-13JIOmeHHbIH BbIrne aJIrOpHTM OTbICKaHHH HanpHmeHHoro COCT
HHHtI
nOJIynOJIOCbI peaJIH3yeM npH nOCTpoeHHH u O.n;HOPO.n;HbIX u perneHHH .n;JI

HeCKOJIbKHX nepBbIX rapMOHHK Ce).l.HHH'IHOH aMnJIHTy.n;OH B HOpMaJIbHOH
HarpY3Ke Ha Toplie. flpH 3TOM cBo6o.n;HbIe 'IJIeHbI B CHCTeMe (8.16) npH-
MYT BH.n;
11 { 2ak (- l)k, n = k, (8.23)
2a n (- 1) 't! = 0, n =1= k.
KOHKpeTfIbIe BbI'IHCJIeHHH npoBe.n;eHbI .n;JIHCJIY'IaueB k = 1,2,3. flpH pe-
rnem1H CHCTeMbI (8.16) .n;JIH Bcex Tpex 3Ha'IeHHH k npHHHMaJIOCb N -
= 20, T = 60. VIHTerpaJIbI BbI'IHCJIHJlHCb no MeTo.n;y CHMncoHa.
B T86JI. 4 npHBe.n;eHbI 3Ha'IeHHH HeCKOJIbKHX nepBblx H u nOCJle.n;HHX
HeH3BeCTHbIX Y
 H3 (8.16) .n;JIH Tpex YKa3aHHbIx 3Ha'IeHHH u k. B no-
CJIe.n;HeH CTpoKe Ta6JIHUbI .n;aHbI 3Ha'IeHHH x (60). .L(aHHbIe 3TOH Ta6JIHUbI
nOK83bIBaIDT eCTecrBeHHYIO TeH.n;eHUHIO K yxy.n;rneHHm Ka'IeCTBa perne-
HHH CHCTeMbI (8.16) C pOCTOM H3MemreMOCTH HarpY3KH IlPH <p HKCH -
pOBaHHoM N H T.


KOTOpbIH B 06llJ,eM CJIY'Iae He npe.n;CTaBJIHeTCn'Iepe3 Ta6YJIHpOBaHHbIe
cneul!aJIbHbIe <PYHKUHH. O.n;HaKO B HaH60JIee He6JIaronpHHTHbIX CJIY'Ia-
SIX Y = 1 HJIH X = 0, T. e. Ha rpaHHue nOJIynOJIOCbI, 3TOT HHTerpaJi
BbIpamaeTCH'Iepe3 xoporno H3Y'IeHHbIe <PYHKUHH, 'ITO n03BOJIHeT3<p-
<peKTHBHO npoBecTH Bbl'IHCJIeHHSI HanpSlmeHHH BO Bcex TO'IKaX KOHTY-
pa, BI\JIm'IaH H yr JIOBb!e.
TaK, paccMaTpHBaH 3Ha'IeHHtI J (x, y) npH x = ° H Y = 1, Haxo.n;HM


J (0, y) = - Ei [- T (l - y)); J (x, 1) = - ci (xT).


(8.19)


I1cnoJIb3Ytl 3TH npe.n;CTaBJIeHHH HHTerpaJIa (8.18), MO>KeM paccMoTpeTb
npe.n;eJIbHbIH nepexo.n; K yrJIOBOn TO'IKe H nOJIY'IHTb He06xo.n;HMbIe £Pop-
MYJrbI .n;JISI BbIqHCJIeHHH HanpHmeHHH.
PaccMoTpHM BHa'IaJIe Te CJIaraeMbIe B (8.17), KOTopbIe nopo3Hb
CO).l.epmaT oco6eHHocTH B yrJIOBOH TO'IKe. flOJIOmHM ).I.JISI onpe.n;eJIeHHO-
CTH, 'ITO no).l.XO.n; K yrJI)' ocyruecTBJHIeTCtI B.n;OJIb nOBepxHocTH y = 1.
Tor.n;a


_.!!!L J (x 1) -
 In (l_e- nx ) =
2n ' 2n


= 2
 (C + InxT)- ;
 In (I_e-1!X) + 0 (x 2 ). (8.20)


OcyruecrBJIHH nepexo.n; K npe.n:eJIY npH x -+ 0, IlOJIY'IaeM
- ;
 ;

 [J (x, 1) + In (l-e- 1tX )) = ;
 (C + In
 ) . (8.21)


3.n;eCb C - nOCTOHHHaSI 9HJIepa.
CyruecTBeHHoe 3Ha'IeHHe npH EbI'IHCJIeHHH HanpHmeHHH B yr JIOBOn
TO'IKe HMeIOT CJIaraeMbIe, nponopUHOHaJIbHbIe 1 - Y H X B BbIpameHHSIx
)J:JIH HOpMaJIbHbIX HanpSImeHHH. Onpe.n:eJIHeMbIe HMH .n:06aBKH B 3Ha'Ie-
HHtlX HOpMaJIbHbIX HanpSImeHHH He MorYT 6bITb B npHHUHne y'ITeHbI npH
fIcnOJIb30BaHHH MeTO)J:a npOCTOH pe.n;YKUHH .n;JISI perneHHSI CHCTeMbI
(8.7). BeJIH'IHHa 3THX .n;06aBoK YCTaHaBJIHBaeTcsr 3JIeMeHTapHbIM npe-
).I.eJIbHbIM nepexo.n;OM. TaK, HanpHMep, paccMaTpHB8SI BbIpa>KeHHe .n;JIH
U x npH x = 0, Haxo.n;HM, 'ITO CJIaraeMOe, co.n;epmallJ,ee x MHOmHTeJIeM,
06paIllaeTCSI B HYJlb. B TO me BpeMSI CJIaraeMOe, npOIIOpUHOHaJIbHOe
a
I - Y rrpH Y -+ 1, HMeeT npelleJI 2
 . TaKoeme 3Ha'IeHHe npe.n;eJIa HMeeT
CJIaraeMOe C MHOmHTeJIeM X npH Y = I H X -+ 0. TaKHM 06P830M, .n;JIH
BbI'IHCJIeHHSI HanpSImeHHH B yrJIOBOH rO'IKe cJre.n;yeT nCnOJIb30Batb
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T

a = _
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n=I


(8.22)


84


I k
n I \
1 2 3
1 -7,181 0,462 -0,473
2 - 1,781 13,71 -0,897
3 - 2,222 1,659 -20.18
. .. . . . . . . . . .
18 - 2,881 3,024 -3,084
19 - 2,883 3,035 -3,107
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x(60) 1- 2,883\ 3,152 I -3,426


85




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COCTaBJI5IIO
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MeH5IeMOCTH Harpy3KH npH nOCT05lHHOH ee aMnJIHTy.ne Ha6JI1o.naeTc5I
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B CB5I3H C oniIcaHHbIMH BbIme pe-
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43


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BhI'JlICJIeHHbIe npH X = 0 3Haq
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20 ax (x. 0)
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o


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He xapaKTep
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89




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TO}!FUHl B OKpeeTHOCTH
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rneHIIH CMernaHHOH 3aLIallH LIJI5I KJIHHa [116, 181]. OCHOBHa5I TPYLIHOCTb
B perneHIIH CMernaHHbIX saLIall LIJI5I TeJI KOHellHbIX pa3MepoB SaKJIFOlIaeTC5I
B TOM, liTO npll HCnOJIbSOBaHHIIJ1FOooro MeTOLIa perneHHH ycnex ooecne-
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HYFO ocooeHHOCTb nOJI5I HanpR>KeHHH Ha 5I3bIKe Tex CliCTeM <j:>YHKIUIH,
KOTopbIe npHBJIeKaFOTC5I LIJIH nOCTpoeHIIH oomero perneHHH. TaKHM 06-
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npH pellIeHHH rpaHHlIHbIX saLIall LIJIH IlHJIHHLIpa HCnOJIbSYFOTCH If LIPY-
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CJIeLIyeT YKasaTb paooTbI 11. 11yroJIJIa [205], A. 11. .nYPbe [1221,
r. c. lllanHpo [1901, B. 11. BJIoxa [371,5I. C. Y<pJIHHLIa [181], 11. CHe.lI.-
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JIllllHbIe CJIYllall LIe<P0pMHpOBaHll5I oec- t.
KOHellHoro ynpyroro UllJIHHLIpa oocY}K-
LIaJIllCb B paooTax 11yroJIJIa [206],
A. H. .nYPbe [1221, r. c. lllamipo
[1911ll LIp.
AHaJIllS nepellHCJIeHHbIX II LIpymx
paOOT, Ha KOTopbIe MO}KHO HaHTH
YKasaHH5I B MOHorpa<j:>lUl [124] II ::to:
oosope [51, BPH.lI. JIll HBJI5JeTCH 3.lI.eCb '\j
ueJIec006pasHbIM. 11JIH Hac Ba}KHO
CeHlIaC TO OOCTOHTeJIbCTBO, 'ITO OJIa-
roLIap5I COLIep}KaIIlHMCH B HllX pe-
3y JIbTaTaM llMeeTCH BOSMO}KHOCTb
nOCTpOllTb TaKHe liaoopbI lIaCTHbIX
perneHllH ypaBHeHHH .naMe, KOTopbIe
saBeLIOMO OOJIaLIaFOT CBOHCTBOM nOJI-
HOTbI OTHOCllTeJIbHO npOHSBOJIbHbIX
rpaHllllHbIX YCJIOBllH B Tpex OCHOBHbIX rpaHllllHbIX saLIallax LIJIH UllJIllH-
.n:pa KOHellHOH LIJIllHbl. CJIeLIyeT saMeTllTb, liTO TaKaH BOSMO}KHOCTb cy-
IlI.eCTByeT B OOIIleM cJIYlIae HeocecllMMeTpHlIIIoro LIe<popMHpOBaHllH UH-
JIllHLIpa, OLIHaKO SHallllTeJIbHaH rpOMOSLIKOCTb COOTBeTCTBYFOIlI.HX BbI-
KJIaLIOK LIeJIaeT ueJIeCOOOp3SHbIM OTLIeJIbHOe pacCMOTpeHlle npaKTlllle-
CKll Ba}KHoro OCeCllMMeTpHlIHOrO Hanp5I}KeHHoro COCTOHHllH.
KaK ObIJIO YCTaHOBJIeHO npll I1sYlleHllH rpaHl!lIHbIX SaLIaq LIJIH npH-
MoyrOJIbHllKa II nOJIynOJIOCbI, LIOCT3TOllHOH rlloKocTbFO OOJIaLIaFOT pe-
meHllH ypaBHeHI1H .naMe, OTBellaFOIlI.lle nepHoAHlIecKoMY sarpY}KeHllFO
nOBepxHocTeH Tex 6eCKOHellHbIX 06JIaCTeH, nepeCelleHHeM KOTOpbIX
HBJIHeTCH paCCMaTpi-lBaeMOe ynpyroe TeJIO KOHellHbIX pasMepoB. B LIaJIb-
HeHmeM oYLIeM TaK}Ke YllHThIBaTb, liTO COOTBeTCTBYFOIlI.He BblKJIaAKH HBJIH-
FOTCH OOJIee Har JI5ILIHbIMll, eCJII1 llcnOJIbSOBaTb 01leBHLIHyFO BOSMO}KHOCTb
pa3LIeJIbHOrO HSYlleHHH CJIYllaeB CHMMeTpHlIHoro H aHTllCllMMeTpHlIHOrO
OTHOCllTeJIbHO nJIOCKOCTll
 = 0 (CM. pllC. 7) HanpH}KeHHbIX COCTOHHHH.
11JIH cHMMeTpH'lHOrO CJIYllaH paLIHaJIbHYIO U H oceBYFO W KOMnOHeH-
TbI BeKTopa cMeIIlemlH, COLIep}KaIIlHe HY}KHYIO ereneHb <PYHKUllOHaJIb-
HOro npOHSBOJIa Ha TOPU3X UllJIHHLIpa, npeLIcTaBHM B BllLIe


o


f



- -r-_..........
/ ----+--- "'
r


PRC. 7.


00


Ut = el (Ajz sh Ai Z + C j ch Al) J l(A/r),


co
Wt=Aoz+
 (1 3m m-;:/ ShAjz-A/zchAjz-CjshAjZ) X
/=1


(9.1)


x J o (A,r),
91




3,neCb Ao, A j, C I - npOH3BOJIbHbIe nOCT05IHHble, J n lx) - <PYHKIIHH
BecceJI5I.
Bxo,n5Illl.He B (9.1) 6eCKOHeqHbIe P5I,[(bI nOJIyqeHbI H3 06lll.ero perneHH5I
,nJI5I CJI05I [175] npH «nepHo,nHqeCKoM» 3arpYR<eHHH ero nOBepxHocTeH_
CJIaraeMOe C Ao ,n06aBJIeHO KaK qaCTb 3JIeMeHTapHoro perneHH5I [1191,
,naIOlll.ero B03MOR<HOCTb nOJIyqHTb TOqHOe perneHHe 3a,naqH ,nJI5I UHJIHH,n-
pa npH paBHoMepHoM 06R<aTHH ero no TopuaM H no 60KOBOH nOBepx-
HOCTH. <PopMaJIbHbIM OCHOBaHHeM ,nJI5I BKJIIOqeHH5I qJIeHa Aoz B 06lll.ee
perneHHe MOR<eT CJIYR<HTb TO 06CT05ITeJIbCTBO, qTO CHCTeMa <PYHKUHH
J o (Ajr) npH J 1 (Ai) = 0 CTaHOBHTC5I nOJIHOH JIllrnb nOCJIe BKJIIOqeHll5I
B Hee e,nllHHubI. OpH BbI60pe B KaqeCTBe Ai KopHeH ypaBHeHH5I J o (A)
= 0 CJIe,nyeT nOJIaraTb Ao = O.
Opll HCC31e,nOBaHHll aHTllcllMMeTpllqHOrO Hanp5IR<eHHOrO COCT05IHll5I
UH31HH,npa B perneHllll (9.1) C31e,nyeT nOMeH5ITb MeCTaMll sh AIZ II ch Aiz.
DOMllMO 3Toro, Tpe60BaHHe n031HOTbI CllCTeMbI <PYHKUllH J o (Air) npll
J 1 (Ai) = 0 3aCTaBJ15IeT BK31lOqHTb B BbIpaR<eHHe ,n3151 W C31araeMOe qeT-
Hoe no Z H He3aBHC5IIlI.ee OT r. Opll 3a)l.aHHll Ha Topue rpaHllqHbIX 3Haqe-
HllH ,n3151 oceBoro CMeIlI.eHll5I TaKHM C31araeMbIM MOR<eT 6bITb npoCTO He-
KOTOpa5I nOCT05IHHaSl. O,nHaKO npll <PoPMY31HpOBKe rpaHllqHbIX YC310BllH
Ha TopUe B Hanp5IR<eHll5IX 3TOH KOHCTaHTbI He,nOCTaTOqHO. B BbIpaR<eHll5I
,n31Sl oceBoro CMelll.eHH5I MOR<HO BK31lOqllTb C31araeMOe C O Z2, KOTopoe caMO
no ce6e He 5IB315IeTC5I perneHlleM ypaBHeHllH J1aMe II Tpe6yeT BBelleHllSl B
m-l
pa,nHa31bHOM nepeMerneHHH C31araeMOro - 2 - Cozr.
m
TaKHM 06pa30M, B C31yqae aHTllCHMMeTpllqHOrO OTHOCHTe31bHO n310C-
KOCTll
 = 0 Hanp5IR<eHHOrO COCT05IHll5I Ta qaCTb 06lll.ero peWeHll5I rpa-
HllqHOH 3a,naqH, KOTopaSl y,nOB31eTBOp5IeT npOll3B031bHbIM rpaHHqHbIM
YC310BH5IM B CMelll.eHll5IX H31ll Hanp5IR<eHll5IX Ha Topue IIll31llHlI.pa. llMeeT
Bll,n


= 0 ,ne<popMHpOBaHll5I npll06peTalOT BHll


00
,,-, r 4(m-l)
U 2 = Bor + f::I Bn mk n '1 (krl) - BnrI o (knr) -
- Dnll (knr) ] cos knz,


00


(9.3)


W 2 =
 [B n rI l (knr) +Dnlo (knr)) sin knz.
n=I


00
m-l

u 1 = - 2 - Cozr -+
 (Alz ch AiZ + C j sh A/) J 1 (A/),
m 1=1


(9.2)


3,neCb Bo, Bm D n - npOll3B031bHble nOCT05IHHble, , n (kll) - <PYHKUllll
Becce3151 qnCTO MHllMoro aprYMeHTa n-ro nOp5I,nKa. LL315I Be31HqllHbI ne-
pllo,na, C KOTOpbIM 3a,naHHa5I Ha nOBepXHOCTH r = 1, - h <= Z <= h
HarpY3Ka npo,n031R<eHa Ha BCIO nOBepXHOCTb 6eCKOHeqHOrO Ull31llH,npa,
MOR<HO, KaK II B C31yqae npSlMoyr031bHHKa, npHH5ITb 311060e 3HaqeHlle
T nn
>- 1, T. e. MOR<HO CqHTaTb k n = Th' ECTeCTBeHHo, qTO npH HeKO-
TOpbIX 3HaqeHH5IX T, HanpHMep T = 2 (n = 2k - 1), MOR<HO n0310-
R<HTb Bo = O.
OpH paccMoTpeHHH aHTHcHMMeTpHqHOrO Hanp5IR<eHHOrO COCT05IHH5I
B peweHllll (9.3) C31e.n.yeT 3aMeHHTb sin kllz Ha cos knz II COS knz Ha
-sin knz. PaccYR<.n.eHH5I, aHa310rHqHble npHBe.n.eHHbIM npll nOCTpoeHHH
BblpaR<eHHH (9.2), n03B0315I1OT YCTaHoBHTb BH.n. q31eHOB, CT05Illl.HX BHe
CYMM, BBe,neHlle KOTOpbIX 06YC310B31eHO Tpe60BaHH5IMll n031HOTbI CllCTeM
<PYHKUllH Ha cooTBeTcTBYIOIlI.llX HHTepBa31ax. B llTore .lI.J15I YKa3aHHoro
c31yqa5I cllMMeTpHH HaXOllllM c31e,nYIOIlI.lle npe.n.CTaB31eHH5I KOMnOHeHTOB
BeKTOpa CMeIlI.eHHH:
m-2
 [
u 2 = - 2 -----;:;z- Bozr +
I Dn l l (knr) + BnrI o (knr)-
4(m-l) ] -
- Bn mk n 1 1 (knr) SID knz,


(9.4)


00
A 2 ""'" (A 3m - 4
WI = 0 + CoZ +
 j m'J.... ChA/Z- AlzshAjz-
j=1 J


\
- C i ch AjZ) JoCA/).


W 2 = Bor2 +
 [DJo (knr) + B ,j rI l (knr)) cos knz.
n=1
AlI.llllTHBHaSl nOCT05IHHa5I B BblpaR<eHHH .n.3151 w 2 onYIlI.eHa B CB5I3H C
reM, qTO 06IlI.ee perneHHe OCHOBHbIX rpaHllqHbIX 3a.n.aq .lI.J15I uH31HH.n.pa
CTpOHTCSl B Blllle


06paTHMC5I Tenepb K nocTpoeHlllO TOH qaCTH 06lll.ero perneHH5I, KG-
TOpa5I y.n.OB31eTBOp5IeT rpaHllqHblM YC310Bll5IM Ha 60KOBOH nOBepXHOCTH
Ull31llH.n.pa. B 06IlI.eM HeoceCHMMeTpH'lHOM C31yqae .n. e <p o pMllpOBaHll5I
HCKOMble BblpaR<eHll5I MorYT 6bITb n031yqeHbI Ha OCHOBe ll3BeCTHoro pe-
rneHll5I .lI.J15I 6eCKOHeqHO .n.31HHHOrO UH31llH.n.pa [206]. ,g3151 H3YQaeMoro
3,neCb ocecllMMeTpHqHOrO Hanp5IR<eHHOrO COCTOSlHll5I 3TO peweHlle MO-
MeT 6bITb nOqTH n031HOCTblO nepeHeceHO ll3 MOHorpa<pllll [1241. Ooc31e
,n06aB31eHll5I C31araeMbIX, OTBeqalOIlI.llX paBHoMepHoMY 06R<aTHIO Ull-
31llH,npa no 60KOBOH nOBepxHocTH, BblpaR<eHllSl ,n3151 KOMnOHeHTOB BeK-
Topa cMeIlI.eHllll B cJIyqae cl1MMeTpHqHOrO OTHOCI1Te31bHO n310CKOCTll S =


u = u 1 + u? W = WI + &/2' (9.5)
aWl B (9.2) TaKYIO nOCTOSlHHYIO co.n.epR<llT.
B .n.aHHoH KHllre ,naHO npllMeHeHlle KaK CHMMeTpllqHbIX (9.1), (9.3),
TaK II aHTl1cllMMeTpllqHbIX BbIpaR<eHllH (9.2), (9.4). BHaqa31e OCTaHOBllM-
C5I no.n.p06HO Ha nepBoH OCHOBHOH rpaHllqHOH 3a,naqe 113151 KopoTKoro
lLHJ1llH.n.pa B c31yqae cllMMeTpHqHoro OTHOCllTe31bHO n310CKOCTH
 = 0
Hanp5IR<eHHOrO COCT05IHll5I. Ha OCHOBe peweHHH (9.2) II (9.4) npoBe,neM
HCC31e)!.oBaHlle cMernaHHoH 3a.n.aqll Teopllll ynpyrocTll, B03HllKalOIlI.eH
npll paccMoTpeHllH ll3rll6a R<eCTKO-3aIlI.eM31eHHOH Kpyr 310H n31llTbI.


92


93




1 u _ m Bo + Ao -
 (Aiz sh 'Aiz +
20 r - m-2 m-2 ;=1
+ C I ch 'A,Z) +J 1('A i r) +
I (Ai'AiZ sh 'Aiz + Ai ; ch 'A,z + Ci'A i ch 'AjZ) x

 ( 3m-2 4(m-l) 1 ( k
x J 0 ('At) +
 l Bn ----;n- 10 (knr) - mknr 1 n r ) -
n=1
- k n rl 1 (knr)') + D'1 (+ 11 (knrJ - kio (knr))] cos knz,
1 m Ao
 (A , h
 +
-us=-B G +- 2 +.-;."J l ZS "", Z
20 m-2 m- i=J



 ' I m-I
X J 1 ('Air) +:::1 B"k"rlo (k,,r) - B.,.2 ---m-/ 1 (knr) +
+ Dnkn1I (knr) ] sin knz,
OOMI1MO CJl.eJIaHHbIX BbIllie npeJl.nOJIOmeHI1H OTHOCIITeJIbHO xapaKTepa
'leTHOCTI1 CPYHKL{IIM B rpaHII'lHbIX YCJIOBI1f1X (9.6), 6YJl.eM C'lIITaTb B Jl.aJIb-
HeHllIeM, 'lTO B paccMaTpIIB3'CMOH KpaeBpH 3aJl.a'le BbmOJIHeHO YCJIOBIIe
napHOCTI1 KaCaTeJIbHbIX HanpflmeHIIH B 'yrJIOBOH TO'lKe, TorJl.a B BbIpa-
meHIIflX (9.1) 1'1 (9.3) II II cooTBeTcTBYIO
IIX IIM HanpflmeHI1f1X (9.7)
MomeM npl1HflTb B K3'leCTBe 'A, KOpHII ypaBHeHIIfI J 1 ('A) = 0, a TaKme
C'lI1TaTb T = 1.
OocJIeJl.YIOllJ,l1e BbIKJIa)J.KII 6y Jl.eM RpOBOJl.IITb, npeJl.CTaBJIfIfI BXOJl.fI
IIe
B (9.6) CPYHKUIIII COOTBeTCTBeHHO pflJl.aMII <I>YPbe II <I>YPbe - I!.IIHII:
w . 00
11 (z) ct: fiO +
 /In c0sk n z. fPI (Z) = 2: fPln sin knz,
n=1 n=1


OYCTb H3npflmeHHoe COCTOflHIIe llIIJIIIHJJ.pa onpeJl.eJIfleTCH cJIeJl.YIO-

IIMII rpaHII'lHbIMII YCJIOBIIflMII:
1 1
20 {Jr (1, z) = 11 (z), 20 Tn (1, z) = fPI (Z),
do uz(r, + h)=f2(r), 2
 Trz(r, + h)= + fP2(r). (9.6)
PaBeHcTBa It (z) = It (-z) II fPI (z) = -fPI (-z) cnp a BeJl.JI IIBbI OTHO-
CIITeJIbHO CPYHKllIIH fl (z) II fPI (z), 'lTO Jl.3eT B03MomHOCTb npII nocTpoe-
HIIII 06
ero pellieHIIfI rpaHII'lHOH 33J1.a'lII (9.6) OrpaHII'lIITbCfI npeJl.CTaB-
JIeHIIflMII BeKTopa CMellJ,eHIIH B BIIJl.e (9.1) II (9.3). 06'beJl.IIHfIfI B COOT-
BeTCTBIIII C (9.5) 3TII npeJl.CTaBJIeHIIfI, HaXOJl.IIM Ha OCHOBe 3aKo
a
rYK3 CJIeJl.ylO
IIe BbIpameHIIH Jl.JIfI KOMnOHeHTOB TeH30pa HanpflmeHIIII:


00


00


(9.8)


f2 (r) = f20 + }: h,J o ('AirJ, fP2 (r) = .... fP2;ld'A , r) ,
i=1 i=1
K09cpcpIIUIIeHTbI KOTOpbIX BbI'lIICJIflIOTCfI no II3BeCTHbIM cpopMYJIaM [68,
1821.
ECJIII CPYHKUIIII, 3aJl.alOllJ,IIe HOpMaJIbHbIe HarpY3KII B rpaHII'lHbIX
YCJIOBIIflX, 06JIaJl.alOT KYCO'lHO-HenpepbIBHbIMII nepBbIMII npOII3BoJl.-
HbIMH, a Jl.JIfI KaCaTeJIbHbIX H3npflmeHIIH B (9.6) 3TIIM CBOHCTBOM 06-
JIaJl.alOT BTopbIe npOII3BOJl.HbIe, TO Jl.JIfI K03cpcpIIUIIeHTOB pflJl.OB B (9.8)
nOJIY'lIIM oueHKH


00
+ c. ch 'A Z)
 J I ('A;r) +
 2.. A, ch 'A,zJ o ('AIr) +
I I r m ;=1
00
+ \' r -D ..!- 1 ( k r ) + B ( 4 (m - I) 11 (knr)-
-- L n r 1" n mknr (9.7)
n=1
- m m 2 10 (kkr»)] cos knz,
12m - 1
 (A m - 1 h

-u =-Bo+-Ao+;t, i2 -c. ""jZ-
20 Z m - 2 m - 2
 m
1=1
- Ai'Alz sh 'Alz - Ci'A j ch 'AiZ) J o ('A,r) +
I [ B"knrliJ. (knr) +
+ Bn
-10 (knr) + D"k,.lo (knr)} cos knz,

 T z =
 ( A.'Ajz ch 'A.z - A. m - 2 sh 'Aiz + C,'Aj sh.'AiZ) X
20 r i=1 I I I m


/In = 0 (
 ) ,
\ n


(9.9)


= _ 2 (- I)" 'PI (h) + 0 (
 )
fPln hk n n 3 '
( 1 ) 2'P3 (I) ( 1 )
hi = 0 1..;" , <PZi = - Ajlo (Ai) + 0 1..;/0 .
OpII'leM II3 YCJIOBIIH napHocTH KaCaTeJIbHbIX HanpflmeHIIH Ha pe6pe
cJIeJl.YeT paBeHcTBo fPl (h) = fP
 (1).
OOJl.CTaHOBKa BbIpameHIIH Jl.JIfI HanpflmeHIIH (9.7) B rpaHII'lHble yc-
JIOBIIfI (9.6) npIIBOJl.IIT K HeKoTopoH CIICTeMe CPYHKUI10H3J1bHbiX ypaB-
HeHIIH, II3 KOTOpOH CJIeJl.yeT onpeJl.eJIIITb COJl.epmallJ,IIeCfI B (9.7) npo-
H3BOJlbHbIe nOCTOflHHble. Y CJlOBIIfI Jl.JIfI KaCaTeJIbHbIX HanpflmeHIIH B CII-
.rry Toro, 'lTO J 1 ('A,) = sin knh = 0, Jl.alOT cJIeJl.YIO
IIe npocTbIe COOT-
HOllieHIIfI Jl.J1f1 HeII3BeCTHbiX BeJIII'lIIH:


( m - 2 ) 'PZi
C/A, = A, --;n- -'Alh dh 'Aih + sh Afh '
Dk =B f 2 tn-I -k 'o(krf) I+

n n " I m
 " (k n ) j 'I (k n ) .
MCnOJlb3YSI npeJI.-CT3BJl.€HIIe rIInep60JIIIII€CKHX CPYHKUIIH TpIIrOHO-
MeTpII'leCKIIMH pflllaMII (5.6), BNleCTO CPYHKIlIIOHaJIbHOrO ypaBHeHIIfI.


(9.lD)


94


95




BhIpmKaIOlll.ero rpaHIlqHOe YCJIOBHe .l{JISI U r , nOJIyqaeM CJIe.l{YlOllI.He aJI-
re6paHqeCKHe paBeHCTBa:


o
m 2 Bo + m Ao 2 +
 :
 J o (A j ) = 110'
m ;=1 '
r Ig (k n ) m - 1 k 1
Bnkn liCk n } -11 (k n ) - 2 mk
 II (n) +
"" 4Ajk
 (_I)n shAjh
 2Aj (-It CP2j
+ L A/o (A;) 2 2 + £..J . Ii: A 2 J o (A;) +
1=1 h (k n + "i)2 ;=1 h ( n + ;)
+
 [ 1 - k 10 (k n ) ] = I
k n n II (k n ) In.
flpHBe)]:eHHe BTOPOro, ypaBHeHHSI, B03HHKalOllI.erO npH y)]:OBJIeTBo-
peHHH YCJIOBHH .l{JISI, U z ' K aJIre6paHQeCKHM COOTHOllleHHSIM OCYllI.eCTB-
JISIeTCSI C nOMOIIlblO pa3JIO)KeHHH


(9.11)


Idk n ) 2k I ( k \. J o (AV)
10 (knr) =2
+n I. n)!-J ( k2 + A
 ) J ( A' ) '
n ,=1 n , 0 1
4
knrI I (knr) = 21 0 (k n ) - k;; II (k n ) +


(9.12)



 00
k 2 I k
 J 0 ( A
 _ 4k 3 I ( k )
 J 0 (Ajr)
+2 n o( J
 ( Jl2. + A
 ) J ( A" ) n I n
 (k2.+A
)2J (A')'
}=I n } 0 1 }=I n , 0 1
COOTBeTCTBYlOllI.He aJIre6paHQeCKHe paBeHCTBa HMelOT CJIe.l{YlOllI.HH BH)]::
2 m - 1 \.
 n 2CPIn

Bo+
Ao+£..J(-I)
=120'
m - m - ;,=,I n
( A ih )
 n 4kn'07J1 (k n )
Ai ch A;h + shA'h +.£..J (- I) Bn (Ii: + A
)2 J 00') +
} n=1 n 1 O}


(9 13)


+
 (- I)" (jJln
 2: n - £P2i cth A,h = f2j-
n=1 ( n +1,;) J o (A;)
CHCTeMa .l{BYX aJIre6paHQeCKI1X ypaBHeHHH )]:JISI Ao H Bo MO)KeT 6bITb
';IpHBe.l{eBa K 60JIee y.l{06HOMY BH.l{Y. flpH 3TOM HCnOJIb3ylOTCSI pa3JIO-
)KeHHSI


00
r = _
 2J 1 (Air) , z = _


 Ai J 0 (A;)
/=1 n=I
C nOMOlll.blO nepBoro H3 HHX HaXO)]:HM
00 1

 %i J ) 1 r ( d \.
 2J 1 (Ajr)
.£..J A .h 0 (Aj = II J (jJ2 r) r r £..J Ai J 0 (A .)
;=1 } 0 }=I }
96


2 (_ I)n
k sin knz.
n


(9.14)


I
= - -*- 5 (jJ2 (r) r 2 dr. (9.15)
o


.


AHaJIOrmHO HaXO.l{HM CYMMY pSI.l{a, CO.l{ep)Kalll.ero QJln,
(9. I 3) .l{JISI Ao H Bo:
00 n

 I n 2CPln 2 S
.1..J (- )
 = - h «PI (Z) zdz.
n=1 0


B ypaBHeHHH


I


(9.16)


TaKHM 06pa30M, nOCTOSIHHhIe Ao H Bo 3JIeMeHTapHhIX pellleHHH,
)]:alOllI.HX B03MO)KHOCTh Y.l{OBJIeTBOpHTb rpaHHQHhIM YCJIOBHSIM Ha no-
BepXHOCTH UHJIHH.l{pa B CMbICJIe npHHUHna CeH-BeHaHa, onpe)]:eJISIIOTCSI
CJIeJJ:YlOllI.eH CHCTeMOH aJIre6paHQeCKHX ypaBHeHHH:
I
m m 2 Bo + m Ao 2 .= Ito + + 5 QJ2 (r)r 2 dr,
o
h
2 m-I 2 S
m-2 Bo + m-2 Ao = 120 + h QJI (z)zdz
o


(9.17)


Y.l{OBJIeTBOpeHHe rpaHHQHhIX YCJIOBHH JJ:JIfI caMoypaBHoBeIIleHHhIX coc-
TaBJISIlOllI.HX BHeIIlHeH HarpY3KI1 CBSI3aHO C pellleHHeM 6eCKOHeQHOH CH-
CTeMhI ypaBHeHHH, 06pa30BaHHOH BTOpbIMH paBeHCTBaMH B (9. 11) H
(9.13). flpHcTynHM K HCCJIe.l{OBaHHIO 3TOH CHCTeMbI.
BBOllSI BMeCTO A. H Bn HOBbIe HeH3BeCTHble no <popMYJIaM


X n = (- It Bn4knII (k n ),


4
Yi = - A; h J o (Ai) A; sh Aih, (9.18)


nOJIYQaeM Y.l{06HYIO )]:JISI HCCJIe.l{OBaHHSI 6eCKOHeQHYlO cHcTeMy
I

 + (!In
X n = t;; £=1 Yi (k
 + A
)2 tn'


(9.19)


00
I
,
y; = s:- !.J X n
1 n=1
3.l{eCb npHHSIT@ 0603HaQeHHfI
:l
_ 1 [ fo (k n ) m - I ] . _ h ( Ajh )
t n - T
 - I - 2
 , SJ - 4 " cth AJh + h 2" . h '
II (k n ) m«n T\" \ S T\,I

 I ( I) n (- I)n r I '0 (k n ) ]
'.V n = In - -l))ln- -k n --
k n ItCk n )


A

} + 1J'i .
(k
 + 1,7)2 SI




 2Aj%j
- J A.
. h (k 2 + A
 ) . o( ,).
/=1 n 1


(9.20)


00


1Tr J " t
 2 (- I)n k n
T i = - 12; 0 (A;) - QJ2;J O ("";) C hAih + "-J QJln
n=1
 + 1,7
PerYJISIpHOCTh CHCTeMbI (9.19) YCTaHaBJIHBaeTCSI Ha OCHOBe oueHOK
CYMM K03<p<pHUHeHTOB K3)KJJ:OH CTpOKH. HCXO.l{SI H3 CBOHCTB paBHOMepHOH


7 7-2
6



97




CXO,AHMOCTH p5I,AOB (5.6) H (9.12), HaXO,AHM cJIe,AYIOmHe paBeHuBa:
1 V Ii;. m+ I
t;; f'=; (k
 + Aj)2 = 1 -1jJ (n), 1)) (n) = 2mk
tn


(9.21)


A

;j
, (
-/- 1,;)2 = I - fP (j), fP (j) = 21,JSj .
B npOQeCce nOJIYIleHH5I paBeHCTB (9.21) YCTaHaBJIHBaeTC5I
O<;:1jJ(n)<I, 0<;: fP(i)< 1 (n, j= 1,2, ...). (9.22)
I1epBoe paBeHcTBo B (9.21) MOMmo paccMaTpHBaTb K3K ell1.e O.n:HO ,AO-
Ka3aTeJIbCTBO OTCYTCTBH5I Bell1.eCTBeHHbIX KopHeH onpe,AeJI HTeJI5I , B03-
HHKaIOmero npH nocTpOeHHI1 o.n:HOpO,AHbIX perueHHH ,AJI5I KopOTKoro
QHJIHH,Apa If paBHoro, C TOIlHOCTbIO ,AO k
,
bIpa}f{eHHIO t n [1241-
3HalleHH5I CYMM KoaqxpmmeHToB 6ecKoHellHoH CHCTeMbI (9.19),
onpe,AeJI5IeMbIe paBeHcTBaMH (9.21), If OQeHKH (9.22) ,AJI5I tPYHKQm'i 1jJ (n)
H fP (j) YKa3bIBaIOT, IlTo paccMaTpHBaeMa5I CHCTeMa Co,Aep}f{I1T nOJIO-
}f{HTeJIbHbIe KoacpcpmlHeHTbI H 5IBJI5IeTC5I perYJIHpHoH. Y CJIoBIie cYll1.e-
CTBOBaHHH OrpaHl1qeHHoro perneHH5I perYJI5IpHoH CHCTeMbI HaJIaraeT
onpe,AeJIeHHbIe orpaHIiqeHH5I Ha xapaKTep y6bIBaHIi5I ee CBo60,AHbIX
IlJIeHoB (3.3). I10JIYlleHHbIe Bblpa}f{eHH5I .n:JI5I 1jJ (n) Ii fP (j) nOKa3bIBaIOT,
qTO orpaHHlleHHoe perneHHe CIiCTeMbI (9.19) CYll1.eCTByeT, eCJIH.n:JI5I <D

H W j CnpaBe,AJII1BbI OI{eHKH <D" = 0 ( n\ ) HP j = 0 ( ;; )- C.n:eJIaHHoe npe.n:-
nOJIo}f{eHHe 0 napHocTH KaCaTeJIbHbIX Hanp5I}f{eHHH Ha pe6pe H OQeHKH
(9.9) KoacpcpIiQHeHTOB pH,AOB tPYHKQHH, 3a,AaIOll1.HX BHeruHHe HarpY3KH,
no3BOJI5IIOT JIerKo nOKa3aTb, IlTo Tpe6yeMbIH nop5I,AoK y6bIBaHH5I CBo-
60,ABbIX IlJIeHoB cymecTByeT.
CHcTeMa (9.19) C orpaHHIleHHbIM perueHIieMBnOJIHe aHaJIorHqHa
CHCTeMe (5.15), nOJ1yqeHBOH npH pacCMoTpeHHIi nepBoH oCHoBHoH 3a-
,AaQH ,AJI5I np5IMoyroJIbHHKa. Ha,AJIe}f{all1.eH 3aMeHOH HeH3BeCTHbIX ee
TaK}f{e MOMma npHBeCTH K BnOJIHe perYJI5IpHOH, noJIYQHB TeM caMbIM
KaK .n:OKa3aTeJIbCTBO e,AHHCTBeHHOCTH ee perueHHH, TaK Ii B03MO}f{HoCTb
OTKa3aTbC5I aT CtPoPMYJIHpoBaHHbIX npli nOJIYQeHIiIi cooTHorueHHH (9.9)
Tpe60BaHHH rJIa,AKOCTH tPYHKQHH B rpaHIiQHbIX YCJIoBHHX (9.6). Ok
HaKo, KaK nOKa3bIBaeT onbIT perueHIIH 3a.n:a'IH .n:JIH npHMoyrOJIbHIiKa,
HaJIo}f{HB HeKoTopbIe orpaHHQeHH5I Ha cpYHKWiH B rpaHHIlHbIX YCJIOBII5IX,
MbI nOJIYIlHJIII Bo3Mo}f{HOCTb YCTaHoBIiTb aCIiMnToTHQeCKHe CBoHCTBa
HeII3BeCTHbIx, 3HaHIIe KoTopbIX 5IBJIHeTC5I OCHoBoH nOCTpoeHH5I 3cpcpeK-
THBHOro MeTo,Aa KoJIHlleCTBeHHoH HHTepnpeTaUIIIi rroJIYQeHHbIX tPoPMYJI.
HaH,AeHHa5I npII paCCMOTpeHIIH .n:ecpopMaQIiH QHJIllH,Apa KOHellHoll
,AJIHHbI 6ecKoHellHa5I CHCTeMa (9.19) HaCTOJIbKo 6JIH3Ka K cooTBeTcTBY-
lOll1.eH CHCTeMe .n:JI5I np5IMoyrOJIbHIiKa, 'ITa CnpaBe,AJIlfBoCTb 3aKOHa
aCHMnTOTH'IeCKHX BbIpa}f{eHHH .n:JI5I HeH3BeCTHbIX x" Ii Y/ OQeBH,AHa.
I1o,Ap06HOe ,AoKa3aTeJIbCTBO :)Toro CBoHcTBa Hell3BeCTHbIX CO,Aep}f{llTCH
B pa60Te [73J. B CB5I3H C 3THM npH tPaKTHlleCKO,M perueHHH CHCTeMbI (9.19)


98


,AJIH KOHKpeTHbIX KpaeBbIX YCJIOBHll 6Y,AeM yqHTbIBaTb, 'ITO
Iim x n = Iim Yj = a o .
n-+oo ;-+00


(9.23)


COBna,AeHHe npe,AeJIbHbIX 3Ha'IeHHH HeH3BeCTHbIX Co3,AaeT npe,AnoCbIJIKH
,AJIH nOJIYIleHH5I 3qxpeKTHBHoro perueHHH C noMoll1.bIO MeTo,Aa JIHMHTaHT.
B CJIe,AYIOmeM naparpatPe npHBe,AeHbI pe3YJIbTaTbI perueHH5I KOHKpeT-
HoH KpaeBoH 3a,AaIlH, nOJIyqeHHbIe 3THM MeTO,AOM.



 10. TenJIOBbIe HanpmKeHHH B
HJIHH
pe
KOHe'lHOii
JIHHbI


BbI60p HepaBHoMepHoro HarpeBa B KalleCTBe tPaKTopa, o6YCJIOBJIH-
BaIOmero HanpH}f{eHHo-,AecpopMHpoBaHHoe COCTOHHHe QllJIHH,Apa, B
HeKoTOpoH CTeneHH o6"bHCHHeTcH CneQHtPHKoH TenJIOBbIX H Boo6ll1.e Bcex
HOCHll1.HX o6"beMHbIH xapaKTep Bo3,AeiIcTBHH. B Tex CJIYllaHx, Koma
HanpH}f{eHHoe CoCTOHHHe B npoCTpaHcTBeHHoM ynpyroM TeJIe Bo3HII-
KaeT no,A ,AeHCTBHeM HepaBHoMepHoro HarpeBa, QeHTp06e}f{HbIX CHJI
H ,ApyrHx aHaJIorH'IHbIX tPaKTopoB, BbI60p ynpOll1.eHHbIx paclleTHbIx
cxeM 6bIBaeT 3aTpY,AHeH. I1pH ,AeHCTBHH 3THX tPaKTopoB BJIHHHHe TOPQa
QHJIHH,Apa Ha HanpH}f{eHHoe COCTOHHHe, OlleBll.n:HO, CKa3bIBaeTCH npH
JII050H era ,AJIHHe. I1oMHMO 3roro, CneQHaJIbHbIM BbI60poM xapaKTepa
TCMnepaTypHoro nOJI5I Mo}f{Ho ,A06HTbCH Taro, qTo o,AHII H Te }f{e pe3YJIb-
TaTbI MorYT TpaKToBaTbC5I KaK OTHoCHll1.HeC5I K CJIYllaIO HepaBHOMepHoro
HarpeBa, TaKll K CJIYllaIO HarpY}f{eHH5I QeHTpo6e)KHbIMH CHJIaMH. I1MeH-
HO TaKoe TeMnepaTypHoe nOJIe H paCCMOTpHM. u
I1YCTb HanpH)KeHHoe COCTOHHHe QHJIHH,Apa KOHe'IHoH ,AJIHHbI o6y-
CJIoBJIeHO IlHCTo pa,AHaJIbHbIM TeMnepaTypHbIM nOJleM
T (r) = To + T 1 r 2 . (1O.l)


I1pe,AnoJIO)KHM, 'ITO nOBepXHOCTb QHJIHH,Apa CBo6o,AHa OT BHeruHHX
HarpY30K, T. e. cnpaBeJJ:JIHBbI CJIe,AYIOll1.He rpaHH'IHbIe YCJIoBH5I:
1 r) I rTJ ) ° I
20 (a r + a
 ) = 0, 20 (1: rz + 1: rz = npH r = ,


(10.2)


I (T) I ITJ ) 0
20 (a z + az ) = 0, 20 (T rz + 1: rz =
3,AeCb HH,AeKCoM (T) OTMelleHbI BeJIHqHHbI, oTBeqaIOmHe llaCTHoMY perne-
HHIO ypaBHeHHH .naMe ,AJIH HepaBHoMepHoro HarpeBa. B CoOTB:rCTBHll
C H3BeCTHbIMH pe3YJIbTaTaMH [1091, llaCTHoe perneHHe ypaBHeHHH .naMe
,AJIH HepaBHOMepHoro HarpeBa Mo)KeT 6bITb HaH,AeHo KaK 'IaCTHOe pe-
rueliHe ypaBHeHHH I1yaccoHa ,AJIH TepMoynpyroro nOTeHQHaJIa:
a 2 ID + 2.... aID + (
2ID = m + I arT, (10.3)
iJr
 r ar az 2 m-l


npH Z = :f: h.


I


I


r,Ae ar- Ko3tPcpHQHeHT JIHHeHHoro paCI1lllpeHH5I MaTepHaJIa QHJIHH,Apa.
.llJI5I TeMnepaTypHoro noJIH (10.1) llaCTHoe perueHHe ypaBHeHH H


7*


99




Be.n:eHbl .n:JIH CJIyqa
 h = 1, m = 3. OKa3aJIOCb, qTO B .n:aHHOM cJIyqae
rrpH paccMoTpeHHH OTBeqaIO/UeH (9.19) KOHeqHOH cH
TeMbI, co.n:ep.>Ka-
llI,eH JIHlllb no .n:eBHTb Hel13BeCTHblX X n H Yb y.n:aeTcH HaHTH Bce HeH3BeC!-
Hble B CHCTeMe (9.19) C TOqHOCTbIO.n:O 2,5%. 3HaqeHHH HeH3BecTHblx:_C
TOqHOCTbIO.n:O MHO.>KHTeJI
 K = m+ I I aTTl npHBe.n:eHbI B Ta6JI. 6.
tn-
B 3TOH Ta6JIHLle qepe3 X", Y i H X n , YI 0603HaqeHbl COOTBeTCTBeHHO
3HaqeHHH HeH3BeCTHblX C H36bITKOM H He.n:OCTaTKOM. nOCJIe.n:H

 cTpoKa
Ta6JIHLlbl co.n:ep.>KHT 3HaqeHHH
JIHMHTaHT, OrpaHHqHBaIO/UHX
cBepxy H cRlmy 3HaqeHH

Bcex HeH3BeCTHblX C HOMepaMH,
60JIblllHMH .n:eB
TH. OpH BblqH-
CJIeHl111 HarrpH.>KeHHH H CMe/Ue.
HHH B LlHJIHH.n:pe B KaqeCTBe
paCqeTHbIX BeJIHqHH K03<p<pHLlH-
eHTOB p
.n:OB eCTeCTBeHHO npH-
HHTb cpe.n:HHe 3HaqeHHH OT rrpH-
Be.n:eHHbIX B Ta6JIHLle. BeJIHqHHa
a o B (9.25) MO.>KeT 6blTb rrpHHHTa
pamlOH 0,7565.
Ope.>K.n:e qeM npHcTyrrHTb
K BblqHCJIeHHIO HanpH.>KeHHH,
He06xo.n:HMO, KaK H B cJIyqae
3a.n:aqH .n:JIH npHMoyrOJIbHHKa, npoBecTH yJIyqllleHHe CXO.n:HMOCTH
cooTBeTcTByIO/UHX 6eCKOHeqHblX PH.n:OB. XapaKTep CBH3aHHblX C
3THM BbIKJIa.n:OK nOKa.>KeM, pacCMaTpHBaH no.n:P06HO Bblpa.>KeHHe .n:JIH
Oz, KOTopoe C yqeTOM npoBe.n:eHHblx 3aMeH .n:JIH Ko3<fxPHLlHeHToB pH.n:OB
B (9.7) npH06peTaeT BH.n:
I _ 2 B m-I A _
20 Oz - m _ 2 0 + m - 2 0
_
 Yi h [ chAiz (1 + A.-hcthA.-h)-A..Z ShAjZ ] Jo(Ajr) +
L.J 41.,. sh A.h I 1 1 sh 'Aih J 0 ('Aj)
i=1 I I
+ "
 (_I)n [ k Jl(knr) +
(
r:!i12-k Jo(knr» ) ' ] COSkZ_
l.J x n 4k n n r /1 (k n ) /1 (knr) \ n /1 (k n ) n
n=1


(10.3) Haxo.n:HTCH 3JIeMeHTapHO H HMeeT BH.n:
CD = :+
 aT (+Tor2 + 1
 T l r 4 ). (104)
OTBeqaIOllI,He 3TOMY 3HaqeHHIO TepMoynpyroro nOTeHLlHaJIa qHCTO Ten-
JIOBble COCTaBJIHIO/UHe TeH30pa HarrpH.>KeHHH BblqHCJIHIOTCH no CPOPMY-
JIaM [l09]


I IT>_ m+I ( I T - I 2 )
20 Or - - m _ I aT T 0 + T Tlr ,
I m m+l ( 1 3 2 )
20 00 = - m _ 1 aT T To + T T1r ,
1 m_ m+1 2
20 Oz - - m _ I aT (To + T1r ).


(10.5)


KaCaTeJIbHbIe HanpH.>KeHHH, OTBeqaIOllI,He TepMoynpyrOMY rrOTeHIll1aJIY
(lOA), rO.>K.n:eCTBeHHO paBHbI HYJIIO.
TaKHM 06pa30M, YCJIOBHH (10.2) npHBo.n:HT K CJIe.n:YIOllI,eM rpaHHqHOH
3a.n:aqe .n:JI
 O.n:HOpO.n:HOro ypaBHeHHH JIaMe:
I m+l II I )
20 or(1,z)= m-I aT\TTo+TT 1 ,
1 _ m+ I 2
20 °2(r, :f::h)- m- I adTo+Tl')' (10.6)
1 1
20 Trz(l,z)= 20 Trz(r, :f::h)=O.
I1crroJIb3YH npe.n:cTaBJIeHl1e


co
r 2 = _ 2 1 + 4
 J o ('Ajr) ( 10.7 )

 'A 2J o ('Ai) ,
,n:JIH CBo60.n:HbIX qJIeHOB B 6eCKOHeqHOH CHCTeMe (9.19) Haxo.n:HM CJIe.n:yIO-
llI,He 3HaqeHHH:


"'" 0 m+ I 4
'Pn= ,qri=-
aTTl2 (10.8)
tn- '),;,
CooTBeTCTBeHHo CHCTeMa ypaBHeHI1H (9.18) .n:JIH 3JIeMeHTapHblX pellleHI1H
rrpl106peTaeT BI1.n:
m:'2 Bo + m Ao 2 = -} :+
 aT(To + -}-Tl) ,
(10.9)
m 2 2 Bo+ : ; Ao= :+: aT (To + -}T 1 ).
PellleHHe CHCTeMbI (9.19) npoBe.n:eM C HCnOJIb30BaHHeM MeTO.n:a, oc-
HOBaHHoro Ha OLleHKe rpaHHIl JIHMHTaHT. XapaKTep BblqHCJIeHHH, CBH-
3aHHbIX C peaJIH3BLlHeH 3TOro MeTo.n:a, no.n:P06HO onHcaH npH paccMoT-
peHHH sa.n:aqH .n:JIH rrpHMoyrOJIbHI1Ka. KOHKpeTHwe BWqHCJIeHHH npo-
100


Ta6J1Hua 6


n. ; I Xn lK I xn lK I Y;!K I Y;!K
1 0,2646 0,2641 1,2059 1,2050
2 0,6046 0,6027 0,8871 0,8848
3 0,7110 0,7077 0,8046 0,8009
4 0,7481 0,7434 0,7764 0,77 II
5 0,7632 0,7570 0,7654 0,7587
6 0,7700 0,7623 0,7608 0,7528
7 0,7732 0,7641 0,7595 0,7500
8 0,7748 0,7644 0,7593 0,7487
9 0,7756 0,7639 0,7597 0,7476
>9 0,7791 0,7439 0,7791 0,7439


il
II
I
I


co 00
_
 -, ch 'AjZ '\. -, J o (k"r) k
qJ2j -J o (A..r) + l... qJln -J- ( k ) COS n Z '
sh'Ajh I ,, = 1 1 n
;=1


(10.10)


"I


OOBe.n:eHHe K03cpcpHLlHeHTOB pH.n:OB B npHBe.n:eHHbIX Bblpa.>KeHI1HX
onpe.n:eJIHeTCH npe.n:eJIbHblM paBeHcTBoM (9.25). B CBH3H C 3THM OTHO-
CI1TeJIbHO xapaKTepa I1X CXO.n:HMOCTH OCTaIOTCH crrpaBe.n:JIHBblMI1 Bce
3aMeqaHI1H, c.n:eJIaHHble rrpl1 aHaJII13e COOTBeTCTBYIOllI,I1X CPOPMYJI B
3a.n:aqe .n:JIH npHMoyroJIbHHKa. HeKoTopble OTJIl1qHH B xapaKTep,e cxo-
.n:I1MOCTI1 Ha6JIIO.n:aIOTc
 JIHlllb B TOqKe r = 0 Ha TOpLle LlHJIHH.n:pa.


101




ECJIH B saJJ;aqe JJ;JIH npHMoyroJIbHHKa B cepeJJ;IfHe CTOpOH CXOJJ;HMOCTb
PHJJ;OB JJ;JIH HOpMaJIbHbIX HanpmKeHHH HMeJIa TaKOH )Ke xapaKTep, KaK H
(-w
CXOJJ;IfMOCTb pHJJ;a C 06IllHM qJIeHOM -n-' TO, paCCMaTpHBaH BbIpa)KeHIfH
JJ;JIH HOpMaJIbHbIX HanpH)KeHHH B (10.10) npH r = ° HZ = h, y6e)KJJ;aeM-
CH, qTO SJJ;eCb CXOJJ;IfMOCTb xapaKTepHsyeTcH pHJJ;OM C 06IllHM qJIeHOM
( _l)n
Yn .
MeJJ;JIeHHaH CXOJJ;HMOCTb PHJJ;OB B (10.10) Ha rpaHHI
e I
HJIIfHJJ;pa H
B6JIHSH ee SaTpYJJ;HHeT HenOCpeJJ;CTBeHHOe BWqHCJIeHHe HanpH)KeHIfH C
nOMOrn.blO YKaSaHHbIX <POPMYJI. 3HaHIfe aCIfMnTOTIfqeCKIfX CBOHCTB
K03<p<PHI
IfeHTOB PHJJ;OB nOSBOJIHeT SHaqHTeJIbHO YJIyqlllHTb If X CXOJJ;H-
MOCTb, a TaK)Ke npHnIfcaTb IlX CYMMaM <pIfSHqeCKH onpaBJJ;aHHbIe SHa-
qeHHH B Tex TOtJKax, B OKpeCTHOCTH KOTOpbIX OTCYTCTByeT paBHOMepHaH
CXOJJ;HMOCTb.
CJIeJJ;yeT OTMeTIfTb TaK)Ke OtJeHb MeJJ;JIeHHYIO CXOJJ;HMOCTb PHJJ;OB,
cOJJ;ep)Karn.HX Cf!2j If Cf!ln B BbIpa)KeHHHX JJ;JIH HOpMaJIbHbIX HanpH)KeHHH.
3HaqeHHe 3THX PHJJ;OB Ha pe6pe Bo06rn.e HeJIbSH HaHTH HenocpeJJ;cTBeH-
HO. aHO MO)KeT 6bITb nOJIyqeHO JIHlllb npeJJ;eJIbHbIM nepexoJJ;oM B BbIpa-
)KeHHH, 06'beJJ;HHHIOIlleM rJIaBHbIe qaCTH CYMM 060HX pHJJ;OB, onpeJJ;eJIHe-
MbIX Ha OCHOBe o
eHOK (9.9).
PaccMoTpHM BHaqaJIe qaCTb BbIpa)KeHHH u l" onpeJJ;eJIHeMYIO pHJJ;OM
no <PYHKI
HHM EeCCeJIH, H ooosHatJHM ee a
e):
00
1 Ce) _
 h [ Ch AjZ
2G U l' -.l..J Yj 4A' ShA'h (1 + AjhcthAjh)-
j=1 / /
-A.z ShAjl. ] Jo(Ajr) ( 10.11)
/ sh Ajh J 0 (Aj) .
3TOT pHJJ; 06JIaJJ;aeT YJJ;oBJIeTBOpHTeJIbHOH CXOJJ;IfMOCTblO BeSJJ;e, KpOMe
TOqeK, 6JIHSKHX K TOpIW L{HJIHHJJ;pa. CqHTaH r > ° H HCnOJIbsYH aCHMn-
TOTHqeCKHe BbIpa)KeHIfH JJ;JIH <PYHK
HH EeCCeJIH BeIlleCTBeHHoro apry-
MeHTa H KopHeH ypaBHeHHH J 1 (A) = ° [681. HaXOJJ;IfM npH 60JIblllHX
SHaqeHIfHX j aCHMnTOTIftJeCKoe paBeHcTBo
F ( ) h l chAjZ (I ShA'2; 1 J (A'r)
j r, z = Y; 4Aj sh Ajh + 'Ajh cth Ajh) - A;Z sh A:h - -lv A;r) :::::::
· a . -nCh-z) ( i+
 ) ( 1 )
:::::::Fj (r, z) =
Or h(- 1)/ e 4 -;- + h-z X
4r r In
. X cos[i:rtr- : (I-.r)]. (10.12)
l{JIH CYMMbI pHJJ;a no j OT F; (r, z) nOJIyqIfM cJIeJJ;YIOrn.ee SHaqeHHe:
00 n

. ah --Ch-Zlf 1 n
J(r,z)=.....Fj(r,z)=-L- e 4 1 --cos-(I-

 4

 4


_ h - Z e -nCh-z) f cos --.::. (I _ r) (cos :rtr + e -nCh-z» +
l) (r. z) 4
+ sin
 (I - r) sin :rtr n '
6 (r, z) = 1+ 2 cos :rtre- nCh - z ) + e-2nCh-z). (10.13)
u Ce)
C HCnOJIbSOBaHHeM 3Toro pesYJIbTaTa BbItJIfCJIeHHe cocraBJIHIOrn.eH a z
MO)KeT OblTb npOBeJJ;eHO CJIeJJ;YIOrn.HM 06pasoM:
M
2
 u
)=
 [Fj(r,z)-Fj(r,z)]+J(r,z). (10.14)
1=1
B 3TOM paBeHCTBe BeJIHtJIfHa M MO)KeT 6bITb JII06bIM tJHCJIOM, Bo06rn.e
rOBopH, npeBbIllialOrn.IfM qIfCJIO HeHSBeCTHbIX Yj' HaHJJ;eHHbIX no pellleHHIO
KOHeqHOH cHcTeMbI, OTBeqalOrn.eH (9. 19). B TaKOM CJIYtJae HeJJ;OCTalOrn.Ife
SHaqeHHH Yj B Fj (r, z) CJIeJJ;yeT npIfHHTb paBHbIMIf a o , T. e. cpeJJ;HeMY
.maqeHHIO JIIfMHTaHT. He06xoJJ;HMOCTb YBeJIHqeHIfH qHCJIa M MO)KeT
BOSHHKHYTb npH BbIqIfCJIeHHHX B TOqKaX, JJ;OCTaTOqHO 6JIHSKHX K TOqKe
r = ° npH z = h.
l{JIH BbIqHCJIeHHH SHaqeHHH HOpMaJIbHbIX HanpH)KeHHH B
eHTpe
TopI
a I
HJIHHJJ;pa paccMoTpHM BbIpa)KeHHe Fj (r. z) npH r = 0, z = h.
YqHTbIBaH


00
Jim (h - z)
 (- Ii VI e-jncn-z) = 0, (10.15)
l'-+h j=1
HaXOJJ;HM, qTO npIf JJ;OCTaTOqHO 60JIblllHX j <PYHKUHH FI (0, h) annpoKCIf-
MHpyeTcH CJIeJJ;YIOrn.HM BbIpa)KeHHeM:
aoh (- I)j
Pi (0, h)::::::: 4 Y2 . YT . (10.16)
CJIeJJ;oBaTeJIbHO, npH BbItJHCJIeHHH HanpH)KeHIfH CXOJJ;HMOCTb pHJJ;OB MO-
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I
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1 n sin nre-nCh-z)

r) In6(r, z) -- sin- (I -r) arctg-
n 4 1 + cas nre-nCh-z)


00
L (-
i =(V2-Ig(f)=-I,460(V2-I), (10.17)
j=1 YI
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00

 Cd) _ \., (_ I)n [ k 11 (knr) +
2G U l' -.:...oJ X n 4k n r I ( k )
n=1 n 1 n
lo(knr) (2 k 10 (k n ) )] k ( 10 18 )
+ 7;. (k;;) - n II (k n ) COS n Z . .
3aTPYJJ;HeHHH npH BbIqHCJIeHHIf SHaqeHIfH 3TOro pHJJ;a BOSHHKalOT JIHlllb
B HenoCpeJJ;cTBeHHoH OJIHSOCTH K nOBepxHocTH r = I H Ha caMOH nOBepx
HOCTH. I1cnOJIbsYH aCHMnTOTHqeCKHe BbIpa)KeHHH JJ;JIH <PYHKI
IfH Eecce-


102


103




00


M [ 2 ( l ) i ( 1 ) - ( 1 + ..!- ) 1t(h-Z} (
_ \' . ch Ai Z J ( A.r ) ' - 't2 e 4 cos J 'nr +
_ fJ!2/ shA.h 0 / T .
r:
i=1 I :n} II r
+
 r-
 )]= :}
ht In (1 +2cos
 ze-
 (l-r) +e- 2 : (l-r»)_
CP2 (l) -
 (h-z) [ :n ( 1 ) I ( 1 + 2 -1t(h-z) + e -21t(h-Z) } +
---;:- e cos""4 -r n cosnre
:nyr
sin we- 1t (h-z) ]
+ sin
 (1 - r) arctg 1 + cos :nre-n1h-z) .


(1O.22)


JHI, HaXO):{HM rJIaBHYIO lIaCTb K03(pqmUHeHTOB TpHroHoMeTpHlIecKoro PH-
,na (10.18): ..
L (r) = x ( 1)'1 f k r ..!. dkn r ) + /0 (knr) ( 2 _ k /0 (k n ) ) ' ]

'I 'I 4k n 'I /} (k n ) 11 (k n ) 'I I} (k n )

= L
 (r) = Go (-Jt e- knO - r ) [ 3r 2 + 8 6r -1 k n (1- r) ] . (1O.19)
4k n r r
TpHrOHOMeTpHlIeCKHH pH):{ C K03<p<pHUHeHTaMH L
 (r) MO:>KeT 6bITb npo-
CYMMHpOBaH TOIlHO:


S (r, Z) =
 L
 (r) cosknz =
'1=1


Goh
8:n y r


3r 2 + 6r-l
8r


( :n -
 (l-r) -2 .!:.. II-r» )
In 1 +2cosT ze h +e h +
:n _...:!. (l-r) -2!!. (1-r)
cos-ze h +e h
h


(1O.20)


YCTaHOBJIeHHble Y):{06HbIC npH npOBe):{eHHH BbIIIHCJJeHHH <POPMYJJbI
(10.14), (10.21) H (10.22) MorYT 6bITb HCnOJJb30BaHbI B JJI060H TOIIKe
06JJaCTH, 3a HCKJIIOlIeHHeM TOlleK OCH H pe6pa UHJIHH):{pa. fIpH BbIIIHC-
JIeHHH HanpH:>KeHHH B ueHTpe TOpua HCnOJJb3yeM 3HalleHHe CYMMbI pH.na
(10.17). <POpMyJIbI .nJIH BbIIIHCJIeHHH HanpH:>KeHHH B yrJJOBbIX TOIlKax
1\10rYT 6bITb nOJJYlIeHbI JJHlllb npe.neJJbHbIM nepexo.noM, npHlIeM B 3TOM
CJJYlIae COOTHOllleHHH (10.14) H (10.21) .nOJJ:>KHbI paccMaTpHBaTbcH COB-
MeCTHO. fIpH TaKOM paCCMoTpeHHH co):{ep:>Kall1.HeCH B HHX JJOrapH<pMH-
lIeCKHe oc06eHHoCTH B3aHMHO noralllaIOTCH. B3aHMHo noralllaIOTCH TaK-
:>Ke oc06eHHocTH, co.nep:>Kall1.HeCH B COOTHOllleHHH (10.22), eCJJH YlIecTb
napHOCTb KaCaTeJJbHbIX HanpH:>KeHHH Ha pe6pe, T. e. fJ!I (h)
 fJ!2 (1).
Ba:>KHO OTMeTHTb, liTO 3.neCb, KaK H B CJJYlIae 3a.nallH .nJIH npHMoyroJIb-
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He 3aBHCHT OT HanpaBJIeHHH, no KOTOPOMY COBeplllaeTCH npe.neJJbHblH
nepexo):{. LIJIH He06xo.nHMOH B COOTBeTCTBHH C (10.10) npH BbIIIHCJIeHHH
U z pa3HOCTH S (1, h) - J (1, h) Haxo.nHM 3HalleHHe
S (1, h) -J (1, h) =

 In h-

 . (10.23)


I
'I
'I


:n -
 (l-r) -2 !!. (l-r)
1+2cos-ze h +e h
h
I1cnOJJb30BaHHe 3Toro pe3YJIbTaTa .naeT B03MO:>KHOCTb npOBo.nHTb BbIIIHC-
JIeHHe COCTaBJIHIOIUeii U
d) no CJIe.nYIOll1.eH <popMyJJe:
N
1 (d)
 .
20 u 2 = ""'" [Ln (r) - Ln (r)] cos knz + S (r, z).
'1=1


+ Go (1 4 - r)


(10.21)


3.neCb N 06bIlIHO paBHo IIHCJIY HeH3BeCTHbIX x'" y.nep:>KHBaeMblx npH
oueHKe rpaHHU JIHMHTaHT.
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Hble npoBe.neHHbIM npH nOJIYlleHHH COOTHOllleHHH (10.14) H (10.21),
n03BOJIHIOT nOJIYIIHTb .nJIH BbIIIHCJIeHHH CJIaraeMblX C fJ!ln H fJ!2j B U z
CJIe.nYIOll1.ee paBeHcTBo:

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= n
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1 (k n hk n yr


-k (l-r) ] k
e 'I cos n Z -


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pa3pbIBOB B KaCaTeJJbHbIX HanpH:>KeHHHX. KaK H3BeCTHO [197], B Mec-
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k '\ ch AjZ .
COS n Z - ..- fJ!2j sh 'A .h J 0 (Air) =
i=1 1


104


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eTC5I B6JIH3H OCH I.{HJIHH.lI.pa. npH K
1
r = 0 yqeT CB060.ll.HblX TOpI.{OB CHH-
}KaeT 3HaqeHHe oceBblX HanpH}Ke-
HHH .lI.JIH Ky6006pa3Horo I.{HJIHH.lI.pa
nOqTH Ha 25 %, B TO BpeMH KaK
Ha nOBepxHocTH BJIHHHHe CBOook
HOro TOpI.{a npOHBJI5IeTCH B .lI.Ba
pa3a cJIa6ee.
HaH60JIblIlHH HHTepec B paccMa-
rpHBaeMoH 3a.ll.aqe npe.ll.CTaBJI5IeT
nOBe.ll.eHHe Hanp5I}KeHHoro COCT05I-
HHH B6JIH3H TOpI.{a IIHJIHH.lI.pa.
3.l1.eCb 06HapY}KHBaeTC5I p5Ill Ka-
qeCTBeHHblX OTJIHqHH B xapaKTepe
pacnpe.ll.eJIeHHH HanpH}KeHHH no cpaBHeHHIO C TeM, KaKoH Ha6JIIO-
.lI.aeTC5I B Cpe.ll.HeH qaCTH I.{HJIHH.lI.pa. 3TH OTJIHqH5I HJIJIIOCTPHPYIOTC5I
pHC. 9 H 10.
Ha pHC. 9 .lI.aHO pacnpe.ll.eJIeHHe HanpH}KeHHH (Je. ECJIH B Cpe.ll.HeM
CeqeHHH (z = 0) KpHBble TOqHOro H npH6JIH}KeHHoro pelIleHHH KOJIH-
HeapHbl, TO Ha TOpI.{e pacnpe.ll.eJIeHHe HanpH}KeHHH KaqeCTBeHHO HHoe.
B qaCTHOCTH, Ba}KHOH oc06eHHOCTbIO B nOBe.ll.eHHH Hanp5I}KeHHH (Je
HBJIHeTCH nOCTOHHCTBO HX 3HaKa .lI.JIH Bcex 0 -<: r -<: 1. CJIe.ll.yeT TaK}Ke
OTMeTHTb, qTO, KaK CJIe.ll.yeT H3 Ta6JI. 8, oKpecTHocTb pe6pa B IIHJIHH.lI.pe
5IBJI5IeTC5I 30HOH nOBbllIleHHblX Hanp5IJKeHHH (Je.
Pacnpe.ll.eJIeHHe pa.ll.HaJIbHblX Hanp5I}KeHHH B CeqeHHHX Z = 0 Ii
Z = h npellCTaBJIeHO Ha pHC. 10. llJI5I HHX TaK}Ke, KaK H .lI.JI5I HanpH}Ke-
HHH (Ja, xapaKTepHo cYIllecTBeHHoe pa3JIHqHe xapaKTepa H3MeHeHHH B
Cpe.ll.HHHOH nJIOCKOCTH H Ha TOpI.{e. O.ll.HOH H3 oco6eHHocTeH nOBe.ll.eHHH
Hanp5I}KeHHH (J r H (Je 5IBJIheTC5I TO, qTO HaH.lI.eHHble no TOqHOMY pelIleHHIO
HX BeJIHqHHbl 60JIblIle, qeM 3HaqeHH5I, OTBeqalOIllHe npH6JIH}KeHHOMY
pelIleHHIO.
3aKaHqHBaH paCCMOTpeHHe KOHKpeTHoro CJIyqa5I .lI.e
OpMHpOBaHH5I
I.{HJIHH.lI.pa, YKa}KeM, qTO OTMeqeHHble BbllIle oc06eHHOCTH nOBe.ll.eHH5I
HanpH}KeHHH Ha TOpI.{e I.{HJIHH.lI.pa nOJIHOCTblO coxpaHHIOTCH H npll YBe-
JIHIIeHHH ero .lI.JIIIHbl. nOCKOJIbKY npll 3TOM a6COJIIOTHble BeJIHqHHbl Ha-
npH}KeHHH Ha TOpI.{e He MeHblile IIX 3HaqeHHH B Cpe.ll.HeM CeqeHHH, TO
B03MO}KHOCTb npeHe6peqb BJIHHHHeM TOpI.{OB npH OI.{eHKe Hanp5I}KeHHO-
ro COCT05IHH5I I.{HJIIIH.lI.pa .lI.0JI}KHa YCTaHaBJIHBaTbC5I He TOJIbKO Ha OCHO-
Be ero reOMeTpHqeCKHX xapaKTepHCTHK, HO H Ha OCHOBe aHaJIH3a xa-
paKTepa HarpY}KeHHH.


YKa3bIBaeT Ha B3allMHoe noralIleHHe 3THX oc06eHHocTeH, MO}KeT pac-
CMaTpHBaTbCH KaK eIlle O.ll.HO .lI.OKa3aTeJIbCTBO cnoc06HOCTH npe.ll.JIarae-
MOro MeTo.n;a pelIleHH5I rpaHHQHblX 3a.ll.aQ .lI.aBaTb .lI.OCTOBepHYIO Ka-
QeCTBeHHYIO KapTHHY Hanp5I}KeHHoro COCTOHHHH ynpyroro TeJIa.
nOCJIe npOBe.ll.eHHoro TIllaTeJIbHoro aHaJIH3a
OPMYJI .n;JIH BblqHCJIe-
HHH Hanp5I}KeHHH H YJIyqlIleHH5I CXO.ll.HMOCTH BXO.ll.HIllHX B HHX p5IllOB,
()z 011
K; 1<,


8


8


4


0
r
-4- -4-


-8


-B


Pnc. 8.


Pnc. 9.


nO.ll.CQeT HanpH}KeHHH B I.{HJIHH.lI.pe He COCTaBJI5IeT TpY.lI.a. Pe3YJIbT8Tbl
BblqHCJIeHHH, npOBe.ll.eHHblX .lI.JI5I paCCMaTpHBaeMoro CJIyq85I TenJIOBblX
Hanp5I}KeHHH, npe.ll.CTaBJIeHbl B Ta6JI. 7-10 B HHX C TOqHOCTblO .lI.O
K G m+ I
MHO}KHTeJIH 1 = 8 m _ I aT T
 npellCTaBJIeHbl COOTBeTCTBeHHO 3Ha-
qeHHH (Jr' (Je, (Jz' Trz, KOTopble nOJIYQeHbl CYMMHpOBaHHeM BeJIHqHH,
onpe.ll.eJI5IeMblX no
opMYJIaM THna (10.10), H qHCTO TenJIOBblX COCTaB-
JIHIOIII;HX (10.5). ECTecTBeHHo, qTO npH rpaHHqHblX YCJIOBH5IX (10.2)
HanpH}KeHHH B I.{HJIHH.lI.pe He 3aBHC5IT OT BeJIHqHHbl To.
npHBe.ll.eHHble B Ta6JI. 7-10 .lI.aHHble nOKa3bIBalOT TaK}Ke CTeneHb
TbqHOCTH Y.ll.OBJIeTBOpeHIi5I rpaHHqHblM YCJIOBHHM. HaH60JIblIla5I no-
rpelilHocTb B Y.ll.OBJIeTBOpeHHH rpaHHqHblM YCJIOBH5IM HMeeT MeCTO B
yrJIOBOH TOqKe. OTHOCHTeJIbHa5I BeJIHqHH8 HeBH3KH B rpaHHqHblX
YCJIOBHHX B 3TOH TOqKe no cYIllecTBY paBHa OTHOCHTeJIbH.oH norpelilHOCTH,
xapaKTepH3YIOIlleH npe.ll.CTaBJIeHHOe B Ta6JI. 6 pelIleHHe 6eCKOHeqHOH CH-
CTeMbl. B OCTaJIbHblX TOqKaX nOBepxHocTH TOqHOCTb Y.lI.OBJIeTBOpeHHH
rp8HHqHblX YCJIOBHH 3HaqHTeJIhHO BbllIIe.
HeKoTOpble oc06eHHocTH Hanp5I}KeHHoro COCT05IHH5I I.{HJIIIH.lI.pa 60-
JIee HarJIH.n;HO MOJKHO npe.ll.CTaBHTb rpa
HqeCKH. Ha npHBe.ll.eHHblX HH}!{e


108


4-


o


-4


Pne. 10.


109




S 11. CMemaHHaH I'paHUqHaH sanaqa
PaccMoTpHM OCHOBHYlO cMeillaHHYIO rpaHHqHYIO 3a.l{aqy .l{JUI llH-
JIHH.l{pa KOHeqHOH .l{JIHHhl npH YCJIOBHH, qTO JIHHHfI pa3.l{eJIa rpaHIlqHhlX
YCJIOBHH COBna.l{aeT C yr JIOBOH oKpymHocThlO. ll.JIfI onpe.l{eJIeHHOCTH,
HMeH B BH.l{Y nocJle.l{YJOll1.ee IICCJle.l{OBaHHe H3rH6a meCTKO-3all1.eMJleHHOH
KpyraBoH nJlHThl, OCTaHOBHMCH Ha CJJyqae 3a.l{aHHH HanpHmeHHH Ha
Topllax llHJlHH.l{pa II CMell1.eHHH Ha era 60KOBOH nOBepxHocTH. ITo.l{06-
Hhle BhIKJla.l{KH npOBe.l{eM .l{JlH cJlyqaH aHTHcHMMeTpIIqHOrO OTHOCHTeJlh-
HO nJlOCKOCTII Z = 0 HanpflmeHHoro COCTOHHHH.
OTBeqmOll1.He aHTHcHMMeTpHqHoMY cJlyqalO .l{ecpopMupoBaHHH Bhlpa-
meHHH .l{JlH KOMnOHeHT BeKTOpa CMell1.eHIIH (9.2) H (9.4) 6hlJlII no-
CTpoeHhl B paCqeTe Ha B03MomHOCTh y.l{OBJleTBopHTh C HX nOMOll1.hlO rpa-
HHqHhlM YCJlOBIIHM Ha cooTBeTcTBYIOll1.I1X qaCTHX rpaHHllhl B CMell1.eHHHX
H RanpHmeHIIHX. B CBH3II C 3THM npH paccMoTpeHHH CMell1.eHHoH rpa-
Hl1qHOH 3a.l{aqH nOCTOHHHaH Bo MomeT 6hITh npHHHTa paBHOH HYJlIO.
CoxpaHHM, O.l{HaKO, BHaqaJle HeKOTOpyJO II36hlTOqHOCTh npOII3BOJla B
06ll1.eM peilleHHH rpaHHqHOH 3a.l{aqII, no.nyqeHHOM Ha OCROBe (9 2) H
(9.4) no COOTHOilleHIIHM (9.5). PaCnOpH.l{HBillIICh JlHlUHeH nocToHHHOH,
B .l{aJlhHeHilleM HeCKOJlhKO ynpocTIIM 6eCKOHeqHYlO cIIcTeMY .l{JlH on-
pe.l{e"leHHH K03cpcpIIllHeHToB pfl.l{OB B (9.2) H (9.4).
OTMeTIIM, qTO coxpaHeHIIe nOCTOHHHhlX Ao, Co, Bo B BhlpameHIIHX
(9.2) H (9.4) IIMeeT CMhlCJI JlIIillh npII onpe.l{eJleHHOM xapaKTepe npo-
.l{OJlmeHIm rpaHIIqHhlX YCJlOBIIH Ha nOBepXHOCTII ynpyrara CJlOH H 6ec-
KOHeqHOra llIIJlIIH.l{pa. ITOCKOJlhKY BapHalll1H cnoco6a npO.l{OJlmeHI1H B
paCCMaTpIlBaeMOM CJJyqae CMeillaHHOH rpaHHqHOH 3a.l{aqII He npIIBO.l{IIT
K KaKOMY-JlII60 ynpOll1.eHIllO COOTHOilleHIIH .l{JlH onpe.l{eJleHHH nOCTOHH-
HhlX Ai' C;, Dn II Bn> TO B .l{aJlhHeHilleM npIIMeM B KaqeCTBe A, KOpHII
ypaBHeHIIH J I 0,) = O. a B KaqeCTBe k,1 nOJlOmHM k n =
'It . ITpe.l{hI-
.l{Yll1. He paccym.l{eHIIH OTHOCHTeJlhHO Ao, Bo H Co KaK pa3 H OTHOCHTCfI
K OTBeqalOll1.eMY BhI6paHHhlM 3HaqeHIIHM k l1 II 'Ai cnoc06y npO.l{OJlmeHHfI
rpaHIIqHhIX YCJlOBHH.
06pall1.aHCh K 3aKOHY fYKa, npe.l{CTaBIlM COOTBeTCTBYIOll1.He CMell1.e-
HHHM (9.2) II (9.4) KOMnOHeHThI TeH30pa HanpHmeHHH B BH.l{e
I
 I
2 G ar = - 2C o z - 2Boz - "'" (Aiz ch At Z + C, sh'A;z) - J 1 (Air) +
,=1 r
+
1 (A,'A,z ch 'A,z +
 A, sh A;Z + C,A i sh A/Z) J 0 (A,r) +
+ n
1 [DII (klllo (knr) - + II (knr») + Bn (knr/l (knr)-
-
 _m - 2 10 (k"r) + 4 m k I Idknr) ) ] sin knz,
m m n , /


- A.A -z ch A.Z - C.A. sh A .z ) J o (A r) -
 f D"k..Io (knr) +
, , , ", , n=I
+ BII
 10 (knr) + Bnknrli (knr) j sin kllz.


o

 ao = - 2C o z - 3Boz +
 (A;z ch A;Z +
2G ;=1
00
I 2

+ C. ShA.Z) - J 1 (A.r) + - "'" A , ' sh A;ZJ o (A,r) +
, , r , m ;=1


(11 1)



 [ I m-2 I
+
 D" ----;-/1 (kllr) + ------;n- BII 0 (knr)-
n=J
- Bn 4 m k Ill (k,l) ] sin k"z,
m n'

-1:rz = - m-l Cor +
Bor + f [ A, (A,zshA;Z-
2G m m ;=1
- m - 2 ch A.;Z ) + C,A., ch A.;z j JdA./) +
 [ D"knli (kllr) +
m _I
m-I J
+ Bnkllrl 0 (kllr) - BII2 ------;n-/I (kllr) cos k.,z.
CO.l{epmall1.IIeCH B BhlpameHHHx .l{JlH CMell1.eHHH H HanpHmeHHH npOH:-
BOJIhHhle nOCTOHHHhle onpe.l{eJlHM H3 CJle.l{YIOll1.I1X rpaHl1qHhIX YCJlOBHH:


I I
2G a z (r, + h) = :f: It (r), 2G 1: r7 (r. + h) = 'PI (r),
u (1, z) = 12 (z), W (1, Z) = 'P2 (z),
r.l{e 12 (-z) = -12 (z). 'P2 (-z) = 'P2 (z). B .l{aJlhHeHillHX BhIKJla.l{KaX
BXO.l{Hll1.He B rpaHHqHhIe YCJlOBI1H CPYHKllHII npe.l{CTaBHM pH.l{aMI1


(11.2)


I
II
II'
I
!


""


Idr) = 110 +
 h,J 0 (A./),
;=1


'PI (r) =
 'PI;J I (A./),
'=1


(11. 3)


00 ""
12 (z) =
 12n sin knz, !j)2 (z) = 'P20 + 1: 'P?n COS k"z.
n=I n-I


""
] m-] 4 \--, ( m-]
2G a z = 2 ------;n- Coz - m Boz +
 2 --m- A, sh }.l z -
;-1


KaK H .l{JlH paCCMOTpeHHhlX Bhlille CJIyqaeB nepBoH OCHOBHOH rpa-
HIIqHOH 3a.l{aqH, HeKOTopble H3 rpaHIIqHhlX YCJIOBHH (11.2) npIIBe.l{eM K
npocThlM CBH3HM Mem.l{y HCKOMbIMH BeJlHqIIHaMH. B .l{aHHOM cJlyqae
raKIIMII YCJlOBIIHMH HBJlHIOTCH YCJlOBHH .l{JlH pa.l{IIaJlhHOrO CMell1.eHHH
u npH r = 1 II .l{JlH a npH Z = h. ITpII npOH3BOJIhHhlX 3HaqeHHHX Co
Z u
H Bo HeKOTOpaH onpe.l{eJleHHaH KOM6HHallHH 3TIIX BeJlHqHH BOH.l{eT B
coorHOilleHIIe, CBH3hIBalOll1.ee D n H Bn H nOJlyqeHHOe Ha OCHOBe YCJlOBHH
.l{JlH u. 1..JT06hl ynpocTIITh 3TO COOTHOilleHHe, Bhl6epeM npOH3BOJlhHO O.l{HY
H3 BeJIHqHH CO HJIH Bo TaK, qT06hl B 06ll1.eM peilleHHH .l{JIH U CTOflll1.He BHe


110


111




3TH ypaBHeHHH npH06peTYT BH

Xn
n = tl Yi

 A1 ( k

 A7 - m m I ) + Co
 -
- qJ2n (- It + f2n (- It


:j +
1 I n (A;)lli h (k

A
) ,
I (I I 9)
y.S. = \" x (
 m - I ) + !to , 2 _
1/ L.J 'I 12 + A
 k 2 + A
 - ---rn:- 2h A

'1=1 'I I 'I , I
co
- . 10(Aj) _ f . 10 (Ai) th
 .h- \" ( _ I ) n f 2kn
qJII A- II A' C ""I l..J 2'1 .2 + 02 '
I 1 '1=1 f<iJ. "'j



 =
 [ 16 (k n ) _ I _ 4 m - I 10 (k n ) ]
'I 4 Ii (k n ) mk n L/ 1 (k n ) ,
h ( A 'h )
Sj = 4A- cth Aih - Sh/A-h .
I I
YpaBHeHHe
JIH onpe
eJIeHHH Ao B HOBblX 0603HaqeHHHX HMeeT BH

C<) 00
Ao=cp20-Bo-C o ;3 + m 2m I
 Y! -Lf2j 10
Ai) . (1I.'I1)
;=1 Ai i=1 Aih
AHaJIH3HpYH paccMOTpeHHble paHee 3a.ll.aqH
JIH npHMoyroJIbHHKa H
I1.HJIHH
pa, 3aMeqaeM, 'ITO ycnex B HaXO>K)J:eHHH
ocToBepHbIx KOJIH-
qeCTBeHHoH H KaqeCTBeHHoH KapTHH HanpH>KeHHOra COCTOHHHH 06y-
CJIOBJIeH ypoBHeM 3HaHHH xapaKTepa nOBe
eHHH HeH3BeCTHbIX B 6ecKo-
HeqHbIX cHcTeMax, nOpO>K)J:aeMblx cooTBeTcTBYIOIlI.HMH rpaHH'IHblMH
3a
aqaMH. PaCCMaTpHBaeMaH 3
ecb CMelllaHHaH rpaHHqHaH 3a
aqa He
npe
CTaBJIHeT B 3TOM CMblCJIe HCKJIIOqeHHH. HCCJIe
OBaHHe nopO}K
ae-
MOH eIO 6eCKOHeqHOH CHCTeMbl (I I .9) Ha'IHeM C
OKa3aTeJIbCTBa perYJIHp-
HOCTH.
LlJIH aHaJIH3a perYJIHpHOCTH CHCTeMbI (I I .9) Heo6xo
HMO OI1.eHHTb
CJIe
YIOIlI.He BeJIHqH
b
 I )
 I k
 - (m - 1) A1 /
Q (J) - 1ST ...... (Il + A
)2 '
1 '1=1 m 'I 1
00 I ..2 2 1
I " f<iJ.-(m-I)A j
P (n) = TI";;T
 m (k 2 + A
)2 '
1=1 'I 1
npH OI1.eHKe BeJIHqHHbI Q (j) McnOJIb3yeM CJIe
YIOIlI.He 3Ha'IeHHH 6eCKO-
HeqHbIX CYMM:
00


3HaKOB CYMMHpOBaHHH CJIaraeMble 06paIll.3JIHCb B HYJIb, T. e.
-2 m-I C -2 m-2 B =0.
mOm 0


(11.4)


BbITIOJIHHH rpaHHqHble YCJIOBHH
JIH U H U z' HaXO
HM CJIe
YIOIlI.He
aJIre6paHQeCKHe COOTHOllleHHH:
m-I 4
2 - Coh - - Boh = fl0'
m m
D = B f 4
- 10 (k n ) ] +
 (11.5 )
'I 'I mk n II (k n ) II (kn> '
C i A i=AI ( 2 m-I - A.hcthA.h ) -

mil sh Ajh .
nepBoe ypaBHeHHe :JTOH CHCTeMbI H ypaBHeHHe (I I _4) 0
H03HaQHO on-
pe
eJIHIOT nOCTOHHHbIe Co H Bo Qepe3 HecaMoypaBHoBellleHHYIO QaCTb
HOpMaJIbHOH HarpY3KH H8 TOpI1.e I1.HJIHH
pa:
C = m-2 , flo B - _ flO ( II .6)
o m - I 2h' 0 - 2h'
fpaHHQHble YCJIOBHH
JIH oceBblX CMeIll.eHHH H KaCaTeJIbHblX HanpH-
>KeHHH B (I I .2) nopO>K
aIOT cHcTeMY tPYHKI1.HOHaJIbHbIX ypaBHeHHH.
LlJIH npHBe
eHHH ee K aJIre6paHQeCKHM COOTHOllleHHHM HC n OJIb3YIOTCH
npe
CTaBJIeHHH mnep60JIHQeCKHX tPYHKI1.HH TpHroHOMeTpHQeCKHMH PH-

aMH (5.6), a TaK>Ke CJIe
YIOIlI.He paBeHCTBa:


r
e


co
Z2=.!!!.. + \" 4(-I)ncoskz
3 - k2 'I (-h
z
h),
'1=1 'I


r = _ y
, /1 (Air)
;::1 Ai IdAi) (0
 r < I),


00 (I I. 7)


/ 1 ( k n r ) = - \
 2Aill (k n 2 ) 11 (Ar) 0
_ .2 (
 r < I),
i=1 (
+ Ai) J 0 (Ai)


00
k rl ( k r ) = _ \' 2Ajk n / o (k n ) 1 1 (J..f) +
nOn
.2 2
i=1 (1<" + A;) /0 (Ai)
\.
 4Ai
/1 (kn> /1 (Air)
+
 2 2 (0
 r < I),
;=1 (kn. + A i )2 10 (Ai)
J1cnOJIb30BaHHe tPOPMY JI (5.6) H (I I. 7) n03BOJIHeT 3aMeHHTb tPYHK-
I1.HOHaJIbHble ypaBHeHHH, BbIpa>KaIOIlI.He rpaHHQHbIe YCJIOBHH
JIH oce-
Bora CMeIll.eHHH H KaCaTeJIbHOro HanpH>KeHHH, aJIre6paH'l.e':KHMH_ C YQe-
TOM COOTHOllleHHH (I1.5) nOCJIe 3aMeHbi HCKOMbIX BeJIH'IHH no tPop-
MYJIaM




 h ( Aih )

 (fi2 + A
)2 = 4Ai cth Aih - sh2 Ai h = Si'
n I


'1=1


X n = 4 (- It B,,11 (k n ), Yj = -
 Aj sh AihJ o (A j )
112


f A1 h ( th A h Ai h 2 )
=1 (
+ A1)2 = 4Aj C i + sh 2 Ajh - Ai h '
113


(11.8)


8 7-2464




I


(1 I. 10)


(I I. 12)


'"


(11.13)


I I




3TH paBeHCTBa JIerKO YCTaHaBJIHBaIOTCH Ha OCHOBe CBOHCTBa paBHOMep-
HOH CXO,n:I1MOCTI1 pH,n:OB B (5.6).
3HaqeHI1e nepBoH CYMMbI B (I1.13) YKa3bIBaeT, qTO Sj > 0 (j =
= 1, 2, ...). B CBH311 C 3TI1M ,n:JIH Q (j) HaXo,n:I1M oueHKY
chA'h-
-2 m-I ( shAjh _
 )
1 sh Ajh m Ajh sh Ajh
Ajh
ch Ajh - Shr:h
I


YCJIOBl1e perYJIHpHOCTI1 ,n:JIH nepBoH rpynnbI ypaBHeHI1H B (I 1.9)
6y,n:eT BbInOJIHeHO, eCJII1 cnpaBe,n:JII1BO HepaBeHcTBo


Q(j)<


< 1. (I 1.14)


2 2 m-I 2 2
'I (k n ) -/ 0 (k n ) +
 10 (k n ) 11 (k n ) + --;-2/1 (k n ) > O. (I 1.19)
n

npl1 m:;? 3 cnpaBe,n:JII1BOCTb ::noro HepaBeHcTBa BbITeKaeT 113 cpaBHe-
mHI BblpameHI1H B ero JIeBOH '1aCTI1 C CYMMOH BTOporo pH,n:a B (11.17).
J
JIH ,n:OKa3aTeJIbCTBa HepaBeHcTBa B o6r.u;eM cJIY'Iae 6y,n:eM I1CXO,n:I1Tb H3
IIpe,n:cTaBJIeHI1H B BH,n:e pH,n:a np0113Be,n:eHI1H <PYHKUI1H DeCCeJIH [67]:
1m (x) In (x) =
m+n co
( X ) '\.-:' (m + n + 2s)1
= 2""" LJ sf (m + s)! (n + s)! (m + n + s)!
5=0
C I1cnOJIb30BaHI1eM 3Toro npe,n:CTaBJIeHI1H <PYHKUI1H B JIeBOH '1aCTI1 He-
paBeHCTBa (11.19) MomeT 6bITb 3anl1CaHa B Bl1l{e CJIe,n:YIOr.u;ero CTeneHHO-
ro pH,n:a:


(
 t.


(11.20)


nOCJIe,n:Hee HepaBeHCTBO HBJIHeTCH CJIe,n:CTBl1eM O'leBI1,n:HbIX HepaBeHCTB:


sh Ajh Ajh 0
-'::;h- - ShAjh > ,
(11.15)
ChAh-
-2 m-I ( shAjh _
 ) 0
j sh A;h m A;h sh Ajh > .
II:JIH 6oJIbIlll1X HOMepoB j 113 (11. 14) nOJIyql;lM
Q (j) < 1 - 2 m m 1 A
h + 0 (e- 2A /,), (I I.I6)
T. e. C pOCTOM j pa3HocTb Mem,n:y Q (j) 11 e,n:I1Hl1ueH CTpeMI1TCH K HYJIlO
H, CJJe,n:OBaTeJIbHO, OTHOCI1TeJIbHO K03<P<PI1Ul1eHTOB BTOpOH rpynnbl ypaB-
HeHI1H B (11.9) CHCTeMa HBJIHeTCH npOCTo perYJIHpHOH.
OUeHKY BeJIl1'1HHbl P (n) npOBe,n:eM C HCnOJIb30BaHI1eM paBeHCTB

 k
 _
 [ I
 (k n ) - 1 -
 J

 ( 1?. + A
 ) 2 - 4 1 2 ( k)' 1?.'
1=1 nIL n n

 A7 _
 [ 1- I
(kn) +
 lo(k n ) J
(;;:1 (
+ 1..7)2 - 4 I
 (k n ) Rn 'dkn) .


(11. 17)


00
'\.' (2s)! [ 2 5
L (SI)3 (I + s)/ (m - 2) s + 2""" (m - 2) s +
5=0
++s+m-2](
n t. (11.21)
Tenepb cnpaBe,n:JIHBOCTb HepaBeHCTBa (II 19) ,n:JIH JIlo6oro m :;? 2 HB-
JIHeTCH O'leBI1,n:HOH.
TaKI1M 06pa30M, YCTaHOBJIeHO, '1TO YCJIOBI1H perYJIHpHOCTI1 BbInOJI-
IIHIOTCH 11 ,n:JIH K03<p<pI1Ul1eHTOB nepBOH rpynnbI ypaBHeHI1H (11.9) H,
CJle,n:OBaTeJIbHO, 6eCKOHe'lHaH CI1CTeMa (11.9) HBJIHeTCH perYJIHpHOH.
PaCCMaTpl1BaH oueHKY ,n:JIH P (n), npl1 ,n:OCTaTO'lHO 60JIbIlll1X HOMepaX
n nOJTyql1M
P (n) < 1 - 2 (
C: ;)
 m + 0 ( :J . (1 1.22)
9TO HepaBeHCTBO nOKa3bIBaeT, '1TO ,n:JIH JII06oro m > 2 OTHOCHTeJIbHO
CI1CTeMbl (11.9) cnpaBe,n:JII1BO 60JIee CI1JIbHOe YTBepm,n:eHl1e, a I1MeHHO:
paCCMaTpl1BaeMaH CI1CTeMa HBJIHeTCH BnOJIHe perYJIHpHoH. AHaJIH3 ee
C!3060,n:HbIX '1JIeHOB nOKa3bIBaeT, '1TO Heo6xo,n:I1MbIH ,n:JIH cyr.u;ecTBoBaHHH
e,n:I1HCTBeHHoro OrpaHi1'leHHOro perneHI15I nopH,n:OK I1X y6bIBaHI1H o6ec-
IIC'II1BaeTCH npe,n:nOJIOmeHl1eM 0 KYCO'lHOH HenpepbIBHOCTI1 <PY HK UI1H,
BXOl{5Ir.u;I1X B rpaHUqHble YCJIOBIUI (11.2).
Y CTaHOBI1B BnOJIHe pery JJ5IpHOCTb CI1CTeMbI (11.9), nOJIY'll1M cyr.u;e-
nB08aHI1e e,n:I1HCTBeHHoro ee peIlleHI1H, OTHOCI1TeJIbHO KOToporo 113-
BeCTHO JII1Illb, 'ITO BeJIl1'1l1HbI X n 11 Yj HBJIHIOTCH OrpaHl1qeHHbIMH. JIerKo
IIOKa3aTb, qTO TaKOH I1H<popMaUlfl1 OTHOCI1TeJIbHO Hel13BeCTHbIX B (11.9)
lIel{OCTaTO'lHO ,n:ame ,n:JIH TOro, '1To6bI YTBepm,n:aTb Cyr.u;eCTBoBaHl1e peIlle-
IIIIH paccMaTpHBaeMoH CMernaHHOH rpaHl1'1HOH 3a,n:a'lH B <POp Me (9.2) If
(9.4). II:JIH 3TOro paCCMOTpI1M, Hanpl1Mep, npl1 r = 1 TY '1aCTb B 8blpa-
IKCIII1If ,n:JIH 1: rz B (11.1), KOTopaH npe,n:cTaBJIHeTCH TpHroHOMeTpl1'1eCKI1M
I'U,n:OM. Y'IHTbIBaH COOTHOIlleHI1H (11.5) H CBH3b Mem,n:y En If X n (11.8),


3TH paBeHCT8a HaXO,n:HM, paCCMaTpl1BaH paBHOMepHO CXO,n:H
l1eCH Ha
HHTepBaJIe 0'" r '" 1 pa3JIOmeHHH (9.12) B TO'lKe r = 1. Y'IHTbIBaH.
m-I 2 2
qTO 4--:;? npH m:;? ,Ha OCH08e cpa8HeHHH BblpameHHH ,n:JIH !1 n co
m
3HaqeHHeM CYMMbI BToporo pH,n:a B (11.17) 3aKJIIO'IaeM, qTO !1 n < 0
,n:JIH Bcex n = I, 2, ... Tor,n:a ,n:JIH OueHKI1 BeJIH'II1HbI P (n) nOJIyql1M He-
. paBeHCTBO


2
=1--.
m


)

 + (m - I) 1..7
P (n) < - 6. n
 m (k2 + A
)2 =
1=1 12 I
2 2 m-I 2 2
II (kn> -/ 0 (kn> + -/ 0 (k n ) '1 (kn> + 2""/ 1 (kn>
k n k n
2 2 m-I
II (k n ) -/ 0 (k n ) + 4 - k 10 (k n ) Idk n )
m n


(1 I.I8)


Wi


tj"'


115


I"




Haxo,n:HM


HCCJIe,n:OBaHIUI 6eCKOHeqnOH CIICTeMbI (11.9), a Ha OCHOBe aHaml3a
<pII3IIqeCKOrO co,n:ep}KaHIISI KpaeBOll 3a,n:a 1 UJ. B CBSI3II C 3THM no OTHOIlle-
HIIIO K HeII3BeCTHbIM B CIICTeMe (11.9) OHII Bblpa}KaIOT nOKa He TO, KaKII-
MII 3'fH HeII3BeCTHble eCTb, a KaKIIMII OHII ,n:OJI}KHbI 6bITb. O,n:HaKO, He-
CMOTpSI Ha 3TO, HaJIHqIIe COOTHOIlleHHH (11.24) II (11.25) 3HaqHTeJIbHO
ynpOIuaeT 3a,n:aqy IICCJIe,n:OBaHHSI CIICTeMbI (11.9). LLeHCTBHTeJIbHO, <p II -
3H4eCKH onpaB,n:aHHble npe,n:nOJIO}KeHIISI 0 nOBe,n:eHIIH HeH3BeCTHblX CBO-
,n:SIT YKa3aHHYIO 3a,n:aqy K npoBepKe CnpaBE',n:JIIIBOCTII oueHOK (11.24)
II (11.25), HaXO}K,n:eHIIIO BeJIHqIIH ex II
 II YCTaHOBJIeHIIIO CBSI3II ME'}K,n:y
nOCTOSIHHbIMII a o II bu.
PaccMoTpIIM aCIIMnTOTH4eCKHe paBeHcTBa. BbITeKaIOw.He H3 CIICTeMbI
(11.9) npII yqeTe CooTHoIlleHIIH (11.24) II (11.25). YqHTbIBaSI, qTO B
COOTHOIlleHIISIX ex < 1 11
 < 1, II coxpaHSISI B CIICTeMe JIHIllb rJIaBHble
qJIeHbI, Haxo,n:IIM


;0 T: z (r = 1) =
= m 2m I n
l (- It X n cos knz +
l k n f2n cos knz. (11.23)
3,n:ecb «*» YKa3bIBaeT Ha TO, qTO paCCMaTpIIBaeTCSI HenOJIHOe Bblpa}KeHIIe
,n:JISI KaCaTeJIbHbIX HanpSI}KeHIIH.
06ru.ee IICCJIe,n:OBaHIIe xapaKTepa CXO,n:llMOCTII pSI,n:OB ,n:JISI HanpSI}Ke-
HHH II cMeru.eHIIH B UIIJIHH,n:pe npoBe,n:eM B cJIe,n:YIOru.eM naparpacpe. ¥13
Bblpa}KeHIISI (11.23) CJIe,n:yeT, 4TO YCJIOBIISI OrpaHIIqeHHOCTII K03<pcpll-
UIIeHTOB x n He,n:OCTaTOqHO ,n:JISI YTBep}K,n:eHIISI B03MO}KHOCTII BblqHCJIeHIISI
KaCaTeJIbHbIX HanpSI}KeHIIH Ha nOBepxHocTll r = 1. 3TO 06CTOSITeJIbCTBO
3aCTaBJISIeT npe,n:npllHSITb 60JIee Tru.aTeJIbHbIH aHaJIII3 CIICTeMbI (11.9)
C ueJIbIO nOJIyqeHIISI 60JIee nOJIHOH IIHcpopMauIIII 0 nOBe,n:eHIIII HeII3BeCT-
HbIX B HeH.
LlTo6bI nOJIyqIITb ,n:aHHble 06 aCIIMnTOTIIqeCKIIX CBOllCTBax HeII3BeCT-
HbIX B CIICTeMe (11.9), 6y,n:eM IICXO,n:IITb II3 cpIl311qeCKIIX oc06eHHocTeH
paccMaTpHBaeMoH 3a,n:aqH. H3BecTHo [181, 277], qTO B cMeIllaHHoH rpa-
HllqH0I1 3a,n:aqe B TOqKaX CMeHbI xapaKTepa rpaHHqHbIX YCJIOBIIH, T. e.
B ,n:aHHOM cJIyqae B yrJIOBbIX TOqKaX, KOMnOHeHTbI TeH30pa HanpSI}KeHIIH
,n:OJI}KHbI IIMeTb oc06eHHocTb. EO"1ee Toro, caM xapaKTep oc06eHHOCTII
MO}KHO CqIITaTb 113BeCTHbIM Ha OCHOBe peIlleHHSI CMeIllaHHOll 3a,n:aqII
,n:JISI qeTBepTbnJIOCKOCTII [181] II 06ru.IIX IICCJIe,n:oBaHIIll xapaKTepa Ha-
npSI}KeHHo-,n:ecpopMHpoBaHHoro COCTOSIHIISI nJIIIT B oKpecTHocTII pe6ep
[11, 105]. 3TO 06CTOSITeJIbCTBO n03BOJISIeT c,n:eJIaTb BbIBO,n:, qTO, Hanpll-
Mep, B (11.23) K03cpcpIIUIIeHTbI ,n:OJI}KHbI y6bIBaTb He 6bICTpee qeM
 (0 <
k

< ex < 1). HaJIIIqlle oc06eHHOCTII JIIIIllb B yrJIOBbIX TOqKaX YKa3-bIBaeT,
qTO aCl1rvmTOTllqeCKOe Bblpa}KeHIIe ,n:JISI X I1 ,n:OJI}KHO IIMeTb BII,n:


3m-4 .
_
4m k Ct + I -
n
Go ( I+B m-I I )
= k f3 + 1 4 . I + B - -2m' . I - B '
n sm
n sm
n
Go b o ( I - a m - I I )
4j.,f3+ 1 = A,Ct+1 . I-a -
'. I-a .
i i 4sm -2- n sm -2-- n
flOJIyqeHHble aCIIMnTOTIIqeCKIIe paBeHcTBa Cyru.ecTByIOT, eCJIII CBO-
60,n:Hble qJIeHbI B 6ecKoHe4HoH CHCTeMe IIMeIOT nopSI,n:oK y6bIBaHIISI
1/kk II I/,,
 COOTBeTCTBeHHO. JIeBble qaCTII paBeHcTB nOJIyqaIOTCSI 3JIe-
MeHTapHo C IICnOJIb30BaHlleM aCIIMnTOTII4eCKIIX Bblpa}KeHIIH ,n:JISI f).n II
Sj' LITO KaCaeTCSI npaBbIX 4aCTeH, TO rJIaBHble qJIeHbI 6eCKOHeqHbIX pSI,n:OB
BblqIICJIeHbI Ha OCHOBe COOTHOIlleHIISI [144]


(] 1.26)


X---'1-.3L
n k

r,n:e b o - HeKoTopaSI nOCTOSIHHaSI nO.ITfJ}KHTeJIbHaSI IIJIH oTpIIUaTeJIbHaSI
BeJIllqllHa.
AHaJIOrII4Hble paccY}K,n:eHIISI, npoBe,n:eHHble OTHOCIITeJIbHO BeJIllqIIH
Yj' ,n:aIOT ,n:JISI HIIX cJIe,n:yIOru.ee aCIIMnTOTIIqeCKOe Bblpa}KeHIIe:


to<ex< I),


(} 1.24)


lim f -}- f (
 ) = S f (x) dx. (11.27)
k.....co n=1 0
H3 paBeHcTB (11.26) npe}l(,n:e Bcero CJIe,n:yeT, 4TO ex
. KpoMe TO-
ro, cyru.ecTBoBaHlle OTJIIIqHbIX OT HyJISI BeJIllqIIH au II b u B03MO}KHO JIIIIllb
,n:JISI 3HaqeHllH ex, 06paw.aIOw.Hx B HYJIb onpe,n:eJIHTeJIb O,n:HOpO,n:HOH
CIICTeMbI (11.26). 3TO YCJIOBHe npHBo,n:HT K cJIe,n:YIOw.eMY TpaHcueH,n:eHT-
HOMY ypaBHeHIIIO ,n:JISI onpe,n:eJIeHIISI ex:
a 3m-4
41< sin 2 """2 n + 4ex 2 - (x + 1)2 = 0, X = m 01.28)
YpaBHeHIIe (11.28) COBna,n:aeT C ypaBHeHIIeM, nOJIY4eHHbIM B {I8I]
npH paccMoTpeHIIII cMeIllaHHoH rpaHIIqHOH 3a,n:aqII ,n:JISI 4eTBepTbnJIOC-
KOCTII. 3TO ypaBHeHlle IIMeeT ,n:Ba Bew.eCTBeHHbIX KOpHSI, MeHbIllHX
e,n:IIHHUbI no a6COJIIOTH0I1 BE'.1HqHHe H OTJIllqmoru.llXCSI 3HaKOM. OTpIIua-
TeJIbHbIH KopeHb npoTIIBOpeqHT npe,n:nOJIO}KeHHSIM, c,n:eJIaHHbIM npII
nOJIyqeHHH cOOTHoIlleHHH (11.26), H B ,n:aJIbHeHIllIIx paccMOTpeHHSIx BO
BHHMaHIIe He npHHIIMaeTCSI.


Yi =

 (O<
< I). 01.25)
1
3HaSI xapaKTep oco6eHHOCTII HanpSI}KeHHOrO COCTOSIHHSI, MO}KeM YKa-
3aTb 3HaqeHHSI ex H
 B COOTHOIlleHllSIX (11.24) II (11.25), o,n:HaKo He 6y-
,n:eM 3Toro ,n:eJIaTb, a non
TaeMCSI HaHTII 3TH 3HaqeHllH npII aHaJIII3e
6eCKOHe t IIlOH CHCTeMbI.
COOTHOIlleHIIH (11.24) II (11.25), B OTJIIIqHe OT aHaJIOm4HblX no co-
,n:ep}KaHHIO CooTHoIlleHIIH (3aKoH aCIIMnTOTHqeCKIIX Bblpa}KeHIIH) B pac-
'CMoTpeHHblx 'BbIIlle 3a,n:a4ax, YCTaHOBJIeHbI He nYTeM <popMaJIbHOrO
116


117




AHaJIII3 BbIpallieHIIH .lI,JIH KOMnOHeHT TeH30pa HanpHllieHIIH B CBH3II
C YCTaHOBJIeHHbIM XapaKTepOM nOBe.lI,eHIIH K03!fJqmu.IIeHTOB COOTBeT-
CTBYlOm IIX 6eCKOHeqHbIX pH.lI,OB 6Y.lI,eT npOBe.lI,eH B CJIe.lI,YlOIlI.eM nap arpa-
.q>e. TaM llie 6Y.lI,eT nOKa3aHo, qTO BbI60p B KaqeCTBe ex nOJIOlliIITeJIbHoro
KOpHH ypaBHeHIIH (11.28) npIIBO.lI,IIT B paccMaTpIIBaeMoH 3a.lI,aqe K
TaKoH llie oc06eHHocTII Ha pe6pe, KaK II B cJIyqae cMemaHHoH 3a.lI,aqII
.lI,JIH qeTBepTbnJIOCKocTII. DOJIbmOe BHIIMaHIIe 6Y.lI,eT Y.lI,eJIeHO TaKllie BO-
npocy YJIyqrneHIIH CXO.lI,IIMOCTII pH.lI,OB .lI,JIH HanpHllieHIIH II cMemeHIIH.
3.lI,eCb llie ocraHoBIIMCH Ha 3a.lI,aqe nocTpoeHIIH 3!fJ!fJeKTIIBHoro aJIropIIT-
Ma pemeHIIH CIICTeMbI (11.9)
CpaBHeHIIe pe3YJIbTaTOB, nOJIyqeHHbIX npII paCCMOTpeHIIII rpa-
HIIqHbIX 3a.lI,aq B HanpHllieHIIHx II CMernaHHOH rpaHIIqHOH 3a.lI,aqII, MO-
llieT CJIYlliIITb nOBO.lI,OM .lI,JIH Toro, qT06bI OTMeTIITb HeKoTopbIe MOMeHTbI,
BalliHbIe .lI,JIH xapaKTepIIcTIIKII pa3BIIBaeMoro B pa60Te MeTO.lI,a II ou.eHKII
'Tpe60BaHIIII, npe.lI,'bHBJIHeMbIX K perneHIIlO B03HIIKalOmIIX npII ero npII-
MeHeHIIII 6eCKOHeqBbIX CIICTeM.
DeCKOHeqHbIe CIICTeMbI, nOJIyqeHHbIe II IICnOJIb30BaHHbIe npII pac-
cMoTpeHIIII nepBoH OCHOBHOH rpaHIIqHOH 3a.lI,aqII, OKa3aJIIICb pery-
JIHpHbIMII. CMernaHHaH rpaHIIqHaH 3a.lI,aqa, paccMoTpeHHaH BbIme,
npIIBeJIa K BnOJIHe perYJIHpHOH 6ecKoHeqHoII CIICTeMe C y6bIBalOmIIMII C
pOCTOM HOMepa HeII3BeCTHbIMII. C TOqKII 3peHIIH TeopIIII 6eCKOHeqHbIX
CIICTeM Bcer.lI,a npe.lI,nOqTlfTeJIbHeH IIMeTb .lI,eJIO C cIIcTeMoH BnOJIHe
perYJ1HpHOH. B 3TOM CJIyqae rOpa3.lI,0 npome pernalOTcH BonpOCbI, CBH-
3aHHbIe C II3yqeHIIeM e.lI,IIHCTBeHHOCTII pemeHffH II ou.eHKII npII6JIIIllieH-
HbIX 3HaqeHIIH HeII3BeCTHbIX, nOJIyqeHHbIX no MeTO.lI,y pe.lI,YKu.IIII. O.lI,-
HaKO eCJIII 06paTIIMcH K ou.eHKe 6bICTpOTbI CXO.lI,IIMOCTII pH.lI,OB .lI,JIH
HanpHllieHIIH II CMeIll.eHIIH, TO B CMemaHHOH 3a.lI,alIe CXO.lI,IIMOCTb pH.lI,OB
3HaqIITeJIbHO XYllie, qeM .lI,JIH nepBOH OCHOBHOH rpaHIIqHOH 3a.lI,aqII.
Eno Tpe6yeT 3HaQIITeJIbHoro YBeJIIIlIeHIIH QIICJIa ypaBHeHIIH B KOHeQHOH
CIICTeMe npII nOCTpoeHIIII npII6JIIIllieHHoro perneHIIH, II npeIIMymecTBa
BnOJIHe perYJIHpHbIX CIICTeM C TOlIKII 3peHIIH perneHIIH KpaeBOH 3a.lI,alIII
OKa3bIBalOTCH HecymecTBeHHbIMII.
1'13 CKa3aHHoro CJIe.lI,yeT, lITO npII pemeHIIII CIICTeMbI (1] .9) BonpOCbI
YJIYQmeHIIH MeTO.lI,a npocTOH pe.lI,YKILIIII IIMelOT HIICKOJIbKO He MeHbmee
3HalIeHIIe, GeM B paCCMOTpeHHbIX paHee rpaHIIQHblX 3a.lI,alJaX B HanpH-
:>KeHIIHX. OCHOBOH .lI,JIH nOCTpoeHIIH 3<p!fJeKTIIBHbIX aJIroplITMOB perne-
HIIH CIICTeMbI (11.9) HBJIHlOTCH II3BeCTHbIe aCIIMnTOTlllIeCKlle CBoHcTBa
HeII3BeCTHbIX, BbIpallieHHble COOTHomeHIIHMIl (1] .24) II (11.25). B COOT-
BeTCTBIIII C 3TIIMII COOTHomeHIIHMII npll nepexo.lI,e OT 6ecKoHerIHoH
'CIICTeMbI K KOHeQHOH C coxpaHeHIIeM N HeII3BeCTHbIX x n II M HeII3BeCT-
HbIX Yj nOJIOllillM


"IiTb KaQeCTBO nOJIYQeHHoro pemeHIIH CIICTeMbI, He 06paIll.aHCb K rpa-
IIIllIHbIM YCJIOBIIHM 3a.lI,aQII. llJIH TaKIIX ou.eHOK MOlliHO IICnOJIb30BaTb
COOTHomeHIIH (11.26), II3 KOTOpbIX CJIe.lI,yeT
( ao ) 2 _ 3m - 4 a + m - 2 (11.30)
b;; - m . a - (m - 2) .
I
C.ml npII nO.lI,CTaHOBKe B 3TO paBeHcTBo BMeCTO b o II a o COOTBeTCTBeHHO
x.\1zr;; II yM'A% OTHOCIITeJIbHaH pa3HOCTb Melli.lI,y JI
BOH II npaB
H QaCTblO
Jll'JIIIKa, TO CJIe.lI,yeT YBeJIIIQIITb QIICJIO ypaBHeHIIII B KOHeQHOII CIICTeMe,
T e. QIICJIa N II M.
B03MOlliHOCTII II3JIOllieHHOro cnoc06a perneHIIH 6eCKOHeQHbIX CIICTeM
II pOIIJIJIlOCTpIIpyeM B CJIe.lI,YlOmeM nap
rpa!fJe nI:II II3YQeHilII HanpH:>KeH-
IIOro COCTOHHIIH llieCTKO-3ameMJIeHHOII KpyrJIOII nJIIITbI.



 12. HanpsmeHHoe COCTOSHHe mecTKo-aa.rqeM.JIeHHOii
KPYl'.JIoii n.JIHTbl


HCnOJIb3YH COOTHorneHIIH (11.5) II (11.8), JIemo npe.lI,CTaBIITb
bI-
IW)KeHIIH .lI,JIH KOMnOHeHT BeKTopa cMemeHIIH II TeH30pa HanpHllieHIIII B
JlII.lI,e pH.lI,OB, K03!fJcpIIlLIIeHTbI KOTOpblX HBJIHlOTCH HeII3BeCTHbIMII B 6ec-
IWHelIHoH CIICTeMe (11.9). flOCKOJIbKY .lI,JIH pa3JIIIQHblX !fJaKTopoB Ha-
JlpmKeHHO-.lI,e!fJopMIIpoBaHHoro COCTOHHIIH 3TII pH.lI,bl O.lI,HOTIInHbl, TO
xapaKTep BbIKJla.lI,OK, CBH3aHHbIX C YJIYQrneHIIeM IIX CXO.lI,IIMOCTII, npo-
JlJIJlIOCTpllpyeM Ha npIIMepe pH.lI,OB .lI.JIH oceBoro CMeIll.eHIIH II pa.lI,lIaJIbHO-
ro HanpHllieHIIH:


XNk

X=-
n k



(n :;;.. N),


yM'A%
Yj=

1


(j :;;.. M). (] 1.29)


_ A + m - 2 . 110 Z2 _ 110 r2--1-
w - 0 m- I 2h 2h I
h 00 [ m - 2 ch 'Ajl 1 J 0 ('Ajr)
+4
 Yj HC(AjZ)---,n--. sh'Ajh j 'A;Jo('A;) +
1=1
+ J...-
 (- It x n [ BC (knr) + 4 m k I . I; (
t; ] cos knz +
4 '2=1 m n 1 n

 f li ch 'Ail \.
 f 10 (knr) k
+ }, -. -Jo(A.r) + :, 2n 1 (k) cos n Z '
f;:::,1 'Ai sh 'Ai h / n=' 1 1Z
I (J tlo' l!... 4 /
 . 00 =1 Y / " [ HS ( A / Z) +
2G r = h (m - I)
+ 2 sh Ail ] Jo ('Air) + l!.-
 y. [ HS (A.Z) +
sh 'Aih J 0 ('Aj) 4 ;::1 / J
2 m- I . Sh'Ai l_ ]
 .
 ('Ajf) +
+ m sh 'Ajh 'Air J 0 ('Ai)
+ + n
1 (- 1)" X n [2 m mI .
:


) + knBC(knr)-
119


(12 1)


T aKoH no.lI,xo.lI, .lI,aeT B03MOlliHOCTb nOJIYQIITb KOHeQHYlO cIIcTeMY,
B KOTOpOH npII6JIIIllieHHO YQTeH BKJIa.lI, rJIaBHbIX QJIeHOB Hell3BeCTHbIX
C HOMepaMIl Bblrne N II M. KpoMe Toro, nOHBJIHeTCH B03MOlliHOCTb ou.e-


118




ch 'AjZ sh 'AjZ
HS (Ajl) = Ajl sh 'A.h - \h cth A;h sh 'A.h '
1 1
sh 'AjZ ch 'AjZ
HC (Ajl) = AIZ sh 'A .h - Ajh cth Ajh sh f..'h '
1 1
BS (knr) = r 10 (f..nr) 10 (kn> I dkn r )
11 (k n ) - 11 (k n ) . 11 (k n ) ,
BC ( k Idknr) 10 (k n ) 10 (knr)
nr) = r 11 (k n ) - 11 (k n ) . 11 (k n ) .
HaH60JIbiliHe TPy.n:HOCTH rrpH llCrrOJIb30BaHHH BbIpa>KeHHH {12.1)
B03HHKalOT rrpH BbIIIHCJIeHHH CMe
eHHH H HarrpH>KeHHH Ha rrOBepXHOCTH
UHJIHH.n:pa. B 3TH BbIpa>KeHHH BXO.n:HT 6eCKOHeIIHbIe pH.n:bI .n:BYX THrrOB.
O,D:HH H3 HHX HMelOT B KaIIeCTBe K03<pqmUHeHTOB HeH3BeCTHbIe H3 6ec-
KOHeIIHOH CHCTeMbI (11.9) 3HaHHe aCHMrrTOTHIIeCKHX cBoiicTB BeJIHIIHH
X n H Yj rr03BOJIHeT YJIYIIIllHTb CXO.n:HMOCTb pH.n:OB H rrOJIYIIHTb y.n:06HbIe
.n:JIH BbIIIHCJIeHHH <POPMYJIbI. CXO.n:HMOCTb p5l.n:OB .n:Pyroro THrra, BXO.n:51-

HX B (12.1), orrpe.n:eJI5IeTCH CKOpOCTblO y6blBaHH5I K03<p<pHUHeHToB
<Pypbe <PYHKUHH, 3a.n:alO
HX CMe
eHHH H HarrpH>KeHHH Ha rpaHHIIHbIX
rroBepxHocTHX UHJIHH.n:pa (11.2). HeKoTopbIe o6
He Tpe60BaHHH K
CTerreHH rJIa.n:KOCTH 3THX <PYHKUHH 6blJIH C<POPMYJIHpOBaHbI BbIIlle
rrpH H3YIIeHHH aCHMrrTOTHIIeCKHX CBOHCTB HeH3BeCTHbIX B CHCTeMe (11.9).
3.n:ecb 3TH Tpe60BaHHH 6y.n:YT KOHKpeTH30BaHW HCXO.n:51 H3 3HaIIeHHH
K03<p<pHUHeHTOB COOTBeTCTBYIO
HX pH.n:OB B (11.3).
C<pOPMYJIHpOBaHHoe BbIIlle Tpe6oBaHHe, IIT06bI cBo6o.n:HbIe IIJIeHbI B
rrepBoH rpyrrrre ypaBHeHHH (11.9) y6bIBaJIH He Me.n:JIeHHee IIeM llk
,
rro OTHOIlleHHIO K CP2 (z) H 11 (r) 3KBHBaJIeHTHO Tpe6oBaHHIO HerrpepblB-
HOCTH 3THX <PYHKUHH Ha COOTBeTCTBYIO
HX HHTepBaJIax. IlpH 3TOM CPZn =
= 0 ( + ) , IIj = 0 (1/A/ 2 ) H pH.n:bI ch, B (12.1) CXO.n:HTCH. K03<P<PHUHeHTbI
. k n
pH.n: OB <I:>YPbe Izn B HailleM CJIYIIae HMelOT r JIaBHYIO IIaCTb BH.n:a ( k 1)n 12 (h).
n
B CBH3H C 3THM C<POPMYJIHpOBaHHoe BbIwe Tpe60BaHHe OTHOCHTeJIbHO
rropH.n:Ka y6bIBaHHH CBo6o.n:HbIX IIJIeHOB B (11.9) npHBo.n:HT K YCJIOBHIO
12 (h) = 0, T. e. rpaHHIIHbIe 3HaIIeHHH .n:.'IH U .n:OJI>KHbI HenpepblBHO rrpo-
.n:OJIlliaTbCH C OTpe3Ka -h < Z < h Ha BCIO rrOBepXHOCTb 6eCKOHeIIHOrO
UHJIHH.n:pa r = 1 pH.n:OM rro sin knz (kn =
'If, ). OTKa3aTbcH OT 3Toro


(12.2)


Tpe60BaHHH MO>KHO rrpH HCnOJIb30BaHHH .n:Pyroro cnocooa rrpo.n:oJI>Ke-
nHH rpaHHIIHbIX YCJIOBHH B o6
eM peIlleHHH (9.4), OTBeIIalO
ero BbI60PY
2n-1 6 u
k n =
 n, T. e. aHaJIOrHIIHO TOMY, KaK 3TO blJIO c.n:eJIaHO BO BTOpOH
rJIaBe rrpH paccMoTpeHHH rpaHHIIHoH 3a.n:aIIH C HaPYIlleHHeM YCJIOBHH
rrapHOCTH KaCaTeJIbHbIX HarrpH>KeHHH B yrJIOBOH TOIIKe. OrpaHHIIeHHH
Ha rropH.n:oK y6bIBaHHH cBo6o.n:HbIX IIJIeHOB BO BTOpOH rpyrrrre ypaBHe-
HHH CHCTeMbI (11.9), rrpHHHTbIe rrpH ee aCHMrrTOTHIIeCKOM aHaJIH3e,
,KBHBaJIeHTHbI Tpe6oBaHHIO HerrpepblBHocTH <PYHKUHH CPI (r).
PaccMoTpHM BbIpa>KeHHH (12.1) Ha rpaHHue uHJIHH.n:pa, T. e. rrpH
r = 1 H Z = h. IlpH r = 1 pH.n:bI, co.n:ep>Ka
He OTHOIlleHHH mrrep6o-
JIHIIeCKHX <PYHKUHH, 06JIa.n:aIOT .n:OCTaTOIIHO xopoilleH CXO.n:HMOCTblO BO
Bcex TOIIKax 3a HCKJIIOIIeHHeM MaJIOH oKpecTHocTH pe6pa. B CBH3H C
1THM B YJIYIIIlleHHH HX CXO.n:HMOCTH HeT Heo6xo.n:HMOCTH. LlocTaToIIHo
JIHilib HCCJIe.n:OBaTb xapaKTep oco6eHHocTH, KOTOpylO HMeIOT 3TH pH.n:bI rrpH
Z -';- h.
TpHroHoMeTpHIIeCKHe pH.n:bI C K03<p<pHUHeHTaMH X n B (12.1) HMelOT
OIIeHb Me.n:JIeHHYIO CXO.n:HMOCTb. AHaJIH3HPYH HX rrpH r = 1, 3aMe-
qaeM, IITO CXO.n:HMOCTb MO>KeT 6bITb YJIYIIIlleHa, eCJIH BOCrrOJIb30BaTbCH
CJIe.n:YIO
HMH pa3JIOllieHHHMH:


I BS (k )1 . k
 flj sh f..jZ I J (
 )
- r n r Sin n l +

. sh f.. .h . ---;- 1 1'.1 -
j=1 1 1
-
 hj S s h h

Z h - Jo(A/) + f hn [ kn lo(knr)
j=1 1\,1 n=1 ,) (kn>
I 11 (knr) ] . k
- r" /1 (k n ) SIn n Z .
3.n:eCb .n:JIH COKpa
eHHH 3arrHCH BBe.n:eHbI 0603HaIIeHHH


Z (h 2 - Z2)'I'-1 'If, = L 1/ k
'If, .
2'1'h '1'-1 r (v) sin V - n=1
2


J I (knh)
'1'+-
2 . k
sin nZ'
'If,
k'l'sin"-
n r 2


(12.3)


(h 2 _ Z2)'I'-1
'If,
2'1'h'l'-zr (v) cos V-
2


co
= L V kn:'If, .
1=


h'l'V 1t
2'1'+lr(v+-})cos v ;
J I (knh)
'1'-2"
COS knz.
'If,
k'l'cos"-
n r 2


IlpH 60JIbIllHX 3HaIIeHHHX n K03<p<pHUHeHTbl 3THX TpHroHoMeTpHIIecKHx
pn.n:oB HMelOT CJIe.n:YIO
He aCHMrrTOTHIIeCKHe 3HaIIeHHH:
J I (knh)
V /zn
'If, . 'I' + 2" 'If,
k'l'sin" -
n r 2
J I (knh)
1/ knt . 'I' - "2 'If,
k'l'cos"-
n r 2


(-It

--
k'l' '
n


(12 4)


(_I)n
-';- - .
k'V
n


120


HaH.n:eM rJIaBHblB IIJIeH aCHMrrTOTHIIeCKoro rrpe.n:CTaBJIeHHH K03<p-
IIJHIJ.HeHTOB TpHrOHOMeTpHIIeCKoro pH.n:a C X n B BblpmKeHHH .'(JIH (Ir
(12.1). I1crrOJIb3YH aCHMrrTomIIeCKHe rrpe.n:CTaBJIeHHH .n:JIH <PYHKUHH
121




BeCCeJIH, HaXOll,HM
(_l)n [ 2 m-l . 'oCk,,) + k BC(k )]
4 x n m '1 (k n ) n n

(-It xN'lN m-2
-
4k
 m
3,lLeCb BeJIH'lHHa N HMeeT CMbICJI TaKOll >Ke, KaK H B paBeHcTBax (11.29).
TaKHM o6pa30M, BbIlfHCJIeHHe 3HalfeHHll cooTBeTcTBYIOIlLero TpHroHo-
MeTpHlfecKoro pHlI,a B a, MO>KeT 6bITb npOBell,eHO CJIell,YIOIlLHM o6pa30M:


( 12.5)


PaCCMOTpHM BbIpa>KeHHe lI,JIH LIt npell,CTaBJUUI 6ecKoHelfHbIll pH,lL B BHlI,e
00 a-I I
Ll = n a - I (h - Z)a-I

 e- k, (12.8)
i=1 I
r,lLe k = lIn (h - z). YfcnOJIb3YH COOTHorneHHe (11.27), npH z
 h
HMeeM


I
 I n [2 m-l lo(kn} k BC k ] . k
T ...::..J (- ) X n -. -r-- ( k ) + n (n) Sin n Z =
n=1 m 1 n

 \
 !{ _ I ) n X 1 2 m - I . 10 (k n ) + k BC ( k )] +


 n m I ( k ) n n
,,=] I n
J I (knh) I
m - 2 a knJm. a+ 2" .
+-xNkN V -. smkz-
4m 2 n n
k
 sin a -
2


00
L a-I {h ) a-I 5 -a -x d
1
 n - z X e X,
o


(12.9)


Z (h? - z2)a-]
n
2 a h a - l r (a) sin a -
2
npH BbIlfHCJIeHHH HOpMaJIbHblX HanpH>KeHHll Ha nOBepxHocTH r = I
HCnOJIb3yeTcH nepBbIll pHlI, (12.3) npH Y = ct. YJIYlfrneHHe CXOll,HMOCTH
cooTBeTcTBYIOIIJ.ero TpHroHoMeTpHlfecKoro pH,lLa B BbIpa>KeHHH lI,JIH oce-
BOro CMeIlLeHHH W B (12.1) lI,OCTHraeTCH HCnOJIb30BaHHeM BTOpOro pHlI,a
(12.3) npH y = ct + l. STOpOll pHlI, (12.3) HCnOJIb3yeTcH TaK>Ke npH
BbIlfHCJIeHHH KaCaTeJIbHbIX HanpH>KeHHll, O,lLHaKO B 3TOM CJIYlfae CJIe,lLyeT
nOJIO>KHTb Y = ct.
PaccMaTpHBaH COOTHorneHHe (12.6) H YlfHTbIBaH xapaKTep oco6eH-
HOCTH <PYHKIJ,Hll, npell,CTaBJIeHHblX TpHroHoMeTpHlfecKHMH pH,lLaMH (12.3),
MO>KeM 3aKJIlOlfHTb, lfTO BbI60p B KalfeCTBe ct KOpHH TpaHCIJ,eH,lLeHTHoro
ypaBHeHHH (11.28) npHBOll,HT K nOHBJIeHHIO TaKOll >Ke oco6eHHocTH Ha
pe6pe ,lLJIH KOMnOHeHT TeH30pa HanpH>KeHHll, KaK H B CJIYlfae 3all,aIJ:H
,lLJIH lfeTBepTbnJIOCKOCTH [ 177!.
lIJIH Toro lfT06bI YCTaHOBHTb OTCYTCTBHe oco6eHHOCTellll,Pyroro nIna,
CJIe,lLyeT paccMoTpeTb npH r = I pHlI,bl C OTHorneHHeM rHnep60JIHlfeCKHx
<PYHKIJ,Hll B (12.1). Ha OCHOBe xapaKTepa nOBell,eHHH K03<P<PHIJ,HeHTOB
(11.29) MO>KeM 3aKJIlOlfHTb, lfTO xapaKTep oco6eHHocTell. coll.ep>KaIlLHXCH
B 3THX pH,lLax, TaKOll >Ke, KaK H B pHlI,aX


m-2 k a
--XN N
4m


(12.6)


T. e. COCTaBJIHIOIlLHe 6ecKoHelfHbIx pH,lLOB G: K03<P<PHIJ,HeHTaMH Yj B (12.1),
r JIaBHblll lfJIeH aCHMnTOTHlfecKoro npell.CTaBJIeHHH K03<p<p H IJ,HeHTOB Y
KOTOpbIX npOnOpIJ,HOHaJIeH lfJIeHaM pH,lLa lI,JIH L]. He BHOCHT HOBblX OCO-
6eHHOCTell B BbIpa>KeHHH lI.JIH HanpH>KeHHll. OlfeBHlI,HO, lfTO pHlI,bI C
rJIaBHbIMH lfJIeHaMH, npOnOpIJ,HOHaJIbHbIMH lfJIeHaM pH,lLa lI,JIH L2' TaK>Ke
He npell.CTaBJIHIOT HOBbIX oc06eHHOCTell.
npH BbIlfHCJIeHHH CMeIlLeHHll H HanpH>KeHHll Ha TOpIJ,eBbIX nOBepx-
HOCTHX CTaJIKHBaeMCH C nJIOXOll CXOll,HMOCTblO pHlI.OB <I>YPbe - lIHHH
B BbIpa>KeHHHX (12.1). OnHcaHHbIll BbIrne B npHMeHeHHH K pHlI.aM <I>YPbe
CTa H lI.apTHbIll npHeM n03BOJIHeT YJIYlfrnHTb HX CXOll.HMOCTb C nOMOIlLblO
CJIell.YIOIIJ.HX pa3JIO>KeHHll:
(l - r2)V-1
2 v r (y) CDS Y



1 =- f J
 ('Ai) Iv (Air)
2 v r (y+l) cos l'
 ;=1 CDS Y 2"" J o (Ai) AJ J o (Ai) ,


(12. 10)


r (l - r2)V-1
2 v r (y) sin y



_
 J 1 + V (Ai) JdAjr)

 J ( ' ) A , '!' In (Ai)
;=1 sin y ; v "i


L =

 e- i1h - z )1t
] ......a '
1=1 I


(l2.7)


TaKHM 06pa30M, 3HaHHe 3HalfeHHH HeH3BeCTHbIX B CHCTeMe (11.9)
n03BOJIHeT npOBeCTH ,lLOCTaTOlfHO 3<p<peKTHBHO BWlfHCJIeHHe Bcex BeJIH-
lfHH, xapaKTepH3YIOIlLHX HanpH>KeHHO-lI.e<POpMHpoBaHHoe COCTOHHHe
IJ,HJIHHlI,pa npH CMernaHHWX rpaHHlfHbIX YCJIOBHHX. TOlfHOCTb perneHHH
6eCKOHelfHOH CHCTeMbI 3aBHCHT OT TOro, HaCKOJIbKO 6bICTpO HeH3BeCTHble
B HeH BbIXOll,HT Ha aCHMnTOTHlfeCKHe 3HalfeHHH, T. e. OT TOlfHOCTH
onpell,eJIeHHH HeH3BeCTHbIX C BbICOKHMH HOMepaMH B COOTBeTCTBHH C co-
OTHorneHHHMH (11.29).
HaJIHlfHe npell,CTaBJIHIOIlLeHCH 1I,0BOJIbHO 3<p<peKTHBHOH MeTOlI.HKH
perneHlfH CMernaHHbIX rpaHHlfHbIX 3all,alf lI,JIH IJ,HJIHHlI,pa KOHelfHOll lI.JIH-
HbI n03BOJIHeT H3YlfHTb HHTepecHbIll Bonpoc 0 xapaKTepe HanpH>KeHHoro
COCTO:<IHHH TOJICTbIX OJIHT B6JIH3H >KeCTKO-3aIlLeMJIeHHbIX KpaeB. KOH-
KpeTHO 6YlI.eT paCCMOTpeHa CJIell,YIOIlLaH rpaHHlfHaH 3all,alfa:
1 1
20 az(r, + h) = :f:: P, 20 .,z(r, + h} = 0;


L (h ) ,--, .-0.+1 -j(h-z)1t
2= -Z...::..JJ e .
)=1


u (1, Z) = W (I, Z) = 0,


(12 II)


I


122


123




T. e. 3aJ.(aIla 06 1I3m6e }KeCTKO-3alll.eM.neHHoH Kpyr.noH TO.nCTOH II.nIlTbI,
HarpY}KeHHOH paBHoMepHo pacrrpeJ.(e.neHHoH HarpY3KoH.
D.,n}! orrpeJ.(e.neHII5I aJ.(J.(IITIIBHOH rrOCT05IHHOH Ao B Bblpa}KeHIIII J.(.n5I
oceBoro CMelll.eHII5I rro.nYIlIlM COOTHOlIleHlIe


D..n5I Hell3BeCTHblX C HOMepaMII N 11 M rro.nYIleHbI B J.(BYX paccMaT-
pllBaeMblx c.nYIla5IX cJIel{YlOllI.lIe 3HaIleHII5I:


A P m - 2 P m + I
 'Ii
o=2il- 6(m-l) h+

 "2.
;=1 ';


XN = -0,07314,
XN = - 0,07869,


YM = 0,2328
YM = 0,3968


(h = 0,5);
(h = 0,25,.


(12.15)


(12 12)


npaBa5I IlaCTb B ypaBHeHlI1I (11.30) rrpll m == 4 11 ex. = 0,745 paBHa


npIl6.nIl}KeHHa5I 3aMeHa 6ecKoHeIlHoH CIICTeMbI (11.9) KOHeIlHOH ocy-
Il(eCTB.n5IeTC5I Ha OCHOBe paBeHcTB (11.29). KOHeIlHa5I CIICTeMa, cOJ.(ep-
}Kalll.a5I N Hell3BeCTHblX x n 11 M Hell3BeCTHblX Yj, rrpll06peTaeT BIIJ.(
M ( 2 )
_ lJi k n m-l a
xn!1 n -
 ,?2 + "'
 k2 + "'
 - -m- + YM"'MSI (M, n) +
}=I n I n I
2 m-2 P
+!;2 m- I h '
n


3m-4 ma+m-2 = 10 16
m ma-m+2 ,.


(12.16)


npIl6.nIl}KeHHble 3Ha'IeHII5I Be.nIlIlIlH a o 11 b o , BbIIlllc.neHHble Ha OCHOBe
.l!.:1HHbIX (12.15), J.(alOT J.(.n5I .neBOH IlaCTII paBeHcTBa (11.30)


Yj8j =
 k

 "'7 ( k

 ""7 - m -;;; I
rJ.(e


(12.13)
) + x N krvS 2 (N, j) +

 .
",.
J


(
: r = 10,21


(h = 0,5);


( ao ) 2
b; = 9,14 (h = 0,25)


(12.17)


m


I
k
 +"'7


] :
 '
I '"
j ----k- .
k n


(12.14)


CpaBHeHlIe J.(aHHbIX (12.16) 11 (12.17) rrOKa3bIBaeT, IlTO B paCCMaTpll-
BaeMblX JaJ.(a'IaX HaHJ.(eHbI J.(OCTaTO'IHO TOIlHble 3HaIleHII5I Bcex He1l3-
BeCTHbIX 113 6eCKOHeIlHOH CIICTeMbI (12.13) 11, c.neJ.(OBaTe.nbHO, rrpll IIX
IICno.nb30BaHIIII MO}KHO O}KIIJ.(aTb BbICOKOH TO'IHOCTII yJ.(OB.neTBopeHII5I
rpaHII'IHbIX YC.nOBI1H. npoBeJ.(eHHble BbI'IHc.neHII5I rrOJ.(TBep}KJ.(aIOT 3TO
rrpeJ.(rro.nO}KeHHe.
B Ta6JI. 11 rrpHBeJ.(eHbI 3Ha'IeHH5I oceBoro CMelll.eHI15I W J.(.n5I U,1I.nIlHJ.(pa
C h = 0,5, a B Ta6.n. 12 - .l!.JI5I U,H.nIlHJ.(pa C h = 0,25. nOCJIeJ.(HlIe CTpO-
KH B 3THX Ta6JIHu,ax KOJIH'IeCTBeHHO xapaKTepH3YlOT TOIlHOCTb YJ.(OB-
.neTBopeHII5I rpaHIIIlHoro YC.nOBH5I W = ° Ha rroBepxHocTH r = 1.
rpaHI1'IHOe YCJIOBlIe OTCYTCTBII5I paJ.(Ha.nbHoro CMelll.eHII5I Ha 3TOH rro-
BepxHocTII yJ.(oB.neTBOp5IeTC5I TOIlHO.
AHa.nH3 J.(aHHbIX Ta6JI. 11 11 12 BCKpbIBaeT HeKoTopble ot06eHHocTlI
rrOBeJ.(eHII5I oceBoro CMelll.eHH5I B TOHKHX 11 TO.nCTbIX rr.nHTax. D.JI5I c.nYIla5I
h = 0,5, rJ.(e 06ll1.aH To.nlll.lIHa rrJIIITbI paBHa paJ.(HYCy, BO BceM ee 061>eMe
oceBble rrepeMelll.eHH5I TOIleK Ha Topu,eBblx rroBepXHOCT5IX 60JIblIle, IleM
rrepeMelll.eHIIH cooTBeTcTBYlOllI.HX TOIleK cpeJ.(IIHHOH rrJIOCKOCTII. B c.nYIlae
rr.naCTHHbI C TO.nlll.HHOH, paBHOH rrOJIOBHHe paJ.(lIyca (h = 0,25), TaKoe
rrOBeJ.(eHIIe oceBoro CMelll.eHH5I IIMeeT MeCTO .nlllIlb J.(.nH 06.nacTII r > 0,5.
OnpeJ.(eJIeHHbIH HHTepec npeJ.(CTaB.nHeT cpaBHeHHe rro.nYIleHHblx
3HaIleHlIH oceBoro CMelll.eHII5I cpeJ.(IIHHOH rr.nOCKOCTII rr.nIlThI C J.(aHHbIMII,
J.(OCTaB.n5IeMbIMII TeoplleH rr.naCTIIH. D..nH 60.nee rrOJ.(XOJ.(HllI.erO K TaKOMY
cpaBHeHHIO c.nYIlaH h = 0,25 3HaIleHlIe rrpom6a cpeJ.(HHHoH rroBepxHoc-
TII onpeJ.(e.nHeTC5I rro iPopMy.ne
w*IP = 4,5 (1 - r 2 ). (12.18)



 [ 2
S (M n =
 k n
1 ,) _ (k2 + ",
)2
,-M+I n I
S ( N . ) =
 [ k
 _ m-l
2,J
 ( k2 + ",
 ) 2 m
n=N+1 l' I


m-I


I
k 2 + "'

n I


npeJ.(CTaBIITb B 3aMKHYTOM BIIJ.(e p5IJ.(bI J.(.n5I S} (M, n) 11 S2 (N, j) He
YJ.(aeTC5I, OJ.(HaKO OHH J.(OCTaTO'IHO 6bICTpO CXOJ.(5ITC5I 11 BbIIlHC.nemre no
HIIM He rrpeJ.(CTaB.nHeT TpYJ.(a.
PelIleHlIe KOHeIlHOH CHCTeMbI (12.13) yJ.(06Ho rrpOBOJ.(HTb MeTOJ.(OM
rroc.neJ.(OBaTe.nbHbIX npH6.nH}KeHHH. CXOJ.(IIMOCTb c.neJ.(yeT H3 YCTaHoB-
.neHHOH paHee BnOJIHe pery.n5IpHOCTH 6eCKOHe'IHOH CIICTeMbI (11.9).
I1cno.nb30BaHlIe MeTOJ.(a rrOCJIeJ.(OBaTe.nbHbIX rrpIl6.nH}KeHHH J.(aeT u,eJIbIH
p5IJ.( rrpellMYlll.eCTB npll pelIleHlI1I CIICTeM Ha 3.neKTpoHHbIX MalIlHHaX.
D.ocTaToIlHo 6blcTpaH CXOJ.(HMOCTb MeTOJ.(a (J.(.n5I rrOJIYIleHHH lIleCTII-ceMII
3HaIlalll.lIX U,HiPP J.(OCTaTOIlHO J.(eC5ITb - J.(BeHaJ.(u,aTb rrpH6JIII}KeHHH)
11 'OTcYTCTBlIe HaJ.(o6HocTH xpaHIITb B naM5ITII MaTpllu,y CHCTeMbI n03BO-
JI5IIOT, rrpll Heo6xoJ.(HMOCTH, JIerKO pelIlaTb Ha 3.neKTpoHHbIX MalIlHHax
cpeJ.(HeH MOlll.HOCTII CHCTeMbI C HeCKo.nbKIIMII COTHHMII HeH3BeCTHblX.
npll pelIlemm CHCTeMbI (12.13) J.(.nH UH.nIlHJ.(pOB KOHKpeTHOH reo-
MeTp1l1l npllHIIMa.nocb m = 4, 11, c.neJ.(OBaTeJIbHO 1181J, ex. = 0,745.
Bce Be.nrrIlIlHbI - HeH3BeCTHble B CHCTeMe (12.13), rrepeMellJ.eHIIH rr Ha-
npH}KeHH5I BbIIlrrc.nH.nIlCb C TOIlHOCTblO J.(O MHO}KHTe.n5I P/2h. KOHe'IHa5I
CIICTeMa ypaBHeHIIH iPopMHpoBa.nacb npll N = 20, M = 40. DbI.nH
paccMoTpeHbI U,H.nIlHJ.(pbI C h = 0,5 H h = 0,25.


ECJIrr J.(JI5I npom6a B u,eHTpe rr.nIlTbI rro.nYIlaeM xopolIlee COBrraJ.(eHlIe
(wiP = 4,54; w*IP = 4,50), TO rrpll r = 0,9 pa3.nIlIlrre CTaHOBIITCH
cYlll.eCTBeHHbIM (wiP = 0,46; w*IP = 0,16).


124


125



I




Ta6JJllua II


CYIl(eCTBeHHoe pa3J1JiIme BeJlHqHH oceBoro CMeIl(eHHH, HaH
eHHbIX
no TpeXMepHOH TeopHH ynpyrocTH H no TeopHH nJiaCTHH, eCTeCTBeHHO
B CBeTe CJle
YIDIl(ero 06CTOHTeJlbCTBa.


I 2/h
, I I I I a,li I l,a
a a,2 a,4 a.6
0 1,360 1,368 1,391 1,423 1,458 1,485
0,2 1,290 1,298 1,322 1,356 1,393 1,423
0,4 1,086 1,094 1,120 1,158 1,202 1,239
0,6 0,769 0,777 0,802 0,843 0,984 0,946
0,8 0,385 0,389 0,403 0,432 0,484 0,554
1,0 0,006 0,006 0,004 -0,002 -0,006 -0,006


Ta6JJllua 12


6r I I
2GP
6r 0,8 0
26P t

h
0 z
11 -2


I z/h
r I I I I a,6 I a,8
a a,2 a,4 a,5
0 4,544 4,542 4,536 4,525 4,503 4,465
0,2 4,271 4,270 4,266 4,258 4,240 4,208
0,4 3,496 3,496 3,498 3,499 3,494 3,478
0,6 2,348 2,352 2,361 2,378 2,394 2,406
0,8 1,050 1,056 1,072 1,107 1,150 1,200
1,0 0,006 0.006 0,004 0 -0,006 -0,002


-0.0


-4


PIIC. 11.


PIIC. .12.


HCXO
H H3 rpaHHqHbIX YCJlOBHH }KeCTKOH 3a
eJlKH, B TeopHH nJlac-
) dw
THH [172 W = 0, liT = 0 npH r = 1 JlerKO nOJlyqHTb Ha KOHType
CJIe
YIOIl(ee COOTHomeHHe:


Ta6JJllua 13


M, a,
----m
Me - ae - .


( 12.19)


I z/h
, I I I 0,6 \ 0,8 I 0,9
0 0,2 0,4
0,2 0,301 0,289 0,252 0,191 0,107 0,056
0,4 0,597 0,575 0,506 0,386 0,215 1,112
0,6 0,858 0,836 0,759 0,602 0,343 0,179
0,8 0,988 0,986 0,971 0,890 0,603 0,328
0,9 0,964 0,971 0,991 1,014 0,920 0,615
1,0 0,894 0,896 0,910 0,943 1,027 1,166


C
pyroH CTOpOHM, npHHHB B KaqeCTBe npocTpaHCTBeHHOrO aHaJlOra
YCJlOBHH }KecrKoH 3alIeJlKH YCJlOBHH OTCYTCTBHH nepeMeIl(eHHH Ha no_
dw
BepXHOCTH r = 1, nOJlyqaeM (ii" = 0 (r = 1). HcnoJlb3YH COOTHomeHHH
3aKOHa rYKa, HaXO
HM


Ta6JJllua 14


.!!!:.... = m- l. (12.20)
ae
CpaBHeHHe COOTHOllleHHH (12.19) H (12.20) nOKa3bIBaeT, qTO Me}K
y
YCJlOBHHMH paBeHcTBa HYJlID CMellleHHH Ha U:HJlHH
pHqeCKOH nOBepxHoc-
d z W
26P


I z/h
r I I I I I 0.9
0 0,2 0.4 0,6 0,8
0,2 0,600 0,576 0,504 0,384 0,216 0,115
0,4 1,200 1,152 1,008 0,768 0,432 0,228
0,6 1,800 1,730 1,512 1,151 0,646 0,337
0,8 2,324 2,252 2,022 1,584 0,899 0,464
0,9 2,296 2,278 2,198 1,960 1,294 0,656
1.0 1,842 1,860 1,904 2,013 2,277 2.490




68
28P
O,



0,,,


.z..
11


o
-0,1


Fllc. 13.


PIIC. 14


126


127


I




Tll II YKa3aHHbIMli BbIIIIe YCJIOBIiSIMIi >KeCTKOH 3a.n;eJIKll B TeOp1i1i IlJIaC-
TIiH HeT aCllMnTOTllqeCKOrO COOTBeTCTBIiSI. 3TO, no CYIII,eCTBY, 03Haqa-
eT, qTO npll JIl060H, CKOJIb yrO.lI.HO MaJIOH TOJIIII,IlHe nJIaCTllHbI pe3YJIb-
TaTbI paCqeTOB no 113JIO>KeHHOH BbIIIIe MeTO.lI.llKe II no QJOPMY JIaM TeopliH
nJIaCTHH 6Y.lI.YT OTJIHqaTbCSJ Ha KOHeqHYlO BeJIHqHHY. 06HaPY>KeHHbIH
QJaKT YKa3bIBaeT Ha Ba>KHOCTb Bonpoca 0 nOCTaHOBKe KOppeKTHbIX rpa-
HHqHbIX YCJIOBHH npH nOCTpoeHllll 60JIee TOqHbIX, no cpaBHeHlllO C
KJIaCCllqeCKOH, MO.lI.eJIeH B TeopHll 060JIOqeK H nJIaCTllH.
TIpe.n;CTaBJIeHHe 0 xapaKTepe nOBe.lI.eHllSI II 0 BeJIllqllHe KaCaTeJIbHbIX
HanpH>KeHllH B paCCMaTpHBaeMbIX nJIHTaX .lI.alOT .lI.aHHbIe Ta6JI. 13 H 14,
OTHOCHIII.lleCSI COOTBeTCTBeHHO K CJIyqaSIM h = 0,5 II h = 0,25.
BbIqllCJIeHllSI Ha TOpU.eBbIX nOBepxHocTSIX, r.lI.e YCJIOBHSJ .lI.JISI HOp-
MaJIbHbIX HanpSI>KeHHH BbmOJIHSIlOTCSI TOqHO, nOKa3blBalOT, qTO rpa-
HllqHbIe YCJIOBHSI .n;JISI KaCaTeJIbHbIX HanpSI>KeHHH Y.lI.OBJIeTBOpHlOTCSI C
TOqHOCTblO .lI.0 .lI.eCHTOH .lI.OJIll npou.eHTa BeJIHqHHN HarpY3Kll P.
LlJISI 06ellx paccMOTpeHHbIx 3a.lI.aq xapaKTepHO HaJIllqlle .lI.OCTa-
TOqHO Y.lI.3JIeHHOH OT 3a.lI.eJIKll 30HbI, B KOTOpOti B Ka>K.lI.OH nJIOCKOCTH
z const Ha6JIlO.lI.aeTCSI B03paCTaHHe KaCaTeJIbHbIX HanpH>KeHllH npo-
nopu.lloHav1bHo pa.n;IlYcy. Pacnpe.lI.eJIeHHe KaCaTeJIbHbIX HanpSI>KeHllH no
TOJIIII,llHe B :noH 30He nOqTH CTpOro CJIe.lI.yeT 3aKoHY KBa.lI.paTHOH na-
pa60JIbI. TIpH no.n;XO.lI.e K 3aIIl,eMJIeHHoti nOBepxHocTH xapaKTep pac-
npe.lI.eJIeHHSI HanpSI>KeHllti no TOJIIII,llHe MeHSIeTCSI. 3HaqHTeJIbHO B03pac-
TalOT KaCaTeJIbHbIe HanpH>KeHllH Ha nJIOIIl,a.lI.SIX, y.n;aJIeHHbIX OT cpe.n;HH-
HOH nOBepxHocTH. B HenOcpe.lI.CTBeHHoH 6JIll30CTll K 3aIIl,eMJIeHHoMY
KpalO 30Ha MaKCHMaJIbHbIX KaCaTeJIbHbIX HanpH>KeHllti CMeIIl,aeTCSI co
Cpe.lI.llHHoti nJIOCKOCTll K TOpu.eBbIM nOBepxHocTHM.
CBoe06pa3eH xapaKTep nOBe.lI.eHHSI B6JIll3ll 3a.lI.eJIKH H .lI.pyrllx KOM-
nOHeHTOB TeH30pa HanpH>KeHllH. Ha pllC. 11 II 12 .lI.aHO pacnpe.lI.eJIeHlle
HanpmKeHHH arB pa3HbIX CeqeHHSIX U.HJIHH.lI.POB COOTBeTCTBeHHO .n;JISI
CJIyqaeB h = 0,5 H h = 0,25.
KpHBbIe Ha 3TllX pHCYHKax HarJIH.n;HO llJIJIlOCTPHPYlOT oc06eHHOCTll
pacnpe.lI.eJIeHHSI pa.lI.llaJIbHbIX HanpSI>KeHHH B >KeCTKO-3aIIl,eMJIeHHbIX
nJIHTax. CJIe.lI.yeT OTMeTllTb, qTO B .lI.OCTaTOqHO Y.lI.aJIeHHOH OT KpaeBoti
30HbI 06JIaCTll Ha6JIlO)laeTCSI CYIIl,eCTBeHHaSI HeJIHHeHHOCTb B pacnpe-
.lI.eJIeHHll HanpH>KeHllH no TOJIIII,HHe. XapaKTepHbIH pe3KllH H3rH6
Ha KpHBbIX npH nO.lI.XO)le K Topu.eBbIM nOBepxHocTSIM COXpaIIHeTC5I )lJISI
Bcex CeqeHllH r< 0,6. B 06JIaCTH, npllJIeralOIII.eH K 3a.lI.eJIKe, )lJIH TOJIC-
TbIX nJIllT Ha6JIlO.lI.aeTCH YMeHbIIIeHHe pa)lHaJIbHbIX HanpSI>KeHHH Ha
npBepxHocTSIX Z = + h no cpaBHeHlllO C BHYTpeHHHMll TOqKaMIi (pllC.
11). LlJISI nJIHTbI C h=0,25 TaKoH 3QJQJeKT 06HapY>KllBaeTCSI npH r>0,9.
B cJIyqae TOJICTOH nJIllTbI (h = 0,5) B6JIll3H 3a.lI.eJIKll .lI.JISI pacnpe)leJIe-
HllSI pa.lI.llaJIbHbIX HanpSI>KeHHH no TOJIIII,HHe xapaKTepH3 nepeMeHa
3HaKa (CM. pllC. 11).
HarJISI.lI.HOe npe)lCTaBJIeHHe 06 oc06eHHOCTSIX nOBe.lI.eHHSI OKPY>K-
HOro HOpMaJIbHOrO HanpH>KeHllSI ae .lI.aeT pliC. 13. Ha HeM H306pa>KeHO
pacnpe.lI.eJIeHHe HanpH>KeHllH as no TOJ1II1,HHe nJIIiTbI C h = 0,5 COOT-
BerCTBeHHO B CeqeHHHX r = 0.6; 0,9; 1,0. KaK .lI.JISI HanpH>KeHllH as,
TaK II .n;JISI HanpSI>KeHllH a r (CM. pliC. 11) B cJIyqae TOJICTOH nJIHTbI (h =


= 0,5) Ha 3aIIl,eMJIeHHOH nOBepxHocTll xapaKTepHa 3HaqHTeJIbHaSI
IIIllpllHa 06JIaCTH, B KOTOpOH 3Tll HanpSI>KeHllSI npaKTllqeCKH paBHbI
HYJIlO.
3TO 3QJQJeKT pa3rpY3KH cpe.n;HeH qaCTll nJIllTbI B6JIll3ll nOBepXHOCTII
3aIIl,eMJIeHllSI npOSIBJISIeTCSI H npH paccMoTpeHHH HanpSI>KeHHH a z .
]-:Ia pHC. 14 .n;aHO pacnpe.lI.eJIeHlle oceBbIX HanpSI>KeHllH a z no TOJIIII,llHe
nJIHTbI B Tpex CeqeHllSIX.
I1cnOJIb30BaHHbIH npll paccMoTpeHllll cMeIIIaHHoti 3a)laqH .lI.JISI
CnJIOiliHOrO U.llJIHH.lI.pa nplleM ll3yqeHllSI aCllMnTOTllqeCKHX CBOHCTB He-
H3BeCTHbIX B 6eCKOHeqHOH CllCTeMe, OqeBll.lI.HO, MO>KeT 6bITb pacnpo-
cTpaHeH Ha Bce pO.lI.CTBeHHbIe paccMoTpeHHoti 3a)laqll. Henocpe.n.-
CTBeHHO MorYT 6bITb TaK>Ke HCnOJIb30BaHbI BbITeKalOIII,He H3 HaH.lI.eH-
HbIX CBOHCTB HeH3BeCTHbIX cnoc06N YJIyqIIIeHllSI MeTO.lI.a pe.lI.YKLUlH II
nOBbIIIIeHllSI CKOpOCTH CXO.lI.HMOCTll 6eCKOHeqHbIX pSI.lI.OB, npe.n;cTaB-
JISIlOIII,llX Bce xapaKTepHcTHKH HanpSI>KeHHO-.lI.eQJOPMllpoBaHHoro co-
CTOSIHHSI.


I


I


128


9 7-2
6





r Jla6a 3
OCECIIMMETPlIqHbIE 3A,lI,AQII ,lI,JIH IJOJIoro I{IIJIIIH,lI,PA
KOHEQHOn ,lI,JIlIHbI


OCHOBHoe BHHMaHHe B 3TOH rJIaBe y.n;eJIeHO paccMoTpeHHfO npaKTH-
4eCKH BaR<HOH nepBoH OCHOBHOH rpaHHIlHOH 3a.n;a4H .n;JIH nOJIoro llHJIHH-
.n;pa KOHellHOH .n;JIHHbI. HapH.n;y C HeH paccMoTpeHbI 3a.n;a'IH .n;JIH 06JIac-
TeH, HBJIHfOIl{HXCH B HeKoTopoM CMbICJIe ITpe.n;eJIbHbIMH ITPH 1f3MeHeHlfH
reoMeTpliIf CITJIOlIlHoro If ITOJIoro llHJIHH.n;pOB KOHe4HOH .n;JIIfHbI.
Iho KacaeTCH MeTo.n;OB pelIleHHH 3a.n;all .n;JIH ITOJIoro llHJIHH.n;pa KOHell-
HOH .n;JIHHbI, TO, 0IleBHgHO, ITYTeM He3Ha4lfTeJIbHOH MO.n;HqmKallHH JIfO-
60H ITO.n;XO.n;, pa3BHTbIH .n;JIH H3Y4eHHH HaITpHR<eHHOrO COCTOHHHH CnJIOlIl-
HOm llHJIHH.n;pa, MOR<eT 6bITb paCnpOCTpaHeH Ha CJIYllaH nOJIOrO
llHJIlm.n;pa. B CBH3I1 C 3THM npH 06Il{eH XapaKTepHCTHKe pa3JIH1lHbIX CITO-
C060B pelIleHIIH rpaHH4HbIX 3a.n;a4 .n;JIH ITOJIoro llHJIHH.n;pa MOR<HO IT04TlI
.n;OCJIOBHO nOBTOpHTb BBe.n;eHHe K ITpe.n;bI.n;YIl{eH rJIaBe. D03TOMY OCTaHO-
BHMCH JIHlIlb Ha XapaKTepHCTHKe HeKOTOpbIX KOHKpeTHbIX pa60T, BbI-
nOJIHeHHbIX C HCnOJIb30BaHHeM pa3JIH1lHbIX ITO.n;XO.n;OB.
3a.n;alla .n;JIH ITOJIOrO llHJIHH.n;pa KOHellHOH .n;JIHHH B paMKaX MeTO.n;a
O.n;HOpO.n;HbIX pelIleHHH, nOR<aJIYH, HaH60JIee nOJIHO paCCMOTpeHa B
pa60Te f331. ABTOPOM 3TOH pa60TbI OITHCaH XapaKTep Tpy.n;HoCTefl,
B03HHKafOIl{HX ITPH BbI'IHCJIeHHH KOpHeH XapaKTepHCTH4ecKoro ypaB-
HeHHH, H ITpe.n;JlOR<eH cnoc06 HX ITpeO.n;OJIeHHH. PaCCMOTpeHHbIe KOHKpeT-
Hble CJ1Y4aH HarpYR<eHHH n03BOJIHfOT C.n;eJlaTb HeKOTOpble 06Il{He Bb!-
BO.n;bI 0 B03MOR<HOCTHX MeTO.n;a.
BaR<HOe npHMeHeHHe O.n;HOpO.n;HbIX pelIleHHH ITepBOH OCHOBHOH rpa-
HHIlHOH 3a.n;a4H .n;JIH ITOJloro llHJIHH.n;pa .n;aHO B pa60TaX f28, 29]. B HHX
YKa3aH CITOc06 IToC'rpoeHHH HeITpOTHBOpe4HBbIx TeopHi'I llHJIHH.n;pHlle-
CKHX 060JIOlleK, IT03BOJlHfOIl{HX Y4ecTb norpaHcJloH JIf060ro nopH.n;Ka.
MeTo.n; ITOCTpOeHHH 06Il{ero pelIleHHH rpaHHIlHbIX 3a.n;all .n;JIH ITO-
JIoro llHJIHH.n;pa, COBITa.n;afOIl{HH C pa3BHTbIM E. n. A6paMHHoM MeTO.n;OM
. .n;JlH CnJIOlIlHoro llHJIHH.n;pa, OITHCaH BO MHorHX pa60Tax. 3.n;ecb cJle.n;yeT
YKa3aTb Ha pa60TY caMoro E. n. A6paMHHa f31, ITpOaHaJIH3IIpOBaB-
lIlero pa3.TJH4HbIe THITbl rpaHH4HbIX YCJIOBHH, He ITpIlBO.n;HIl{HX K 6ecKo-
He4HbIM CIICTe1\1aM. AHaJIOnr4Hble CJIY4aH .n;ecpopMlIpoBaHHH paccMoT-
peHH B pa60Tax 149, 154, 1551. OCHoBHaH CMelIlaHHaH rpaHII4HaH 3a.n;a-
4a .n;JIH ITOJIoro llJlJIIIH.n;pa KOHe4HOH gJIHHbI paCCl\1OTpeHa B CTaTbe f27]
ABTOpaMH YCTaHOBJleHa perYJIHpHOCTb 6ecKoHe4HbIx CHCTeM .n;JIH CJIY-
4aH 3a.n;aHlIH CMeIl{eHHH no llHJIHH.n;pH4ecKHM ITOBepXHocTHM H HaITpH-
R<eHHH Ha TOpllax.


130


Cpe.n;H 3apy6eR<HbIX aBTOpoB 60JIblIlOH 06'beM HCCJIe.n;oBaHHH npH
paccMO'rpeHIIH 3a.n;all .n;J1H CnJIOlIlHOrO H nOJIoro llHJIHH.n;pOB BbIITOJIHeH
I1eHrapoM II coaBTopaMH f222, 225, 226]. B 3THX pa60Tax He ITpe.n;npH-
HHMaJlaCb nonbITKa 060CHOBaTb cpopMaJIbHble noC'rpoeHIIH .n;OKa3aTeJIb-
CTBOM pa3pelIlIlMOCTH 6eCKOHe1lHbIX CHCTeM MeTO.n;OM p
.n;YKllHII. DpH
coxpaHeHHH He60JIblIlOrO KOHe4Horo 4HCJIa ypaBHeHIIH B CHCTeMax
paCCMOTpeH pH.n; KOHKpeTHbIX 3a.n;all YITpyroCTH H Te
MoynpyrocTH.
04eHb Me.n;JIeHHaH CXO.n;HMOCTb pH.n;OB .n;JIH HaITpHR<eHHH na rpaHHlle
llIlJIHH.n;pa CHHR<aeT lleHHOCTb nOJlY4eHHbIX YKa3aHHbIM cnoc060M KO-
JIHlleCTBeHHbIX OlleHOK HaITpHR<eHHOCTH.
OrMe4eHHbIH He.n;OCTaTOK ITpHBe.n;eHHbIX BbIlIle pa60T xapaKTepeH
TaKR<e .n;JIH 60JIblIlOrO 4HCJIa HCCJle.n;OBaHHH, BbIITOJIHeHHbIX HITOHCKHMH
aBTopaMH f245, 259, 260]. Pa60Ta f259] co.n;epR<HT 061I1HPHYfO 6H6JIIfO-
rpacpHfO .n;pyrHx HITOHCKHX ITy6JIHKallHH H HHTepecnyfO .n;HCKYCCIIfO 0
B03MOR<HOCTHX HCnOJIb30BaHHoro aBTopaMH MeTo.n;a H 06 OlleHKax
KalleCTBa nOJlYlleHHoro pelIleHHH. DpHBe.n;eHHbIe B HeH pe3YJIbTaTbI pac-
4eTOB Haxo.n;HTCH B HBHOM ITpOTHBOpellHH C TeM CITOC060M, KOTOpbIM 3TH
pe3YJIbTaTbI nOJlYlleHbI. PaCCMaTpHBaH pa3JIH4Hble THnbI pa3pbIBHbIX
HarpY30K H y.n;ep1KHBaH B pH.n;ax .n;JIH HaITpHR<eHHH KOHe4Hoe 4HCJIO
4JIeHOB (nopH.n;Ka 15), HeB03MOR<HO 06HapYR<HTb HaJIH4He CKallKa B
OKPVR<HbIX HaITpHR<eHHHX H ero BeJlHIlHHY. u
CJIOR<HOCTb 3a.n;a4H cTpororo y.n;oBJIeTBOpeHHH rpaHH4HbIX YCJIOBHH
Ha BceH nOBepxHocTH llHJIHH.n;pa KOHe4HOH .n;JIHHH 06YCJIOBHJIa pa3pa-
60TKY pH.n;a npH6JIHR<eHHbIX no.n;xo.n;OB. Do.n;p06HbIH aHaJ1H3 pa60T
COBeTCKHX aBTopoB MOR<HO HaHTH B 0630pe no JIHHeHHOH TeopHH yn-
pyrocTH {I 27]. HeKoTopble H3 ITpH6JIHR<eHHbIX MeTo.n;OB B HaCTOHIl{ee
BpeMH ITOTepHJIH CBoe 3Ha4eHHe, o.n;HaKO .n;pyrHe, pa3BHTble, HanpHMep,
B pa60Tax f36, 41, 1671, H ceH4ac lIlHpOKO HCITOJIb3YfOTCH npH paCCMOT-
peHHH npIlKJIa.n;HbIX 3a.n;a4. I-b 3apy6eR<HbIX pa60T CJIe.n;yeT OTMeTHTb
ny6J1HKallHH XOpBH f219, 2201, B KOTOpbIX Ha npHMepe 3a.n;a
 .n;JI51 nOJIy-
nOJIOCH H ITOJIy6ecKoHe4Horo llHJIHH.n;pa .n;an BapHallHoHHHH no.n;xo.n; K
nOCTpoeHHfO pelIleHHH rpaHH4HbIX 3a.n;all .n;JIH TeJI KOHe4H
Ix pa3MepoB.
3a.n;a4a .n;JIH ITOJIoro KpyroBOro llHJIIiH.n;pa KOHeqHOH .n;JlHHbI HC-
CJIe.n;OBaJIaCb C nOMOIl{bfO MeTo.n;a, OCHoBaHHoro Ha aHaJlOrIlH MeR<.n;y
ITJIOCKHMH H ocecHMMeTpHIlHbIMH nOJIHMH fIa7, 146, 187]. HCnOJIb-
30BaHHe TaKoro no.n;xo.n;a .n;aJIO B03MOR<HOCTb JIerKO paccMoTpeTb Te
cJlYllaH rpaHHIlHbIX YCJIOBHH, KOTopble He npHBo.n;HT K 6eCKOHe1lHbIM C
-
CTeMaM B paMKax MeTo.n;a E. n. A6paMHHa H O.n;HOpO.n;HbIX pelIleHHH.
B cJlyqae OCHOBHbIX rpaHH4HbIX 3a.n;a4 HCnOJlb30BaHHe 0606Il{eHHbIX
aHaJIHTH4eCKHX CPYHKllHH He npHBo.n;HJIO K KaKHM-JIH60 ynpOIl{eHHHM.
DOMHMO 3a.n;a4 .n;JIH nOJIoro llHJIHH.n;pa KOHeqHOH .n;JIHHbI H nOJIy6ec-
KOHe4Horo llHJIHH.n;pa B 3TOH rJIaBe paCCMaTpHBafOTCH oceCHMMeTpH4Hble
rpaHH4Hble 3a.n;a4H .n;JIH CJIOH C llHJIHH.n;pHlleCKIfM OTBepCTII
M H aHaJIO

rI14HaH 3a.n;alla .n;JIH nOJIynpocTpaHCTBa C llHJIHH.n;pH4eCKOH ITOJlOCTbfO.
L(JlH CJIYllaH MaJIOH TOJIIl{HHbI ITJIHTbI B pa60Te {I4] .n;aHO aCHMITTOTH4e-
CKoe pelIleHHe ITepBOH H3 YKa3aHHblx 3a.n;a4 Ha OCHOBe MeTo.n;a O.n;HO-
pO.n;HbIX pelIleHHH. 9TOT MeTO.n; HCITOJIb3yeTcH TaKR<e B pa60Te f9] ITPI1
H3Y4eHHII nanpmKeHHoro COCTOHHHH CJIOH C KpyroBbIM OTBepcrHeM


g*


131




06JIa.naIOmHX o£Ipe.neJIeHHbIMH cBoikTBaMH CHMMeTpHH OTHOCHTeJIbHO
Koop.nHHaTHbIX oceH. E.nHHCTBeHHbIM MOTHBOM, 06YCJIOBJIHBaIOmHM
TaKoe pa36HePIHe o6mero rrOJI51, 5lBJI51eTC51 CTpeMJIeHHe H36e2KaTb H3-
JIHIIIHeH rpOM03.nKOCTH BbIKJIa.nOK. ECTeCTBeHHO, qTO B CJIyqae rro
oro
UHJIHH.npa KOHeqHOH .nJIHHW pa3.neJIbHOe H3yqeHHe CHMMeTpHqHOH H
aHTHCHMMeTpHqHOH OTHOCHTeJIbHO rrJIOCKOCTH
 = ° qaCTeH Harrp51-
2KeHHOrO COCT051HH51 5lBJI51eTC51, C 3TOH (
TOqKH 3peHH51, ern.e 60JIee Heo6xo.nH-
MbIM.
Bce BbIKJIa.nKH, CB513aHHble C rrOCT-
poeHHeM o6ruero peIIIeHH51 rpaHHqHbIX
3a.naq H £Io.nqHHeHHeM ero o£Ipe.neJIeH-
HOH CHCTeMe rpaHHqHbIX YCJIOBHH, 6y.neM
BeCTH B 6e3pa3MepHbIX Koop.nHHaTax,
BbI6Hpa51 B KaqeCTBe XapaKTepHoro pa3-'

Mepa BHeIIIHHH pa.nHYC UHJIHH)I.pa, T. e. p
P
 H Pl
r=-, Z=-, h=-, E -.
Po Po
 Po
B COOTBeTCTBHH C rrpHMeH51eMbIM B
pa60Te MeTo.nOM rrOCTpOeHH51 o6rn.ero
peIIIeHH51 rpaHHqHbIX 3a.naq B paCCMaT-
pHBaeMOM CJIyqae rrOJ10ro II.HJIHH.npa
KOHeqHOH .nJIHHbI pa.nHaJIbHa51 U H oce- Pt
Ba51 W KOMrrOHeHTbI BeKTOpa rrepeMern.e-
HHH rrpe.nCTaBJI51IOTC51 B BH.ne CYMMbI: PIIC. 15.
U = u 1 + u 2 , W = WI + W 2 . (13.1)
DepBble CJIaraeMble 3.neCb CTp051TC51 B COOTBeTCTBHH C rroTpe6HocTbIO
HMeTb B KOJIbII.e E < r < 1 rrpH Z = h .nBe rrpOH3BOJIbHble <fJYHKUHH B
Bblpa2KeHH51X .nJI51 U H W HJIH a z H'Tn, BbI6opOM KOTOpbIX M02KHO y.nOBJIe-
TBOpHTb rrpOH3BOJIbHble rpaHHqHble YCJIOBH51 B CMern.eHH51X HJIH Harrp51-
2KeHH51X Ha Topue II.HJIHH.npa.
ECJIH rrpH paCCMOTpeHHH rpaHHqHbIX 3a.naq )I.JI51 CrrJIOlliHoro II.HJIHHJJ:-
pa rrpOH3BOJIbHble B Kpyre r < 1 <fJYHKII.HH rrpe.nCTaBHTb B BH.ne p51.nOB rro
J o (AI) H J 1 (Ajr), TO B .naHHOM CJIyqae M02KHO HCrrOJIb30BaTb orrpe.ne-
JIeHH
Ie KOM6HHaII.HH <fJYHKII.HH EecceJI51 H HeHMaHa. Bblpa2KeHH51 .nJI51
TOJIrn.HHHbIX <fJYHKII.HH (<fJYHKII.HH Koop.nHI-IaTbI z) OCTaIOTC51 TaKHMH 2Ke,
KaK H B CJIyqae crrJIOlliHoro II.HJIHH.npa. B CB513H C 3THM )I.JI51 CHMMeTpHq-
Horo OTHOCHTeJIbHO rrJIOCKOCTH Z ,0 Harrp512KeHHoro COCT051HH51 rrep-
Bble CJIaraeMble B (13.1) rrpe.ncTaBHM B BH.ne
co
,-,
U 1 =
 (Ajz sh 'Ajz + C i ch 'A/) N] (AIr),
;=1 0:1.2)
WI = Aoz +
 (A. 3m
4 ShAjZ-AIzcht'iz-CishA;2' No (A;r).
;=1 I m I
3.necb, KaK H B (9.1), Ao, A j' C j - rrpOH3BOJIbHble rrOCT051HHble, a
No (A/) ---= J 1 (Ai) Yo (A;r) - Y 1 (A) J o (Ajr), (13.3)
N 1 (Air) = J 1 (Ai) Yl (Air) - Y 1 (Ai) J 1 (Air).
133


B cJIyqae C2KaTH51 ero .n8BJIeHHeM, paBHoMepHo pacrrpe.neJIeHHbIM £Io
KOHII.eHTpHqeCKoMY KOJIbII.Y BOKpyr OTBepCTII51. DOJIHMOMeHTHa51 TeopH51
TOJICTbIX rrJIHT, rr03BOJI51IOrn.a51 paccMoTpeTb 06ruHH CJIyqaH .ne<fJ0PMH-
pOBaHH51 CJI051 C oTBepcTHeM, rrpHBe.neHa B pa60Tax [94, 951.
MeTO.nOM, aHaJIOrHqHbIM HCrrOJIb3yeMoMY 3.neCb, B pa60Te [196J
H3yqaJIaCb 3a.naqa 0 Harrp512KeHHOM COCT051HHH rroJIynpocTpaHcTBa C
II.HJIHH.npHqeCKOH rrOJIOCTbIO. 06rn.He Bblpa2KeHH51 .nJI51 KOM£IOHeHTOB
BeKTopa cMern.eHHH rrpe.ncTaBJIeHbI HHTerpaJIaMH <I>ypbe H Be6epa.
DOJIyqeHHa51 rrpH y.noBJIeTBopeHHH rpaHHqHWX YCJIOBHH CHCTeMa HH-
TerpaJIbHWX ypaBHeHHH peIIIaJIaCb MeTo.nOM npocToH pe.nYKUHH.
l(aK H B rrpe.nbI.nymeH, TaK H B 3TOH rJIaBe rrpH o630pe JIHTepaTypbI B
OCHOBHOM 6bIJIH YKa3aHbI pa6oTbI, OTHOC51rn.HeC51 K CJIyqaIO ocecHMMeTpHq-
HOH .ne<fJopMaII.HH. B paMKaX MeTo.na o.nHOpO)I.HbIX peIIIeHHH H MeTo.na,
HCrrOJIb3yeMoro B pa6oTe, rrepexo.n OT ocecHMMeTpHqHOrO CJIyqa51
K HeCHMMeTpHqHOMY He BbI3bIBaeT 3aTpy.nlJ:eHHH Ha 3Tarre rrOCTpOeHH51
o6rn.ero peIIIeHH51. O.nHaKo rrOCJIe.nYIOrn.He BbIKJIa.nKH, CB513aHHble C <fJaK-
THqeCKHM y.noBJIeTBopeHHeM rpaHHqHbIX YCJIOBHH, qpe3BblqaHHO rpo-
M03.nKH. HeKoTOpble Ba2KHble CJIyqaH HeocecHMMeTpHqHbIX 3a.naq pac-
cMoTpeHbI B rr51TOH r JIaBe. CJIe.nyeT YKa3aTb 3.neCb TaK2Ke .nBe pa60TbI
3. H. EaH.nbI [30, 311, B KOTOpbIX paccMaTpHBaeTC51 06ruHH CJIyqaH
HeocecHMMeTpHqHOH .ne<fJ0pMaUHH rrOJIoro II.HJIHH.npa KOHeqHOH .nJIHHW.
Ba2KHbIH pe3YJIbTaT rrpH paccMoTpeHHH 3TOH 3a.naqH rrOJIyqeH
B. A. I1BaHoBbIM [1031. OH rrOKa3aJI, qTO aCHMrrTOTHqeCKHe CBOHCTBa
HeH3BeCTHblX B 6eCKOHeqHbIX CHCTeMax B CJIyqae HecHMMeTpHqHOH )I.e-
<fJ0pMaII.HH TaKHe 2Ke, KaK H B oceCHMMeTpHqHOM CJIyqae.
OCHOBOH 3<fJcpeKTHBHoro peWeHH51 rpynrlbI paccMoTpeHHblx HH2Ke
3a.naq 5lBJI51eTC51 3HaHHe aCHMrrTOTHqeCKHX CBOHCTB HeH3BeCTHblX B
6eCKOHeqHbIX CHCTeMaX. 3TH CBOHCTBa He Bcer.na MorYT 6bITb YCTaHOBJIe-
HbI rrYTeM <fJopMaJIbHoro aHaJIH3a COOTBeTCTBYIOrn.HX ypaBHeHHH. Do-
JIyqeHHble B £Ipe.nbI.nyrn.HX rJIaBaX pe3YJIbTaTbI rr03BOJI51IOT, O)I.HaKO,
CCPOPMYJIHpOBaTb cpH3HqeCKH orrpaB.naHHble rrpe)I.rrOJI02KeHH51 0 CBOH-
CTBax HeH3BeCTHblX H 3aTeM rrpoBepHTb HX HerrpoTHBOpeqHBOcTb. TaKoH
rro)I.xo.n HCrrOJIb30BaH B pa60Tax [75, 781, B KOTOpbIX paccMaTpHBaeTC51
3a.naqa .nJI51 rrOJIoro II.HJIHH.npa KOHeqHOH )I.JIHHbI. L{pyrnM, BKJIIOqeHHbIM
B 3TY rJIaBY 3 a .naqaM, rrOCB51rn.eHbI CTaTbH [74, 76, 911.



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B rrpe)I.bI.nyrn.eH r JIaBe 6bIJIH paccMoTpeHbI OCHOBHbIe rpaHli'lHble
3a.naqH )I.JI51 CrrJIOWHOro II.HJI1m.npa KOHeqHOH .nJIHHbI. He3HaqHTeJIbHa51
Mo.nH<fJHKaII.H51 HCrrOJIb30BaHHbIX rrpH 3TOM £Ipe)I.CTaBJIeHHH )I.JI51 KOMrro-
HeHT BeKTopa cMern.eHHH rr03BOJIHJIa rrOJIyqHTb peweHHe ypaBHeHHH
yrrpyroro paBHOBeCH51 C .nocTaTOqHOH CTerreHbIO rrpOH3BOJIa .nJI51 y.n OB -
JIeTBOpeHH51 rpaHHqHWX YCJIOBHH Ha rroBepXHOCT51X nOJIoro UHJIHH.npa
KOHeqHOH .nJIHHbI (pnc. 15).
Bblwe, rrpH paccMoTpeHHH rpaHHqHbIX 3a.naq .nJI51 rrp51MoyroJIbHHKa
H CrrJIOWHOrO II.HJIHH.npa KOHeqHOH .nJIHHbI, rrOCT051HHO HCrrOJIb30BaJIHCb
B03M02KHOCTH pa3)I.eJIbHOrO H3yqeHH51 rrOJIeH Harrp512KeHHH H cMern.eHHH,


132


I


I




fIpH TaKOM BbI60pe KOM6HHaQHH CPYHKQHH BeCCeJIH B Ka'leCTBe na-
paMeTpa 'Aj CJIeJJ.yeT B3HTb KOpHH ypaBHeHIfH N I (I.B) = O. fIpIf 3TOM
CIfCTeMbI 4>YHKQIfH N I ('Ajr) If No ('A/r) CTaHOBHTCH opTOroHaJIbHbIMIf B
KOJIbQe B < r < I C BeCOM r. CIfCTeMa <PYHKQIfH N I ('Ajr) 06pa3yeT nOJI-
HYIO cHcTeMY B YKa3aHHOM KOJIbQe, a CHCTeMa <PYHKQIfH No ('Ajr) CTa-
HOBIfTCH nOJIHOH nOCJIe JJ.onOJIHeHIfH ee eJJ.IfHIfQeH. C 3TIfM CBOHCTBOM
CIfCTeMbI cpYHKQIfH No ('Air) H CBH3aHO nOHBJIeHHe CJIaraeMoro Aoz
B BblpameHIfIf .l\JIH WI'
fIOJIHble CIfCTeMbI <PYHKQIfH MorYT 6bITb nOJIY'leHbI If C IfCnOJIb30Ba-
HIfeM .lIPyroH KOM6IfHaQIfIf <PYHKUMH BeCCeJIH. CBOHCTBaMIf nOJIHOTbI
H opTOrOHaJIbHOCTIf B KOJIbQe B < r -<: I 06.na.l\aIOT
No ('A/) = J o ('A;) Yo ('Ajr) - Yo ('A;> J o ('Ajr),
N I ('Air) = J o ('A j ) Y I ('Air) - Yo ('A j ) J I ('Ajr),
r.l\e 'Aj - KOpHH ypaBHeHHH N" (AB) = O. fIpH 3TOM nOCTOHHHaH Ao
B BblpameHIfIf .l\JIH WI CTaHOBIfTCH JIHIIIHeH If MomeT 6bITb onYIQeHa.
B 3TOi1 rJIaBe IfCnOJIb3YIOTCH KOM6IfHaQHH <PYHKQIfH BeCCeJIH (13.3).
Co06pameHIfH, npHBO.l\.SIIQHe K nocTpoeHIfIO nOJIHbIX B KOJIbQe B -<:
-<: r < I CHCTeM <PYHKQHH B BH.l\e (13.3) H (13.4), H .l\OKa3aTeJIbCTBO HX
nOJIHOTbI MomHO HaHTIf B MOHorpa<pHH 3. 11. THT'lMapliia fI 73].
B aHTHcIfMMeTpHIlHoM OTHOCHTeJIbHO nJIOCKOCTH Z = 0 cJIYllae .l\e-
<POPMHpoBaHHH QHJIHH.l\pa BblpameHHH COCTaBJIHIOIQHX U I H WI B (13. I)
no aHaJIOrHH C (9.2) 3anHIIIeM B BH.l\E'
m-I

U I = - 2 m Cozr + ;::1 (A/z ch 'Aiz + C j sh AiZ) N i ('A/r),


- 4 m - 1 Q] (kllr) " cos knz,
mk n


(13.6)


n
w 2 = -
 rro (kllr) + rQI (kllr)]
in kllz.
11=1
3':{ecb BBe.l\eHbI CJIe.lIYlOIQHe KOM6HHaUHH MO.l\H<pHQIfpoBaHHbIX <PYHKQHH
EeCCeJIH:


(13.4)


f 0 (knr) = 'V"lo (kllr) + 611Ko (krr),
Qo (k'l r ) = qJo (kllr) + p"Ko (kllr),
1 dl'o (kilT) k r =
 dQo (kilT) ,
r](k"r) = k;; dT ,QI( II) k n dT


(13.7)


6 - n p OH3BOJIbHble nOCTOHHHble.
B KOTOpbIX 'VII' no qn H Pn rtv- p MH p OBaHHH
fIpH HCCJIe.l\OBaHHH aHTHCHMMeTpHIlHoro CJIYllaH .l\e

nOJIoro QHJIHH.l\pa B KalleCTBe BTOpbIX COCTaBJIHIOIQHX B (13.1) CJIe.l\yeT
IICnOJIb30BaTb BblpameHHH


U1, = - 2 m:;; 2 Bozr -
I r f 1 (k"r) + rQo (krr)-
- 4 m -I QI (kllr) ] sin kllz,
mk n


(138)


W2 = Bor2 + Do In r -
 [r 0 (K"r) + rQI (knr)] cos knz,
n=1


(13.5)


WI = Ao + C O Z2 +
 I A, 3':1>- 4 ch 'Ajz - A,z sh AjZ-
=, I
- C j sh 'Aiz j No ('Ajr).
fIPH'lHHbI, n06Y*JJ.alOIQHe BKJIIOIlHTb B 3TH BblpameHHH CJIaraeMble c
Ao H Co, TaKHe me, KaK If H3JIOmeHHble npH nocTpoeHIfH BblpameHIfH
(9.2) npe.l\bI.l\YIQeH rJIaBbI.
fIpIf nocTpoeHIfIf BTOpbIX CJIaraeMbIX B COOTHOllieHIfHX (13.1) IIfOmHO
HCnOJIb30BaTb BblpameHIfH (9.3) If (9.4) If3 npe.l\bI.l\Yl.QeH rJIaBbI. CO.l\ep-
:>KaIQerOCH B HHX npOIf3BOJIa .l\OCTaTOIlHO .l\JIH Y.l\OBJIeTBopeHHH rpaHIfIl-
HbIM YCJIOBHHM Ha QIfJIIfH.l\pIflleCKoH nOBepxHocTIf r = I.
llOnOJIHHTeJIbHbIH He3aBIfcIfMbIH Ha60p 'laCTHblX pellieHIfH, Y.l\OBJIe-
TBOpHIOl.QIfH rpaHIfIlHbIM YCJIOBIfHM Ha BHYTpeHHeH nOBepxHocTIf UHJIHH-
JJ.pa, MomHO nOJIYIlIfTb H3 (9.3) H (9.4) nYTeM CHCTeMaTIflleCKOH 3aMeHbI
<PYHKQIfH BeCCeJIH nepBoro p0.l\a <PYHKUHHMH BeCCeJIH BToporo pO.l\a.
ITOJIYlleHHble npH 3TOM Bblpa:>KeHHH, eCTeCTBeHHO, CJIe.l\yeT .l\onOJIHHTb
cooTBeTcTBYIOl.QHMH CJIaraeMbIMIf OC06bIX Ha OCM 3JIeMeHTapHblx pellieHIfH.
B HTore nOJIYllaeM CJIe.lIYIOl.Qee npe.l\CTaBJIeHIfe KOMnOHeHTOB BeKTopa
CMel.QeHIfH B cJIYllae cHMMeTpHIlHoH .l\e<p0pMaUIfIf:
u 2 = Do + Bor +
 I f. (k.-,r) + rQo (k"T) __
T 11=1,


fIOCTpoeHHe npe.l\CTaBJIeHHH (13.6) H (13.8) BbillOJIHeHO B npe.l\noJIo:
meHHJi 'no npJiHHT TaKoH cnoc06 npO.l\OJImeHHH rpaHJiIlHbIX YCJIOBJiH
C YllaC
KoB UJiJIHH.l\pHlleCKHx nOBepxHocTeH -h < z -<: h Ha BClO
nOBepxHocTb nOJIoro 6ecKoHellHoro UJiJIHH.l\pa, npJi KOTOPO

k =!!:!!:.. ECJIH nOJIOmHTb k = (2n - 1 )n/2h, TO nOJIHOTa peweHIfIf
n h . n 6
(13.6) H (13.8) Ha QHJIHH.l\pHlleCKHx nOBepxHocTHX coxpaHHeTcH H e3
npHBJIelleHHH 3JIeMeHTapHblx peweHHH C npOIf3BOJIbHbIMIf nOCTOHHHbIlIIlJ
Bo H

'JIb30BaHHe 3aKOHa fYKa .l\aeT B03MomHOCTb JIerKC: 3anHcaTb
cooTBeTcTBYIOIQHe nocTpoeHHbIM BblpameHHHM .l\JIH CMeIQeHIf
 3HalleHHH
IWMnOHeHTOB TeH30pa HanpHmeHHH. ITOCKOJIbKY B .l\aJIbHelJWeM nOL\-
06Hble BbIKJIa.l\KH npOBO.l\HTCH JIIfWb .l\JIH cIfMMeTpHIlHoro OTHOCHTeJIb-

o nJIOCKOCTIf Z = 0 CJIYllaH .l\e<popMHpoBaHHH QIfJIHH.l\pa, TO BbIllHIIIei\I
BblpameHHH ,UJIH HanpHmeHHH, COOTBeTCTBYIOIQHe JIHWb 3TOMY CJIYllalO:
I
1 _ mBo+Ao _ Do _
 ( A ZShA.Z +C.ch'Aiz)-N I ('Ajr) +
-0... - 2 2
 I I I r
2G... m - r ;=1
+
I (AjAjZ sh 'Aiz +
 A; ch "Ail + Ci'A i ch 'AIZ) No ('Air) +
135


134




,co [ ]
+ 2:1 knr 0 (knr) - r r I (knr) + knrQI (knr) +
m-] 3m-2 ]
+ 4 k QI (knr) - Qo (knr) cos knz,
mnT m
mBo + Ao DO
 (A h 1 C h I
m - 2 + T2 + £;1 jZ S I\,jZ + j C AjZ) r N I (Air) +



 f 4. AHamm 6eCI\OHe'IHbIX CIlCTeM
B nepBoii OCHOBHOii I'paHH'IHOii 3a
a'Ie


]
:'0 ae =


PaCCMOTpliM CJIe
YlOIllYlO rpaHlIqHYlO SalIaqy LlJIH TOJICTOCTeHHoro

llJIlIHLlpa KOHeqHOH LlJIlIH
:
2
 a, (r, + h) = fl (r), do 'frz (r, + h) = + fJ!I (r);


+

rr


00

 AjchAjzNo(A;r) +
 [+rdkn r ) + m;;;2
J=I '1=1
4 m-] J ,
- k . QI (knr) cos knz,
'IT m


Qo (knr) -


2
 aT (8, z) = f2 (Z),
]
20 aT (1. Z) = fa (z),


I .
20 'f rz (8, Z) = fjJ2 (z),
I
20 'f TZ (1, z) = fjJg (z).


(14.1)


00
] 2Bo m - 1 A
 (2 m - 1 A
20 a, = m'-2 + tn=2 0 +.LJ -m- jChAjZ-
j=1
- AjAjZ sh AjZ - CjAj ch AjZ) No (Ajr) -
1 [ knf 0 (knr) +
+ ; Qo (knr) + knrQI (knr)] cosknz,
]
 I m-2 )
20 'fTZ=
1 AjAjZchAjZ- m AishAl+CjAjshAiZ NdAjr)-
00 [

 m-l
-
 ktli (knr) + knrQo (knr) - 2 QI (knr) ] sin knz.
n=1 m
Ha OCHOBe aHaJIliSa rrpliBeLleHHbrx Bbr11le Bbrpa)f{eHlIH LlJIH KOMrro-
HeHT BeKTopa cMeIlleHlIH 1I TeRsopa HarrpH)f{eHlIH MO)f{HO JIerKO YKasaTb
pHLl crreU:lIaJIbHbrX CMernaHHblX rpaHlIqHbrX SaLlaq, npll lICCJIeLlOBaHlili
KOTopbrx COLlep)f{aIllHeCH B o6IlleM perneHlIli rrpOliSBOJIbHbre nOCTOHHHbre
onpeLleJIHlOTCH 6es perneHlIH 6eCKOHeqHbIX CliCTeM. TaKlie rpaHlIqHbIe
SaLlaqll BosHliKalOT, HanpliMep, npll saLlaHlili Ha
lIJIlIHLlPllqeCKp.x no-
BepxHocTHX lIJIli Ha Topu:e U:lIJIlIHLlpa HOpMaJIbHOH K rpaHlIqHOH no-
BepxHocTli COCTaBJIHlOIlleH BeKTOpa rrepeMeIlleHlIH 1I KaCaTeJIbHoro KOM-
nOHeHTa TeHsopa HanpH)f{eHlIH. OLlHaKo Ha1l6oJIbrnliH npaKTlIqeCKliH
lIHTepeC rrpeLlCTaBJIHlOT OCHOBHbIe rpamIqHbre SaLlaqll LlJIH nOJIoro U:"-
JIlIHLlpa KOHeqHOH LlJIlIHbI. OCHoBHbre TPYLlHOCTlI, BosHliKalOIlllle npll
OTbrCKaRliIt perneHlIH 3TlIX SaLlaq, CBHsaHbr C nOCTpoeHlIeM 3qxpeKTlIBHoro
aJIropliTMa perneHlIH 6eCKOHeqHbIX CliCTeM. B 3TOH rJIaBe rrOLlpo6HO 6y-
LleT paccMoTpeHa nepBaH OCHOBHaH rpaHlIqHaH SaLlaqa, npll lICCJIeLlOBa-
Hlill KOTOpOH YLlaeTcH npllMeHlITb pa3BlITbre B npeLlbrLlYIllHx rJIaBaX cno-
C06bl perneHlIH 6eCKOHeqHbrX CHCTeM.


(13.9)


3LleCb f2 (z) H fa (z) rrpeLlrrOJIaralOTCH qeTHbrMH, a 'P2 (z) 1I fjJa (z) - He-
qeTHbrMH <PYHKU:lIHMH aprYMeHTa z. DOMHMO 3TOro CqllTaeM, qTO B pac-
CMaTpliBaeMOH KpaeBOH SaLlaqe BbIIlOJIHHlOTCH YCJIOBHH rrapHOCTH Kaca-
TeJIbHbrX HarrpH)f{eHlIH B yrJIOBbrX TOqKaX, T. e. 'PI (1) = 'Pg (h),
fjJt (8) = fjJ2 (h). CLleJIaHHbre LlorrYIlleHlIH rrOSBOJIHlOT lICrrOJIbSOBaTb rrpll
lIsyqeHHH HanpH)f{eHHOro COCTOHHHH
HJIlIHLlpa C I rrOBepXHOCTHbrMli
HarpysKaMH (14.1) Bbrpa)f{eHHH LlJIH HarrpH)f{eHlIH (13.9).
Dpou:eLlypa rrOJIyqeHlIH aJIre6pallqeCKliX COOTHorneHlIH LlJIH HellS-
BeCTHbIX BeJIlIqllH B o6meM rrpelICTaBJIeHlIli KOMrrOHeHTOB TeHsopa Ha-
rrpH)f{eHlIH (13.9) BrrOJIHe aHaJIOrllqHa TeM, KOTopbIe peaJIliSOBaHbr paHee
B SaLlaqax LlJIH npHMoyroJIbHlIKa 1I CnJIOrnHoro
HJIlIHLlpa. Y CJIOBliH LlJIH
KaCaTeJIbHbIX HarrpH)f{eHlIH B COOTBeTCTBlili C Bbr6paHHbrMli SHaqeHlIHMlI
LlJIH 'A/ 1I k n rrpliBOLlHT K TaKliM CBHSHM Me)f{LlY lICKOMbIMli BeJIlIqllHaMlI:
( m - 2 \ 'PIi
CiAJ = Aj m Ajh dh AJh + sh Ai h '
knrt(k n ) =2 tn-I QtCk,.)-knQo(kr)-fjJ3n, (14.2)
m


m-l
knf I (k n 8) = 2 m QI (k n 8) - k n 8Qo (k n 8) - tj)2n.


3LleCb BBeLleHbI CJIeLlYlOIlllle 060SHaqeHlIH LlJIH K03<p<p1IU:lIeHToB 6ecKo-
HeqHbrX pHLlOB, rrpeLlCTaBJIHlOIllllX <PYHKUlIlI, orrpeLleJIHlOIlllle pacrrpeLle-
JIeHlIe KaCaTeJIbHbIX HanpH)f{eHlIH Ha rroBepxHocTHX UliJIlIHLlpa:
1
'Plj = Do (
jo e) 5 fjJi (r) N I (A /) rdr,
E
1 0 2 2 ( 1
Do (Ai' 8) = 2 [No (AJ - 8 No 1\,;8)],


(14.3)


h
2 j ' .
'PIn =
 fjJl (Z) '.In knzdz (l = 2, 3).
,/
o
LlJIH npeo6pasoBaHliH K aJIre6pallqeCKHM COOTHorneHlIHM ypaBHe-
HlIH, Bbrpa)f{alOIllero rpaHlIqHOe YCJIOBlie LlJIH Oz Ha Topue U:lIJIlIHLlpa,
Heo6xoLlllMO rrpeLlcTaBlITb 6eCKOHeqHbr;.:1t1l PHLWMH no No (Air) <PYHKUlili


136



3i




ro (knr), Qo (knr), knr Q1 (k n , r). fIoJIY'IHTb 3Ha'IeHHH K03cp4mUHeHTOB
3THX paSJIOLKemn'i npH npOHSBOJIbHbIX 1'n' {jn' Pn H qn ,D.OBOJIbHO JIerKO,
eCJIH HCnOJIbSOBaTb ,D.HcpcpepeHUHaJIbHbIe ypaBHeHHH, KOTOpblM Y,D.OBJIe-
TBOpHlOT YKaSaHHble KOM6HHaUHH cpYHKUHH DeCCeJIH. fIpH 3TOM ,D.JIH
Qo (knr) nOJIY'IaeM CJIe,D.YlOllJ.HH PH1\:
Q ( k ) = 2 [QI (kn> - eQI (kne)] -L
o n r k n (I _ 102) ,


+
 knN o (1cj) QI (kd - Rn FN " (1cje) QI (klle) N (
. )
...:;;. 2 2 0 "'/.
j=1 D" (1cj, e) (k n + 1c j )
PH,D. ,D.JIH r 0 (knr) nOJIY'IaeM saMeHOH Q Ha r B npe.D.CTaBJIeHHH pH,D.OM
Qo (knr), a paSJIOLKeHHe k n rQ1 (knr) - ,D.H<pcpepeHUHpOBaHHeM pH.D.a
(14.4) no kn-
I1cnOJlbSYH COOTHOrneHHH (14.4) H (14.2), HS rpaHH'IHOrO YCJIOBHH .D.JIH
(J2 Ha Topuax UHJIHH,D.pa nOJIY'IaeM cJIe.D.YlOllJ.He aJlre6paH'IeCKHe paBeH-
CTBa:


(14.4)


00
2Bo m - I
 2 (- I)n _
m - 2 + m - 2 Av +.... k ll (I _ 82) (e<P21l - <P3n) - f10'
,=1
( 1c'h )
 n 2kn1c7 [No (1cj) QI (k n ) - eN I (1cj8) QI (kne)]
A. ChA.h + -1- =""'" (- 1) +
I I sh 1cjh n=1 Dti (1c I , e) (k
 + 1c7)2
00 k [N ( A' ) '" -ION (J..'e)cp ]
+1 . + .cthA.h-
 ( _ l ) n n 0 l .t 3n "I 2n . (14.5)
II <PII J.-J D ( 1c. e ) ( Il- + 1c
 )
n=1 0 I' n I
3,D.eCb 110' It j - K03cp<pHUHeHTbl paSJIOLKeHHH B pH.D. no No (Ajr) cpYHKUHH
It (r) B (14.1). KOTOpble Bbl'IHCJIHlOTCH no TaKHM cpopMYJlaM:
I
2 ('
llO = 1- 102 J II (r)rdr,
E


(14.6)


I
flj = D (L .) S Idr) No (A;r)rdr.
o I' e E
fIpH Y.D.OB.neTBopeHHH rpaHH'IHbIX YCJIOBHH .D.JIH HOpMaJIbHbIX Ha-
npHR<eHHH Ha 60KOBblX nOBepxHocTHX nOJIY'IaeM ypaBHeHHH, BKJllO'Ia-
10llJ.ue Bce 'IeTblpe nOCTOHHHbIe 'I'n. (jn' Pn H qn. I1cnOJIbsYH COOTHorneHHH
(14.2), JIlO6ble .D.Be HS HHX MOLKHO HCKJllO'IHTb. O,D.HaKO BH.D. BTOpOH rpyn-
nbI ypaBHeHHH B (14.5) YKasbIBaeT, 'ITO HaH60JIee Y.D.06HO B .D.aJIbHeH-
rnHX BbIKJIa,D.KaX npHHHTb B Ka'IeCTBe HeHSBeCTHblX BeJIH'IHH TaKHe KOM-
6HHaUHH Pn H qn' KOTopble 06pasYlOT Q1 (k rl ) H Q1 (kne).
D.ml cpYHKUHH Qo (k n ) H Qo (kne) Ha OCHOBe HX onpe.D.e.neHHH JIerKO
HaXO,D.HM cJIe,D.YlOllJ.He BbIpaLKeHHH:
Qo (k l1 ) =
I Q1 (k n ) - k

 QI (kne),
Qo (k l1 e) = -
2 QI (kne) + k:e Q1 (k n ),


(14.7)


138


I


r,D.e


/). = KI (krle) 11 (k n ) - KI (k,.J I) (hI/E),
/).1 = K I (k n 8) 10 (k n ) + /1 (k l1 e) Ko (k n ),
/).2 = Ko (kne) It (k n ) + /0 (kne) K 1 (k n ).


(14.8)


COOTHOmeHHH (14.7) MorYT 6bITb HenOCpe.D.CTBeHHO npHMeHeHbI npH
onpe,D.eJIeHHH ro (knr) 'Iepes r 1 (knr) npH r = 1 H r 8.
I1cnOJIbSOBaHHe paSJIOLKeHHH rHnep60JIH'IeCKHX cpYHKu:HH no TPH-
rOHOMeTpH'IeCKHM (5.6) nOSBOJIHeT npHBecTH ypaBHeHHH, BOSHHKalOllJ.He
npH nO.D.CTaHOBKe BblpaLKeHHH .D.JIH a r HS (1J.9) B rpaHH'IHble YCJIOBHH
(14.1), K c.ne.D.ylOllJ.HM aJIre6paH'IeCKHM COOTHomeHHHM:


00
mBo + Ao _ D +
 fPlj N ( A. ) = f
m - 2 0 """ 1c.h V I;JO'
j=1 1


<x>
mBo + Ao _ Do '
 fPlj N ( A e ) = f . .
_ 2 8 2 """ 'A.h 0 ] 20'
m 1=1 I


(14.9)


k"Q1 (k n ) + 2 m
;" I Qdk n ) - k n
I Qo (k n ) +
 Qo (kne) =
_ _ ,-, _ n 41cj
 sh 1cjhN o (1cj) _ i; (_ 1)" . 2AjN o (t I) +
-
 Aj (1) h ( Il- + 1c2 ) 2 '-1 <Plr h ( k 2 n + 1c J
)
r=1 n j 1-
fP2n (
l I )
+ f3n - kn
 + <P3n . T - k; ,

 (14.10)
k"eQ1 (kne) + 2 :-;;:.: Q1 (kne) + kneT Qo (k"e)-
I _
 _ n 41cjk
 sh]"jhN o ('Aje) _
- 8
 Qo (k n ) - -
 Aj (1) h (k2 + 1c
)2
1=1 n ]
00 n 21cjN o (1cje) f fP3n (
2 + I )
-
 (- 1) <Plj h (J?. + 1c
) + 2n + kne
 - <P2n T kne '
1=1 n ]
K03<P<PHUHeHTbI pH.D.OB cDYPbe cpYHKUHH 12 (z) H ';J (z) B (14.1), 060SHa-
'IeHHble B (14.9) H (14.10) COOTBeTCTBeHHO 'Iepes hn H fan, onpe,D.eJIH-
lOTCH no HSBeCTHbIM cpopMy.naM.
fIepBoe paBeHCTBO B (14.5) H .D.Ba ypaBHeHHH (14.9) 06pasYlOT CHCTe-
MY .D.JIH onpe,D.eJIeHHH nOCTOmIHbIX Ao, Eo H Do 3JIeMeHTapHbIx perneHHH.
3Ha'IeHHH 3TH X nOCTOHHHbIX saBHCHT He TOJIbKO OT HecaMoypaBHoBerneH-
HbIX COCTaBJIHlOllJ.HX HOpMaJIbHbIX HarpYSOK Ha 60KOBbIX nOBepXHOCTHX
II Ha Topue, HO H OT KaCaTeJIbHbIX HanpHLKeHHH. BJIHS:.HHe BeJIH'IHHbI H
xapaKTepa pacnpe.D.eJIeHHH KaCaTeJIbHbIX HanpHLKeHHH, npHJIOLKeHHbIX
no rpaHH'IHbIM nOBepXHOCTHM UHJ1HH,D.pa, BbIHBJ1HeTCH 60JIee HarJ1H,D.HO,
ec.nH npOCYMMHpOBaTb 6eCKOHe'IHbIe pHlI.bl, BXO,D.HllJ.He B ypaBHe.
HHH (14.9) H nepBoe ypaBHeHHe (14.5). fIpH 3TOM COOTHOrneHHH ,D.JIH
139




onpe,!{eJIeHH5I 3JIeMeHTapHbIX peIlI
HHH npH06p
TaroT BHA
h
2Bo m -] A 2 r
m -2 + . m-2 o
 he] _82) J [E(jJ2 (Z)
 (jJa(Z)] zdz = flU'
o


k n Q1 (kJ = - (-ltYn'
xiI)
A. A,j sh 'Ajh = f =
. I h NflfAf)


knEQl(knE) = (- It zn.
x
21
1
No ('Af8) -


04.12)


_
 (jJ1j 2'AjN o ('Ai)
j=I h(
 +A
 '
n' (14.14f
n <r3n (-1) n ( /).2 I
cD 2n = f2n (- l) + k n £/). - (jJ2n(- I)T + k n 8) -

 2'AfN 0 ('Af8)
-
 «PIj h (
 + 'A
 ) ,
/=1 . n 1
F i = - ftfN o (Ai) - «PIj cth AjhN o (Ai) +
+
 (_I)n kn[N
('Ai)<r3n-£No2('Af8)
o(Aj)CfJ2nl .
n=1 Do ('Ab £) (k n + 'A j )
PaCCMaTpHBaeMa5I 3,!{ecb nepBa5I OCHOBHa5I rpaHHqHa5I 3a,!{aqa ,!{JI5I
nOJIoro I(HJIHH,!{pa KOHeqHOH ,!{JIliHbI 13 CMbICJIe ocopeHHoCTeH HanpmKeH-
HOro COCT05IHH5I, nopO.>KJJ:aeMbIX yrJIOBblMH TOQKaMH, po,!{cTBeHHa pac-
CMOTJ)eHHOHBbIIlIe 3a,!{aQe ,!{JI5I CnJIOiliHoro IJ.HJIHH,!{pa. I1cCJIe,!{OBaHHe
peCKOHeQHbIX cHcTeM, nOJIYQeHHbIX ,!{JI}J CnJIOIllHoro UHJIHH,!{pa. no-
3BOJIHJIO YCTaHoBHTb CB5I3b MeLK,!{Y xapaKTepoM Hanp5I)KeHHoro
COCT05IHH5I. B oKpeCTHocTH yr,JIOBbIX TOQeK H aCHMnTOTHQeCKHMH
CBoHcTBaMH HeH3BeCTHblx 13 pecKoHetJHblx CllCTeMaX. STOT pe3YJIbTa'I
II OQeBH,!{Ha5I aHa,JIOrIIH Me)Kj{Y JJ;BYMH YKa3aHHbIMII 3a,!{aqaMII n03BO-
,JI5I1OT no,!{oiITII K IICCJIe,!{OBaHHro peCKOHeQHOll CHCTeMbI (14.13) C HeKOTO-
pbIMII np
,!{nOJIOLKeHII5IMH 0 xapaKTepe nOBe,!{eHH5I HeH3BeCTHblX 13 Hell.
PaCCMOTpIIM, HanpHMep, TY qaCTb B Bblpa)KeHIIH (13.9) ,!{JI5I pa,!{H-
aJIbHOrO Hanp5I)KeHH5I, KOTOpa5I npe,!{cTaB,JI}JeTC5I Ha nOBepXHOCT5IX
r = to H r = 1 TpHroHOMeTpHqeCKHMH p5I,!{aMH. C yqeTOM cooTHoilieHIIll
(14.2) H (14.12) HaXO,!{HM
I.. .
 n [ ( ' Ai m - I I )
20 a, ( r = l) = """ (- 1) !I, .. J A .2 -1- 2 1?:.
o - k
A2 . +
n=1 m n ,,0
+ ::/1 (

 -
2 )] cos knz ,
2
 a; (r = to) = n
1 (-I)fI[Zn(l-
! + 2 :

 +
8
A2) +
+ Y n (

 -
2 ) k
/1 ]cosknZ. (14.15)
.D:JIH K03ctJq:mUHeHToB npH zn H !In 13 3THX Bblpa)KeHH5IX cnpaBe,!{.T1HBbI
CJIeAyro
lle aCHMnTOTHQeCKIIe Ol(eHKH:
/).i _1
2 m-2 _
 . = O ( --L ) .
/).2 m1?:. 1?:./).2 k n '
n n
:I
1- A2 +2 m-l -
=O (
 ) (14.16)
tJ,2 mk
82 /!;.82/12 k n '
k
/)' . (

 -
2 ) = 0 (e-kn<J-EJ).
141


mBo + Ao
m-2


1
Do - h (1
 82) S (r 2 - 8 2 ) <PI (r) dr = fso. (14.11)
It
I

: + h (I
82) S (1- r 2 ) (jJ1 (r) dr = too.
e


mBo + Ao
m-2


BeCKOHeqHYro CHCTeMY YPBBHeHIIll AJI.5I onpe,!{eJleHII5I nOCT05IHHbIX A},
Pn II qn' 06pa30BaHHYro BTOpOH rpynnOll paBeHCTB 13 (14.5) II COOT-
HOIllemI5IMH (14.10), npeo6pa3yeM K 60JIee y,!{06HOMY BII,!{Y. BBO,!{5I HO-
BbIe HeH3BeCTHbIe COOTHOIlIe.HH5IMII


npIt 3TOM 6eCKOHeQHB5I ClICTeMa OTHOCllTeJIbHO Y n . Zn H Xi 3anItCbIBa-
eTC}J B BH,!{e


Yn (
! - 1 - 2 m -I
mJi;


xJ1>Si =

n=J




2 ) + zn (
l -
2 ) k
A =
co x<.I)f?.
-4
 In +CD
-
 ( 1?:. + ,,-2 ) 2 In.
j=1 .. n . i
m -I .. I ) ( /11 . . /).2 ) 1
m
t;.2 +
£2A2 + Y n 8/1 -/1 kn/). =
co x<.2)

-4
 "/ n +CD
- !-J (
 + ,3 ) 2 211,
/=1 n 1
00 2'A
 [YnN
 (Af) + ZnNo ('Aj) NfJ ('A}8)]
2 + Fj

DfJ ('Ai> 8) (k
 + 1-.j)
X(2') = X
I) No ('Aj8)
} 1 No ('Af) .


(14.13)


( /).2
Zn 1 - /).; + 2


3,!{ecb BBe,!{eHbl 0OO3HaQeHH5I
Sj = :j (cth Ajh + stj
jh ) .
<D In = f 3 ..
 ( . . _ l } n _ <r2n (_l)n + .. ( l ) n ( /). .. 1 1 )
J .. k n /1 <p3n - T - k n

140




CpaBHeHl1e Ha OCHOBe 3THX Ol.{eHOK BblpmKemIH (14.15) C pH.n:OM .n:JJH
U z B (10.10), co.n:epx{allJ,HM x n ' IIoKa3bIBaeT, qTO aCI1MIITOTl1qeCKl1e CBOH-
CTBa K03<p<p I1 l.{HeHTOB Y n 11 zn B (14.15) .n:OJJ2KHbI oIIpe.n:eJJHTbCH paBeH-
CTB3MI1


aCHMI1TOTl1qeCKHMH BbIpa2KeHI1HMH. O.n:HaKo Ba2KHO OTMeTI1Tb, qTO .n:JJH
r.1JaBHbIX qaCTeH HeH3BeCTHbIX X<I> H x(
) K03<P<PHl.{HeHTbI I1 p l1 Yo 11 Zo
I I
II COOTHoweHHSlX (14.19) HBJJHfOTCH TOqHbIMI1. 9TO 06CTOSlTeJJbCTBO 06-
JlerqaeT .n:aJJbHeHwyfO IIpoBepKY IIpe.n:IIOJJ02KeHHH (14.17).
DpH aCHMIITOTHqeCKOM aHaJJH3e ypaBHeHHH .n:JJH Y n H zn B (14.13) Ha
OCHOBe COOTHoweHHH (14.19) HCIIOJJb3yeM 3HaqeHHH CYMM CJJe.n:YfOllJ,HX
pH.n:OB:
V k;,Ng (Ai) =
 ( I'.i _ 1 _
 ) _ 2
......0 ( A" 1:: )( k2 + A2 ) 2 2 1'.2 k 2 1'.2 k 2 ( I-E2 ) ,
(=1 tJ It rz , IZ IZ
f k;'N o (Ai) No (AlE) =
 (
 _
 )
 _ 2 (14.20)
-" D ( A E ) ( k2 + A2 ) 2 2 \ EI'. I'. k"I'. k 2 ( 1 _ ,,2 ) ,
j=l \1 J' n I n
"\.
 Iif,Ng (AlE) 1 ( I'.
 I ) 2
L.J D (A E) (k 2 -+- A2)2 ="2 1 - 1'.2 -+ k 2 1:: 2 !'J.2 -- 12 (1 _ 102) .
i=] 0 I' n', n '2


!im Yn = YO' lim Zn = Zo,
n-+co n-+co


(14.17)


r
e Yo H Zo - HeKOTopbIe IIOCTOHHHbIe. 3HaqeHI1H 3TH X nOCTOHHHbIX
He CBH3aHbI Me2K.n:y C060H.
Y CTaHOBJJeHHbIH .n:JJH K03<p<pI1l.{l1eHTOB x n H Yi B (10.10) 3aKOH aCHMn-
TOTHqeCKI1X BbIpa2KeHI1H YKa3bIBaJl, qTO npl1 no.n:xo.n:e K yrJJOBOH TOqKe
B CnJJOWHOM l.{HJJI1H.n:pe B3HTbIe nopo3Hb Tpl1I'DHOMeTpHqeCKl1e pH.n:bI 11
pH.n:bI CPypbe - D.HHI1 06JJa.n:afOT JJOrapH<pMl1qeCKOH oc06eHHoCTbfO.
C<>9THoWeHI1H (14.17) HBJJHfOTCH CJJe.n:CTBHeM npe.n:nOJJ02KeHHH, qTO Ta-
KOI1 2Ke oco6eHHocTbfO 06JJa.n:afOT TpHI'DHOMeTpHqeCKHe pH.n:bI .n:JJH Ha-
npSl2KeHI1H (13.9) B cJJyqae nOJJOI'D l.{HJJHH.n:pa.
PaBeHcTBa (14.17) onHCbIBafOT npe.n:nOJJaraeMbIe CBoHcTBa He113Be-
CTHbIX B 6eCKOHeqHOH CHCTeMe (14.13), YCT3HoB.'IeHHbIe Ha OCHOBe aHa-
JJOrl1H Me2K.n:y 3a.n:aqaMI1 .n:JJH nOJJO!'o H CnJJOWHOI'D l.{I1JJI1H.n:pOB KOHeQHOH
.n:JJI1HbI. DpoBepKa 3TOI'D npe.n:nOJJ02KeHI1H M02KeT 6bITb oCYllJ,ecTBJJeHa
Ha OCHOBe aCHMnTOTI1QeCKOrO aHaJJH33 CooTHoweHHH 6eCKOHeQHOH CI1-
CTeMbI (14.13).
PaCCMOTpl1M Te ypaBHeHHH 6eCKOHeQHOH CHCTeMbI (14.13), KOTopbIe
n03BOJJHfOT Haxo.n:I1Tb 3HaQeHI1H xi Qepe3 Y n 11 Zw I1cnOJJb3YSl aCHMnTO-
THQeCKl1e CBoHCTBa BeJJI1QI1H Y n 11 zn (14.17), Haxo.n:I1M, QTO IIpl1 60JJbWHX
3HaQeHHHX HOMepoB j cnpaBe.n:JJHBO aCHMnTOTI1QeCKOe paBeHcTBo
(I) 1 Ng (Ai) Y
 No (Ai) No (AiE) Ai
xi = 2 DdAi, l::) 0 + 2 Du (Ai. E) Zo + Fil!' (1-U8)
Dp e .n:nOJJ02KeHHe 0 TOM, QTO cB060.n:HbIe QJJeHbI F. B ypaBHeHI1SJX .n:JJH
x(}J CI1CTeMbI (14.13) y6bIBafOT He Me.n:.'IeHHee QeM 11"-;, n03BOJJHeT onpe-
.n:eJJHTb BeJJI1QI1HbI XI npl1 60JJbWI1X 3HaQeHIIHX HOMepa 113 COOTHoweHIIH
(\) 1 Ng (Ai) Yo +
 NdAi) No (AiE)
Xi = 2 Do (Ai, E) 2 Do (Ab E) Zo,
(2) 1 No (Ai) No (AiE) Y 1 Ng (Ai)
Xi = 2 Do (Ab 1::) 0 + 2 Do (Ai, E) ZOo
Dpe.n:nOJJ02KI1B .n:JJH Y n H zn cnpaBe.n:JJIIBOCTb aCI1MnTOTHQeCKI1X pa-
BeHCTB (14.17) H HCIIOJJb3YH cooTBeTcTBYfOllJ,l1e ypaBHeHI1H 6eCKOHeQHOH
CHCTeMbI (14,13), Haxo.n:HM rJJaBHbIe QaCTH HeH3BeCTHbIX xi' ECJJH Tenepb
B pe3YJJbTaTe aCHMnTOTI1QeCKOrO aHaJJI13a I1epBWX .n:BYX rpynn ypaB-
HeHHH B (14.13) Ha OCHOBe paBeHcTB (14.19) He I1pII.n:eM K npOTHBOpeQHfO
C rrpe.n:nOJJ02KeHHeM (14,17), 3HaQHT 6eCKOHeQHaH CHCTe/vla (14.13)
o6JJa.n:aeT peWeHl1eM, aCHMnTOTI1QeCKl1e CBoHcTBa KOTOpOI'D Orrl1CbIBa-
fOTCH paBeHcTBaMH (14.17) H (14.19).
ECTecTBeHHo, QTO npl1 60JJbWI1X 3HaQeHI1HX j BXo.n:HllJ,l1e B K03<P<P H -
l.{l1eHTbI COOTHoweHHH (14.19) <PYHKl.{HH I3ecCeJJH MorYT 6bITb 3aMeHeHbI


npH CYMMHpoBaHHH :jTHX pH.n:OB I1CIIOJJb3YfOTCH pa3JJ02KeHI1H (14.4)
PaccMoTpHM I1epBYfO rpynn) ypaBHeHHH CHCTeMbI (14.13). Do.n:-
CTaBJJHH BMeCTO x(l} HX aCHMnTOTI1QeCKHe Bblpa2KeHI1H H3 (14.19) H
I
IICI10JJb3VH paBeHcTBa (14.20) H aCHMI1TOTI1QeCKl1e Ol.{eHKH (14.16),
lIaxo.n:I1M, QTO rJJaBHaH QaCTb CTOHllJ,eI'D cnpaBa BbIpa2KeHI1H B ::nOH
rpynne ypaBHeHHH .n:onycKaeT aCHMnTOTHQecKYfO Ol.{eHKY
-- x\\lk 2 I
4 L (k/+
2p + <DI"-+
 Yo + <DIn. (14.21)
i=1 t! I


ECJJI1 npe.n:nOJJ02KHTb, QTO cB06o.n:HbIe QJJeHbI <DIn, <D 2n C pOCTOM n y6bI-
BafOT He Me.n:JJeHHee QeM 1/k 2 , TO H3 HaH.n:eHHbIX Ol.{eHOK 11 Ol.{eHOK, BbI-
n
TeKafOIII.HX H3 BTOpOH rpynnbI ypaBHeHI1H B (14.13), CJJe.n:yeT
Iim Yn = Yo, \im zn = Zo0 (14.22)


n-.=


n_o


T aKHM 06pa30M, c.n:eJJaHHOe npe.n:nOJJ02KeHl1e (14.17) OTHOCHTe.r.bHO
xapaKTepa I10Be.n:eHI1H HeH3BeCTHbIX B CI1CTeMe (14.13) no.n:TBep2K.n:aeTCH
I3ecKOHeQHaH CI1CTeMa (14.13) .n:eHCTBI1TeJJbHO HMeeT peWeHl1e, aCI1Mn-
TOTHQeCKHe cBoikTBa KOTOpOI'D OI1HCblBafOTCH paBeHcTBaMH (14.17) H
(14.19). KaK nOKa3bIBafOT pe3YJJbTaTbi npoBe.n:eHHOI'D Bblwe aHaJJI13a
peweHHH 3a.n:aQ .n:.'ISI npHMoYI'DJJbHHKa H CnJJOWHOI'D l.{HJJHH.n:pa, HMeHHO
3TO peweHHe 6eCKOHeQHOH CI1CTeMbI H HY2KHO I1PH H3YQeHHH HanpSl2KeHHO-
.n: e <p o pMl1pOBaHHOI'D COCTOHHHH nOJJOI'D l.{HJJHH.n:pa KOHeQHOH .n:JJI1HbI.
Dpl1 I1CCJJe.n:OBaHI1I1 aCHMnTOTI1QeCKHX CBOHCTB HeH3BeCTHbIX B CI1-
CTeMe (14.13) 6bIJJH npl1HHTbI HeKoTopbIe He06xo.n:HMble .n:JJH .n:OKa3aTeJJb-
CTBa cnpaBe.n:JJI1BOCTH CooTHoweHHH (14.17) II (14.19) npe.n:nOJJ02KeHI1H 0
x3paKTepe y6bIBaHHH cB060.n:Hb1X QJJeHOB <DIn, <D 2n H Pi" J1erKo no-
Ka33Tb, QTO .n:JJH 06eCneQeHHH HY2KHOI'D rropH.n:Ka y6bIBaHHH 3TI1X Be-
JJUQUH nOMHMO 06bIQHbIX npe.n:nOJJ02KeHHH 0 r.'Ia.n:KOCTH <PYHKl1.HH B
143


(14.19)


142


I




(14.1) Heo6xo.D:HlVIO Y.D:OBJleTBOpeHI1e YCJlOBI1H napHOCTI1 KaCaTeJlbHblX
HanpmKeHI1H B yrJloBblx TOqKax.
BbIllle 6blJlO yKa3aHo, qTO .D:JlH aHaJlI13a HanpmKeHHoro COCTOHHI1H
nOJloro UI1J1I1H.D:pa KOHeqHOH .D:JlI1HW Heo6xo.D:I1MO I1CnOJlb30Ba!b TO pe-
rneHI1e 6eCKOHeqHOH CI1CTeMbl (14.13), anIMnTOTI1qeCKI1e CBOI1CTBa KO-
TOpOro onpe.D:eJlHIOTCH COOTHOrneHI1HMI1 (14.17) II (14.19). B C
H3H C
3TI1M I1CnOJlb3yeMb1H npI1 OTbICKaHI1I1 npH6J1I1)KeHHblx 3HaqeHI1I1 HeI13-
BeCTHblX B (14.13) aJlropI1TM .D:OJl)KeH o6eCneqHBaTb Bbl60p HY)KHOrO pe-
rneHHH .D:a)Ke B TOM cJlyqae, eCJlH 6eCKOHeqHaH CI1CTeMa (14.13) HMeeT
He e.D:I1HCTBeHHoe perneHHe.
OCHOBOH .D:JlH nOCTpOeHI1H KoppeKTHoro, B 3TOM CMbICJle, cnoco6a
yceqeHI1H 6eCKOHeqHOH CI1CTeMbl HBJlHeTCH 3HaHHe aCHMnTOTI1qeCKHX
CBOHCTB I1CKOMbIX BeJlHqHH. I1cnoJlb3YH 3TH CBOHCTBa npI1 paCCMOTpeHHI1
CI1CTeMbl (14.13), nOJlaraeM
Yn = YN, zn = Ztv (n> N);
x}lJ = x<1J (j = M, M + 2, .. .), x
lJ = X

_I (j = M - 1,
M + 1, .. .). (14.23)
ITocJle 3TOro 6eCKOHeqHaH CHCTeMa 3aMeHHeTCH KOHeqHOH, cO.D:ep)Kaw.eH
2N M HeI13BeCTHbIX, peweHHe KOTOpOH BMeCTe C paBeHCTBaMI1 (14.23)
.D:OCTaBJlHeT npH6J1H)KeHHble 3HaqemIH Bcex HeI13BeCTHblX B CI1CTeMe
(14.13), 06J1a.D:alOw.Hx HY)KHbIMH npH H3yqeHHH HanpH)KeHH?- .D:e<pop-
MI1pOBaHHoro COCTOHHI1H UHJlHH.D:pa aCI1MnTO
HqeCKI1MI1 CBOHCTBaMI1.
AHaJlOmqHblH npHeM yceqeHHH 6eCKOHeqHOH CI1CTeMbl 6blJl HCnOJlb-
30BaH npI1 paccMOTpemm nepBoH rpaHHqHOH 3a.D:aQH .D:JlH np
MoyroJlb-
HI1Ka. ITpI1 3TOM OKa3aJlOCb, QTO HeH3BeCTHble B 6eCKOHeQHOH CHCTeMe
.D:OBOJlbHO 6blCTpO npH6J1I1)KalOTCH K aCHMnTOTI1QeCKI1M 3Ha;eHHHM,
QTO n03BOJlHeT OrpaHI1QHTbCH npI1eMJleMWM C BWQI1CJlI1TeJlbHOI1 TOQKI1
3peHI1H nopH.D:KOM KOHeQHOH CI1CTeMbI. I1cCJle.D:oBaHI1e 3Toro Bonpoca
.D:JlH paCCMaTpI1BaeMOH 3.D:eCb rpaHHQHOH 3a.D:aQHu6Y.D:eT npoBe.D: eHO B
cJle.D:YIOw.eM naparpacpe npI1 peWeHI1I1 KOHKpeTHOI1 3a.D:aQH. u
KOJlHQeCTBO ypaBHeHI1H B KOHeQHOII CHCTeMe, nOJlYQeHHOH I13 6ec-
KOHeQHOH Ha OCHOBe paBeHcTB (14.23), eCTecTBeHHO cJle.D:yeT onpe.D:eJlHTb
B COOTBeTCTBHI1 C Tpe6yeMoH TOQHOCTblO Y.D:OBJleTBopeHHH rpaHI1QHbl

YCJlOBHH. O.D:HaKO HeKoTopylO I1H<popMaUI11O 0 KaQeCTBe nOJlYQeHHOI1
annpoKCHMaUI1I1 peweHHH 6eCKOHeQHOH CHCTeMbl MO)KHO nOJlYQHTb,
oueHHBaH CTeneHb 6mI30CTI1 3HaQeHHII YN, ZN H x<1J K npe.D:eJlbHblM 3Ha-
QeHI1HM HeH3BeCTHblX B CI1CTeMe (14.13). TaKaH oueHKa MO)KeT 6bITb
nOJlYQeHa Ha OCHOBe CooTHOWeHI1H (1 4.19), I13 KOT OpblX CJle.D:yeT
I , (i) Yo + Zo I
1m xi ="T=8 r
 (1 _ ) ,
2/- 00 J1 I> I>
, . (I) Yo Zo
1m Xi =-- .
2i+ I - 00 I - I> VI> (1- 8)
ECJllI 3HaQeHHH npaBbIX QaCTeH 3Tl1X paBeHCTB IIp II 3aMeHe Yo H Zo
Ha YN II ZN 6J1I13KI1 COOTBeTCTBeHHO K x<,U II X ,J
I (M CQHTaeM QeTHbIM),


TO CJle.D:yeT npe.D:nOJlO)KI1Tb, QTO HaH.D:eHO .D:OBOJlbHO 6J1I13KOe K TOQHOMY
peWeHI1e 6eCKOHeQHOH CI1CTeMbl, 06eCneQI1BalOIIJ;ee BblCOKYIO TOQHOCTb
Y.D: OBJl eTBOpeHI1H rpaHHQHbIX YCJlOBI1H.
TpY.D:HOCTI1, B03HHKalOIIJ;He rrpI1 npOBe.D:eHI1I1 cpaKTI1QeCKHX BbIQHCJle-
HI1H C I1CnOJlb30BaHI1eM Bblpa)KeHI1H .D:JlH HarrpH)KeHI1H (13.9), 06YCJlOB-
JleHHbIe aCI1MnTOTI1QeCKI1MI1 cBoikTBaMI1 I1X K03<p<pI1UI1eHTOB-HeI13BeCT-
HblX B CI1CTeMe (14.13), HMelOT TaKOH )Ke xapaKTep, KaK II B paCCMOT-
peHHblX paHee rpaHI1QHbIX 3a.D:aQaX .D:JlH npHMoyroJlbHI1Ka H CnJlOWHOrO
U I1J1I1H .D:pa. AHaJlOmQHbI II crroco6bl npeO.D:OJleHI1H 3TI1X TPY.D:HOCTeH.
CneUI1<pI1QeCKHe oco6eHHOCTH HCnOJlb30BaHI1H 3TI1X cnoco60B, CBH3aHHblX
C YCJlO)KHeHHeM CTPYKTYPbl 6eCKOHeQHblX pH.D:OB, paCCMaTpI1BaIOTCH B
CJle.D:YIOIIJ;eM naparpa<pe.


PI (M) =


L
/-M.M+2....


k 2
n
(k
 + "7)2


(k
 + "7P


(n = I, 2, ... , N),


(15.1)



 15. 1JuC.JIeHHRJI HHTepnpeT3l"(HJI 06m;HX pemeHHH
I1cnOJlb30BaHI1e COOTHOWeHI1H (14.23) npH peWeHI1I1 6eCKOHeQHOH
CI1CTeMbl (14.13) rr03BOJlHeT npI16J1H)KeHHO 3aMeHI1Tb ee KOHeQHOH CI1-
CTeMbH. OT CI1CTeMbI, B03HI1KalOIIJ;eH rrpH I1CnOJlb30BaHHI1 MeTO.D:a npOCTOH
p e .D:YKUI1I1, nOJlYQeHHaH Ha OCHOBe (14.23) KOHeQHaH CI1CTeMa OTJlI1QaeT-
CH JlI1Wb K03<P<PI1UI1eHTaMI1 npI1 YN, ZN, x<
 II x
. 1JJlH BbIQI1CJleHI1H KO-
3cpcpI1UI1eHTOB npI1 3THX HeI13BeCTHblX Heo6xo.D:HMO rrpOCYMMHpOBaTb
6eCKOHeQHble pH.D:bl
00 ,,

P (N)
 2 2 (1 ' = 1, 2, ..., M ) ,
= n
 (k n -: "i)2


P 2 (M) =


L
i=M-l.M+l....


(n = I, 2, .,. , N).


lITO KacaeTCH nepBoro I13 YKa3aHHblX pH.D:OB, TO nOJlYQI1Tb rrpe.D:CTaB-
JleHI1e .D:JlH ero CYMMbl P (N) .D:OBOJlbHO JlerKO Ha OCHOBe pa3J10)KeHI1H B
pH.D: <I>YPbe mnep60JlHQeCKI1X CPYHKUI1H (5.6). CBOIICTBO paBHoMepHoH
CX0.D:HMOCTI1 3TI1X pH.D:OB .D:aeT B03MO)KHOCTb HaHTH .D:JlH BbIQI1CJleHI1H P (N)
CJIe.D:YIOIIJ;ee BbIpa)KeHI1e:
h ( ,,'h 2 N-l ,,

P (N) = _ 4 ' dh Aih + h :' h - --.,-- h ) -
 2 / 2 . (15.2)
hi S hi hi .l..J (k n +".)2
n=l 1


( 14.24)


ITOJlYQHTb aHaJlOmQHble <PopMYJlbl .D:JlH BblQI1CJleHI1H PI (M) II P 2 (M)
HenOCpe.D:CTBeHHO He Y.D:aeTCH. OCHOBOH .D:JlH OTbICKaHI1H KOHeQHbIX BbI-
pa)KeHHH .D:JlH 3TI1X BeJlI1QI1H MO)KeT CTaTb TOT cpaKT, QTO B 6eCKOHeQHbIX
pH.D:aX (15.1) BemfQI1Ha M 06biQHO HBJlHeTCH .D:OCTaTOQHO 60JlbWOH H B
CBH3I1 C 3TI1M TOQHble 3HaQeHI1H KopHelI TpaHCUeH.D:eHTHOro ypaBHeHI1H
N I (A8) = 0 MorYT 6bITb 3aMeHeHbl I1X aCHMnTOTHQeCKHMI1 3HaQeHHHMH.


144


10 7-2464


145




6eCKOHeqHble pH.nbl, nocpe.nCTBOM KOTOpblX BbIpa>KaIOTCH BCe <jJaKTOpb!
HanpH>KeHHO-.ne<jJOpMnpoBaHHoro COCTOHHnH u.nJInH.npa, 06JIa.naIOT XO-
pOIlleH cxo.nnMOCTbIO n CYMMnpOBaHue nx He npe.nCTaBJIHeT Tpy.n a .
npn no.nxo.ne K nOBepXHOCTn cxo.nnMOCTb 6eCKOHeqHblX pH.nOB yxy.n-
IllaeTCH n .nOCTOBepHble KOJInqeCTBeHHble .naHHbIe no HnM MO>KHO nOJIY-
qUTb JInIllb nOCJIe Bbl.neJIeHUH n npe.nCTaBJIeHnH B 3aMKHYTOM Bn.ne UX
Me.nJIeHHO cxo.nmu.HXCH qaCTeH.
nOCKOJIbKY onepau.nH no YJIyqIlleHnIO cxo.nnMOCTU pH.nOB B paCCMaT-
pHBaeMOM 3.neCb CJIyqae nOJIoro u.nJInH.npa KOHeqHOH .nJIUHbI, C O.nHOH
CTOpOHbl, BO MHOroM aHaJIOrUqHbl no.nP06HO onncaHHbIM BblIlle npn pac-
CMOTpeHnu 3a.naq .nJIH CnJIOIllHOrO U.UJIUH.npa U npHMoyroJIbHUKa, a C
.npyroH CTOpOHbl, CBH3aHbl C .nOCTaTOqHO rpOM03.nKnMU BbIKJIa.nKaMn,
TO OrpaHnqUMCH 3.neCb cpaBHUTeJIbHO no.nP06Hb1M paCCMOTpeHneM JIUIllb
Bblpa>KeHnH .nJIH a z B (13.9). npu 3TOM B nOJIHOM 06'beMe BCKpbIBaIOTCH
Te OTJIUqnH OT YKa3aHHblX BblIlle 3a.naq, KOTopbIe CBH3aHbI C UCnOJIb3Y-
eMblMU B (13.9) CUCTeMaMn <jJYHKu.nH.
Bbl.neJIeHne U nOCJIe.nYIOIll.ee npe.nCTaBJIeHUe B 3aMKHYTOM Bn.ne Me.n-
JIeHHO CXO.nHru.UXCH qaCTeH 6eCKOHeqHbIX pH.nOB B (13.9) OCHOBbIBaIOTCH
Ha 3aMeHe Bxo.nHru.nx B Hnx <jJYHKU.UH n K03<jJ<jJnu.ueHToB ux aCUMllTOTU-
qeCKHMU Bblpa>KeHUHMU. Henocpe.ncTBeHHoe IfCnOJIb30BaHne HaH.neHHblx
npu peIlleHnIf KOHeqHOH CHCTeMbI BeJIIfqnH YN, ZN, x,J
1' xl» B KaqeCTBe
aCUMnTOTIfqeCKnX 3HaqeHIfH K03<jJ<jJIfu.ueHToB paccMaTpnBaeMbIx 6ec-
KOHeqHblX pH.nOB npnBe.neT K TOMY, qTO oc06eHHocTn B yrJIOBblX TOqKax
He 6y.nYT noraIllaTbCH. B CBH3n C 3TIfM B KaqeCTBe npe.neJIbHblX 3Haqe-
HnH .nJIH YKa3aHHblX K03<jJ<jJnu.IfeHTOB CJIe.nyeT npUHHTb cpe.nHIfe BeJIIf-
qnHbl, onpe.neJIHeMble COOTHOIlleHIfHMn
Xlh=
_ Zo
 [ x.w l+
- ZN J
1 - 8 -V 8 (l - 8) 2 - I - 8 -V 8 (I - 8) ,
(15.6)
ZN ]
-V 8 (l - e) .


3.neCb X<p n X<
) - rJIaBHble qaCTn K03<p<pnu.neHToB xlY cooTBeTcTBeH-
HO .nJIH HeqeTHblX If qeTHblX HOMepoB j.
PaCCMOTpIfM BHaqaJIe TY qaCTb 06ru.ero Bblpa>KeHnH .nJIH a z ' KOTopaH
npe.ncTaBJIeHa pH.nOM no No ('AIr). C yqeTOM npoH3Be.neHHblx BbIIlle 3aMeH
.nJIH npOIf3BOJIbHblX nOCTOHHHblX no.nJIe>KaIll.nH CYMMIfpoBaHnIO pH.n npe.n-
CTaBIfM B Bn.ne

 a(l) = nh
 X(I) ( 'A.Z sh All _ ch AjZ _
2G z 2 i=I' , sh Aih sh Alh
ch AIZ )
- 'Ajh efh 'Ajh sh Ai h No ('Air).


[{JIH onpe.neJJeHuH KopHeH
nCnOJIb30BaHa <jJopMYJIa [68]
8'Aj = l)j +
j +


YKa3aHHoro ypaBHeHnH MO>KeT 6blTb


q_p2 + r-4pq+2p3
13: {j



(15.3)


r.ne


s: jn 3 84 (p3 - 1)
u i = P _ 1 ' P = 8P' q = - (Bp)3 (p - I)
32. 1899 (p6_ I) 1
r = 5 (8p)6 (p - I) ,P = e .
TOqHOCTb, .nOCTIfraeMaH npIf IfCnOJIb30BaHnn <jJOPMYJIbI (15.3), TaKOBa,
qTO nOTpe6HocTb B YTOqHeHnU nepBblx KopHeH B03HnKaeT JInIllb npn
8 < 0,4. YTOqHeHIfe 3HaqeHIfH nepBblx KopHeH JIerKO .nOCTnraeTCH C
nOMoru.bIO 06blqHblX npneMoB Ha OCHOBe XOpOIllO pa3pa60TaHHbIx aJI-
ropIfTMoB BblqnCJIeHUH <jJYHKU.IfH EeCCeJIH [2181.
I1cnoJIb3YH Bblpa>KeHIfe (15.3), Bcer.na JIerKO YCTaHOBUTb HOMep j,
HaqIfHaH C KOToporo npIf BbIqnCJIeHIfn 'Aj MO>KHO OrparmqHTbCH nepBblM
CJIaraeMblM B npaBoH qaCTU. T aK, .nJIH paccMaTpnBaeMoro HIf>Ke CJIyqaH
8 = 0,5 BTopoe CJIaraeMOe B (15.3) npu j > 40 YTOqHHeT JInIllb IlleCTYIO
3Haqaru.yIO u.n<jJPY KOpHH. B CBH3If C 3TnM If MeeT CMbICJI paCCMOTpeTb
pH.nbl.nJIH PI (M) H P 2 (M), 3aMeHnB B Hnx 'A J . Ha 8
 = 1 jn . nOCJIe Ta-
l -8
KOH 3aMeHbl .nJIH BblqHCJIeHHH PI (M) U P 2 (M) nOJIyqaeM KOHeqHble
Bblpa>KeHHH CJIen:YIOru.ero Bn.na:
P ( M ) - I - 8 [ th k ( I ) --I-- k n (l - 8) 2 +
I -
 C n - 8 . sh2 k n (l _ 8) k n (I _ 8)
1 k ( I ) cth k n (l - 8) ]
+ sh k n (I - 8) + n - 8 sh k n (I - 8) -
I M-2 k 2
- 4k 2 - _ L (
+n6
2)2 '
11 1=2.4,_.. I
P ( M ) 1 - 8 [ t h k (I ) kn (l - 8) 2
2 = 8
 C n - 8 + sh2 k n (l - 8) - k n (l _ 8)
I


k ( 1 cth k n (I - 8) ] --I-- I
 n
- sh k n (l - 8) - n - 8) sh k n (l - 8) . 4
 -. "'-' (k
 + {j
2)2 .
1=1.3.... ,
npn BbIBO.ne 3TUX COOTHOIlleHHH qHCJIO M .nJIH Onpe.11eJIeHHOCTU Cqn-
TaJIOCb qeTHblM n nCnOJIb30BaJInCb pa3JIO>KeHHH (5.6)
npHMeHHH nOJIyqeHHbIe 3HaqeHHH .nJIH CYMM 6eCKOHeqHblX pH.nOB
(15.2) H (15.5), JIerKO c<POpMupOBaTb KOHeqHYIO CIfCTeMy ypaBHeHnH,
OTBeqaIOIll.YIO 6eCKOHeqHOH cnCTeMe (14.13). nOCJIe OTblCKaHHH ee pe-
IlleHHH H npe.11BapnTeJIbHOH ou.eHKn ero TOqHOCTn Ha OCHOBe paBeHcTB
(14.24) MO>KHO npoBecTH BbIqHCJIeHne 3HaqeHnH KOMnOHeHTOB BeKTopa
CMeIll.eHnH H TeH30pa HanpH>KeHnH B JII060H TOqKe nOJIOro U.HJInH.npa.
D:JIH .nOCTaTOqHO y.naJIeHHblx OT rpaHnqHblX nOBepxHocTeH TOqeK


(15.4)


X
!)=
+ Zo =
 [ x<J)--I--
+
1 - e JI8 (I - 8) 2 . 1 - 8


(15.5)


(15.7)


D:JIH .nocTaTOqHO 60JIbIllHX HOMepoB j 3aMeHHeM KOM6nHau.nIO <jJYHKu.nH
EeCCeJIH No ('AIr) If BeJInqHHbl 'Ai aClIMnTOTIfqeCKHMH Bblpa>KeHnHMU


146


10*


147




II


No (Ajr)
 1t Y
 hi [COS Aj (1- r) - 3

/ Sill Aj (1 - r) 1,


1 "-' J<
 _ J1t
""o-U / __
1- I-B.
Tor.n;a C yqeTOM CBOHCTB K03cpqmll.lIeHTOB x(
) p5!.n;a (15 7) .n;JIH BblqllC-
v I
JIeHlIH era 3HaqeHlIlI B TOqKaX, 6JIlI3KlIX K TOpll.aM lI.lIJIlIH.n;pa, nOJIyqaeM
cJIe.n;YIOllJ.ee paBeHcTBo:
M

 u(I) =
 r 1th xP) ( AoZ sh hjZ _ ch hjZ . _ A h th A h ch hjZ ) '
2G z
 2 I I sh h l oh sh h l oh j C j sh h 0h X
1=1 I
( X1 1 » ) h . . -I)
 (h-z). ]
X No (Ajr) + (I) .
r (1 + t5; (Il - z» e I cos t5 j (1 - r) _
X 2 l5 i f r
h X (I) (1 S . (I) ·
-
r [ 1 1- z) ( 1 - SI) + XI (S3 - Sa) +
2 f r
+ X
I) (h - z) (Sl + S;) + X
I) (S3 + S;)]. (15.9)
Pa3JIlIqlle rrpe.n;eJIbHbIX 3HaqeHlIH K03cp<plIlI.lIeHTOB x(9 M5! 'leTHbIX II
v I
He'leTHbIX 3Ha'leHlIlI lIH.n;eKca 06YCJIOBJIlIBaeT pa3Hble 3Ha'leHlIH MHO)KlI-
TeJIH BO BTOpOM CJIaraeMOM B Bblpa)KeHlIJI rro.n; 3HaKOM CYMMbI B (15.9).
npll He'leTHbIX j CJIe.n;yeT 6paTb Xc:), a rrpll 'leTHbIX - X(
). 3Ha'leHlI5!
cpYHlill.lIH r II Z, 0603Ha'leHHbIX B (15.9) 'lepe3 Sk II orrpe.n;eJIHeMbIX 6ec-
KOHe'lHbIMlI pH.n;aMlI, BbI'lllCJIHIOTCH rro cpopMYJIaM
S
 -I)
(h-z) JI. (I cosl5; (I_ r ) -t(Z)
] = ...:;.J e I cos Ui - r) = t (z)
j=1 I - 2t (z) cos 15; (I - r) + t 2 (z) ,


( 15.8)


J1crrOJIb3Y5! rrpe,ncTaBJIeHHble B KOHeqHOM BlI.n;e CYMMbI p5!.n;OB (10.20),
npllra.n;Hoe .n;JIH BbI'lllCJIeHJIH Bblpa)KeHlIe BTOpOH COCTaBJI5!IOllJ.eH Ha-
npH)KeHlIH B (13.9) npe.n;cTaBlIM B BlI.n;e
N
1 (2) { - 2 -
2G U Z = - n
1 k n fro (knr) - ro (knr)] + m [Qo (knr) - Qo (knr)] +
+ k n , [Ql (knr) - Ql (knr)]} cos knz +
+ Y I -! t ( r COSklz+i(r) Y
 ( I +6r-3r2 _
o Y r ) I + 2t (r) cos k1z + t 2 (r) + °2k1 Yr \ 8r
--
 ) In [ 1 + 2t (r) cos k1z + t 2 (r)] + Zo
=- ( B2 +
rB - 3r 2
2k 1 YrB fB
--{-) In [1 + 2t(r) cosk1z + t 2 (r)] + Zo ;r re B t (r) X
X coskrz+t(r)
I + t (r) cos krz + t 2 (r) ,
t (r) = e-k,(!-r).
3.n;eCb 3HaKOM «......» OTMe'leHbI aClIMrrTOTlI'leCKlIe Bblpa)KeHlI5! M5! COOT-
neTcTBYIOllI.lIX cpYHKlI.lIH.
PaC'leTHble Bblpa)KeHlI5! .n;JI5! BbI'lllCJIeHlI5! Hanp5!)KeHlIH B yr JIOBOH
ro'lKe CJIe.n;YeT nOJIY'laTb C nOMOllJ.bIO rrpe.n;eJIbHOrO nepexo.n;a. XapaK-
Tep 3Toro npe.n;eJIbHOrO nepexo.n;a rrpOllJIJIIOCTpllpyeM paCCMoTpeHlIeM
I1blp8)KeHlIH (15.9) II (15.11) .n;JI5! U z rrpJI r = 1, z -+ h.
B COOTHOIlleHlIlI (15.9) CJIaraeMble KOHeqHOH CYMMbI He 06JIa.n;aIOT
KaKlIMlI-JIlI60 oco6eHHOCT5!MlI B yrJIOBOH TO'lKe II BbI'lllCJIeHlIe 3TOH
CYMMbI MO)KeT 6bITb npOBe.n;eHO no 3Ha'leHlI5!M ee CJIaraeMbIX, B35!TbIX npll
r = 1, z = h. PaccMaTpllBa5! CT05!llJ.lIe BHe 3HaKa CYMMbI COCTaBJI5!IO-
IUlIe Bblpa)KeHlI5! .n;JI5! u(
) B (15.9), Haxo.n;lIM, 'lTO npll r = 1 II Z -+ h
onpe.n;eJI5!eMa5! lIMlI .n;06aBKa B 3Ha'leHlIe Hanp5!)KeHlIH u(
), KOTOPYIO
0603Ha'lllM u(
), lIMeeT cJIe.n;YIOllJ.lIH BlI.n;:
I - (I) h h h
2G UZ =-Yo+-Yoln[I-t(z)]+
r_ Zoln2. (15.12)
1t 1t 1tfB
3aMeHa npe.n;eJIbHbIX 3Ha'leHlIH K03<pcpllll.lIeHTOB x(
) Ha npe.n;eJIbHble
I
3Ha'leHlI5! Y n II zn rrpll rrOJIY'leHlIlI 3Toro COOTHOIlleHlI5! oCYllJ.eCTBJI5!eT-
CH Ha OCHOBe paBeHCTB (15.6).
CJIe.n;yeT 06paTlITb BHlIMaHlIe Ha pOJIb nepBoro CJIaraeMOro B Bblpa-
iKeHlIlI (15.12). Ero n05!BJIeHlIe 06YCJIOBJIeHO OTCYTcTBlIeM paBHoMep-
HOH CXO.n;lIMOCTlI cOOTBeTCTBYIOllJ.ero 6eCKOHe'lHOrO p5!.n;a B (13.9) JI
6e3 ero yqeTa .n;ocTOBepHble 3Ha'leHlI5! Hanp5!)KeHlIH B yrJIOBOH TO'lKe
He MorYT 6bITb rrOJIyqeHbI.
nOBe.n;eHlIe CT05!llJ.lIX BHe 3HaKa CYMMbI COCTaBJI5!IOllJ.lIX a(;) B
(15. II) rrpll r = 1 II Z -+ h MO)KHO OrrJICaTb CJIe.n;YIOllJ.JIM COOTHOIlleHlIeM:
1-(2) h ( 1tZ )
2G U Z = 1t Yo In 2 cos 2h . (15.13)


(15.11)


00 .
S . ,\1 ( i-I) ° (h-z) ..
I =
 - 1) e I cos t5 j (1 - r) =
1=1
..
= -t (z) cos 15 1 (l -:- r) + t(z)
I + 2t (z) cos 6 1 (I - r) + t 2 (z)


00
S
 I -I)
(h-z) ·
3 =
 ---;;:- e I cos t5 i (1 - r) =
j=1 15 1


(15.10)


I .
= - 2iT In [1- 2t (z) cost5 1 (1 -r) + t 2 (z)],
1
S .
 (- 1)/ -I):(h-z) ·
a =

 e I cos t5 j (1 - r) =
j=1 15 1


I 1 ·
= - -".- n [1 + 2t (z) COS t5 1 (1 - r) + t 2 (z»).
2uI
,[LJI COKp . .f. -I). (h-z)
51 allJ.eHlIf! 3anllClI 3.n;eCb BBe.n;eHa 't'YHKlI.lIH t (z) = e 1 .
148


149




YqliTblBaSI


JIyqeHHbiX .lI.JISI HIIX BbIpmKeHIIH YCJIOBIISI OTCYTCT
IISI HanpSI
eHIIH Ha
nOBepXHOCTSIX UIIJIIIH.lI.pa npIIBO.ll.SIT K CJIe.ll.YID
eII rpaHIIqHOII Sa.ll.aqe
.lI.JISI O.ll.HOpO.ll.HbiX ypaBHeHIIll JIaMe:
do O'r (r = e) = 4

\) cxTa (2 In e - I),
1 m+ 1
20 O'r{r= 1) =- 4(m-l) CXTa,
(1S.16)
1 1 1 0
2G T rz (r = e) = 20 Trz (r = 1) = 2G T rz (z = + h) = .
1 m+l I
- 0' (z = + h) = 1 cxTa n r.
20 z - m-
LlJISI KOHKpeTlIsaUIIII CB060.ll.HbiX qJIeHOB 6eCKOHeqHOll CIICTeMbi (14.13)
II CIICTeMbi (14.ll) He06xo.ll.IIMO npe.ll.CTaBIITb <PYHKUlliO In r pSI.lI.OM no
No (A,r). K03<p<pIIUIIeHTbi 3Toro pSI.lI.a JIerKO BbIqIICJISIIDTCSI:


!:..- Y I . I J - t (z) _ h Y I h
o 1m n - - 0 n - ,
n z_h 2 nZ:n I - 8
COS 2h
npIlXO.ll.IIM K SaKJIIDqeHIIID, qTO BbIpmKeHIISI (13.9) .lI.OCTaBJISIIDT .lI.JISI Ha-
npmKeHIIll B yrJIOBbiX TOqKaX BnOJIHe onpe.ll.eJIeHHbie KOHeqHbie SHa-
qeHIISI. CJIe.ll.yeT TaK}!(e nO.ll.qepKHYTb, qTO BII.lI. paCqeTHbiX <POPMYJI.lI. JISI
HanpSI}!(eHIIll He IISMeHSIeTCSI npii paCCMOTpeHIIII npe.ll.eJIbHOrO nepeXO.ll.a
no TOPUY UHJIIIH.lI.pa, T. e. npii Z = h r __ 1.
3aKaHqIIBa51 aHaJIIIS nOBe.ll.eHII51 6eCKOHeqHbiX pSI.lI.OB .lI.JISI HanpSI}!(eHHll
II paCCMOTpeHIIe BonpOCOB, CBSIsaHHbiX C YJIyqWeHIIeM IIX CXO.ll.IIMOCTII,
CJIe.ll.yeT 06paTIITb BHIIMaHIIe ern.e Ha O.ll.HO 06CT051TeJIbCTBO. B npIIBe-
.lI.eHHbiX BbipameHIISIX .lI.JISI O'z (IS.9) II (IS.II) 3HaqeHIISI npe.ll.eJIOB
CYMMIIpOBaHII51 B KOHeqH
X CYMMax COBna.ll.aIDT C KOJIIIqeCTBOM Y.lI.ep-
}!(aHHbiX npii peweHIIII 6eCKOHeqHOll CIICTeMbi HeIISBeCTHbiX xII) II
,
Yn' Zn COOTBeTCTBeHHO. O.ll.HaKO 3TO He 5lBJI51eTC51 06SISaTeJIbHbIM. YKa-
saHHbie npe.ll.eJIbi CYMMIIpOBaHIISI MorYT 6bITb BbI6paHbi 60JIbWIIMII YKa-
saHHbiX BeJIIIqIIH, npii 3TOM K03<p<pIIUIIeHTbi p51.ll.0B C HOMepaMII 60JIbWe
M II N nOJIaraIDTC51 paBHbiMII BbIqIlCJIeHHbiM Ha OCHOBe COOTHoweHIIll
(IS.6) npe.ll.eJIbHbiM SHaqeHIISIM. Heo6XO.ll.IIMOCTb B YBeJIIIqeHIIIl KOJIII-
qeCTBa CJIaraeMbiX B KOHeqHbiX CYMMax BOSHIIKaeT npii paCCMOTpeHIIII
OqeHb TOJICTbIX (e -- 0) IIJIII OqeHb TOHKIIX (e __ I) UIIJIIIH.lI.POB. B 3TIIX
CJIyqaSIX MO}!(eT OKasaTbCSI, qTO saMeHa Ai nepBbiM qJIeHOM aCIIMnTOTII-
qeCKOro BbIpa}!(eHIISI, a TaK}!(e IICnOJIbSOBaHIIe aCIIMnTOTIIqeCKIIX BbI-
pameHIIll .lI.JISI Q(knr) II r (knx) npii j = M II n = N npIlBO.ll.IIT K
SHaqIITeJIbHbiM norpeW'HOCT51M BbIqIICJIeHIIll.
OpOBe.ll.eHHbill aHaJIIIS 060IIX peweHIIll .lI.JI51 nOJIoro UIIJIIIH.lI.pa KO-
HeIJ:HOll .lI.JIIIHbi COS.lI.aeT He06xo.ll.I1Mbie npe.ll.nOCblJIKII .lI.JISI nOCTpoeHIISI
3<p<peKTIIBHoro aJIrOpliTMa OTbiCKaHII51 KOJIIIqeCTBeHHbiX xapaKTepIIcTIIK
HanpSImeHHoro COCT051HIISI. 3TO nOJIO}!(eHIIe nO.ll.TBep}!(.lI.aeT paccMaTpII-
BaeMbill HII}!(e KOHKpeTHbill CJIyqall TepMOHanpSI}!(eHHOCTII nOJIOro UII-
JIIIH.lI.pa, BOSHIIKWIIll npIl o6pa60TKe pesYJIbTaTOB 3KcnepIIMeHTaJIbHbiX
IICCJIe.ll.oBaHIIll TepMOnpOqHOCTII UIIJIIIH.lI.pIIqeCKIIX 06pasuoB HS xpyn-
KHX MaTepIIaJIOB.
Cnoc06 HarpeBa II TepMOIISOJI51UIISI TOpueBhlx nOBepxHocTell B 3KC-
nepIIMeHTe 06eCneqIIBaJIII npaKTIIqeCKH qIICTO pa.ll.HaJIbHOe pacnpe.ll.eJIe-
BIIe TeMnepaTypbI no TOJI
IIHe CTeHKH. PaspyweHIIe HacTynaJIO npii
YCTaHoBHBweMcSI TeMnepaTypHoM nOJIe. B CBSISII C 3TIIM pacnpe.ll.eJIeHIIe
TeMnepaTYP
 B UHJIHH.lI.pe Y.ll.oBJIeTBOpIITeJIbHO onHC
BaJIOCb CJIe.ll.YID-

ell <pYHKUHeH:


(15.14)


2 ( 82 8 2 1 . )

Inr=- ---lne-- +
1 - 8 2 4 2 4.
}=I


NOCAi)-NoCAjB) No (Ajr).
AJDo (Ab 8)


(IS.17)


B CBSISH C OTCYTCTBIIeM KaCaTeJIbHbiX HanpSI}!(eHIIll Ha BceH nOBepXHOC-
TH nOJIoro UIIJIIIH.lI.pa II caMoypaBHoBeweHHbiX HOpMaJIbHbiX HanpSI}!(e-
HIIll Ha UIIJIIIH.lI.pIIqeCKIIX ero nOBepXHOCTSIX onpe.ll.eJISIeMbie paBeHCTBa-
MII (14.14) cB060.ll.Hbie qJIeHbi <DIn II <D 2n 6eCKOHeqHOH CIICTeMbi (14.13)
06pa
aJOTCSI B HYJIb. LlJISI F i B (I4.14) C yqeTOM paSJIO}!(eHIISI (IS.17)
IIMeeM CJIe.ll.YID
lIe SHaqeHIISI:
F. = - K N
(Aj)-;No(Ai)No(AjB) (1S.18)
1 Aj Do (Ab 8)


AHaJIIISIIpySI 3TO BbIpa}!(eHIIe, MO}!(eM SaKJIIDqIITb, qTO .lI.JISI paCC!'IaTpIIBae-
MOH KpaeBOll Sa.ll.aqII (IS.16) CB060.ll.Hbie qJIeHbi 6eCKOHeqHOII CHCTeMbI
(14.13) Y.ll.OBJIeTBOpSIIDT He06xo.ll.HMbiM YCJIOBIISIM .lI.JISI cnpaBe.ll.JIIIBOCTH
aCIIMllTOmqeCKIIX paBeHCTB (14.17) II (14.19).
PaCCMOTpIIM peweHIIe 6eCKOHeqHOH CIICTeMbi (14.13) npii CJIe.ll.YID
IIX
SHaqeHIISIX reoMeTpHqeCKIIX napaMeTpOB H qIICJIa OyaccOHa: e = 0,37;
h = 0,37; m = 6. Opii saMeHe 6eCKOHeqHOH CHCTeMbI, KOHeqHOH Ha oc-
HOBe paBeHCTB (14.23), npIIHIIMaJIOCb N = M = 40. PaHee 6bIJIO iyKa-
saHO, qTO KpHTepIIeM .lI.JISI BbI60pa SHaqeHHH N II M B (14.23) SIBJI51eTCSI
Tpe6yeMaSI TOqHOCTb Y.lI.OBJIeTBOpeHHSI rpaHIIqH
M YCJIOBIISIM. OpIIHSI-
Tble npii paCCMOTpeHHH KOHKpeTHOll Sa.ll.aqII SHaqeHIISI N = M =._40
06YCJIOBJIeHbi He CTOJIbKO Tpe60BaHIISIMH .lI.OCTH}!(eHHSI <pHSIIqeCKII ue-
JIeC006paSHOll TOqHOCTII Y.ll.OBJIeTBOpeHIISI rpaHHqHbiM YCJIOBHSIM (IS.16)
(.lI.JISI 3TOro, nO-BH.lI.HMOMY, OKaSaJIOCb 6bI .lI.OCTaTOqHblM II CIICTeMbi B
neCKOJIbKO pas MeHbwero nopSI.lI.Ka), CKOJIbKO eCTeCTBeHHbiM CTpeMJIe-
PIIeM opIIeHTIIpOBaTb nporpaMMIIpyeMbIH .lI.JISI 3BM aJIropIITM Ha He-
6_'1arOnpIiSITHbIH CJIyqaH. TaKIIM He6JIaronpiiSITHbiM CJIyqaeM MO}!(eT
151


T-T o = a Inr.


{IS. IS)


3.l1.eCb nOCTOSIHHbie To II a. OqeBH.lI.HO, onpe.ll.eJISIIDTCSI SHaqeHIISIMH
TeMnepaTypbI Ha BHYTpeHHeH H BHeWHell nOBepXHOCTSIX UIIJIIIH.lI.pa.
4IICTO TenJIOBbie KOMnOHeHTbI BeKTopa nepeMe
eHIIll H TeHsopa Ha-
npSI}!(eHIIll, T. e. qaCTHbie peweHIISI ypaBHeHIIH ynpyroro paBHOBeCIISI,
)J.JISI )J.aHHoro TeMnepaTypHoro nOJISI JIerKO Onpe.ll.eJISIIDTCSI. C yqeTOM no-


150


I:
I
1




'""'"


6bITb, HarrplIMep, KpaeBa5I 3a.n:aqa C pa3pbIBHbIMlI 3HaqeHII5IMII CPYHKQIIH
B rpaHlIqHbIX YCJIOBII5IX. DplI HenpepbIBHOH arrrrpOKCIIMaQlI1I OTpe3-
KaMII COOTBeTCTBYIOm.IIX TplIroHoMeTplIqeCKIIX p5I.n:OB II p5I.n:OB rro CPYHK-
QII5IM EecceJI5I QeJIeCOO6pa3HO y.n:ep:>KaTb B HIIX no HeCKOJIbKO .n:eC5IT-
KOB qJIeHOB.
DpIIBO.n:IITb nOJIHbIe .n:aHHbIe 0 3HaqeHII5IX HelI3BeCTHbIX B KOHeqHOH
CIICTeMe, OTBeqaIOm.eM 6eCKOHeqHOM CIICTeMe (14.13), HeQeJIeCoo(pa3-
HO. I1HTepec rrpe.n:cTaBJI5IIOT JIlIlllb 3HaqeHII5I HeCKOJIbKIIX rrepBbIx II
rrOCJIe.n:HIIX HeII3BeCTHbIX. AHa-
T a 6 JI II U a 15 JIlI3 rrOBe.n:eHII5I HelI3BeCTHbIX C
BbICOKIIMII HOMepaMIi .n:aeT B03-
MO:>KHOCTb YCTaHOBIITb CKO-
pOCTb, C KOTOpOH OHII npH6JIII-
:>KaIOTC5I K aCIIMrrTOTlIqeCKHM;
3HaqeHII5IM, a cpaBHeHlIe FIX C
BeJIlIqlIHaMIi rrepBbIX HeII3BeCT-
HbIX .n:aeT B03MO:>KHOCTb KOJIIIGe-
CTBeHHO o60cHoBaTb QeJIeco-
06pa3HoCTb npoBe.n:eHHoro Y JIyq-
llleHII5I CXO.n:IIMOCTII 6eCKOHeq-
HbIX P5I.n:OB.
B Ta6JI. 15 C TOqHOCTblO
.n:o MHO:>KIITeJI5I K = m + l aTa
m-I
rrpIIBe.n:eHbI 3HaqeHII5I HelI3BeCTHbIX .n:JI5I paCCMaTplIBaeMOM KOHKpeTHOM
3a.n:aqlI. LLaHHbIe .'nOM Ta6JIlIQbI rrOKa3bIBaIOT, qTO qeTbIpe 3Haqam.lIe QIICP-
pbI B nOCJIe.n:HIIX qeTbIpex 3HaqeHII5IX HelI3BeCTHbIX Yn II Zn o.n:IIHaKOBbI.
XapaKTeplICTIIKa CKOpOCTII CTpeMJIeHII5I 3TlIX BeJIlIqlIH K aCIIMnTOTlIqe-
CKIIM 3HaqeHII5IM 6y.n:eT 60JIee rrOJIHOM, eCJIIi YKa3aTb, qTO, HaqIIHa5I c
n = 15, 3HaqeHII5I Yn II zn OTJIlIqaIOTC5I JIIIlllb rrOCJIe TpeTbero 3HaKa.
TaKlIe :>Ke CBoMcTBa xapaKTepHbI II .n:JI5I XC}) C HeqeTHbIMH 3HaqeHlIHMlI
HOMepoB j. Dpli qeTHbIX j CTpeMJIeHlIe xiI) K rrpe.n:eJIbHbIM 3HaqeHII5IM
I
HeCKOJIbKO 60JIee Me.n:JIeHHOe, o.n:HaKO caMII xC
) nplI 3TOM rro BeJIIIqIIHe
I
Ha .n:Ba nop5I.n:Ka MeHbllle 3HaqeHlIH coce.n:HIIX x(
) C HeqeTHbIMII IIH.n:eK-
]
caMIi.
Dp e .n: Ba pIITeJIbHa5I OQeHKa KaqeCTBa rrOJIyqeHHOrO pellleHII5I KOHeq-
HOM CII, CTeMbI COrJIaCHO COOTBeTCTBIiIO HaM.n:eHHbIx 3HaqeHlIM HeII3BeCT-
HbIX TOqHbIM 3HaqeHII5IM B 6eCKOHeqHOM CIICTeMe, MO:>KeT 6bITb rrpoBe.n:e-
Ha Ha OCHOBe aCIIMrrTOTlIqeCKIIX paBeHcTB (14.24). Do.n:cTaBJI5I5I B IIX
rrpaBYIO qaCTb BMeCTO Yo II Zo COOTBeTCTBeHHO 3HaqeHII5I YN II ZN, Ha-
XO.n:IIM


h, i I Yn I Zn I (I)
x.
1
I -1,195 0,5467 -3,854
2 -1,177 0,6439 0.3299
3 -1,161 0,6694 -3,662
4 -1,155 0,6781 0,04185
.. . . .. .. .
37 -1,150 0,6816 --3,603
38 -1,150 0,6816 -0,05068
39 -1,150 0,6816 -3,603
40 -1,150 0,6816 -0,05023



 + ZN
I = - 0,048,
-8 VE(l-8)
YN ZN
1 -V = - 3,603.
---=8- 8 (1-8)


(15.19)


--


I,


]lIaqeHIIe rrepBoro 1I3 BbIqlICJIeHHbIX BbIpa:>KeHIIM .n:OCTaTOqHO
JIII3KO
I ' X l a qeTbI p e 3HaKa BTo p oro COBrra.n:aIOT co 3HaqeHlIeM X:. Eho
\ 40'
o(jn05ITeJIbCTBO CJIY:>KIIT .n:OCTaTOqHbIM OCHOBaHlIeM .n:JI5I Toro, qT06bI
JI(),nyqIITb BbICOKYIO TOqHOCTb y.n:OBJIeTBOpeHII5I rpaHlIqH
IM YCJIOBII5IM
111 m HCrrOJIb30BaHlIlI K03cpcplIQlIeHTOB p5I.n:OB .n:JI5I CMem.eHIIIi II Harrp5I:>Ke.
IIUH .n:aHHbIx B Ta6JI. 15. DplI rrpaKTIIqeCKIIX BbIqIICJIeHII5IX rrplI6JIII-
il(eHHbIe rrpe.n:eJIbHbIe 3HaqeHII5I K03cpcpIIQIIeHToB orrpe.n:eJI5IIOTC5I rro co.
OTHOllleHII5IM (15.6).
DpoBe.n:eHHbIe BbIqIICJIeHII5I .n:JI5I Harrp5I:>KeHIIM U z II u r rrOKa3bIBa1OT,
liTO Ha rpaHlIlie 06JIaCTli IIX 3HaqeHII5I He rrpeBbIlllaIOT + 0,001 2GK. Ta-
Ta6 JIHua 16


I zJn
r I I 0,4 I 0,6 I 0,8 I I
0 0,2
0,370 0,653 0,642 0,610 0,554 0,469 0,374
0,496 0,274 0,267 0,245 0,207 0,151 0,072
0,622 0,063 0,057 0,041 0,013 -0,027 -0,081
0,748 -0,067 -0,071 -0,084 -0,106 -0,136 -0,175
0,874 -0,178 -0,182 -0,193 -O,211 -0,236 -0,267
1.00 -0,256 -0,258 -0,259 -0,284 -0,307 -0,338


I\HM 06pa30M, nplI onpe.n:eJIeHIIII Hanp5I:>KeHIIM BO BHYTpeHHIIx TOqKaX
CJIe.n:yeT paCCqlITbIBaTb Ha rrOJIyqeHlIe Tpex BepHbIx 3HaKOB nOCJIe 3a-
mITOM.
BbIcoTa QIIJIIIH.n:pa, TepMOHarrp5I:>KeHHOe COCT05IHlIe KOTOpOro 1I3Y-
lJaeTC5I, OTHOCIITeJIbHO HeBeJIIIKa, qTO 06YCJIOBJIliBaeT 3HaqlITeJIbHOe
CHII:>KeHlIe YPOBH5I Harrp5I:>KeHIIM U z B ero cpe.n:IIHHOM rrJIOCKOCTII rro
t"paBHeHIIIO C IIX 3HaqeHlI5IMII B aHaJIOrlI'
,HO HarpeToM .n:JIIIHHOM QIIJIuIIH.n:-
pe. HalI60JIblllee 3HaqeHlIe Harrp5I:>KeHIIII U z IIMeIOT Ha BHYTpeHHelI rro-
nepxHocTII QIIJIIIH.n:pa nplI Z = 0, r.n:e u/2GK = 0,120. 3Ta BeJIlIqlIHa
t"OCTaBJI5IeT MeHee 20% OT TOM, KOTOpa5I orrpe.n:eJI5IeT 3HaqeHlIe Harrp5I.
iKeHIIM U z B aHaJIorlI'IHOM TOqKe .n:JIIIHHOro QIIJIIIH.n:pa C OCB060:>K.n:eHHbI-
MIl no CeH-BeHaHY TOPQaMlI.
HalI60JIbllllIM IIHTepec II HalI60JIblllee 3HaqeHlIe B paCCMaTplIBaeMOM
CJIYIIae TepMOHarrp5I:>KeHHOrO COCT05IHII5I QIIJIIIH.n:pa IIMeIOT Hanp5I:>KeHII5I
Uo. Y Ta6JI. 16 .n:aHbI 3HaIIeHII5I 3TlIX Harrp5I:>KeHIIM C TOqHOCTbIO .n:o 2GK.
Pacrrpe.n:eJIeHlIe rro pa.n:IIYcy Harrp5I:>KeHIIM ue u B .n:BY
 xapaKTepHbIx
CeIIeHII5IX rrOKa3aHO Ha plIC. 16, r.n:e lllTplIXOBOII JIIIHlIeli .n:aHO pacrrpe:
.l1.eJIeHlIe Harrp5I:>KeHIIM Ue, HaM.n:eHHbIx 6e3 paccMoTpeH u lI5I 6ecKoHeIIHolI
t"H, CTeMbI T. e. C yqeTOM JIlIlllb 3JIeMeHTapHbIx pellleHlI1I B 06m.eM npe.n:-
naBJIeH1I1I KOMnOHeHTOB TeH30pa Harrp5I:>KeHIIM (13.9).
HCrrOJIb30BaHlIe rrplI y.n:oBJIeTBOpeHIIII rpaHlIqHbIM YCJIOBII5IM JIlIlllb
::IJJeMeHTapHbIx pellleHIIM B paccMaTplIBaeMoH 3a.n:aqe rr03BOJI}IeT y.n: OB -
.,leTBopIITb rpaHIIIIHbIe YCJIOBII5I Ha Top
e JIlIlllb B CMbICJIe rrplIHQlIrra
CeH-BeHaHa. HeCMOTp5I Ha TO qTO TaKOII rro.n:xo.n:, KaK OTMeqaJIOCb BbI-
me, rrplIBo.n:IIT K cym.eCTBeHHbIM rrorpelllHOCT5IM rrplI orrpe.n:eJIeHIIM
153




oceBbIX HanpRm:eHI1H, OH BCe JKe nOSBOJISIeT nOJIYIJI1Tb .n.OCTOBepHbIe SHa-
IJeHI1SI Hal160JIbIlII1X CJKI1MalOml1X 11 paCTHnlBalOml1X OKPYJKHbIX Ha-
npHJKeHI1H.
HOpMaJIbHbIe HanpHJKeHI1SI Or B paCCMaTpIlBaeMOH sa.n.aIJe no Be-
JII1IJI1He SHaIJI1TeJIbHO MeHbIlIe (B 5-6 pas) OKPYJKHbIX HanpHJKeHI1H
11 I1MelOT npaKTI1IJeCKI1 O.n.I1H H TOT JKe xapaKTep pacnpe.n.eJIeHI1H B pas-
JII1IJHbIX CeIJeHI1HX z = const. Dpl1 BbIIJI1CJIeHl111 SHaIJeHI1H Or BnOJIHe
.n.OCTaTOIJHYIO TOIJHOCTb .n.aeT no.n.xo.n.,
KOTOpbIH y.n.OBJIeTBOpSIeT rpaHI1IJ-
HbIM YCJIOBI1HM Ha TOpue B CMbICJIe
npl1Hl.ll1na CeH-BeHaHa. KaCaTeJIb-
HbIe JKE' HanpSIJKeHI1H npaKTI1IJeCKI1
OTCYTCTBYIOT BO BCeM 06'beMe LlI1-
JII1H.n.pa (B 30 pas MeHbllle Ha1160JIb-
o ! lllero SHaIJeHI1SI (6). 3TI1X cBe.n.eHI1H
1 r 0 HanpHJKeHI1HX or 11 T rz BnOJIHe
-0,2 .n.OCTaTOIJHO, IJT06bI nOJIYIJI1Tb nOJIHOe
npe.n.CTaBJIeHl1e 0 HanpSIJKeHHOM CO-
-0.4 CTOSIHI1H UHJIHH.n.pa.
Pnc. 16. PaCCMOTpeHHaSI sa.n.aIJa TepMoyn-
u PyrocTH !lJIH nOJIoro UI1JIHH.n.pa KO-
HeIJHOH .n.JIHHbI .n.OCTaTOIJHO nOJIHO XapaKTepHsyeT 06'beM BbIIJHCJIeHHH II
06mHE' BOSMOJKHOCTH MeTo.n.a, I1CnOJIbSOBaHHOrO npH ee pellleHI1H.
C TOIJKI1 SpeHHSI MexaHHKH, nOJIYIJeHHaSI KapTHHa HanpSIJKeHHOrO CO-
CTOHHHSI I1HTepeCHa TeM, IJTO .n.OCTaTOIJHYIO TOIJHOCTb npl1 onp e .n. e JIeHHI1
HanpSIJKeHHH (sa HCKJIIOIJeHHeM Oz) .n.aIOT npOCTbIe BbIIJHCJ'JeHI1SI Ha OC-
HOBe 3JIeMeHTapHbIx peIlIeHHH. ECTeCTBeHHO, nOJIYIJeHHbIH peSYJIbTaT
He CJIe.n.yeT I1CTOJIKOBblBaTb KaK KOJIHIJeCTBeHHoe .n.OKaSaTeJIbCTBO BOS-
MOJKHOCTH paCIlIHpHTb ccpepy npl1MeHHMOCTI1 npl16JIHJKeHHbIX no.n.xo-
!lOB K aHaJIHSY 1epMOHanpHJKeHHOCTH, nOCKOJIbKY I1SMeHeHHe xapaKTe-
pa TeMnepaTypHoro nOJISI MOJKeT npl1BeCTH npH TOH JKe reOMeTpHI1 UH-
JII1H.n.pa K SHaIJI1TeJIbHOMY I1SMeHeHI1IO XapaKTepa pacnpe.n.eJIeHHSI
HanpSIJKeHHH.


PaCCMOTpHM nepBYIO OCHOBHYIO rpaHHIJHYIO sa.n.aIJY TeopHH ynpyroc-
TI1 .n.JIH nOJIy6ecKOHeIJHOrO KpyroBoro UI1JIHH.n.pa. CIJHTaeM, IJTO ero
HanpSIJKeHHOe COCTOSIHl1e onpe.n.eJIHeTCH CJIe.n.YIOmHMI1 rpaHI1IJHbIMH
YCJIOBI1SIMI1:


1 1
20 or (1, z) = 'l\Jdz), 20 Trz (1, z) = 'IJ12 (z),
1 1
20 °z (r, 0) = II (r), 20 T rz (r, 0) = [2 (r).


(16.1)


3.n.eCb 11 B .n.aJIbHeHIlIeM I1CnOJIbsYIOTCH 6espasMepHbIe Koop.n.HHaTbI r =
p t )
= R' z = R (pl1e. 17.
<l>YHKUI1H, Bxo.n.HIlI,He B rpaHHIJHbIe YCJIOBHH (16.1), ecreCTBeHHO
npe.n.noJIaraeM npe.n.cTaBHMhIMH CXO.n.HmHMHCSI HHTerpaJIaMI1 <l>YPhe H
pH.n.aMI1 <l>yphe - 11I1HH. 110noJIHHTeJIhHbIe
OrpaHHIJeHI1SI, CBHsaHHwe C Tpe60BaHHeM
OTCYTCTBHSI HanpHJKeHHH Ha 6eCKOHeIJHOC-
TH 11 YCJIOBI1SIMI1 o.n.HOSHaIJHOCTH cpOpMbI F
06mero peIlIeHHH rpaHHIJHOH sa.n.aIJH,
6y.n.YT CcpOPMYJIl1pOBaHhI HHJKe. R
DpH BhI60pe cpOpMbI 06mero peIlIeHI1H
rpaHHIJHoH sa.n.aIJI1 (16.1) cJIe.n.yeT HCXO.n.HTb
I1X xapaKTepa nOBe.n.eHHH KaCaTeJIhHbIX ?
HanpSIJKeHHH B oKpecTHocTH yrJIOBOH OK-
PYJKHOCTH. KOHKpeTHbIe BbIKJIa.n.KI1 npoBe-
.n.eM .n.JISI CJIYIJaSI, Kor.n.a B yrJIOBhIX TOIJ- Pnc. 17.
Kax BbInOJIHHeTCSI YCJIOBHe napHOCTI1 Ka-
CareJIhHbIX HanpSIJKeHHH. Dpl1 3TOM pa.n.l1aJIhHaSI U 11 oceBaSI W KOMno-
HeHTbI BeKTOpa cMemeHHH npe.n.cTaBJIHIOTCSI CJIe.n.YIOIlI,HMI1 BhIpaJKeHI1SI-
MH, y.n.oBJIeTBOpSIIOIlI,HMH ypaBHeHI1SIM JIaMe:
U = - f + [-BiAi + D, ( 3m ,;4 Aiz)]e-AjzJdAl)-
1=1 1
- {- f {A (s) II (rs) + C (s) r rI 0 (rs) - 4 m;;;; 1 II (rs)]} cos szds,
o 00 (16.2)
W =
 (B.+Diz)e-AlJo(A/)+
 I [A (s) 10 (rs) +
;=1 / :n; 0
+ C (s) r 11 (rs» sin sz ds,
r.n.e A. - KOpHI1 ypaBHeHI1SI J 1 (A) = O. BeJIHIJHHbI Bi' Di' A (s) H
C (s)
BJIHIOTCSI npOI1SBOJIhHhIMH H .n.OJIJKHbI 6bITh BbI6paHbI HS YCJIOBI1H
(16.1).
BJII1HHIIe pasphIBa KaCareJIhHbIX HanpHJKeHHH B yrJIOBOH TOIJKe MOJK-
HO HSYIJI1Tb C nOMOmhIO peIlIeHHH ypaBHeHI1H JIaMe B BH.n.e (16.2), ec-
JIH B KaIJeCTBe Ai BSSITh KOpHH ypaBHeHHSI J o (A) =011 saMeHHTh TpHro-
HOMeTpHIJeCKHe MHOJKHTeJII1 B HHTerpaJIax <l>YPhe no CJIe.n.YIOlIJ;eMY npa-
BI1JIY: cos sz-+ -sin sz, sin sz -+ cos sz. I1cnOJIhsySI 3TI1 peIlIeHI1D.


S 16. 3aAaIJH AJIH nOJIy6ecRoHeIJHOl'O n;HJIHHApa,
CJIOH C OTBepCTHeM H nOJIynpocTpaHCTBa
C :qHJIHHAPHIJeCROH nOJIOCThlO
3aKOHIJHM paccMOTpeHl1e ocecHMMeTpl1IJHhIX sa.n.aIJ TeOp1111 ynpyroc-
TI1 .n.JIH 06JIaCTeH, rpaHIIUa KOTOpbIX 06pasyeTcSI IJaCTSIMI1 Koop.n.HHaT-
!fbIX nOBepxHocTeH uHJIHH.n.pHqecKoH CHcreMbI Koop.n.HHaT. 06cy.n.II!',!
KpaTKo BOSMOJKHOCTH paSBI1BaeMOro B pa60Te MeTo.n.a npHMeHI1TeJIhHO
K rpaHI1IJHhIM sa.n.aIJaM .n.JISI TeJI, KOTophIe MOJKHO paccMaTpHBaTb KaK
HeKoTopbIe npe.n.eJIhHbIe cJIYIJal1 npH I1SMeHeHl111 reOMeTpHIJeCKHX xa-
paKTepl1CTHK CnJIOlliHOrO 11 nOJIoro UI1JIHH.n.POB.
eJIec006pasHocTh
paccMOTpeHI1SI TaKHX sa.n.aIJ CBSIsaHa, KaK OTMeIJaJIOCh npH nOCTaHOBKe
sa.n.aIJH .n.JISI nOJIynOJIOChI, C 60JIee 111l1pOKI1MI1 BOSMOJKHOCTHMH npaKTH-
IJeCKOrO HCnOJIbSOBaHHSI KaJK)J.oro .n.OBe.n.eHHoro .n.o IJHCJIa pellieHIISI.


1M


155




HaH)J:eM, KaK H B c.nyqae n.nOCKOH 3a)J:aqH .1J:JI5I np5IMoyro.nbHHKa, pac-
CMOTpeHHOH B r.n. 1, qTO HapyrneHHe napHocTH KacaTe.nbHbIX Hanp5I.>Ke-
HHH BbI3bIBaeT n05IB.neHHe Ha pe6pe LLH.nHH)J:pa .norapH<pMHqeCKOH
oc06eHHocTH B 3HaqeHHH yr.na nOBopoTa B n.nOCKOCTH rOz.
[JO)J:qHH5I5I HaH)J:eHHbIe H3 (16.2) Ha OCHOBe 3aKOHa fYKa BbIpa.>KeHI15I
)J:.n5I KacaTe.nbHbIX Hanp5I.>KeHHH cooTBeTcTBYIO
HM YC.nOBH5IM B (16.1),
HaXO)J:I1M


npHBO)J:HM ee K BH)J:y


82 co Ajl!) "
j)
X (s) 11 (s) =
I Yj (S2 + "';)2 +
I f2j S2 + A;
+ 1JJ2 (s) [ i:
:
 - +] ,


00


+ 'l/Jl (s) +


I
I


(16.3)


3
Ajds + ..! r 2sAj'I)J2 (s) ds + 2'A 1 0 ('A.) (f2j - hj); (16.7)
( S2 + A
)2 11: J S2 + ,,
 I /
10/
1 [ I
 (s) m - 1 ]
11 (s) = - - - 1 - 2 .
4 Ii (s) ms 2
CBOHCTBa K03<pcpI1LLHeHTOB npl1 HeH3BeCTHbIX CHCTeMbI (16.7) aHa-
.nomqHbI CBoHcTBaM CHCTeMbI (8.16), no.nyqeHHOH npl1 paccMOTpeHHH
3a)J:aql1 )J:.n5I no.nyno.nOCbI. 3)J:ecb TaK.>Ke HMeeT c.nyqaiI BbIpO.>K)J:eHHoH
pery.n5IpHOCTI1, nOCKo.nbKY


00


A (s) = - C (s) [ '0 (s) - 2
II (s)
m-2
B.'A. = 2 D. - f2 / '.
1 / m /


m - 1 ] + \
2 (s)
ms s/l (s) ,


Yj =

 X (s)
o


C yqeTOM no.nyqeHHbIX 113 rpaHHqHbIX YC.nOBHH )J:.n5I KacaTe.nbHbIX
Hanp5I.>KeHHH COOTHorneHHH (16.3) ypaBHeHH5I, BbIpa.>KalO
He rpaHHqHbIe
YC.nOBH5I )J:.n5I HopMa.nbHbIx Hanp5I.>KeHHH, npHBo)J:5ITC5I K BH)J:y
C (s) sl (s) [ I
 (s) __ 1 _ 2 m -1 ] =
 D. 2S 2 AjJ o (Aj) +
1 I
 (s) ms 2 j=1 1 (S2 + A;)2
+
 f AjJo (Aj) + .h ( s ) + .h ( s ) [ 10 (s) -
 ] , ( 16.4 )
.-J 2j S2 + A
 '1'1 '1'2 II (s) s
j=1 1
_ 2
 oo 4SA;/l (S) 2
 oo 2S1jJ2 (s) ds
D -- C (s) ds + - + t f
j -- 11: 0 J o (Aj) (S2 + A
)2 11: 0 J o O'j) (S2 + 1.7) 2/ - Ij.


00


A

/ ds = 1,
(S2 + A
)2
J


(16.8)





o


00



 S2 _l1s_ m+1
t=1 (S2 + 1..;)2 - () 2ms 2


[Jo.nyqeHl1e 3TOH CHCTeMbI paBeHcTB conp5I.>KeHO C 06bIqHbIMI1 npe06pa-
30BaHH5IMH <PYHKLLHoHa.nbHbIx ypaBHeHHH C Hcno.nb30BaHHeM pa3.n0.>Ke-
HHH B p5I)J:e <PYHKLLHH Eecce.n5I MHHMoro aprYMeHTa (9.12) H npe)J:cTaB.ne-
HI1H HHTerpa.naMH <l>ypbe 3KcnOHeHLLl1a.nbHbIX <PYHKLLHH (8.9). K03<p-
<pHLLHeHTbI p5I)J:OB H n.nOTHOCTI1 HHTerpa.nbHbIx npe)J:cTaB.neHHH )J:.n5I <PYHK-
LLHH, 3a)J:alO
HX HopMa.nbHbIe Hanp5I.>KeHI15I B (16.1), BbIqHC.n5IIOTC5I no
H3BeCTHbIM <PopMy.naM.
B CB5I3H C npe)J:no.nO.>KeHHeM a paBeHCTBe HY.n1O Hanp5I.>KeHHH Ha
6eCKOHeqHOCTH 3.neMeHTapHbIe perneHH5I B npe)J:cTaB.neHHH 06
ero pe-
rneHH5I KpaeBOH 3a)J:aqH (16.2) 6bI.nH ony
eHbI. ECTecTBeHHo, QTO npl1
Ha.nl1ql1H HecaMoypaBHoBerneHHoH coCTaB.n5IIO
eH B Hanp5I.>KeHH5IX U:z
Ha TOpLLe LLH.nHH)J:pa ({10 =1= 0), yc.nOBJIe paBeHCTBa HY.n1O Hanp5I.>KeHI1H Ha
6eCKOHeqHOCTH MO.>KeT 6bITb BbIno.nHeHO .n11l11b 3a CqeT oceBOH CI1.nbI,
)J:ocTaB.n5IeMOH pacnpe)J:e.neHHbIMH no 60KOBL If nOBepXHOCTI1 KacaTe.nb-
HbIMH Hanp5I.>KeHH5IMH. CooTHorneHHe, OTpa.>KalO
ee 3TOT <paKT, no.ny-
QaeTC5I npl1 BbIBO)J:e paBeHCTB (16.4) H I1MeeT BH)J:


Tpe60BaHI1e 06pa
eHI15I B Hy.nb x (s) npH s = 0, BbITeKalO
ee H3
YC.nOBHH paBeHcTBa HY.n1O Hanp5I.>KeHHH Ha 6eCKOHeqHOCTH, He npOTHBO-
peQI1T nepBOMY ypaBHeHHIO B CHCTeMe (16.7), 11, c.ne)J:OBaTe.nbHO, aHa-
JIlI3 aCI1MnTOTHQeCKHX cBoikTB HeH3BeCTHbIX B HeH MO.>KeT 6bITb Bbmo.nHeH
B npe)J:no.nO.>KeHHH a KOHeQHOCTH Be.nHQHHbI x (s) B HeKoTOpoH KOHeQ-
HOH 8-oKpecTHocTH TOQKI1 s = O. CQI1Ta5I 3a)J:aHHbIMI1 3HaQeHH5I x (s)
B HeKoTOpoH Ma.noH oKpecTHocTH TOQKI1 s = 0, BTOPYIO rpynny ypaBHe-
HI1H B (16.7) npe)J:cTaBHM B BH)J:e
00 .3 8
4 ) fl.. 4 S
y. =- X(S) /2 ds+ - x(s)
J :J1 8 (S2 + A j ) 2 :J1 n


A

1 ds+
(S2 + A;)2


00


+
 r 2SA/ljJ2 (s) ds + 2'A/0 ('Ai) (h, - hj).
"J S2 + A

o I


(16.9)


flo = ..i.. r 'l)J2 (s) ds = 2 r 'l/J2 (z) dz.
:J1 J s J
o 0
BBO)J:5I B CHcTeMY (16.4) HOBbIe Hel13BeCTHbIe
x (s) = 4S/ 1 (s) C (s), Yj = 2'A/0 ('A j ) Di'
156


( 16.5)


B COOTBeTCTBI1H C nepBbIM ypaBHeHl1eM CHCTeMbI (16.7), paccMaTpHBae-
MbIM npl1 8 < s< 00, ypaBHeHI15I (16.9) 06pa3YlOT CI1CTeMY COOTHorne-
HHH.1J:JI5I onpe)J:e.neHH5I x (s) (8 < S < 00) 11 Yj (j = 1,2, ...). [JOCKO.nb-
KY I1HTerpa.nbI


00


(16.6)


\
 A

2 ds,
. (S2 +lq2
o 1


8 A

r X (s) / ds
J (S2 + A;)2
157




YObIBalOT KaK lI'A j , TO npH TaKHX }Ke npe.n.nOJIO}KeHH5IX 0 .n.HQxPepeHUII-
aJIbHbIX cBoikTBaX <PYHKUHH B rpaHlIqHbIX YCJIOBHHX, KaK H B 3a.n.aqe
.n.JIH nOJIynOJIOCbI, .n.JIH HelI3BeCTHbIX B 3TOH CIICTeMe CnpaBe.lI.JIHB 3aKOH
aCIIMnTOTHqeCKHX Bblpa}KeHIIH:
lim X (s) =
im Yj = a o =1= O. (16.10)
5-+00 1-+00


MOCTH B TOqKe , = 0, Z = 0 nOJIyqlIM CJIe.n.YIOIIlee OTHOIlIeHHe HanpH}Ke-
HIIH:



=
+
.
U z 2 m
LLJIH CJIyqaH m = 3 3TO OTHOIlIeHHe paBHo 0,83.
I1cnoJIb3YH .n.aHHble TaOJI. 17, Haxo.n.lIM ;G = 1.02 P. YqlITbIBaSl
J ° (0) = 1, nplIxo.n.HM K 3aKJIlOqeHIIIO, qTO .n.JIH nOJIy6ecKOHeqHOrO
UHJIIIH.n.pa B ueHTpe TOpua u/u z = 1,02. TaKHM 00pa30M, B oceCHM-


(16.14)


AHaJIII3 ypaBHeHIIH .n.JIH X (s) B (16.7) npH 0 <= S <= 8 nOKa3bIBaeT, qTO
nOJIyqeHHble aCHMllTOTHqeCKlIe paBeHCTBa He npOTHBOpeqaT npe.n.nOJIO-
}KeHIIIO 0 KOHeqHOCTII X (s) B OKpeCTHOCTII TOqKII S = O.
KOHKpeTHble BblqlICJIeHII5I HanpH}KeHHOra COCTOHHH5I nOJIyoecKo-
HeqHOrO UIIJIHH.n.pa npOBe.n.eM .n.JIH CJIyqaH HarpY}KeHIIH era Topua HOp-
MaJIbHbIMH caMoypaBHoBeIlIeHHbIMII HarpY3KaMII. 113 Bcex B03MO}KHbIX
pacnpe.n.eJIeHIIH U z no TOPUY BbIoepeM TO, KOTopoe XapaKTeplI3yeTcH
HalIMeHbilieH 1I3MeHHeMOCTblO, T. e. nOJIO)f{IIM B (16.1)
I
2G U Z (z = 0) = PJo ('AI'). (16.11)
lI.eJIeCOOOpa3HoCTb paccMoTpeHHH rpaHlIqHOH 3a.n.aqH (16.1), Kor.n.a
3HaqeHIIe U z Ha TOpue onpe.n.eJIHeTCH no <popMYJIe (16.11), a Bce OCTaJIb-
Hble KOMnOHeHTbI TeH30pa Hanp5I}KeHIIH paBHbI HYJIIO Ha rpaHlIue nOJIY-
UHJIHH.n.pa, BbITeKaeT H3 pe3YJIbTaTOB peilieHIIH 3a.n.aqII .n.JISl nOJIynoJIO-
CbI. npii aHaJIII3e xapaKTepa HanpH}KeHHOrO COCTOHHIIH nOJIynOJIOCbl
ObIJIO YCTaHOBJIeHO, qTO HalIOOJIbIlIIIH 3<p<peKT «CBooo.n.Horo KpaH» .n.JIH
HanpH}KeHHH U y B ueHTpe Topua oy.n.eT npii HalIMeHbilieH H3MeHHeMOC-
TII caMoypaBHoBeIlIeHHOH HOpMaJIbHOH HarpY3KH Ha Topue nOJIynOJIO-
CbI. KaqeCTBeHHO TaKoe }Ke nOJIO}KeHlIe BII.n.IIM II .n.JIH oceCIIMMeTplIqHOH
3a.n.aqlI. KOJIlIqeCTBeHHble oueHKII nOJIyqlIM npii paCCMOTpeHIIH nOCTaB-
JIeHHOH rpaHlIqHOH 3a.n.aqlI.
nOCTpoeHlIe KOHeqHOH CIICTeMbI, OTBeqalOIIleH CIICTeMe (16.7), B co-
OTBeTCTBlI1I C (16.10) npOBo.n.IIM Ha OCHOBe paBeHcTB
Yj = YN (j> N), x(s) = X (T) (s> T). (16.12)
KOHKpeTHble BblqHCJIeHIIH npoBo.n.HJIIICb npH N = 20, T = 20 ,lI.JIH Ma-
TepHaJIa C m = 3. B TaOJI. 17 nplIBe.n.eHbI C TOqHOCTblO .11.0 MHO}KIITeJIH P
3HaqeHIIH .n.Ba.n.uaTH BeJIlIqlIH Yj. lLnH X (T) nOJIyqlIM 3HaqeHIIe 1,296.
B KaqeCTBe npIIOJIII}KeHHOra 3HaqeHHH a o B (16.10) npii npoBe.n.eHHII BbI-
qlICJIeHIIH npHHHTO


Ta6J1Hlla 17
Yj Yj Y/ Yj
I 3,473 6 1,084 11 1,186 16 1,222
2 0,691 7 1,117 12 1,196 17 1,226
3 0,866 8 1,141 13 1,204 18 1,230
4 0,971 9 1,160 14 1,211 19 1,234
5 1,038 IO 1,174 15 1,217 20 1,237


MeTpHqHbIX 3a,lI.aqaX 3<p<peKT CBOOO.n.HOH nOBepxHocTH TaK}Ke npOHBJIH-
eTCH .n.OBOJIbHO CYlll,eCTBeHHO, nplIBo.n.H K nOBbIIlIeHlI1O :HaqeHIIH HanpH-
}KeHHH U, B ueHTpe TOpua Ha BeJIHqlIHY nopH.n.Ka 25 Yo OT aMnJIIITy.n.bI
nepBoH rapMoHIIKII B 3a.n.aHHbIX Ha rpaHlIue HanpH}KeHIIHX U z . lLnH 00-
JIee BbICOKIIX HOMepOB rapMoHHK BJIHHHHe 3Toro 3<p<peKTa, KaK H .n.JIH
nOJIynOJIOCbI, OCJIaOeBaeT. u
'ECJIII pa.n.HYC BHeIlIHeH nOBepxHocTII nOJIoro UIIJIIIH.n.pa KOHeqHOH
.n.JIIIHbI .n.OCTaTOqHO BeJIIIK, TO aHaJIII3 HanpH}KeHllOrO COCTOHHHH BOJIII-
311 BHYTpeHHeH era nOBepXHOCTII y.n.OOHee npOBo.n.lITbu Ha OCHOBe peIlIe-
HIIH rpaHlIqHOH 3a.n.aqII .n.JIH CJIOH C UIIJIHH.n.plIqeCKOH
OJIOCTblO. 3Ha-
qlITeJIbHOe YMeHbIlIeHHe rpOM03.n.KOCTH BbIKJIa.n.OK B 3TOII 3a.n.aqe TaK}Ke
.n.OCTlIraeTCH nYTeM pa3.n.eJIbHOrO paCCMOTpeHHH CHMMeTpHqHOrO H aHTH-
CIIMMeTpHqHOrO OTHOCHTeJIbHO cpe.n.HHHOH nJIOCKOCTH CJIOeB HanpH}KeH-
HbIX COCTOHHHH. OCTaHOBIIMCH no.n.pOOHO Ha CHMMeTpHqHOM CJIyqae.
PaCCMOTplIM OCeCIIMMeTpHqHOe HanpH>KeHHOe COCTOHHlIe OCJIaOJIeH-
Horo UHJIHH.n.pHqeCKOH nOJIOCTblO CJIOH (pHC. 18), nOpo}K.lJ.aeMOe HeKO-
TOpOH clIcTeMoH pacnpe.n.eJIeHHbIX no ero nOBepxHocTH HarpY30K:
1 1 1
- U = oh ( z ) - 'rz = '1\12 (z) npii r = ,
2Gr't'I' 2G


I
a O = "2 fY20 + X (20)] = 1,266.


(16.13)


BblqHCJIeHlIe HanpH}KeHHH B nOJIyulIJllm.n.pe no 3HaqeHHHM K03<p-
<pIIUHeHToB TaM. 17 nOKa3bIBaeT, qTO norpeiliHocTb y.n.OBJIeTBOpeHHH
rpaHlIqHbIX YCJIOBIIH .n.JIH HOpMaJIbHbIX HanpH>KeHIIH, B03HHKalOllI,aSl
3a CqeT IICnOJIb30BaHHH COOTHOilieHIIH (16.12), He npeBbIIlIaeT 2% OT
BeJIlIqlIHbl P. fpaHHqHble YCJIOBIIH .n.JIH KaCaTeJIbHbIX HanpH}KeHIIH y.n.OB-
JIeTBOpHIOTCH TOqHO.
B cJIyqae ocecHMMeTpHqHOrO .n.e<popMlIpOBaHH51 nOJIynpocTpaHcTBa
HOpMaJIbHOH nepHo.n.HqeCKOH HarpY3KoH C JIIOOOH CTeneHblO H3MeHHe-
158


(16.15)


1 1 t ( ) n p H Z = + 11
2G U Z = f1 (r), 2G 'rz = ::i: 2 ,
3.n.ecb , = p/po, Z =
/po, h = H!po, a <PYHKUHH '1\11 (z) H '1\12 (z)
TaKlIe, qTO 'l/J1 (z) = '1\11 (-z), 'l/J2 (z) = - 'l/J2 (-z).
OOlll,ee peilieHlIe rpaHHqHOH 3a.n.aqII (16.15) COCTOHT H3 .n.BYX qac-
TeH. nYCTb nepBaH qaCTb oOlll,ero peIlIeHHH onpe.n.eJIHeTCH Tpe6oBaHHH-
MH nOJIHOTbI ero Ha UHJIIIH.n.pHqeCKOH nOBepxHocTH r = 1. Tor.n.a COOT-


159


Iii
,


I
I
'I
I
I.
I
I




neTCTBYIOIlIJle BblpameHIUI .l(JUI KOMrrOHeHTOB BeKTOpa CMelII;eHllM JIer-
KO rrOCTpOllTb, .l(OrrOJIHllB perneHlle .l(JIH nepllO.l(llqeCKll .l(e<POpMllpOBaH-
HOM BHernHOCTll UllJIllH.l(pllqeCKOM rrOJIOCTll 3JIeMeHTapHbIM perneHlleM:
U I = Do +
 { _ DnKI (knr) + Bn [ rKo (knr) +
r n=I
+ 4 mm
 I K 1 (knr) ]} cos knz,


WI = -
 [DnKo (knr) - B n rl(l (knr)] sin knz.
n=J
t OqeBll.l(HO, qTO 3Tll BblpameHllH .D:JIH
pa.l(llaJIbHOM II oceBOM KOMnOHeHT
BeKTopa nepeMelII;eHllM nOJIyqaIOTCH
ll3 (13.6) npll Bo = 1'n = qn =
= 0, 6n = Dn, Pn = Bn-
BTopaH qaCTb o6
ero perneHllH
.l(OJImHa cO.l(epmaTb .l(OCTaTOqHYIO
CTerreHb <!JYHKUllOHaJIbHOrO npOll3Bo-
JIa .l(JIH Y.l(oBJIeTBOpeHllH rpaHllqHbIX
YCJIOBllM Ha rrJIOCKOCTHX CJIOH BHe OT-
BepcTllH. ITocTpoeHlle Y.l(oBJIeTBOpH-
Puc. 18. IO
llX 3TOMY Tpe6oBaHllIO Bblpame-
HllH .l(JIH KOMrrOHeHT BeKTopa CMe-

eHllM ocy
eCTBJIHeTCH C rroMO
bIO BeKTopHoro npeo6pa30BaHllH
Be6epa.
BbIBO.l( cOOTBeTCTBYIO
llX <!JOPMYJI, OTHOCH
llXCH K 06
eMY CJIY-
[laIO HeocecllMMeTpllqHOrO .D:e<POpMllpOBaHllH CJIOH, CO.l(epmllTcH B r JI. 5.
I1crrOJIb30BaHlle llX .l(JIH paccMaTpllBaeMoro 3.l(eCb ocecllMMeTpllQHOrO
CJIyqaH rr03BOJIHeT rrpelICTaBllTb KOMrrOHeHTbI BeKTopa cMe
eHllM B Bll.l(e
U 2 = - r [A (t) chi z + C (i) ( 3m;;: 4 ch iz + iz sh iZ)] Nr(ir) di,
00 0 (16.17)
W 2 = I [A (f) sh fz + C (i) fz ch iz] No (fr) df.
o


npHmeHllM 6Y.l(YT Y.l(OBJIeTBOpeHbI, ecJIll nOJIOmllTb
[ Ko(kn) 0 m-I J 'lJJ2n
Dnkn = Bn k n K 1 (k n ) +,£ m + K 1 (knl '
A (i ) = - C (i) ( 2 m - I + i h cth ih ) _

m shih '


'I
I'
\1
I
II,
I
,


(16.19)


r.l(e


(16.16)


n
'ljJ2n = + s 'ljJ2 (z) sin knzdz,
o


(16.20)


00


f2 (i) = S f2 (r) [JI (ir) Y I (f) - Y I (fr) J I (i)) rdr.
I
.LI.JIH npeo6pa30BaHHH K Y.D:06HOMY BH.l(Y <PYHKUHOHaJIbHbIX ypaBHe-
HHM, rropOm.l(aeMbIX rpaHllqHbIMH YCJIOBllHMH .l(JIH HOpMaJIbHbIX HarrpHme-
HHM B (16.15), llCrrOJIb3YIOTCH rrpe.l(CTaBJIeHHH TpHroHoMeTpllqecKHMll
pH.l(aMH rHrrep60JIllqeCKHX <!JYHKUllM (5.6), a TaKme CJIe.l(YIOIliHe rrpe.l(-
CTaBJIeHHH <!JYHKIJ.HM MaK.l(OHaJIb.l(a llHTerpaJIaMH Be6epa:


2 S No (fr)
1( 0 ( knr ) = - - knl(I (k n ) 2 . di,
n k + f2
o n
k rl( ( k r ) = -
 k 2 OO s [ Ko (k n ) + 2k n Kl (k n ) ] N ( fr) df.
n 1 n n n k2 + i2 (k2 + f2)2 0
o n n
rpaHHqHOe YCJIOBlle .l(JIH HOpMaJIbHbIX HarrpHmeHllM C yqeTOM
BeHCTB (16.19) rrpllBO.l(HTCH K CJIe.l(YIO
eM CHCTeMe COOTHorneHHM:


(16.21)


pa-


H N I (fr)


00
Do - + s f2 (i) No (f) di = -'ljJ10'
o
00 4 ( _ I ) n fk
 "h fh
B k l:1 ( k ) I( ( k ) - r C ( f ) n
- n n n 1 n - d h (k
 + f2)2
+ (- 1)" a n l1 (k n ),
- C (f)
 sh ihS (f) +

 (- 1)" B kniKI (k n )
h n n=1 n (k
 + f2)2


I
II
I:


No (f)idf +


(16.22)
= S (i)
 (i).
nt


3.l(eCb A (f) II C (f) - rrpOH3BOJIbHble <!JYHKUllH, a No (lr)
0603HaqaIOT CJIe.l(YIOlII;He KOM6HHaUHH <!JYHKUllM BeCCeJIH:
No (ir) = 10 (fT) Y I It) - Yo (iT) J 1 (f)
Ji (f) + Yi(t) ,
N (fr) = J 1 (fT) Y 1 (f) - Y I (fT) J 1 (f)
1 Ji (f) + Yi (t) .
BblpameHHH .If.JIH HanpHmeHHM, OTBeqaIO
lle CM
eHllHM U = U I +
+ U 2 , W = WI + W 2 , MorYT 6bITb JIerKO rrOJIyqeHbI C HCnOJIb30BaHHeM
3aKOHa rYKa. I1cnOJIb3YH HX, Ha OCHOBe paBeHcTB sin knh = 0,
N I (i) = 0 HaXO.l(HM, qTO rpaHHqHble YCJIOBHH .l(JIH KaCaTeJIbHbIX Ha-


3.l(ecb rrpHHHTbI CJIe.l(YIO
lle 0603HaqeHmI:
00
n 2 r fWo(f) [ I Ko(k n ) ]
(- 1) l:1 (k n ) an = 1jJln - h d 12 (f) 1?" + r; df - 'ljJ2n Ii; + K I (k n ) ,
S (t)
 (t) = f (f) + f (f) cth fh _

 (- I)n 'lJJ 2n k n
nt 1 2 n L.J f ( k 2 + t 2 ) ,
n=J n
h h
1JJIo = + s 1JJI (Z) dz, 'ljJln = + 5 'ljJI (z) cos knzdz, (16.23)
o 0


(16.18)


160


11 7-2464


161




co
II (i) = .\ II (r) [J O (ir) Y I (t) -Yo (ir) J I (t)] rdr,
1


B COOTBeTCTBIUI C Tpe60BaHIHIMH nepBoro YC.lIOBHH CnpaBe
.lIHBOCTH
3aKOHa aCllMnTOTHqeCKHX BbIpmKeHHH Jl.lIH Hell3BeCTHbIX B (16.25) He-
06XOJlHMO BbInO.lIHeHlle HepaBeHCTB
l <=
(

) <= L (t> n), 1 <= a
:) <= L (n >i) (16.28)
npJi npOll3BO.lIbHbIX CKO.lIb yrO
Ho 60.lIbIlIHX n H t. 3HaqeHHH K031lJcjm-
I1.HeHTOB an (t) H bn(t) H3 (16.26) n03BO.lIHIOT YCTaHOBHTb
2 b n (I) (t )
( 1];2 ) 2 <=
 <= 2 > n ,
1+/i2
(16.29)
4mh 2 <= an (/l <= (2m) (n > i).
1];2 (m- 1) {h 2 + 1];2)2 [Jr (I) + Yi (I)] <Pn 1]; m - 1
npH nO.lIyqeHHH OI1.eHOK
.lIH K03q I IlJHI1.HeHTOB an (t) llCnO.lIb30BaHO TO
06CTOIITeJIbCTBO, qTO q,YHKI1.HH i lJi (t) + Yi (t)l HB.lIHeTCH MOHOTOHHO
y6bIBaJOIlleH (67] Ii BCerJla n > 1. u
HepaBeHcTBa (16.29) nOKa3bIBa1OT, qTO npH npOH3BO.lIbHOli KOHeq-
HOM: TOJIIl{HHe C.lIOH H npH .lII060M 3HaqeHHH qllC.lIa nyaccoHa m MomHO
YKa3aTb KOHKpeTHbIe 3HaT.J:eHHH nOCTOHHHbIX 1 H L B (16.28). npoBepKa
Breparo YC.lIOBMH cnpaBe
JrllBocTH 3aKOHa aCllMnTOTHT.J:eCKHX BbIpame-
HHM B .QaHHOM c.lIyqae H3.lIHIlIHHH, nOCI<O.lIbKY IlJYHKI1.HH (fin H (J) (t)
y6bIBalOT Ha 6eCKOHeqHOCTH KaK lIn H 1/t.
CYIllecTBoBaHHe KOHeT.J:HbIX 3HaqeHHH 1 H L B (16.28) YKa3bIBaeT, qTO
aCHMnTOTHqeCKHe CBOHCTBa HeH3BeCTHbIX B (16.25) Onpe
eJIHIOTCH pa-
BeHCTBOM


A (k n ) = I K
 (k n ) + 2 mm
 1 , S (i) = 7- (cth th + s
Z Ih ) .
K I (k n ) n
fl..lIH npHBe
eHHH CHCTeMbI (16.22) K KaHOHHqeCKoMY Bll
Y BBe
eM HO-
BbIe HeH3BeCTHbIe npH nOMoIllH CooTHoIlIeHllH
(- l)n II" x (I) h
Bn = - knKI (k n ) , C (i) = - 1];21 sh Ih . (16.24)
HOBbIe HeH3BeCTHbIe x (i) H Yn HB.lIHIOTC5I peIlIeHHHMH C.lIe
YIOIlleH CHCTe-
MbI:


co
Yn = S an (t) X (i) di + an,
o


(16.25)


co


x(i) =
 b n (i) Yn +
 (i),
n=1


rJle


8k
 1
an (i) = 2tJ. (k ) I 12 2 2 '
n n (n+/2)2[JI{/)+YI(/)]
b ( i ) _
 . 1 2
n - S (I) (k
 + 12)2
I1CnO.lIb30BaHlie paBHOMepHOH CXO
HMOCTH PH.QOB'B (5.6) H HHTerpa-
.lIOB B (16.21) n03BO.lIHeT YCTaHOBHTb, qTO MH nOJIOmHTe.lIbHbIX K031lJ-
IlJHI1.HeHTOB CHCTeMbI (16.25) cnpaBe
.lIHBbI paBeHCTsa


(16.26)


lim Yn = lim x (i) = a o . (16.30)
n-'JoOO 1-"00
C.lIe
OBaTe.lIbHO, npH npaKTHqeCKOM peIlIeHllH CHCTeMbI (16.15) MomeM
nO.lIaraTb


co
S an (t) di = 1 - qJn' i; b n (i) = 1 - (J) (t);
o n=1
m- 1 I 2
qJn = 2 m/i;, tJ. (k,,) ' (J) (i) = 1 2 S (I) ,
KOTopbIe YKa3bIBalOT Ha TO, 'ITO CHCTeMa (16.25) HB.lIHeTCH perY.lIHpHoH.
XapaKTep y6bIBaHHH IlJYHKI1.HH qJn H (J) (t) C pOCTOM n H t TaKoH, 'ITO
JIH
CYIlleCTBOBaHHH OrpaHH'IeHHOrO peIlIeHHH CHCTeMbI (16.25) JlOCTaTOq-
HO npe
nO.lIOmeHHH 0 napHOCTH KaCaTe.lIbHbIX HanpHmeHHH B yr.lIOBbIX
TO'IKaX H 06bI'IHbIX npeJlnOJIOmeHHH 0 KYCOqHOH HenpepbIBHOCTH Ha co-
oTBeTcTBYIOIllHX HHTepBa.lIaX nepBbIX npOH3BOJlHbIX OT IlJYHKI1.HH, 3a-

aIOIllHX BHelliHlO1O HarpY3KY
OTHOCllTe.lIbHO aCHMnTOTHqeCKHX CBOHCTB HeH3BeCTHbIX B CHCTeMe
(16.25) eCTeCTBeHHO OmHJlaTb, qTO C pOCTOM n H t IlJYHKI1.HH Yn H X (i)
cTpeMHTcH K 06ll1,eMY OT.lIHqHOMY OT HY.lIH npe
e.lIY. <f>opMa.lIbHOe
o-
Ka3aTeJIbCTBO 3TOro llJaKTa .lIerKO nO.lIyqHTb npoBepKoH He06xoJlHMbIX
YC.lIOBHH, npHBeJleHHbIX B nepBoH r .lIaBe.


(16.27)


Yn = YN (n > N), X (i) = X (T) (i> T).


(16.31)


KOHKpeTHbIe BbIqllC.lIeHl1H npoBe
eHbI
.lIH C.lIOH, Y KOTOporo
Ha-
MeTp OTBepCTHH paBeH TOJIIllHHe (h = 1). 4HC.lIO nyaccoHa MaTepHa.lIa

J1H C.lIOH npHHHTO paBHbIM qeTbIpeM (m = 4). Hanp5ImeHHoe COCTOHHHe
06YC.lIOB.lIeHO npH.lIOmeHHbIMH K nOBepxHocTH nO.lIOCTH pa
Ha.lIbHbIMH
HarpY3KaMH, KOTOpbIM OTBeqaeT IlJYHKI1.HH qJl (z) = -p cos :nz. Oc-
Ta.lIbHbIe IlJYHKI1.HH B rpaHHqHbIX YC.lIOBHHX (16.15) npHHHTbI paBHbIMH
HY.lIJO.
B Ta6JI. 18 C TOqHOCTblO
o nOCTOHHHoro MHOmHTeJIH P
aHbI Be.lIH-
'IHHbI K03l}JqmI1.HeHTOB Yn H3 CHCTeMbI (16.25), HaHJleHHbIe npll N = T =
= 20 B (16.31). fl..lIH HCKOMOH IlJYHKI1.HH x (i) npH t = 20 nO.lIY'IHM
X (20) = 1,158 P.
XapaKTep nOBe
eHHH K031lJIlJHI1.HeHTOB Yn C B03paCTaHHeM HOMepa,
a TaKme
OCTaTOqHO xopOIlIee COBna
eHHe 3HaqeHHH Y20 H x (20) n03-
BOJUllOT Ha
eHTbCH Ha nO.lIyqeHlle
OCTaTOqHO BbICOKOH TOqHOCTH Y
OB-


162


11*


163


II
I'
II


,I
I'


I


I
I
I


II




JIeTBOpeHJHI rpaHHqHbIX YCJIOBHH, eCJIH rrpH BblqHCJIeHHHX HCrrOJIb30-
1
BaTb ,AaHHble Ta6JI. 18, a B KaqeCTBe a o B (16.30) rrpHHHTb ao = 2 [Y20 +
+ x (20)] = 1,139 P. OpOBe,AeHHble BblqHCJIeHHH rrOKa3bIB2IOT, qTO
HaH60JIbmaH OTHOCHTeJIbHaH rrorpemHocTb Y,AOBJIeTBOpeHHH rtJaHHqHbIX
T a 6 JI H Ll a 18


3aMeHa B pemeHHH ,AJIH CJIOH C OTBepCTHeM pSI,AOB 4'YPbe HHTerpaJIaMH
4'YPbe rr03BOJIHeT rrpe,ACTaBHTb o6LQee pemeHHe OCHOBHbIX rpaHHqHbIX
3a,Aaq ,AJIH YKa3aHHOH 06JIaCTH B CJIe,AYIOLQeM BH,Ae:
u = - S [ A (i) + C (i) ( 3m;;: 4 + iZ)] e- tz N 1 (ir) di +
o
00
+
 J {- D (s) K 1 (rs) + B (s) [rKo (rs) +
+ 4 m- 1 K ( rs )]} ( C?S SZ ) ds,
ms 1 - SIn SZ


n I {In II n I Yn II n I Yn II n I Yn
1 2,827 6 1,004 11 1,084 16 1,110
2 0,622 7 1,031 12 1,091 17 1,1l3
3 0,805 8 1,050 13 1,097 18 1,116
4 0,904 9 1,064 14 1,102 19 1,118
5 0,965 10 1,075 15 1,106 20 1,120


(16.32)


00


w = -
 [A (i) - C (i) iz) e-tW o (ir) dt +
o
2 00 5 (
na )
+ - 1- D(s) Ko (rs) + B (s)rK I (rs)] ds.
:rt COS SZ
o


YCJIOBHH KaK ,AJIH Up TaK H ,AJIH U z 6Y,AeT B yrJIOBOH TOqKe. O,AHaKO OHa
He rrpeBblmaeT O,AHOro rrpOlleHTa aMrrJIHTY,AbI BHemHeH HarpY3KH P.
HaJIHqHe BblqHCJIeHHbIX C ,AOCTaTOqHOH TOqHOCTbIO, Ha qTO YKa-
3bIBaeT rrpOBepKa Y,AOBJIeTBOpeHHH rpaHHqHbIM YCJIOBHHM, BeJIHqHH
Yn B CHCTeMe (16.25) rr03BOJIHeT rrpoBecTH BbI'IHCJIeHHe Bcex <paKTOpOB
HarrpHLKeHHo-lIecpopMHpoBaHHoro COCTOHHHH CJIOH. Orrpe,AeJIeHHbIH HH-
Ta6J1H
a 19


npH Y,AoBJIeTBOpeHHH rpaHHqHbIX YCJIOBHH M02f\HO HCrrOJIb30BaTb O,AHO-
BpeMeHHo JIH60 BepxHHe, JIH60 HHLKHHe 3HaqeHHH TpHrOHOMeTpHqeCKHX
MHOLKHTeJIeH B HHTerpaJIax 4'YPbe.
npOlle,Aypa rrO,AqHHeHHH BblpaLKeHHH (16.32) HJIH rrocTpoeHHblx rro
HHM BblpaLKeHHH ,AJIH HarrpHLKeHHH rpaHHqHbIM YCJIOBHHM ,AOCTaTOqHO HC-
Ha. n03TOMY He 6Y,AeM OCTaHaBJIHBaTbCH Ha rrO,Ap06HOCTHX, a 06paTHM-
CH cpa3y K paCCMOTpeHHIO CHCTeM HHTerpaJIbHbIX ypaBHeHHH, rrOJIyqaIO-
LQHXCH rrpH Y,AOBJIeTBOpeHHH rpaHHqHbIX YCJIOBHH B CMell.{eHHHX. CJIY-
qaH rrepBOH OCHOBHOH rpaHHqHOH 3a,AaqH, a TaKLKe rrO,Ap06Hble BbIKJIa,A-
KH, OTHOCHLQHeCH K paCCMaTpHBaeMOMY CJIyqaIO 3a,AaHHH Ha rrOBepx-
HOCTH CMell.{eHHH, MOLKHO HaHTH B pa60Te 1761-
Y,AOBJIeTBOpHH rpaHHqHbIM YCJIOBHHM B CMeLQeHHHX BblpaLKeHHHMH
(16.32) rrpH HCrrOJIb30BaHHH BepxHHx 3HaqeHHH TpHroHoMeTpHqeCKH

MHOLKHTeJIeH, nOJIyqaeM CJIe,AYIOLQYIO napy HHTerpaJIbHbIX ypaBHeHHH
OTHOCHTeJIbHO HeH3BeCTHblX <PYHKllHH Y (s) H x (t):
r 2t 2 sN 0 (t) N; (t) tdt
Y (s) s6. (s) = J x(i) (S2 + t 2 )2 + ids),
o
3m 4 2r 2ts 2
X (i) - = - J Y (s) (t2 + S2)2 ds + i2 (i),
m :rt 0


z/h I U z /20P II z/h I U z /20P II z/h I U z /2GP " z/h I U z /20P
0 -0,965 0,3 -0,677 0,6 -0,111 0,85 0,083
0,1 -0,930 0,4 -0,490 0,7 0,023 0,90 0,059
0,2 -0,830 0,5 -0,292 0,8 0,086 1,00 0,009


Tepec B paccMaTpHBaeMoH 3a,Aaqe rrpe,ACTaBJIHeT BeJIHqHHa H xapaKTep
pacrrpe,AeJIeHHH HarrpHLKeHHH U z Ha rroBepxHocTH oTBepcTHH (Ta6JI. 19).
XapaKTep rI3MeHeHHH HarrpHLKeHHH U z rro TOJIll.{HHe B paccMaTpHBae-
MOH ocecHMMeTpHqHOH 3a,Aaqe TaKOH LKe, KaK H HarrpHLKeHHH u y pemeH-
HOH paHee 3a,Aaqe ,AJIH rrCJIyrrOJIOCbI. BOJIee JII06orrbITHbIM, oD;HaKo, HB-
JIHeTCH CJIe,AYIOll.{ee 06CTOHTeJIbCTBo. TIpH paccMoTpeHHH 3a,AaqH ,AJIH
rrOJIyrrOJIOCbI 6bIJIO YCTaHOBJIeHO, qTO O,AHHM H3 CJIe,ACTBHH HaJIHqHH CBO-
60,AHbIX nOBepxHocTeH B rrJIOCKOH 3a,Aaqe HBJIHeTCH cymecTBeHHoe (rro-
pH,AKa 25%) rrOBblmeHHe HanpHLKeHHH u y rro cpaBHeHHIO C HX 3HaqeHHH-
MH ,AJIH nOJIyrrJIOCKOCTH. ECJIH aHaJIOrHqHO rrpoBecTH cpaBHeHHe 3Ha-
qeHHH U z rrpH Z = 0 rI3 Ta6JI. 19 co 3HaqeHHeM HX B EITOH LKe TOqKe rre-
pJIO,AHqeCKH 3arpYLKeHHoro rrpoCTpaHcTBa C llHJIJIH,ApJIqeCKOH rrOJIOCTbIO,
TO 06HapYLKHBaeM, qTO HarrpHLKeHJIH B Cpe,AHHHOH rrJIOCKOCTH CJIOH Ha
BeJIHtfHHY rropH,AKa 25% BbIllIe, q€M B rrpOCTpaHcTBe C rrOJIOCTbIO.
B 3aKJIIOqeHHe rJIaBbI KpaTKO paccMoTpHM OCeCJIMMeTpJIqHble rpa-
HHqHble 3a,AaqJI ,AJIH rrOJIynpoCTpaHcTBa C llI1JIHH,ApJIqeCKOH rrOJIOCTbIO.
164


(16.33)


r,Ae


_ K
(s) _ 1 + 4 m-s Ko(s)
f),. (s) - 2 ms K (s) ,
Kt (s) 1
N
 (ir) = J 1 (i) Yo (ir) - Y 1 (i) J o (ir),
a <PYHKllJIH il (s) H i2 (i) orrpe,AeJIHIOTCH KOHKpeTHbIM BJI,AOM rpaHJIqHbIX
165




YCJiOBHH. I1CXO.n;SI


HS SHa'IeHHH HHTerpaJiOB
""
S 2ts2 .n
([2 + S2)2 ds = 2 '
o


t = 0 B BH.n;e


S "" 2stWo (t) N; ([)tdt = s [ K
 (S) _ ]
 K 0 (S) J
o ([2 + S2)2 Ki (S) + S Kl (S) ,
.nerKO y6e>K.n;aeMCSI B nOJiHOH perYJiSlpHOCTH CHCTeMbI (J 6.33).
K HHTepecHoMY pesYJibTaTY npHxo.n;HM npH HCnOJibSOBaHHH aHTH-
cHMMeT
H'IHoro npo.n;OJi>KeHHSI B 06JiaCTb oTpHUaTeJibHbIX z rpaHH'lHbIX
YCJiOBHH Ha nOBepxHocTH nOJiOCTH (peIIleHHe C HH>KHHM SHa'leHHeM TPH-
rOHOMeTpH'leCKoro MHO>KHTeJiSl B (16.32». B 3TOM cJiY'lae CHCTeMa om-
rYJiSlpHbIX HHTerpaJibHbIX ypaBHeHHH, BOSHHKalOIltaSl npH y.n;oBJieTBO-
peHHH rpaHH'IHbIX YCJiOBHH B CMeIlteHHSlx, npe.n;cTaBJiSleTCSI B BH.n;e


(16.34)


co
W = - 3m ;;:4 .\ X (i) No (ir) N
(i) di. (16.37)
o
ECJiH Y'leCTb SHa'IeHHe X (i) HS (16.36), TO BH.n;HM, 'ITO no.n;bIHTerpaJib-
lIaSl <PYHKUHSI B BbIpa>KeHHH (16.37) .n;JiSl W HMeeT HeHHTerpHPyeMYI?
uco6eHHOCTb B HYJie. 3TO YKasbIBaeT Ha TO, 'ITO peIIleHHe O.n;HOpO.n;HOH
CHCTeMbI (16.35) He CBSlsaHO C y.n;OBJieTBOpeHHeM <pHSH'IeCKH onpaB.n;aH-
HbIX YCJiOBHH.
ECTeCTBeHHO, npHBe.n;eHHbIe paCCY>K.n;eHHSI HHKaK He peIIlalOT Bonpoc
o TOM, KaKHM 06paSOM .n;OJi>KHO Haxo.n;HTbCSI HY>KHOe peIIleHHe CHCTeMbI
(16.35). He HMeSl BOSMO>KHOCTH OCTaHOBHTbCSI Ha 3TOM no.n;po6HO, YKa-
}KeM, 'ITO npe.n;JiO>KeHHbIe BbIIIle no.n;xo.n;bI K HCCJie.n;OBaHHIO H peIIleHHIO
CHCTeM C BbIpO>K.n;eHHoH perYJiSlpHOCTbIO (noJiynoJioca, nOJiy6ecKOHe'l-
IlbIH UHJiHH.n;p) YKasbIBaIOT nYTb, no KOTOPOMY cJie.n;YeT H.n;TH npH OTbIC-
IWHlUI HY>KHOro peIIleHHSI CHCTeMbI (16.35). 3TO peIIleHHe, O'leBH.n;HO,
.n;OJI>KHO y.n;oBJieTBOpSlTb onpe.n;eJieHHbIM YCJiOBHSlM B TO'lKaX uS = i = O.
ECJiH 3TH YCJiOBHSI C<PoPMYJiHpOBaHbI Ha OCHOBe Tpe60BaHHH o6ecne'Ie-
HHSI onpe.n;eJieHHOrO y6bIBaHHSI <PYHKuHH B rpaHH
HbIX YCJiOBHSlX,
TO OTHOCHTeJibHO CHCTeMbI HHTerpaJibHbIX ypaBHeHHH .n;JiSl HeHSBeCT-
IIbIX X (i) (8 <;: i < (0) H Y (s) (8 <;: s< (0), nOJiY'laeMOH HS (16.35),
cnpaBe.n;JiHBbI Bce TeopeMbI TeopHH pery.nSlpHbIX 6eCK?He'lHbIX CHCTeM H
ee peIIleHHe C HY>KHbIMH CBOHCTBaMH MO>KeT 6bITb HaH.n;eHO.


""
/). S f 2t2
Y (s) (s) = 0 X (i) (S2 + [2)2


2 m- 2 I J .
+ m [2+S2 No(i)No(i)idi+
+ f3 (s), (16.35)


C<J
(i 3m - 4 2 S . f 2[2 m - 2 1 J
x) mt = no y (s) (S2 + [2)2 + 2 m [2 + S2 ds + f4 (i).
OueHKa perYJiSlpHOCTH CHCTeMbI (16.35) .n;OJi}KHa npOBOlIHTbCSI , a oc-
HOBe CJie.n;YIOIltHX paBeHcTB:


Sf 2[ m-2 1 ] .
([2 + 2 S2 )2 + 2 m [2 + S2 No (i) No (t) idt = /1 ('8),
o


C<J
Sf 2t +2 m-2 1 ]d = 3m-4

 ([2 + 12 S2 )2 m [2 + 62 S mt .
o
I1s 3THX paBeHCTB, O'leBH.n;HO, CJie.n;YeT CYIlteCTBOBaHHe HeTpHBHBlIlbHOI'C>
peIIleHHSI O.n;HOpO.n;HOH CHCTeMbI, OTBe'laIOIlteH (16.35). 3TO peIIleHHe HMe-
eT BH.n;


I
I!
I


y (s) = X (i) = const. (16.36)
3.n;ecb CJie.n;yeT OTMeTHTb, 'ITO C no.n;o6HbIM 06CTOSlTeJibCTBOM BCTpe-
'IaeMCSI TaK>Ke npH paccMoTpeHHH B paMKax HCnOJib3yeMoro MeTo.n;a
sa.n;alIH .n;JiSl 'IeTBepTbnJiOCKocTH. O.n;HaKo KaK H B paccMaTpHBaeMoM
cJiY1Iae, TaK H B cJiY'lae sa.n;a'lH .n;JiSl 'IeTBepTbnJiOCKOCTH JierKO nOKasaJ'b,
'ITO peIIleHHe THna (16.36) npOTHBOpe'lHT npe.n;nOJiaraeMoMY xapaK-
Tepy noae.n;eHHSI Ha 6eCKOHe'IHOCTH c\JYHKUHH, 3B.n;BIOIltHX rpaHH'lHbIe
YCJiOBHSI.
PaccMOTpHM, HanpHMep, Ty qaCTb BbIpa>KeHfm (16.32) .n;JiSl CMeIlte-
HHSI W, KOTopaSl npe.n;cTaBJiSleTCSI IIHTerpaJiOM Be6epa. YqHTbIBaSl CBSl31i:
.Me)K)l,Y <PYUKltHSlMH C (J
, A (i) II .cc (t), npe.n;CTaBHM BbIpa
eHlfe npII
.t66


I !


II




II


F.Ilaea 4
OCECIIMMETPllqHOE HAIIPHiKEHHOE COCTOHHIIE
YCEqEHHOrO KOHYCA


UMeeT xapaKTep nOBe.n:eHIUI KaCaTeJIbHbIX Hanp5DKeHuH B oKpecTHoc-
TU yrJIOBHX TOqeK.
I1cnOJIb30BaHue npu nocTpoeHuu 06lII,ero perneHIUI c06cTBeHHblx
<PYHKIJ.UH 3a.n:aqU lllTypMa - JIUYBUJIJIH Ha HenOJIHOM uHTepBaJIe H3-
MeHeHHH MepH.n:HoHaJIbHOH Koop.n:HHaTbI YCJImKHHeT 3a.n:aqy HaXOJK.n:e-
IIHH c06cTBellllblx qHCeJI. O.n:llaKo B03I1HKalOlII,He npH 3TOM 3aTpy.n:He-
HHH .n:OCTaTOqHO JIerKO npeo.n:OJIeBaIOTcH, nOCKOJIbKY COOTBeTCTBYIOlII,He
TpaHCIJ.eH.n:eHTllble ypaBHellHH <POPMYJIHPYIOTCH B xoporno H3yqeHHbIX
<PYHKIJ.HHX (<pYHKIJ.HH JIeJKall.n:pa) [561.
rpallHqHHe 3a.n:aqH .n:JIH KpyroBoro KOHyca OKa3HBalOTCH ropa3.n:0
CJIOJKllee, qeM 3a.n:aqH .n:JIH C<pePbI. DepBoH ny6JIHKaIJ.HeH, OTIIOCHlII,eHcH
K cJIyqalO ocecHMMeTpHqlloro .n:e<PoPMHpoBaHHH KOllyca, HBJIHeTCH pa-
60Ta B. A. CTeKJIOBa [1661. 011 nOCTpOHJI pernell He ypaBHellHH JIaMe,
n03BOJIHIOlII,ee HCnOJIb30BaTb npH y.n:OBJIeTBOpeIlHH rpallHqHbIX YCJIO-
BHH Ha KOIIHqeCKHX nOBepXHOCTHX CBOHCTBa pH.n:OB <1>ypbe no pa.n:HaJIb-
1I0H Koop.n:HHaTe.
B nOCJIe.n:YIOru.eM ony6JIHKOBall0 pH.n: pa60T [164, 189, 238, 243],
B KOTOpbIX 06cYJK.ZJ:aIOTCH qaCTllble CJIyqaH HarpYJKellHH KOllyca. 06-
lII,HH CJIyqaH ocecHMMeTpHqHOH .n:e<}JopMaIJ.HH paccMoTpeH B pa60Tax
[242, 2651. OpHrHllaJIbHbIH MeTO.n: HCCJIe.n:OBaHHH ocecHMMeTpHqlloro
lIanpHJKeHHOrO COCTOHHHH KOHyca, oCHoBaHHbIH Ha YCTaHOBJIeHHbIX
A. 51. AJIeKcaH.n:poBbIM <PoPMYJIaX CBH3H MeJK.n:y 06lII,HMH perneHHHMH
ocecHMMeTpHqHbIX H nJIOCKHX 3a.n:aq, .n:aH B ero pa60Te [171.
DOJIHoe perneHHe 06ru.eH HeocecHMMeTpHqHOH 3a.n:aqH 0 paBHoBe-
CHH KpyrOBoro KOHyca .n:aHO B pa60Tax A. <1>. YJIHTKO [178, 179] Ha
OCHOBe MeTo.n:a c06cTBeHHblx BeKTopHbIX <PYHKIJ.HH. Pe3YJIbTaTbI 3TH X
pa60T Henocpe.n:CTBeHHO HCnOJIb3YlOTCH npH nocTpoeHHH 06lII,ero pe-
rneHHH rpaHHqHOH 3a.n:aqH .n:JIH yceqeHHoro KOHyca. DOJIyqeHHble B HHX
BblpaJKeHHH .n:JIH KOMnOHeHTOB BeKTopa CMeru.eHHH B BH.n:e KOMnJIeKCHbIX
HHTerpaJIOB PHMaHa - MeJIJIHHa npe06pa3YlOTCH K Beru.eCTBeHHbIM HH-
TerpaJIaM <1>YPbe.
4aCTHMH Koop.n:HHaTHbIX nOBepXHOCTeH c<PePHqeCKOH CHCTeMbI MOJK-
HO 06pa30BaTb .n:OCTaTOqHO 60JIbrnoe qHCJIO pa3JIHqHbIX 06JIaCTeH.
YBeJIHqeHHe qHCJIa KOHHqeCKHX H C<PePHqeCKHX yqaCTKOB rpaHHIJ.bI
COOTBeTCTBYIOlII,ero ynpyroro TeJIa, eCTeCTBeHHO, npHBo.n:HT K 60JIee rpo-
M03.n:KHM BblpaJKeHHHM .n:JIH 06lII,ero perneHHH rpaHHqHOH 3a.n:aqH. Or-
paHHqHMCH npOCTeHrnHM B 3TOM CMbICJIe ynpyrHM TeJIOM, paCCMaTpHBaH
oceCHMMeTpHqHYIO 3a.n:aqy .n:JIH CnJIOrnHoro yceqeHHOrO KOHyca. DpH
perneHHH 3TOH 3a.n:aqH BCKpbIBalOTCH OCHOBHble oco6eHHOCTH npHMeHe-
HHH pa3BHBaeMoro B pa60Te MeTo.n:a .n:JIH HCCJIe.n:OBaHHH HanpHJKeHHOCTII
ynpyrHx TeJI, OrpaHHqeHHbIX qaCTHMH C<pepHqeCKHX H KOHHqeCKHX
nOBepXHOCTeH.
DepBaH OCHOBHaH rpaHHqHaH 3a.n:aqa .n:JIH yceqeHHOrO KOHyca C HC-
nOJIb30BaHHeM MeTo.n:a o.n:HOpO.n:HbIX perneHHH HBJIHJIaCb npe.n:MeTOM HC-
CJIe.n:OBaHHH pH.n:a aBTOpoB [128, 137, 267 H .n:p.]. B pa60Tax [128,
137] OCHOBHoe BHHMaHHe y.n:eJIeHO H3yqCHHIO CBOHCTB KopHeH xapaKTe-
pHCTHqeCKoro ypaBHeHHH. D.aHHble, OTHOCHlII,HCCH K BblqHCJIeHHIO KOp-
HeH 3Toro ypaBHeHHH .n:JIH KOHYCOB pa3JIHqHOH reOMeTPHH, a TaKJKe K
169


, I
!
II


B npe.ZJ:bI.n:YlII,HX rJIaBaX 6bIJIH paccMoTpeHbI nJIOCKHe H OCeCHMMeT-
pHqHble rpaHHqHble 3a.n:aqH TeopHH ynpyrocTH .n:JIH TeJI KOHeqHbIX pa3-
MepOB
 rpaHHIJ.a K
)TOPbIX COCTOHJIa H3 qaCTeH Koop.n:HHaTHblx nOBepx-
HOCTeH .n:eKapToBolI H IJ.HJIHH.n:pHqeCKOH CHCTeM Koop.n:HHaT. B TOH qac-
TH, KOTOpaH OTHOCHJIaCb K nocTpoeHHIO 06rnero perneHHH rpaHHqHOH
3a.n:aqH .n:JIH KOHKpeTHoro ynpyroro TeJIa, npe.n:H.n:YlII,He BbIKJIa.n:KH
nOKa3bIBaJIH, qTO He06xo.n:HMaH nOJIHOTa perneHHH JIerKO 06eCneqHBa-
JIaCb npH HaJIHqHH 06lII,HX npe.n:cTaBJIeHHH BeKTopa cMeru.eHHH .n:JIH 06-
JIaCTe
 C rpaHHIJ.eH, cOBna.n:alOlII,eH C O.n:HOH H3 Koop.n:HHaTHblx nOBepx-
HOCTeH.
B - :HOH rJIaBe npe.n:MeTOM HCCJIe.n:OBaHHH 6y.n:YT oCeCHMMeTpHqHble
rpaHHqHble 3a.n:aqH .n:JIH yceqeHHoro KOHyca. B CBH3H C 3THM nocTpoe-
IIHe 09lII,ero perneHHH TaKHX 3a.n:aq 6y.n:eT OCHOBbIBaTbCH Ha HCnOJIb3D-
BaHHII!__perneHHH .n:JIH ocecHMMeTpHqHO .n:e<popMHpoBaHHblx C<pepH H
6eCKOHeqHOrO KOHyca.
Pac
MoTpeHHlo nepBoH H BTOpOH OCHOBHbIX rpaHHqHbIX 3a.n:aq .n:JIH
ynpyro
 c<}JepbI nOC
HlII,eHa 06rnHpHeHrnaH JIHTepaTypa, XOTH YJKe
B nepBoH, OTHOCHlII,eHcH K 3TOMY Bonpocy ny6JIHKaIJ.HH r. JIaMe [233J,
nOJIyqeHO, no Cyru.ecTBY, HCqepllbIBalOlII,ee HX perneHHe. B IJ.eJIOM pH.n:e
nocJIe.n:YIOlII,Hx pa60T pe3YJIbTaThI r. JIaMe OTqaCTH nOBTOpHJIHCb [201,
214,274] HJIH JKe npe.n:cTaBJIHJIHCb B .n:pyroH <popMe [61, 202, 217 H .n:p.L
B YKa3aHHHx ny6JIHKau.HHx 06cYJK.n:alOTcH BonpocH, CBH3aHHHe C
H3yqeHHe
 HanpHJKeHHOrO COCTOHHHH 3aMKHYThIX B nOJIlOcax CnJIOrnHOH
HJIH nOJIOH c<pep. B CBH3H C 3THM npH nocTpoeHHH nOJIHOH CHCTeMbI BeK-
TOpHbIX <PYHKIJ.HH Ha c<PePHqeCKHX nOBepxHocTHX HCnOJIb30BaHbI
c06cTBeHHble <PYHKIJ.HH 3a.n:aqH lllTYPMa - JJHYBHJIJIH Ha nOJIHOM HH-
TepBaJIe H3MeHeHHH MepH.n:HoHaJIbHOH Koop.n:HHaTbI. DOJIyqeHHble TaKHM
06pa30M perneHHH MorYT 6bITb HCnOJIb30BaHbI H npH paccMoTpeHHH rpa-
HHqHbIX 3a.n:aq .n:JIH ynpyrHx TeJI, rpaHHqHaH nOBepxHocTb KOTOpbIX 06-
pa30BaHa qaCTHMH c<PePHqeCKHX H KOHHqeCKHx nOBepxHocTeH. O.n:Ha-
KO aHaJIH3 perneHHH paHee paccMoTpeHHblx 3a.n:aq nOKa3bIBaeT, qTO
y.n:oBJIeTBOpHTb rpaHHqHbIM YCJIOBHHM Ha BceH nOBepxHocTH TaKoro yn-
pyroro TeJIa JIerqe, eCJIU HCnOJIb30BaTb c06cTBeHHble <PYHKIJ.HH TaKHX
3a.z:aq, KOTopble Haxo.n:HTCH B onpe.n:eJIeHHOH CBH3H co CBoHcTBaMH <}JYHK-
IJ.HH, 3a.n:alOlII,HX BHernHHe HarpY3KH.
B qaCTHOCTH, npH BbI60pe <poPMH perneHHH cYlII,eCTBeHHoe 3HaqeHHe
168


: I




pernemuo HeKOTOpbIX KOHKpeTHbIX rpaHIfqHbIX sa.n;aq, cO.n;ep}KaTCSI B
pa60Te (270). 3.n;ecb OTMeqeHbI 3HaqlfTeJIbHbIe BbIqIfCJIIfTeJIbHbIe TPY.n;-
HOCTH, ,?03HIfKalOillIfe npH lfJaKTHqeCKOM y.n;OBJIeTBOpeHHH rp8HHqHbIX
YCJIOBHH.
Jl.nSl qaCTHOro cJIyqaSl reOMeTpIfIl yceqeHHoro KOHyca - nOJIynpocT-
paHCTBa C nOJIyclfJepHqeCKOH BbIeMKOH - 3a.n;aIJa OTbICKaHIIH KopHeH xa-
paKTepIfCTlIqecKoro ypaBHeHHH MeTo.n;a O.n;HOpO.n;HbIX perneHHH 3HaqH-
TeJIbHO ynpo
aeTcSI. 3TO cnoc06cTByeT TOMy, qTO nOJIyqeHHbIe.n;JIH :;)TOH
3a.n;aqH KOJIHqeCTBeHHbIe oueHKIf HanpH}KeHHoro COCTOHHIISI 60JIee nOJI-
HbI. OcecIfMMeTpIfqHbIH CJIyqaH .n;e!fJopMlIpoBaHHH paccMoTpeH B pa60Te
(208). Pe3YJIbTaTbI 3TOH pa60TbI 6y.n;YT 06cY}K.n;eHbI HH}Ke. HanpH}KeH-
Hoe COCTOHHIfe nOJIynpoCTpaHCTBa c nOJIyclfJepIfqeCKOH nOJIOCTblO npH
O.n;HOOCHOM paCTH}KeHIIII II3yqaeTCH B pa60Te [256]


PaCCMOTpIfM nepBYlO rpaHIIGHYlO 3a.n;aqy .n;JIH yceqeHHorO KOHyca
(pIlC. 19). rpaHIfu,? 06JIaCTII, 3aHHTOH ynpYrHM TeJIOM, o6pa3YeTCH qac-
THMH clfJepHqeCKOH (p = Po, 0 <.: it <.: it o ) II KOHlIqeCKOH (it = it o ' P>
o > Po) nOBepxHocTeH. I1oCJIe.n;YlO
ee HSJIO}KeHHe Be-
.n;eTCH B 6e3pasMepHbIX Koop.n;HHaTax r = p/Po.
B COOTBeTCTBHH C c
eCTBOM npHHHToro B pa6o-
Te MeTo.n;a, 06
ee perneHHe rpaHlIqHOH 3a.n;aqll .n;JIH
yc:qeHHOrO KOHyca B KOMnOHeHTax BeKTopa CMe
e-
HHH 3anIfCbIBaeTCSI CJIe.n;YlO
HM ,o6pasoM:
U r = u
C) + U
K), (17.1)
Uf} = uW) + u
).
I1epBbIe CJIaraeMbIe B 3THX paBeHCTBax .n;QJI}KHbI
y.n;OBJIeTBOpHTb ypaBHeHIIHM paBHOBeCIfH JIaMe H, KpO-
Me Toro, .n;OCTaBJIHTb nOJIHbIe CHCTeMbI IfJYHKUlfH .n;JISI
y.n;OBJIeTBOpeHIfH rpaHlIqHbIX YCJIOBIfH B CMe
eHIIHX
z HJIII HanpH}KeHHSlX Ha c<PePllqeCKOH qaCTH rpaHHUbi
PRC. 19. (r = 1, 0 <.: it <':u it o )' PerneHIfH ypaBHeHIIH JIaMe,
OTMeqeHHbIe 6YKBOH «K» B (17.1), Bbr6HpalOTCSI IfCXO-
.'(H liS Tp
oBaHHH nOJIHOTbI BbIpa}KeHHH .l{JISI KOMnOHeHTOB BeKTopa
CMe
eHIIH Ha nOBepxHOCTH KOHyca npIf r > 1.
ClfJepllqeCKaH COCTaBJIHlO
aH 06
ero perneHl:IH (17.1) MO}KeT 6bITb
JlerKO nOCTpoeHa Ha OCHOBe If3BeCTHoro perneHIIH .n;JIH BHeWHOCTH .oce-
CHMMeTpIfqHO .n;elfJopMlIpoBaHHOH clfJepllqeCKOH nOJIOCTH [124). LlJISI 3Toro
. .l{OCTaTOqHO lfJopMaJIbHO saMeHHTb B YKasaHHOM perneHHH ueJIO-
'IIfCJIeHHbIe CTeneHII IfJYHKUIfH JIe}KaH.n;pa Ha HeKOTOpylO nOCJIe.n;OBaTeJIb-
HOCTb nOJIO}KIfTeJIbHbIX qllceJI Vk' I1pll 3TOM BbIpa}KeHIfH .n;JIH KOMnOHeH-
TOB BeKTopa CMe
eHHH, y.n;OBJIeTBOpHlO
He ypaBHeHIISlM paBHOBeCHSI


JIaMe, npIf06peTalOT BII.n;
u(C)=_.-5L+
 [
 (
+ 3m-4 ) _
r r 2
 'Yk 'Yk m'Yk
k=I r ,


dk
r'Yk+ 2



x
'Yk



 17. TIocTpoeHHe o6m;ero perneHHJI
rpaHH1JRbIX sa,J:{a1J ,J:{JIJI yce1JeHHOro ROHyca


X(I+
k )]P'Yk(COSit), (17.2)
- 1 dP (cos it)
u(c) _
 [
 ...!- ( 4 m - 1 _ I) + dk
k_
{} -
 'Yk 'Yk mVk . 2 'Yk+ 2 dit .
k=I r I 'Yk'
BbIpa}KeHIIH .n;JIH U
C) II U
) B (17.2) C npOll3BOJIbHbIMlI nOCTOHHHbIMIf
do, Ck' d k y.n;OBJIeTBOpHlOT JIIfrnb nepBOMY 1I3 .n;BYX OCHOBHbIX Tpe6oBa-
HIIH, npe.n;'bHBJIHeMbIX K u
C) II u
) B (17.1), a HMeHHO y.n;oBJIeTBOpHlOT
ypaBHeHIfHM paBHOBeCIIH naMe. BTopoe Tpe60BaHlIe, .n;ocTaBJIHTb nOJI-
HYlO clIcTeMY IfJYHKUIIH .n;JIH y.n;OBJIeTBOpeHIfH rpaHlIqHbIX YCJIOBIfH B
cMe
eHIfHx IfJIIf HanpH}KeHIIHX Ha clfJepllqeCKOH qaCTIf rpaHl:IUbI yce-
qeHHorO KOHyca, MO}KeT 6bITb BbInOJIHeHO npH Ha.n;JIe}KaIlleM BbI60pe
qllceJI 'V k .
B KaqeCTBe SHaqeHIIH v k MO}KHO BbI6paTb nOCJIe.n;oBaTeJIbHOCTb nOJIO-
}KIITeJIbHbIX KopHeH o.n;Horo 1I3 ypaBHeHIIH
dP", (cos it) I . _
dit f}=t}.-O, P'Y(cosit o ) =0. (17.3)
I1pll 3TOM BbIpa}KeHIfH (17.2) npH06peTalOT He06xo.n;IfMbIe CBoHcTBa nOJI-
HOTbI Ha nOBepxHocTII r = I, 0 <.: it <: it o' CJIe.n;yeT TaK}Ke OTMeTl:ITb,
'ITO npll BbI60pe B KaqeCTBe V k KopHeH BToporo ypaBHeHIIH B(17.3) npo-
H3BOJIbHYlO nocToHHHYlO do B (17.2) CJIe.l{yeT nOJIO}KHTb paBHoH HYJIlO.
Bonpoc 06 0.n;H03HaqHOM BbI60pe 3HaqeHlIH Vk pernaeTcH Ha OCHOBe
aHaJIIf3a CBOHCTB IfJYHKUIfH B rpaHlIqHbIX YCJIOBIfHX. MCnOJIb30BaHlIe
BToporo ypaBHeHIIH B (17.3) UeJIeC006pa3Ho npH BbI.n;eJIeHIIII OC06eHHOC-
TeH H8npH}KeHHOrO COCTOHHHH B CJIyqae HapyrneHIIH YCJIOBHH napHoc-
TII KaCaTeJIbHbIX HanpH}KeHIfH B yr JIOBbIX TOqKax. HH}Ke B KaqeCTBe V k
BbI6IfpaeM KOpHII nepBoro ypaBHeHIIH B (17.3), opIfeHTlIpYH TeM caMbIM
o6
ee perneHIfe rpaHlIqHOH 3a.n;aqH (17.1) Ha rpaHlIqHbIe YCJIOBl:ISI, B
KOTOpbIX YCJIOBHe napHocTII KaCaTeJIbHbIX HanpH}KeHHH B yr JIOBOH '1'0'1-
Ke BbInOJIHHeTCH.
B nOCJIe.n;YlO
Hx BbIKJIa.n;Kax y.n;06HO l:ICnOJIbSOBaTb BMeCTO r HO-
BYlO pa.n;lIaJIbHYlO Koop.n;IfHaTY t, BBO.n;HMYlO no 1fJ0pMYJIe
t=.lnrJr=ei). (17.4)
BbIpa}KeHIfH .n;JISI cMe
eHIfH (17.2) tIIepenllrneM B HOBbIX Koop.n;IIHaTax.
a HaH.n;eHHbIe no HIIM Ha OCHOBe 3aKOHa rYKa HanpH}KeHIIH npe.n;CTa-
BHM CJIe.n;YlOIUIIM 06paSOM:
8/.t
 [ - ( 'Yk- .L ) 1
.!!.-- dC) = 2d o e-'I.t +
 akcke 2 +
2G r k=\


'170


( 3 )
2 1 - 'Yk+ -- t
+ (1 + v;) (I + Vk )tlke 2] P'Yk (costt),
'd.71




e

t oW = _ d o e- 3M _
 {[ 1'k C k e - ('I'k- {-) t +
k=1
+ (1 +
k r dke- ('I'k+
 ) t] P'I'k (cost}) + [6 k c k e - ('I'k-i) t +
-('I'k+ n t ] ctgit dPv (CDS it) }
+ dke '\I
 k dit , (17.5)
/I.t
 [ - (vc ...!... ) t
2G T
<ij = f::.1
kcke . 2 -
_ ( 1 +
 ) dke- ('I'k+ n t ]
 dP vk (CDS it)
'\Ik ) '\Ik dit '
i l . t
2(1 (a
C) + a
) + a<
» =


+ (.2 + +) ] + Tb
l) ('t) - + bf) ('t») -
1: (f.t) 1 a
O (T) r 2 : =
 X
X ({-- T2) + (T 2 + +)J _a
2) (T)4 :=; T+
+ + bg) (T) + 'tb
2) (T) )] sin Tt} dT, (17.7)


co
u
)ell.t = - 2 S ([ ex
 (f.t) (a
l) (T) 2
;

) - a
2) (T) m -: 2 + b
l) (T») +
o
+ I3df.t) (a
l) (T) m -: 2 + a
) (T) 2
;

) - b
2) (T))] cos 'tt +
+ [ ex' (f.t) ( a(O (T)
- + a(2) (TJ 8 -7m + b
2) (T» ) _
1: 0 m-2 0 2
-

-

 (f.t) ( a
1) (T) 2
;

) a
2) (T) m -: 2 - b
O (T) ) ] sin Tt} dT.
B 3TlIX BbIpa:>KeHlIHX co.n.ep:>KaTCH 4eTblpe np01l3BOJIbHble iPYHKQlIlI na-
paMeTpa lIHTerplipOBaHliH T: ab O (T), af) ('t), bg) (T) II b
) (T). l1.JIH iPYHK-
1.{1IH yrJIOBOH KOOp.n.lIHaTbl t} npHHHTbI CJIe.n.YIOIIJ.lIe 0603Ha4eHlIH:
sin it dP -1/.+i1: (ft)
ex1: (f.t) = cos t}P -1/.+i1: (f.t) + - -
2(17 2 ++) dit


3.n.eCb


co ( I )
= _ 2 m + I '"" I ( I ) - vk- - t
m .tt - '\Ik 2 - V; cke 2 P'I'k (cos t}).
npllHHTbI CJIe.n.YIOIIJ.lIe 0603Ha4eHlIH:
ex k = -1-
 + 2 -;-,
k = 1- 2. +
....!..-
'\Ik m '\I k
 m
 '
k k


17 sin it
I
17 2 +_
4
, sin it dP _1/.+h (ft)
ex1: (f.t) = -
 P -'/.+h (f.t) + cos t} dit

: (f.t) = T sin t}P _1/.+h (f.t) (f.t = cos t}),


dP -'/.+h eft)
dit


(17.6)
'V =-1 I
 +
_

 51 4 4 1
k --r- 'I' 2 2' Uk = - I + - - --.
k '\Ik m '\I" '\Ik m '\Ik
06IIJ.ee perneHlIe OCHOBHblX rpaH1I4HblX 3a.n.a4 .n.JIH KOHyca nOJIY4e-
HO B pa60Tax fI78, 179J B iPopMe KOMnJIeKCHbIX lIHTerpaJIOB PlIMaHa-
MeJIJIlIHa. DOJIaraH B HliX S =
 + iT, T. e. Bbl611paH nYTb lIHTerpll-
pOBaHliH B.n.OJIb npHMOH, napaJIJIeJIbHOH MHlIMOH OCli II OTCTOHIIJ.eH OT
I
Hee Ha paCCToHHlIli c = 2' npe.n.CTaBliM KOMnOHeHTbl BeKTopa CMeIIJ.e-
HliH .n.JIH paccMaTpliBaeMoro 3.n.eCb ocecliMMeTp1l4Horo CJIY4aH B BlI.n.e
BemeCTBeHHblX lIHTerpaJIOB cI>YPbe:


!31:(f.t) =


(17.8)


r.n.e P -'/.+i1: (cos t}) - npllcoe.n.lIHeHHaH KOH1I4eCKaH iPYHKl.{lIH MeJIe-
pa [641.
Ha KOH1I4ecKoH nOBepxHocTli t} = t}o Bblpa:>KeHlIH .n.JIH CMeIIJ.eHlIH
(I 7.7) II COOTBeTCTBYlOmliX lIM HanpH:>KeHlIH 06JIa.n.a1OT 1I36blT04HblM
np01l3BOJIOM, nOCKOJIbKy co.n.ep:>KaT lIHTerpaJIbl cI>YPbe C np01l3BOJIbHOH
nJIOTHOCTblO KaK no cos T t, TaK II no sinT t. cI>opMaJIbHO 3Ta 1I36bITOQ-
HOCTb yCTpaHHeTcH TeM, 4TO, paCnOpH:>KaHCb .n.BYMH 1I3 4eTblpex, co.n.ep-
:>KaIIJ.lIXCH B (17.7), np01l3BOJIbHblMlI iPYHKQlIHMlI 't, .n.0611BaIOTCH 06pa-
IIJ.eHlIH B HYJIb Ha nOBepxHocTlI nJIOTHOCTli lIHTerpaJIa cI>YPbe no sin T t
RJIli no cos T t. B rpaH1I4HblX 3a.n.a4ax Teoplill ynpyroCTli .lI.JIH yce4eHHoro
KOHyca lIMelOIIJ.HHCH np01l3BOJI CJIe.n.yeT lICnOJIb30BaTb .n.JIH nocTpoeHlIH
perneHlIH, .n.alOIIJ.ero B03MO:>KHOCTb Ha1l60JIee npocTo onlicaTb oc06eH-
HOCTlI HanpH:>KeHHOrO COCTOHHlIH B oKpecTHocTli yrJIOBblX T04eK. Dpll
nocTpoeHlIlI ciPepllQeCKOH COCTaBJIHIOIIJ.eH 06IIJ.ero perneHlIH ee iPopMa
BbI611paJIaCb npllMeHlITeJIbHO K 3a.n.a4aM C BbmOJIHeHHbIM YCJIOBlieM nap-


u
K)e'J.f = 2 J {[ ex1: (f.t) ( ag) (T) [2 : =
 ({- - T2) + (T 2 + + ) ] _
- a
2) (T) 4 ;;:
 T - + bg) (T) - Tb
2) (T» ) . +
1: (f.t) ( a(lJ (T) 4 m - I T +
o m-2
+ a
2)(T) [2 :
 ({-- T2) + (T 2 + +)J -Tbh')(T) + + b
2) (T») 1 X
X COSTt + [ ex1: (f.t) (a&l) (T) 4 m -1 T + a(2) (T) [ 2 m -1 (
 _ T2 ) +
\ m-2 0 m-2 4
172


173


I
I




HO
TH KaCaTeJlbHbIX Hanp5DKeHHH B yrJlOBbIX TOqKax. OpHeHTHpY5ICb
Ha TaKHe 3a
aqH, cJle
yeT OCTaBHTb OTJlHqHbIMH OT HYJl51 Ha rpaHHLle
HHTerpaJlbI «I>ypbe no cos T:t
Jl5I Hanp5I
eHHM aft H no sin T:t
Jl5I Ka-
CaTeJlbHbIX Hanp5I
eHHM 1: r ft.
Cnoc06 HCnOJlb30BaHH5I 06II.{ero perneHli5I (17.1), c<pepHqeCKa5I H KO-
HHqeCKa5I COCTaBJI5IIOII.{He KOTOporo OlIpe
eJlHIOTC5I <popMYJlaMH (17.2)
H (I 7.7), lIPH perneHHH rpaHHqHbIX 3a
aq M5I yceqeHHOrO KOHyca TaKoM

e, KaK H BO Bcex paccMoTpeHHblx paHee 3a
aqax. O
HaKo B 06II.{eM cJlY-
qae rpaHHqHbIX YCJlOBHM He06xo
HMble BbrKJIa
KH qpe3Bb,qaMHO rpo-
M03
KH. B CB5I3H C 3THM paccMoTpHM B HeKoTopoM CMbICJle qaCTHb'M cJlY-
'laM Harpy
eHH5I yceqeHHOro KOHyca, lIPH KOTOpOM KaCaTeJlbHble
Hanpj!f
eHH5I Ha KOHHqeCKOM lIoBepxHocTH OTCYTCTByroT. fIOCKOJlbKY Ka-
CaTeJlbHble HalIp5I
eHH5I OT c<pepHqeCKOM COCTaBJl5IIOII.{eM 06II.{ero perne-
HH5I lIPH lIpHH5ITbIX 3HaqeHH5IX v k 06paII.{aIOTC5I Ha lIoBepxHocTH tt =
=tt o B HYJlb, YKa3aHHoe OrpaHHqeHHe lI03BOJl5leT CB5I3aTb
Be OCTaBrnHe-
C5I B (17.7) lIpOH3BOJlbHble <PYHKLlHH O
HOpO
Hb'M aJlre6paHqeCKHM co-
OTHorneHHeM. I1ClIOJlb3Y5I :lTO COOTHorneHHe H 0603HaQa5I 'lepe3 p (-r)
lIpOH3BOJlbHYlO lIJlOTHOCTb HHTerpaJla «I>YPbe lIO cos T:t
Jl5I HalIp5I
eHHM
aft Ha KOHH'leCKOH lIOBep, XHOCTH
Jl5I OTBeqalOII.{HX CMeII.{eHH5IM (17.7)
KOMlI0HeHTOB TeH30pa HalIp'5DKeHHM IIOJIYQaeM cJle.n:YIQII{Me Bblpa
eHH5I:
3{ t 00
e2
 a
K) = 2 S P (1:) {[KI: (f.t) ll; -I L-,; (f.t) ll;] cos T:t +
o


Y
06HOH lIPH lIOCJle
YIOII.{eM pemeHHII rpaHHqHOH 3a
aqH:
K-,; (f.t) = [6
1: (f.t) + 1]-,;(3-,; (f.t)] a;2 (f.to) - [1]i
1: (f.t) -
- 6-,;(31: (f.t)] d
 (f.to) - (1: 2 + -{-) [a1: (f.t) a;, (f.to) - (3-,; (f.t) a;l (f.to)],
L-,; (f.t) = [61: a -,; (f.t) + 1]-,;(31: (f.t)] d
 (f.to) + [1]
-,; (f.t) - 6-,;(31: (f.t)] a
 (f.to)-
- (1'2 + -{-) [a1: (f.t) a;, (f.to) + (3-,; (f.t) a;l (f.to)J,
p 1: (f.t) = a;l (f.t) a;2 (!-to) - a;, (f.t) a 22 (f.to) - a;2 (f.t) a;l (f.to) - d;2 (f.t) d;l (f.to).
Q1: (f.t) = a;, (f.t) a;2 (f.to) + a;, (f.t) d;2 (f.to) - d;2 (f.t) a;l (f.to) - a;2 (f.t) a;, (f.to),
R1: (J.-l) = a;l (f.t) a
 (!-to) - d
l (f.t) a;2 (/10) - a;2 (f.t) a;l (J.-lo) + a;2 (J.-l) a;l (f.to).
8 I: (f.t) = d;l (f.t) a;2 (f.to) + a;1 (f.t) a;2 (f.to) - a;2 (f.t) a;1 (f.to) - a;2 (f.t) a;l (f.to).
M-,; (f.t) = [2T: 2 a1: (f.t) -1:/31: (f.t)) a
2 (f.to) + [T:a1: (f.t) + 2T: 2 (31: (f.t)J a;2 (f.to).
N-,; (f.t) = f2T: 2 a1: (f.t) - 1'{3, (f.t)ld;2 (fl;o) - [1:a1: (f.t) +
h2(31: (f.t)J a;2 (f.to) ,
(17.10)
a;1 (f.t) = )"J,u.-,; (f.t) + Pt(31: (f.t) + ctg tt [f.tTa
 (f.t) + x1:(3
 (f.t)].
a;l (f.t) = P1: a 1: (11) - 1.1:(31: (f.t) + ctg tt [x1:a
 (f.t) - f.t-,;(3
 (f.t)).
a;2 (f.t) = (-{- - 1: 2 ) at (f.t) + 1:/3-,; (f.t) + ctg tta
 (f.t).
d;2 (f.t) = (-{- - T: 2 ) (31: (ft) - 1'a-,; (f.t) + ctg tt(3
 (f.t).
. . I .
a;1 (f.t) = v1:a
 (fA;) + %1:(31: (f.t), a22 (f.t) = Tat (f.t) - 1"(31: (f.t).
" . I'
a;1 (f.t) = %-ra
 (f.t) - V1:(3
 (f.t).
2 (f.t) = 1'a-r (f.t) + 2 (31: (f.t).
ll
 = a;, (!-to) a;2 (f.to) - a;1 (f.to) a;2 (f.to) - a;2 (f.to) a;1 (f.to) -
- a;2 (/-to) a;. (f.to),
ll; = d;l (f.to) a;2 (f.to) + all (f.to) a
 (f.to) - a;2 (f.to) a;1 (f.to)-
- a;2 (f.to) a;l (f.to),
-,; 8 - m (7 - 36.2) -,; _
. 8 - m (15 - 4.2)
6 = 8 (m - 2) · 1] - 4 m - 2 ·
8 - m (7 + 4.2) %1: = l' m
v-,; = 4 (m - 2) m - 2
8 - m (7 - 4.2) . 8 - m (9 + 4.2)
"''I: = 8 (m _ 2) P't = 4 . m - 2
8-7m
f.t-'; = 2 (m - 2) .
175


+ [L-,; (f.t) ll
 - 1(-,; (f.t) ll:J sin T:t} d.
[Lli)2 + [Ll;)2
e

t at;) = 25 peT:) {[P1:(f.t)ll
 + Q-,; (f.t)ll;J COST:t +
o
. " . d.
+ [Q-,; (f.t) ll-,; - P1: (f.t) 111:J SIn T:t} . .. .
[Ll1:)2 + [A.:)2
e'f.t c;; . "
2ll T:

 = 2 j P (1') ([R1: (f.t) ll-,; + S-,; (f.t) ll-,;J cos T:l +
o


(17.9)


+ [8-,; (f.t) ll
 - R't (f.t) ll
J sin Tt} d.
[Ll
12 + [Ll;;)2
¥t 00
e2
 (a
K) + at;' + a
) = 4 :

 S P (1:) ([Mdf.t) ll
 +
o


+ N1: (f.t) ll;J cos T:t + [NT (f.t) /11: - M1: (f.t) ll
J sin Tt} d.
[Ll
)2 + [Ll;)2
PacrnH<pPOBKa BBe
eHHblx B (17.9) 0603HaQeHHH
aeT B03MO
HOCTb
COCTaBHTb lIpe
CTaBJleHHe 06 06"beMe Bb'KJJa
OK. KOTopwe He06xoJJ;HMO
npO
eJIaTb C U;eJIbIO nOJlyqeHH5I BblpalKeHHH
Jl5I HanpSlJKeHHH B <POp Me,
174


,
I


,I


I,


I
II
I


II
I


I
I


I II




I
I


Co.n:epLKa
IIeCH B (17.5) II (17.9) npOlI3BOJIbHble nOCTOHHHble C k ,
d k II HelI3BeCTHaH nJIOTHOCTb p ('r) .n:OJILKHbI 6bITb onpe.n:eJIeHbI B COOT-
BeTCTBlI1I C YCJIOBHHMH Ha rpaHlIqHbIX nOBepxHocTHX yceqeHHOrO KOHY-
ca:


1
2(;a r = f(tt),
1
2G aft = 'lj) (t),


1
2G Trft = 'P (tt)
1
2G Trft = 0


npli t = 0;


m 1 "
do = - ,p (1;) Do(T, !--to) dT + Tfo,
i5


l
II


C HCnOJIb30BaHlieM 3TIIX COOTHollIeHHH H3 YCJIOBHH .n:JIH a r II Trft Ha-
XO.n:IIM


npli tt = tt o .


(17.ll)


(XkCk+(I+
k )(I+
k )dl<=-2rp(T)Dk(T,!--to)dT+fk' (18.3)
" 0


<I>YHKQIIII f (l't), 'P (l't) H 'lj) (t) B .n:aJIbHeHllIeM npe.n:nOJIaraIOTCH Henpe-
pbIBHbIMIi II OrpaHHqeHHbIMII, nplIqeM e 3 f:t 'lj) (t) --+ 0 npH t --+ 00 TaK,
qTO BbIllOJIHHIOTCH He06xo.n:IIMble YCJIOBIIH .n:JIH npe.n:cTaBJIeHHH
noro
npolI3Be.n:eHIIH KOCIIHyc-IIHTerpaJIOM <I>YPbe.


(3k C k - (I +
k ) d k = - 2 r P (T) 0" (T, /-to) dT + 'Pk'
o


r.n:e


ft.
S P
k (cos tt) P

, (cos tt) sin l'tdtt =
o


(18.1)


2Vk + 1
Dk (T, !--to) = - sin tt P (p.) P(l) (/-I ) [(
')2 + (
")2] X
o v k 0 v k O"t "t
ft.
X , r K-r: (cos tt) 111: + L1: (cos l't) 111:] P Vk (cos l't) sin l'tdtt:
o
2Vk + 1
Ok (T, !--to) = - vk (Vk + t) sin ttoP v (p.o) P
) (p.o) [(

)2 + (
;)2] X
k k
ft
X f r R1: (COS tt)
1: + S1: (COS l't)

l P

 (cos l't) sin ttdtt; (18.4)
o ft
f =- " 2Vk+l. r.f(tt)Pv (costt)sinttdtt;
k Sin tt oP v (/-10) p(l) (p.o) J k
k Viz 0
ft.
- - ( 2V k + I) 5 m (it) p(I) ( cos tt ) sin ttdtt.
'Pk - " (I) 't' "k
Vk (Vk + 1) sin ttoP v (p.o) P v (/-10) 0
k k
TIpli BblqlICJIeHHH Do (T, !--to) H fo CJIc.n:yeT nOJIaraTb V o = 0 H yqeCTb,
"(1) t tto
qTO Po (!--to) = - g 2""'
CBoHcTBa HHTerpaJIa <I>ypbe n03BOJIHIOT npe06pa30BaTb ypaBHeHHe
.n:JIH aft B (17. II) K cJIe.n:YlOlileMY BH.n:y:
P (T) = doa o (T) +
I r ,\VkaVk-d T ) +
+ (I +
k )" dka vk (T) J P Vk (!--to) + 'lj) (T), (1
.5)



 18. Onpe
eJIeHHe npOH3BOJILHbIX nOCTOJIHHbIX
H .pYHRQHH B 06IQeM pemeHHH


BbI60p B KaqeCTBe v k KopHeH nepBoro ypaBHeHIIH B (17.3) II CTPYKTY-
pa BblpaLKeHIIH .n:JIH 'Lrft B (17.9) 06YCJIOBJIHBaIOT aBTOMaTlIqeCKOe BbI-
nOJIHeHlIe YCJIOBIIH .n:JIH KaCaTeJIbHbIX HanpHLKeHIIH Ha KOHHqeCKOH no-
BepxHocTIi. YCJIOBIIH .n:JIH HOpMaJIbHbIX II KacaTeJ1bHbIX HanpHLKeHIIH
Ha c<peplIqeCKOH qaCTli rpaHIIQbI B (17. II) nopoLK.n:aIOT TPH <PYHKQHO-
HaJIbHbIX ypaBHeHIIH .n:JIH onpe.n:eJIeHIIH c k , d k II P (T).
TIpe06pa30BaHlie nepBblx .n:BYx ypaBHeHIIH CHCTeMbI, OTBeqaIO
IIX
YCJIOBHHM (17. II) npH t = 0, ocy
eCTBJIHeTCH Ha OCHOBe CBOHCTB OPTO-
rOHaJIbHOCTli II nOJIHOTbI CIiCTeM <PYHKQIIH P V /? (cos tt) II P

 (cos it)
Ha HHTepBaJIe 0 <= tt <= l't o . CooTHollieHIIH, oTpaLKaIOllllIe CBOHCTBO op-
ToroHaJIbHOCTH YKa3aHHblx <PYHKQIIH, HMeIOT BII.n:
ft.
S P Vk (cos tt) P Vk ' (cos tt) sin ttdtt =
o
sin tt o P tt p(1) ( A
- 2Vk + I Vk (cos 0) V k cos U"o),



I


k=k';


0,


k =1== k';


I . A VkCVk+l) P ( t!: )p "(I/ ( A ) k =k" ,
= - SIn U"o 2Vk + 1 Vk COS 0 Vk COS U"o ,
O. k =1= k' ,
r.n:e TOqKOH 0603HaqeHa npolI3BoJJ:HaH <PYHKQIIH neLKaH.n:pa no HH.n:eKcy:
" (1) ( OP(l» )
P Vk (cos tt) =
. (18 2)
ov V=Vk


r.n:e


1
aVk(T) = n


3
Vk + 2""" 3
3 2 ,a o (T) = 2Jt
( Vk + 2") +..2
'lj) ('L) = + I e 3M 'lj) (t) cos'tt dt.
o


1
I 3 ) 2 ,
(2 + l2""" (18.6)


I


176


12 7-2464


177


I I




I1HTerpaJIbI, BXO,lJJIII\He B BblpmKeHHH .n:JIH D" (L, f-lO) H Ok (., f-lO)'
MO)KHO BbI4HCJIHTb B 3aMKHYTOM BH.n:e. llJIH 3TOro He06xo.n:HMO, npe)K.n:e
BCerO, HaHTH K03cpq:mIWeHTbI pa3JIO)KeHHH a" (.) H
R (.) no cpYHKllHHM
JIe)KaH.n:pa,n:JISI a-r (cos it) H
-r (cos it) H3 (17.8). HBHble Bblpa)KeHHH .n:JIH
3THX K03cpcpHllHeHToB MO)KHO HaHTH C HCnOJIb30BaHHeM YCJIOBHH
opTOroHaJIbHOCTH (18.1) H ypaBHeHHH .n:JIH P Vk (cos it) H P'/,:f=i-r(COS it):
(2Vk + I) {[ Vk (Vk + I) + 't 2 - +] a
 hl O ) + 2.

 (f.!O)}
ak(L) =-
p

 (f.!o) {[ Vk (Vk + I) +.2 -+ r + 4't 2 }


(18.7)


T aKHM 06pa30M, 3a.n:a4a 0 BbI60pe 3Ha4eHHH npOH3BOJIbHbIX nOCTOHH-
HbIX H cpYHKllHH co,n:ep)Karn.HxcH B 06II\eM peIIleHHH, no 3a,n:aHHbIM rpa-
HH4HbIM YCJIOBH
M CBe.n:eHa K peIIleHHlO CHCT:MbI CooTHo
eHHH (18.3)
H (18.5). Ycnex B peIIleHHH paccMaTpHBaeMoH rpaHH4HoH 3a,n:a4H ,n:JISI
yce4eHHoro KOHyca B 3Ha4HTeJIbHOH Mepe CBH3aH co 3HaHHeM aCHMn

TOTI14eCKHX CBOHCTB HeH3BeCTHblX Ck' d k H P (.). Hal160JIee npOCTOH
nYTb, Be.n:YII\HH K YCTaHOBJIeHHlO 3THX CB
HCTB, nO-BH
HMOMY, COCTOHT
B TOM, 4T06bI, OCHOBbIBaSICb Ha 04eBH,n:HOH cpH3H4eCKOH po,n:CTBeHHOCTH
paCCMaTpHBaeMOH 3.n:eCb 3a.n:a4H H 60JIee npOCTbIX HCCJIe.n:OBaHHbIX upaHee
3a.n:a4, CcpOPMYJIHpOBaTb npe.n:nOJIO)KeHHe 06 O)KH.n:aeMbIX CBOHCTBax
HeH3BeCTHbIX, a 3aTeM npoBepHTb cnpaBe.n:JIHBOCTb 3TOro npe,n:noJIo)Ke-
HHSI.
HaJIH4He B ynpyroM TeJIe yrJIOBbIX T04eK npHBo.n:HT, KaK nOKa3bl-
BalOT pe3YJIbTaTbI peIIleHHH rpaHH4HbIX 3a.n:a4 ,n:JIH npHM
yrOJIbHHKa H
llHJIHH,n:pa, K nOHBJIeHl1lO JIOrapHcpMH4ecKHx oc06eHHocTeH B HOpMaJIb-
HbIX HanpSI)KeHHHX ,n:JIH B3SITbIX nop03Hb COCTaBJIHlOII\HX 06II\ero pe-
IIleHHSI. B CBH3H C 3THM eCTeCTBeHHO npe,n:nOJIO)KHTb, HanpHMep, 4TO B
paccMaTpHBaeMoM CJIY4ae nOBe,n:eHHe cpYHKllHH p (.) Ha 6ecKoHe4Hoc-
TH 06YCJIOBHT HaJIH4He JIOrapHcpMH4ecKoH oc06eHHocTI1 npH t - 0 B
HHTerpaJIe


,
,


( 2V k + I) {[ vk (Vk + I) + 't 2 - +]

 (f.!o) - 2'ta-r (f.!O)}

k (L) = - {[ 3 ] 2 .
p
: (f.!o) Vk (Vk + I) + 't 2 - 4" + 4't 2 }
DocJIe 3TOro BbI4HCJIeHHe HHTerpaJIOB, BXO.n:HII\HX B BbIpa)KeHHSI .n:JIH
Dk (., f-lo) H Ok (T, f-l ), He npe,n:CTaBJISIeT Tpy.n:a. llJIH 3TOro .n:OCTaT04HO
B cpYHKllHHX K-r (cos tt), L, (cos it), R, (cos it) H S, (cos it), KOTopble on-
pe.n:eJIHlOTCH cpopMYJIa1\1H (17.10), 3aMeHHTb a[ (cos it),
,(cos it) H HX
npoH3Bo,n:HblecooTBeTcTBYlOII\HMH pH.n:aMH no P v " (cos it) H P
k(cOS it).
OpoBe.n:H Ha OCHOBe COOTHOIIleHHH opTorOHaJIbHOCTH BbI4HCJIeHHe HH-
TerpaJIOB, Haxo.n:HM
Dk (., f-lo) =
= (",
;k
't:"';)2 ([ £,a 'l ) (f-lo) - Than (f-lo) - (L" + +) a 'l'l (f-lo)] L1 T +
+ [£-ra;'l (
to) + 1'J-r a 'l'l (f-lo) - (L 2 + + ) a
1 (f-lo) 1.1-r} +
+ ("'
)

':"';)2 {[ ;-ra
'l (f-lo) + 1'J,a
'l (f-lo) + (L 2 + + ) a'll (f-lo) ] .1, -
- [ £-r a 'l'l (f-lo) - 1'J-r a 'l'l (f-lo) + (L 2 + -{-) a'll (!-to) 1.1
} ,
(18.8)
Ok (L, f-lo) = . ak (.) u { [ V-ra
n (f-l o ) - X-ra
n (!-t o ) - _ 2 1 an i (f-lo) +
("'-r)2 + ("'-r)2 _L " L
+ La'll (!-tV>].1-r + [X-rU 'l'l (f-lo) + V-rU 22 (f-lo) - La'll (f-lo) - + a'll (f-lo) 1.1
} +
+ [:Ik (.) {f ( . I. J .1
("'
)2 + ("'
)2 X-ran f-lO) + V-rU 2'l (!-to) + La'll (f-lo) +""2 a'll (f-lo) -r -
- [V-r
'l (f-lo) - X-rU 'l2 (f-lo) + + a 21 (f-lo) - La 21 (f-lo) J .1
}.
178


""


J (t) = .\ p (L) cos Lt d1:,
n


(18.9)


(k) u
KOTOpbIH nponopllHOHaJIeH HanpH)KeHHSIM at} Ha KOHH4eCKOH nOBepx-
HOCTH.
113 CYII\eCTBOBaHHH JIOrapHcpMH4eCKOH oc06eHHOCTH y cpYHKllHH J (t)
npH t _ 0 CJIe.n:yeT, 4TO nOBe.n:eHHe nJIOTHOCTH HHTerpaJIa <I>YPbe B
(18.9) npH ,n:OCTaT04HO 60JIbIIlOM 1: onHCbIBaeTCH paBeHcTBoM
p (L) =
o + 0 ( + ), a o =1= O. (18.10)
3HaHHe xapaKTepa nOBe.n:eHHH cpYHKllHH p (T) n03BOJ1HeT nOJIY4HTb
OlleHKH npH 60JIbIIlHX 3Ha4eHHSIX k HHTerpaJIOB B npaBblx 4aCTHX pa:
BeHCTB (18.3). DpH 3TOM ,n:JIH nOCTpoeHHH aCHMnTOTH4eCKHX Bblpa)KeHHH
cpYHKllHH Dk (1:, f-lo) H Ok (., f-lo) HCnOJIb3yeM CJIe.n:YlOII\ee npe,n:CTaBJIe-
HHe KOHH4eCKHX cpYHKllHH MeJIepa npH 60JIbIIlHX . [96]:
I
P (rn, ( 0 _<I. ) '" T,rn-'/. eTt} [ 1- m 2
"'4 ct g it + 0 (
 2 \ ]
-'/.+i't csu =Y2nsinit 't .!
(0 < tT < 11:). (18.11)
aTO COOTHOIIleHHe .n:aeT B03MO)KHOCTb npH .n:OCTaT04HO 60JIbIIlOM . BbI-
4HCJIHTb 3Ha4eHHH 3JIeMeHTOB onpe.n:eJIHTeJIeH .1
 H .1; H caMH 3TH onpe-
,n:eJIHTeJIH no cpopMYJIaM


e'tt} ( 15 cas it )
a
l (f-l) = - XL 3 V2msin it sin it - 8. '


12*


179




21: s:n ,'I- ),


MbI (18.3). fIpe.n:cTaBIIM era B CJIe.n:YIOIJJ:eM BlIlle:
ro N
- 2 I p ('r) Dk ('t, /-10) d't = - 2 I P ('t) Dk ('t. /-10) dt-
o 0


e'ti} ( -&
a;1 (/-1) = - %'t 3 cosit + CDS ctg-& _ 2-4 sin 2 -& )
y 2n1: sin -& 81: 1: sin it '


D:ByqJleHHbIX aCIIMnTOTlIqeCKIIX rPOPMYJI .n:JlH KOHlIqeCKIIX rPYHKUIIH

eJlepa (18. II ) OKa3aJlOC? He.n:ocTaTOqHO .n:JlH nOJlyqeHIIH TOqHOra 3Ha-
'I,eH
H rJlaBHOrO qJleHa Llt (/-1). O.n:HaKO .n:JlH aHaJllI3a aCIIMnTOTlIqeCKIIX
CBOIICTB HelI3BeCTHblX C k , d k II P ('r) B COOTHOIIIeHIIHX (18.3) H (18.5)
BnOJlHe .n:OCTaTOqHO 3HaHlIH JllIlllb nopH.n:Ka BeJllIqlIHbI Ll; (/-1).
D I1cxo.n:H 1I3 COOTHOllleHHH (18.12), Haxo.n: IIM , qTO .n:JlH rPYHKUHH
k ('t, /-10) II Ok ('t, /-10) B (18.8) cnpaBe.n:JlIIBbI CJle.n:YIOIJJ:lIe oueHKII:
Dk (1:, /-10) = _ 2 (2
k + I) Vk (Vk + I) [1: + 0 (I )]
p
1J (flo) [Vk (Vk + I) + 1: 2 12 '
k
Ok (1:, /-1) = _
2 ( 2V k + I) Vk (Vk + I) 0 (
 ) .
p

 (flo) [Vk (Vk + I) + 1:2]2 \.


a;2 (J-t) = - 't 2 e'ti} ( cos'it _ 3 cos -& ctg-&
V 2m sin -& 81:
e'ti1 (
d;2 (/-1) = - 't 2 sin it _ 3 cos -& )
Y 2m sin -& 81:'
a' (11) = -%'t 3
 (cosit- 3+sin 2 it
21 y 2m sin -& 31: sin it
a;, (/-1) = %'t 3 _e
 (sin it + 9cos-& )
Y2m sin -& 81:'
e'ti1 (
a
 (/-1) = - 't 2 sin it _ 3 C
 -& ) ,
V2msin -&


'ti}
a;2 (/-1) = 't 3 e ( cos'it _ 3 cos it ctg -& )
V 2m sin -& 81:'


. e 2 'ti}
Ll't (/-1) = 2%'t 4
2n1: sin it


( I,
o -:t),


Ll
 (/-1) = 2%'t 4


e 2 'ti}
2n1: sin it


[I +0(+)]


( m
%-
- m-2 '


O<it<n) .


sin -& )
1: '


- 2 \ P (1:) Dk ('t, /-10) d't,
'tv
r.n:e N - HeKOTOpoe .n:OCTaTOqHO 60JlbIIIoe qIICJlO.
D:JlH nepBOro CJlaraeMorO B npaBOH qaCTlI paBeHCTBa (18.14)
(IIHTerpaJl B KOHeqHbIX npe.n:eJlax) C yqeTOM BbIpa)KeHIIH .n:JlH D k (1:, /-10)
Haxo.n:IIM oueHKY


(18.14)


(18.12)


N
- 2 I p('t) Dd1:, /-10) d1: = 0 (V;3/2).
o
fIplI nOJlyqeHHH oueHOK BToporo CJlaraeMOro IICnOJlb3yeM BWpa)KeHIIH
(18.10) .n:JlH p (1:) II (18.13) .n:JlH Dk (1:, /-10) nplI 60JlbIIIHX 3HaqeHlIHX 1:.
BWnOJlHHH IIHTerplIpoBaHHe, Haxo.n:IIM .n:JIH 60JlbllIlIX k cJle.n:ylOIJJ:ee pa-
BeHCTBO:


( 18.15)


ro I k+1 Y2n sin it
- 2 P ('t) Dk ('t, !lo) d't = (- I) V 0 na o + 0 (V;'/2), (18.16)
Vk-&o
N


nplI nOJlyqeHlI1I KOToporo IICnOJlb30BaHO aCIIMnTOTlIqeCKOe COOTHOllle-
HlIe
P
 k ) (/-10) = (- I)k+lvkit o V -2
 [I + 0 (v;I)]. (18.17)
nVk SIn 0
PaccMaTplIBaH TpeTbe ypaBHeHlIe B CIICTeMe (18.3) C IICnOJlb30BaHII-
eM oueHKII nopH.n:Ka y6bIBaHlIH <PYHKUIIII Ok (1:. (.to) (18.13), Haxo.n:IIM


ro
- 2 ,\ P (1:) Ok ('t, /-10) d't = 0 (V;8/2).
o


(18.18)


(18.13)


CYMMIIPYH pe3YJlbTaTbI oueHOK IIHTerpaJlOB B npaBbIX qaCTHX
ypaBHem:Ii%(18.3) II npe.n:nOJlaraH, qTO CBOHCTBa rPYHKUIIH f (it) II cp (it),
3a.n:alOIJJ:IIX HanpH)KeHIIH Ha CrPeplIqeCKOH nOBepXHOCTII yceqeHHOrO KO-
Hyca, 06eCneqlIBaIOT y6bIBaHlIe K03rPrPlIUlIeHTOB fk II ([Jk KaK v;3i., nOJl)'-
qaeM CJle.n:YIOIJJ:lIe aCIIMnTOTlIqeCKlIe CBOHCTBa nOCTOHHHbIX C k II d k :
I
Vk [( I ) k+1 Y 2n sin -&0 + ( - 2 )]
C k = - 2 -

 na o 0 Vk ,
r Vk-&O
1
d = -
 f( - I) k+l Y2nsin-&0 + ( -2 )1
k 2
r- nao 0 Vk .
r Vkito
06paTmdCH Tenepb K COOTHOllIeHlI1O (18.5), onpe.n:eJlHIOIJJ:eMY 3Ha-
qeHIIH rPYHKUlI1I p ('t) qepe3 C k II d k . 113yqlIM, IICXO.n:H 1I3 3Toro COOT-
HOllIeHIIH, nOBe.n:eHlIe rPYHKUlI1I p (or) nplI 60JlbililIX 3HaqeHlIHX 't.


(18.19)


f!pe.n:nOJlO)KelllIe 0 xapaKTepe nOBe.n:eHIIH rPYHKUHII p (1:) (18.10)
II HaHJI:eHHbIe oue
KII .n:JlH Dk(1:, /-10) II Ok (1:, /-10) n03BOJlHIOT 1I3yqHTb aCIIM-
nTOTlIqeCKlIe CBOIICTBa nOCTOHHHbIX C k II d k B (18.3). D:mI 3TOro pac-
CMOTplIM BHaqaJle IIHTerpaJl B npaBoH qaCTII BToporo ypaBHeHIIH ClIne..
180


181




y!3 nepBoro ypaBHeHlUl B (18.3) npH c,n:eJIaHHblX npe,n:nOJIO>KeHHflX 0
XapaKTepe nOBe,n:eHHfI p ('r) (18.10) CJIe,n:yeT OrpaHHqeHHOCTb nOCTOflH-
HOH do. OTHOCHTeJIbHO <PYHKUHH 'IjJ{t) B (18.6), 3a,n:mOlll.eH 3HaqeHHfI Ha-
npfl>KeHHH Uti Ha KOHl14eCKOH nOBepxHocTH, npe,n:nOJIaraeM TaKYIO CTe-
neHb rJIa,n:KOCTH, KOTopafl o6ecne4l1BaeT y6bIBaHlie 'IjJ ('r) C pOCTOM T KaK
'{-2. lJJI5! ::noro ,n:OCTaTOqHO, 4T06bl npH cooTBeTcTBYIOllI.eM xapaKTepe
y6bIBaHHfI Ha 6eCKOHe4HOCTli <PYHKUIiH 'IjJ (I) I1MeJIa KycOqHO-Henpe-
PbIBHYIO nepBYIO np0I13Bo,n:HYIO. TaKIiM o6pa30M, ,n:JIH ,n:BYX CJIaraeMblX
B npaBoH qaCTIi paBeHCTBa (18.5) HMeeM npH 60JIbIllHX T
doG o ('r) + 'IjJ (T) = 0 (T- 2 ). (18.20)
PaccMoTpHM Tenepb nOBe,n:eHHe npl1 60JIbIlll1X T <PYHKUHI1, onpe,n:eJIHe-
MOH 6eCKOHe4HblM pfl,n:OI\! B (18.5). C 3THM pH,n:OM MO>KHO 6blJIO 6bI no-
cTynHTb TaK >Ke, KaK npH H3Y4eHHl1 CBOHCTB I1HTerpaJIa (18. I 4), onycTIiB
B noc.1e,n:YIOIUeM aHaJIH3e HeKoTopoe KOHe4Hoe ql1CJIO ero nepBblx 4JIe-
HOB, KOTopble I1MeIOT 60JIee BblCOKHH nopH,n:oK y6bIBaHliH no T, 4eM CYM-
Ma OCTaBIllerocfI 6eCKOHeqHOrO pfl,n:a. O,n:HaKo, nOCKOJIbKY 3TO He npHBO-
,n:HT K nmlBJIeHHIO oco6eHHOCTeH, MO>KHO coxpaHHTb B HeM CYMMHpOBa-
HHe OT e,n:HHI1Ubl, HCnOJIb3YH OTMeqeHHYIO B03MO>KHOCTb 3aMeHHTb B ero
qJIeHaX <PYHKUHH HOMepa HX aCHMnTOTl14eCKl1MH Bblpa>KeHHHMH. ITpH
3TOM HCnOJIb3yeM C.1e,n:YIOllI.He aCHMnTOTHqeCKHe Bblpa>KeHHfI ,n:JIfI <PYHK-
UHH JIe>KaH,n:pa H KopHeH nepBoro ypaBHeHHfI B (17.3) v k :
Pvk(cosito)=(-l)k 2. ( 1-- 4 1 ) ,
I 2nvk SIn {to Vk
(Vk++)ito=(k++)n- 3t\c
g{to . II (]8.21)
k+T
C TOqHOCTbIO ,n:o BeJIH4HH nopH,n:Ka V;-2 B cpaBHeHHH C e,n:HHHueH H3
BTOpOro ypaBHeHHfI CHCTeMbl (18.3) CJIe,n:yeT
I ) 2
(l + V;; d k = C k + 0 (V;;B'.). (19.22)
3TO COOTHOIlleHHe ,n:aeT B03MO>KHOCTb npe,n:cTaBHTb CYMMY 6eCKO-
He4Horo pfl,n:a B (18.5) B BH,n:e


. C YCTaHOBJIeHHblMH paHee (18.20)
06'he,n:HHflH nOJIyq
HHblH pe3YJIbTa
 n HXO,n:HM K 3aKJIIOqeHHIO, qTO
OlLeHKaMH <PYHKUHH do Go (T) H 'IjJ ( ),,1-, P H P (T) B (18.5) onpe,n:eJIH-
IIPH 60JIbIllHX 3HaqeHHflx T nOBe,n:eHHe 't'YHKUH
CTCH paBeHCTBOM


P ('t') =
o + 0 ('t'-l).
T O>K,n:eCTBeHHOCTb paBeHC
: l y 1
;;
H H;(
0H;:I3b:

n

o

H:T

' 1 8 10 ) 0 xa p aKTepe nOBe,n:eH 't' 6 eHH
\ . . (18 3) H (185) ,n:eHcTBHTeJIbHO 0 JIa,n:aeT pem -
CIiCTeMa COOTHOIlleHHH . MnTO;HqeCKHMH CBoHCTBaMH. KaK H B pac-
CM C cooTBeTcTBYIOllI. HMH aCH ax 3a,n:aqaX 3TO 06CTOflTeJIbCTBO C03-
cMoTpeHHblx B npe,n:bl,n:YllI. HX rJIaB K ,n:JIH docTpoeHHfI 3<p<peKTHBHoro
,J.aeT ,n:OCTaTOqHble npe,n:noCblJIKH. Ka C d H P ( T ) TaK H
HH 3HaqeHHH HeH3BeCTHblX k' k ,
aJIrOpHTMa OTblCKaH x KOJIHqeCTBeHHblX xapaKTepHcTHK HanpH-
..'I.JIH nOJIyqeHHH ,n:ocToBepHbl
>KeHHOCTH yceqeHHOrO KOHyca.


(18.26)



 19. rp3HH'IHbIe 33A3'1H Ann nonynpocTp3HCTB3
C nonyciPepH'IeCKHM BbIpe30M


J (T) =
 CkPVk (f!O)[1'kGVk-2 (T) + Gv k ('t')J + 0 (T- 2 ). (18.23)
k=l
K TaKoMY 3aKJII04eHHIO npHxo,n:HM Ha OCHOBe 04eBH,n:HOH oueHKH CYM-
MbI pfl,n:a


B cJIyqae nOJIynpoCTpaHcTBa c nOJIyc<pepHqeCKHM Bblpe30M (it o =
JL _ it = 0 ) ,1-,O P M Y JIbI npe,n:bI,n:Ylll.HX naparpa<p oB 3HaqH-
- - f!o - cos 0 't'
- 2 ' K H aBHeHHfI (17.3) 6y,n:YT UeJIblMH qeTHblMH
TeJIbHO ynpOlll.aIOT cfI. 2k oP ( : y g 1 2 ...). B COOTBeTCTBHH C 3THM H
qHCJIaMH T. e. V k - , , , .
<POPMYJIb
 (18.8) MO>KHO 3anHcaTb B 0603pHMOM BH,n:e.
(4k + I) P 2 k (0) P _1/.+t't (0)
Dk ('t', 0) = 2 P(1). (0) X
_1/.+1'1
2 ) m+I ( 2 I )
2k (2k+ I) (4172_3 + -,;- +2 m 17 + ""'4
(2k(2k+ I) +17 2 - + r + 417 2
7 2 )
(4k + I) P 2k (0) P _1/.+l't (0) (317 2 + 4"" ---;:;;
0 " ('t', 0) = - 3 ) 2 }
P
'/.+l't (0) {[ 2k (2k + I) +17 2 - "4 + 417 2


X


(19.1)


00

 = o (r2J.
t: Vk (v
I+ 17 2 )
OCYlll.eCTBJIfIfI npHBe,n:eHHe K 06ll1.eMY 3HaMeHaTeJIIO ,n:po6eH B
KBa,n:paTHbIX cKo6Kax B (18.23) 11 npeHe6peraH 3aTeM B K03<p<P H uHeHTax
-I .
pflJ/.a BeJIH'lHHaMH nopfl,n:Ka Vk B cpaBHeHHH C eJ/.HHHueH, nOJIY'lael\l
a
J ('t') = -f- + 0 (r 1 ). (18.25)


(18.24)


6 COOTHomeHHH CBfl3bIBaIOllI.HX nOCTOflHHble
d qHH
e y HHHJu
:

)
;106Ho npOBo,n:HT
, BBe,n:fI npeJ/.BapHTeJIbHO HO-
Ck' k 't'
Bble HeH3BeCTHble no <popMYJIaM
2d o = -xo'
l ak c 2k + ( 1 + + ) (1 + 2 I k) d 2 k J P 2 k (0) = - + X2k.
183


182




[ Pk C2k - ( 1 + - k l ) d2k J P2k (0 ) = 3 Y
k(2k-l) 2k,
( T2 +
 )
- '/'+lT; lO)
4 p(l) . ( 0 ) P IT) = Z (T).
C -'/2+1'1:
YlleTOM 3aMeHbI ,n:JUI HeH3BeCTHbIX CHCTeMa ypaBHeHHH (18.3) H (18.5)
npe06pa3YeTCH K BH,n:y


3
2k+""2
a2k (T) = h ll = - 2

 ,(19.4)
(2k++r +1;2 ' k(l+k)
2k
3kak 2
h 12 = - 2k (k + I)
2k h 21 = - M 2k
3(2k+l) ( 1 )
h 22 = (2k)2
2k ' l:1 2k = cx k + .1 + 2il Pk'
XapaKTep CooTHoIIIeHHH (19.2), KOTOpbIMH C IIeJIbIO y,/J,06CTBa npH
BbIlIHCJIeHHHX BBO,n:HTCH HOBbIe HeH3BeCTHble B CHCTeMY paBeHcTB (18.3)
H (18.5), B 3HallHTeJIbHOH Mepe npe,n:onpe,/J,eJIeH pe3YJIbTaTaMH npOBe,/J,eH-
HOro B npe,n:bI,/J,YIIIeM naparpacpe aHaJIH3a aCHMnTOTHlIeCKHX CBOHCTB
HeH3BeCTHblX. K03cpqmIIHeHTbI npH HeH3BeCTHblX B (19.2) BbI6paHbI CJIe-
,n:YIOI.IIHM 06pa30M:
Xoo = Yoo = z(oo) = Co. (19.5)
Cnoc06 llCnOJIb30BaHllH 3Toro CBoHcTBa npH OTWCKaHllH npll6JIH-
:meHHoro peIIIeHHH CHCTeMbI (19.3) nOJIHOCTbIO aHaJIOrHlIeH onHcaHHoMy
npH paccMoTpeHHH 3a,n:all .lI.JlH npHMoyrOJIbHHKa H IIHJIHH,/J,pa. OH 3a-
KJIIOlIaeTCH B TOM, liTO, HallHHaH C HeKOToporo HOMepa k > K H 3Halle-
HHH aprYMeHTa . >N, Bce Hell3BeCTHble npHHHMaIOTCH paBHbIMH ,/J,pyr
,n:pyry. DOJIYlleHHaH TaKHM 06pa30M KOHellHaH CHCTeMa pelliaeTCH lIllC-
JIeHHO. 0 TOllHOCTH nOJIYlleHHoro npH TaKOM nO,/J,xo,/J,e peIIIeHHH MOLKHO
CY,/J,HTb, C O,/J,HOH CropOHbI, no xapaKTepy CTpeMJIeHHH HeH3BeCTHbIX K
npe,/J,eJIbHOMY 3HalleHHIO, a C ,/J,pyroH - no rollHOCTH y,/J,OBJIeTBOpe-
HHH rpaHllllHbIX YCJIOBHH. Pe3YJIbTaTbI HCnOJIb30BaHHH TaKoro cnoc06a
ycelleHHH 6eCKOHellHbIX CllCTeM B paCCMOTpeHHbIX paHee 3a,/J,allaX no.zr:-
TBepLK,/J,aIOT ero 3cpcpeKTllBHOCTb. 0 ero B03MOLKHOCTHX npH peIIIeHHH
rpaHHlIHbIX 3a,/J,all ,/J,JIH ycelleHHoro KOHyca MOLKHO CY,/J,HTb no npllBe,/J,eH-
HbIM HHLKe pe3YJIbTaTaM peIIIeHHH ,/J,BYX KOHKpeTHbIX rpaHHlIHbIX 3a-
,n:all ,n:JIH nOJIynpocTpaHCTBa C nOJIyccpepHlIeCKHM Bblpe30M.
3aoatta 1. HanpHLKeHHoe COCTOHHHe nOJIynpocTpaHcTBa no,/J, ,/J,ell-
CTBHeM paBHOMepHO pacnpe,/J,eJIeHHOrO ,/J,aBJIeHllH no nOJIyccpepHlIeCKO-
MY Bblpe3Y. B 3TOM CJIYllae rpaHHlIHble YCJIOBHH (17.11) npll06peTaIOT
BH,n:
I 1
20 or = - P = const, 20 'rit = 0


(19.2)


00


Xo= S
o


75 0 (1;, 0)
1 Z (T) d. - to,
1;2+_
4


00


X2k = 2k (k + -{-) [P 2 k(0)]2 S Dk (1;,
)
o 1;2 + -
4
Y2k = + (2k)4 (1- dk ) (1 + 4
 ) (P 2k (0)]2 X


k
Z (T) d. - 2 P 2k (0) tk'


00
xS
o


Ok (1;, 0) 2 1
1 z (.) d. + ""3 k 2 (1- 2il ) P2k (0) fJJ k ,
1;2 + _
4


(19.3)


I ( 1;2 + + ) P -1/2+iT: (0) {
Z (T) = - 1 -
;It p(l). ( 0 ) - _ 2 xoao (T) +
-'/.+l't
+
 [( (I + 2
 r h ll a 2k (.) +
lV2ka2k-2 (T») X2k +
+ (( 1 + dk ) 2 hl
a2k (T) + h22V2
2k-l (T) ) k
2
 ]} +
2


( 1;2 + +) P -'/2+1'1: (0)
.+ pm 'IjJ (.).
-'I.+h
3,n:ecb .zr:JIH COKpaIIIeHHH 3anHCH BBe,n:eHbI 0603HalleHllH:
2k (2k + I) [4 (1;2 + +) _ 2 2m;:: 1 ] +2 m m + 1 ( ,;2 + _ 4 1 )
Dk (" 0) =
r 2k (2k + I) + ,;2 - + r + 4,;2
3 ( ,;2 + + ) + m;; 2
Ok (T, 0) =
[2k(2k+ I) +,;2_ : r +4,;2 '
184


O<it<
 ),


(t= 0,
(it =
 '


1 I
20 Oft = 20 Trit = 0


O<t<oo).


B COOTBeTCTBllH C 3THM ,n:JIH CB060,/J,HbIX lIJIeHOB CHCTeMbI (19.3) nOJIY-
lIaeM
to = - p, fJJk = h = 0 (k> 1), 'IjJ (T) = O. (19.6)
PeIIIeHHe nOCTaBJIeHHOll 3a,n:allH CTpOHM B ,/J,Ba 3Tana. BHallaJIe B03b-
MeM lIaCTHoe peIIIeHHe CllCTeMW ypaBHeHHll (19.3) B BH,n:e
X
2) = p; x

 = y

 = Z(2) (.) = 0 (k> 1, 0 <.< 00). (19.7)
185


Ii
Ii I





 U(2) = _ P e-31
20 r ,


I I 1
- d 2 ) = _ U(2) = _ P e-31
20 {} 20 Q) 2 '



1:(21_0
20 r{}- .


[ 40
Y2k = + (2k)4 (1 - 2 1 k ) (1 + 4
 ) [P 2k (0)]2 J


Ok (1:, 0) Z (1:) d1: +
1
1: 2 +-
4


, I
I,
I,
II


Hanp5DKemUI, COOTBeTCTBYIOllll1e 3TOMY pernemuo, C yqeTOM 3aMeHbI .1J.JIH
HeH3BeCTHblX (19.2), BblqI1CJHIIOTC5I no cpopMYJIaM (17.5), (17.9) H paBHbI


(19.8)


+ z (40) f,
(40)]'
( 1:2 -f -
 ) P 1 / + " (0) {
4 -. l't 1-
- 2 xoa o (1:) +
nP!.:>'/.+i't (0)


HeTPY.D.HO 3aMeTHTb, qTO onpe.1J.eJI5IeMOe 3Tl:!MI1 <j:;opMYJIaMH Hanp5lJKeH-
Hoe COCT05IHl1e COBna.1J.aeT C Hanp5lJKeHHbIM COCT05IHl1eM ynpyroro npo-
C'TpaHCTBa co ccpep/1lleCKOH nOJIOCTblO, HarpYJKeHHOH paBHOMepHbIM .1J.aB-
1
JIeHl1el'l'1 20 u r = - p (3a.D.aqa naMe).
PerneHHe 6eCKOHeqHOH CHCTeMbI (19.3) l:!I.qeTC5I Tenepb .1J.JI5I H3MeHeH-
HbIX rpaHHqHbIX YCJIOBl:!H, a HMeHHO:
do U r = 2 1 0 T{}r = 0 (t = 0, 0 < it <
 ) ,


Z (T) =


(19.9)


+
 l (( 1 + ik r h ll a 2k (T) + 'V2kh 21 a 2k-2 (T) ) X2k +
(( I ) 2 - - ) Y2k J (19.12)
+ 1 + 2k h 12 a 2k (T) + 'V2k h 22 a 2k-2 (1:) 1 +
k--
2


I _ I -3t
20 ut} - - 2 pe ,


1
20 Trt} = 0


( it =

2 '


O<t<oo).


t k = C{Jk = 0


(k>- 0),


3p
W(T) =--
4n


9
1: 2 +_
4


(19.10)


I
1: 2 + 4
9
1: 2 + 4
3HaqeHl:!H HHTerpaJIOB I\k) (40), I
) (40) l:! CYMMbI S't (21) BblqHCJIH-
IOTCH no cpopMYJIaM
1\0) (40) = m
 I ( + 1t _ 2 ar
tg 80 + + arctg
o ) ,
I\k) (40) = + [2k (2k + 1) + m
 1 ]( C/
-I
I 40 1 40 ) 2m - I 2 k ( 2k
---arctg I + +1 arctg +1 - m +
a- I C(- a a
+ 1 ) 1t ( 2a + b + c -
 a arctg 80 _
a+l a-I n
2 b t g 40 _ ..!.. C arctg 40 )
- n a + I arc a + 1 n a-I a-I'
3 1 40
t k ) ( 40 ) = - ( n arctg a-I +
2 4a a 2 - I a - I
I 40 ) m - 2 ( b (19 13)
+ a + I arctg a + 1 + 2m 1t 2a + a + 1 + .
C 4 2 b 40
+ a-I - n a arctg 80 - n a + I arctg a + I
2


} 3p
+ x 4 0S't (21) - 4n


P -1/. +i't (0)
-(1)- .
P -',.+iT (0)


CB060.1J.Hble qJIeHbI CHCTeMbI (19.3) .1J.JIH 3TOro CJIyqa5I HarpyJKeHl:!H paBHbI


Lh06bI He 3arpOMOJK.1J.aTb nOCJIe.1J.YIOI.qHe CPOPMYJIbI, HeH3BeCTHble 6ec-
KOHeQHOH CHCTeMbI (19.3), OTBeQalOI.qHe rpaHHqHOH 3a.1J.aQe (19.9), l:!
cooTBeTcTBYIOI.qHe HM HanpHJKeHHH He 6Y.1J.eM OTMeqaTb LlOnOJIHl:!TeJIb-
HbIMH HH.1J.eKcaMH.
DpH nOCTpoeHHH KOHeqHOH Cl:!CTeMbJ, OTBeQalOI.qeH COOTHOrneHl:!HM
(19.3), Ha OCHOBe YCTaHOBJIeHHbIX aCHMnTOTl:!qeCKHX CBOHCTB HeH3BeCT-
HbIX nOJIaraeM


I



X2k = x 40 , Y2k = Y40 (k >- 20), Z (1:) = Z (40) (T>- 40). (19.11)
B pe3YJIbTaTe CHCTeMa ypaBHeHl:!H (19.3) npl:!06peTaeT BH.1J.
40
Xo = S Do (-.;'
) Z (T) dT + Z (40) 1\0) (40),
o -.;2+ 4""


[ 40
X2k = 2k (k + +) [P 2k (0)]2 J


Dk (1:, 0)
I
1: 2 +_
4


Z (T) <h + Z (40) Ii" (40) ] ,


--
n


C arct g
- )
a-I a-I'


186


187




Jt m; 2 20
S'{(21)=-cth---- """'
'[ 2 '[2

k=1 k 2 + (+ r


CaTeJIbHblX HanpH:m:eHlfii 'iri) saMeTHM, qTO OTKJIOHeHHe HX OT HYJIH HH
B O.n.HOH TOqKe nOBepXHOCTH c<pepHqeCKoro Bblpesa He npeBbIIIIaeT 0,1 %
::>TOH BeJIHqHHbI.
I1HTepecHo cpaBHHTb BblqHCJIeHHble SHaqeHHH HanpH:m:eHHH ai} H
ac:p B HeKoTOpbIx xapaKTepHblx TOqKaX C pemeHHeM JIaMe .n.JIH saMKHy-
TOH c<pepHqeCKOH nOJIOCTH B ynpyroM npoCTpaHcTBe, Harpy:m:eHHo H
Ta6JIlIua 20


I
I
I!


Jt2 I
So (21) =6-
k2'
k=1


r.n.e


(19.14)


I Or I 0i} I 0c:p II 'i} I Or I 0'i} I 0c:p
'i} 2Gp 2Gp 2GP 2Gp 2Gp 2GP
0 1,000 0,477 0,477 :n;
0,6 2 1,000 0,258 0,400
:n; -1,000 0.471 0,475
0.1 2 :n; 0,178
0,7 2 -1,000 0,367
Jt -1,000 0,453 0.469
0,2 2 :n;
0,8 2 -1,000 0,089 0,326
Jt -1,000 0,424 0,459
0,3 2 Jt
0,9 2 -1,000 0,006 0,275
:n; -1,000 0,382 0,445
0,42
-1,000 -0,034 0,217
Jt -1.000 0,326 0,425
0,5 2


a=2k+
 a= I
2' 2(k+I)(2k-I)2k(2k+l)
k
4 (k + I) (2k + + ) 2k (2k + I)
-(2k+ I)
I .
4 (2k - I) (2k +""2) 2k (2k + 1)


b=


c=


qHCJIeHHOe pemeHHe CHCTeMbI ypaBHeHHH (19.12) npH m = 4 npo-
Be
eHo MeTO.n.OM nOCJIe.n.OBaTeJIbHbIX npH6JIH:m:eHHH. Co.n.ep:m:aw.HecH B
HeH HHTerpaJIbI OT z (-r) SaMeHHJIHCb KOHeqHbIMH cYMMaMH no MeTO.n.y
raycca. DpI1 3TOM HHTepBaJI HHTerpHpoBaHHH [0, 401 paS6HBaJICH Ha
.n.Ba HHTepBaJIa [0, 201 H [20, 40] H B Ka:m:.n.OM HS HHX no.n.bIHTerpaJIbHaH
<PYHKIJ.HH annpoKCHMHpOBaJIaCb no ee SHaqeHHHM B 24 YSJIOBbIX TOq-
Kax. Y.n.BoeHHe KOJIHqeCTBa TOqeK annpOKCHMaIJ.HH He npHBo.n.HJIO K
cyw.eCTBeHHbIM HSMeHeHHHM SHaqeHHH HCKOMbIX BeJIHqHH.
DpOBe.n.eHHble BblqHCJIeHHH 06Hapy:m:HJIH HHTepeCHYIO oc06eHHOCTb B
nOBe.n.eHHH HeHSBeCTHbIX B CHC1;,eMe (19.12) JI.JIH .n.aHHoro BH.n.a Harpy:m:e-
HHH nOJIynpoCTpaHcTBa. OKaSaJIOCb, qTO npe.n.eJIbHble SHaqeHHH BeJIH-
qHH X2k, Y2k HZ (-r), no MeHbmeH Mepe, Ha .n.Ba nopH.n.Ka MeHbme, qeM
SHaqeHHH HX JI.JIH MaJIbIX k H 17. Do CYW.ecTBY, y:m:e JI.JIH k > 10 H 17 > 20
B npaKTHqeCKHX BblqHCJIeHHHX MO:m:HO nOJIaraTb X2k, Y2k H Z (17) paB-
HbIMH HYJIIO. B CBHSH C 3THM ynpow.aeTcH sa.n.aqa BblqHCJIeHHH HanpH:m:e-
HHH no <popMYJIaM (17.5) H (17.9), B KOTOpbIX MO:m:HO OrpaHHqHTbCH KO-
HeqHbIMH cYMMaMH no k H HHTerpaJIaMH B KOHeqHbIX npOMe:m:yTKax,
He HcnOJIbSYH aCHMnTOTHqeCKOe paBeHcTBo (19.5) lI.JIH CBOpaqHBaHHH
MeJI.JIeHHO CXO.n.HW.HXCH qaCTeH pH.n.OB H HHTerpaJIOB.
o CTeneHH TOqHOCTH nOJIyqeHHOrO peweHHH KpaeBoll 3a.n.a'lH MO:m:HO
Cy.n.HTb no .n.aHHbIM Ta6,:IT. 20, B KOTOpOH npHBe.n.eHbI cYMMapHble HanpH-
:m:eHHH Ha c<pepHqeCKOH nOBepXHOCTH t = 0, COOTBeTCTBYIOIllHe rpaHHq-
HbIM YCJIOBHHM (19.5).
llaHHble Ta6JI. 20 YKasbIBalOT, qTO MaKCHMaJIbHaH norpemHocTb B
y.n.OBJIeTBOpeHHH rpaHHqHbIX YCJIOBHH .n.JIH HOpMaJIbHbIX HanpH:m:eHHH
Ha6JIIO.n.aeTcH B TOqKaX yr JIOBOH OKpy:m:HOCTH t = 0, tt =
 H no BeJIH-
qHHe He npeBocxo.n.HT 3,5% BeJIHqHHbI sa.n.aIiHbIX Ha c<pepHqeCKOH no-
nepxHocTH HOpMaJIbHhIX HanpmKeHHH 2
 a r = -po OTHOCHTeJIbHO Ka-


188


BHYTpeHHHM .n.aBJIeHHeM. B sa.n.aqe JIaMe BO Bcex TOqKaX Ha nOBepxHoc-
TH nOJIOCTH 2
 ai} = 2
 ac:p = 0,5 p. DpH tt = ° B HaH.n.eHHoM peme-
I I 77 u
HHH - ai} = - ac:p = 0,4 p, T. e. OTJIHqHe OT SHaqeHHH HanpH:m:e-
20 20
HHH B sa.n.aqe JIaMe HeSHaqHTeJIbHo. PaSJIHqHe SHaqeHHH HOpMaJIbHbIX
HanpH:m:eHHH B paCCMaTpHBaeMOH sa.n.aqe H B sa.n.aqe JIaMe YBeJIHqHBa-
eTCH no Mepe npH6JIH:m:eHHH K CB060.n.HOH rpaHHIJ.e nOJIynpoCTpaHCTBa.
Jt I
B TOqKaX 3KBaTopa (tt = 2) HanpH:m:eHHH ai} paBHbI HYJIIO, a 20 acp =
= 0,217 p, T. e. B .n.Ba pasa MeHbme, qeM .n.JIH saMKHYToH nOJIOCTH.
DpH BblqHCJIeHHH HanpH:m:eHHH B paccMoTpeHHoll sa.n.aqe no <POPMY-
JIaM (17.5) H (17.9) OrpaHHqHBaJIHCb KOHeqHbIMH oTpesKaMH pH.n.OB H
HHTerpaJIaMH B KOHeqHbIX npe.n.eJIax. OCHOBOH .n.JIH TaKoro yceqeHHH 6ec-
KOHeqHbIX pH.n.OB H Hec06cTBeHHblx HHTerpaJIOB nOCJIY*HJIa MaJIOCTb
npe.n.eJIbHoro SHaqeHHH K03Q:JQ:JHIJ.HeHTOB pH.n.OB H nJIOTHOCTeH HHTerpa-
JIOB. ECTecTBeHHo, qTO TaKoH no.n.xo.n. 060CHoBaH, eCJIH BOSHHKalOIllHe
B yr JIOBbIX TOqKaX B CBHSH C aCHMnTOTHqeCKHMH paBeHcTBaMH (18.19)
H (18.26) oc06eHHocTH B HanpH:m:eHHHx cQ:JepHqeCKOH H KOHHqeCKOH co-
CTaBJIHIOIllHX 06Illero peweHHH BsaHMHO norarnalOTCH. .llJIH 06w.ero CJIy-
qaH .n.eQ:JopMHpoBaHHH yceqeHHOrO KOHyca .n.OKaSaTeJIbCTBO 3Toro MO:m:HO
JIerKO nOJIyqHTb, npOaHaJIHSHpOBaB C 3TOH TOqKH speHHH caM npo-
IJ.ecc nOJIyqeHHH paBeHcTB (18.19) H (18.26). B cJIyqae nOJIynpocTpaH-
189




I
2Gp aq> = 1 + 2 (7m _ 5) r<13m - 20)e- 31 - 3mr 5t ],
2
P a, = 1 - 7m
 5 [(13m - 20) r JI - 6me-5 t ]
llJIH paccMaTpHBaeMoH 06JIaCTlI Bblpa>KeHHH (I 9.] 6) YllOBJIeTBOp51-
lOT HYJIeBbIM rpaHH4HbIM YCJIOBHHM Ha nOJIYCQ:Jep1I4eCKOM Bblpe3e II YC-
JIOBlIla paBem':TBa HYJIIO KaCaTeJIbHbIX Hanp51>KeHlIH Ha rpaHlIue nOJIY-
npOCTpaHCTBa, HO YCJIOBHe OTCYTCTBIfH HopMaJIbHbIX HanpH>KeHIfH Ha
rpaHHue nOJIynpoCTpaHCTBa OKa3bIBaeTCH He BbmOJIHeHHbIM. flo:noMY
Ha BTOpOM 3Tane peweHlIH 3a11a4If He06xo11IfMO HaHTH pacnpelleJIeHlIe
Hanp51>KeHHH B nOJIynpoCTpaHcTBe C nOJIyc<}JepH4ecKHM Bblpe30M npH
CJIe11YIOIll.IfX rpaHIf4HbIX YCJIOBlI51X:


CT
a C nOJIycQ:JeplIqeCKlIM BbIpe30M B3alIMHoe noraweHlIe oc06eHHOC-
Tell B HOpMaJIbHbIX HanpH>KeHlIHX YCTaHaBJIlIBaeTCH HlI>Ke Henocpe11-
CTBeHHbIM aHaJIIf30M Bblpa>KeHlIH 11JIH HanpH>KeHIfH.
3aiJa'la 2. K0!'lueHTpauIfH HanpH>KeHlIH Y nOJIycQ:JepIf4ecKoro BbIpe-
3a Ha cB06011HOIf rpaHlIue nOJIynpocTpaHcTBa (3a1!.a4a P 106aHKca
(208]) .
3a11aqa CTaBIfTCH CJIe11YIOIll.lIM 06pa30M: Ha 6ecKoHe4HocTlI 3a11aHO
nJIOCKOe ocecIfMMeTpIf4Hoe nOJIe pacTHrlIBalaIll.IfX HanpH>KeHIfH
I I
2G ax = 2G a y = p,


a z = 7: xy = 7: yz = 7: xz = O. 09.15)
a rpaHlfIla nOJIynpocTpaHcTBa it =
J1
=-2-' t,> 0 If nOBepxHocTb nOJIOCTH
t = O. 0 < it < -% cB06011HbI OT
HanpH>KeHIfH (pHC. 20).
KaK If B npe11bI11YIll.eM cJIyqae,
PHC. 20. peweHIfe CQ:JOPMYJIIfpOBaHHoH 3a-
11a41I CTpOlIM B 11Ba 3Tana.
Ha nepBOI\1 3Tane BOCnOJIb3yeMc51 If3BeCTHbIM [124] peweHIfeM 3a11a-
'Ill 0 KOHueHTpaUIfIf HanpH>KeHIfH OKOJIO cQ:Jep1I4ecKoH nOJIOCTlI B ynpy-
rOM npOCTpaHCTBe, HarpY>KeHHOM Ha 6eCKOHeqHOCTIf HanpH>KeHIfHMIf
(19.15). <l>OPMYJIbI 11JIH HanpH>KeHIfH Ha c<}JepIfqeCKOH nOJIOCTlI (t = 0)
If B 3KBaTOpIfaJIbHOH DJIOCKOCTII (it = .::) lIMelaT BlI11
2


npIf t = 0, 0 < it < ;


a, = 7:,fi = 0


(t = 0,


n
O<it<T)'


I 3m
a o ( r3t - 3e- ol ), 7:rI) = 0
2GTJ = - 2(7m-5)


(19.17)
(it =
 ' t'>O)


B npe11bIllYllleM npHMepe nOJIY4eHo peweHHe 3a11aqlI B CJIY4ae If3Me-
HeHlIH HOpMaJIbHbIX Hanp51>KeHHH Ha rpaHlflle nOJIynpocTpaHCTBa no 3a-
KOHY e-3 1 . CJIellOBaTeJIbHO, Tenepb 1l0CTaT04Ho HaHTH peweHIfe npIf
rpaHH4HbIX YCJIOBIfHX
I * _ -51 0 ( ,1>. _ n 0)
2GTJ afi - e , 7:,fi = u - 2 ' t>-'.
9m
flOCT051HHbIH MHO>KIfTeJIb 2 (7m-5) 6Y11eT y'neH B OKOH4aTeJIbHbIX
Q:JopMYJIaX npH BblqHCJIeHIfH HanpH>KeHlfI1.
PerneHHe rpaHWlHOH 3a11a4If (19.18) npIfBo11lITCH K CHCTeMe aJIre6pa-
lIqeCKIfX ypaBHeHHH (19.12) C TeM JIIfWb pa3JIIf4HeM, qTO Tenepb CBO-
6011HbIH 'weH B nOCJIe11HeM ypaBHeHIfIf If MeeT BIfll
I
'(2+_
7 4
2n 9
'{2+_
4


(19.18\


a, = 7:,fi = 0,


(19.19)


I 2 { I I
2Gp afi =""3 1 + 7m _ 5 (23m - 10) P2 (cos it) +
+ (4m - 5) ctg it

2 ]},

pafj)=+{2- 7m
5 [(8m-25)P 2 (COSit) +
+ (4m - 5) ctg it
2 ]};
npIf it =
 t"'- 0
2' <9'


P -'I.+iT (0)
(1) .
P _l/.+iT (0)


(19.16)


B pe3yJIbTaTe qIfCJIeHHOrO peweHHH CIfCTeMbl aJIre6paH4eCKIfX co.
oTHoweHHH (19.12) llJIH HeIf3BeCTHblX X2k, Y21l HZ ('t) nOJIyllIfM 3HalleHIfH,
npHBelleHHble B Ta6JI. 21.
(.1cnOJIb3YH HaH11eHHble 3HaqeHIfH HeIf3BeCTHbIX, 3Ha4eHIfH Hanp51>Ke.
HIfH B npOIf3BOJIbHOH T04Ke nOJIynpoCTpaHcTBa C nOJIycQ:JeplIllecKIfM Bbl-
pe30M Bbl4HCJIHeM no Q:JopMYJIaM (17.5) H (17.9). npIf 3TOM cYlllecTBeH-
HbIM OKa3bIBaeTC51 3HaHIfe npelleJIbHorO 3Ha4eI-lIf51 HeIf3BeCTHblX (19.5),
KOTopoe npH6JIH>KeHHO BbI6paHo paBHbIM Co = 0,07.
flpe>Klle Bcero 06paTIfMcH K Q:JopMYJIaM llJIH HanpH>KeHIfH Ha OC060H
:n:
JIlIHHlI rpaHHubl {) = 2"' t = O. B llaCTHOCTIf, npoBepHM YllOBJIeT-
BopeHHe rpaHIfllHbIX YCJIOBIfH 11JIH HopMaJIbHbIX HanpH>KeHIfH ao np"


I 3m
afi _ 1n-3t _ 3 n-5t ),
lGp - 2(7m-5) \<- <-


7:,fi = 0,


191


190




nOJ{xoJ{e K yrJIOBbIM TOqKaM BJ{OJIb nOBepXHOCTU nOJIyn OCTpaHCTBa
(t = 10 -- 00). ITo tPopMYJIaM (17.5) U (17.9) C yqeTOM 3aMeH J{JIH He-
U3BeCTHbIX (19.2) nOJIyqUM
e 3 f.E /I.E x [ 1 (
2G at) = 2G (a
) + a
» = T - tt (1 + 2k r h ll X 2k +
+ h 12 k

 ) + "f 2k (h 21 X 2k + h 22 k
2
 )]-
2 2
co 40
( 2'
-Co
 T-4e)e-2k8+2 \
k=2I ii


YKa3bIBaeT, qTO OTBeqalOw:aH
ctPepUqeCKOH qaCTU o6w:ero pe-
meHUH COCTaBJIHIOw:aH B Ha-
npmKeHuHx afJ HMeeT B yrJIo-
BOH TOqKe oco6eHHoCTb. AHa-
JIOrHqHYIO oco6eHHOCTb, KaK CJIeJ{yeT U3 paBeHcTBa


00
S COS 't8
't d't = - In 210 - In 20 - 0,5772 '" + 0 (e),
40
UMeeT KOHHqeCKaH COCTaBJIHIOw:aH o6w:ero pemeHuH. OqeBHJ{Ho. qTO
B UeJIOM B tPopMYJIe (19.20) 9TII oco6eHHocTH B3aUMHO noramalOTCH.
CJIeJ{yeT OTMeTHTb, qTO B CHJIY paBeHcTBa
00 I
Hm 10
 e- 2kE = - (19.23)
&+0 k=I 2


00


+ 2C o .r
40
Ta6JIHu;a 21


k I x2k I 1/2k I I z (1;1
0 0,1795 0 0,048 0,0603
1 0,0110 0,0300 0,253 0,0618
2 0,0464 0,0585 0,617 0,0687
3 0,0612 0,0709 1,136 0,0844
4 0,0680 0,0770 1,800 0,1050
5 0,0712 0,0795 2,599 0,1209
6 0,0726 0,0804 3,519 0,1263
7 0,0729 0,0803 4,546 0,1229
8 0,0728 0,0796 5,662 0,1148
9 0,0724 0,0788 6,849 0,1054
IO 0,0718 0,0778 8,089 0,0965
11 0,0711 0,0768 9,359 0,0886
12 0,0704 0,0759 10,64 0,0820
13 0,0697 0,0748 11,91 0,0767
14 0,0690 0,0740 13,15 0,0724
15 0,0684 0,0731 14,34 0,0691
>16 0,07 0,07
15 0,07


f92


p(1) (
-1/.+[1: 0)
P -'/.+i1: (0)


B paccMaTpuBaeMoM cJIyqae yceqeHHoro KOHyca B o6w:ux BbIpameHuHx
J{JIH HanpHmeHUH TaKme npHcYTcTBYIOT COCTaBJIHIOw:ue. KOTopbIe He MO-
rYT 6bITb yqTeHbI npu UCnOJIb30BaHUU J{JIH pemeHUH CHCTeMbI (19.3)
MeTOJ{a npoCToH peJ{YKuuu. AHaJIOrHqHOe nOJIOmeHue U y J{pyrHx KOM-
nOHeHTax TeH30pa HanpHmeHHH Ha OC060H JIUHUU rpaHuUbI.
B Ta6JI. 22 npHBeJ{eHo pacnpeJ{eJIeHHe HanpHmeHuH Ha nOBepxHoc-
TU ctPepHqeCKoro BbIpe3a. KOTopoe cooTBeTCTByeT rpaHUqHbIM YCJIOBHHM
(19.15) npH 3HaqeHUH qHCJIa ITyaccoHa m = 4. ITepBbIe J{Be CTpOKU 3TOH
Ta6JIHu;a 22


III
II


Z (T)
1
T2+_
4


d't +


t)
(J I 0.2
 I 04
 I 06
 I O,8
 I n
0 ""2
2 ' 2 ' 2 2
(Jr/ 2G p 0,0003 0,0002 0,0001 0,0000 -0,0002 -0,0018
T r t)/2Gp 0,0000 -0,0001 0,0000 0,0000 0,0000 0,0000
(Jt)/2Gp 2,088 1,844 1,231 0,506 0,014 0,0040
(J ",/2Gp 2,088 2,026 1,875 1,700 1,598 1,676


COS't8
't


d't.


(19.20)


AHaJIH3 nOJIyqeHHoro BbIpa-
meHUH J{JIH at) nOKa3bIBaeT, qTO
B cJIyqae yceqeHHoro KOHyca
UMelOT MeCTO Bce OTMeqaBmH-
eCH B paHee paccMoTpeHHbIx
3aJ{aqax oco6eHHocTH CXOJ{H-
MOCTU pHJ{OB U HHTerpaJIOB B
tPopMYJIaX J{JIH HanpHmeHUH.
T aK, 3HaqeHHe CYMMbI 6ecKo-
HeQHoro pHJ{a


Ta6JIuUbI YKa3bIBalOT Ha J{ocTaTOqHO BbICOKYIO TOqHOCTb YJ{OBJIeTBOpe-
HUH rpaHUqHbI \ YCJIOBHH, a J{Be J{pyrHe JlalOT pacnpe}leJIeHHe HanpHme-
HUH at} U a", Ha nOBepxHocTH BbIpe3a.
MaKCHMaJIbHOrO 3HaqeHHH HanpHmeHHH at} H a", J{ocTHralOT B TOqKe
i} = 0 Ha nOBepxHocTH BbIpe3a, rJ{e
I 1
2Gp at} = 2Gp a", = 2,088. (19.24)
lI.JIH cpaBHeHuH YKameM, qTO B cJIyqae 3aMKHYToH c<PepUqeCKOH nOJIOC-
TH MaKCHMaJIbHOe 3HaqeHHe 3TUX HanpHmeHHH B nOJIlOcax i} = 0,
t} = 1t paBHo


00

 - :_e- 2kB = _ 21n (1- e- 2E )
k=l


(19.21)


I 1
2Gp a[J = 2Gp a", = 2,087.


09.25)


(19.22)


Ha OCHOBe conOCTaBJIeHHH 3TUX 3HaqeHUH npHxoJ{HM K BbIBOJ{y, qTO pa3-
JIuque B K03tPtPuUHeHTax KOHueHTpauuH J{JIH 3aMKHYToHctPepHqeCKOH
nOJIOCTH U nOJIyctPepHQeCKoro BbIpe3a COCTaBJIHeT Bcero 0,05%.
P. 106aHKc [2081, KOTOpbIH BnepBbIe paccMaTpHBaJI 3TY 3aJ{aqy,
npl1IueJI K OUm60qHbIM pe3YJIbTaTaM. Y Hero MaKCHMaJIbHbIe HanpHme-
HUH 2
P at} = 2
P a", = 2.23. Y1cTOqHHK 3TOH omH6KH, no BceH BHJ{H-
MOCTU, COCTOHT B TOM, qTO npu pemeHUH 6eCKOHeQHOH CUCTeMbI aJIre-
6paHqeCKHX ypaBHeHHH, nOJIyqeHHOH Ha OCHOBe MeTOJ{a OJ{HOpOJ{HWX
pemeHUH, OH OrpaHHqHJICH KOHeqHOH CHCTeMOH H3 26 ypaBHeHuH H He
npOU3BeJI aCHMnTOTHqeCKOrO HCCJIeJ{OBaHHH nOBe.n.eHUH oT6pomeHHbIX
HeH3BeCTHbIX. BCJIeJ{CTBue 3Toro, CJ{eJIaHHbIe P. lO6aHKcoM BbIBOJ{bI H3
conOCTaBJIeHUH pemeHHH J{aHHOH 3aJ{aqU C pemeHHeM P. MaYHCeJIJIa
[2361 J{JIH nOJIynJIOCKOCTU G BbIpe30M no nOJIYKpyry TaKme HBJIHIOTC5I
OmU60qHbIMU.


13 7-2
6





r.ftaea 5
HEOCECIIMMETPIPIHblE rp AHllqHblE 3AJJ:A qlI
)],JIH CJIOH C RPyrOBbIM OTBEPCTIIEM


HOM COCTOHHlUI O)l.HOOCHO paCTflrImaeMOH ynpyroH nJIOCKOCTIi, oCJIao-
JIeHHOH KpyroB
M oTBepcTIieM, KOTopaH HBIiJIaCb Ha
aJIOM HOBoro Ha-
npaBJIeHIiH B MexaHIiKe TBep)l.OrO )l.e<P0pMIipyeMoro TeJIa, CBH3aHHoro C
Ii3y
eHIieM KOHueHTpaI!IiIi HanpmKeHIiH. 3a oOJIee
eM nOJIYBeKoBoH npo-
MeMYToK BpeMeHII 3TO HanpaBJIeHIie O<P0pMIIJIOCb B caMoCTOflTeJIbHbIH
pa3)1.eJI Teopllll ynpyrocTIi, xapaKTepH
H KaK nocTaHoBKoH 3a)l.a
, TaK
Ii cneuIi<pIiKoH MeTO)l.OB IiX perneHIiH. OOCTOHTeJIbH
H 0030p pe3yJIbTa-
TOB, nOJIy
eHH
x B 3TOM pa3)1.eJIe MexaHIiKIi, MOMHO HaHTIi B paooTax
[157, 244].
DOJIbrnoe
IICJIO IICCJIe)l.OBaHIiH no KOHueHTpaUIiIi HanpflMeHIiH B
ynpyrIix TeJIaX B
nOJIHeHO Ha OCHOBe ynporualOruHx rIinOTe3 Teoplllf
H oOOJIO
eK. 4TO KacaeTCH OCHOBHoro HanpHMeHHoro COCTOHHIiH, TO
TaKOH no)l.XO)l. 3a
aCTYIO He BHOCHT cyruecTBeHHbIX norpernHOCTeH B
3Ha
eHIiH HanpHMeHHH. O)l.HaKO ero IICnOJIb30BaHIie npH ornCKaHIiIl
B03MyrueHHoro nOJIH HanpHMeHIIH, KOTopoe xapaKTepH3yeTcfI OOJIbrnOH
H3MeHHeMOCTblO, He )l.OCTaTO
HO OOOCHOBaHO Ii MOMeT npIiBeCTIi K 3Ha-

IiTeJIbHbIM norpernHOCTHM. I1HTepecHO OTMeTIITb, qTO HeOOXO.iI.HMOCTb
OOJIee ToqHOrO nO)l.XO)l.a K H3y
eHIiIO HanpHMeHHoro COCTOflHHfI UOJIH3H
OTBepCTIIH CJIe)l.yeT Ii3 aHaJIIi3a caMaro npIiOJIIiMeHHoro perneHHfI [172].
B nOCJIe)l.Hlle ro)l.
 Ka
eCTBeHHble IiCCJIe)l.OBaHIiH xapaKTepa Hanpfl-
MeHHH BOJIlI311 oTBepCTHH [10,57] nOKa3aJIIi,
TO MeM)l.y perneHIiHMH no
npIiKJIa)l.H
M TeOpIiflM H perneHlIeM npoCTpaHcTBeHHoH 3a)l.a
Ii TeopIiIi
ynpyrocTH HeT aCIiMnTOTHqeCKOrO COOTBeTCTBHfI. 9TO OOCTOflTeJIbCTBO
C)l.eJIaJIO ell!e OOJIee HeOOXO)l.HM
M nOJIy
eHlIe KOJIH
eCTBeHH
X oueHOK
TO
HOCTH npHOJIHMeHH
x perneHHH XOTfI ObI )l.JIfI HeKOTOp
X TIinIi
HbIX
3a)l.a
 0 KOHueHTpaUHIi HanpHMeHIiH. TaKHMIi 3a)l.a
aMH H HBJIflIOTCfI,
B qaCTHOCTH, paCCMaTpHBaeMbIe 3)1.eCb 3a)l.aqa KHprna H 3a)l.a
a UIi-
JIIiH)l.pH
eCKoro H3rHoa CJIOH C OTBepCTIieM.
PaCCMOTpeHHIO YKa3aHHbIX B
rne 3a)l.a
 0 KOHueHTpaU IiIi HanpflMeHHH
nocBflrueHO cpaBHHTeJIbHO HeOOJIbrnOe KOJIH
eCTBO paOOT. XpOHOJIOrIi-

eCKH nepBOH HBJIHeTCH paOOTa A. E. fpIiHa [215], nocBHrueHHaH 3a)l.a-

e KIiprna B TpexMepHoH nOCTaHOBKe. B HeH nOCTpoeHo )l.OCTaTO
HO
ooruee perneHHe rpaHH
HoH 3a)l.a
Ii TeopIiIi ynpyrocTH, nO)l.
HHeHIie KO-
Toporo rpaHH
HbIM YCJIOBIiflM npIiBO)l.IiT K oeCKOHe
HbIM CHCTeMaM
ypaBHeHHH. O)l.HaKO npIi npaKTHqeCKIiX B

HCJIeHIiHX A. E. fpIiH or-
paHHqHJICfI JIIirnb rneCTblO rapMoHHKaMIi no TOJIruHHe nJIHrn. DpHBe)l.eH-
HbIe HM oueHKH TO
HOCTIi y)l.OBJIeTBOpeHHfI rpaHIi
H
M yCJIOBHflM )l.JIfI
HOpMaJIbHbIX K rpaHHqHblM nJIOCKOCTflM CJIOH HanpHMeHHH cKopee IiJI-
JIIOCTpHpylOT TO
HOCTb B

IiCJIeHHfI HeKoTop
X IiHTerpaJIOB,
eM
Ka
eCTBO perneHIiH rpaHH
HoH 3a)l.a
Ii. Do)l.xo)l. A. E. fpHHa 6
JI pa3-
BHT, cY)l.fI no yKa3aHHflM pfl)l.a aBTOpoB [172, 278], B paooTe Ll. B. AJIo-
JIaca, C KOTOpOH HaM He y)l.aJIOCb n03HaKOMIiTbCfI.
9HepreTlI
eCKIiH MeTO)l. perneHHfI TpeXMepHOH 3a)l.a
H KIiprna npe)l.-
JIaraJICfI B paooTe 9. llhepHoepra H M. A. Ca)l.OBcKoro [266], O)l.HaKO
nOJIy
eHH
e KOJIH
eCTBeHH
e pe3yJIbTaTbI, no npIi3HaHHIO O)l.HOro H3
aBTopoB [263], HeHa)l.eMH
. Dpe)l.JIaraBrnIiHcfI 9. lllTepHoeproM H
P. MYKIi [264] no)l.XO)l. K YKa3aHHoH 3a)l.a
e, oCHoBaHH
H Ha pa3JIO-
MeHHH HCKOMoro perneHIifl B pfl)l. no CTeneHflM K03<P<pIiUHeHTa Dyacco-


\ I :
!I


r JIa

 C
:
 %;H



 MeTO)l., C nOMorublO KOToporo B npe)l.
)l.YruHx
3a)l.a
 TeopHH ynp
rocnI:
rn
:::a

:
H
eOCKHX II ocecHMMeTpH
H
X
3aHH
X
aCTH
X pe npHro.zr.eH KaK )l.JIfI YKa-
cJIy
afl oorueH Heo




e:e


:HHpOBaHHfI ynpymx TeJI, TaK H )l.JIfI
xapaKTepa <p H 3HqeCKHX nOJI
H cMe

oCMaullll. O)l.HaK
 YCJIOMHeHHe

aHX oTpaMaeTcH B YCJIOMHeHHIi CT H HanpflMeHHH B 3THX CJIY-
H
X 3a)l.a
 H cnocoooB nO)l.
IIHeHIi:


rp p
e
orull
 perneHHH rpaHH
-
HH
H
M YCJIOBHflM. eHIiIi KOHKpeTH
M rpa-
B 3TOH rJIaBe paccMoTpIi
:O=M

t


::/7fJ;

O

;
:;:;J
?





o



:




=




 p

)l.a
y 0 UHJIHH)l.pH
ecKoM H3rHoe CJI

CT



:=

 Pa


::=:
:
)I.
BalOruHM nocTaHoBKy 3THX 3a)l.a
 B aM u '
flBJIfleTCfI YCTaHOBJIeHHafl Ii n OHJIJIIOCT p Kax )l.aHHOIi paoorn,
Bax cnOCOOHOCTb pa3BHBaeMcfro MeTon!H y P JI O a B B aHHafl B npe
)l.yruHx rJIa-
KaJI <P<P '" JIHBaTb caMble TOHKlle JIO-


::=:
3


: p

rn
i






=JI=
;:oH

:B

:e

=;e
;:
:







u HfI YKa3aHHbiX 3a)l.a
 0 KOHueHTpaUIiIi Han fI

IiIi, npIi paCCMOTpeHIiH KOTOpbIX, no MeTKOMY 3aMe
aHIiIO 9 uf Me-
epra, «ynycTHTb H3 BIi)l.a MeCTH
e 3<p<peKT
 _ no Bce a . TepH-
THTb H3 BH)l.a cyruecTBo 3a)l.a
» [263, c. 11. P BHO qTO onyc-
JIeT
O



:e






Tf
;

:
a



2T

e

OKOMnOHeHTHOey)l.OB-
yc
oMHeHHHM COOTHorneHIiH )l.JIfI npOH3BOJIbHbIX


:
:H:

3
HrnHHIiM

IiIi B oorueM perneHHH u rpaHIiqHOH 3a)l.a
lI. E)l.IiH
H no)l.XO)l. K yn

ru


K
:n
:




T:e



:IIH



:


)I.:(::K



)l.e:::;
H
OM:aIiH:UIiH Ii3
TeJIa COCTaBJIHeT CyruHOCTb MeTO)l.a coocTBeHH
x BeKTOP

X <pf::

:



flP;
=IiH 3a)l.aq TeopMH ynpyrocTH. PaccMaTpHBaeMbIe 3a)l.aqM )l.JIfI


II:o
rnH 3::0




b n


:=


Iin
;::===


Tco;
B:
T

:par;;:




bKO npll perneHIiIl HeocecMMMeTpH
HbIX 3a)l.aq )l.JIfI U HJIIi H n pa
qTO OqeBM)l.HO HO II n p H. '" ,
, paccMoTpeIlIiIi HeocecMMMeTpllqHbIX 3a)l.aq )l.JIfI
TeJI
{oHe
H
x pa3MepoB B c<pepH
ecKllx KOOp)l.HHaTax
KOHue XIX CT. r. KlIprn 1230] onYOJIHKOBaJI pao
TY 0 HanpflMeH-
194


13*


195




Ha, BrrOCJIe
CTBlm rrplf3HaH aBTopaMlf HerrpaKTlf'lHbIM [263]. B pH
e pa-
60T aHaJIOflf'lHaH 3a
a'la 06 lf3rI:16e paCCMaTpI:lBaeTCH Ha OCHOBe
Y'lI:lTbIBaIOII1.eM C
Blff TeOplfI:l PeMCHepa I:I C rrOMOII1.bIO MeTo
a pa3JI01Ke-
HI:IH rro TOJIII1.lfHe [251, 253 I:I
p.]. OCHOBHOM He
OCTaTOK 3TI:IX pa60T
c.oCTOI:lT B OTCYTCTBI:Ilf oueHOK TO'lHOCTI:Ilf rrpe
eJIOB rrplfMeHI:IMOCTI:I rroJIY-
'"IeHHbIX pe3YJIbTaTOB.
113 lfCCJIe
OBaHlfM, OTHOCHII1.I:IXCH K paCCMaTpI:lBaeMbIM 3a.n:a'laM, oco-
60 CJIe
yeT BbI
eJIlfTb pa60TY [278], B KOTOpOM I:I3Y'laeTCH HarrpH1KeHHOe
COCTOHHI:Ie rroJIyrrpocTpaHcTBa C UI:IJII:IH
plf'leCKlfM oTBepCTI:IeM, BbI3-
BaHHoe 'lI:lCTbIM C
BlfroM Ha 6eCKOHe'lHOcTlf. B 3TOM pa60Te rryreM aHa-
JIlf3a 'lI:lCJIeHHbIX pe3YJIbTaTOB perneHlfH lfHTerpaJIbHOrO ypaBHeHI:IH,
B03HlfKaIOII1.erO rrplf y
OBJIeTBOpeHI:Ilf rpaHlf'IHbIX YCJIOBlfM, YCTaHOBJIe-
HbI aCI:IMrrTOTI:I'IeCKI:Ie CBoMcTBa lfCKOMOM <PYHKUlflf. ITplf
OCTaTO'lHO
TO'lHOM orrpe
eJIeHI:Ilf rrpe.n:eJIbHOrO 3Ha'leHlfH YKa3aHHoM <PYHKUI:Ilf CJIe-

YeT 01Klf
aTb BbICOKOM TO'lHOCTI:I Y
OBJIeTBOpeHlfH rpaHI:I'lHbIX YCJIOBI:IM
Ha BceH rroBepxHocTli yrrpyroro TeJIa, BKJIIO'IaH If yrJIOBble TO'lKI:I.
CpaBHeHI:Ie 'lI:lCJIeHHbIX
aHHbIX pa60TbI [2781 C rrOJIY'leHHbIMlf HaMI:I
fipOBO
I:ITCH HI:I1Ke. 3
ecb JII:Irnb CJIe
yeT oTMeTlfTb, 'lTO aBTopbI YKa3aH-
ROH pa60TbI He rrpOBeJII:I .n:OCTaTO'IHO TII1.aTeJIbHbIM aHaJII:I3 BbIpa1KeHI:IH
J{JIH HarrpH1KeHI:IH, B pe3YJIbTaTe 'lero B HeH CO
ep1KlfTCH HeBepHoe YT-
sep 1K .n:eHI:Ie 0 pa3pbIBHOCTI:I <PYHKUI:IH HOpMaJIbHbIX HarrpH1KeHI:IH B yrJIO-
BOH TO'lKe.
DOJIbrnI:le B03M01KHOCTI:I .n:JIH I:ICCJIe.n:OBaHI:IH KOHueHTpaUlflf HarrpH-
>KeHlfH B
OCTaTO'lHO TOJICTbIX rrJII:ITaX rrpe
CTaBJIHeT MeTO
 O
HOpO
HbIX
perneHI:IH. CYII1.eCTBeHHbIe pe3YJIbTaThI C lfCrrOJIb30BaHlfeM 3Toro MeTo
a

OJIY'IeHbI B pa60Te [13], r
e, B 'laCTHOCTlf, rro.n:P06HO paccMoTpeHa 3a-

a'la 0 UlfJIlfH
plf'IeCKOM I:I3rI:l6e CJIOH C oTBepcTlfeM.


B COOTBeTCTBI:Ilf C MeTo.n:OM c06cTBeHHbIx BeKTopHbIX <PY
KUlfH pe
:-
HI:IH 3a
a'l Teoplflf yrrpyrocTlf [1801, BeKTop rrepeMeII1.eHlfI:l TO'leK -
oceclfMMeTplf'lHO-
e<p0pMlfpoBaHHoro CJIOH rrpe.n:CTaBJIHeTCH B Bli.n:e
u (r, z, 8) = ;:n;
 I [W k (A, z) L
) (r. 8) + Uk (A, z) M
) (r, 8) -f--
k=-oo 0
+ v k (A, z) N
) (r, 8)] AdA. (20.2)
lh06bI rrpe
CTaBJIeHHbIH TaKlfM 06pa30M BeKTop rrepeMeII1.eHlfH y
OB-
JIeTBOpHJI ypaBHeHI:IHM JIaMe, CJIe.n:yeT cooTBeTCTB
HHO orrpe

JIlITb
rro
bIHTerpaJIbHbIe <PYHKUlflf. qTO KacaeTCH <PYHKUlfI:l TOJIII1.I:IHHOlf KO-
op
lfHaTbI Z, TO lfX CJIe.n:yeT BbI6I:1paTb CJIe.n:YIOII1.I:IM 06pa30M:
w k (A, z) = Ak (A) e'A.z + Bk (A) e- M +C k (A) ze'A.z + Dk (A) ze-'A.Z',
A 2 U k (A, z) = AAk (A) e'A.z - ABk (A) e-'A.z +
+Cd A ) ( 3m;;;4 +Az)e'A.z+Dk(A)( 3m;;;4 _AZ)e-'A.Z, (20.3)
A 2 V k (A, z) = M k (A) e'A.z + N k (A) e-'A.z,
r
e Ak (A), Bk (A), C k (A), Dk (A), M k (A) I:I N k (A) (k = 0, ::i:: J.
::t: 2, ...) - rrpOlf3BOJIbHbIe <PYHKUI:II:I rrapaMeTpa A.
CI:ICTeMa c06CTBeHHbIX BeKTopHbIX <PYHKUI:IH B UI:IJII:IH
pI:l'leCKlfX KO-
op
I:IHaTaX orrpe.n:eJIHeTCH rro <POPMY JIaM
L
) (r, 8) = kS
) (r, 8),
iJSf;) ee iJS<{:)
M
) (r, 8) = grad S
) (r, 8) = er---aT + r (jfJ , (20.4)
iJS(k) iJS(k)
k (k) e, "- "-
Nk) (r, 8) = mt[kS" (r, 8)] = r as- -ee----ar '
k e e - O p TbI cooTBeTcTByIOII1.I:IX oceii UI:IJII:IH.n:pI:l'leCKOH clfcTe7"
r
e , T' e (k)
MbI KOOp
I:IHaT a CKaJIHpHbIe <PYHKUI:II:I S,,- (r, 8) HBJIHIOTCH perneHI:IeM
ypaBHeHI:IH re'.nbMrOJIbUa Ha rrJIOCKOCTI:I (1 .:::; r < 00, 0 -< e -< 2n}:
iJ2S 1 as +
 a 2 s + A2S = O. (20.5)
iJr 2 + r ar r 2 iJ6 2
Ha
JIelKaII1.lfM BbI60pOM perneHlfH ypaBHeHI:IH reJIbl\lrOJIbua (20.5)
M01KHO
06I:1TbCH TOro, 'lTO BeKTopHoe cooTHOrneHI:I
 (20.2) If BbITeKaIO-
II1.ee I:I3 Hero rrpe
CTaBJIeHI:Ie BeKTopa HarrpH1KeHI:II:I Ha rrOBepXHOCTHX
CJIOH z = ::t:h
orrYCKaIOT rrpOCToe 06paII1.eHlfe rro.n:06HO aHaJIOrI:l'lHbIM
COOTHorneHlfHM B CKaJIHpHbIX 3a
a'lax.
PerneHI:Ie ypaBHeHlfH reJIbMrOJIbua (20.5) CTpOI:lTCH MeTO

M pa3
e-
JIeHlfH rrepeMeHHblx. Y'lI:lTbIBaH cBoHCTBa 06JIaCTlf, B KOTOpOI:l 3TO pe-
rneHI:Ie I:III1.eTCH, rrpe
CTaBlfM ero B CJIe
YIOII1.eM BI:I.n:e: _
S
) (r, 8) = [ak (A)J k (Ar) - b k (A) Y k (Ar)] e tke . (20.6)
3.n:ecb ak (A) I:I b k (A) - rrpOI:l3BOJIbHbIe <PYHKUllI:l.
197



 20. Co6cTBeHHbIe BeKTOpHbIe .pYHKII;HH B 3ana'le
o HanpHmeHHOM COCTOHHHII
HeocecHMMeTpll'lHO- Ae.popMllpoBaHHol'o CJIOH
OCHoBHoe BHlfMaHlfe y
eJIlfM paccMoTpeHlfIO 3a
a'l 0 HeoceClfMMeT-
pI:l'lHOH
e<p0pMaUI:Ilf yrrpyroro CJIOH C UI:IJIlfH
pH'IeCKI:IM oTBepCTI:IeM.
CorJIaCHO nplfHHTOMY MeTo.n:y rroCTpoeHlfH o6II1.ero perneHHH rpaHlf'l-
HbIX 3a
a'l .n:JIH TeJI KOHe'lHbIX pa3MepoB, O
HlfM H3 ero 3JIeMeH-
TOB B 3TI:IX CJIY'laHX
OJI1KHO 6bITb TaKoe perneHI:Ie ypaBHeHlfH JIaMe,
KOTopoe co.n:eplKlfT .n:OCTaTO'IHYIO CTerreHb rrpOlf3BOJIa
JIH y
OBJIeTBOpe-
fiI:lH rpaHlf'IHbIX YCJIOBlfH Ha rrJIOCKI:IX rpaHI:IuaX CJIOH BHe oTBepcTHH (CM.
pI:lc. 18). Tpe60BaHHe rrOJIHOTbI y
OBJIeTBOpHeTcH, eCTecTBeHHo, H 06II1.I:IM
perneHlfeM
JIH CnJIOWHOrO CJIOH, O
HaKO I:ICrrOJIb30BaHHe ero YCJIOIKHH-
eT 3a
a'lY y.n:oBJIeTBOpeHI:IH rpaHlf'lHbIM YCJIOBI:IHM Ha UI:IJIlfH
plf'leCKOH
IJOBepXHOCTI:I P = Po.
nOCJIe
YIOII1.He BbIKJIa.n:Kli rrp, OBe.n:eM BBe.n:H 6e3pa3MepHbIe UI:IJIlfHk
pI:l'leCKI:Ie KOOp
I:IHaTbI:


r = p/Po, Z =
/po, h = H/po
(1 ,:::;r<CXJ, -h':::;z':::;h).
196


(20.1)


II
I





nOJIHOTa Ha60pa qaCTHbIX peIlIeHHH (20.6) B OTHOIlIeHHH yrJIOBOH
KOOp.n.HHaTbI e CJIe.n.yeT H3 H3BeCTHbIX CBOHCTB KOMnJIeKCHOro p5l.n.a
<PYPbe. 4TO KaCaeTC5I KOM6HHaL{HH <lJYHKL{HH EecceJI5I B (20.6), TO no
HeH MOLKHO npe.n.cTaBHTb npOH3BOJIbHYIO <lJYHKL{HIO pa.n.HaJIbHOH KO-
op.n.HHaTbI r, eCJIH nOJIOLKHTb
a k (A) = sin exAY
 (A) + (cosex + + sin ex) Y k (A),
b k (A) =sin exAJ
 (A) + (cos ex + + sin ex) J k (A).
npH 3TOM JI.JIH npOH3BOJIbHoro ex cnpaBe.n.JIHBbI CJIe.n.YIOIL{He <lJOPMYJIbI
()6061L{eHHoro HHTerpaJIbHOrO npe06pa30BaHHH Be6epa [173]:


(20.7)


napy <lJOPMYJI (20.2) H (20.10) eCTeCTBeHHO Ha3BaTb BeKTopHbIM npe-
06pa30BaHHeM <Pypbe - Be6epa. CnpaBe.n.JIHBOCTb <lJOPMYJIbI 06palL{e-
HHH (20.10) MOLKeT 6bITb .n.OKa3aHa Henocpe.n.cTBeHHoH npoBepKoH co
rJIaCHO reopeMe pa3JIOLKeHHH <PYPbe H COOTHOIlIeHHHM (20.8). B caMOM
.n.eJIe, OTHOCHTeJlbHO HopMaJlbHoro KOMnOHeHTa BeKTOpa CMelL{eHHH <lJop-
MYJIbI (20.2) H (20.10) HBJIHIOTC5I npHMbIM CJIe.n.CTBHeM <lJOPMYJI (20.8)
.H cooTBeTcTBYlOlL{HX COOTHoIlIeHHH TeopHH KOMnJIeKCHbIX pH.n.OB <PYPbe.
n03TOMY npH .n.OKa3aTeJlbCTBe cnpaBe.n.JIHBOCTH paBeHcTBa (20.10) .n.u-
cTaTOqHO OrpaHHqHTbCH paccMoTpeHHeM KOMnOHeHTOB BeKTopa ClVIe
lL{eHHH JIeLKalL{HX B nJIOCKOCTH z = const.
VI.c
OJIb3YH peKyppeHTHbIe <lJOPMYJIbI .n.JI5I <lJYHKL{HH EeCCeJIH [6
]
dZk (Ar) A Z
-dr- = ""2 [ k-l (Ar) - Zk-j-I (Ar)],
k A
-,,-Zk (Ar) = ""2 [Zk-l (Ar) + Zk+l (Ar)],


(20.1 i)


Ip (A) = Sf (r)[J k (Ar) a k (A) - YdAr) b k (A)] rdr,
o


fer) = r Ip(A) Jk('Art a...k(
) -Yk(ArLbk('A) AdA
J [uk (A)]2 + [b k (A)J2 .
o
TIpH .n.OKa3aTeJIbCTBe npe.n.nOJIaraeTC51, qTO f (r) 51BJIHeTC5I <lJYHKL{HeH
OrpaHHqeHHOH BapHaL{HH B JII060M KOHeqHOM npOMeLKYTKe (1, R) H
"" dr
S If (r) I Yr < 00. (20.9)
o
YKa3aHHbIH BbIllie npOH3BOJI B BbI60pe 3HaqeHHH ex B HeKOTOpOM CMbIC-
JIe 3KBHBaJIeHTeH TOMY, KOTOpbIH OTMeqaJICH npH nOCTpoeHHH 061L{HX
peIlIeHHH paCCMOTpeHHbIX paHee 3a.n.aq. TaM OH B03HHKaJI B CB5I3H C
B03M
LKHOCTblO pa3JIHqHOrO npo.n.oJILKeHH5I 3a.n.aHHbIx rpaHHqHbIX yc-
JIOBHH Ha BCIO KOOP.n.HHaTHYIO nOBepxHocTb. KOHKpeTH3aL{H5I <lJOpMbI 06-
ll(ero peIlIeHH5I rpaHHqHOH 3a.n.aqH oCYIlleCTBJIHJIaCb npH 3TOM B COOTBeT-
CTBHH C xapaKTepoM rpaHHqHbIX YCJIOBHH (BbInOJIHeHbI HJIH HeT YCJIOBH5I
napHocTH KaCaTeJIbHbIX Hanp5lLKeHHH B yrJIOBbIX TOqKax) H Tpe60BaHH5I-
MH npoCTOTbI <lJYHKL{HOHaJIbHbIX ypaBHeHHH, nOJIyqeHHbIX npH y.n.OB-
JIeTBOpeHHH rpaHHqHWX YCJIOBHH.
npH paccMoTpeHHH KOHKpeTHbIx 3a.n.aq HCnOJIb3yeM B03MOLKHOCTb
npOH3BOJIbHoro BbI60pa ex B (20.7). 3.n.ecb LKe npH nocTpoeHHH <lJOPMYJIbI
()6paIlleHH5I npe.n.cTaBJIeHH5I (20.2) 6y.n.eM opHeHTHpoBaTbcH Ha peIlIeHHe
ypaBHeHH5I feJIbMrOJIbL{a (20.6), B KOTOpOM ex = o.
np e .n.noJIara5l, qTO CHCTeMbI BeKTopHbIX <lJYHKL{HH Lf'), M
) H N
> B
(20.2) 06pa3YIOT nOJIHbIe CHCTeMbI B TIJlOCKOCTH 1 -< r < 00, 0 -< e -<
-< 2n, H HCXO.n.51 H3 Toro, qTO e.n.HHcTBeHHbIH KOMnOHeHT BeKTopa L
k)
06pa3yeTc5I C nOMOlL{blO c06cTBeHHbIx <lJYHKL{HH CKaJI5IpHOH 3a.n.aqH,
MOLKeM 3anHcaTb


(20.8)


r.n.e Zk (Ar) - npOH3BOJIbHaH <lJYHKL{H5I EecceJI5I BeIII.eCTBeHHoro ap-
rYMeHTa, H3 06m.ero npe.n.cTaBJIeHHH BeKTopa CMeIII.eHHH (20.2) Haxo.n.HM
u r + ius = 2

 J N k+l (Ar) A [- Uk (A, z) + iV k (A, z)] eiksAdA,
11=-""0


(20.12)


00 00
u r - ius = 21:rr,

 N k - l (Ar) A [Uk (A. z) + iv" (A, Z)J eiksAdA.
11=-""0
3.n.ecb .n.JI5I COKpalL{eHH5I 3anHCH 0603HaqeHO
Nk:tl (Ar) = Jk:tI (Ar) Yk(A) - Yk:tI (Ar) Jk(A). (20.13)
PaccMaTpHBaH paBeHcTBa (20.8) B qaCTHbIX CJlyqa5lx ctg ex = + k +
I
+ 2' nOJIyqaeM


""


Ip(A) = S f(r) Nk:tl (Ar)rdr,
1


(20.14)


00 N ( Ar )
- r ( A k:tl AdA
f (r) - J Ip ) y2 (A) + J2 (A) .
o k k


{[JdA)]2 + [Yk (A)]2} X {A 2 [e,uk (A, z) + esv k (A, z)] + kWh (A, z») =


TIoCJIe.n.H5I5I rpynna CooTHoIlIeHHH n03BOJI5IeT peIlIHTb ypaBHeHH5I (20.12)
()THOCHTeJlbHO <lJYHKL{HH Uk (A, z) H v k (A, z). no H3BeCTHbIM 3HaqeHH5IM
pa.n.HaJIbHOH H OKpYLKHOH KOMnOHeHT BeKTopa CMeIII.eHHH 3TH <PYHKL{HH
onpe.n.eJl5lIOTC5I CJIe.n.YIOIL{HM 06pa30M:
A [Y
 (A) + J
 (A») [- Uk (A, z) + iV k (A, z)) =


231: ""
= S S (uzLf'>* + urMfJ* + uANf'>*) rdrd8,
o 1
r.n.e «*» 0603HaqaeT KOMnJIeKCHOe COnp5lLKeHHe.


(20.10)


231: 00
= 5 5 (Uk + ius) N k+l (Ar) e-ikSrdrde,
o 1


198


199


I
"I
I
II




....


III


A [Y
 (A) + J
 (A)j [Uk (A, z) + iV k (A, z)J =


(20.15)


IJeCKH .necpopMHpoBaHHoro ynpyroro npOCTpaHCTBa C lI.HJIHH.npHIJeCKOH
nOJIOCTblO.
B paMKax MeTo.na co6CTBeHHbIX BeKTopHbIX cpYHKlI.HH HCKOMoe npe.n-
CTaBJIeHHe BeKTopa ynpyrHx CMeIlI.eHHH sanHIIleM B BH.nef179]


211: co
= S S (u r - ius) Nk-l (lvr) e- ikS rdrd8.
o I
TIpH nOJIYIJeHHH Bblpa)KeHHH (20.15) MbI HCXO.nHJIH JIHIIlb HS o6IlI.ero
npe)l.CTaBJIeHH5I BeKTopa CMeIlI.eHHH B CJIoe (20.2) H HCnOJIbSOBaJiH CBOM-
CTBa nOJIHOTbI cpYHKlI.HM S
) (r, 8). B CB5ISH C 3THM CnpaBe.lI.JIHBOCTb cpop_
MYJIbI 06paIlI.eHH5I (20.10) 6y.neT YCTaHOBJIeHa, eCJIH )l.JI5I KOM6HHaUHM
cpYHKUHM Uk (A, z) H v k (A, z) HS Hee BbITeKalOT Bblp3)KeHH5I, H.neHTHIJ-
Hble (20.15).
PaccMaTpHBa5I npoeKUHH BeKTopHoro paBeHcTBa (20.10) Ha OCH c
opTaMH e r H ea, Haxo.nHM
2 2 211: co ( iJS(k). U iJSCk). )
1.,2 [Y k (A) + J k (A)] Uk (A, z) = S S U r
 +

 rdrd8
o I iJr r iJ6 '
2 2 211: co ( iJS;k). iJS(k). )
1.,2 [Y k (A) +Jk(A)jVk(A, Z) = S S U

-us
 rdrd8.
r r iJ8 iJr
o I
(20.16)


1 1 co co ( L ln) (8 )
U ( r z 8 ) = - . -

 [Ukn r) k ,z +
" 2n 2h


k=-co n=-co


+ Vkn (r) Mk n ) (8, z) + Wkn (r) N
n) (8, z)].


(20.18)


3.neCb co6CTBeHHble BeKTopHble cpYHKUHH onpe.neJI5IIOTC5I CJIe.nYIOIlI.HM
06paSOM:


L
n) (8, z) = erS
n) (8, z),


iJS<[:) (6, z)
M
n) (8, z) = es iJ6


Cn) iJS
n) (6, z)
Nk (8, z) = k iJz '


(20.19)


I1cnOJIbSOBaHHe peKyppeHTHblx COOTHOIIleHHH (20.11) nOSBOJI5IeT
nOJIYIJHTb Bblpa)KeHH5I .nJI5I COOTBeTCTBYIOIlI.HX KOM6HHaUHM cpYHKUHM
Uk (A, z) H v k (A, z), nOJIHOCTblO COBna.nalOIlI.HX C paBeHCTBaMH (20.15).
3THM SaKaHIJHBaeTC5I .nOKaSaTeJIbCTBO cnpaBe.nJIHBOCTH cpOPMYJIbI06pa-
IlI.eHH5I (20.10).
Bblpa)KeHHe BeKTopa Hanp5I)KeHHM Ha nJIOIlI.a.nKax, napaJIJIeJIbHbIX
rpaHHIJHbIM nJIOCKOCT5IM CJI05I, Ha OCHOBe saKOHa rYKa sanHIIleM B
BH.ne paSJIO)KeHH5I
Od _00

F =
 '-' S[( m-l dWk -A 2 Uk ) L(k)+
20 z 2n
 m-2 dz m-2 A
k=-co 0
+
 ( dUk + W ) MCk) + _
 dVk N Ck ) ]
 d

2 dz k A 2 dz A /\, /\'.


r.ne e.nHHHIJHble BeKTOpbI en es H k HMelOT TOT )Ke CMbICJI, IJTO H B (20.2),
a HBJI5IIOIlI.a5IC5I nepHo.nHIJeCKHM no z If 8 peIIleHHeM ypaBHeHH5I reJIbM-
rOJIbua Ha nOBepxHocTH r = 1 cpYHKUH5I S
n) (8, z) paBHa


S In) (8 ) _ i(kS-an'Z)
k ,z -e .


(20.20)


(20.17)


B Bblpa)KeHHH .nJI5I S
n) napaMeTp an Heonpe.neJIeH H BbI6HpaeTC5I B
COOTBeTCTBHH C ueJIec006pasHbIM B Ka)K.nOM KOHKpeTHoM CJIYIJae cnoco:
60M npo.nOJI)KeHH5I sa.naHHbIX npH -h <: z -< h rpaHHIJHbIX YCJIOBHH
Ha BCIO UHJIHH.npHIJecKYIO nOBepxHOCTb r = 1. ILBa BOSMO)KHbIX BapH-
aHTa BbI60pa SHaIJeHHM an paccMaTpHBalOTC5I HH)Ke npH aHaJIHSe KOHK-
peTHblx sa.naIJ.
Bblpa)KeHHe (20.18) .nJI5I BeKTOpa CMeIlI.eHHM y.noBJIeTBOp5IeT ypaB-
HeHHIO JIaMe B UHJIHH.npHIJeCKHX Koop.nHHaTax H YCJIOBH5IM OTCYTCTBH5I
Hanp5I)KeHHM Ha 6eCKOHeIJHOCTH, eCJIH BbI6paTb co.nep)KaIlI.HeC5I B HeM
pa.nHaJIbHble cpYHKUHH B BH.ne


HaJIHIJHe cpOPMYJIbI oQpaIlI.eHH5I (20.10) nOSBOJI5IeT )l.OBOJIbHO npocTO
.nOBeCTH .no KOHua B 06IlI.eM cJIYIJae .necpopMHpOBaHH5I CJI05I peIIleHHe KaK
nepBoM, TaK H BTOpOM rpaHHIJHbIX sa.naIJ. CJIe.nyeT saMeTHTb, IJTO npH
3TOM peIJb H.neT JIHIIlb 0 rpaHHIJHbIX YCJIOBH5IX Ha nOBepXHOCT5IX z =
= + h. YCJIOBH5I Ha UHJIHH.npHIJeCKOM nOBepxHocTH r = 1 OCTaIOTC5I
HeBbITIOJIHeHHbIMH H peIIleHHe (20.2) He co.neP)KHT Heo6xo.nHMoro npo-
HSBOJIa .nJI5I HX BWnOJIHeHH5I.
ILJI5I nOJIYIJeHH5I o6IlI.ero peIIleHH5I OCHOBHbIX rpaHHIJHbIX sa.naIJ .nJI5I
CJI05I C UHJIHH.npHIJeCKHM OTBepCTHeM K Bblpa)KeHH5IM (20.2) CJIe.nyeT npH-
coe.nHHHTb peIIleHHe ypaBHeHHM JIaMe, .nOCTaBJI5IIOIlI.ee Heo6xo.nHMYIO
CTeneHb npOHSBOJIa Ha UHJIHH.npHIJeCKOM nOBepxHocTH r = 1. T aKHM
peIIleHHeM, OIJeBH.nHO, 5IBJI5IeTC5I o6IlI.ee peIIleHHe sa.naIJH L\JI5I nepHo.nH-
200


, 2 "
Ukn (r) = anAknKk (anr) + rB kn [anKk (anr)-
3m-4 a;' 2 c 1 K ( )
- ---E.... Kk (anr)] + k 1m - k (X,zr ,
m r r


1 [ ,
Vkn (r) = A kn r Kk (anr) + B kn aJ(k (anr) -
mIl ] ,
-4 - -Kk(anr) +anCknKk(anr),
m r


(20.21)


I
I
,I
I
!I


Wkn (r) = AknKk (anr) + anBknrK
 (anr).


201




.


3
eCb A kn , Bkn. C kn - np0l1SBOJIbHbIe nOCT05IHHbIe, a lUTpHX YKaSbI-
BaeT Ha
HcpcpepeHUHpOBaHHe no nOJIHOMY aprYMeHTY.
D.JI5I nOJIYIIeHH5I CHCTeMbI aJIre6pal1'IeCKHX ypaBHeHHH OTHOCHTeJIb-
HO nOCT05IHHbIX Aim, B kn H C kn npH sa
aHHH Ha nOBepXHOCTH r = 1
SHaIIeHHH BeKTopa nepeMelU.eHHH HCnOJlbsyeM cpOPMYJlY 06paw.eHH5I
erUkn (r) + ka
Vkn (r) + eok 2 Wkn (r) =
231 h
= S S [ULknd* + VM
n" + wN1 n' *] d8dz. (20.22)
o -h
OOCKOJIbKY BeKTOp Hanp5I)KeHHH Ha UHJIHH
pHIIeCKOH nOBepXHOCTH
r = const TaK)Ke npe
CTaBJl5IeTC5I no COOCTBeHHbIM BeKTopaM L
n) .
M (n) N (n),+, 6 u
k H k, TO 'f'0PMYJIY 0 palU.eHH5I HCnOJlbsyeM H B CJlYIIae nepBOH
OCHOBHOH rpaHHIIHOH Sa
aqH.
HeCMOTp5I Ha OTHOCHTeJlbHYIO npOCTOTY nOJlYIIeHH5I perneHHH Sa
aq

Jl5I CJlO5I H BHernHOCTH UHJIHH
pHIIeCKOH nOJlOCTH, HCCJIe
OBaHHe Ha-
npmKeHHOrO COCT05lHH5I CJlO5I C OTBepCTHeM nYTeM cpOpMaJlbHOrO HC-
nOJlbSOBaHH5I OnHCaHHOH BbIrne CXeMbI paCCMOTpeHH5I TaKoro KJlaCCa
Sa
aq MO)KeT npHBeCTH K HeonpaB
aHHOMY YCJlO)KHeHHIO COOTHOrneHHH

Jl5I npOHSBOJlbHbIX nOCT05lHHbIX H cpYHKUHH B 06lU.eM perneHHH. YIIeT
CneUHcpHKH rpaHHIIHbIX YCJIOBHH, OC06eHHO B CJlYIIa5IX, KOr
a Ha nJloc-
KOCT5IX Z= + h OTCYTCTBYIOT KaCaTeJlbHbIe Hanp5l)KeHH5I,
aeT BOSMO)K-
HOCTb ynpocTHTb KaK CTPYKTYPY perneHH5I
Jl5I CJI051 (20.2), TaK H BHl!
cpYHKUHOHaJlbHbIX ypaBHeHHH, nop0)Kl!aeMbIX rpaHHIIHbIMH YCJlOBH5IMH.

 21. 06u
ee perneHHe TpeXMepnoii 38):1;8'IH RHprn8
OOCTpoeHHoe paHee perneHHe OCHOBHbIX rpaHHQHbIX sa
aII
JIH CJI05I
C UHJlHHl!pHIIeCKHM OTBepCTHeM npeo6pasyeM K BHl!Y,Yl!o6HOMY npM
paCCMOTpeHHH B TpeXMepHOH nOCTaHOBKe HSBeCTHOH 3al!aIIH KHprna
0YCTb Hanp5l)KeHHOe COCT05IHHe CJlO5I C UHJlHH
pHIIeCKHM OTBepCTH-
I
eM onpel!eJI5IeTC5I Ol!HOOCHbIM paCT5I)KeHHeM Ha 6eCKOHeQHOCTH 2G a =
= P = const Bl!OJIb OCH 8 = 0, Z = O. Ha l!OCTaTOQHOM Yl!aJIeHHH OT
oTBepCTH5I OTJIHIIHbIe OT HYJI5I KOMnOHeHTbI TeHsopa Hanp5l)KeHHM B
UHJII1H
pHIIecKHx Koop
HHaTax npH 3TOM 6Yl!YT HMeTb CJIel!YIOlU.He 3Ha-
'1eHH5I:
I -10) I P(1 28)
 (0) -
 P(I -' 28)
2G v;. = 2 + cos , 2G a fJ - 2 cos,


HanpmKeHHoe COCT05IHHe), KOTopoe 6bIJIO nOJIYIIeHo r. KHprneM [230].
n
AHaJIHS 3TOro perneHH5I nOKaSaJI, qTO B TOKax 8 = 2 Ha rpaHHue
OTBepCTH5I OKpY)KHbIe Hanp5l)KeHH5I ao B TPH pasa npeBOCXO
51T SHa-
'1eHHe Hanp5I)KeHHM Ha 6ecKOHeqHOCTH. OCHOBHOM ueJIblO npel!npHH5I-
Toro ToqHOrO HCCJIel!OBaHH5I Sal!aIIH KHprna 51BJI5IeTC5I HSYIIeHHe xa-
paKTepa pacnpel!eJIeHI151 Hanp5l)KeHHH: ao no TOJIlU.HHe CJI051 H KOJIHIIeCT-
BeHHa5I oueHKa SHaQeHHH Tex Hanp5l)KeHHH (B IIaCTHOCTH, a z ), KOTOpbIMH
B nJIOCKOH sal!aIIe npeHe6peraJIH. OpHIIeM HaH60JIbrnHM HHTepec npel!-
CTaBJI5IIOT Te CJIYIIaH l!e<l>opMHpOBaHH5I, B KOTOpbIX l!HaMeTp.oTBepCTH5I
COHSMepHM C TOJIlU.HHOH CJI05I.
XapaKTep Hanp5l)KeHHoro COCT05lHH5I CJI051 npH Ol!HOOCHOM paCT5I-
)KeHHH nOSBOJI5IeT SHaIIHTeJIbHO ynpocTHTb COCTaBJI5IIOlU.He o6lU.ero pe-
rneHH5I paccMaTpHBaeMoH rpaHHIIHoH sal!aIIH (20.2) H (20.18). OCHOBOH
l!JI5I TaKoro ynpOlU.eHH5I 51BJI5IeTC5I, npe)Kl!e Bcero, cHMMeTpH5I nOJI5I Ha-
np5l)KeHHH OTHOCHTeJIbHO Cpel!HHHOH nJIOCKOCTH. YIIeT 3Toro 06CT05l-
TeJIbCTBa nOSBOJI5IeT OrpaHHQHTbC5I B perneHHI1 (20.18) JIHrnb Bew.eCT-
BeHHoH QaCTblO KOMnJIeKCHbIX P5ll!OB <I>YPbe no Koopl!HHaTe z npH
a - nn CooTHOrneHH5I (20.19), onpel!eJI5IIOlU.He cnoco6 nocTpOeHH5I
n - h'
co6CTBeHHbIX BeKTopHbIX <l>YHKUHM l!JI5I BHernHOCTH UHJIHHl!pHIIeCKoro
oTBepCTH5I, YKasbIBalOT, IITO B paccM
TpHBaeMoM CJIYIIae MO)KHO
orpaHHIIHTbC5I TaK)Ke JIHrnb BelU.eCTBeHHOH IIaCTblO B p5ll!ax no OKPY)K-
HOM Koopl!HHaTe.
TaKHM o6pasoM, HCXOl!51 HS npel!cTaBJIeHH5I (20.18
, nOJIYIIaeM CJIe-

YIOlU.He BbIpa)KeHH5I KOMnOHeHTOB BeKTopa CMelU.eHHH, Yl!OBJIeTBOp5I-
IOlU.He ypaBHeHH5IM J1aMe npH npOHSBOJIbHOM k l!JI5I Ka)K
OrO n:
I I =
U(l) = - UkO (r) cos k8 + - h
 Ukn (r) cos k8 cos anz,
4nh 2n n=l
v(1) = -
 VkO (r) sin k8 - 2 \
 Vkn (r) sin k8 cos anz, (21.2)
4nh n n=1
1 =
w(1) = - -
 Wkn (r) cos k8 sin anZ.
2nh n=l


I
,i!



 ( O)__
 P . 28
2G 't rO - 2 sIll .
D.JI5I onpel!eJIeHH5I Hanp5l)KeHHOrO COCT05lHH5I CJI051 Heo6xol!HMO no-
CTpOHTb perneHHe ypaBHeHHH ynpyroro paBHOBeCH5I, l!alOlU.ee HCqeSa-
IOlU.He Ha 6ecKoHeIIHocTH SHaIIeHH5I KOMnOHeHTOB TeHsopa Hanp5l)KeHHH H
B CYMMe C BbIpa)KeHH5IMH (21.1) o6ecneIIHBalOlU.ee OTCYTcTBHe Hanp5I-
)KeHHH Ha nJIOCKOCT5IX CJI051 z = + h H Ha UHJIHHl!pHqeCKOM nOBepx-
HOCTH oTBepCTH5I r = 1.
OOCTaBJIeHHa5I TpeXMepHa5I Sal!aIIa COl!ep)KHT npOCTOe npH6JIH)KeH-
Hoe perneHHe B paMKax nJIOCKOH S
aIII1 TeopHH ynpyrocTH (nJIOCKOe


(21. 1)


3l!eCb BbI1!eJIeHbI CJIaraeMbIe, OTBeIIalOlU.He nJIOCKol!e<l>opMHpoBaHHoMY
COCT05lHHIO npocTpaHcTBa C UHJIHH
pHQeCKHM OTBepcTHeM. 3HaIIeHH5I
<l>YHKUHH pal!HaJIbHOH Koop
HHaTbI B 3THX CJIaraeMbIX onpe
eJJ5IIOTC5I
Ha OCHOBe npe
eJIbHoro nepexol!a npH an -+ 0 B (20.21):
UkO (r) = b k (k - 2 + 4 m;;; 1 ) r- k + 1 + crr-k-I.
( m - 1 ) k + 1 1 -k-I
VkO(r) = b k 4 mk 1 r- -Tck r .
r
e b k H C" - npOHSBOJIbHbIe nOCT05IHHbIe. u
COOTHOrneHH5I saKOHa rYKa nOSBOJI5IIOT HaHTH cooTBeTcTBYIOlU.H

CMelU.eHH5IM (21.2) BbIpa)KeHH5I M5I KOMnOHeHTOB TeHsopa Hanp5l)KeHHH


(21.3)


202


203




- 2
h
1{An( :: K k -
n K
)+Bn[
n (k 2 +4 m;;;1 )K
-
- (
 + m-2 ex? +4 m-I
 ) K J
T 2 m n m r2 k+
k 2 ( , 1 )}
+ C n r anKk - ,Kk coskS cosanz,


(21.4)


3,neCb ,nJUI cOKpa
eH1I5I sanlIClI ony
eH aprYMeHT <PYHKl.{lIH MaK-
,nOHaJIb,na Kk (an r), a lIITpllX, KaK II paHblIIe, YKasblBaeT Ha ,nllqxpe-
peHl!lIpOBaHlIe no nOJIHoMyaprYMeHTY.
BS5ITble nplI k = 2 BblpmKeHlIH (21.4) nOSBOJIHlOT y,nOBJIeTBOpIlTb
OO
IlM rpaHlIqHblM YCJIOBlIHM B Hanp5l2KeHll5lX Ha l.{lIJIlIH,nplIqeCKOH no-
BepxHocTII r = 1, 1I3MeHHeMOCTb KOTOpblX B OKPY2KHOM HanpaBJIeHlIlI
onlICblBaeTCH BTOpOH rapMoHIIKOH. XapaKTepHoH lIX ocooeHHocTblO
HBJIHeTCH TO, qTO nplI Z = + h KaCaTeJIbHble HanpH2KeHlIH T

) II T


OOpalIl,alOTCH B HYJIb. 3TO OOCT05lTeJIbCTBO, C yqeTOM Toro, qTO B pac-
CMaTplIBaeMOH sa,naqe KlIplIIa TpeoyeTcH paBeHCTBO HYJIlO KaCaTeJIbHblX
HanpH2KeHlIH Ha nOBepXHOCTHX CJI051 Z = + h, n03BOJIHeT ynpoCTlITb Ty
qaCTb oo
ero pelIIeH1I5I, KOTopaH OTBeTCTBeHHa 3a y,nOBJIeTBOpeHlIe rpa-
HlIqHblX YCJIOBlIH Ha nJIOCKlIX rpaHlIl.{aX CJIOH.
KOMnOHeHTbl BeKTopa Hanp5l2KeHlIH Ha nJIO
a,nKax, napaJIJIeJIb-
HblX rpaHlIqHblM nJIOCKOCTHM CJIOH, ,nalOTCH COOTHOlIIeHlIHMlI (20.17).
OTMeqeHHble ocooeHHocTlI Bblpa2KeHlIH .ZJ:JIH KaCaTeJIbHblX HanpH2KeHlIH
B (21.4) II xapaKTep rpaHlIqHblX YCJIOBlIH B paccMaTplIBaeMoH sa,naqe
YKa3bIBaeT Ha TO, qTO Heooxo,nlIMaH CTeneHb OO
HOCTII pelIIeHlIH oy,neT
coxpaHeHa, eCJIlI B (20.17) orpaHlIqlITbCH JIlIllib BTOpOH rapMoHlIKOH
k = 2 II, KpoMe Toro, nOJI02KlITb
dUk 0 dVk 0 h
.
 + w k =, ----cIZ = nplI Z =:f:: . (21.5)
3Tll COOTHOlIIeHlIH YKaSbIBalOT, qTO B ,naHHOM cJIyqae <PYHKl.{1I5I v k (A, z)
B {20.3) M02KeT ObITb nplIHHTa paBHoH HYJIlO, a qlICJIO npOlISBOJIbHblX
<PYHKl.{lIH OT A B w k (A, z) II Uk (A, z) M02KHO YMeHblIIlITb ,no ,nBYX. Ec-
JIll nplI 3TOM elIl,e yqeCTb clIMMeTplIlO nOJI5I HanpH2KeHlIH OTHOClITeJIbHO
cpe,nlIHHoH nJIOCKOCTlI CJI05l, TO Bblpa2KeHlIH .ZJ:JIH <PYHKl.{lIH Koop,nlIHa-
Tbl Z B (20.3) nplIOOpeTYT BlI,n
( m 1 '
W k (A,Z)=2C k (A) zchA.z-2 ;;;;., ShAZ-hcthAhshAZ),
(21. 6)
A2Uk (A, z) = 2C k (A) (AZ sh 1>.z + m ;;;2 ch AZ - Ahcth Ahch AZ).
OTCYTCTBlIe KaCaTeJIbHblX HanpH2KeHIIH Ha nJIOCROCTHX CJIOH nplI-
BO,nHT K TOMY, qTO y,noBJIeTBOpeHlIe rpaHlIqHblX YCJIOBlIH nplI Z = :f::h, KaK
CJIe,nyeT lIS (20.17), aHaJIOrlIqHO pelIIeHlIlO CKaJI5IpHOH sa,naqlI Teopllii
nOTeHl.{lIaJIa. B CBH31I C 3TlIM B Bblpa2KeHlIlI ,nJIH S!/:> (r, 8) (20.6) M02K-
HO OrpaHlIqlITbCH JIlIllib Be
ecTBeHHoH qaCTblO. I10JIaraH 2C k (A) =
= C (A), npe,ncTaBlIM KOMnOHeHTbl TeHsopa HanpH2KeHlIH ,nJIH np01I3-
BOJIbHOH k-H rapMoHlIKlI no OKPY2KHOH Koop,nllHaTe CJIe,nYlO
lIMlI Bbl-
pa2KeH1I5IMlI:
1 I r
20 u?> = - 2:n J C (A) IjJ (AZ) N k'Ad'A cos k8 +
o


(npll sanllCII 3TIIX Bblpa2KeHIIH oy,neM onycKaTb IIH,neKC k y npOllSBOJIb-
HbIX nOCT05IHHbIX):
1 1
2ll U
I) = 4:nh [(1 - k) (2 + k) br- k - c (1 + k) ,.-k-2) cos k8 +
+ 2
h
I {An[(a
+ :: )Kk-
n K
] +Bn[an(a
r+ :2 +
+ 4m - 4 ) K' _ (
 + 3m - 2 a2 + 4m - 4 k 2 ) K . J
mT k r 2 m n m r2 k+
+ C n :2 (anK
 - -+- Kk) 1 COS kS cos anz,

u(I) =
. 4(1 -k) b,.-kcoskS-

 [ A a2K +
20 z 4:nh m 2:nh.""-l n n n
n=1
+ a
Bn (
 Kn + anrK
) J cos kS cos anz,
1 1
20 U<J) = 4:nh [(k - 1) (k - 2) b,.-k + c (1 + k) ,.-k-2) cos ke-


---.!...... (l)__

 { A 2 K ' +B 2 [( k2 )
20 "C rz - 2:nh =1 l1. an k nan anr + anT K k -
-2 m;;;1 K
J+ k

n CnKk}coskSsinanz,
1 (1) 1
 [ A k K B 2 ( ' m - 1 k
20 "C ze =- 2:nh n7:::1 nan, k + nan kKk- 2 m anT Kk) +


ka 2 ' J
+ C n + Kk sin ke sin anz,


1 1
20 "C

 = 4:nh [k (1- k) b,.-k - c (1 + k) ,.-k-2)sin k8-
- L2
h
1{An+-(anK
--+-Kk)+Bn[(a
+ :: +4 m;;;./ )Kk-
_ an 5m-4 , J C . [( a
 k 2 ) K an ' J}
T m Kk + n ""2 + 72 k - r Kk sin kS Cos anz.


+ d:n r CA
A) [1jJ(AZ)-
 ChAZ]( :: Nk-+N
)AdACoskS,
u


204


205




I I


2
 0
2) = 2
 r C (A)[cp (AZ) - 2 ch AZ] NkAdA cos k8,
o



 22. AHaJIII3 6eCKOHe'IHbIX CIICTeM
DpOH3BOJIbHble 1I0CTOHHHble H <PYHKUHH B BblpameHHHx .n:JIH HallpH-
:>KeHHH (21.4) H (21.7) .n:OJImHbl ollpe.n:eJIHTbCH H3 CJIe.n:YIOIIJ.eH CHCTeMbl
<PYHKUHOHaJIbHbIX ypaBHeHHH:

 0(1) (1 z) +
 0(2) (1 Z ) +
Pcos28 = 0
20 r ' 20 r ' 2 '
;0 T

(l,Z)+ ;0 T

(1,Z)-+Psin28=0,
I (I) ( 1 ) + I (2) ( 1 0
20 T,z ,Z 20 T rz ,z) = ,


I
II
I



O<2)=
 r C(A) l m(AZ)-
ChAZ ](
N
-
20 e 211: J A 2 't' m r
o
k 2 ) I 2"':
-(2"N k AdAcosk8- 211: . m
 C(A)chAzNkAdAcosk8, (21.7)
o

 T(2) =
 7 C (A) [ dip (AZ) _ 2A sh A Z ] N
 AdA cos k8
20 rz 211:J A dz '
o

 (2) _ _
 . . l!.-. r 0 (A) [ dip (}.z)
20 T2'e - 2n r J A,2 dz
o
I k r C (A) [ 2 I ( I A" )
20 T

 = 211: J
 cp (AZ) -mChAZ £2"N k -,N k AdA sin k8.
o
3.n:ecb IIpHHHT
 CJIe.n:YIOIIJ.He 0603Ha'IeHHH:
N - N ( A ) _ J k (Ar) ak (A) - Y k (Ar) b k (A)
k - k r - [ak (A)]2 + [bk (A)]2 ,
cp (Az) = ch AZ - Ah dh Ah ch AZ + AZ sh AZ.
IIlTpHx Y N k YKa3bIBaeT Ha .n:HcpcpepeHUHpOBaHHe 110 1I0JIHOMY aprYMeH-
TY. cI>YHKUHH ak (A) H b k (A) Ollpe.n:eJIHIOTCH paBeHCTBaMH (20.7).
KaK CJIe.n:yeT H3 (21.1), O.n:HOOCHoe paCTHmeHHe Ha 6eCKOHe'IHOCTH
nopom.n:aeT OTJIH'IHYIO OT HYJIH oceCHMMeTpH'IHYIO COCTaBJIHIOIIJ.yIO no-
JIH HallpH:>KeHHH. B CBH3H C 3THM 06w.ee perneHHe TpexMepHOH 3a.n:a'IH
KHprna .n:OJImHO co.n:epmaTb, 110M HMO 1I0CTpOeHHblX Bblrne HeOCeCHMMeT-
PH'IHblX CJIarae
X, perneHHe OCeCHMMeTpH'IHOH 3a.n:a'IH .n:JIH BHernHOCTH
3arpymeHHorO paBHOMepHblM .n:aBJIeHHeM UHJIHH.n:pH'IeCKOrO OTBepCTHH.
3ro perneHHe HMeeT BH.n:
p
U(3) = - W(3) = 0,
2r '


(22.1)


2A sh AZ] NkAdA sin kfJ,


do 0
1) (r, + h) + ;0 0
2) (r, + h) = o.


I P
_0(3)=__
20 r 2r 2 .



 (3)_

20 u e - 2r 2 '


I
_T(3) = 0
20 rz .


(21.9)


HYJIeBble YCJIOBHH .n:JIH KaCaTeJIbHblX HallpHmeHHH Ha nJIOCKOCTHX
Z = + h CJIOH Y.I10BJIeTBOpHIOTCH 3a C'IeT Bbl60pa <pOpMbl perneHHH.
Dpoue.n:ypa npHBe.n:eHHH paBeHcTB (22.1) K aJIre6paH'IeCKHM co-
OTHOUJeHHHM. COrJIaCHO IIpe.n:bI.I1YIIJ.eMY H3JIOmeHHIO, .n:OCTaTO'IHO O'IeBH,l{-
Ha H OCTaHaBJIHBaTbCH no.n:P06HO Ha HeH HeueJIeco06pa3Ho. CJIe.n:yeT
JIHUJb OTMeTHTb, 'ITO npIJ npoBe.n:eHIJIJ cooTBeTcTBYIOW.HX BbIKJIa.n:OK
He06xo.n:HMO HcnOJIb30BaTb He TOJIbKO pa3JIOmeHHe rHllep60JIH'IeCKHX
<PYHKUHH B TpHrOHOMe'rpH'IeCKHe pH.n:bl, HO H IIpe.n:CTaBJIeHHH <PYHKUHH
MaK.n:OHaJIb.n:a HHTerpaJIaMH Be6epa. 3TH npe.n:CTaBJIeHHH npH npOH3-
BOJIbHOM 3Ha'IeHHH a B (20.7) IJMeIOT CJIe.n:YIOIIJ.HH BH.n::
j "'; Nk (Ar) N: (A)
Kk (anr) = - anKk (an) 2 AdA,
a +A 2
o n
, 00 * [ k2+a

arlKk(anr) = - r N k (Ar) N k (A) 2 . Kk (a n )-

 an + A 2
2a
 . ]
- 2 Kk (an) AdA.
(an + A2)2
3.n:ecb N k (Ar) onpe.n:eJIHeTCH BblpameHHeM (21.8), a
N; (Ar) = J k (Ar) a k (A) - Y k (Ar) b k (A).


(22.2)


(21.8)


II


(22.3)


TaKHM 06pa30M, nocTpoeHbl Bce COCTaBJIHIOw.He BeKTopa nepeMew.e-
HHH, B03HHKaIOIIJ.HX B C.Hoe C oTBepcTHeM IIPH CHHTHH C 1I0BepxHocTH
r = 1 HallpHmeHHH o
o) H T

 OCHOBHoro HallpHmeHHoro COCTOHHHH
(21.1). OcecHMMeTpH'IHaH 'IaCTb 06w.ero peUJeHHH (21.9) Bbl6paHa cpa-
3Y B BH.n:e, 06eClle'IHBaIOw.eM y.n:oBJIeTBOpeHHe cooTBeTCTBYIOw.eH 'IaCTH
rpaHH'IHblX YCJIOBHH. DpOH3BOJIbHble nOCTOHHHble H <PYHKUIJH, co.n:epma-
w.HeCH B .n:BYx .n:PyrHx COCTaBJIHIOIIJ.HX (21.4) H (21.7), .n:O.'1mHbl 6bITb Bbl-
6paHbl H3 rpaHH'IHblX YCJIOBHH, H3MeHeHHe KOTOpblX B.n:OJIb oKpymHoH
Koop.n:HHaTbl onHCbIBaeTCH BTOpOH rapMoHHKOH.


PaCCMaTpHBaH CHCTeMY (22.1), MomHO 3aMeTHTb, 'ITO HeKoToporo
ynpOIIJ.eHHH ee MomHO .I10CTH'lb Ha.n:.1emaw.HM Bbl60pOM a B (20.7). y!3
BblpameHHH .n:JIH 't

 CJIe.n:yeT, 'Iro CHCTeMa (22.1) YllpOIIJ.aeTCH, eCJIH
N
 (A) = O. KOM6HHaUHH <PYHKUHH 5eCCeJIH, 110 KOropOH CTpOHTCH npe-
06pa30BaHHe Be6epa, 6y.n:eT 06JIa.n:aTb TaKHM CBOHCTBOM, eCJIH B (20.7)
nOJIO:>KHTb


I. 0
cosa + Tsma = .


(22.4)


206


201




co
2m I r x (A) * C
m-2 Po = m J
N2CA) N 2 (A) AdA, 2nh = P,
o


OnbIT perneHlUI paccMOTpeHHbIx paHee 3aJI.aIJ JI.JISI ynpyrIIx TeJI C
yrJIOBbIMII TOIJKaMII YKa3bIBaeT, IJTO 9
cpeKTIIaHocTb perneHlISI CIIcreMbI
(22.5) II nOCJIeJI.YIOrn;erO BbIIJIICJIeHIIH B cyrn;ecTBeHHOH Mepe onpeJI.eJIH-
eTCH 3HaHIIeM aCIIMnTOTIIIJeCKIIX CBOHCTB HeII3BeCTHbIX BeJIIIIJIIH Pn'
qn II X (A). JIorIIIJHbIM npeJI.nOJIo:meHIIeM, Ha OCHOBe KOTOpOro MO:mHO
npOBeCTII IIccneJI.OBaHIIe CBOHCTB HeII3BeCTHbIX B (22.5), SIBJISIeTCSI TO,
IJTO nOBeJI.eHIIe pHJI.OB II IIHrerpaJIOB B
opMYJIaX (21.4) II (21.7) JI.JISI HOp-
ManbHbIX HanpH:meHIIH 6YJI.eT TaKIIM :me, KaK II B paccMoTpeHHbIx paHee
ocecIIMMeTpIIIJHbIX 3aJI.aIJax.
HMeSI B BIIJI.Y YKa3aHHoe npeJI.nOno:meHIIe, paccMoTpIIM, HanpIIMep,
BbIpa:meHIIe JI.nH Oz II3 (21.7). YIJIITbIBaH CBSI3II Me:mJI.Y
YHKUIIHMII
C (A) II x (A) B (22.6), 3HaIJeHIISI HanpH:meHIIH Oz BbIIJIICnSIIOT no <i>0pMyne


DocJIe JI.OBOJIbHO rpoMo3JI.KIIX BbIKJIaJI.OK CIICTeMa <i>YHKUIIOHaJIbHbIX
ypaBHeHIIH (22.1) npIIBOJI.IITCSI K CJIeJI.YIOrn;IIM COOTHorneHIIHM:


f a
 *
Pnal (an) + g"b 1 (an) = J x (A) (a
 + A2)2 N 2 (A) N 2 (A) 'AdA,
o


co 2
Pnadan) + gn b 2 (an) = - 2 r x i
) 2 an N; (A) N 2 (A) AdA +
J (a + A2)2
o n
+ 2 co
 x (A) N; (A) N 2 (A)
m
 n AdA, (22.5)
a +A"
o n


co

 0(2) = !:- cos 28 \" x
A) [ <p (AZ) - 2 S C h h
 h z ] N 2 (Ar) N; (A) AdJ..
2G z 4 .J "- sh Ah "-
o


(22.8)


X (A) = C(A) 2AS
 Ah , t =
 K 2 (an)
nhN 2 (A) an K; (an) ,
( a
+4 m-l )
An = -En t -2 m -2tC n .
an an
1J.JIH K09<i><i>IIUHeHTOB npII HeII3BeCTHbIX Pn II gn nOJIYIJHM cneJI.YIOLqHe
BbIpaMeHIIH:
a 1 (an) = + [ 1- t 2 (a
 + 4) _ 6 m -1 ] ,
ma

a2(an)= (
+2 m-21 _t2 a
+4
an man a

b d a n )=(t-l- :
 ), b 2 (a n )=(++2 la;t


(22.6)


DonaraH B 9TOM BbIpa:meHlIH r = 1 H HCnOJIb3yH aCIIMnTOTHIJeCKOe pa-
. 2
BeHCTBO N 2 (A) N 2 (J.) = nA ' <i>OPMYJIY (22.8) B TOIJKaX U HJI 11 HJI.P II-
IJeCKOH nOBepxHocTH r = 1, 6JIH3KIIX K yrJIy, 3aMeHHM CneJI.YIOrn;eii:
T
2
 0
2)(1, z) =
 cos28
 X (I.) [


i
o
_ :n cos 28 I X (A) I + e-"-(h-Z) + (h - z) e-A(h-Z)] dJ., (22.9)
r
rJI.e T - JI.oCTaTOIJHO 6onbllioe IJl1cno.
AHaJIII3 BbIpa:meHI1H (22.9) nOKa3bIBaeT, IJTO II HrerpaJI, npeJI.CTaB-
JIHIOrn;HH HanpH:meHHe 0Z' o6naJI.aeT B yrnoBoH TOIJKe norapH<i>MIIIJeCKOH
oco6eHHocTbIO, ecnH nOBeJI.eHHe <i>YHKUHH X (A) npH 60JIblliHX A onpeJI.e-
JIHeTCSI aCHMnTOT'IIIJeCKHM paBeHcTBoM
lim X (A) = a o =1= O.
A_CO


chAZ ] *
2 shAh N 2 (A)Nd A )dA-


h Ah co
X (A) 4A (cth J.h + sh2 Ah ) =
 [Pn
n=I


A 2
(a
 + A2)2 +


+ gn ] 8 Po
a
+1..2 - m-2 y-
HeII3BecTHbIe B CHcreMe (22.5) CBSI3aHbI C npOH3BOJIbHbIMII nOCTOSIH-
HbIMII H <i>YHKUHSIMH B (21.4) II (21.7) paBeHCTBaMII
3 n' B
Pn = a" (- 1) K2 (an) nh .
gn=-a
(-l)"K2(an) ;;; ' Po=

: b,


(22.10)


3TO paBeHcTBo II HBnHeTCSI TeM HCXOJI.HbIM npeJI.nOJIo:meHIIeM 0 nOBeJI.e-
HHH He113BeCTHbIX B (22.5), cnpaBeJI.JIHBOCTb KOTOporo cneJI.yeT npoBe-
pHTb nYTeM aHanH3a CI1CTeMbI.
HCXOJI.51 H3 npeJI.CTaBneHHH I1HrerpaJIaMII Be6epa <i>YHKUIIH MaK-
JI.OHaJIbJI.a (22.2), JIerKO HaHTH 3HaIJeHHSI cneJI.YIOrn;l1X HHrerpaJIOB:


-2 m-/ t ) , (22.7)
man
:
t ).


00 Ct.? 1
r 2 n N 2 (A) N; (A) J.dA = _ 2 [1 - t 2 (a
 + 4)],
j (an + A2)2
o


r a 2 * I ( 1 ) I
j A2(a

A2)2 N 2 (A)NdJ.)AdJ. = a
 2+ t - 2a
 [1-
o


208


14 7-2
6



209




- t 2 (o:
 + 4)],


(22.11)


HCXO,n:HbIM nYHKTOM npH HsyqeHHH cBoticTB HeHSBeCTHbIX B CUCTeMe
(22.5) SIBJISIJIOCb c,n:eJIaHHOe Ha OCHOBe aHaJIUSa perneHuH oceCUMMeT-
PUqHbIX sa,n:aq npe,n:nOJIOJKeHUe (22.10). CYIlLecTBoBaHue OTJIUqHOrO aT
HYJISI npe,n:eJIbHOrO SHaqeHUSI clJYHKII.UU x (A) U K03clJclJ U II.UeHTOB Pn
TaK JKe, KaK U B ocecuMMeTpUqHbIX sa,n:aqax, nopOJK,n:aeT oco6eHHocTII
B yrJIOBbIX TOqKaX B OT,n:eJIbHO BSSITbIX BbIpaJKeHUSIX ,n:JISI HOpMaJIbHbIX
HanpSIJKeHUH us (21.4) II (21.7). O,n:HaKo COBna,n:eHue npe,n:eJIbHbIX SHa-
qeHUH x (A) U Pn npuBo,n:uT K BsauMHoMY HX norarneHHID H HanpSIJKeHHSI
HMeIDT B yrJIax BnOJIHe onpe,n:eJIeHHble KOHeqHbIe SHaqeHHSI. Cnoco6


00


.\


').2 (a
I + ').2) N 2 (A) N; (A) AdA =

 (-} + t) .


o
Ha OCHOBe 3THX paBeHcTB H aCHMnTOTHqeCKHX CBOHCTB clJYHKII.HH x (A)
npHxo,n:HM K SaKJIIDqeHHID, qTO nOBe,n:eHHe HeHSBeCTHblX Pn H gn B
(22.5) C pOCTOM HOMepa MOJKHO onpe,n:eJIHTb, paccMOTpeB CJIe,n:YIDIlLYID
cHcTeMY ,n:BYX ypaBeHHH:
Pnal (O:n) + gnbl (O:n) = 2 ao + 0 (0:;1),
an
Pna2 (O:n) + gnb2 (O:n) = 0 (0:;2).
npOBO,n:SI aCHMnTOTHqeCKHH aHaJIHS ee perneHHSI npH 60JIbrnHX n, Ha-
XO,n:HM
Pn = aD + 0 (1), g
 = 0 (0:;2). (22.13)
3TH paBeHCTBa YKaSbIBaIDT, qTO HeHSBeCTHble Pn B (22.5) HMeIDT TO JKe
npe,n:eJIbHOe SHaqeHHe, qTO H clJYHKII.HSI x (A). LIJISI BTOpOH rpynnbI He-
HSBeCTHblX gr, y,n:aeTCSI yCTaHoBHTb JIHrnb 06IlLHH XapaKTep nOBe,n:eHHSI
C YBeJIHqeHHeM HOMepa.
CJIe,n:yeT no,n:qepKHYTb, qTO nOJIyqeHHble OII.eHKH .D:JISI HeHSBeCTHbIX
Pn H gn SIBJISIIDTCSI CJIe,n:CTBHeM npe,n:nOJIOJKeHHSI (22.10) a nOBe,n:eHIHI
clJYHKII.HH x (A). Era HenpoTHBOpeqHBOCTb COOTHorneHHSIM CHCTeMbI
(22.5) Tenepb MOJKeT 6bITb YCTaHOBJIeHa nYTeM aCHMnTOTHqeCKOrO aHa-
JIHSa ypaBHeHHSI, onpe,n:eJISIIDIlLerO SHaqeHHSI x (A) qepes Pn H gn" Pac-
cMaTpHBaSI 3TO ypaBHeHHe npH 60JIbrnHX SHaqeHHSIX A Ha OCHOBe pa-
BeHCTB


(22.12)


Ta6JIHua 23
n I -Pn I gn a
 II n I -Pn I gn a

1 0,4414 1,212 9 0,6912 3,911
2 0,5772 2,448 10 0,6921 3,948
3 0,6350 3,063 11 0,6927 3,977
4 0,6618 3,392 12 0,6930 4,001
5 0,6755 3,588 13 0,6930 4,020
6 0,6829 3,719 14 0,6932 4,037
7 0,6871 3,800 15 0,6932 4,051
8 0,6897 3,864 >15 0,6932 4.051


(X) ').2
2: (a 2 + ').2)2
n=1 n


h ( 'Ah ) 1
4'). cth M+ sh2'Ah - 2').2 ,


npe,n:eJIbHOrO nepexo,n:a npu BblqHCJIeHUU SHaqeHUH HanpSIJKeHUH B yr-
JIOBOH TOqKe, a TaKJKe npueMbI YJIyqrneHUSI CXO,n:UMOCTU pSI,n:OB U HHTe-
rpaJIOB B paCCMaTpUBaeMOH sa,n:aqe, OqeBH,n:Ho, HaCTOJIbKO 6JIHSKU K
no,n:po6HO onHcaHHbIM npH perneHHH oCeCHMMeTpHqHbIX sa,n:aq, qTO BnOJI-
He ,n:onycTHMo He npHBo,n:HTb s,n:eCb COOTBeTCTBYIDIlLHe ,n:OBOJIbHO rpOMOS,n:-
KHe BbIKJIa,n:KH.
HSBeCTHble aCHMnTOTHqeCKHe CBOHCTBa HeHSBeCTHbIX B CHCTeMe
(22.5) SIBJISIIDTCSI OCHOBOH ,n:JISI nOCTpoeHHSI 3
eKTHBHoro MeTO,n:a ee
perneHHSI. OCHoBbIBaSIcb Ha COOTHorneHHSIX (22.16), npH nepexo,n:e K KO-
HeqHOH CHCTeMe nOJIaraeM




n=1


').2 = 0 (A -I-E)
a
 (a
 + ').2)2


(0< E< 1),


(22.14)


Pn = PN,


a

gn = gN
, x (A) = x (T)
an


(22.17)



 I _ O ( A- 2 )

 a
(a
+').2) - ,


00


x (A) = aD + 0 (A -E). (22.15)
TaKHM 06paSOM, YCTaHOBJIeHO, qTO CHCTeMa ypaBHeHUH (22.5) ,n:eHCTBH-
TeJIbHO 06JIa,n:aeT perneHHeM, aCUMnTOTUqeCKUe CBOHCTBa KOToporo
onHCbIBaIDTCSI COOTHorneHHSIMU
lirn x (A) = lirn Pn = aD =1= 0,
h-'CO n-+CXJ
gn = 0 (0:;2).


(22.16)


(n;;;:"N, A;;;:..T).
npu 3TOM perneHHe KOHeqHOH CHCTeMbI ,n:OCTaBJISIeT npU6JIHJKeHHble SHa-
qeHHSI He TOJIbKO HeCKOJIbKUX nepBbIX HeUSBeCTHbIX, HO U BeJIUqUHbI
aD B (22.16). 3TO 06cToSITeJIbcTBo oco6eHHo BaJKHO C TOqKU speHuSI BOS-
MOJKHOCTU YJIyqrueHuSI CXO,n:UMOCTU pSI,n:OB U UHTerpaJIOB, a TaKJKe HMe-
eT cYIlLecTBeHHoe SHaqeHUe npu BblqUCJIeHUU HanpSIJKeHUH B yrJIOBbIX
TOqKax. KpuTepueM ,n:JISI BbI60pa KOHKpeTHblx BeJIUqUH NuT B (22.17)
SIBJISIeTCSI Tpe6yeMaSI TOqHOCTb y,n:oBJIeTBOpeHuSI rpaHUqHbIM YCJIOBUSIM.
HSJIOJKeHHaSI npoII.e,n:ypa perneHuSI HeocecuMMeTpUqHOH rpaHUqHOH
sa,n:aqu ,n:JISI CJIOSI C II.HJIHH,n:pUqeCKUM oTBepcTueM qUCJIeHHO peaJIUsyeT-
CSI C nOMoIlLbID 3BM. OCTaHoBuMcSI Ha aHaJIUSe pesyJIbTaTOB, nOJIyqeH-
HbIX npII paccMoTpeHIIII ,n:BYX KOHKpeTHblx CJIyqaeB ,n:eclJopMupoBaHHSI.


nOJIyqaeM


210


14*


211




I13YllMM Hanp5DKeHHOe COCTOHHMe O!(HOOCHO paCTHrllBaeMOrO CJ10H
C OTHOCMTeJIbHbIMM TOJIIII.MHaMll, paBHbIMll O!(HOMY II lleTbIpeM !(llaMeT-
paM OTBepCTIIH, T. e. h = 1 M h = 4. GHCJIO TIyaccoHa MaTepllaJIa npll-
HHTO paBHbIM lleTbIpeM (m = 4). B 060HX CJIYllaHX npH nepexo!(e K KO-
HellHOH CllCTeMe N = 15, T = 40.
B Ta6JI. 23 npHBe!(eHbI 3HalleHIIH BeJIllll11H Pn II gna
 !(JIH h = 1
C TOllHOCTblO !(O MHO:>KIITeJIH P. KaK Bll!(HO, Hell3BeCTHbIe Pn II gn B
CllCTeMe (22.5) !(OCTaTOllHO 6bICTpO npH6JIH:>KalOTCH K aCHl'vInTOTlllleCKIIM
3HalleHHSlM. 3TOT BbIBO!( nOJIHOCTblO coxpaHHeTcH II npll aHaJIll3e !(aH-
HbIX !(JIH h = 4.


Ta6JIHua 24


I zlh
h I 0.1 I 0.2 I I I I I I I I
0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1 3,08 3,08 3,08 3,08 3,06 3,05 3,04 3,00 2,96 2,86 2,76
4 3,04 3,04 3,04 3,04 3,04 3,05 3,05 3,05 3,04 2,97 2,63


I1MeH 3HalleHIIH Hell3BeCTHbIX Pn II gn> a TaK:>Ke 3HalleHHH <j:>YHK
1l1l
x (A) Ha BceM IIHTepBaJIe 113MeHeHIIH ee aprYMeHTa, MO:>KeM BbIll11CJIIITb
Bce BeJIllll11HhI, xapaKTepll3YlOIII.He HanpH:>KeHHO- !(e<lJopMHpoBaHHoe co-
CTOHHlle CJIOH. TIpll .3TOM 3a!(alla ll11CJIOBOH peaJIll3a
1l1l TaKoH B03MO:>K-
HOCTII cYIII.eCTBeHHo ynpOIII.aeTCH 6JIaro!(apH npllMeHeHlllO 113BeCTHbIX
MeTO!(OB yJIYll111eHHH CXO!(IlMOCTH PH!(OB II IIHTerpaJIOB, npe!(CTaBJIHlO-
III.IIX HCKOMbIe <j:>YHK
IlIl.
Hall60JIblllllH IIHTepec, eCTeCTBeHHO, B paccMaTpllBaeMoH 3a!(alle
npe!(CTaBJIHeT llCCJIe!(OBaHHe BeJIHllMHbI M xapaKTepa pacnpe!(eJIeHIIH
u 8 n
no TOJIIII.MHeoKpY:>KHbIX HanpH:>KeHMM a8 npM = 2"" Ha nOBepxHocTM
oTBepcTMH. CooTBeTcTBYlOIII.Me !(aHHbIe JlJIH 060MX paClleTHbIX CJIYllaeB
npMBe!(eHbI B Ta6JI. 24. BbIllMCJIeHHbIe no TpexMepHoMY pellleHlllO 3Ha-
lleHMH HanpH:>KeHMH YKa3bIBalOT, llTO onpe!(eJIeHHbIH Ha OCHOBe pellleHMH
nJIOCKOH 3a!(allM KO.3<lJ<lJM
MeHT KOH
eHTpaIlMM !(aeT !(OCTaTOllHO XOpO-
lllYlO KOJIMlleCTBeHHYlO o
eHKY !(JIH HanpH:>KeHMH a 8 !(a:>Ke B CJ1Yllae
BeCbMa MaJIoro OTBepCTMH (h = 4).
LLaHHbIe Ta6JI. 24 !(JIH h = 4 n03BOJIHlOT npoBecTM cpaBHeHMe nOJIY-
lleHHbIX 3HalleHMH OKPY:>KHbIX HanpH:>KeHIIH C pe3YJIbTaTaMM 3. IIlTepH-
6epra M K. IOHr!(aJIa [278] JlJIH nOJIynpoCTpaHcTBa C
MJIMH!(pMlleCKIIM
oTBepcTMeM. GTO KacaeTCH SHalleHIIH HanpH:>KeHMH B!(aJIM OT yrJIOBbIX
TOlleK, TO 3!(eCb Ha6JIlO!(aeTCH xopOlllee COrJIaCOBaHMe !(aHHbIX Ta6JI. 24
M cooTBeTcTBYlOIII.ero rpa<j:>MKa B [278] !(JIH m = 4. CJIe!(yeT OTMeTMTb,
lITO YKa3aHHbIH rpa<j:>MK TaK:>Ke OTMellaeT TeH!(eH
MlO K HeKOTopOMY YBe-
JIMlleHMlO OKpy:>KHbIX HanpH:>KeHMH Ha paCCTOHHMM nOpH!(Ka pa!(Myca
OTBepCTMH OT yrJIOBOH TOllKM. O!(H.aKO npM nO!(Xo!(e K yrJIOBbIM TOIIKaM
M oco6eHHO B caMMX TOIIKax paCXO:>K!(eHMe YBeJIMllMBaeTCH. I1CXO!(H MS
pa60ThI [278], B yrJIOBOH TOllKe npll 8 =
 nOCJIe HeCJIO:>KHOrO nepe-


212


.


ClleTa Ha CJ1YlIaH O!(HOOCHOro paCTH:>KeHMH nOJIYlIaeM no !(aHHbIM
1 1
pa60ThI [278] 2GP Of) = 2,25 npM 2GP a8 = 2,63 B Ta6JI. 24. OTCYT-
CTBMe B pa60Te [278] onMcaHIIH MeTO!(a HaXO:>K!(eHMH npe!(eJIbHbIX SHa-
lleHMH HeM3BeCTHbIX M cnoco6a McnOJIb30BaHMH llX npM BbIllMCJ1eHMM
HanpH:>KeHMH He n03BOJIHeT YKa3aTb npMllMHY OTMelleHHoro paCXO:>K!(e-
HIIH pe3yJIbTaTOB.
B paCCMOTpeHHbIX 3a!(allaX onpe!(eJIeHHbIH MHTepec npe!(CTaBJIHeT
XapaKTep nOBe!(eHIIH M KOJIMlleCTBeHHbIe o
eHKM !(JIH HanpH:>KeHMH a z M


Ta6JIHlla 25


.


h=1 h=4
z I I
h U z Tz8 U z Tze
2GP CDS 28 2GP sin 2B 2GP CDS 28 2GP sin 28
0 0,292 0 0,495 0
0,1 0,282 0,017 0,494 0,006
0,2 0,274 0,039 0,492 0,012
0,3 0,268 0,055 0,487 0,020
0,4 0,256 0,082 0,479 0,030
0,5 0,223 0,098 0,465 0,044
0,6 0,175 0,104 0,442 0,064
0,7 0,148 0,126 0,400 0,093
0,8 0,085 0,125 0,323 0,134
0,9 0,070 0,125 0,180 0,177
1,0 0,002 ° 0,002 °


.


'tz8 Ha 'nOBepXHOCTM OTBepCTMH r = 1. 3HalleHMH MX B 060MX pac-
CMOTpeHHbIX CJ1YllaHX no TOJIIII.MHe CJ10H npMBe!(eHbI B Ta6JI. 25.
AHaJIM3MpYH !(aHHbIe Ta6JI. 25, npe:>K!(e Bcero CJ1e!(yeT OTMeTMTb, lITO
B CJIYllae MaJIbIX OTBepCTIIH 3Hall11TeJIbHOH BeJIMllMHbI !(OCTMralOT HOp-
MaJIbHbIe HanpH:>KeHMH Ha nJIOIII.a!(HX, napaJIJIeJIbHbIX rpaHMllHbIM nJIOC-
KOCTHM CJIOH. XapaKTepHOH oco6eHHOCTblO B pacnpe!(eJIeHMM Kaca-
TeJIbHbIX HanpH:>KeHMH 'tz8 HBJIHeTCH CMeIlieHMe 06JIaCTll HaM6oJIblllerO
MX 3HalleHMH K yr JIOBbIM TOllKaM. LLOnOJIHHH !(aHHbIe Ta6JI. 25, YKa:>KeM,
llTO B CJ1Yllae h = 4 HaM60JIblllee 3HalleHMe HanpH:>KeHMH T z 8 paBHo
0,183 M !(OCTMraeTCH TaK :>Ke, KaK M !(JIH h = 1, Ha paccToHHIIM nOpH!(Ka
0,3 pa!(Myca OTBepCTMH OT yrJIOBOH TOllKM.



 23. KVH.qeHTpa.qoH HanpHmeHOD: B cJlve
C .qIlJIHH1I:pO'leCKOM OTBepCToeM npo o3m6e
PaccMoTpHl'v1 3a!(allY 0 KOH
eHTpa
MM HanpH:>KeHMH B CJIoe, oCJIa6-
JIeHHOM KpyroBbIM oTBepcTlleM, npM lIaCTOM M3rM6e. cI>opMaJIbHO .3Ta
sa!(alla HMlleM He OTJIMllaeTCH OT paCCMOTpeHHOH BbIllle TpeXMepHOH 3a!(a-
11M KMpllla. O!(HaKO C TOllKM 3peHIIH oco6eHHOCTeH HanpH:>KeHHOrO CO-
CTOHHMH npe!(CTaBJIHeT 3HallMTeJIbHbIH MHTepec. AHaJIM3 npM6JIM:>KeHHoro
pellleHMH .3TOH 3a!(alIll B paMKaX TeopMM KMpxro<j:>a - JIHBa 06HapY)i{H-
213




BaeT 3HaqHTeJIbHYIO rrOrpeIllHOCTb rrOJIyqaeMblX pe3YJIbTaTOB .l(JIR MaJIblX
oTBepCTHH [172]. YTOqHeHHe KOJIHqeCTBeHHblX OIJ.eHOK HarrpR2KeHHH Ha
OCHOBe TeopHH PeHccHepa rrpHBO.l(HT K TOMY, qTO K03<pqJHIJ.HeHT KOH-
lJ.eHTpalJ. HH OKa3bIBaeTCR paBHblM TpeM, T. e. TaKHM 2Ke, KaK H B rrJIOCKOH
3a.l(aqe KHpIlla. DOJIyqeHHble rrpH paccMOTpeHHH TpexMepHoH 3a.l(aqH
KHpIlla 3HaqeHHH OKPY2KHbIX HarrpH2KeHHH B yrJIOBOH TOqKe YKa3bIBa-
IOT Ha B03M02KHOCTb 3aBblIlleHHH pe3YJIbTaTOB paCqeTOB rro TeopHH PeHc-
cHepa.
PaccMoTpHM yrrpyrHH CJIOH, HaxO.l(HllJ.HHCH B O.l(HOpO.l(HOM HarrpH-
2KeHHOM COCTOHHHH, orrpe.l(eJIHeMOM e.l(HHCTBeHHbIM OTJIHqHblM OT HYJIH
l\OMrrOHeHTOM TeH30pa HanpH2KeHHH: -
1 (O)- M
2G ax - oz.


(23.1)


B IJ.HJIHH.l(pHqeCKOH CHCTeMe KOOp.l(HHaT HanpH2KeHHOe COCTOHHHe CJIOR
orrHCbIBaeTCH cJIe.l(YIOllJ.HMI1 3HaqeHHHMH HanpH2KeHHH:/
1 (0) _ 1 M + 1 M 28
2G a r - 2 oZ 2 oZ cos ,


I (0) _ 1 M 1 M 28
2G a e - 2 oZ-2 oZcos ,

 T(O) = -
 M o z sin 28
2G re 2 .


(23.2)


Iho6bl 113yqHTb B03MYllJ.eHHe HanpH2KeHHoro COCTOHHHH qHCTOro
H3rH6a (23.2), Bbl3bIBaeMoe KpyroBblM oTBepCTl1eM, Heo6xo.l(HMO nOJIY-
qHTb peIlleHl1e rpaHl1qHOH 3a.l(aqH
1 I I
2G at (1, z) = -2 M Oz- 2MoZ cos 28,
;G Tre (1. z) = -} Moz sin 28, Tn (1, z) = 0, (23.3)
a z (r, ::i:: h) = Tn (r, + h) = Tze (r, :f: h) = °


.AJIH CJIOH C OTBepCTl1eM rrpH yCJIOBHI1 paBeHCTBa HYJIIO HanpH2KeHHH Ha
6eCKOHeQHOCTI1.
PeIlleHl1e rpaHl1qHOH 3a.l(aQI1 (23.3) COCTaBI1M H3 HeCKOJIbKI1X CJIa-
raeMblX. Dpe2K.l(e Bcero, eCTeCTBeHHO, nOJIYQHM OT.l(eJIbHO oceCHMMeTpI1Q-
HYIO qaCTb HanpmKeHHoro COCTOHHI1H CJIOH, nop02K.l(aeMoro rpaHl1q-
HblMH YCJIOBI1HMH (23.3). COOTBeTCTBYIOllJ.He 3HaQeHHH KOMnOHeHT
BeKTOpa CMelIJ.eHHH H TeH30pa HarrpH2KeHHH JIerKO HaXO.l(HM B BH.l(e
1 2 1
U(I) = - M o - w(1) = - - M o In r
2 r' 2'


I 1 2
-a(l)=--M o -
2G r 2 r2 I


(23.4)


1 (I) _ 1 M 2
2G a e - 2 0 -;:2 .


OCTaJIbHble KOMrrOHeHTbl TeH30pa HarrpH2KeHHH B 3TOM CJIYQae 6Y.l(YT paB-
HbI HYJIIO.


214


,


DOCTpoeHHe Bblpa2KeHI1H .l(JIR KOMrrOHeHT TeH30pa I.;arrpH2KeHHH,
Heo6xo.l(HMblX lJ.JIR y.l(OBJIeTBOpeHHH Heocecl1MMeTpHqHOH QaCTI1 rpa-
HI1QHblX YCJIOBHH, CBO.l(HTCH JII1Illb K He3HaQHTeJIbHblM npeo6pa30Ba-
HI1HM <POPMYJI (21.4) 11 (21.7). TaK, 3aMeHHH B (21.7)shAz Ha chAz H
ch AZ Ha sh AZ, nOJIyqHM cJIe.l(YIOllJ.l1e Bblpa2KeHHH:

 a(2) = -
 CO s C (A) !VI (AZ) N kAdA cos k8 +
2Gr 2n
o


+ in S c:.
) [!VI(AZ)-
 shAz](
: Nk-+N
)AdACOSk8,
o


co



 a
2) =
 S C (A) [<vI (Az) - 2 sh AZ] N kAdA cos k8,
2G 2n 0


f


co
1 (2) Ire (I..) l 2 ] ( 'A'
2G ae = 2n J
 !VI (AZ) - m sh AZ r N k -
o


co


k 2 ) I 2 r
--;2 N k AdAcos k8 - 2n . m J C (A) shAzNkAdAcosk8,
o


co


.



 (2) _
 r c (I..) l dipi (1..2) _ 2A ch AZ ] N
AdA cos k8, (23.5 )
2G Trz - 2n J I.. dz
o


co


2A ch AZ] N kAdA sin k8,



 (2) _ _
 . ..!!..- r c (I..) l dipi (t..z)
2G Tz6 _. 2n r J 1.. 2 dz
o
2
 T
J = :n I C

) [!Vd AZ ) -
 sh AZ] ( :2 N k -+ N;) AdA sin k8.
o
3.l(eCb !VI (AZ) = sh AZ - Ah th Ah sh AZ + AZ ch AZ. aN Il (Ar) orrpe.l(eJIH-
eTCH <POPMYJIOH


.


1 k (I..r) y
 (I..) - Y k (w) I; (I..)
N k ( Ar ) = . .
[Y k (1..)]2 + [I k (1..)]2
Dpe.l(cTaBJIeHHH (21.7) .l(JIH HanpmKeHHH fIOJIYQeHbl KaK QaCTHblH
CJIyqai'l o6llJ.ero peIlleHI1H .l(JIH CJIOH C yqeTOM Toro, qTO KaCaTeJIbHble
HanpmKeHHH Ha rrJIOCKOCTHX Z = + h paBHbl HYJIIO. CJIe.l(oBaTeJIbHO, Bbl-
pa2KeHHH (23.5) TaK2Ke MorYT 6bITb HCrrOJIb30BaHbl JIHlllb B TOM CJIYQae,
Kor.l(a OTJII1QHblMH OT HYJIH Ha nJIOCKI1X rpaHHlJ.aX CJIOR RBJIHIOTCH Ha-
rrpH2KeHHH a z . B CBH3H C 3THM rrpH nOCTpOeHHI1 TpeTbeH COCTaB
HIOllJ.eH
o6mero pellleHI1H CJIe.l(yeT He TOJIbKO 3aMeHHTb cas (J.,nZ Ha -sm (J.,nZ H
2n-1
sin (J.,nZ Ha cos (J.,nz, HO 11 B KaQeCTBe (J.,n npl1HHTb (J.,n = 2h n. DpH


215




9TOM COXpamHOTCSl He06xoJI.lfMble CBOHCTBa nOJIHOTbl 9TOH COCTaBJISlIO-

eH Ha nOBepXHOCTlf r = ] nplf -h < z < h, If, KpOMe TOro, KaCa-
TeJIbHble HanpSl:>KeHlf T r 2 If T28 nplfHlfMalOT Ha nJIOCKOCTSlX Z = + h
HYJIeBble 3Ha'IeHlfSl.
KaCaTeJIbHble HanpSl:>KeHlfSl T

 B (23.5) nplf r = 1 06pallJ,a1OTCSI
B HYJIb. B CBSl3lf C 9TlfM rpaHlf'IHble YCJIOBlfSl JI.JISl Tn B (23.3) .lI.alOT B03-
MO:>KHOCTb lfCKJIIO'IlfTb OJI.HY nOCJIeJI.OBaTeJIbHOCTb npOlf3BOJIbHblX no-
CTOSlHHblX If npe.lI.CTaBJiTb, lfCX0JI.Sllf3 (2] .4), KOMnOHeHTbl TeH30pa HanpSl-
:>KeHlfH TpeTbeH COCTaBJISlIOllJ,eH 06llJ,ero perneHlfSl B BlfJI.e



 (3) =

 { 2B [K (
 t k2(a
+k2) +2 m-I k 2
2G (JT 21th
 an n k T2 2 m 2 +
n=1 an anT 2
2 k2 ) . ( a2 + k 2
+t(a n + k 2 )+ 1 +
 -anKh r+t n 2 +
+
anT a,.[ a
T


)] [ ( 2 ) ]}
m - 1 2 1 k 2 an a' t
+2 ma
T +kC n Kk -;:2+ 2(2t+""2 t -+Kk(l + 2) X
X sin anz cos ke.
1 (3) 1
'2 B 2 1 2 .
2G (J2 = - 21th
1 ,an nl K " (tk + ta n - ) - anrKk) +
2 1
2 an
+ k C n 2" tK k Sin anz cos ke,


CXJ
1 (3) _ 1 "" { lK ( m - 2 2 t 2 2 2 k 2
2G (J8 - - 21th L.J Bn k ---;:;z-a n + T2 k (k + an) + ----;:2 +
n=1
+ 2 m - 1 .
 ) _ an K
 ( tk2 + ta') + 2 m - 1 + k2 ) ] +
m
 T n m
+ k 2 C n l-}Kk(l +
2 t)-
n K
tI + {-)]} sin anzcoske, (23.6)
1 (3) 1
 { 2B [ ( k2 ) 22 ]
2G T r2 = 21th ;:1 an n - K" an' + anT + Kkt (k + an) -
- k 2 C n
n [+ K" - antK
]} COS anz COS ke,
2
 T
J = 2
h
1 {a
Bn[ - kK"t a
a: k 2 + kK
]-
- kC n r Kkt k

n -

 Kk]} COS anz Sin ke,
1 (3) k
f2 [ ( a
+k2 +2 m-1 X
2G TrS =- 21th

anBn -K k 1 +t 22m
n=1 l anT


216


f


...


1 k 2 ) . ( a
 + k 2 m - 1 1 )]
X
+
 +anK" t 2 +2 m +- -
anT anT anT a
T a
T
-kCn[K" (;
 +
 + t 2
2 ) -

n tK
]}SinanZSinke.
3JI.eCb BeJIlf'IJiHa t Onpe.lI.eJISleTCSl COOTBeTCTBYIO
lfM paBeHCTBOM B
(22.6). O.lI.HaKO, KaK Y:>Ke 6blJIO YKa3aHO Bblrne, nplf paCCMOTpeHlflf 3aJI.a-
6 2n-l
'Ilf 0 lf3rlf6e CJIOSl CJIeJI.yeT nOJIaraTb an = 2h n.


II
I,
I


Ta6JIHua 26






h=1 h=4
2
II Os I °2 I
2S Os I °2 I T2S
2GMo 2GMo CDS 2S 2GM O sin 2S 2GMOh 2GMoh cDs 28 2GMOh sin28
,
0 0 0 0,295 0 0 0,082
0,1 0,25 0,026 0,295 0,31 0,062 0,083
0,2 0,50 0,049 0,295 0,62 0,124 0,086
0,3 0,74 0,071 0,294 0,92 0,185 0,091
0.4 0,98 0,088 0,292 1,23 0,240 0,099
0,5 1,22 0,099 0,288 1,53 0,288 0,111
0,6 1,44 0,101 0,282 1,83 0,326 0,129
0,7 1,66 0,092 0,269 2,12 0,340 0,158
0,8 1,85 0,070 0,242 2,39 0,310 0,198
0,9 2,00 0,033 0,184 2,57 0,194 0,237
1,0 2,09 0,002 0 2,43 0,003 0


.


Bblpa:>KeJUISl (23.5) If (23.6) nocTpoeHbl TaKJiM 06pa30M, 'Iro6bl HY-
JIeBble rpaHliqHble YCJIOBliSl .lI.JISl KaCaTeJIbHblX HanpSl:>KeHJii't Ha nJIOC-
KOCTSlX CJIOSl Z = + h Ii YCJIOBlfe .lI.JISl T r2 nplf r = ] B (23.3) YJI.OBJIeTBOplf-
JIlfCb TO:>K.lI.eCTBeHHO. OCTaJIbHble rpaHlfqHble YCJIOBlfSl MorYT 6blTb Bbl-
nOJIHeHbl nYTeM HaJI.JIe:>KallJ,ero Bbl60pa 3Ha'IeHlfH npOlf3BOJIbHblX no-
CTOSlHHblX Bn If C n B (23.7) If npOlf3BOJIbHOH <PYHKlI.JiJi C (A) B (23.5).
HenOCpe.lI.CTBeHHaSl nO.lI.CTaHOBKa Bblpa:>KeHliH (23.5) If (23.6) B COOTBeT-
CTBYIOllJ,lfe COOTHorneHlfSl (23.3) nopo)KJl.aeT ClfCTeMY Tpex <PYHKlI. lfO -
HaJIbHblX ypaBHeHlfH, KOTopaSl nOCJIe OqeBlfJI.HblX npe06pa30BaHlfH nplf-
BOJI.lfTCSl K CJIe.lI.YIOllJ,eH ClfCTeMe paBeHcTB:


+


CXJ l ]
A h th Ah 'Ah 'A 2 1
X ( ) 4'A ( - ch2 'Ah ) = L Pn ('A 2 + 2)2 + gn 'A 2 + 2 '
n=1 an an
CXJ
 a 2
Pna] (an) + gn b ] (an) = X (A) n 2
o ('A 2 + a n )2
CXJ
Pn a 2 (an) + gnb2 (an) = - 2 r X (A) 1 2

 'A 2 ('A 2 + an)
X N 2 (A) N; (A) AdA-
o .
anh


N .
 3Mo
2 (A) N 2 (A) AdA +
 .
anh


( a
 _ - m l ) X
'A 2 + a



(23.7)


217




CHCTeMa (23. 7)aHaJIOrH1.JHa nOJIY1.JeHHOH BbIllie npH paccMoTpeHHH
TpexMepHoH 3a.D.a1.JH KI1p1II8 CI1CTeMe (22.5). 3.D.eCb TaK)Ke K03<jJ<jJHLl.l1eH-
TbI npH Hel13BeCTHbIX Pn H gn onpe.D.eJUHOTCH <jJopMYJIaMH (22.7). CBH3b
Me)f{.D.Y HeH3BeCTHbIMH B 9TOH CI1CTeMe H np0l13BOJIbHbIMI1 nOCTOHHHbIMH
H <jJYHKUHHMI1 B BbIpa)f{eHHHX (23.5), (23.6) YCTaHaBJIHBaeTCH COOTRO-
meHI1HMI1


(- I)n+1 < .
Pn = nh a"Kdan) Bm


(- It' 2", , C
gn = 1th anI\j! (an) m


(23.S)


x (A.) = _ C (A.) 21.. c
 I..h
nhN 2 (A)
Ta 6 J] Ii U a 27 KOHKpeTHbIe BbI1.JHCJIeHHH, KaK j
H B CJIY1.Jae 3a.D.a l IH Kl1prna, npl1Be- I
.D.eHbI .D.JI
 .D.BYX 3Ha1.JeHI1H OTHOCH-
TeJIbHOH TOJIll1.I1HbI CJIOH: h = 1 II
h = 4. O.D.HaKO .D.JIH Y.D.06CTBa
cpaBHeHI1H pe3YJIbTaTOB paClIeTa
C .D.aHHbIMI1 TeopHH KHpxro<jJa H
PeHcCHepa [I721 npl1HHTO m = 3.
KOHellHaH CI1 CTeM a , OTBeqaI0ll1.aH
CI1CTeMe (23.7), <jJOpMHpOBaJIaCb Ha
OCHOBe 113BeCTHbIX aCHMnTOTl1l1e-
CKIIX CBOHCTB HelJ3BeCTHbIX, KOTO-
pbIe B .D.aHHOM CJIYlIae TaK)f{e onH-
CbIBalOTCH COOTHorneHHHMI1 (22.16).
B Ta6JI. 26 nplJBe.D.eHbI 3HaqeHHH OTJIl1l1HbIX OT HYJIH KOMnOHeHTOB
TeH30pa HanpH)f{eHHH Ha nOBepxHocTH UHJIHH.D.pHlIecKoro oTBepCTI1S1
.D.JISI .D.BYX paCqeTHbIX CJIyqaeB. 3HaqeHHe OKPY)f{HbIX HanpSl)KeHHH
as .D.aHbI B CeqeHHI1 8 =
 , r.D.e OHH MaKClJMaJIbHbI. BblqHCJIeHHbIe npH
z = h 3HaqeHI1H HanpH)f{eHHH a z YKa3bIBalOT Ha BbICOKYIO TOqHOCTb
Y.D.OBJIeTBOpeHHH rpaHHqHWX YCJIOBHH.
B COOTBeTCTBl111 C .D.aHHbIMH Ta6JI. 26 .D.JIH h = I K09<jJ<jJHuHeHT KOH-
ueHTpaUHH OKPY)f{HbIX HanpH)f{eHI1H k paBeH 2,09. KmlCcHlIecKaH Teo-
pHH lJ3rH6a nJIaCTHH .D.aeT .D.JIH JIlO6oro OTHoweHHH TOJIll1.HHbI CJIOH K
pa.D.HYcy oTBepcTHH k* = 1,80. Do .D.aHHWM TeOpHlJ PeHcCHepa, Ii =
= 2,30.
DpHlIHHOll 3HaqlJTeJIbHOrO pa3JIHqHH 3HalleHHH K03<jJQJHuHeHTa KOH-
ueHTpaUHH, Bblql1CJIeHHbIX no TOqHOMY pelIIeHHIO 11 Ha OCHOBe TeopHH
PellCCHepa, HBJIHeTCH, nO-BH.D.HMOMY, TO 06CTOHTeJIbCTBO, qTO nOJIO)f{eH-
HbIe B OCHOBY .:'HOll TeOpl1H npe.D.nOJIO)f{eHHH 0 xapaKTepe pacnpe.D.eJIeHHSI
no TOJIll1.HHe HOpMaJIbHbIX H KaCaTeJIbHbIX HanpH)f{eHHH He COrJIaCYIOTCH
C .D.eHCTBHTeJIbHbIM HX pacnpe.D.eJIeHHeM B paCCMaTpl1BaeMOM CJIyqae
.D.e<jJOpMHpOBaHHH CJIOH. JIHHeHHaH 11 KBa.D.paTliqHaH annpOKCI1MaUHH
)I.JIH HanpH)f{eHHll at) 11 'tz(j COOTBeTCTBeHHO no TOJIll1.HHe CJIOH B CJIYlIae
n = 1 HBJIHIOTCH HeY.D.OBJIeTBOpHTeJIbHbIMli.


2 I us I Uz I
z8
h 2GMoh 2GMOh CO
 28 2GMOh sin 28
0,91 2,58 0,177 0,240
0,92 2,59 0,159 0,244
0,93 2,59 0,138 0,246
0,94 2,59 0,116 0,244
0,95 2,58 0,094 0,236
0,96 2,56 0,065 0,218
0,97 2,54 0,042 0,188
0,98 2,51 0,023 0,144
0,99 2,48 O,Oll 0.083


218


'B CJIyqae h = 4 npe.D.nOJIO)f{eHHH TeopHH PellCCHepa HaX0.D.SlTCH B
HBHOM npOTHBOpeqHH C .D.ellCTBHTeJIbHbIM pacnpe.D.eJIeHHeM HanpH)f{eHHH
as H 'tzs no TOJIll1.HHe. DpH TaKoll OTHOCHTeJIbHOll TOJIll1.HHe CJIOH Ha-
npH)f{eHHH as HMelOT HaH60JIbrnee 3HaqeHHe Y)f{e He B TOqKaX yrJIOBOll
OKPY)f{HOCTH. Pacnpe.D.eJIeHHe HanpmKeHHH TzS TaK)f{e .D.OBOJIbHO CBoe-
06pa3HO H xapaKTepl13yeTcH HaJIHqHeM MaKcHMYMOB B6JIH3H nOBepx-
HOCTell Z = + h.
LlJIH nOJIHOH xapaKrepHcTHKH HanpH)f{eHHOrO COCTOHHHH CJIOH B
cJIyqae h = 4 .D.aHHbIX Ta6JI. 26 He.D.OCTaTOqHO. DOJIee nOJIHYIO KapTHHY
HanpH)f{eHHOCTH B oKpecTHocTH yrJIOBOH OKpy*HOCTH .D.alOT 3HaqeHHH
HanpH)f{eHHll, npIIBe.D.eHHble B Ta6JI. 27.


I: '
I
I




r.ltaea 6
YCTAHOBIIBIIIIIECH IWJIEBAHIIH ITPHMOyrOJIbHOn ITPII3MbI
II RPyrOBbIX D;IIJIIIH,[J;POB


HOH npl1SMbI H KpyroBoro UHJIHH,lI,pa (cnJIOIl1HOrO H nOJIoro) KOHeq-
HOH ,lI,JIHHbI.
B paMKax HCnOJIbsyeMoro B pa60Te MeTO,lI,a paccMoTpeHHe sa,lI,aq
,lI,JIfI YKasaHHbIx ynpyrl1x TeJI KOHeqHbIX pasMepoB cYIlLecTBeHHbIM 06-
pasoM 6asI1pyeTcfI Ha HSBeCTHbIX peIl1eHHflX ,lI,JIfI CJIOfl (nJIaCTHHKH)
II 6eCKOHeqHoro UI1JIHH,lI,pa. TIepBbIe HCCJIe,lI,OBaHHfI, nOCBflIUeHHbIe BO-
npocaM pacnpocTpaHeHHfI npO,lI,OJIbHbIX HOpMaJIbHbIX BOJIH B CnJIOIl1HOM
KpyroBOM UHJII1H,lI,pe, BbInOJIHeHbI JI. TIoxraMMepoM [2481 H 11. KpIl
[201 I. I1MI1 6bIJIO nOJIyqeHO ,lI,HCnepCI10HHOe ypaBHeHl1e, onpe,lI,eJIflIO-
IUee xapaKTepHCTI1KH pacnpocTpaHflIOIUHxCfI HOpMaJIbHbIX BOJIH B
UHJIHH,lI,pe. XOTfI YKasaHHbIe pa60TbI ony6JII1KOBaHbI ,lI,aBHO, ql1CJIeHHble
pesYJIbTaTbI peIl1eHHfI ,lI,HcnepCHoro ypaBHeHI1f1 nOJIyqeHbI JIHIl1b B no-
CJIe,lI,Hee BpeMfI. 3,lI,eCb CJIe,lI,yeT YKasaTb pa60TbI [204,240, 2471, B KOTO-
pbIX ,lI,OCTaTOqHO nO,lI,p06HO HCCJIe,lI,OBaHO nOBe,lI,eHHe cneKTpa qaCTOT
npO,lI,OJIbHbIX HOpMaJIbHbIX BOJIH. BblBO,lI, ,lI,HCnepC110HHoro ypaBHeHI1f1
,lI,JIfI nOJIoro UI1JIHH,lI,pa 11 HeKoTopbIe pesYJIbTaTbI ql1CJIeHHOrO aHaJIHSa
noro ypaBHeHI1f1 cO,lI,ep2KaTCfI B pa60Te [2131.
TIp06JIeMbI, CBflsaHHbIe C peIl1eHl1eM ,lI,I1HaMHqeCKI1X sa,lI,aq n;JIfI CJIOfl,
TaK2Ke ,lI,OCTaTOqHO XOp0Il10 OCBeIlLeHbI B cOBpeMeHHoH JIHTepaType [42,
110, 1431. TIepBbIe HCCJIe,lI,OBaHHfI OTHOCflTCfI eIUe K LL. PeJIeIO [2501 II
f. JI3M6y [2311. B CBflSH C 3TI1M ,lI,l1cnepCl10HHbIe ypaBHeHHfI, onpe,lI,eJIfI-
IOIUHe pacnpocTpaHflIOIlLHecfI B CJIoe HOpMaJIbHble BOJIHbI, HOCflT HX
HMfI. TIepBbIe ql1CJIeHHbIe pesYJIbTaTbI,lI,JIfI cpasoBbIx CKopocTeH HOpMaJIb-
HbIX BOJIH HaHHHSIl1HX HOMepoB nOJIyqeHbI f. JI3M60M [2311. B nOCJIe,lI,y-
IOIUeM qHCJIOBOH aHaJIHS ypaBHeHI1H PeJIefl - JI3M6a 6bIJI npe,lI,MeTOM
MHoroql1CJIeHHbIX I1CCJIe,lI,OBaHHH [239, 246, 2711, B pesYJIbTaTe npOBe,lI,e-
HI1f1 KOTOpbIX BonpocbI nOBe,lI,eHI1f1 HOpMaJIbHbIX BOJIH B 6eCKOHeqHOM
CJIoe (nJIaCTI1HKe), KaK H B cJIyqae 6eCKOHeqHoro KpyroBoro UHJIHH,lI,pa,
TaK2Ke XOp0Il10 I1syqeHbI.
I1CCJIe,lI,OBaHl1e ,lI,HCnepCI10HHbIX ypaBHeHI1H B cJIyqae CJIOfl 11 UHJIHH,lI,-
pa npen;CTaBJIfleT C060H ,lI,aJIeKO HeTpIlBl1aJIbHYIO sa,lI,aqy. Ba2KHOCTb 3TO-
ro I1CCJIe,lI,OBaHI1f1 onpe,lI,eJIfleTCfI He TOJIbKO nOTpe6HOCTbIO nOSHaTb
xapaKTepHbIe oc06eHHOCTI1 YCTaHOBI1BIl1HXCfI BOJIHOBbIX ,lI,BI12KeHHH B
UI1JII1H,lI,pe 11 CJIoe, HO H BOSM02KHOCTbIO HCnOJIbSOBaTb nOJIyqeHHbIe pe-
SYJIbTaTbI npH peIl1eHHH Il1HpOKOro Kpyra sa,lI,aq. B qaCTHOCTI1, SHaHl1e
cneKTpa HOpMaJIbHbIX BOJIH He06xo,lI,I1MO npH l13yqeHHH pacnpOCTpaHe-
HHfI HMnYJIbCa [1291. LLpyroe Ba2KHOe npHMeHeHl1e pesYJIbTaTOB aHaJII1Sa
,lI,I1CnepCHoHHWX ypaBHeHI1H BOSHI1KaeT npl1 y,lI,OBJIeTBOpeHHH rpaHHq-
HbIX YCJIOBI1H Ha nOBepXHOCTflX cooTBeTcTBYIOIUI1X ynpyrl1x TeJI KO-
HeqHbIX pasMepoB (npflMoyroJIbHafl npl1SMa, UHJIHH,lI,p KOHeqHOH llJII1-
HbI).
TIlLaTeJIbHOe HCCJIe,lI,OBaHHe xapaKTepHCTI1K pacnpocTpaHflIOIlLl1xCfI
H HepacnpocTpaHflIOIlLl1xCfI npO,lI,OJIbHbIX BOJIH B UHJII1H,lI,paX H nJIaCTH-
Hax ,lI,aJIO BOSM02KHOCTb nOCTpOI1Tb npl16JII12KeHHbIe TeopHH ,lI,JIfI onHca-
HHfI ,lI,I1HaMl1qeCKOrO nOBe,lI,eHI1f1 nJIaCTI1H, I1MeIOIlLl1e 60JIee Il1HpOKYIO
06JIaCTb npI1MeHI1MOCTI1, qeM TeOpl1H TUna KHpxrocpa [239, 2401.
O,lI,I1H I1S BOSM02KHbIX MeTO,lI,OB HsyqeHI1f1 YCTaHOBHBIl1HXCfI KOJIe6a-
HI1H TeJI KOHeqHbIX pasMepoB flBJIfleTCfI eCTeCTBeHHbIM 0606IlLeHI1eM


3a,lI,aqa 06 YCTaHOBI1BIl1HXCfI BbIHY2K,lI,eHHbIX KOJIroaHllflx ynpyroro
TeJIa CBO,lI,HTCfI K peIl1eHHIO HeKoTopoH KpaeBoH Sa,lI,aqH ,lI,JIfI CHCTeMbI
ypaBHeHI1H 3JIJIHnTllqeCKoro THna. fJIy60Kile paSJIHqHfI B cpllsHKe flB-
JIeHHH CTaTl1qeCKOrO 11 ,lI,I1HaMl1qeCKOrO ,lI,e<popMl1pOBaHI1f1 ynpyrHx
TeJI HaXO,lI,flT oTpaLKeHl1e B TOM, qTO peIl1eHl1e YKasaHHOH Sa,lI,aqH SaBI1-
CHT OT napaMeTpa - qaCTOTbI KOJIe6aHI1H.
TIpl1MeHeHHbIH BbIIl1e npH I1syqeHHI1 CTaTl1qeCKOrO ,lI,ecpOpMl1pOBaHHfI
ynpyrHx TeJI KOHeqHbIX paSMepOB MeTO,lI, nOCTpoeHHfI 06IUI1X peIl1eHHH
KpaeBbIX sa,lI,aq, OqeBH,lI,HO, M02KeT 6bITb I1CnOJIbSOBaH H npl1 paCCMOTpe-
HI1I1 sa,lI,aq 06 YCTaHOBHBIl1HXCfI KOJIroaHHflX. ECTecTBeHHo, qTO B 3TOM
cJIyqae y,lI,OBJIeTBOpeHl1e rpaHl1qHbIX YCJIOBHH TaK2Ke npI1BO,lI,I1T K 6eCKO-
HeqHbIM CHCTeMaM ypaBHeHI1H OTHOCHTeJIbHO npOI1SBOJIbHbIX nOCTOflH-
BbIX B BbIpa2KeHI1f1X ,lI,JIfI cMeIlLeHI1H 11 Hanpfl2KeHHH.
XapaKTepHoH oc06eBHOCTbIO sa,lI,aq ,lI,I1BaMI1KI1 ,lI,JIfI TeJI KOHeqBWX
pasMepoB flBJIfleTCfI TO, qTO aMnJIHTY,lI,HbIe xapaKTepHCTI1KH Hanpfl2KeH-
HO-,lI,ecpopMl1pOBaBBoro COCTOflBI1f1 CJI02KHbIM 06paSOM SaBI1CflT OT qaCTO-
TbI. <PI1Sl1qeCKafl CYIlLHOCTb paCCMaTpHBaeMoro flBJIeHHfI YKaSbIBaeT, qTO
npH HeKOTopbIX SBaqeHI15IX qaCTOTbI aMnJII1TY,lI,bI Hanpfl2KeHHH H cMeIlLe-
HHH 11, eCTeCTBeHBO, BeHSBeCTHbIe B 6eCKOHeqHbIX CI1CTeMaX ,lI,0.12KHbI
CTaBOBHTbCfI 6eCKOBeqHO 60JIbIl1HMH.
3TO 06cToflTeJIbCTBO npHBO,lI,HT K TOMY, qTO 06blqBbIe MeTO,lI,bI pe-
rneHHfI 6eCKOHeqHbIX CI1CTeM, OnHp3[{)IUl1eCfI Ha npe,lI,Bapl1TeJIbHOe
,lI,OKaSaTeJIbCTBO HX perYJIflpHOCTI1, He npl1MeHI1MbI npH aBaJIHSe
YCTaHOBI1BIl1I1XCfI KOJIe6aBI1H, nOCKOJIbKY C caMoro BaqaJIa flCHa HeBOS-
M02KHOCTb nOJIyqeHI1f1 oueHOK perYJIflpHOCTI1, paBHopMeBbIX OTHOCH-
TeJIbHO qaCTOTbI KOJIe6aBI1H.
TIpl1 paCCMOTpeHl111 CTaTI1QeCKI1X rpaBI1QHbIX sa,lI,aQ B npe,lI,bI,lI,YIlLHX
rJIaBaX OCHOBOH 3cpcpeKTHBHoro peIl1eHHfI 6eCKOHeQHWX CI1CTeM 6bIJIa
YCTaHOBJIeHHafl CBflSb Me2K,lI,y aCHMnTOTI1QeCKI1MI1 CBOHCTaMI1 HeI1SBeCT-
HbIX H oc06eHHocTflMH nOJIfI Hanpfl2KeHI1H, nop02K,lI,aeMbIX HaJIl1qHeM
yr JIOBbIX TOQeK. OKaSblBaeTCfI, QTO TaKoro 2Ke THna CBflSb HMeeT MeCTO
H B CJIyqae sa,lI,aq ,lI,HHaMHKH. B CBflSH C 3THM nOflBJIfleTCfI BOSM02K-
BOCTb paSBHTb ,lI,OCTaTOQBO 3cpcpeKTHBHbIe cnoc06bI KOJII1QeCTBeHHoro
aHaJIHSa qaCTOTHbIX xapaKTepHCTllK H Hanpfl2KeHBOCTH ynpyrHx TeJI
KOHeqHbIX paSMepOB. HH2Ke 3Ta BOSM02KHOCTb 6Y,lI,eT peaJIHSOBaHa npH
paCCMOTpeHI1H sa,lI,aQ 06 YCTaHOBHBIl1HXCfI KOJIroaHHflx npflMoyroJIb-
220


221


Ii
III
II
II
II




MeTO,lI,a O,lI,HOpO,lI,HbIX perneHl:lii B CTaTH'leCKHX Sa,lI,a'laX TeOpHH ynpyroc-
TH Ha BOSM02KHOCTb TaKOrO 0606IlLeHHfI 6bIJIO BnepBbIe YKaSaHO
K. K3pTHCOM [1291. TIpH 3TOM ,lI,HCnepCHoHHbIe ypaBHeHHfI B sa,lI,a'lax ,lI,H-
HaMHKH HrpaIOT TY 2Ke pOJIb, 'lTO H ypaBHeHHfI ,lI,JIfI c06CTBeHHbIX SHa-
'leHHH B sa,lI,a'lax CTaTHKH. I1cnOJIbSOBaHHe MeTO,lI,a O,lI,HOpO,lI,HbIX perne-
HHH nOSBOJIHJIO nOJIY'lHTb pfl,lI, HHTepecHbIx KOJII1'leCTBeHHblX 11 Ka'leCT-
BeHHbIX BbIBO,lI,OB 0 cneumpHKe BOJIHOBorO npouecca B6JIHSI1 TOpUOB
nOJIynOJIOCbI [272, 2731 H nOJIYUHJIHHn;pa [2791.
LLpyroH nYTb I1CnOJIbSOBaHI1f1 perneHI1H ,lI,JIfI 6eCKOHe'lHbIX UHJIHH,lI,-
pa 11 CJIOfl npl1 HSY'leHI1I1 BbIHY2K,lI,eHHbIX KOJIe6aHHH CBflsaH C npHMeHe-
I-IHeM paSBI1TOrO npH peIl1eHHI1 sa,lI,a'l CTaTI1KH MeTO,lI,a JIaMe. TIO-BH-
,lI,I1MOMY, nepBbIe pesYJIbTaTbI S,lI,eCb npI1Ha,lI,JIe2KaT A. 30MMepcpeJIb,lI,y
[2611, paccMOTpeBIl1eMY sa,lI,a'lY 0 KOJIe6aHI1f1X napaJIJIeJIenHne,lI,a. O,lI,-
HaKO B CBOHX HCCJIe,lI,OBaHHflX OH orpaHI1'lHBaJICfI JII160 cneUHcpH'leCKH-
MI1 rpaHI1'lHbIMH YCJIOBHflMI1 (sa,lI,aHbI HOpMaJIbHbIe CMeIlLeHl1f1 H Kaca-
TeJIbHble Hanpfl2KeHI1f1), JIH60 'laCTHbIM CJIY'laeM ,lI,ecpopMl1pOBaHHfI
(061)eMHOe paCIl1l1peHl1e paBHo HYJIIO.)
TIoCJIe,lI,YIOIlLl1e Ba2KHbIe pa60TbI, CBflsaHHbIe C HCnOJIbSOBaHHeM Me-
TO,lI,a JIaMe npH I1syqeHHH KOJIe6aHHH napaJIJIeJIenHne,lI,a 11 UI1JII1H,lI,pa
KOHeqHOH ,lI,JII1HbI, npI1Ha,lI,JIe2KaT C. KaJIHCKoMY [227, 228, 2291. B ero
pa60Tax paCCMOTpeHbI He TOJIbKO cneUHcpl1qeCKl1e CJIyqal1 rpaHl1qHbIX
YCJIOBI1H, He npl1BO,lI,flIlLHe 06bIqHO K 6eCKOHe'lHbIM cHCTeMaM, HO H
OCHOBHbIe rpaHl1qHbIe Sa,lI,a'lH ,lI,JIfI napaJIJIeJIenHne,lI,a H UI1JIHH,lI,pa.
B 3THX sa,lI,aqax y,lI,OBJIeTBOpeHHe rpaHHqHbIX YCJIOBHH eCTeCTBeHHO npl1-
BO,lI,HT K 6eCKOHe'lHbIM CHCTeMaM JII1HeHHbIx ypaBHeHHH.
B CBOI1X HCCJIe,lI,OBaHHflx C. KaJII1CKHH OCHOBHoe BHI1MaHHe y,lI,eJIl1JI
,lI,OKaSaTeJIbCTBY perYJIflpHOCTH 6eCKOHeqHbIX CI1CTeM. TIpH TaKOM no,lI,-
xO,lI,e npHXO,lI,HTCfI CYIlLeCTBeHHO OrpaHI1'lI1BaTb ,lI,l1anaSOH paCCMaTpHBae-
MbIX qaCTOT, qT06bI HCKJIIOql1Tb peSOHaHCHbIe 06JIaCTH. Bonpoc 06 OTbIC-
KaHl111 c06cTBeHHbIx qaCTOT ynpyroro TeJIa B YKasaHHbIx pa60Tax He
CTaBI1TCfI. TIpe,lI,nOJI02KeHHble C. KaJIHCKI1M MeTO,lI,bI peIl1eHI1f1 6eCKOHeq-
HbIX CHCTeM He nOSBOJIflIOT nOJIyqHTb ,lI,OCTOBepHble KOJIl1qeCTBeHHbIe
OueHKI1 ,lI,JIfI c06CTBeHHWX qaCTOT H xapaKTepHCTHK cpOpM KOJIe6aHHH
paCCMOTpeHHbIX TeJI KOHeqHbIX paSMepOB.
TIepBoH pa60TOH, B KOTOpOH ,lI,aHO ,lI,OCTaTOqHO qeTKOe I1SJI02KeHHe
H,lI,el1 0606IlLeHI1f1 MeTO,lI,a JIaMe Ha CJIyqaH sa,lI,aq 06 YCTaHOBHBIl1HXCfI
KOJIe6aHHflx ynpyrl1x TeJI KOHe'lHbIX pasMepoB (npflMoyroJIbHafl nJIac-
THHa, KOHeqHbIH UHJIHH,lI,p), flBJIfleTCfI pa60Ta A. r. BJIaCOBa [551.
B nOCJIe,lI,Hee BpeMfI TaKOH no,lI,XO,lI, I1CnOJIbSOBaH npH paccMOTpeHI1H pfl,lI,a
KOHKpeTHbIx rpaHl1qHbIX sa,lI,aq [15,65, 66, 1211. 06IlLeH xapaKTepHoH
qepTOH HaSBaHHbIX pa60T flBJIfleTCfI TO, qTO peIl1eHHe Sa,lI,aqH ,lI,OBO,lI,I1TCfI
JIHIl1b ,lI,0 nOJIyqeHI1f1 6eCKOHe'lHbIX CHCTeM H He ,lI,aeTCfI KaKHX-JII160
peKOMeH,lI,aUHH OTHOCHTeJIbHO TaKoro nO,lI,XO,lI,a K HX peIl1eHHIO, KOTOpbIH
Y'lHTblBaJI 6bI cpl1SHqeCKHe oc06eHHOCTH paCCMaTpHBaeMbIX sa,lI,a'l (Ha-
JIl1qHe peSOHaHCOB).
B 3TOH rJIaBe paCCMOTpHM pfl,lI, sa,lI,aq 06 YCTaHOBHBIl1HXCfI KOJIe6a-
HHflX ynpyrHx TeJI KOHe'lHbIX pa3MepoB. TIpH peIl1eHHH 3TI1X sa,lI,a'l
nOKa2KeM, qTO oCHoBaHHbIH Ha npHBe,lI,eHHH rpaHHqHbIX sa,lI,a'l K 6ecKo-
222


HeqHbIM CHCTeMaM MeTo,lI, HCCJIe,lI,OBaHHfI YCTaHOBI1BIl1HXCfI KOJIroaHHH
CTaHOBHTCfI 3cpcpeKTI1BHbIM JII1Il1b Tor,lI,a, Kor,lI,a y,lI,aeTCfI YCTaHoBHTb
aCHMnTOTl1qeCKHe CBOHcTBa Hel1SBeCTHbIX B 6eCKOHeqHbIX CI1CTeMax.
TIpl1 3TOM YKa2KeM 3cpcpeKTI1BHble cnoc06bI OTbICKaHHfI c06cTBeHHbIx
qaCTOT H n;HHaMl1qeCKOH Hanpfl2KeHHocTH.
KaK 11 B cJIY'lae CTaTI1K 11 , nepBOH paccMoTpHM nJIOcKYIO sa,lI,aqy
06 YCTaHOBI1BIl1I1XCfI KOJIe6aHI1f1X npflMoyrOJIbHOH npHsMbI (nJIaCTHH-
KH). 3Ta HaH60JIee npocTafl HS paCCMaTpl1BaeMOro Kpyra sa,lI,aqa ,lI,aeT
BOSM02KHOCTb I1SJI02KI1Tb cYIlLecTBo ,lI,OCTaTOqHO 06IlLero MeTO,lI,a peIl1eHI1f1
,lI,I1HaMl1qeCKI1X KpaeBbIx sa,lI,aq TeOpHI1 ynpyrocTI1. TIpH 3TOM BCKpbIBa-
eTCfI xapaKTep TPY,lI,HOCTeH, BOSHI1KaIOIUI1X npl1 nOnbITKe nOJIY'lI1Tb
KOJIl1qeCTBeHHYIO KapTI1HY ,lI,I1HaMI1'leCKOH Hanpfl2KeHHOCTH H qaCTOT-
HbIe xapaKTepHCTI1KI1 KOHKpeTHoro ynpyroro TeJIa.
TIOMI1MO sa,lI,aql1 0 nJIOCKI1X KOJIe6aHHflX npflMoyrOJIbHOH npl1SMbI
paCCMOTpl1M oceCI1MMeTpl1qHbIe KOJIe6aHI1f1 cnJIOIl1HorO H nOJIoro UHJII1H,lI,-
pOB KOHeqHOH ,lI,JIHHbI. 3cpcpeKTI1BHOe peIl1eHHe 3TI1X sa,lI,aq YKaSbIBaeT,
'lTO paSBI1BaeMbIH MeTO,lI, ,lI,aeT BOSM02KHOCTb nO,lI,OHTI1 K I1CCJIe,lI,OBaHI1IO
KOJIe6aHHH nO'lTH Bcex ynpyrl1x TeJI, ,lI,JIfI KOTOpbIX B npe,lI,bI,lI,YIlLHX
rJIaBaX nOJIyqeHbI peIl1eHI1f1 CTaTHqeCKHX Sa,D,aq. I1CKJIIOqeHl1e COCTaB-
JIflIOT S,lI,eCb Sa,lI,a'lH ,lI,JIfI KOHeqHbIX ynpyrl1x TeJI, rpaHH'lHafl nOBepx-
HOCTb KOTOpbIX 06paSOBaHa qaCTflMI1 Koop,lI,I1HaTHbIX nOBepXHOCTeH
ccpepl1qeCKOH CI1CTeMbI Koop,lI,I1HaT. PeIl1eHl1e 3TH X sa,lI,aq OCJI02KHfleTCfI
TeM, qTO peIl1eHHe ,lI,I1HaMl1qeCKOH Sa,lI,a'lH ,lI,JIfI 6eCKOHeqHOrO KpyroBoro
KOHyca HaCTOJIbKO rpOMOS,lI,KO H MaJIO I1syqeHO, qTO I1CnOJIbSOBaHHe
ero npl1 y,lI,OBJIeTBOpeHHI1 rpaHHqHWX YCJIOBHH Ha cooTBeTcTBYIOIlLHX
nOBepxHocTflX TeJI KOHeqHbIX pasMepoB saTpY,lI,HeHo.
TIoCTpoeHHe H aHaJIHS 06IlLHX peIl1eHHH B 3TOH r JIaBe conpOB02K,lI,a-
IOTCfI pfl,lI,OM npl1MepOB paCqeTa, ueJIbIO KOTOpbIX flBJIfleTCfI HJIJIIOCTpaUHfI
BOSM02KHO u CTel1 MeTO,lI,a, a He aHaJII1S qaCTOTHbIX xapaKTepHCTHK H cpOpM
KOJIe6aHI111 KOHKpeTHWX ynpyrl1x TeJI.
06CY2KD;eHl1e xapaKTepHbIX oco6eHHOCTeH B ,lI,I1HaMHqeCKOM nOBe-
,lI,eHI1I1 paCCMaTpl1BaeMbIX Tl1nOB ynpyrl1x TeJI C HCnOJIbSOBaHl1eM npe,lI,-
JI02KeHHoro MeTO,lI,a M02KHO HaHTH B pa60Tax [79-81, 931. B 3TI1X 2Ke
pa60Tax cO,lI,ep2KI1TCfI ,lI,OBOJIbHO nOJIHbIH nepeqeHb ny6JII1KaUI1H, nOCBfI-
IlLeHHbIX, B qaCTHOCTH, TaKoMY HHTepecHoMY flBJIeHHIO, KaK KpaeBoH
pesoHaHc.



 24. TIOCTpOCHHC o6rn;cro pCrnCHHJI 3a):1;a1JD
o BhIHym):1;CHHhIX KOJIc6aHHJlX npJlMoyrOJIbHOii npH3MhI


PaCCMOTpHM sa,lI,aqy 06 YCTaHOBI1BIl1HXCfI BbIHY2K,lI,eHHbIX KOJIe6aHHflx
6eCKOHeqHOH npflMoyroJIbHOH npl1SMbI B nJIOCKOCTI1, nepneH,lI,HKYJIflp-
HOH K ee OCI1 [731. 3a,lI,a'la SaKJIIOqaeTCfI B onpe,lI,eJIeHI1H HanpflLKeHHo-
n;ecpopMl1pOBaHHoro COCTOflHI1f1 npHsMbI npl1 paSJII1'lHbIX SHaqeHI1f1X 'lac-
TOTbI BblHYLK,lI,eHHbIX CI1JI, a TaK2Ke B OTbICKaHHH c06CTBeHHblx qaCTOT.
TIocJIe OT,lI,eJIeHHfI rapMOHl1qeCKOrO BpeMeHHoro MH02KHTeJIfI e icu ( B
BeJIl1ql1HaX, xapaKTepl1sYIOIlLI1X COCTOflHl1e ynpyroro TeJIa, npHXO,lI,I1M
223


III


I,
II
11




K KpaeBOH Sa
aqe
iJ
u iJ 2 u
iJx 2 + iJ y 2 +
iJ 2 w iJ 2 w
ax2 + iJ y 2 +


m
 + pro2a2 -0
m - 2 ax G U - ,


(24 1)


m
m-2


iJe + pro2a
 W = 0
iJy G


B npHMoyrOJIbHOH o6JIacTH (pHC. 21). Eha CHCTeMa ypaBHeHHH C<POPMY-
JIHpOBaHa OTHOCHTeJIbHO 6espasMepHblx (oTHeceHHblx K a) aMnJIHTYlI-
HbIX xapaKTepHcTHK KOMnOHeHTOB BeKTopa cMeIlleHHH U = ui + wj.
" iJu iJw
KpoMe TOro, npH sanHCH ypaBHeHHH (24.1), r
e e = iJx + iJy ,
BBe
eHbI 6espasMepHble KooplIHHaTbI x = x*/a, Y = y*/a, 'I'] = b/a.
y" rpaHHqHble YCJIOBHH, KOTOpbIM
lIOJILKHbI 6blTb nO
qHHeHbI y
OB-
JIeTBOpHIOIllHe ypaBHeHHHM (24.1)
KOMnOHeHTbI BeKTopa cMeIlleHHH,
onpe
eJIHIOTCH cnoco6oM Bos6YLK
e-
HHH KOJIe6aHHH H BblpaLKaIOTcH
COOTHorneHHHMH JIli60 OCHOBHbIX rpa-
HHqHbIX sa
aq, JIH60 paSJIHqHOrO TH-
na CMernaHHblX sa
aq. npe
noJIara-
Pnc. 21. eTCH, qTO HSMeHeHHe BO BpeMeHH BCeX
Sa
aHHbIX B rpaHHqHbIX VCJIOBH-
HX BeJIHqHH onpe
eJIHeTCH MHOLKHTeJIeM elm!. - -
B npOllecce perneHHH rpaHHqHbIX 3a
aq
JIH CHCTeMbI (24.1) B npH-
MoyrOJIbHHKe MOLKHO
OCTHqb 3HaqHTeJIbHorO ynpow.eHHH BbIKJIa
OK,
eCJIH 06IllHH CJIyqaH
e<p0pMHpoBaHHH npe
CTaBHTb B BH
e HaJIOLKeHHH
nOJIeH
e<popMallHH C onpe
eJIeHHbIMH CBOHcTBaMH CHMMeTpHH. OCTa-
HOBHMCH no
po6HO Ha paccMOTpeHHH cHMMeTpHqHorO OTHOCHTeJIbHO
06eHX Koop
HHaTHbIX oceM nOJIH
e<popMallHM B npH3Me
B COOTBeTCTBHH C npHHHTbIM B pa60Te MeTO
OM nocTpoeHHH perneHHH
rpaHHqHbIX 3alIaq TeopHH ynpyrocTH, 06Illee perneHHe OCHOBHbIX rpa-
HHqHbIX 3a
aq
JIH CHCTeMbI (24.1) KOHCTpYHpyeTcH B BH
e CYMMbI
BYX
COCTaBJIHIOw.HX. XapaKTep Ha6opoB qaCTHblX perneHHH B KaLK
oH H3
COCTaBJIHIOIllHX onpe
eJIHeTCH Tpe60BaHHHMH nOJIHOThI HX Ha npoTHBo-
nOJl0LKHbIX cTopoHax npHMoyrOJIbHHKa (CM. pHC. 21)
Onpe
eJIHM nepBYIO cocTaBJIHIOw.yIO o6Illero perneHHH Ha OCHOBe
Tpe60BaHHH ero nOJIHOTbI Ha CTopOHax x = + 1 npH3MbI.3TOMY Tpe-
60BaHHIO y
OBJIeTBOpHIOT CJIelIYIOw.He B
paLKeHHH
JIH KOMnOHeHTOB
BeKTopa CMeIlleHHH:
u I = Ao sin 1'X +
 (An sh PIX + Bn sh P2 X ) cas anY,
n=1


co


(24 2)


WI=-
 (An an chPIx+Bn
chP2x)sinanY.
n=1 PI an
roa nn 2
l' = _ V ,V D - CKOpOCTb BOJIH pacrnHpeHHH, an = -, PI =
D

224


3
eCb


IIII
. I


2 2 2 2 2 m- I
= an - ,\" P2 = an - m _:! 1'2, Ao, Am Bn - npOH3BOJIbHbIe no-
CTOHHHbIe.
CJIe
¥eT OTMeTHTb, qTO 3a HCKJIIOqeHHeM OT
eJIbHbIX
HCKpeTHbIX
3HaqeHHH qaCTOTbI BblpaLKeHHH (24.2) n03BOJIHIOT y
OBJIeTBOpHTb npo-
H3BOJIbHbIM YCJIOBHHM B cMew.eHHHX HJIH HanpHLKeHHHX Ha CTopOHax
X = + 1, -'I']«.Y «. 'I'] npHMoyrOJIbHHKa. Bxo
Hw.He B BblpaLKeHHH
JIH
u l H WI 6eCKOHeqHbIe pH
bI HBJIHIOTCH Ha6opOM pa3JIHqHbIX HOpMaJIbH
X
BOJIH B 6eCKOHeQHOM CJIOe (nOJIOce) - 1 «. X -< 1. He3aBHCHw.ee OT Y
CJIaraeMOe
06aBJIeHO B COOTBeTCTBHH C Tpe60BaHHHMH nOJIHOTbI CHCTe-
MbI <PYHKllHH cos anY Ha COOTBeTCTBYIOIlleM HHTepBaJIe. .
Tpe60BaHHH nOJIHOTbI o6w.ero peweHHH Ha CTopOHax Y = + 'I'] npH-
MoyrOJIbHHKa 3aCTaBJIHIOT BbI6paTb BTOpyIO ero COCTaBJIHIOIllYIO B
CJIe
YIOw.eM BH
e:
u 2 = - f (C k fl q k ch qIY + Dk
2 ch q2Y) sin PkX,
k=1 1 pk


co


(24.3)


W 2 = Fo sin 1'Y +
I (C k sh qIY + Dk sh q2Y) COS
kX,
r lI e (.!. k 2 (.!.2 2 2 (.!.2 2 m - I ..2 F C
Pk = :n, qI = pk - 1', q2 = pk - m _ 2 1'; o. k' Dk - npo-
H3BOJIbHble nOCTOHHHble. 3TH B
pa>KeHHH 06JIa
aIOT TeMH LKe B03MOLK-
HOCTHMH
JIH y
OBJIeTBOpeHHH rpaHHqHbIX YCJIOBHM Ha nOBepXHOCTHX
Y = + 11, QTO H BblpaLKeHHH (24.2) Ha rpaHHx x = + 1.
TaKHM o6pa30M, o6w.ee perneHHe OCHOBHbIX rpaHHqHbIX 3a
aq
JIH
npH3Mb
 nocTpoeHo. I1cnOJIb3yeMble npH y
OBJIeTBOpeHHH rpaHHqHbIX
YCJIOBHH BblpaLKeHHH KOMnOHeHTOB BeKTopa cMeIlleHHB onpe
eJIHIOTCH
no <popMYJIaM
u = U 1 + u 2 ' W = WI + W 2 . (24.4)
CooTBeTcTBYIOW.He 9THM cMew.eHHHM KOMnOHeHTbI TeH30pa HanpHLKelIHH
HaXO
HTCH C nOMow.bIO cooTHorneHHH 3aKOHa rYKa H npeACTaBJIHIOTCH
B BH
e


I I
I,


I
fi


I m-I A + Fo'V
 [ A 2 2
2G ax = m _ 2 01' cos 1'x m _ 2 cos 1'Y +

 (an + P2) X
n=1 PI
X ch PIX + BnP2 ch P2x 1 cos anY +
 [ Ck ( qf - flk Pk ) ch qIY

k=1 ql m-2
- D k q2 ch Q2Y] cas Bkx,
) Ao'V m - I

 r A ( p2 - a 2
- a - c a s
'X + Fo
' cas
' Y + --.!!:.- I n
2G IJ- m-2 r m - 2 r r "-I
'1=' p, m-2


15 7-2l..64


225




- a
) ch PtX - BnP2 ch P2X] COS anY +

 [ Ck 2 2 ]
+ £=1 2ql «(3,. + q2) ch qlY + D k q2 Ch-q2Y COS (3k X ,
I 00 [
20 Txy = -
 Ana
 sh PtX + n
n (p
 + a
 sh P2X ] sin anY -
n=1 LM n
-
 [ C k (3k sh qIY + 2
k «(3
 + q
) sh qfl.Y 1 sin (3t'x.
k=1 1-'''


(24.5)


COBMeCTHbrM aHaJUl3 Bblpa.>KeHItH
JIH HanpHmeH'tIi'l 11 cMern.eHI1H no-
Ka3bIBaeT, tlTO ItCnOJIb30BaHI1e nOCTpoeHHoro pellleHrfH YP3BHeHI1H (24.1)

aeT B03MomHOCTb JIerKO 113YtlI1Tb KOJIe6aHI1H npl13MbI npl1 YCJIOBI1HX
rJI3
Koro KOHTaKTa ee rpaHeH C a6COJIlOTHO TBep
IM TeJIOM. B 3TOM
CJIYllae rpaH
tlHble YCJIOBI1H Ha K3}K)1.0H rpaHI1 cpOPMYJII1PYIOTCH OTHO-
CI1TeJIbHO HOpMaJIbHbIX K nOBepXHOCTI1 KOMnOHeHTOB BeKTopa CMeIII,eHI1H
11 KaCaTeJIbHbIX HanpSlmeHI1H, tlTO n03BOJISIeT onpe
eJII1Tb npOH3BOJIb-
HbIe nOCTOSlHHbIe B (24.5) 113 CHCTeMbI aJIre6pal1t1eCKI1X ypaBHeHI1H BTO-'
poro nopSl
Ka. YpaBHeHHe
JISI c06CTBeHHbIX tlaCTOT TaKme 06pa3yeTcH
KOHetlHbIM onpe
eJIIiTeJIeM.
3a,l\atla Onl1CaHHSI ,l\I1HaMHtleCKOH HanpSlmeHHOCTH 11 OTbICKaHHSI qac-
TOTHbIX xapaKrepl1CTI1K npl13MbI 3HatlltTeJIbHO ynporn.aeTCSI B CJIytlae
3a,l\aHHSI YCJIOBHH rJIa,l\KOro KOHTaKTa Ha O,l\HOH 113 nap npOTI1BOno-
JIOmHbIX CTOpOH. He 6Y,l\eM OCT3HaBJIltB3TbCH Ha I1CCJIe,l\OBaHI1H YKa3aH-
HbIX rpaHHtlHbIX 33,l\atl, a 06paTI1MCSI K O,l\HOMY 1t3 Hal160JIee CJIOmHbIX
CJIytlaeB - K nepBoH OCHOBHOH rp3HHtlHOH 3a,l\aqe.
uCJIOmHOCTI1, B03HHKalOIII,l1e npl1 H3Y'leHHI1 BbIHym,l\eHHbIX KOJIe6a-
HI1H npl13MbI, BbI3BaHHblX Sa,l\aHHbIMH Ha rpaHHLl.e HanpSlmeHItSlMH, CBSI-
saHbI C Te
, tlTO y,l\OBJIeTBOpeHHe rpaHI1t1HblX YCJIOBI1H C nOMorn.blO
Bblpa
eHHI1 (24.5) npItBO,l\HT K 6eCKOHetlHbIM CHCTeMaM JII1HeHHbIx ypaB-
HeHHH OTHOCI1TeJIbHO co
epmarn.HxcSl B HI1X npOH3BOJIbHbIX nOCTOSlH-
HbIX. YCJIOBHe paBeHcTBa HYJIIO 6eCKOHeQHoro onpe,l\eJIItTeJISI :;noH CHC-
TeMbI ,l\aeT ypaBHeHHe ,l\JISI c06CTBeHHblx qaCTOT.
OTMeQeHHbIe oc06eHHOCTI1 CJIytlaSl nepBoH OCHOBHOH rpaHHtlHOH 3a-
,l\atlH YKa3bIBalOT Ha TO, tlTO 3<p<peKTHBHOCTb npe,l\JIaraeMOro nO,l\XO,l\a
K ee perneHl110 TeCHeHIIlI1M 06pa30M CBSl3aHB C B03MomHOCTbIO YKa3aTb
cnoc06 KoppeKTHoro ycetleHI1Sl 6eCKOHeqHOH CI1CreMbI KaK npl1 113ytle-
HI1H HanpSlmeHHOCTI1, TaK H npl1 OTbICKaHHI1 c06CTBeHHbIX QaCTOT.
B CBSl311 C 3TI1M BO 1136emaHI1e H3JIHlllHeH rpQM03.QKOCTI1 BbIKJI3,!1,OK OCTa-
HOBI1MCSI Ha Hal160JIee npOCTOH no cpopMe nepBOH OCHOBHOH rpaHHtlHOH
3a
alle:


J
- u = t elmt
?{J x 0 ,


I
20 TXIJ = 0


(x = :!: I),


I I
20 u y = 2(]" Tx.<i = 0


(24.6)


(y = :!: 1'J),


226


.


T. e. paCCMOTpHM 3a
allY 0 BbIHym,AeHHbIX KOJIet'5aHlfHx npHMoyrOJIb-
HOH npHBMbI, BbI3BaHHbIX paBHOMepHO pacnpe
eJIeHHbIM nepHO
HtleCKH
H3MeHSlIOrn.HMCSI BO BpeMeHI1 HOpMaJIbHbIM
aBJIeHHeM no rpaHSIM x =
= ::I:: 1. OTJII1t1Hble OT (24.6) rpaHI1t1HbIe YCJIOBHSl H3MeHSIT JIHlllb CBO-
60
Hble tlJIeHbI nOJIytlaeMblx CHcreM, tlro SlBJISIeTCH Hecyrn.ecTBeHHbIM
npH pa3pa60TKe MeTO
OB I1X pellleHHSI.



 25. HCCJleAOBaHlle 6ecKoHellHbIX CHCTeM
DO,l\CTaBJISISI BblpameHI1S1
JISI HanpSlmeHHH (24.5) B rpaHHtlHble
YCJIOBItH (24.6), npHXO
ItM K CJIeAYIOrn.HM CPYHKLl.HOHaJIbHblM ypaBHe-
HHHM:
:=
 AoYcosy+ m
Y2 cosyy+
 [ ;
 (pi- m
2 )ChPl +
n=1

 BnP2 ch P2 ] cos anY + f (- l)k [ - Ck ( (3
 + m
 2 ) ch qlY-
k=1 ql
- D k q2 ch q2Y] = to,
2a
Ansh Pt + Bn (P
 + a
) Shp2 = 0,
m - 1 A y i'-, [ Ck ( 2 y2 )
m-2 FoYcosy1'J+ m
2 cosyx+ L -----q;\ql- m-2 chqt1'J+
k=1


+ Dkq2Chq21'J ] COS(3k X + i: (_I)n [ _ An ( a
+ m
2 ) chP1x-
n=1 Pt
- BnP2 ch P2X] = 0, (25.1)
2(3
Ck sh ql1'J + Dk«(3
 + q
)sh q21'J = O.


YCJIOBHSI )1JISI KaCaTeJIbHbIX HanpSlmeHI1H
alOT BTopoe H tleTBep-
Toe COOTHOllleHI1S1 B :noi'l CI1CTeMe. XapaKTep CBSl3H Mem
y nOCTOSlHHbIMH
An' Bn H C k , Dk' BbITeKalOIII,ei'l 1t3 3TI1X ypaBHeHHH, pa3JIHtleH npl1 pa3-
JII1t1HbIX 3HatleHHSlX tlaCTOT. TaK, Hanpl1Mep, B cJIYQaSlX, Kor
a sh qt1'J
HJII1 sh Pt 06pamaFOTCSI B HYJIb, Dk = 0 I1JIH Bn = 0 COOTBeTCTBeHHO,
a nOCTOSlHHbIe Ak It C" OCTalOTCSI npOH3BOJIbHbIMIt. B nOCJIeJLYIOrn. ItX
BbIKJIaJLKaX nplt ynpo
eHHIt CHCTeMbI (25.1) He 6YJLeM OCTaHaBJIItBaTb-
CSl Ha 3THX tlaCTHbIX 3HatleHltSiX QaCTOT, KOTopbIe, KaK t'5YJLeT BH,l\HO, He
Tpet'5YIOT KaKHX-JIH60 H3MeHeHHH B cnoc06e OTbICKaHUSI HeU3BeCTHbIX
B (25.1). H I1me Bea,n.e npltHHM3eTCSI CJIe;!I.YIOrn.BSl CBSl3b MemJLY ItCKOMbIMH
BeJIItQUHaMIt:


p
+a

An = - Bn 2
2cx
n


13
 + q

C k =
 D " sh q2'1l . (25.2)
213f sh Ql'1l


Sbp2
shpi '


15*


227




, _ sin I' + '
 [ sin (tlk -+ V) + sin (tlk - V) I R
cos yx - - 1..J A + -------.:\- COS Pk X ,
I' k=1 ....k. I' ....k - V
e yqeTOM CooTHoIlleHllii (25.2) nOJIyqaeM BMecTO CllCTeMbI clJYHKIl.IIOHaJIb-
HbIX ypaBHeHllii (25.1) CJIe.D.yIOru.ylO 6eCKOHeqHYlO cIIC'feMY JIIIHeiiHblx
aJIre6pallqeCKIIX ypaBHeHllii:
2 2 co
sin Y'l I' YI
' Xk
Yo- + (m-l)xocosY---r l.J
 = fo(m-2),
VfJ bl tlkqi
. 2 2 co
SIn V . I' 1'1 ,-, Yn
Xo-+ (m--l)yocosY'l-
..;..J
 = 0,
V n=1 anPi


(25.4)


Bblpa:>KeHIIH .D.JIH I).n (p) H I).k (q) COBna.D.aIOT C BblpameHHHMH onpe.D. eJIll -
TeJIH PeJIeH - JI3M6a COOTBeTCTBeHHO .D.JIH nOJIOC IllllpllHOii, paBHoii e.D.ll-
HHIl.e H 1').
I1HclJopMaIl.IIH, KOTOPYIO He06xo.D.IIMO nOJIyqHTb npH paCCMoTpeHllll
CllCTeMbI (25.4), .D.OBOJIbHO 06IllllpHa. C o.D.Hoii CTOpOHbI, .D.JIH aHaJIll3a
HanpHmeHHocTH He06xo.D.HMO nOJIyqHTb 3HaqeHIIH HeH3BeCTHbIX, KOTO-
pble HBJIHIOTCH K03clJclJHIl.lleHTaMIl 6eCKOHeqHbIX pH.D.OB B BblpameHIIHx
.D.JIH cMeru.eHllii H HanpHmeHllii. C .D.pyroii CTOpOHbI, He MeHbIllHM HHTepec
npe.D.CTaBJIHeT OTbICKaHHe Tex 3HaqeHHii qaCTOTbI, npll KOTOpbIX KOHeq-
Hoe peIlleHlle CHCTeMbI (25.4) He Cyru.eCTBYeT. 3Tll qaCTOTbI HBJIHIOTCH
c06CTBeHHbIMll qaCTOTaMH ynpyroro TeJIa.
CIICTeMa ypaBHeHHM (25.4) no cyru.eCTBY 3a.D.aqll, npllBe.D.Illeii K Heii,
He MomeT 6bITb perYJIHpHOM npH npOll3BOJIbHOM 3HaqeHllll qaCTOTbI.
B CBH3H C 3TIIM paCCMOTpllM Bonpoc 0 ee KBa311peryJIHpHOCTll, npe.D.no-
JIaraH npll 3TOM, qTO .D.Hana30H 113MeHeHIIH qaCTOTbI BbI3bIBalOru.llX KOJIe-
6aHHH YCHJIHM OrpaHHqeH HeKOTopbIM 3HaqeHlleM Yo (I' < Yo)'
CHCTeMa ypaBHeHHii .D.JIH onpe.D.eJIeHIIH HeH3BeCTHbIX X k (k >- K) H
Yn (n >- N), BbITeKalOru.aH H3 (25.4) npll YCJIOBHH, qTO 3HaqeHIIH Xu.
Yo, X k (k < K) H Yn (n < N) 113BeCTHbI. MOmeT 6bITb JIerKO 3anllcaHa.
.llJIH llCCJIe.D.OBaHIIH ee perYJIHpHOCTll, OqeBll.D.HO, CJIe.D.yeT OIl.eHIITb
cJIe.D.YIOru.lle BblpameHIIH:
co I 2 2 I
1 1'2 '"' 2mPk 1'1
J - (k >- K),
1 =. I f:,k (q) I m - 2 n
 (a
 -+ qj) (a
 + q
) a
 (a
 + q
)
co I 2 2 I
1 '1'2
 2ma n 1'1
J 2 = I f:,n (p) I m - 2 k":: (tI
 -+ PI) (tI
 + p
) - 13
 (tI
 -+ p
) (n>- N).
(25.7)


I1cnoJIb3YH npe.D.cTaBJIeHHH rllnep60JIllqeCKIIX clJYHKIl.IIM TpHroHO-
MeTpHqeCKHMH pH.D.aMll 113 CooTHoIlleHIIM (5.6) BTOpOM rJIaBbI ll.D.onOJIHHH
llX pa3JIOmeHIIHMll


_ sin YfJ +
 '; [ sin (an -+ 1')fJ + sin (an - 1') fJ ]
COS
' Y -- cosa Y
I YfJ n=1 fJ (an -+ 1') (an - 1') fJ n ,
(25.3)


Y I). (p) - 1'2
 X k [ 2 2 y
 2
n n m - 2 k=1 tlk (l3k -+ PI)
+ YoLn _ °
m-2 '


2ma 2 ]
(tI
 -+ P
) (13
 -+ p
) +


xl). (q) - 1'2
 Yn[ 2 t' 2
k k m - 2 =1 an (an -+ ql)
+ XoEk = °
m-2 .


2mB 2 ]
k +
(a
 -+ qi) (a
 -+ q
)


(25.5)


BbI6epeM KIlN H3CTO.nbKO 60JIbIllIIMH, I-IT06bI npll 3a.D.aHHOM 3Ha-
qeHllll Yo (CMbICJI 3TOM BeJIllqllHbI pacIllHclJpoBaH BbIIlle) qllCJIa Y
/
k H
YI/a
 6bIJIll.D.OCTaTOqHO MaJIbIMll no cpaBHeHlllO C e.D.HHIIIl.eii. Tor.D.a qJIe-
HbI 6eCKOHeqHbIX pH.D.OB B (25.7) cTaHYT nOJIOmllTeJIbHbIMll II BeJIHqllHbI
JIll J 2 6Y.D.YT .D.OnYCKaTb npOCTYIO OIl.eHKy. T aK, HanpllMep, .D.JIH J 1
no.nyqllM


HeH3BeCTHble 3TOM CIICTeMbI CBH3aHbI C Hell3BeCTHbIMll (25.1) co-
OTHOIlleHIIHMIl


kin
xk=(-l) D k -shq21'), Yn=(-I) Bn sh P2' AoY=x o , FoY=Yo'
fJ


KpoMe TOro, B (25.4) npllHHTbI CJIe.D.YIOru.lle COKparu.eHIIH:
[ (a
 -+ p
)2 J
I).n (p) = P2 cth P2 - cth PI 2 '
4a n PIP2
[ (tI
 -+ q
)2 ]
I).k (q) = Q21') cth q21') - cth Ql1'J 2 '
4t1kql q2


1'2
 2mtl

J 1 < I f:,k (q) I m - 2 =1 (a
 -+ qI){a
 -+ q
)
_ I f:,P
 ) I [ -.!L ( cth q21\-
 ) - --.:!L ( cth ql1') -
 )] . (25.8)
k q q2 q2fJ ql qlfJ


(25.6)


YqHTbIBaX, qTO k >- K, r.D.e K - .D.OCTaTOqHO 60JIbIllOe qllCJIO, C TOq-
HOCTblO )1.0 ue.nllqllH nOpH.D.Ka 1/
k HaXO.D.IIM


y
 = 2 m -1
,2,
m-2 I


Ln = 21' sin yfJ
fJ (1'2 - a
) ,


Ek = 21'sinV
V2_p
 .'


"1 < l--i-
....kfJ
229


(k> K).


(25.9)


228




2
J 2 < 1 - -
an


(n > N).


(25..10)


B 06meM, OTJUillHOM OT (24.6), CJIYll8e H8rpY>KeHlHl npll3MbI nOBe-
.neHlle BeJIlltIHH F n H <D" 6y.neT 38BHceTb OT xap8KTepa <PYHKUHH, 3a.n8-
IOWHX rpaHHtIHbIe YCJIOBHH. B =nOM CJIYllae cnpaBe.nJIHBOCTb oueHOK
(25.12) CBH3aHa C TaKHMH >Ke npe.nnOJIO>KeHHHMH OTHOCHTeJIbHO rJIa.n-
KOCTH YKa3aHHbIx <PYHKUHH, KaK II B 3a.natIaX CTaTHKH. B llaCTHOCTll,
.nOJIJKHbI BbmOJIHHTbCH YCJIOBHH napHocTH KaCaTeJIbHbIX HanpH>KeHHH B
yrJIOBbIX TOIlKax.
3JIeMeHTapHbIe npe06pa30BaHllH n03BOJIHIOT npe.ncTaBHTb cHcTeMY
(25.11) B BH.ne


K aHaJIOrllllHOMY pe3YJIbT8TY npllxo.nllM npH paccMoTpeHllll BbIpaJKeHllH
j(JIH J 2:


COrJIaCHO nOJIYtIeHHbIM oueHK8M .nJIH J 1 H J 2' npe.nnOJIO>KeHHe 0 KOHell-
HocrH nepBbIX N + K HeH3BeCTHbIX B 6eCKOHetIHOH cHcreMe (25.4) n03-
BOJIHeT YK83aTb cnoc06 HaXO>K.neHHH ocraJIbHbIX HeH3BeCTHbIX, KOTO-
pble B 3TOM CJIYll8e TaK>Ke HBJIHIOTCH KOHe'lHbIMH. DpnlleM KOJmlleCTBO
nepBblx HeH3BeCTHbIX, OTHOCHTeJIbHO KOTOpbIX npe.nnOJIaraeTCH KOHell-
HOCTb, cywecTBeHHo 3aBHCHT OT paccMaTpHBaeMoro .nHana30Ha H3MeHe-
HllH llaCTOTbI.
DpoBe.neHHble BbIKJIa.nKH n03BOJIHIOT c.neJIaTb .nBa Ba>KHbIX BbIBo.na,
xapaKTepH3YlOWHX HCnOJIb3yeMbIH MeTO.n HCCJIe.nOBaHHH KOJIe6aHHH.
Bo-nepBbIx, H3 .nOKa3aTeJIbCTBa perymlpHocTH .nJIH HeH3BeCTHblX
C BbICOKHMH HOMepaMH CJIe.nyeT, tITO Bonpoc 0 cymecTBoBaHHH peIlJeHHH
6ecKoHe t IHoH CHCTeMbI (25.4) B npHHUHne MO>KeT 6bITb CBe.neH K Bonpo-
cy 0 pa3penlHMocTH KOHeIJHOH CHCTeMbI. 11, BO-BTOPbIX, nopH.noK TaKoH
KOHellHoH CHCTeMbI cywecTBeHHO 3aBHCHT OT BeJIllllHHbI HHTepBaJIa
H3MeHeHHH llaCTOT, BbI3bIBalOmHX KOJIe6aHHH B03.neHcTBHH. K CO>K8JIe-
HHIO, YCTaHoBHTb B HBHOM BH.ne onpe.neJIHTeJIb YKa3aHHoH KOHeIJHOH
CHCTeMbI He y.naeTCH, o.nHaKO, KaK 6y.neT nOKa3aHO HH>Ke, MO>KHO no-
CTpOHTb rJIaBHYIO llaCTb 3TOro onpe.neJIllTeJIH, YCJIOBHe paBeHCTBa HYJIIO
KOToporo .nOCTaBJIHeT ypaBHeHHe .nJIH c06CTBeHHbIX IJ8CTOT ynpyrora
TeJIa.
BepHeMCH CHOBa K CHCTeMe YP8BHeHHH .nJIH HeH3BeCTHbIX C BbICOKH-
MIl HOMepaMH (k > K, n > N), BbITeKalOweM H3 cllcreMbI (25.4) B
npe.nnOJIO>KeHHH 0 KOHellHOCTH nepBblx N + K HeH3BeCTHblX. OCYWecTB-
JIHH B HeM nOCJIe.noBaTeJIbHOe oT6paCbIBaHHe BeJIHIlHH nopH.nKa l'i/a

11l'i/

 (n > N, k> K) no cp8BHeHHIO C e.nHHHueM, nOJIYIJHM cJIe.nylOWlle
COOTHOllJeHHH:


4a
 ( I )
Yn +
 x" 2 2 = 0 - ,
"=K (an +
,,)2 an
ex> 3

 4
k = O (
 )
X" + rtf:k Yn TJ (a
 +

)2 tlk'
CTer:eHb 6JIH30CTH 3TOH CHCTeMbI H CllCTeMbI (5.15) n03BOJIHeT c.neJIaTb
BbIBO.n 0 TOM, IlTO aCHMnTOTHlleCKHe cBoiICTBa HeH3BeCTHbIX B (25.13)
onHCbIBalOTCH paBeHCTBOM
lim X k = - limYn = a o .
k-+ 8 n-+-oo


ex>


(25.13)


(25.14)


F n = 0 (

 ) , <Dk = 0 ( ;
 ) .
230


(25.12)


TaKEM 06pa30M, OTHOCHTeJIbHO CHCTeMbI (25.4) YCTaHOBJIeHO, IlTO B
CJIYIJae, Kor.na OrpaHHIJeHHOe perneHlle 3TOH CHCTeMbI cymeCTBYeT,
aCHMnTOTHlleCKHe CBOHCTBa HeH3BeCTHblX onpe.neJIHIOTCH paBeHCTBOM
(25.14). Dpl1 =nOM cywecTBeHHO HCnOJIb30BaHO npe.nnOJIO>KeHHe 06
orpaHnlleHHOCTH .nHana30Ha H3MeHeHHH llaCTOT B03MymalOwlix CHJI.
3HalleHHe YCT8HOBJIeHHOrO <paKTa C TOIlKH 3peHHH YJIYIJllJeHHH cxo-
.nllMOCTH 6eCKOHellHbIX pH.nOB .nJIH HanpH>KeHHH H CMeweHHH BnOJIHe
MO>KHO oueHHTb Ha OCHOBe 8HaJIH3a perneHHM CTaTHIJeCKHX 3a.nall.
3.neCb CJIe.nYeT OCTaHOBHTbCH Ha BbIHCHeHHH era pOJIH npH perneHHll
o.nHoro H3 rJIaBHbIX BonpocoB B 3a.nallax 0 KOJIe6aHHHx TeJI KOHellHblX
pa3MepoB, a I1MeHHO npH OTblCKaHHH c06cTBeHHbIx llaCTOT.
ECJIH npH perneHHH CHCTeMbI (25.4), OCHOBbIBaHCb Ha paBeHcTBe
(25.14), nOJIO>KHTb
x k = XK (k> K), Yn = YN (n> N). (25.15)
TO npH.neM K KOHellHOM CliCTeMe JIHHeHHbIX ypaBHeHHH. TIpH 3TOM no-
HBHTCH B03MO>KHOCTb He TOJIbKO nOJIYIlHTb .nOCTaTOIlHO 6JIH3KHe K TOIl-
HbIM 3HalleHI1H Bcex Hell3BeCTHbIX H3 6eCKOHellHOH CHCTeMbI (25.4), HO
II o.nHoBpeMeHHo KOHellHbIH llacToTHbIH onpe.neJIllTeJIb, KOTOpbIH BKJIIO-
llaeT B ce6H rJIaBHYIO llaCTb 6eCKOHeIJHOrO onpe.neJIliTeJIH HCXO.nHOH CHC-
TeMbI (25.4).
TaKHM o6pa30M, HaJIHlllle COOTHorneHlHl (25.14) n03BOJIHeT IJaCTHIl-
HO cBepHYTb 6ecKoHellublH onpe.neJIHTeJIb CHCTeMbI (25.4) .nJIH HaXO>K-
.neHHH c06cTBeHHblx IJaCTOT. ECTecTBeHHo, B03HHKaeT Bonpoc 0 BbI60pe
Heo6xo.nHMbIX 3HalleHHH K H N .nJIH nOJIYlleHllH .nocToBepHblx 3Halle-
HHH co6cTBeHHbIx IJaCTOT B HeKoTopoM KOHeIJHOM HHTepBaJIe. TIpH perne-
HllH ,noro Bonpoca B03MOJKeH CJIe.nYIOIUllH no.nxo.n. BbI6paB, PYK030.U.-


C>O 2a

Yn 2
 +
 Xk =Fn (n> N),
n k=K (a
 + tI
)2
2t1
 (25.11 )
x" 2
k +
 Yn = <Dk (k:>- K).
(a
 +

)2
'=N


CBo6o.nHble IJJIeHbI B 3TOM CHCTeMe 3aBHCHT OT X k (k = 0,1, ..., K - 1)
H Yn (n = 0,1, ..., N - 1). 5IBHoe Bblpa>KeHHe 3THX BeJIHIJHH JIerKO
MO>KeT 6bITb nOJIYlleHo H3 ypaBHeHHH (25.4), o.nHaKO CeMIJaC Ba>KHO
JIHrnb TO, IlTO H3 npe.nnOJIO>KeHllH 0 KOHellHOCTH nepBbIX K + N
HeH3BeCTHbIX .nJIH CBo6o.nHbIX IlJIeHOB B CllCTeMe (25.11) CJIe.nyeT
oueHKa


231




CTBYHCb BbIlllell3JIO:>KeHHbIMll C006pa:>KeHllHMll, HeKOTOpbIe 3HaqeHllH
N II K, paCCMOTpllM 3a.n.aqy 0 BbIHY:>K.n.eHHbIX KOJIe6aHllHX yrrpyroro
TeJIa rrpll pa3JIllqHbIX 3HaqeHllHX qaCTOTbI B03MYIll,eHHbIX YCllJIllH. Opll
qaCTOTaX, .n.OCTaTOqHO y.n.aJIeHHbIX OT C06CTBeHHbIX 3HaqeHllH KOHeqHOH
MaTpI1IJ,bI, KpllTeplleM .n.JIH BbI60pa N Ii K CJIY:>KllT TOqHOCTb y.n.OBJ1eTBO-
peHllH rpaHllqHbIX YCJIOBllH. Opll6JIll:>KeHlle B03MYIll,alOIll,eH qaCTOTbI K
C06CTBeHHOH rrpllBe.n.eT K POCTy BJIllHHllH oT6pOllleHHbIX Ha OCHOBe
(25.15) CJIaraeMbIX ll, CJIe.n.OBaTeJIbHO, K 60JIbllleH rrorpelllHOCTll B
y.n.oBJIeTBOpeHllIi rpaHllqHbIX YCJIOBHH. 3a.n.aBlllHcb ll3 <pll3HqeCKHX co-
06pa:>KeHllH HeKoTopoH .n.orrYCTllMOH rrorpelllHocTblO 6 B y.n.oBJIeTBOpe-
HHH rpaHllqHbIX YCJIOBllH, Haxo.n.HM, qTO Ha HeKoTopoM HHTepBaJIe
qaCTOT OT WI .n.o W 2 , BKJIlOqalOIll,eM c06CTBeHHYlO qaCTOTY KOHeqHOH CHCTe-
MbI, 0lllH6Ka rrpH y.n.OBJIeTBopeHHll rpaHIiqHbIM YCJIOBHHM rrpeBbIlllaeT
6. CqHTaeM, qTO c06cTBeHHaH qaCTOTa yrrpyroro TeJIa 3aKJIlOqeHa B llH-
TepBaJIe (WI' ( 2 ). ECJIll HaH.n.eHHbIH HHTepBaJl OKa3bIBaeTCH CJIll111KOM
lllHpOK, 3HaqllT Heo6xo.n.HMO YBeJIHqHTb 3HaqeHHH N II K.
XapaKTep rrOBe.n.eHllH Hell3BeCTHbIX B CllCTeMe (25.4), orrpe.n.eJIHe-
Ml:JjH paBeHcTBoM (25.14), 06YCJIOBJIllBaeT pH.n. CrreIJ.H<pHqeCKHX oc06eH-
HocTell B rrOBe.n.eHHll pH.n.OB .n.JIH HarrpH:>KeHllll, BbIqllCJIHeMbIX rro 1iJ0PMY-
JIaM (24.5).
B qaCTHOCTll, CJIe.n.yeT OTMeTllTb, qTO B oKpecTHocTH yrJIOBbIX TOqeK
pH.n.bI .n.JIH BbIqHCJIeHHH HanpH:>KeHllH He 06J1a.n.alOT paBHOMepHOH CXO-
.n.HMOCTblO Ii .n.OCTOBepHbIe 3HaqeHHH HanpH:>KeHllH MorYT 6bITb nOJIyqeHbI
JlHlllb nYTeM rrpe.n.eJIbHOrO nepexo.n.a. AHaJlOrHqHbIll npOBe.n.eHHOMY B
CJIyqae 3a.n.aq CTaTHKll aHaJIll3 06Ill,llX 1iJ0PMYJI rrOKa3bIBaeT, qTO HCnOJIb-
30BaHHe MeTo.n.a npOCTOH pe.n.YKIJ.llH npll pellleHllll 6eCKOHeqHOH CHCTe-
MbI (25.4) TaKLKe npHBo.n.HT K HeycTpaHHMoll nYTeM nOBblllleHHH nopH.n.Ka
KOHeqHOH CllCTeMbI norpelllHOCTll npH OIJ.eHKe HanpH:>KeHHH B yrJIOBbIX
TOqKax.
Cnoc06bI yqeTa YKa3aHHblx oc06eHHOCTeH npH npoBe.n.eHHIi BbIqHC-
JIeHHH .n.OCTaTOqHO no.n.p06HO onHcaHbl npH paccMOTpeHllH CTaTHqe-
CKHX 3a.n.aq. 003TOMY He 6y.n.eM Ha HllX OCTaHaBJIHBaTbCH, a 06paTHMCH
K paCCMOTpeHHlO KOHKpeTHblX npllMepOB, rr03BOJIHlOlIJ.llX y6e.n.llTbCH B
3<PliJeKTHBHOCTll pa3BllTOrO MeTO.n.a pellleHHH .n.HHaMHqeCKHX 3a.n.aq .n.JIH
TeJI KOHeqHbIX pa3MepOB.



 26. AH3JIH3 pe3YJIbT3TOB p3CQeT3


113JIO:>KeHHbIH B rrpe.n.bI.n.YIll,eM naparpaliJe aJIrOpllTM pellleHllH 3a.n.a-
'Ill 0 BbIHY:>K.n.eHHbIX KOJIe6aHHHx npHMoyrOJIbHOH npH3MbI peaJIH30BaH
Ha - 3BM .n.JIH HeCKOJIbKllX KOHKpeTHblx reOMeTpHqeCKllX pa3MepoB B
.n.Hana30He qaCTOT 0 <.: l' <.: 5,5. fpaHHqHble YCJIOBHH Ha rrOBepXHOCTll
npH3MbI orrpe.n.eJIHlOTCH paBeHcTBaMH (24.6). OCTaHoBHMcH Ha aHaJIH3e
pe3YJIbTaTOB .n.JIH KBa.n.paTHoH H npHMoyrOJIbHOll ('I'] =0,5) npll3M li3
MaTepHaJIa C qHCJIOM OyaccoHa m =- 3,33. Opll paCqeTaX B (25.15)
Ha BceM .n.llana30He H3MeHeHHH qaCTOT npllHllMaJIOCb N 15, K - 30.
Opll BbI60pe nopH.n.Ka KOHeqHOll Cll, CTeMbl cooTBeTcTBYIQIll,ell CllCTeMe


232


I
!


'




(25.4), 1'ila'1 <.: 0,16. 3TO 3HaqllT, qTO BbI6paHHbIe 3HaqeHlIH N II K
cooTBeTcTBYlOT paccMoTpeHHoMY .n.llarra30HY qaCTOT.
AHaJIll3 pe3YJIbTaTOB pellleHllH KpaeBoll 3a.n.aqll .n.JIH KBa.n.paTHoll
rrpll3MbI B YKa3aHHoM .n.llarra30He ll3MeHeHllH l' C lllarOM 0,05 .n.aeT llleCTb
pe30HaHCHblX qaCTOT (1,15-1,20), (1,40-1,45), (1,70-1,75), (3,85-
3,90), (4,35-4,40), (4,75-4,80). llJlH nepBblx Tpex pe30HaHcoB
BbJIIOJIHeHbl 60JIee rro.n.p06Hble paCqeTbI (60JIee MeJIKllll lllar rro 1'),
llJIJIlOCTpllPYlOlIJ.lle B03MO:>KHOCTll rrpe.n.JIaraeMOro rro.n.xo.n.a .n.JIH YTOqHeH-


I
I


T a6JJIIua 28


'I' I 0(1) j 0(0) II 'I' I 0(1) I 0(0)
x x x x
1,\75 0,9965 35,\5 1,710 0,9986 35,96
1,\80 0,9943 55,07 \,715 0,9973 66,73
1,185 0,9863 128,1 1,720 0,9827 411,9
1,190 1,0420 -383,4 1,725 1,0043 -101,3
I 1,195 1,0086 -76,50 1,730 1,0020 -45,62
1,200 1,0048 -42,41 1,735 1,0013 -29,66


Horo onpe.n.eJIeHIiH c06cTBeHHbIx qaCTOT. B Ta6JI. 28 rrpllBe.n.eHbl C TOq-
HOCTblO .n.o to 3HaqeHllH HarrpH:>KeHHll Ox B IJ.eHTpe CTOpOHbl x = 1 (o
l)
II Ha OCll npll3Mbl (o
O) rrpll npOXO:>K.n.eHHll nepBoro H TpeTbero pe30HaH-
COB. OpHBe.n.eHHble 3HaqeHllH HanpH:>KeHllll o
1) xapaKTepll3YlOT TOq-
HOCTb y.n.OBJIeTBOpeHHH rpaHllqHbIX yCJIOBHll. CJIe.n.yeT 3aMeTHTb, qTO
BO Bcex OCTaJIbHblX TOqKaX rpaHllIJ.bI YCJIOBHH .n.JUI HOpMaJIbHbIX Ha-
rrpH:>KeHllll y.n.OBJIeTBOpHlOTCH C TaKOll :>Ke BbICOKOll TOqHOCTblO. YCJIOBllH
.n.JIH KaCaTeJIbHbIX HanpH:>KeHllll, OqeBH.n.HO, y.n.OBJIeTBOpHlOTCH TOqHO.
OpllBe.n.eHHbIe B Ta6JI. 28 .n.aHHble 0 nOBe.n.eHIill HarrpH:>KeHIiH nOKa-
3bIBalOT OTMeqeHHbIe BbIllle oc06eHHOCTll pellleHllH 3a.n.aqH 0 BbIHY:>K.n. eH -
HblX KOJIe6aHllHX
npll3MbI. KaK H npe.n.nOJIaraJIH, Ha6JIlO.n.aeTcH HeKO-
Topoe yxy.n.llleHlle KaqeCTBa pellleHllH rpaHllqHOll 3a.n.aqH rrpII no.n.xo.n.e
K c06cTBeHHoll qaCTOTe. O.n.HaKo 3HaHlle aCHMnTOTHqeCKllX 3HaqeHllll
HeH3BeCTHblX ll3 6eCKOHeqHOll CllCTeMbl .n.aeT B03MO:>KHOCTb rrOJIyqHTb
.n.ocTaTOqHYlO TOqHOCTb BbJIIOJIHeHllH rpaHHqHbIX YCJIOBHllII B Herrocpe.n.-
cTBeHHoll 6JIll30CTll K pe30HaHcy. OTMeTHM TaK:>Ke, qTO rrpII paccMoTpe-
HllH 3a.n.aqH )J.JlH KBa.n.paTHoll rrpll3MbI rrpoBo.n.llJIOCb pellleHIIe 6eCKOHeq-
HOll CllCTeMbI (25.4) MeTo.n.OM rrpocTOll pe.n.YKIJ.Hll. ECJIll B oKpeCTHOCTll
rrepBoro pe30HaHca TaKOll rro.n.xo.n. eIll,e 06eCrreqHBaeT OTHOCHTeJIbHO
TOqHOe OrrllCaHlle HarrpH:>KeHHOCTll B TOqKaX, y.n.aJIeHHbIX OT yrJIOBOll,
TO Y:>Ke B OKpeCTHOCTll BToporo pe30HaHca TOqHOCTb pe3KO rra.n.aeT H He
y.n.aeTCH rrOJIyqllTb .n.a>Ke COBrra.n.eHllH 3HaKOB HarrpH:>KeHIIll, BblqHCJIeH-
HbIX II 3a.n.aHHbIX B rpaHHqHblX YCJIOBHHX.
B paCCMOTpeHHOll 3a.n.aqe orrpe.n.eJIeHHbIll llHTepeC rrpe.n.CTaBJIHlOT
TaK:>Ke .n.aHHble 06 ll3MeHeHHH xapaKTepa rrOJIell HarrpH:>KeHHll H CMeIll,e.
Hllll C H3MeHeHlleM qaCTOTbl.


I


233




B c,nyqae KBa,lJ.paTHOH npH3MbI paCnpe.ll.e,neHHe JJ,HHaM!JqeCKHX
HanpmKeHHH npH06peTaeT pH.lI. XapaKTepHbIX OC06eHHOCTeH Y/Ke npH
nO.ll.XoJJ,e K nepBOMY pe30HaHCY. l'lg HHX CJIeJJ,yeT OTMe1HTb, npeJKJJ.e BCe-
ro, 3HaIJ:HTe,nbHYIO HepaBHOMepHOCTb HanpH/KeHHH 0 x no CeIJ:eHHIO
x = const. TaK, BeJIHIJ:HHa HanpmKemfii Ox B LleHTpe nOnepeqHOrO ce-
IJ:eHHH npH3MbI (x = 0, y = 0) nOIJ:TH B TpM pa3a npeBblIUaeT BeJIMqHHY
3THX HanpH/KeHHH B cepeJJ,HHe 60KOBOH rpaHH (x = 0, y = 1). C,neJJ,yeT
TaK/Ke YKa3aTb Ha TO, IJ:TO B c,nyqae KBaJJ,paTHOH npM3MbI HanpH/KeHHSI
6x ° x M ° Ii B LleHTpe nOnepeIJ:HOrO ce-
2G1o qeHlU1 npaKTMIJ:eCKH COBnaJJ,alOT no
4 BeJJMIJ:HHe. OTMeIJ:eHHble 06CTOSlTeJIb-
r= 1.1 CTBa MO/KBO CIJ:HTaTb npHqHHOH Toro,


2Gfo
4
I
2


2


r=J.4


-L
),1


0,5'!J


o


-2


D


-4


PHC. 22.


PHC.


qTO B 3TOM CJIYIJ:ae «CTep/KHeBaSl» MOJJ,eJIb .lI.aeT 3aBbIIlleHHOe Ha BeJIHqHHY
nopSlJJ,Ka 15% 3HaIJ:eHHe nepBoH co6cTBeHHoH qaCTOTbI. OJJ,HaKo YMeHb-
IUeHHe IUHpHHbI npH3MbI ('f) <: 1) 6bICTpO npHBoJJ,HT K TOMY, qTO HOJIH-
IJ:eCTBeHHO OTMeqeHHble BbIIUe 3<P<PeKTbl CTaHOBHTCH MeHee BbIpa/KeH-
HbIMH. TaK, Y/Ke npM 'f) = 0,5 «cTep/KHeBaH» MOJJ,eJIb JJ,aeT B03MOJKHOCTb
onpeJJ,e,nHTb nepBYIO c06cTBeHHYlO qaCToTY C norpeIUHoCTblO nOpH,l{Ka
3 %. HanpSlJKeHHH a y B 3TOM cJIYIJ:ae B LleHTpe nOnepeqHoro CeqeHHH JJ,O-
CTHralOT no BeJIHIJ:HHe Bcero JIHIUb 15% HanpHJKeHMH U x- HepaBHoMep-
HOCTb nOCJIeJJ,HHX no BbICOTe BblpaJKeHa He3HaIJ:HTeJJbHO.
KaIJ:eCTBeHHYIO Kapnmy 3aBHCHMOCTH xapaKTepa HanpHJKeHHOro
COCTOHHHH npH3MbI OT IJ:aCTOTbI MOJKHO nOJIYIJ:I1Tb Ha OCHOBe BHaJIH3a
BbIpaJKeHHH JJ,JISI HanpHJKeHHII (24.5). 113 3THX BbIpBJKeHHH CJIeJJ,yeT,
IJ:TO npH JJ,OCTaTOIJ:HO MaJIbIX IJ:aCTOTax e.ll.HHCTBeHHbIMH CJIaraeMbIMH,
OTBeqalOIll.HMH 3a nepeJJ,aqy 3HeprHH OT rpaHHLlbI BO BHYTpeHHHe 06.hac-
1M npH3MbI, HBJISIIOTCH CJIaraeMbIe C Ao H F o .l1pyrHe npOH3BOJIbHble no-
CTOSlHHbIe npH yJJ,OBJIeTBOpeHHH rpaHUIJ:HbIX YCJIOBI1H, HO CTOHIll.He npM
HHX tPYHKLlMM KoopJJ,HHaT 3KcnOHeHLlHaJIbHO y6blBalOT BHYTPb 06JIaCTH.
C pOCTOM IJ:aCTOTbi npOHCXOJJ,HT nepeXO.ll. HeKOTopbIX rl1nep60JIHIJ:eCKMX
tPYHKLlHH B TpHroHoMeTpHIJ:eCKHe H B KaqeCTBeHHOM CMbICJIe ypaBHHBa-
eTCH pOJIb CJIaraeMbIX, CTOHIll.MX B (25.4) BHe 3HaKOB CYMM, H nepBblx c,na-
raeMblX H3 6eCKOHeqHbIX pHJJ,OB. DpoBeJJ,eHHble paCIJ:eTbI nOKa3bIBalOT,
IJ:TO npH onpeJJ,eJIeHHbIX IJ:aCTOTax cooTBeTcTBYlOll1.He CJIaraeMbIe 113 6ec-
KOHeqHbIX pHJJ,OB CTaHOBSI fCH JJ,OMHHHPYIOIl1.HMH KOJIHIJ:eCTBeHHO H
HMeHHO HX xapaKTep H3MeHeHHH onpeJJ,eJIHeT M3MeHHeMOCTb HanpSlJKe-
HHH B CeqeHHHX x = const H Y = const. 3TO 06cToHTe,nbcTBo n03Bo,nH-
eT C.lI.eJIaTb 3aKJIIOIJ:eHHe 0 TOM, qTO pacnpe.ll.eJIeHHe HanpHJKeHHH BO


234


1


BHYTpeHHHX TOIJ:KaX 06JJaCTH onpeJJ,e,nHeTCH B OCHOBHOM 3HaIJ:eHHeM
IJ:aCTOTbI B03MYIll.alOIll.eH CHJJbI. a He BHJJ,OM tPYHKLlHH. 3aJJ,alOIll.HX ee 3a-
BHCHMOCTb OT KoopJJ,HHaT.
Ha pHC. 22 H 23 npHBeJJ,eHbI KpHBbIe. onHCbIBalOIll.He pacnpeJJ,e,ne-
HHe HanpHJKeHHH U x H u y B CeIJ:eHHH x = ° JJ,JIH CJIyqaH 'f) = 0,5 npH
pa3JIHqHbIX 3HaQeHHHX IJ:aCTOTbI. l1JIH,\, =- 1,1 T. e. .lI.JIH IJ:aCTOTbI, MeHb
IUe nepBoH pe30HaHcHoH (} ,3 <: '\'
) <: 1,35), pacnpeJJ,eJIeHHe HanpHJKe
Ta6JIHua 29


1


x=O x= 0.4 x=0.6 , x =0.8
y ax 1 CJ y CJ x I a y ax I CJ y ax I a y
,
° -0,20 I -0,80 0,12 0,21 0,48 0,85 0,74 1,14
0,1 -0,33 -0,71 0,08 0,21 0,52 0,79 0,81 1,04
0,2 -0,67 -0,48 ° 0,22 0,61 0,62 0,99 0,78
0,3 -1,l0 -0,20 -0,10 0,20 0,72 0,40 1,22 0,45
0,4 -1,50 -0,01 -0,17 0.14 0,82 0,17 1,39 0,15
0,5 -1,78 0 -0,12 ° 0,85 ° 1,39 0


,
j


HHH a no CeIJ:eHHIO nOIJ:TH paBHOMepHoe, a HanpH JKeHHH o!J no BeJIHIJ:HHe
x _

3HaQHTeJIbHO MeHbIlle. DpH nOBbIIUeHHH IJ:aCTOTbI (2,30 <: '\'/ "'" 2,35)
Ha6JIIOJJ,aeTcH HBHO BbIpaJKeHHaH HepaBHoMepHocTb B pacnpeJJ,eJJeHHH
HanpHJKeHHH a x' a HanpHJKeHMH a y no BeJIHqHHe ctaHoBHTCH Bno,nHe
cOH3MepHMbIMH C HHMM.
XapaKTepHoH oc06eHHocTbIO JJ,HHaMHIJ:eCKoro nOBeJJ,eHHH npH3Mbl
S1BJIHeTCH TaKJKe HaJIHIJ:He JJ,Hana30HOB IJ:aCTOT, npH KOTOpbIX Ha6JIIOJJ,a
eTCH 3HaqHTeJlbHOe YMeHbIUeHHe HanpHJKeHHH Ox no cpaBHeHMIO CO
CTaTHIJ:eCKHM CJIyQaeM BO BHYTpeHHHX TOqKaX 06JIacTH. HanpHJKeHHH
U npH 3TOM TaKJKe He3HaIJ:HTeJIbHbI. KOJIHqeCTBeHHO 3TO 06CTOHTeJIb
y 29 -
CTBO HJIJIIOCTpHpyeTcH JJ,aHHbIMH Ta6JI. ,COCTaBJIeHHOH no pe3YJIb-
TaTaM paCIJ:eTa HanpHJKeHUOCTH B CJIyqa '\' = 3,6; 'f) = 0,5. 3HaIJ:eHHH
HanpHJKeHHH B 3TOH Ta6JIHLle npHBeJJ,eHbI C TOqHOCTblO JJ,O nOCToHHHOro
MHOJKHTeJIH 2 Glo.
DOCTpoeHHoe peIlleHHe JJ,HHaMHIJ:eCKOH 3aJJ,aIJ:H JJ,JIH npH3MbI n03BOJIH-
eT npoBecTH TaKJKe nOJJ,p06HbIH KHHeMaTHIJ:eCKHH aHaJIH3 npoLlecca JJ,e
ct>OpMHpoBaHHSI. l1JIH 3TOro CJIeJJ,yeT HCnOJIb30BaTb BbIpaJKeHHH .lI.JIH KOM.
nOHeHTOB BeKTOpa CMeIll.eHHH (24.2) H (24.3). CYMMHpoBaHHe COOT BeT-
CTBylOIll.HX 6eCKOHeQHbIX pHJJ,OB JJ,JIH CMeIll.eHHH B CHJIY HX JIyqIUeH CXO-
JJ,HMOCTH BbI3bIBaeT MeHbIUHe 3aTpYJJ,HeHHH, GeM npH BbIIJ:HCJIeHMH Ha-
npHJKeHHH.
B 3aKJIIOIJ:eHHe CJIeJJ,yeT OTMeTHTb, IJ:TO H3JIOJKeHHbIH aJIrOpMTM H3Y-
IJ:eHHH KOJIe6aHHH npH3MbI npaKTM'IeCKH 6e3 H3MeHeHHH MOJKeT 6bITb HC-
nOJIb30BaH npM aHaJIH3e KOJIe6aHHH npHMoyrOJIbHOH nJIaCTHHKH B ee
nJIOCKOCTH (nJIOCKOe HanpmKeHHoe COCTOHHl-le). AUaJIOTHIJ:HbIM noJJ,xoJJ,
MOJKeT 6bITb peaJIH30BaH H npM H3YIJ:eHHH H3rH6HbIX KOJIe6aHHII nJIac-
THHKH C pa3JIHIJ:HbIMH YCJIOBHHMH 3aKpenJIeHHH. B 060MX 3THX


235




CJIyqaHX ,l1;I1anaSOH HSMeHeIHfH qaCTO Bos6Yllil1.aIOIlI.HX CHJI eCTeCTBeHHO
OrpaHHqHBaeT
H Tpe60BaHHeM MaJIOCTH TOJIIJJ.HHbI nJIaCTHHKH no cpaBHe-
HHIO C ,n:JIHHOH BOJIHbI.


::x:
(\.,j


4


Ha nepBOM 3Tane peweHHH sa,n:aqH 06 YCTaHoBHBwHXCH KOJIe6a-
HHHX KOHeqHOrO mlJI
lH,n:pa He06xo,n:HMO nOCTpOHTb TaKoe perneHHe
ypaBHeHHH (27.2), KOTopoe ObI saBe,n:OMO y,n:OBJleTBOpHJIO ,n:OCTaTOq-
HO 06IlI.HM YCJIOBHHM B CMelJJ.eHHHX HJIH HanpHllieHHHX Ha rpaHHqHbIX
nOBepXHOCTHX. OpH nOCTpoeHHH TaKoro peWeHH5I 6y,n:eM HCXO,n:HTb HS
B03MO*HOCTH COKpaIlI.eHH5I 06"beMa BbIKJIa,n:OK pas,n:eJIeHHeM 06IlI.erO
nOJIH HanpH*eHI1H B LlHJIHH,n:pe Ha CHMMeTpHqHYIO H aHTHCHMMeTpHq-
HYIO OTHOCHTeJIbHO nJIOCKOCTH Z = ° qaCTH. PeIllaH YKa3aHHYIO Bblwe
KOHKpeTHYIO sa,n:aqy, no,n:po6HO OCTaHOBHMCH Ha aHTHCHMMeTpHqHOM
CJIyqae .
HCnOJIb3YH H3BeCTHble peweHHH ,n:HHaMHqeCKHX 3a,n:aq ,n:JIH 6ec-
KOHeqHbIX CJIOH H LlHmlH,n:pa, npe,n:CTaBHM HCKOMbIe KOMnOHeHTbI BeKTO-
pa nepeMelJJ.eHHH B BH,n:e
u=
 [ Al
 1 /I(Plr)+Bl.!!!..-/I(P2r) j 'sinklZ-
1=1 1 P2
C<J
I ( Di -.!:L sh qIZ + C j r
 sh q2Z) Jd'A-/),
i=1 ql 1


(27.3)



 27. OcecHMMeTpH'IHbIe KO.JIe6aHH 11
KPYI'OBOI'O .QH.JIHH
pa KOHe'IHOii
JlHHbI


COrJIaCHO uCYIlI.eCTBOBaHHIO XOpOillO H3yqeHHbIX perneHHH ypaBHeHHH
,n:HHaMH'IeCKOH TeopHH ynpyroCTH ,n:JIH 6eCKOHeqHbIX CJIOSJ H LlHJIHH,n:pa,
paccMoTpHM B paMKax npe,n:JIaraeMOro MeTo,n:a 3a,n:aqy 0 BbIHY*.ll:eHHbIx
KOJIe6aHHHX KpyroBoro LlHJIliH,n:pa KOHeqHOH ,n:JIHHbI. HCnOJIb-
SOBaHHe 3TH}( peweHHH n03BOJISJeT
nOJIY'IHTb TaKHe BbIpallieHHH ,n:JIH
KOMnOHeHTOB BeKTopa CMelJJ.eHHH
H TeH30pa HanpHllieHHii, KOTopbIe
y,n:oBJIeTBopSJIOT ,n:OCTaTOqHO npoH3-
BOJIbHble rpaHH'IHbIe YCJIOBHH B Tpex
OCHOBHbIX rpaHHqHbIX sa,n:aqax Ha
r' BceH nOBepxHocTH LlHJIHH,n:pa KOHe'I-
HOH ,n:.rIHHbI.
KaK H B cJIyqae paccMoTpeHHoH
BbIwe nJIOCKOCTH sa,n:a'IH 06IlI.ee pe-
weHHe HCnOJIb3yeM npH paccMoTpe-
HHH KOHKpeTHoH rpaHH'IHOH 3a,n:a-
qH, B Ka'IeCTBe KOTOpoii BbI6epeM
3a,n:a'IY 06 H3rH6HbIX KOJIe6aHHHx
KpyrJIoii TOJICTOH nJIHTbl no,n: ,n:eHcTBHeM nepHOD.H'IeCKH H3MeHHIOIJJ.HXCH
JIHHeHHO pacnpe,n:eJIeHHWX HOpMaJIbHWX HanpSJllieHHH Ha ee 60KOBOH
nOBepxHocTH (pHC. L4) AHa.nH3 YC.nOBHH cYIlI.eCTBoBaHHH peIlleHHH B
06IlI.eM c.nY'Iae sarpy*eHHH HBH.nCH 6bl, no cYIlI.ecTBY, nOBTopeHHeM
cooTBeTcTBYIOIlI.HX BbIK.naD.OK D..nH CTaTH'IeCKHX 3aD.aq.
HanpHllieHHO-D.e<popMHpoBaHHoe LOCTOHHHe n.nHTbI no.nHOCTbIO sa-
,n:aeTCH BeKTopoM ynpyrHx CMeIlI.eHHH, KOTOpbIH onpeD.eJIHeTCH ero npo-
eKLlHHMH Ha OCH LlH.nHH,n:pHqeCKOH CHCTeMbi KOOpD.HHaT (CM. pHC. 24):
u = ue, + wk (27. I)
oe3pa3MepHbIe (oTHeceHHbIe K paD.HYcy n.nHTbI R) aMn.nHTYD.HbIe 3Ha-
qeHi1H KOMnOHeHTOB BeKTopa CMeIlI.eHHH D.O.nlliHbl y.n:oB.neTBopHTb B pac-
cMaTpHBaeMoM ocecHMMeTpH'IHOM c.nyqae c.ne.n:YIOIlI.eH CHCTeMe .n:H<p<pe-
peHLlHa.nbHwx ypaBHeHHH:
iJ 2 u I iJu U iJ 2 u m iJS pro" R2
(ff2 + r a;--"72 + iJz2 + m- 2 ar + ----o--u = 0,
iJ 2 w I dw iJ 2 w m iJS pro 2 R2
(ff2+-;-ar +--al2 + -m-2 az+ G w = 0,
r,n:e 0 - 06"beMHOe pacwHpeHHe, a 6e3pa3MepHbIe Koop.n:HHaTbI onpe,n:e-
,,1}i!OTCH paBeHcTBaMH r = r fR, z = z'/R. -


z'


K
o


C<J






w = Ao cos yz +
 rAz10 (PIr) + BJo (P2r)] cos klz +
Z=I


PRC. 24.


+
 (D, ch qIz + C i ch q2Z) J o ('An.
1=1
Ka*,n:oe OT.n:e.nbHO B3HToe CJIaraeMOe B 3THX BbIpallieHl1HX y.n:oB.neTBopH-;
eT ypaBHeHHSJM (27.2) lip H npOH3BOJIbHblX k l H 'Ai' He06xoD.HMbIe CBOH-
CTBa no.nHOTbI npe.n:CTaB.neHHH BeKTopa CMeIlI.eHHH H COOTBeTCTBYIOlJJ.e-
ro TeH30pa HanpSJllieHHH Ha nOBepxHoCTSJX LlH.nHH.n:pa 06eCne'IHBaIOTCH
B c.nyqae, KorD.a k l = (2/ - I) nf2h (h = H f R) H J I ('Ai) = O. 3Ha-
qeHHSJ k{ H 'Ai' KaK H B sa.n:aqax CTaTHKH, Heo.n:H03HaqHO onpe.n:e.nHIOTCH
Tpe60BaHHSJMH no.nHOTbI COOTBeTCTBYIOIlI.HX CHCTeM <PYHKLlHH. YKa3aH-
HbIe 3Ha'IeHHH y,n:OOHbI npH y.n:OB.neTBopeHHH rpaHI1'IHbIX YCJIOBHH.
LI.nSJ no.nHOH onpe.n:eJIeHHOCTH c.ne.n:yeT YKa3aTb, qTO B BbIpa*eHHHx
(27.3) Ao, A" Blo Di' C i - npoH3Bo.nbHbIe nOCTOHHHbIe H, KpoMe TOro,


2 m-2
y = 2 (m - 1) .


pm 2 R2
G


ro 2 R'
-
,
V D


(27.2)


pi = kf - y2, P
 = k7- 2 m-l y2,
m-2
2 'A2 2 q
 = 'A
 - 2 m-l y2.
q]= i-V, m-2


(27.4)


npH HSMeHeHHH 'IaCTOTbI Ko.ne6aHHH BeJIH'IHHbI P H q, BXo.n:HIlI.He B
BbIpallieHHH (27.3), MorYT 6bITb KaK BeIlI.eCTBeHHbIMH, TaK H MHHMbIMH.
npH 3TOM lI..nH onpe.n:eJIeHHH 3HaqeHHH <PYHKLlHH oecce.nH H mnep60.nn-


236


237




lleCKHX CPYHKI!JfH HCnOJIb3YIOTCH COOTHOrneHHH
10 (iX) = J O (x), II (ix) = iJ I (X),
ch (ix) = COSX, sh{ix) = isinx.


(27.5)


AHaJIOrflllHble npe06pa30B8HHH n03BOJISIIOT npHBeCTH CPYHKIIHOHaJIbHOe
ypmmeHlle, OTBellaIOIII.ee YCJIOBHSIM Jl.JISI 0' r B (27.6), K CJIeJl.YIOIII.eMY BH-
Jl.Y:


COOTHOllleHHSI 3aKOHa rYKa n03BOJISIIOT JIerKO 3anHcaTb BblpmKeHHSI
Jl.JISI HanpmKeHlIH, oTBellaroIIl.He CMerrreHHSIM (27.3). XapaKTep Jl.aJIbHeH-
lllHX BbIKJI8Jl.OK, CBSI3aHHbiX C YJl.OBJIeTBOpeHHeM rpaHH'lHbiX YCJIOBHH.
Jl.OCTaTO'lHO SIceH. D03TOMY MO}KHO npHBecTH 3.neCb JIHlllb 1fX KOHellHbIH
pe3YJIbTaT. DpH 3TOM CIlHTaeM, liTO rpaHH'lHble YCJIOBHSI .nJISI aMnJIHTY.n-
HblX XapaKTepHCTHK HanpSI}KeHHH onpe.neJISIIOTCSI CJIeJl.YIOIllHMH paBeHCT-
BaMH:


B I k [ /0 (P2) (p
 + kf)2 I II (PI) _
 ] +
I I (P2) I /1 (P2) - 4k;ptP2 /1 (PI) 2k;P2


(27.6)


cOSI'
 h('::'-'2) i (_1)/+IC;chq2hJO(Ai) X
1'2- I 1=1
[ yi 2m lii ] 2 (- 1)1+1 (27.9)
X 2 I? cii . ? 2 2 2 = hk 2 to'
"'i ( I + I) (RI + q]) (k l + q2) I
. 2 2 m-I 2
B paBeHcTBax (27.8) H (27.9), KaK H paHbllle, npHHSIToy] = m- 2 y.
ECTecTBeHHoH 38MeHoH HCKOMblX BeJIH'lHH B (27.8) H (27.9) no cpop-
MYJIaM


21'2 (- 1)/+1
+ Ao h (m - 2)


1 1
20 a r = foz, 20 Tn = 0
1 1
20 a z = 20 '1: r2 = 0


npH r= I,


npH z = h.


rpaHH'lHble YCJIOBHSI .nJISI KaCaTeJIbHblX HanpH}KeHHH B CHJIY Bbl60pa
3Ha'leHHH k , H A; npHBo.nSIT K CJIe.nYIOIllHM CBH3SIM Me}K.ny 3Ha'leHHSIMH
npOH3BOJIbHbiX nOCTOSIHHblX B (27.3):
Al = - Bl P
 + kf
2PIP2
D C "'
 +
 ch qjlh
i = - I 2",
 . chqlh .
1


C.
Yi = -t ch q2 hJ O (A).


(- 1)/+1
XI = B,k,!. (P2)
P2


(27.10)


/1 (P2)
/1 (PI) ,


npHl
e.ll.eM 3TH COOTHOllleHHSI K CJIe.n.YIOIlleH 6eCKOHe'lHOH CHCTeMe JIHHeH-
HblX aJIre6paH'leCKHX ypaBHeHHH:


""


(27.7)


. 1'4 ""XI 0
Ao Y SIn yh + 2 m _ 2 L..J ---:?2 = ,
1=1 RlPI
1'2
 [ Yi
x,!:i, (p) - m.:::"2
 Yi A
 (I? + q2)
1=1 ,I 1
A 21'2 cas I'h 2fo
+ U h (m - 2) y2 -lij = hkl '


2mlij ]
(kf + qi)(,q +
) +


Y}Ke caMa cpopMa 3anMCH 3THX CooTHollleHHH pa3pernaeT He paCCMaT-
pHBaTb CneIIHaJIbHO Te 'laCTOTbl, npH KOTOpblX PI = iAj HJIH ql h =
= if
 + k:rt).
DocJIe HeCJIO}KHbIX npe06pa30BaHHH C HcnOJIb30BaHHeM npe.n.CTaB-
JIeHHH pSI.n.aMH no J 0 (A{) CPYHKIIHH 5eCCeJISI MHHMoro aprYMeHT3 H
COOTHOllleHHH (27.7) Ha OCHOBe rpaHH'lHbIX YCJIOBHH Jl.JISI 0'2 B (27.6)
nOJIY'lHM cJIe.nYIOrrrylO cHcTeMy paBeHcTB:


(27.11)


1'2
 [ Yi
y;!:i; (q) - m _ 2 L.J XI k2 ( ",
 + P 2 )
1=1 I 1 1


2ml.
 ]
1 -- 0
(",
 + pi) (1.7 + p
) - .


A oy sin Y h + m 2y4 2
 (- 1)1+1
1..J 2 B,II (P2) = 0,
1=1 klPIP2
C;q2 ch q2h ( th q2h ("'
 ;-
)2 th qlh ) -
4A;Qlq2
00
y2 L (- 1)1+1
 B /1 (P2) ( Yi
m - 2 1=1 P2 [Jo (loJ) kl (pi + "'
)
2mt.,
 J
1 _ 0
,.2 2 2 2 -.
If'1 + Ai) (P2 + AJ)
238


3.n.eCb
!:i [ 10 (P2)
,(p) = P2 /1 (P2) -
!:i; (q) = q2h [ th q2h


(
 + k;)2 .
 PI) ] _ m - 1
4klPIP2 II (PI) m - 2
(",2 1 , + q
)2 ]
th qlh
4"'7 Q lqg


1'2
J?'
I


(27.12)


(27.8)


SIBJIHIOTCfJ COOTBeTCTBeHHO onpeJl.eJIHTeJIHMH OoxraMMepa H Pe.1eSI -
J13M6a.
OCHOBOH 3<pcpeKTlIBHoro aJIropHTMa pellleHHH CHCTeMbJ (27 11) MO}KeT
CJIY}KHTb 3HaHHe aCI1MnTOTH'leCKI1X CBOHCTB lIeH3BeCTHbiX XI H Yj" BHLl
COOTIIOllJellHH (27_1 I) npH cpaBHeHlIH HX C nOJIy'leHHbIMH paHee B CJIY-
(me 3a.n.a'lH .nJISI npSIMO}TOJIbHHKa (25.4) YKa3bIBaeT lIa TO, 'lTO 3.n.eCb
239




TaK:>Ke CJIe:nyeT paCC'IIITbIBaTb Ha cym:eCTBOBaHIIe paBeHCTBa
Jim Yi = - lim Xl = a o . (27.13)
;_00 1-40:.5
I10,ll,XO,ll" IICrrOJIb30BaHHbIH npII aHaJIII3e CooTHoUleHIIH (25.4), n03BOJIH-
eT JIerKO y6e,ll,IITbCH B 3TOM.
I1YCTb ,lI,IIana30H II3MeHeHIIH 'IaCTOTbI KOJIe6aHIIH OrpaHII'IeH HeKO-
TOpbIM KOHe'IHbIM 'IIICJIOM 1'0 (0 < I' < 1'0). BbI6epeM B COOTBeTCTBIHl co
3Ha'IeHIIeM 1'0 TaKIIe ,lI,OCTaTO'IHO 60JIbUlIIe 'IIICJIa L II N, 'ITo6bI BeJIII'IlIHb1
I'
/"A'iv II I'ilk'iv 6bIJIlI ,lI,OCTaTO'IHO MaJIbIMlI no cpaBHeHlIIO C e,ll,lIHHLLeH.
PaccMoTpHM 6eCKOHe'IHYIO cHcTeMY ypaBHeHHH ,lI,JIH onpe,ll,eJIeHHH He-
H3BeCTHbIX Xl (l > L) II Yj (j > N), BbITeKaIOm:yIO H3 (27.11) npH
YCJIOBHH, 'ITO 3Ha'IeHHH HeH3BeCTHblX C HOMepaMH, MeHbUlHMH L 11 N,
H3BeCTHbI. ::ha CHCTeMa MO:>KeT 6bITb 3anHcaHa B TaKOM BH,lI,e:
X/)'l (p) = m
 2 ! Yj [ "7 (k; + qi) - (k7 + q
7
!7 + q
) ] +


OTHOCHTeJIbHO K03<p<pIILLIIeHTOB nepBOH rpyrrrrbI paBeHCTB B (27.15)
rrOJIY'IIIM cJIe,ll,YIOm:yIO OLLeHKY (L < 1 < 00):
2mv. L oa" k7 2mv. L k7
J 1 = 2 < - 2 2

l (p) (m - 2) j=N (k7 + "j).
l (P) (m - 2) j=1 (k l + A j ).
mv. [ 16 (kl) -1-
 ] =1-2 m+ I +
2
l (p) (m - 2) Ii (kl) k7 mkl


+O(
 ).


(27.17)


AHaJIOrII'IHO OLLeHIIM CyMMY K03<pcjJIILLlIeHTOB BO BTOpOH rpynne pa-
BeHCTB:


J 2 =


2mv.
 "1 4'Aj [
 "7 "7 J -
",. (q) (m - 2 )
 (k 2 + ,,2). < h
 (k 2 + ,,
 ) . - ( J? + ,,
 ) . -
1 I=L 1 i 1=1 I I I I
4
 4

I <1- I +
h (kr + "7). h (kr + "7).


+ o(
 ),


'Ajh
= th "Ajh - ch. "jh


(27.14)


v.
 [ vi 2m/wi )
y;6. j (q) = m- 2 ..:. XI k 2 (,,2. 2) - ( ,,2 + 2 ) ( ,2 + ...2 ) +
l=L 1 j -r PI j PI Fo j /12
+ 0 ( ,,
 ) .


+ o( :7 ).


(27.18)


v.
2fT'
1


(27.16)


OLLeHKII ,lI,JIH K03<p<pIILLlIeHToB CIICTeMbI (27.15) MO:>KHO rrpe,ll,CTaBIITb
B CJIe,ll,YIOll(eM BII,lI,e:
J 1 =1-0(+), J 2 =1-0(
; ). (27.19)
TaKaH 3anIICb YKa3bIBaeT rrpe:>K,lI,e Bcero Ha TO, 'ITO CIICTeMa (27.15) HB-
JIHeTCH perYJIHpHoH. KpoMe Toro, y'IIITbIBaH KOHe'IHOCTb 'IIICeJI L II N,
3aMe'IaeM, 'ITO JII06bIe CKOJIb yrO,ll,HO TO'IHble OLLeHKII CYMM K03<p<pIILLIIeH-
TOB B CIICTeMe (27.15) He II3MeHHT OLLeHOK CKOpOCTII, C KOTOpOH BeJIII'III-
HbI J 1 II J 2 CTpeMHTCH K e,ll,IIHIILLe. 3TO 06CTOHTeJIbCTBO O'IeHb Ba:>KHO,
rrOCKOJIbKY KaK pa3 Ha cpaBHemIlI CKOpOCTII y6bIBaHHH <PYHKLLIIH, xa-
paKTeplI3YIOm:HX peryJIHpHOCTb CIICTeMbI, C ee K03<p<pIILLIIeHTaMII OCHO-
BbIBaeTCH 3aKJIIO'IeHIIe 0 CrrpaBe,ll,JIIIBOCTH ,lI,JIH HeII3BeCTHbIX 3aKOHa aCIIM-
IITOTlI'IeCKIIX Bblpa:>KeHIIH (CM.
 3). XapaKTep K03<p<pIILLIIeHToB CIICTeMbI
(27.15) II OLLeHKII (27.19) YKa3bIBaeT Ha TO, 'ITO ,lI,JIH (27.15) crrpaBe,ll,JIIIB
3aKOH aCHMnTOTlI'IeCKIIX Bblpa:>KeHIIH, KOTOpbIH C Y'IeTOM 3HaKOB B
COOTHOUleHIIHX (27.15) BbIpa:>KaeTCH cJIe,ll,yIOm:IIM paBeHcTBoM:
Jim Xl = - Jim Y; = a o . (27.20

1...00 ;-+00
I1pII rrepexolI,e OT CIICTeMbI (27.14) K CIICTeMe (27.15) 6bIJIII OT6po-
UleHbI BeJIH'IIIHbl nOpH,lI,Ka l/ki II 1I"A7. B CBH3II C 3TIIM 3aKJIIO'IaeM, 'ITO
CIICTeMa (27.14) TaK:>Ke HBJIHeTCH perYJIHpHOH II ,lI,JIH CyMM K03<p<pIILLIIeH-
TOB B cTpoKax CHCTeMbI CrrpaBe,ll,JIIIBbI OLLeHKII (27.19). OTCIO,lI,a CJIe,ll,yeT
II 60JIee Ba:>KHOe 3aKJIIO'IeHIIe 0 TOM, 'ITO HeII3BeCTHbIe B CHCTeMe (27.14)
nO,ll,'IHHHIOTCH npe,ll,eJIbHoMY paBeHcTBY (27.20).


KOHKpeTHoe Bblpa:>KeHHe ee CB06o,ll,HbIX 'IJIeHOB B ,lI,aJIbHei1UleM Hecy-
m:eCTBeHHO, a OLLeHKH ,lI,JIH HHX CJIe,ll,YIOT HenOCpe,ll,CTBeHHo H3 npe,ll,no-
JIO:>KeHHH 0 KOHe'IHOCTH HeH3BeCTHblX C HOMepaMII, MeHbUlHMH N H L.
PaccMoTpHM HapH,lI,Y C cHcTeMoH (27.14) TaKYIO, KOTopaH CJIe,ll,yeT
II3 Hee npII rrOCJIe,ll,OBaTeJIbHOM oropacbIBaHIIH BeJIII'IHH I'ilkl H I'V"A'iv no
cpaBHeHHIO C e,ll,HHHLLeH. 3TY cIIcTeMY MO:>KHO npe,ll,CTaBIITb B cJIe,ll,YIOm:eM
BII,lI,e:


Xl=- 2mv. L Y; k7 . +O (
 ) ,

l (p) (m - 2) i=1V (kl + "7). . kl
Yj=- 2mv. LXI 2 "i 2 +0(+ ) .

j (q) (m - 2) '=L (k 1 + Ai). , I
.l\cllMIITOTII'IeCKHe (c TO'IHOCTbIO lI,0 BeJIII'IHH nOpH,lI,Ka I'ilkl
3Ha'IeHHH onpe,ll,eJIHTeJIeH I10xraMMepa H PeJIeH - JI3M6a B
OTHOUleHHHX onpe,ll,eJIHIOTCH no <popMYJIaM
- V. m
6., (p) "" 6. 1 (p) = 2 (m - 2) kl


(27.15)


II I'i/"A'iv)
3TIIX CO-


-- V 2m
6. / " (q) "" 6. / ' (q) = J.
2 (m - 2) i
240


16 7-246q


241




ECJJH Tenepb OOpaTHTbCH K HCXO.n:HOH OeCKOHeIIHOH CHCTeMe (27.11),
KOTOpylO TOJIbKO 11 Ha.n:o pernaTb npH paccMOTpeHHI1 KOHKpeTHblx rpa-
HHIIHblX sa.n:aII, TO npe.n:bl.n:Y
He BbIKJIa.n:KH nOSBOJISJIOT c.n:eJIaTb CJle.n:ylO-
rn;ee SaKJIIOIIeHl1e: eCJIH cy
eCTBYIOT orpaHHIIeHHble perneHI1H CHCTeMbl
{27 .11), TO .n:JIH o.n:Horo HS HHX cnpaBe.n:JIHBO npe.n:eJIbHOe paBeHcTBo
(27.20). 3TH CBe.n:eHHSJ 0 nOBe.n:eHHH perneHHSJ oeCKOHeIIHOH CHCTeMbl
(27.11) npe.n:CTaBJIHIOT OOJIbrnylO IJ.eHHOCTb C TOIIKH speHHH nocTpoeHHH
perneHHH rpaHHIIHblX sa.n:aII, nOCKOJIbKY HMeHHO perneHl1e CI1CTeMbl
(27.11) C YKasaHHblMH aCHMnTOTHIIeCKHMH CBoHcTBaMH .n:aeT HCKOMoe
perneHHe rpaHHIIHoH sa.n:aql1. Yoe)K.n:aeMcH B 3TOM, OOblIIHO, nYTeM npo-
BepKH Y
03JIeTBOpeHHH rpaHI1IIHbIX YCJIOBHH.
HCnOJIbSOBaHHe aCHMnTOTI1IIeCKHX paBeHcTB THna (27.20) B paHee
paccMoTpeHHblx sa.n:aIIax CTaTHKH .n:aBaJIO BOSMO)KHOCTb YKasaTb 9CP-
cpeKTHBHblH aJIrOpHTM .n:JISJ HaXO)K.n:eHHH Bcex HeHSBeCTHblX HS oeLKo-
HeIIHblX CHCTeM B CJIYIIae sa.n:aII .n:HHaMHKI1 CBOHCTBO HeI1SBeCTllbIX,
Dblpa)KeHHOe paBeHcTBoM (27.20), .n:aeT BOSMO)KHOCTb nocTpOHTb 3CP-
<peKTHBHbIH aJIrOpliTM OTbICKaHHSJ npHHHToro perneHHH oecKoHeIIHoH
CIICTeMbI, a TaK)Ke IIaCTHIIHO cBepHYTb 6ecKoHeIIHblH IIacToTHblH onpe-
.n:eJllfTeJIb .n:JIH coocTBeHHblx IIaCTOT. Bce 9TI1 BOSMO)KHOCTH peaJllf3YIOT-
CSJ npH yceqemlH oecKoHeIIHoH CHCTeMbI (27.11) C nOMOrn;blO paBeHcTB
XI = XL (I> L), Yj = YN (j> N). (27.21)
He oy.n:eM npHBo.n:HTb s.n:eCb BH.n: KOHeIIHOM CHCTeMbI, nOJIYIIeHHoM C
H
nOJIbSOBaHHeM paBeHcTB (27.21). YKa)KeM JIHrnb SHaIIeHHH CYMM He-
KOTOpblX oecKoHeIIHblx pSJ.n:OB, HY)KHbIX .n:J1SJ BbIIII1CJIeHHH Ko9<fJ<PHIJ.HeH-
TaB npH XL H YN B KOHeIIHOH CI1CTeMe


TOT (1'0 nopH.n:Ka 15). f1pl1 OnHCaHl111 pesYJIbTaTOB BblIIHCJleHI1H OCTa-
HOBI1MCSJ npe)K.n:e Bcero Ha Tex MOMeHTax, KOTopble He HarnJIH OTpa)Ke-
HHH npl1 paCCMOTpeHl111 sa.n:aIII1 a KOJIeOaHI1HX npHMoyrOJIbHOH npl1SMbl.
PernaH sa.n:aIIY 0 KOJIeOaHI1SJX npHMoyrOJIbHOH npl1SMbI, Mbl He OCTa-
HaBJII1BaJII1Cb Ha OIJ.eHKe pOJII1 SHaHI1SJ aCI1MnTOTI1IIeCKI1X CBOHCTB He-
I1SBeCTHblX C TOIIKI1 SpeHI1H YJIYIIrneHI1SJ KaIIeCTBa perneHI1SJ oecKoHeII-
HaM CI1CTeMbI. B .n:I1HaMI1IIeCKI1X sa.n:aIIax OCHOBOH .n:JIH yceIIeHI1H oec-
KOHeIIHblx CI1CTeM SJBJISJIOTCH paBeHcTBa Tl1na (27.21), KOTopble B CBOIO
Ta6J1Hu:a 30


I I . I I I .
I Xl "I i Yi y.
J
I 0,5268 0,5260 I 0,2569 0,2564
2 0,5369 0,5323 2 -1,084 -1,082
3 0,5362 0,5244 3 -0,6543 --0,6527
4 0,5344 0,5111 4 -0,5825 -0,5806
" . .. . .. . .. . .. . .. .
16 0,5301 0,2894 36 -0,5293 -0,4760
17 0,5300 0,2763 37 -0,5293 -0,4739
18 0,5300 0,2641 38 -0,5294 -0,4718
19 0,5299 0,2528 39 -0,5294 -0,4697
20 0,5299 0,2423 40 -0,5294 -0,4677
>20 0,5296 ° >40 -0,5296 °


00 [ 2

 1'1
L 2 2 2
1=1 k l ('A j + PI)


\ 00, I I I h \
- = -- l - th "h - h 2 /
.-.. ._?.,2 2 2 I ,
1=1 "jP, I' I' I
2m'A
 ] 1'2 ( h )
- J =
 h 2 _ _ th h-
('A
 + p
) ('A
 + p
) 2q
, qi ql
'A
 (m - 2) ( h h )
J 2 - th q2h - - th qIh ,
I' q2 qi


(27.22)


OIIepe.n:b SJBJISJIOTCSJ npSJMblM C.'1e.n:CTBl1eM npe.n:eJIbHOrO paBeHcTBa
(27.20). B TaOJI. 30 .n:JIH l' = 1,9 C TOIIHOCTblO.n:O to I1S (27.6) npHBe.n:e-
Hbl SHaIIeHI1H Hel1SBeCTHbIX XI 11 Y J , nOJIYIIeHHbIe npl1 O.n:HI1X 11 Tex )Ke
YKasaHHbIx Bblrne BeJII1III1HaX L 11 N Ha OCHOBe paBeHcTBa (27.21) H no
MeTo.n:y npoCToM pe.n:YKIJ.I1H, KOTopble OTMeIIeHbl sBes.n:oIIKoH. BbIIIHC-
JIeHI1SJ npoBe.n:eHbl .n:JISJ OTHOCI1TeJIbHO TOJICTOM KpyrJIOM nJII1Tbl C h =
= 0,2.
llaHHble TaOJI. 30 YKaSbIBaIOT, npe)K.n:e Bcero, Ha .n:OCTaTOIIHO BbICO-
KYIO CTeneHb OJII1S0CTI1 SHaIIeHI1M X 20 11 Y40' 3TO OOCTOHTeJIbCTBO HB-
JISJeTCSJ KOCBeHHblM YKaSaHl1eM Ha TO, IITO npH yceIIeHl111 6eCKOHeIIHOM
CI1CTeMbI OCTaBJIeHO .n:OCTaTOIIHOe KOJII1IIeCTBO ypaBHeHI1M .n:JISJ aHaJII1Sa
.n:HHaMI1IIeCKOro nOBe.n:eHI1H KpyrJIOH nJII1TbI npl1 .n:aHHOM SHaIIeHl111 IIac-
TOTbl. HHTepeCHOM ocooeHHOCTblO perneHI1H, nOJIYIIeHHoro no MeTOn.y
npOCTOIJ pe.n:YKUI1H, HBJISJeTCSJ SHaIII1TeJIbHOe paSJII1IIl1e B CTeneHI1 TOII-
HOCTI1 SHaqeHI1H BeJII1III1H X; 11 y;. Hel1sBecTHble Y; npH TaKOM IIo.n:xo.n:e
onpe.n:eJISJIOTCH TOIIHee, IIeM X;. T aKoe nOJ10)KeHHe, nO-BH.n:IiMOMY, CBSJ-
saHO He TOJIbKO C TeM, IITO B KOHeIIHYIO CHCTeMY BOBJIeKaJIOCb paSJIIiII-
Hoe KOJIHIIeCTBO HeliSBeCTHblX X; H Y;, HO Ii .CO cneUHcpHKOH paCCMaTpH-
BaeMOH rpaHIiIIHOH sa.n:aIIIi.
f1pOBO.n:H Ha OCHOBe .n:aHHbIX TaOJI. 30 cpaBHHTeJIbHblH aHaJIIiS B03-
MO)KHOCTeH paSJIIiIIHbIX cnOCOOOB perneHIiH oecKoHeIIHbIx CIiCTeM, CJIe-
.n:yeT HMeTb B BIi.n:y, IITO .n:Ba IiCnOJIbSOBaHHblX no.n:xo.n:a npHHIJ.linHaJlbHO
pa3JlliIIalOTCSJ Me)K.n:y COOOM. 3TO paSJIIiIIHe OTpa)KeHO B IIOCJIe.n:HeY..




 [ I'
 2mkl J y
 [ I I '0 (PI) +
f=. 'A7 (kl + q
) - (kl + q
)(k¥ + q
) = pi S - - 2PI I] (PI)
+
 ] _ 2(m-2)k] [ I lo(P2) _
_
 '0 (PI) +
 ]

 1'2 2P2 11 (P2) P
 2p] I] (PI) Vr'
HSJIO)KeHHbIH aJIrOpl1TM onpe.n:eJIeHHH COOCTBeHHblX IIaCTOT H .n:H-
HaMIiIIeCKOH HanpH)KeHHOCTH npH oceCHMMeTpliIIHblX KOJIeOaHHSJX UH-
JIIiH.n:pa KOHeIIHOH .n:JIHHbl peaJIH30BaH C nOMO
bIO 3BM. f1pH npOBe.n:e-
HHIi BblIIHCJIeHlil1 npHHHMaJIOCb L = 20, N = 40, m = 3. TaKoM Bbl-
oop IIIiCeJI L Ii N nOSBOJISJeT HSYIIIiTb .n:HHaMHIIecKYIO HanpSJ)KeHHOCTb
QHJIIiH.n:pOB, y KOTOpblX h < 1 B .n:OCTaTOIIHO rnlipoKoM .n:lianaSOHe IIac-


242


16*


243




CTpOKe Ta6JI. 30. 3.n.ecb 3HaqeHHH BeJIHqHH Xl H Yj C HOMepa."dH, 60JIbIIIH-
MH 20 H 40, onpe.n.eJIeHbI KaK cpe.n.Hee apH<pMeTHqeCKOe a6COJIIOTHbIX 3Ha-
qeHHM x 20 H Y40, B TO BpeMH KaK no MeTO.n.y pe.n.YKlLHH X; = u7 = °
(1 > 20, j > 40). 3HaHHe 3HaqeHHM Bcex HeH3BeCTHbIX H3 6eCKOHeq-
HOM CHCTeMbI cYIlLecTBeHHo npH npoBe.n.eHHH BbIqHCJIeHHM Ha rpaHHlLe
ynpyroro TeJIa, qTO .n.OCTaTOqHO HarJIH.n.HO 6bIJIO npOHJIJIIOCTpHpoBaHo
npH paccMoTpeHHH 3a.n.aq CTaTHKH.
OKOHqaTeJIbHaH npoBepKa npHro.n.HOCTH HaM.n.eHHbIx H3 6eCKOHeq-
!HOM CHCTeMbI BeJIHqHH Xl H Yj .n.JIH aHaJIH3a HanpHmeHHOCTH lLHJIHH.n.pa
OCYIlLeCTBJIHeTCH nYTeM npOBepKH rpaHHqHbIX YCJIOBHM. DpOBe.n.eH-
HbIe BbIqHCJIeHHH .n.a}Ke .n.JIH .n.OCTaToqHO 6JIH3KHX K pe30HaHCHbIM qac-
TOT nOKa3aJIH, qTO B .n.aHHOM CJIyqae norpelliHOCTb y.n.OBJIeTBopeHHH rpa-
HHqHbIX YCJIOBHM He npeBbIlliaeT 1 % MaKCHMaJIbHbIX 3HaqeHHM 0 Ha
rpaHHlLe. YIMeHHo C TaKoM norpelliHocTblO annpoKcHMHpyeTcH JIH
eM-
HaH <PYHKlLHH .n.Ba.n.lLaTblO qJIeHaMH pH.n.a no sin klz.
LIJIH paccMaTpHBaeMoM 3a.n.aqH 06 H3rH6HbIX KOJIe6aHHHx KpyrJIOM
DJIHTbI C h = 0,2 BbIqHCJIeHHH npoBe.n.eHbI B .n.Hana30He .n.BYx nepBbIx
c06cTBeHHbIx qaCTOT. LIJIH HHX nOJIyqeHbI CJIe.n.YIOIlLHe OlLeHKH: 0,7 <
< y
l) < 0,8; 2,2 < 'I'
2) < 2,3. ECJIH B KaqeCTBe paCqeTHbIX npHHHTb
cpe.n.H He 3HaqeHHH, TO cpaBHeHHe BbIqHCJIeHHblX 3.n.eCb 3HaqeHHM C COOT-
BeTCTBYIOIlLHMH .lI.aHHbIMH [168], nOJIyqeHHbIMH Ha OCHOBe rHnOTe3bI
KHpxro<pa (.y,.), .n.aeT cJIe.n.YIOIlLHe pe3YJIbTaTbI:
-(1) -(2)
Y
I) = 1,2; Y;2) = 1,7. (27.23)
Y r Y r
OTcIO.n.a CJIe.n.yeT, qTO .n.JIH cB060.n.HOM nJIHTbI C OTHOCHTeJIbHOM TOJIlI.LH-
HOM 2h = 0,4 rrrnOTe3a KHpxro<lJa .n.aeT 3aBbIIIleHHoe Ha BeJIHqHHY no-
pH.n.Ka 20% 3HaqeHHe nepBoM c06cTBeHHoM qaCTOTbI. LIJIH 60JIee BbICO-
KHX qaCTOT pacxom.n.eHHe, eCTeCTBeHHO, YBeJIHqHBaeTcH.
YI3JIOmeHHbIM no.n.xo.n. K H3yqeHHIO .n.HHaMHqeCKOM HanpHmeHHocTH
11 c06cTBeHHbIx qaCTOT KpyrJIbIX nJIHT .n.aeT B03MomHOCTb nOJIyqHTb TaK-
}Ke HeKoTopbIe BamHbIe xapaKTepHcTHKH <pOpM KOJIe6aHHM. Ha pHC. 25
.n.aHO pacnpe.n.eJIeHHe nporH60B Cpe.n.HHHOM nOBepxHocTH nJIHTW B OK-
pecTHocTH nepBoro H BToporo pe30HaHcoB. PaccMoTpeHHe npHBe.n.eH-
HbIX KpHBbIX n03BOJIHeT nOJIyqHTb npe.n.CTaBJIeHHe 0 <POp Max KOJIe6aHHH
DJIHTbI H .n.OBOJIbHO TOqHO onpe.n.eJIHTb nOJIOmeHHe y3JIOBbIX oKpymHoc-
TeM B cpe.n.HHHOM nJIOCKOCTH. LIJIH paCCMaTpHBaeMoro CJIyqaH (h = 0,2)
Haxo.n.HM, qTO npH nepBoM <popMe KOJIe6aHHM pa.n.Hyc Y3JIOBOM oKpym-
HOCTH paBeH 0,68. OKpymHocTH C r 1 = 0,40 H r 2 = 0,85 HBJIHIOTCH
Y3JIOBbIMH npH BTOpOM <p0pMe.
Onpe.n.eJIeHHbIM HHTepec B paccMaTpHBaeMoH 3a.n.aqe npe.n.CTaBJIH-
eT H3yqeHHe xapaKTepa H3MeHeHHH HanpHmeHHM 0, no TOJIIII.HHe nJIHTbI.
Ll.JIH Tpex xapaKTepHbIx 3HaqeHHM qaCTOTbI pacnpe.n.eJIeHHe HanpH}KeHHM
(Ir npH r = 0,6 npe.n.cTaBJIeHO Ha pHC. 26. ECJIH B oKpecTHocTH nepBoro
pe30HaHca (y = 0,8) OTKJIOHeHHH OT JIHHeMHoro 3aKOHa HecYIlLecTBeH-
IHbI, TO yme B OKpeCTHOCTH BToporo pe30HaHca (y = 2,3) KpHBaH, xa-
paKTepH3YIOIlLaH H3MeHeHHe HanpHmeHHM, 3HaqHTeJIbHO OTKJIOHHeTCH


OT npHMoM. TpeTbH KpHBaH .n.JIH CJIyqaH 'I' = 2,7 YKa3bIBaeT Ha HaJIH-
qHe TeH.n.eHlLHH K pe3KOMY H3MeHeHHIO xapaKTepa pacnpe.n.eJIeHHH Ha-
npH}KeHHM 0, no TOJIIlLHHe C pOCTOM qaCTOTbI.
Dpe.n.nOJIOmeHHe 0 JIHHeMHOM 3aKOHe H3MeHeHHH HanpHmeHHM 0, no
TOJIIlLHHe nJIHTbI COCTaBJIHeT O.n.HO H3 OCHOBHbIX .n.onYIlLeHHM TeopHH
nJIaCTHH KHpxro<pa. HecMoTpH Ha TO qTO B OKpeCTHOCTH nepBoro pe30-
HaHca pacnpe.n.eJIeHHe pa.n.HaJIbHbIX HanpHmeHHM .n.OBOJIbHO 6JIH3KO K
JIHHeMHOMY, npH6JIHmeHHaH TeOpHH
.n.aeT B 3TOM CJIyqae 3HaqeHHe nep- z
BOM c06CTBeHHOM qaCTOTbI CO 3Ha-
'IHTeJIbHOM norpelliHOCTblO. O.n.HOM H3
O.ffi
npHqHH 3Toro, nO-BH.n.HMOMY, HB-
JIHeTCH HaJIHqHe 3HaqHTeJIbHbIX no


0.12


o


I
dr
cS;=


0,08


PIIC. 25.


PIIC. 26.


244


BeJIHqHHe (nOpH.n.Ka 30% OT 3a.n.aHHbIX B rpaHHqHbIX YCJIOBHHX or)
oceBbIX HanpH}KeHHM B OKpeCTHOCTH UHJIHH.n.pHqeCKOM nOBepXHOCTH
r = 1.
Dpe.ZJ.bl.ll.YIlLHe BbIKJIa.n.KH H BbIqHCJIeHHH nOKa3aJIH, qTO peIIIeHHe nep-
BOM OCHOBHOM rpaHHqHOM 3a.n.aqH 06 YCTaHOBHBIIIHXCH KOJIe6aHHHX KO-
HeqHOrO lLHJIHH.n.pa MomeT 6bITb npe.n.CTaBJIeHO B BH.n.e (27.3). lLnH onpe.n.e-
JIeHHH co.n.ep}KaIlLHXCH B BbIpa}KeHHHX (27.3) npOH3BOJIbHbIX nOCTOHHHbIX
no rpaHHqHbIM yCJIOBHHM YKa3aH 3<p<peKTHBHbIM aJIropHTM. 3TO n03BO-
JIHeT OCTaHOBHTbCH eIIl.e Ha O.n.HOM MOMeHTe, CBH3aHHOM C H3yqeHHeM
B03MO}KHOCTeM npH6JIH}KeHHbIX no.n.xo.n.OB K OlLeHKe .n.HHaMHqeCKOM Ha-
npH}KeHHOCTH nJIaCTHH.
DpH OlLeHKe npe.n.eJIOB npHMeHHMocTH HH}KeHepHbIx TeopHM KOJIe6a-
HHM nJIaCTHH 06bIqHO HCXO.n.HT H3 aHaJIH3a TOqHbIX peIIIeHHM 3a.n.aq .n.JIH
CJIOH [131, 1421. B CBH3H C 3THM BbICKa3bIBaJIHCb npe.n.nOJIO}KeHHH [l08]
o TOM, qTO yqeT KOHeqHOCTH ynpyroro TeJIa MomeT nOTpe60BaTb cy-
IlLeCTBeHHoe YTOqHeHHe OlLeHOK, nOJIyqeHHbIX H3 peIIIeHHH .n.JIH CJIOH.
P
CCMOTpHM BbIpa}KeHHH (27.3), npe.n.CTaBJIHIOIlLHe KOMnOHeHTbI
BeKTopa nepeMeIlLeHHM B 06IlLeM cJIyqae nepBoM OCHoBHoMrpaHHqHOM
3a.n.aqH. B 3THX BbIpameHHHx BbI.n.eJIeHbI .lI.Be COCTaBJIHIOIlLHe, onpe.n.eJIHe-
MbIe COOTBeTCTBeHHO cYMMaMH no 1 H j. EeCKOHeqHbIe pH.n.bI, co.n.ep}Ka-
IlLHe npOH3BOJIbHbIe nOCTOHHHbIe D i H C i , HBJIHIOTCH 06IlLHMH peIIIeHHH-
MH OCHOBHbIX rpaHHqHbIX 3a.n.aq .n.JIH CJIOH.LIJIH y.n.oBJIeTBOpeHHH rpaHHq-
HbIX YCJIOBHM Ha lLHJIHH.n.pHqeCKOM nOBepxHocTH r = 1 B BbIpameHHH
245





nH KOMnOHeHTOB BeKTOpa CMeweHHH BKnIDqeHW 6eCKOHeqHWe rH
W
C npOH3BOnhHhIMH nOCTOHHHhIMH Al H BI. XapaKTep 3THX pH
OB TaKOB,
qTO
nH Bcex qaCTOT, npH KOTOphIX PI H P2 BeweCTBeHHhI, BnHHHHe
ycnoBHH Ha uHnHH
pHqeCKOH nOBepxHocTH nOKanH3yeTcH B6nH3H
3TOH nOBepxHocTH H He OKa3hIBaeT MHHHHH Ha xapaKTep nonH HanpHLKe-
HHH BO BHYTpeHHHX TOqKaX 06naCTH. 3TO 06CTOHTenhCTBO YKa3hIBaeT
Ha TO, qTO oueHKH npe
MOB npHMeHHMOCTH npH6nHLKeHHhIX TeoplIH,
nonyqeHHhIX Ha OCHOBe pellleHIIH
nH cnOH, cnpaBe
nlIBhI B
OCTaTOqHO
lllHpOKOM
lIana30He qaCTOT.


BTOPYID COCTaBnHIDWYID o6wero pellleHHH 6y
eM pa3hIcKHBaTh B
BII
e
112 = Aoj I (I'r) + BoY I (I'r) + L [ + QI (Plr) + k
 QI tP2 r ) 1 X
/==1 1 P2
k ( 28.2)
x cos /Z,


U I = - L -& (D;'A; ch qlz + C 1 ch q2Z) N I (/\;/r),
1==1 1


(28.1)


W 2 = -
 [Qo (Pl r ) + Qo (p
)J sin klz,
1==1
nonaraH k 1 = in/h. 3TH BhIpaLKeHIIH 6Y
YT Y
OBneTBopHTh ypaBHeHH-
HM
BIILKeHHH (27.2) B UlInHH
pHqeCKHX Koop
HHaTax. ecnH pa
lIanh-
HhIe CPYHKUIiIi B HHX nO
qIIHHIDTCH COOTHOllleHHHM
dQd
pr) = QI (pr),
d 2 Qo (pr) +
. dQo lpr) _ p2Q (pr) = O.
dr2 , dr 0
4 To6hI He YCJIOLKHHTh BhIKJIa
KH BBe
eHlIeM KOMnJIeKCH03HaqHhI'(
CPYHKUIIH, BhI6epeM pellleHHe ypaBHeHHH
JIH Qo (pr) B BH
e
Qo (pr) =
ILo (pr) + 6 1 M o (pr), (28.4)
r
e
I H 6 1 - npOH3BOJIbHhIe nOCTOHHHbIe, a CPYHKUHH pa
l1yca
{ Io (pr) (p2 > 0),
Lo (pr) = j 0 (\ p I r) (p2 < 0);
{ Ko (pr) (p2 > 0),
Mo (pr) = Yo <I pi r) (p2 < 0).
npli TaKOM BhI60pe CPYHKUHM Qo (pr) BhIpa>KeHHH (28.2) y
OBJIeTBOpH-
IDT ypaBHeHIIHM
BHLKeHHH (27.2) II, KpoMe TO
O, co
epLKaT
Be nOCJIe-

OBaTeJIbHOCTH TpHroHOMeTpl1qeCKHX CPYHKUHII, n03BOJIHIDWHe BhI6o-
pOM 3HaqeHHH
l H 6 1 BhmOJIHIITb rpaHlIqHbIe ycnoBHH B CMeweHHHX
IIJIH HanpHLKeHIIHx Ha UHJIIIH
pHqeCKHX nOBepxHocTHX. npH 3TOM CJIe-

yeT IIMeTh B BH
y, qTO B BbIpaLKeHl1HX
Qo (plr) =
P) Lo (Plr) + 6P) M 0 (plr),
Qo (P2r) = 13/)L o (P2 r ) + 6
2J Mo (P2r)
Bce npOlI3BOJIbHbIe nOCTOHHHbIe He3aBHCHMbI.
nOcJIe
YIOWHe BbIKJIa
KI1, CBH3aHHbIe C y
OBJIeTBOpeHHeM rpaHHq-
HbIX YCJIOBI1H, qpe3BbIqaHHO rpoMo3
KH. n03TOMY 6y
eM onyCKaTb He-
KOTopbIe npoMeLKYTOqHbIe pe3YJIbTaThI npli paccMoTpeHHII CJIe
yID-
llleM npoCTeHlllei'I rpaHlIqHoi'I 3a
aqlI:
I I 0
2G OZ = fo, 2G 't,z =


(28.3)


S 28. BbIHY
\AeHHbIe ROJIe68HIIH nOJIoro n:IIJIIIHAP8
ROHeQHOH AJIllHbI


npH nOCTpoeHlIH 06WHX pellleHHH rpaHHqHbIX 3a
aq 06 YCTaHOBIIB-
llllIXCH KOJIe6aHlIHx npHMoyroJIbHOH npH3MbI H cnnolllHOro UHJIHH
pa
IICXO
HM H3 TOH LKe H
elI, KOTopaH HCnOJIb30BaJIaCb B paHee paccMOT-
peHHbIx 3a
aqaX CTaTHKII. npH :!TOM aHaJIOrliH MeLK
y 3a
aqaMH CTaTH-
KH II
HHaMHKII He 3aKaHqHBaeTCH Ha 3Tane nOCTpoeHHH o6wero pe-
llleHIIH, a npOCJIeLKHBaeTcH Ha Bcex 3Tanax ero Hcnonb30BaHliH npH
y
oMeTBopeHHH rpaHHqHbIX YCJIOBHH BnJIOTb
o H
eHTHqIIOCTII xapaK-
Tepa BbIqlICneHHH npli KOJIHqeCTBeHIIbIX oueHKax HanpHLKeHIIOCTIi. BaLK-
HeHllllIM MOMeHTOM B YCTaHOMeHHOH aHaJIOr1i1i HBJIHeTCH COBna
eHHe
aCHMnTOTlIqeCKHX CBOHCTB K03cpcpHUHeIITOB 6eCKOHeqHhIX pH
OB, KO-
TOpbIMIi npe
CTaBJIeHbI HanpH>KeHHH H CMeweHHH. l1cnOJIb3YH 3TO
06CTOHTeJIbCTBO, OqeBH
HO, MOLKHO nOJIyqHTb B paMKax pa3BHBaeMoro
l\IeTO
a H pellleHHe 3a
aqH 06 YCTaIIOBHBlllHXCH KOJIe6aHHHX nOJIoro
UHJIHH
pa KOHeqIIOH
JIHIIbI. <PaKTHqeCKOe nOCTpoeHHe 3TOro pellleHIIH
ocywecTBHM
m:1 CJIyqaH CHMMeTpHqHbIX OTHOCHTeJIbHO cpe
Hero ce-
qeIIHH CPOpM KOJIe6aHHH.
nepBaH COCTaBJIHIDwaH 06wero npe
CTaMeHIIH KOMnOHeHTOB BeK-
Topa CMeIUeIIHH
JIH nOJIoro UIiJIHII
pa.
OCTaBnHIDIUaH He06xo
HMbIH
npOlI3BOJI
JIH y
OBJIeTBOpeHHH rpaHllqHbIX YCJIOBHH Ha TOpuax, 3a-
flHCbIBaeTCH OqeHb npOCTO:


(28.5.)


WI = Co sin I'z +
 (D/qI sh qlz + C, sh Q2 Z ) No (A/).
1==1 Q2
OT cooTBeTcTBYIDIUeH qaCTH B 06meM pellleHlI1i (27 .3)
nH cnnOlllIIoro
UHJIHH
pa 3TlI BbIpaLKeHIIH OTJIHqaIDTCH HeCKOJIbKO nepecTpoeHHhlMH C
yqeTOM cHMMeTpHqIIOCTH 3a
aqH BblpaLKeHHHMH
JIH CPYHKUHH oceBOH
Koop
HHaTbl H BBe
eHHeM BMeCTO CPYHKUHH EeCCeJIH IIX KOM6HIIaUHH
No (Air) H Nt (Ai r), onpe
eJIeIIIIbIX COOTHOWeHHHMII (13.3). nOcJIe
o-
BaTeJIbHOCTb qHCeJI Ai onpe
eJIHeTCH H3 ypaBIIeHHH Nt (A fo) = 0,
r
e E - oTHOCmeJIbHblH BHYTpeIIHHH pa
IIYc UHJIHH
pa. Ee3pa3Mep-
Hble Koop
HIIaThI 3
eCb BBe
eIIbI TaKHM LKe o6pa30M, KaK H npH pac-
cMoTpeIIHH CTaTHqeCKoi'I 3a
aqH
JIH 110JIoro UHJ]HH
pa (CM.
 13).
246


(28.6)


npH Z = + h,


(28 7)


I I
2G a, = 2G T rz = 0
247


npH r = E, 1.




Y J].OBJIeTBOpHH C nOMOIl{blO HaHJ].eHHbIX Ha OCHOBe COOTfIOlIleHJII1 3aKD-
Ha rYKa BbIpameHHH AJIH HanpHmeHHH YCJIOBHH J].JIH KaCaTeJIbHbIX Ha-
npHmeHHH B (28.7), nOJIyqHM cJIeJ].YIOIl{YIO CBH3b MemJ].Y HCKOMbIMH Be-
JIHqHHaMH:


'A; + q
 sh q2 h
2'Ayqlq2 sh qlh '

+kj
2 2 QI (p,J,
P 2
p
+k}
2 QI (P2 8 ).
2P 2
TaKaH 3anHCb COOTHOlIleHHH YKa3blBaeT, Ha TO qTO B J].aHHOH 3aJ].aqe
CJIeJ].yeT OTJ].eJIbHO paCCMaTpI4BaTb Te CJIyqal:1, KOrJ].a qlh = 0 + ikn HJIH
paBHO HYJIIO OJ].HO H3 qHCeJI PI' P2. HHKaKHX npHHUHnHaJIbHbIX 3aTpyJ].-
HemiH npH 3TOM He B03HHKaeT.
Tpe60BaHHH rpaHHqHbIX YCJIOBHH (28.7) OTHOCHTeJIbHO HOpMaJIhHbIX
HanpHmeHHH npHBOJ].HT K 6eCKOHeqHOH CHCTeMe JIHHeHHbIx ypaBHeHHH.
fJpH BbIQHCJIeHHH K03<pqmUHeHToB 3TOH CHCTeMbI HCnOJIb3YIOTCH <POPMY-
JIbI pa3JIOmeHHH rHnep60JIHQeCKHX <PYHKIWH no TpHroHOMeTpHQeCKHM
(5.6) H <PYHKUHH Qo (pr) no No ('Aj r). OnycKaH Bce npoMemYTOQHbIe
BbIKJIaJ].KH, 3anHIIleM YKa3aHHYIO CHCTeMy B BHJ].e


D i = - C 1


QI (PI) = -


(28.8)


QI (PI8) = -


C m - I h A 2 J I ('I') - Ell ('1'15) + B 2
o m _ 2 'V COS 'V + 0 m _ 2 I _ 152 0 m _ 2 X


00
Y I (V) - EY I ('1'15) 2 m - I '1'2 ,'" Xl - EYI _ j:
X I _ 15 2 (m _ 2)2 ' I _ 1;2 "-' k 2 2 - 10'
1=1 IPI
C sinvh A[ I m-l j
· h(m-2) + 0 -e- Jd 'V 8 )+'V m-2 J o ('V 8 ) +
[ I m- I J
+ Bo --;- Yd'V 8 ) + 'V m-2 Yo ('V 8 ) -


4 m-l
 No ('AjE) Zj 0
- 'V (m - 2)2
 No ('Ai) 'A
 q 2 = ,
1=' 1 I


CO h(
n

) +Ao!-Jd'V)+'V :=; Jo('V)}+Bo[-YdV)+
00
m-IY ()J 4 m-l \" Zj 0
+ 'V m - 2 0 V -V (m - 2)2
 JY2 = ,
;=1 iqI
(P 2 2 + 1? 1 )2 A ( ) A ( )
Lli PI p2 Ll i P2
4kl to. (pJ - 2 to. (P2)


C 2'1'2 sin V h
- 0
h (m - 2) pi


+X l [


248


m-I L ] Y [ (p
+ k])2
m-
 k 2 + / 4k 2
/ /
'1'2
Z [ vi
m - 2
 ; 'A 2 ( k2 + q 2 )
1=1 I / 1
_ C 2'1'2 sin V h + X [ (p
 + k})2 to. 3 (p,1 2 to. 3 (P2) ] +
o h (m _ 2) pi / 4kj to. (PI) - P2 !1 (P2)
+ Y [ (p
 + kj)2 to. 4 (PIL _ 2 to. 4 (P2) _ I m - I L ] _
/ 4kj to. (Pt) P2 to. (P2) 15 m - 2 ki
'1'2
 NoCAje) Z [ vi _ 2kmj ] =0,
- m - 2
 No ('A;) ; 'A7 (k7 + qf) (kj + qf)(kj + q
)
1'2 No ('Aj) N 'A J
m-2 qiDo(J'-j.l;) {Ao [8Jdv8) No ('A j 8)-Jdv) o( j) +
+ Bo [8Y I (V8) No ('Aj8) - Y 1 (V) No ('A j )]} +
r (q
 + 'A
)2 ] '1'2
+ Zjq
 cth q
 - 2 / cth qI h m _ 2 X
4'A;qlq2

 X/N
 ('Ai) - eY/N o ('Ai) No ('Ail',) [ vi _ 2m J ] = o.
X £=1 2Do ('Aj. e) k'7 (Ay + pi) ('Ay + pi) ('A7 + p
)
HeH3BeCTHbIe BeJIHQHHbI B .:nOH CHCTeMe CBH3aHbI C K03<p<PHlIHeHTa-
MH 6eCKOHeQHbIX pHJ].OB J].JIH CMeIUeHHH (HanpHmeHHH) CJIeJ].YIOIUHMH
COOTHOlIleHHHMH:


to. 2 (PI) _ p2 to. 2 (P2) ] _
to. (PI) 2 to. (P2)


2mkj J - 0 (28.9)
(k} + qi) (ki + q
) - ,


1


XI = (- 1)1 k
 QI (P2)'
P 2
Y I = (_ 1)1 k
 Qdp#),
P2
sh q2h
ZI = No ('A;)---q;j! C i .
OpHHHTbIe npH 3anHCH K03<p<pHUHeHToB CHCTeMbI 0603HaQeHHH paCIIlH<p-
pOBbIBalOTCH TaKHMH paBeHcTBaMH:
1'1 (p) = M 1 (p) LI (p8) - M 1 (p8) LI (p),

l (P) = M 1 (p8) Lo (p) - L I (p8) Mo (p),
1'1 2 (p) = Mo (P) LI (P) - M 1 (P) Lo (p), (28.11)
1'13 (p) = M I (p8) Lo (p8) - LI (p8) Mo (p8),
1'14 (p) = LI (p) Mo (p8) - M 1 (p) Lo (p8),
Do ('Ai> 8) = + [N6 ('Ai) - 8W6 ('A;8)).


(28.10)


17 7-24
4


249








Dpe)J.nOJIO>KeHl1e 0 CYIll.eCTBOBaHHl'I peWeHHH C KOHeqHbIMH 3HaqeHH-
HMH HeH3BeCTHbIX C HOMepaMH l < L, j < N n03BOJIHeT C nOMOlll.bIO
nOllxO)J.a, HCnOJIb30BaHHOrO npH paCCMOTpeHHH CTaTHlIeCKHX 3a)J.aq
(rJI. 3), ycnmoBHTb cYIll.eCTBOBaHHe peweHHH CHCTeM (28.9) co CJIellYIO-
lll.HMH aCHMnTOTHlIeCKHMH cBoikTBaMH HeH3BeCTHbIX:
lim Xl = XO' lim Y l = YO'
l....r:iO i_oo


Ha6JIIO)J.aeTCH npaKTHlIeCKH paBHoMepHoe pacnpe)J.eJIeHHe HX no Bce-
MY LI.HJIHHllPY. DpH nO)J.xo)J.e K nepBoMY pe30HaHcy (1,00 < 'i'
l) <
< 1,05) 06HapY>KHBaeTCH 3HallHTeJIbHOe OTKJIOHeHHe OT paBHoMepHoro
pacnpe)J.eJIeHHH, liTO OTpa>KeHO COOTBeTCTBYIOIll.HMH KpHBbIMH Ha
r


lim Z.=-
2i-= '


Xo
I-tO


Yo
Ve(l- E} ,


(28.12)


, : Z _ _ Xo + Yo
1m . - 1 .
2i+l-+= I - EVE (l - f)
3TH COOTHoweHHH n03BOJIHIOT 3<p<peKTHBHo CTpOHTb KOHellHbIe CH-
CTeMbI, OTBellaIOIll.He 6eCKOHellHOH CHCTeMe (28.9), C yqeTOM rJ1aBHbIX
qaCTeH HeH3BeCTHbIX C 60JIbWHMH HOMepaMH. DpeHMYlll.eCTBa TaKoro
nOllXO)J.a no cpaBHeHHIO C MeTOllOM npocTOH pellYKLI.HH KaK npH onpelle-
.neHHH c06CTBeHHbIX qaCTOT, TaK 11 npH BbIqHCJIeHlIH HanpH>KeHHH 1l0C-
TaTOllHO nOllp06HO onHcaHbI BbIwe. HecMoTpH Ha TO qTO nOJIyqeHHbIe B
pe3YJIbTaTe peweHHH KOHellHOH CHCTeMbI 3HalleHHH Bcex HeH3BeCTHbIX no-
3BOJIHIOT I1CnOJIb30BaTb H3BeCTHbIe MeTO)J.bI YJIY4weHltH CXO)J.HMOCTH PH-
[(OB, OpraH113aLI.HH IWHKpeTHbIX BbIlIHCJIeHHH B paCCMaTpHBaeI\1OH 3a)J.a-
qe [(OCTaTOqHO CJlO>KHa. He I1MeH B03MO>KHOCTH OCTaHOBHTbCH Ha 3TOM
Bonpoce, paccMoTpHM HeKoTopbIe pe3YJIbTaTbI paClIeTOB, n03BOJIHIO-
lll.HX Ha KOHKpeTlloM npHMepe npolloJ1>KHTb H3Y4eHHe 3aBI1CHMOCTH xa-
paKTepa HanpH>KeHHO-[(e<p0pMHpoBaHHoro COCTOHHHH ynpyroro TeJIa
OT lIaCTOTbI KOJIe6aHHH.
DpHBe)J.eHHbIe HH>Ke pe3YJIbTaTbI pac4eToB OTHOCHTCH K rpaHHqHOii
3alla4e (28.7). DpH npoBelleHHH BbI4HCJIeHHH B KOHeqHOH CHCTeMe Y)J.ep-
>KHBaJIHCb HeH3BeCTHbIe C HOI\lepaMH l <: 25, j <: 14. HeH3BecTHbIe C
60JIee BbICOKI1MH HOMepaMH yqI1TbIBaJIHCb B KOHeqHOH CHCTeMe Ha or
HOBe paBeHcTB t!!"
Xl = X 25 , Y l = Y 25 (l > 25),
Z/=Z13 (j= 15,17, ...), (28.13)
Z, = Z14 (j = 16, 18, .. .).
TaKHM 06pa30M, HH<popMaLI.HH 0 Hell3BeCTHbIX BeJIHqHHaX B 6eCKOHeq-
HOCI CHCTeMe (28.9) 6bIJIa nOJIY4eHa Ha OCHOBe peweHHH KOHellHOH CH-
CTeMbl weCTHlleCHTH ceMH ypaBHeHHH. TaKOH nopH)J.OK KOHeqHOH
CHCTeMbI n03BOJIHeT H3yqaTb llHHaMH4eCKYIO HanpH>KeHHOCTb nOJIoro
LI.HJ1HH)J.pa B 1l0CTaTOqHO WHpOKOM [(Hana30He lIaCTOT. <I>aKTH4e-
CKHe BbIlIHCJIeHHH BbInOJIHeHbI Ha HHTepBaJIe (0 < 'i' <: 5). DpH 3TOM
llJIH reOMeTpH4eCKI1X xapaKTepl1CTHK LI.HJIHH)J.pa H 411CJ1a DyaccoHa
6blJ1H npIfHHTbI cJIellYIOIll.He 3Ha4eHHf!: Iz = 1; f: = 0,5; m = 3.
06paTI1MCH K paccMoTpeHHIO oceBbIX HanpH>KeHHH B uH.nJtH[(pe,
KOTopbIe no yCJ10BI1IO (28.7) paBHOI\lepHO pacnpe)J.eJIeHbI no TOpLI.aM,
It npOCJIe)J.I1M 3a HX 113MeHeHHeM B cpellHeH nJIOCKOCTH 1.1I1JIHH)J.pa z =
= 0 C pOCTOM lIaCTOTbI. LI.J1H OTHOCHTeJIbHO MaJIbIX qaCTOT (1' < 0,8)
250


-1


o


6 z f
/6 z max I


PHC. 27.


pItC 27 4 T06bI HMeTb B03MO>KHOCTb nOKa3aTb 3HaK HanpH>KeHHH,
Ha pRC. 27 npl1BelleHO 3HaqeHHe OTHoweHHH ojl a
ax I )J.JIH pa3JIH4HbIX
4aCTOT.
llaJIbHeHWee nOBbIweHHe qaCTOTbI npl1Bo)J.HT K "'1I060nbITHoMY <p aK -
TY - pacnpelleJIeHHe oceBbIX HanpH>KeHHH B LI.HJIHH)J.pe CHOBa CTaHOBHT-


PHC. 28.


PHC. 29.


CH npaKTH4eCKH paBHoMepHbII\1. TaKoe nOJIO>KeHHe Ha6JIIO)J.aeTcH B
)J.Hana30He 4aCTOT 1,4 < 'i' < 2,0. llJIH 60JIee BbICOKHX qaCTOT HpKO
BbIpa>KeHa HepaBHoMepHocTb B pacnpelleJIeHHH oceBbIX HanpH>KeHHH_
HeKoTopoe npellCTaBJIeHHe 0 <popI\1ax KOJIe6aHI1H LI.HJIHH)J.pa MO>KHO
nOJIYlIHTb Ha OCHOBe llaHHbIX 0 pallHaJIbHbIX nepeMeIIJ.eHI1HX T04eK BHeru-
HeH (r = 1) 11 BHYTpeHHeH (r = B) nOBepXHOCTeH. llJIH pH)J.a 3HaqeHHH
11 a CTOTbI KOJIe6aIll1H xapaKTep H3MeHeHI1H pa)J.HaJIbHbIX nepeI\1eIll.eHHH
BllOJIb 06pa3YIOIll.HX npe)J.CTaBJIeH Ha pHC. 28 11 pHC. 29. 4T06bI


17*


251




...


npe,L\CTaBHTb Ha O.nHOM lfepTe:IKe SHalfeHHH nepeMemeHHH npH paSHbIX lfaC-
TOTaX, HX BbIlfHCJIeHHbIe SHalfeHH5I COOTBeTCTBeHHO npOHopMHpyeM, npH-
lfeM B KalfeCTBe I u Imax BbI6epeM MaKCHMaJIbHOe SHalfeHHe pa.nHaJIbHbIX
nepeMemeHHH .nJI5I Ka:IK.nOH lfaCTOTbI H .nJI5I Ka:IK.nOH nOBepXHOCTH. l1MeH-
HO n03TOMY Bce npHBe.neHHbIe KpHBbIe HMeIOT TOlfKY C op.nHHaTOH + 1.
3HalfeHH5I pa.nHaJIbHbIX nepeMemeHHH npH '\' = 1 nOSBOJI5IIOT npe.n-
CTaBHTb xapaKTep KOJIe6aHHH B oKpecTHocTH nepBoro pesoHaHca.
f1paKTHlfecKoe COBna.neHHe KpHBbIX ,lI.JI5I 3TOH lfaCTOTbI Ha pHC. 28 H
pHC. 29 YKasbIBaeT Ha TO, lfTO B OKpeCTHOCTH nepBoro peSOHaHca KOJIe-
6aHH5I npOHcxo.n5IT 6es saMeTHoro YBeJIHlfeHH5I TOJImHHbI lLHJIHH.npa.
lI.Pyroi'J xapaKTepHoH oc06eHHocTbIO KOJIe6aHHH npH '\' = 1 5IBJI5IeTC5I
TO, lfTO pa.nHaJIbHbIe nepeMemeHlJ5I TOlfeK B6JIHSH cpe.nHero CelfeHH5I Z =
= 0 SHalfHTeJIbHO 60JIbllle, lfeM Y TOpLlOB LlHJIHH.npa.
C YBeJIHlfeHHeM lfaCTOTbI KOJIe6aHHH xapaKTep pacnpe.neJIeHH5I CMe-
meHHH CTaHOBHTC5I 60JIee CJIO:IKHbIM. B lfaCTHOCTH, 06bIlfHbIM CTaHOBHT-
C5I TaKoe nOJIO:IKeHHe, Kor.n.a pa.n.HaJIbHbIe nepeMemeHH5I COOTBeTCTBYIO-
mHX TOlfeK BHYTpeHHei'J H BHelllHeH nOBepxHocTei'J lLHJIHH.npa HMeIOT
npOTHBOnOJIO:IKHbIe <paSbI.
B KalfeCTBe npHMepa, HJIJIIOCTpHpYlOmero OomHe nOCTpOeHH5I B sa-
.nalfe 06 YCTaHoBHBlllHXC5I KOJIe6aHH5IX nOJIoro UHJIHH.n.pa KOHelfHoH
.nJIHHbI, paccMoTpeHa rpaHHlfHa5I sa.n.alfa .nJI5I .n.OBOJIbHO TOJICTOrO LlH-
JIHH.n.pa (8 = 0,5). B CB5ISH C 3THM CJIO:IKHOCTb KapTHHbI Hanp5I:lKeHHO-
.n.e<P0pMHpoBaHHoro COCT05IHH5I 06HaPY:lKHBaeTC5I Y:lKe npH cpaBHHTeJIb-
HO HHSKHX lfaCTOTaX. YMeHbllleHHe TOJImHHbI, eCTeCTBeHHO, paclllHpHT
.nHanaSOH lfaCTOT, B KOTOpOM BOSMO:IKHO onHcaHHe COCT05IHH5I LlHJIHH.n.pa
Ha OCHOBe fHnOTeS THna KHpxro<pa - J15IBa. f1pe.neJIbI npHMeHlIMocTH
TaKoro no.nxo.n.a KaK npH OTbICKaHHH c06cTBeHHbIx lfaCTOT, TaK H npH
HSYlfeHHH Hanp5I:lKeHHOCTH MorYT 6bITb YCTaHOBJIeHbI Ha OCHOBe nOCT-
poeHHoro BbIllle <pHsHlfecKH TOlfHOro pellleHH5I.


t


1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
II.
12.
13.
14.
15.
16.
.
17.






JIIiTEPATYPA


A6paMRH E. JI. K 3a.nalJe ocecHMMeTpHlJHOll .ne<l>opMauHH KpyrJIOro UHJIlIH.npa.-
LlOKJI. AH APMCCP, 1954, 19, N2 I, C. 3- 12.
A6paMRH E. J7. K nJIOCKOll 3a.nalJe TeopHH ynpyrocTH .nJIH npHMoyrOJIbHHKa.-
DPHKJI. MaTeMaTHKa H MexaHHKa, 1957,21, BbIn. I, c. 89-101.
A6paMRH E. JI. HeKoTopble 3aAalJH paBHoBecHH KpyrJIOro I
HJIHH.npa.- LlOKJI
AH APMCCP, 1958,26, N2 2, c. 65-72.
A6paMRH E. JI. 06 O.nHOM cJIYlJae nJIOCKOll 3a.nalJH TeopHH ynpyrocTH .nJIH npHMo-
yrOJIbHHKa.- LlOKJI. AH APMCCP, 1955, 21, N2 5, c. 193-198.
A6paMRH E. JI., A:zcKcaHiJpoB A. fl. OcecHMMeTpHlJHbIe 3a.nalJH TeopllH ynpyroc
TH.- Tpy.nbI II BcecOI03. C'be3LIa no TeOpeT. H npHKJI. MexaHHKe. MexamlKa TBep-
.noro TeJIa. M., 1966, c. 7-37.
A6paMRH E. JI., Ea6AORH A. A. K H3rH6y TOJICTbIX KpyrJIbIX nJIHr ocecHMMeTpHlJ
HOll HarpY3Koll.- I.13H. AH ApMCCP. Cep. <l>H3 -MaT. HayK, 1958. 11. N2 4,
c. 95-106.
A6paMRH E.!J Ea6AORH A. A. 06 O.nHOll 3a.nalJe ocecHMMeTpHlJHOll .ne<l>opMauHll
nOJIoro UHJIHH.npa KOHelJHOll .nJIHllbI.- 113B. AH ApMCCP. Cep. <l>H3-MaT. HaYK,
1962, 15, N2 2, c.87-99.
A6paMRI-l E. JI., Mal-lYKfIH M. M. PerneliHe nJIOCKOll 3a.nalJll TeopHn ynpyrocfII
.nJIH npHMoyroJIbHHKa B nepeMellleHHHx.- LlOKJI. AH APMCCP, 1957, 25, N2 4.
c. 177-184.
A u'leH6cpi! 0.. 10.. Wanupo r. C. 0 nepe.nalJe n:aBJIeHHH lJepe3 CJIOll, HMelOlllHll Ull.
JIHIl.npHlJeCKOe oTBepcTHe.- l1H>KeHepHbIll C60pIlHK, 1950, BbIn. 7, c. 65-69.
AKccHmRH O. K. 0 KOHueHTpauHH llanpHMellHll B TOJICTbIX nJIHTax.- DpHKJI
MaTeMaTHKa H MexaHHKa, 1966, 30, BbIn. 5, c. 963-970.
AKccHmRI-l O. K. Oc06ellHocTH HanpHMeHllo-.ne<l>opMHpoBaIiHoro COCTOHIlHH nJllITbI
B oKpecTliocTH pe6pa.- DPHKJI. MaTeMaTHKa H MexaHHKa, 1967, 31, BbIn. I,
c. 178-186
AKCeHmRH O. K., BOpOBUft H. H. HanpHMeliHoe COCTOHHHe nJIHTbI MaJIOll TOJIlliH
HbI.- DPHKJI. MaTeMaTHKa H MexaHHKa, 1963, 27, BbIn. 6, c. 1057-1074.
AKCCHmRH O. K., Koco.wnoB A. H. HeKoTopble 3a.nalJH KOllueHTpauHH llanpH1KeHllii
B TOJICTbIX nJIHTax.- 113B. AH CCCP. MexallHKa TBep.noro TeJIa, 1971, N2 3.
c. 142-150.
AKccHmRH O. K., l1l.cnKuH r. r. 113rH6 TOJICTOll nJIHTbI C rnapHHpHo 3ai\penJIeH
HbIM OTHepCTlIeM.- TpY.f(bI VII BcecolO3. KOIl<l>. no TeopHH 060JIOlJeK H nJIaCTHHOK
M., 1970, c. 33-37.
AAcKcaHiJpoB A. M., DoiJaAKo6 B. B. BTopaH KpaeBaH 3a.nalJa nJIOCKOll TeopHH
ynpyrocTH 06 YCTaHoBHBrnHxcH KOJIe6aHHHx npHMoyrOJIbHHKa.- LlOKJI. HaYlJ.-
TeXH. KOH<l>. no HTOraM HaYlJ.-HCCJIeA. pa60T 3a 1968-1969 rr. M., 1969, c. 3-9
AAcKcaHiJpoB A. fl. HeKoTopble 3aBHCHMOCTH Me1K.ny perneHHHMH nJIOCKOll H oce-
cHMMeTPlllJHOll 3a.nalJ TeopHH ynpyrocTH H perneHHe oceCHMMeTpHlJHbIX 3a.nalJ npn
nOMO!llll aHaJIHTHlJeCKHX <l>YHKUHll.- LlOKJI. AH CCCP, 1959, 129, N2 4, c.
754-757.
AAcKcaHiJpoB A. fl. HeKoTOpble COOTBeTCTBHH MeM.ny <l>YHKUHHMH HanpHMeHHll
oCeCHMMeTpHlJHOll H nJIOCKOll 3a.nalJH TeopHH ynpyrocTH H perneHHe oceCHMMeTpHlJ'
Hall "a.nalJH .nJIH 6eCKOHelJHOrO nOJIoro THMe.l\orO KOHyca.- 113B. CO AH CCCP.
Cep. TeXH. HaYK, 1962. N2 2, c. 15-24


253




--


18. AlleKCGHopoB A. fl. OperneHHH OCHOBHhIX TpexMepHbIx 3a,lJ,aq TeopHH ynpyrocTH
,lJ,JUI TeJl npOH3BOJlbHOll <jJOPMbI.- AHHoTauHH ,lJ,OKJI. Ha XlII Me}!\Ji,YHap. KOHrpec-
ce no Teop. H npHKJI. MexaHHKe. M., 1972, c. 27.
19. AlleKcaHopoe A. fl. PerneIIHe npocTpaHcTBeHHbIx 3a,lJ,aq TeopHH ynpyrocTH ,lJ,JlH TeJl
Bp
UleHHH npH nOMOUlH aHaJlHTHqeCKHX H 0606UleHHbIx aHaJlHTHqeCKHX <jJYHK-

H
. 3 l PY ,lJ,bI HOBOCH6. HH-Ta HH>KeHepOB >K.-,lJ,. TpaHcnopTa, 1970, BbIn. 96,
20. AlleKCQHopoe A. fl., BOllbnepm B. C. 0 npHMeHeHHH O,lJ,HOrO MeTO,lJ,a perneHHH oce.
cH
MeTpHqHbIX 3a,lJ,aq TeopHH ynpyrocTH K 3a,lJ,aqe 0 rnape H npoCTpaHcTBe C rnapo-
BOH nOJlOCTbIO.- 113B. AH CCCP. MexaHHKa H MarnHHocT p oeHHe 1961 "'0 6
c. 106-109. ' ,.'
'
21. AlleKCQHopoe A. fl., COIlOebeBlO. H. O,lJ,Ha <jJopMa perneHHH npocTpaHCTBeHHbIx oce-
cHMMeTpHqHblX 3a,lJ,aq TeopHH ynpyrocTH npH nOMOUlH <jJYHKUHll KOMnJleKCHOro
nepeMeIIHoro H perneHHe 3TH X 3a,lJ,aq ,lJ,JlH c<jJepbI.- TIPHKJI. MaTeMaTHKa H MexaIIH-
Ka, 1962,26, BbIn. 1, c. 138-146.
22. Ea61lo;:H A. .4. K 3a,lJ,aqe oCeCHMMeTpHqHOll ,lJ,e<jJopMauHH KpyrJIOro UHJIHH,lJ,pa KO
HeqHOH ,lJ,JIHHbl H3 TpaHCBepCaJIbHO H30TpOnIlOro MaTepHaJIa.- 113B. AH
ApMCCP. Cep. <jJH3.-MaT. HaYK, 1961, 14, N2 4, c. 61-70.
23. Ea61l0RH A.. A. K H3rH6y TOJICTbIX KpyrJIbIX nJIHT npOH3BOJIbHOll HarpY3KoiL-
I1H>KeHepHbIH >KYPH., 1964, 4, N2 4, c. 750-759.
24. Ea61l0RH A. A., rYIlKaHRH H. O. 06 o,lJ,Hoii CMernaHHOll 3a,lJ,aqe ,lJ,JIH npHMoyrOJIb-
HHKa.- 113B. AH ApMCCP. Cep. <jJH3.-MaT. HaYK, 1969, 22, N2 I, c. 3-17.
25. Ea61l0RH A. A., MellKoHRH A. n. 0 ,lJ,BYX CMernaHHblX oceCHMMeTpH'IHbIX 3a,lJ,aqaX
TeopHH ynpyrocTH.- 113B. AH A p MCCP. Ce p . mH3.-MaT. Ha"K 1 9 69 2 2 "'0 5
c.3-16. 'I' J' -, ,.'
 ,
26. Ea61l0RH A. A.. CaaKRH C. M. 0 ,lJ,BYX 3al\aqaX 0 paBHOBeCHH npHMoyrOJIbHOrO na-
paJIJIeJIeIlHne,lJ,a co CMemaHHblMH rpaHH'JHblMH yCJIOBHHMH.- 113B. AH ApMCCP.
Cep. <jJH3.-MaT. IIayK, 1964, 17, N2 6, c. 27-46.
27. Ea61l0RH A. A.. TOHORH B. C. 06 O,lJ,HOH 3a,lJ,aqe ,lJ,JIH nOJIoro KOHeqHoro UHJIHH,lJ,-
pa.- 113B. AH ApMCCP. MexaHHKa, 1970,23, N2 6, C. 3 -II.
28. Ea3apeHKO H. A., BOpOBU'I. H. H. AHaJIH3 TpexMeplloro HanpH)J\( HHom H ,lJ,C<jJopMH-
pOBaHllOro COCTOHHHH KpyroBblx UHJIHH,lJ,pH'JecKHx o60JIo'JeK.- TIpHK.
. MaTeMa.
THKa H MexaHHKa, 1969, 33, BbIn. 3, c. 495-511.
29. Ea3apeHKo H. A., Bopoeu'I. 11. H. ACHMnTOTHqeCKOe nOBe,lJ,eHHe pemeHHH 3a,lJ,aqH
TeopHH ynpyrocTH ,lJ,JIH nOJIoro UHJIHH,lJ,pa KOHeqHOI
I ,lJ,JIHHbl npH MaJIOll TOJIUlH-
He.--:;- TIPHKJI. MaTeMaTHKa H MexaHHKa, 1965,29, BbIn. 6, c. 1035-1053.
30. Eauoa 3. H. 061l\ee pemeIIHe 3a,lJ,aqH 06 ynpyro-,lJ,e<jJopMHpoBaHHoM COCTOHHHH
H30TponHoro L\HJIHlI,lJ,pa KOHeqHOii ,lJ,JIHHbI.- 17-51 Hayq. KOH<jJ. npo<jJ.-npen. COCTaBa
JIeIIHHrp. HH>K.-CTpOHT. HH-Ta. CeKuHH MaTeMaTHKH, conpOTHBJIeHHH MaTepHaJIOB,
Teop. MexaHHKH, <jJH3HKH, XHMHH, 3JIeKTpOTexHHKH. JI., 1959. c. 32-37.
31. Eauoa 3. H. 06Ulee pemeHHe 3a,lJ,aqH ynpyroro ,lJ,e<jJopMHpoBaHHoro COCTOHHHH
CnJIOIlIHOrO H nOJIoro UHJIHH,lJ,pOB.- Ha y q. ,lJ,OKJI. Bblcmell mKOlIbl 1959 "'0 2
c. 37-41. .', ,.'C,
32. Eauoa 3. H. 061l\ee pemeIIHe ypaBHeHHH paBHoBecHH H 3a,lJ,a'Ja 0 napa.
JIeJIenHne-
,lJ,e ,lJ,JIH aHH3OTpOnIlOrO H H30TponHoro TeJI.- L(OKJI. 16-ii Hayq. KOH<jJ. npo<jJ.-
npen. COCTaBa JIeIIHHrp. HH>K.-CTpOHT. HH-Ta, JI., 1958, c.611-614.
33. Ealla6aHoBJJ. M. O,lJ,HOp0,lJ,Hble perneHHH H y,lJ,OBJIeTBOpeIIHe rpaHH'JHbTX YCJIOBHii Ha
TOpL\aX B 3a,lJ,aqe 0 paBHoBecHH nOJIoro TOJICTOCTeHHoro H30TponHoro UHJIHH,lJ,-

 pa.- 113B. AH CCCP. MexaHHKa TBep,lJ,Om TeJIa, 1968, N2 I, c. 95-101.
34. EelloHocoe C. M. MaTeMaTHqecKHe np06JIeMbi TeopHH ynpyrocTH ,lJ,JIH 06JIaCTeH C
yrJIaMH.- TpY,lJ,bI 11 BcecoI03. C"be3,lJ,a no TeopeT. H npHKJI. MexaHHKe. MexaHHKa
TBep,lJ,OrO TeJIa. M., 1966, c. 48-60.
35. EepecHeB A. H., lllenemee A. H. OcecHMMeTpHqHaH 3a,lJ,aqa TepMoynpyrocTH ,lJ,JIH
CnJIOrnHOrO KpyroBoro uHJIHH,lJ,pa KOHe'JHoH ,lJ,JIHHbl H3 TpaHCBepCaJIbHO H30Tpon-
Horo MaTepHaJIa.- YqeH. 3an. KeMepoB. ne,lJ,. HH-Ta, 1969, N2 19, c. 9-14.
36. EuoepMaH B. JJ. PaC'JeT CHMMeTpHqHO IIarpY>KeHHbIX L\HJIHH,lJ,pH'JecKHx ,lJ,eTaJIeH.-
B KH.: OCHOBbl cOBpeMeHHblx MeTO,lJ,OB paCqeTa Ha npOqHOCTb B MarnHHOCTpoeHHH.
T. 2. M., 1958, c. 315-385.
37. Ellox B. H. TeopHH ynpyrocTH. XapbKoB, M3,lJ,-BO XapbK. yn-Ta, 1964. 483 c.
254


.t.


38. DIlOX B. H. <1>YHKUHH HanpR>KeHHR B TeopHH ynpyroCTH.- TIPHKJI. MaTeMaTHKa H
MeX3HHKa, 1950, 14, BbIn. 4, c. 415-423.
39. EOHoapeHKo n. C. K Bonpocy 0 e,lJ,HHCTBeHHOCTH ,lJ,JIH 6eCKOHe'JHbIX CHCTeM JIH-
HellHblX ypaBHeHHH.- MaT. C60PHHK, 1951, 29, N2 2, c. 403-418.
40. EoxHep C. JIeKuHH 06 HHTerpaJIax <1>YPbe. M., <1>H3MaTrH3, 1962.360 c.
41. EORpwuHoe C. B. PaCqeT TOJICTOCTeHHblX nOJIblX l\H.
HH1!pOB, HaXO,lJ,HIl\HXCH no,lJ,
,lJ,ellCTBHeM npOH3BOJIbHOll OCecHMMeTpH'JHOll IIarpy3KH.- B KH.: PaCqeTbI H3
npOqHOCTb, >KeCTKOCTb H nOJI3yq€CTb 9JIeMeHTOB MamHHccTpOHTeJIbHblX KOHCTPYK-
UHll. M., 1953, c. 106-124.
42. EpexoecKux JJ. M. BOJIHbl B CJIOHCTblX cpe,lJ,ax. M., 113,lJ,-BO AH CCCP, 1957.502 c.
43. EyxapuHoe r. H. I1ccJIe,lJ,oBaHHe no 3a,lJ,aqe TI. <1>. TIanKoBH'Ja B cJIyqae oceCHMMeT-
pH'JHOH ,lJ,eljJopMauHH uHJIHH,lJ,pa.- B KH.: TIpo6JIeMbi MexallHKH TBep,lJ,OrO ,lJ,e<jJop-
MHposaHHoro TeJla. JI., 1970, c. 89-91.
44. EyxapuHoB r H. OcecHMMeTpH'JHaH ,lJ,€<jJopMaUHH UHJIHH,lJ,pa KOHeqHOii ,lJ,JIHHbl.-
BecTH. JIeHHlIrp. YH-Ta, 1956, 7, c. 77-87.
45. Eaum},!eH r.. 3poeuu A. BblcmHe TpaHcUeH,lJ,eHTHble <jJYIIKL\HH. T. 1. rHrepreo
MeTpn'JecKaH <jJYHKL\HH. <1>YHKUHH JIe>KaH,lJ,pa. M., "HayK3», 1965. 294 c.
46. EaumMeH r.. 3poeuu A. BblcmHe TpaHCL\eH,lJ,eHTHble <jJyHKL\HH. T.2. <1>YHKUHH
EeCCeJIH, <jJYHKL\HH napa60JIH'JecKoro UHJIHHI1pa. opTOrOHaJIbHble MHorO'J.
eHbl.
M., «HayKa», 1966. 295c.
47. EaumMeH r" 3poeuu A. TaooHubl HHTerpaJIbHblX npeC6pa30lJaHHll. T. 1. TIpe-
06pa30 B 3IIHH <1>YPbe, JIanJIaca, MeJIJIHHa. M., "HaYKa», 1969. 343 c.
48. EaiimMeH r.. 3poeiiu A. Ta6JIHUbi HHTerpaJIbHblX npe06paaOlJaHHH. T. 2. TIpe06-
pa30BaHHH EeCCeJIH, HHTerpaJIbl OT crreuHaJIbHbIX <jJYHKL\HH. M., «!-layKa», 1970.
327 c.
49. BailoB r. M. KOHTaKTllaH 3a,lJ,aqa 06 ynpyroll ocecHMMeTpH'JHoll ,lJ,e<jJopMauHH
cn.l!0rnHoro H nOJIoro KpyroBoro uHJIHH,lJ,pa.- TpY,lJ,bl CH6Hp. MeTaJIJIypr. HH-Ta.
TIPHKJI. MaTeMaTHKa H MexaHHKa, 1957, 4/A, c. 19-28.
EO. Balloe r. M. 060,lJ,HOll OCHOBHOll cMemaHHoll 3a,lJ,aqe TeopHH ynpyrocTH ,lJ,JIH npH'
MoyroJIbHHKa.- 113B. AH CCCP. MexaHHK3 H MamHHocTpoeHHe. 1961, N2 3,
c. 133-142.
'jl. Balloe r. M. 06 OcccHMMeTpHqHOH ,lJ,eq,opMaL\HH CnJIOIIJHoro KpyroBoro UHJIHH,lJ,pa
KOHe'JHoH ,lJ,JIHHbl.- TIPHKJI. MaTeMaTHKa H MexaHlIKa, 1962, 26, BbIn. 4, c. 650-
668.
52. BailoS r. M. OCeCHMMeTpHqHaH 3a,lJ,a'Ja 0 C>KaTHH ynpymro KpyroBom UHJIHH,lJ,pa,
nOKOHIl\erOCH Ha rJIal!KOM >KeCTKOM OCHOB3I!HH.- 113B. AH CCCP. MexaHHKa H
MamHHocTpoeIIHe, 1961, N2 6, c. 151-154.
53. Balloe r. M. TIepBaH OCHOBHaH rpaHH'JHaH 3a,lJ,a'Ja TeopHH ynpyrocTH ,lJ,JIR npHMo,
yrOJIbHoro napaJIJIeJIenHne,lJ,a.- TIPHKJI. MaTeMaTHKa H MexaHHKa, 1966, 30,
BbIn. 6, c. 1057-1071.
54. Bamam r. H. TeopHH 6eCCeJIeBbiX <jJYHKUHll. T. 1. M., 113,lJ,-BO HHOCTp. JIHT., 1949.
798 c.
55. Bilacoe A. r. MeTo,lJ, nepeonpe,lJ,eJIeHHblX pH,lJ,OB B HeKOTopblX KpaeBblx 3a,lJ,aqaX
MaTeMaTHqeCKOll <jJH3HKH.- Bonp. ,lJ,HHaM. TeopHH pacnpoCTpaHeHHR cellCM.
BOJIH, 1959. BbIn. 3, c. 403-463.
56. BoeK 11. B., rpllH'l.enKO B. T. 0 CC6CTBCHHbiX 3H3'JCHI-!llX B HeKoTopblx'cceUJMMCT
PHqHblX 3a,lJ,a'Jax TeopHH ynpyrocTH H aKycTHKH. TIPHKJI. MexaHHKa, 1971, 7,
c. 95-97.
57. BOpOBll'l. 11. H. HeKoTopble MaTeMaTHqeCKHe BonpOCbl TeopHH nJIaCTHH H 060JIO-
'JeK.- TpY,lJ,bI II BcecolO3. C"be3,lJ,a no TeOpeT. H npHKJI. MexaHHKe. MexaHHKa TBep-
,lJ,oro TeJIa. M., 1966, c. 116-136.
58. Bopoell'l. H. H. 0 nOae,lJ,eHHH pemeHHll OCHOBHblX Kp3eBblx 3a,lJ,aq nJIocKoii TeopHH
ynpyrocTH B oKpecTHOCTH OC06blX TO'JeK rpaHHL\bl.- AHHOTaL\HH ,lJ,0K.,'l. III Bce-
COI03. C"be3,lJ,a no Teop. H npHKJI. MeX3HHKe. M., 1968, c. 80.
59. BopofJLl'I. H. 11., Koeallb'l.YK B. E. 0 6a3HcHbiX CBollcTBax O,lJ,HOH CHCTeMbl O,lJ,HOpO,lJ,-
HblX pemelmll.- TIPHKJI. MaTeM3THKa H MeX3HHKa, 1967, 31, BbIn. 5, c. 861-870.
60. Bopoeu'I. H. H., KonaccHKo B. B. HeKoTopbIe 3a,lJ,a'JH TeopHH ynpyrocTH,lJ,JIH nOJIY-
nOJIOCbl.- TIPHKJI. MeTeMaTHKa H MexallHKa, 1966, 30, BbIn. 1. c. 109-116.
61. rallepKuH E. r. PaBHoBecHe ynpyroH c<jJepHqeCKOll 060JIO'JKH.- TIPHKJI. MaTe-
MaTHKa H MexaHHK3, 1942, 6, BHn. 6, C. 487-497.


.






.


255




62. Fa/lUH JJ. A. 1(0HTaKTHblc 3a)\a'l1l TCOpll1l ynpyrocTII. M., rOCTCXII3)!aT, ] 963.
264 c.
63. Fa/l£/1aHflH fl. 0., lJo6aHflH K. C. PcrncHllc O)!HOH KOHTaKTHOH 3Ma'lll )!JIH npHMo-
yroJIbHIIKa.- TIPIIKJI. MaTCMaTIIKa II McxaHIIKa, 1966, 30, BbIll. 3, c. 569-578.
64. Fo6coH E. B. TCOpllH C<pCpll'lCCKIIX II 3JIJIllnCOII)!aJIbHbIX <pyHKItIlH. M., H3Jl-BO
IIHOCTp. JIIIT., 1952. 476 c.
65. FO/l06UH O. A. 0 BbIHY}K)!CHHbIX npO)!OJIbHbIX KOJICOaHIIHx ItllJIIIH)!pa.- H3B.
AH ApMCCP. McxaHIIKa, 1970,23, Ng 3, c. 43-49.
66. FO/lOI3UH O. A. 0 BbIHY}K)!CHHbIX npo,IJ.OJIbHblX KOJICOaHIIHx ItllJIIIH,IJ.pa npll 3a)!aH-
HbIX Ha nOBcpxHOCTII HanpH}KCHIIHX.- H3B. AH ApMCCP. McxaHIIKa, ]971, 24,
Ng 4, c. 69-76.
67. FpaaUlmeuH H. C., Pbl3ICUK H. M. TaOJIIIllbI IIHTCrpaJIOB, CYMM, PH)!OB II npoH3-
Be)!CHIIH. M., <I>1I3MaTr1l3, 1962. 1100 c.
68. rpeu 3., MemblOs r. E. <I>YHKItIlIl ECCCCJIH H HX npllJIO}KCHHH K <p1I311KC H McxaHH-
KC. M., H3,IJ.-BO IIHOCTp. JIIIT., ]953. 371 c.
69. rpuH6epe r. A. H30paHHbIC BonpocbI MaTCMaTH'ICCKoi'r TCOpHII 3JICKTpH'ICCKIIX II
MarHHTHbIX HBJICHlli'r. M., H3,IJ.-BO AH CCCP, ]948. 727 c.
70. rpUli6epe r. A. 0 pcmcHIIII nJIOCKOH 3a,IJ.a'lH TCOpll1l ynpyrocTII II 3a,IJ.a'lH 06 H3rH-
OC TOHKOH nJIHTbI C 3aKpCnJICHHbIM KOHTypOM.- .LLoKJI. AH CCCP, 195], 76, Ng 5,
c. 661-664.
71. rpuH6epe r. A., JJe6eiJee H. H., Y£/1/lRHa fl. c. MCTa)! pcmCHHH OOWCH 6HrapMo-
HII'ICCKoi'r 3a)!a'lll ,IJ.JIH npHMoyroJIbHOH OOJIaCTH npH 3a,IJ.aHIIH Ha KOHTYpC 3Ha'lC'
HMI'i <PYHKItHII IICC HOpMaJIbHOH npOIl3BO,IJ.HOH.- TIPHKJI. MaTCMaTHKa H McxaHIIKa,
1953, 17, BbIll. 1, c. 73-87.
72. FpUHJieHKO B. T. 3a,IJ.a'la TcpMoynpyroCTII ,IJ.JIH nOJIynOJIOCbI.-- TCnJIOBbIC HanpH-
}KCHHH B 3JICMCHTax KOHCTPYKItIlH, 1973, BbIll. 13, c. 86-91.
73. rpUH'ieHKO B. T. HanpH}KCHHOC COCTOHHIIC KpyrJIoro TOJICTOrO ,IJ.HCKa B nOJIC ItCHT-
POOe}KHbIX CIIJI.- Tpy,IJ.bI IV BCCCOI03. KOH<p. no TeopllH OOOJlO'lCK II n.
aCTHH.
EpcBaH, 1964, c. 423-430.
74. rpUH'leHKO B. T. OCCCHMMCTpH'IHaH 3a)!a'la T€OPHII ynpyrocTII ,IJ.JIH nOJIyoecKoHc'I-
HOro KpyrOBoro ItllJIJIH,IJ.pa.- TIPHKJI. MCXaHHKa, ]965,1,,N!! I, c. 109-119.
75. rpUH'leHKO B. T. OCCCIIMMCTpll'lHaH 3a,IJ.a'la TCOpHII ynpyrocTH ,IJ.JIH TOJICTOCTCH-
HOro ItllJIIIH,IJ.pa KOHC'IHOi'r ,IJ.JIIIHbI.- TIPHKJI. McxaHIIKa, 1967, 3,,N!! 8, c. 93-103.
76. rpUH'leHKO B. T. OCCCIIMMCTpll'lHoe TcpMoynpyroc COCTOHHHC nOJIynpocTpaHCTBa c
ItHJIIIH,IJ.pH'ICCKOJ'i nOJIOCTblO. - TCnJIOBbIC HanpH}KCHIIH B 3JICMCHTax KOHCTpyK-
Itlli'r, 1972, BbIll. 12, c. 90-95.
77. rpUH'leHKO B. T. CTaItHOHapHbIc TCnJIOBbIC HanpH}KCHHH B cn.
omHOM ItHJIHH,IJ.pe
KOHe'lHOJ'i ,IJ.JIIIHbI.- TCnJIOBbIC HarrpH}KCHHH B 3JICMCHTax TYPooMawHH, 1962,
BbIll. 2, c. 41-49.
78. rpUH'leHKO B. T. TcpMOHanpHJKCHHoe COCTOHHIIC TOJICTOCTCHHOro ItllJIJIH,IJ.pa KO-
HC'IHOH ,IJ.JIIIHbI.- TCnJIOBbIC HanpH}KCHIIH B 3JICMCHTax KOHCTPYKItlli'r, 1967,
BbIll. 7, c. 147-155.
79. rpUH'leHKO B. T., KOMuccapoea r. JJ. 0 COOCTBCHHbIX 'IaCTOTax OCCCIIMMeTpH'IHO
KOJICOJIlOwci'rCH TOJICTOH nJIIITbI.- TIPHKJI. McxaHIIKa, 1974, to, Ng 11, c. 81-87.
80. rpUH'leHKO B. T., Me/leUlKO B. B. BbICOKO'laCTOTHbIC OCCCIIMMCTpll'lHbIC KOJICOa-
HIIH KpyrJIbIX ,IJ.IICKOB.- TIPHKJI. McxaHIIKa, 1976, 12, Ng 12, c. 60-68.
81. FpUH'leHKO B. T., Me/leUlKO B. B. 0 KpacBoM pC30HaHce npll nJIaHapHbIX KOJICoa-
HIIHX npHMoyroJIbHbIX nJIaCTIIH.- TIPIIKJI. McxaHIIKa, 1975, 11, ,N!! 10, c. 52-
58.
82. FpiH'leHKO B. T., Jl/limKO A. T. TIpo OCOOJlIIBOCTi HanpY}KCHoro CTaHY }KOpCTKO-
3aKpinJICHOl KpyrJIOl nJIHTH.- .LLonoB. AH YPCP. Cep. A, 1970, Ng 2. l. 162-
164.
83. rpUH'leHKO B. T., Y/lUmKO A. <P. AHaJIII3 ,IJ.HHaMII'IccKqH HanpH}KCHHOCTII II 'lac-
TOTHbIX xapaKTcpllcTIIK KpyrJIOH nJIHTbI B paMKax TpCXMCpHOi'r TCOpll1l ynpyroc-
'1'11.- Tpy,IJ.bI VIII BCCCOI03. KOH<p. no TCOpllH OOOJIO'leK H nJIaCTIIH. M., 1973,
c. 436-442.
84. rpUH'leHKO B. T., Y/lUmKO A. <P. .LLHHaMII'IecKaH 3Ma'laTeOpHH ynpyrocTII )!JIH
npHMoyrOJIbHOH npIl3MbI.- TIPIIKJI. McxaHHKa, 1971, 7,,N!! 9, c. 50-57.
is. rpUH'leHKO B. T., Jl /lUmKO A. <P. 3a)!a'lH TcpMoynpyrocTII ,IJ.JIH OOJIaCTci'r,orpa-


256


.,..


.L


HII'IeHHbIX nepneH)!HKYJIHpHbIMII rpaHII'IHbIMII nOBepxHocTHMII.- TenJIOBbIC Ha-
npH}KeHIIH B 3JICMCHTax KOHCTPYKItIlH, 1969, BbIll. 8, c. 110-125.
86. rpUH'leHKO B. T., Jl/lUmKO A. <P. I(paeBaH 3a)!a'la TepMoynpyrocTII ,IJ.JIH npH-
MOyroJIbHOH nJIaCTIIHbI.- TCnJIOBbIe HanpH}KCHIIH B 9JIeMCHTaX KOHCTPYKItllii,
1965, BbIll. 5, c. 138-146.
87 rpUH'leHKO B. T., Y/lUmKO A. <P. 00 113rlloe }KCCTKO-3aweMJIeHHOH Ima)!paTHOH
nJIaCTIIHKII.- TIPIlKJI. McxaHIIKa, 1965, 1, Ng 9, c. 134-137.
88. rpUH'leHKO B. T., Y/lUmKO A.<P. 0 TO'lHOM pcmcHIIH OCCCIIMMCTpll'lHOH 3Ma'lH
TCOpll1l ynpyrocTII ,IJ.JIH KpyrJIOH }KeCTKO-3aweMJICHHOii nJIHTbI.- H3B. AH
ApMCCP. Cep. <p1I3.-MaT. HaYK, 1963, 16, Ng 5, c. 125-132.
89. rpUH',eHKo B. T., Y/lUmKO A. <P. PacnpC,IJ.eJIeHIIC HanpH}KCHIIH B 3aKpenJICHHOH
rpaHII HcpaBHoMepHo HarpeToH 'IeTBCpTbnJIOCKOC'l'II.- TIpIlKJI. MexaHIIKa, 1965,
1, Ng 6, c. 33-38.
90. rpUH'leHKO B. T., Y/lUmKO A. <P.CMcIIIaHHaH oceCHMMeTpll'lHaH 3a)!a'laTCOpHII
ynpyroCTlI ,IJ.JIH ItHJIIIH)!pa KOHe'lHOH ;:!JIHHbI.- COnpOTIIBJIeHIIC lZarepllaJIOB II TCO-
pllH cOOPYA{CHIIH, 197]. BbIll. 15, c. 3-8.
91. rpUH'leHKO B. T., Y/lUmKO A. <P. TO'lHoe peIIICHlIe 3a;:!a'lH 0 pacnpC)!CJICHIIII Ha-
npHJKCHllii OKOJIO KpyroBoro OTBepCTIIH B ynpyroM CJIOC.- TIPIIKJI. MexaHHK3,
]968, 4, Ng 10, c. 38-46.
92. FPUH'leHKO B. T., KOea/leHKO A.l1.., YlwmKO A. <P. AHaJIII3 HanpHJKCHHOro coc-
TOHHIIH }KCCTKO-3aWCMJICHHOH nJIaCTIIHbI Ha OCHOBC pcrneHIIH npOCTpaHCTBeHHOH
3a)!a'l1l TCOpll1l ynpyroCTII.- TpY)!bI VII BceC0103. KOH<p. no Teopllll 06oJIo'lCK II
nJIaCTIIHOK. M., 1970, c. 205-211.
93. rpUH'leHKO B. T., Kap/laUl B. JJ., MelleUlKO B. B., Y/lUmKO A. <P. HCCJIC)!OBaHIIC
nJIaHapHbIx KOJICoallHH npHMoyroJIbHbIX nbc30KCpaMII'ICCKIIX nJIaCTIIH.- TIPIIKJI.
MexaHlIKa, 1976, 12, Ng 5, c. 71-78.
94. Fpysoee 10. A., flpoKonoe B. K. I( 3aJ(3'1C 113rlloa TOJICruH TIJIIITbI.- TIPIIKJI.
McxaHlIKa, 1970, 6, ,N!! 5, c. 3-9.
95. rpyBiJe£J 10. A., flpoKonoe B. K. TIOJIIIMOMCHTHaH TCOpliH paBHOBeCIIH TOJICTbIX
nJIIIT.- TIPIIKJI. MareMaTlIKa 1I McxaHIIKa, ]968, 32, BbIll. 2, c. 344-352.
96. ryceuH-3aae M. H. HCOCCCIIMMCTpll'lHaH )!c<p0pMaItIlH ynpyroH TOJICTOii nJIIITbI.-
HH}KCHCpHbIH }KypH., 1965, 5, Ng 2, c. 275-283. .
97. o.aHufloecKafl B. H. TIpllJIO}KCHIIC Bapll3ItlloHHoro Ha'laJIa l(acTlIJIbHHO K nJIOCKOli
3Ma'iC TcpMoynpyroCTH.- TIPIIKJI. MexaHIIKa, ]968, 4,,N!! 12, c. 33-40.
98. o.eee B. M. f1JIocKaH 3a;:!a'la TCOpllll ynpyrocTlI )!JIH npHMoyrOJIbHOii nOJIOCbI Ko-
HC'IHOH )!JIIIHbI.- ConpOTIIBJICHIIC MaTCpHaJIOB II TCOpllH cOOPY}KCHIIi'r, 1971,
BbIll. 13, c. 46-51.
99. o.emuHKo <P. M. I( 3a)!a'lC oceclIMMCTpll'lHoro paBHoBcclIH ynpyroro UliJIIIH)!pa KO-
HC'IHOi'r')!JlIIHbI.-H3B. AH CCCP. McxaHIIKa TBCP)!oro TCJIa, 1970, Ng6, c.142-]49.
100. )/(YPuHa M. H., KapAwsuHa JJ. H. TaOJlIIUbI II <POPMYJIbI )!JIH c<pep 'IeCKIIX <PYHK-
Ulli'r P 1 (x). T. 1. M., BbI'IlICJIIITCJIbHbIi'r CHTp AH CCCP, 1962. 319 c.
-2" +i1:
101. 30MMep£/1e/lba A. llH<p<pepcHUllaJIbHbIC ypaBHcHliSI B 'IaCTHblX npOIl3BO)!HbIX <p1I311-
KII. M., H3,IJ.-BO IIHOCTp. JIlIT., 1950. 456 c.
102. 30MMep£/1eflba A. McxaHIIKa ,IJ.e<tJopMlIpyeMbIx CPC)!. M., 113J{-BO IIHOCTp. JIlIT.,
1954. 486 c.
103. HeaHoe B. A. TpcxMcpHaH 3a)!a'la TCOpHli ynpyrocTII )!JIH TOJICTaCTCHHOro UlI-
JIIIH,IJ.pa KOHC'IHOH )!JIlIHbI.- TIPHKJI. McxaHlIKa, 1970, 6, NQ 9, c. 16-22.
104. HOHoe B. H. PaBHoBcclIC ynpyroro UIIJIIIH)!pa KOHe'lHOH )!JIlIHbI.- I1cCJIC)!OBa-
HIIH no TCOpll1l cOOPY}KCHHii, 1957, BbIll. 7, c. 413-437.
105. Ka/laHaUfl A. H. 3aMC'IaHliH 00 OC06CHHOCTII ynpyrnx pcrncHlIi'r BOJIII311 yrJloB.-
flpllKJI. MaTCMaTIIKa 1I MCXaHIIKa, 1969,33, BbIll. I, c. 132-135.
106 KaHmOpOeU'l JJ. B., Kpblfloe B. H. TIp.1I0JIII}KCHHbIC MeTO.l!bI BbICWCro aHaJIlI3a.
M.- JI., <I>1I3MaTrH3, ]962. 695 c.
107. KanUlUIJUU O. O. flpo P03B'H3aHHH OCeCIIMeTpll'lHo'i 3Ma'li Tcopii npY}KHOCTi
)!JIH lIJapy 3 UIIJIiH)!PIl'lHOIO nOp0}KHIIHOIO.- BieH. I(lIlB. YH-TY, 1961, 1, ,N!! 4,
c. 96-106.
108. KU/lb'leBCKUU H. A. OCHOBbI aHaJIHTII'IecKoii McxaHIIKlI 060JIO'lCK. K. YI3){-BO
AH YCCP, ]963. 353 c.
109. KOea/leHKO A.D,. OcHOBbI TcpMoynpyrocTH. 1(., «HaYK. ,IJ.yMKa
, 1970.23!J c.
257


"




110. KOllbCIWii r. BOJlHbI Hanp5l>KeHll51 B TBeplIblX TeJlaX. M., H31I-BO lIHOCTp. JlllT..
1955. 192 c.
Ill. KOflIlOeU'i E. M. Y!cCJlelIOBaHHe 0 6eCKOHe'JHbIX CllCTeMax JlllHeHHblX aJlre6pall-
'JeCKllX ypaBHcHllH.- I13B. <1>ll3.-MaT. llH-Ta llM B. A. CTCKJlOBa, 1930, 3, c.
41-167.
112. KOflIlOeU'i E. M. K Teopllll 6eCKOHelJHblX CllCTeM JllIHeHHblx ypaBHeHllH.- Tpy-
lIbI <1>1I3.-MaT. llH-Ta llM. B. A. CTeKJlOB3, 1932,2, Ng 4, c. 1-/6.
113. KOflIlOeU'i E. M. 06 OlIHOM ypaBHeHllll C lJaCTHblMll npcll3BollHbIMlI 'JeTBepTOro
nOp5llIKa. Cn6., 1902. 182 c.
114. KpY11lKOO /0. A. T eH30p <PYHKl.\lIH mmp5l>KeHlIii II 06mlle pemeHll51 B CTaTHKe Teo-
pllll ynpyrocTll. M.- JI., I13lI-BO AH CCCP, 1949. 199 c.
115. KY3bMUft P. O. K TeopHll 6eCKOHC'JHbIX ClICTeM JlHHeHIIblX YP3imeHllH.- TpYlIbI
<1>ll3.-MaT.- lIH-Ta llM. B. A. CTeKJIOBa, 1931,2. Ng 2, c. 18--41.
lib. KYPIUUH JI. M. CMemaHHa51 nJIOCKa51 3alIa'Ja Teopllll ynpyrOl'Tll )(.151 KBalIpaH-
Ta.- DpllKJI. MaTeMaTHKa II MexaIIHKa, 1959,23, Bbm. 5, c. 981-984.
117. JIe6eoee H. H. Cnel.\lla.%IIbIe <PYHKl.\lIll II llX npllJlO>KCHlUf. M., <!>H3M3TrH3,
1963. 379 c.
118. JIe6eoeB H. H., CKall'JcKufl fl. fl.. Ycp_n'lfto H. C. C60pHlIK 3alI3"1 no MaTCMaTlI-
'JecKoll <p1I311Ke. M., rOCTex1I3lIaT, 1955.420 c.
119. JIeii6eft3<Jft JI. C. Kypc Teopllll ynpyrocTlI. M.- J1., rOcTex1I3lIaT, 1947.464 c.
120. JlucU/{':Jm E. IH. 06 OlIHOM MeTOlIe peI! eHlI5I 3alIa l ! Teopllll ynpyrocTlI.- DpllKJI.
MexaHlIKa, 1967, 3, Ng 4, c. >j5-92.
121. JIucmoe r. H. )1l1HaMlI'IeCKa51 3alIa'Ja Teopml ynpYIOcTlI 06 YCTaHoBlIBmllxc5I
KOJle6aHlI5IX np5lMoyroJlbHoH 06J1aCTll.- I13B. AH CCCP. MexaHllKa TBeplIoro
TeJIa, 1968, .Nil I, c. 116-122.
122. JIYPbe A. H. K TCOpllll TOJlCTblX nJlllT.- DpllKJI. MaTeMaTlIKa II MexaHlIKa, 1942,
6, Bbm. 2-3, c. 151-169.
123. JIypbe A. fl. Hanp5l>KeHHOe COCT05lHlIe B ynpyroM LlllJlllHlIpe, HarpY>KeHHOM no
60KOBOH nOBepxHocTlI.- HH>KeHepHblll C60PHlIK, 1953, Bbm. 17, c. 43-59.
124. JIypbe A. fl. DpoCTpaHcTBeHHbIe 3alIa"lll TCOpIlll ynpyroCTlI. M., rCcTex1I31I3T,
1955. 491 c.
125. JIflB A. MaTeMaTlI"IeCKa51 TCOpll51 ynpyrocTlI. M.- JI., 06"belI. HaY"l.-TeXH. 1I3lI-
BO, HKTD CCCP, 1935. 674 c.
126. MellKOftflH A. fl. 06 OlIHOH CMemaHHOH rpaHlI'JHoll 3alIa'Je lIJl5I 1.\1IJ1l1HlIpa KOHe"l-
HOH lIJllIHbI.- H3B. AH ApMCCP. MexaHlIKa, 1971, 24, Ng 2, c. 3-16.
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258


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259




r


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