Бутковский А.Г. Методы управления системами с распределенными параметрами

P

:">

/

~?

:,'..

:,/

338

, ,

C

VnPABAEHME

nnA3MOA

Jicnonb3yH

aaKOH

nonHoro

TOKa,MOJKHO

nOKa9aTb,

liTO

none

H

o

B06naCTH

VI

paBHO

HynIO,

a B

06naCTH

V O){HOPO){HO. none

Eo

o

saBHCHT

TonbKO

OT

f.

.LJ:anee, YlJHTbIBaH, liTO

Bce

BoaMy~eHHH

no

BpeMeHH HOCHT

rap-

MOHHlJeCKHH

xapaKTep,

.MOJI<HO

3anHcaTb:

E = [X(jlE

o

(f)

+eeJ

e/lilot,

H =

(XzH

o

+

eh)

e/

lilot

, H

o

= °

npH

f >f

l'

J =

(X(jlJ

0 +8j)

e/lilo

t

,

r){e

e=

X(jle'!'

(f,

Z,

t),'

(1)

h=x,h'(r,

z,

t)+X~Z(f,

Z,

t),

(2)

j = X(jlj(jl (z,

t)

(3)

H

X,.

X(jl'

Xz-OPTbI

KpyroBoA

qHnHH){pHlJeCKOA

CHCTeMbI KOOp){HHaT

f,

<p,

z.

nO){CTaBnHH

3TH

BblpaJKeHHH

B

CHcTeMy

KBasHcTaqHOHap_

HbIX

ypaBHeHHH

MaKCBenna

){JIH

06JIaCTeH

V H VI'

KOTopbIe

HMe-

lOT

BH){

o

rot

E = -

!J.o

~~

,

rot

H = 0,

div

E = 0,

div

H = 0,

(3a)

H

npHpaBHHBaH

MeJK){y

c060A

BeJIHlIHHbI

O){HOro

nOpH){Ka

MaJIOCTH

no

e,

nOJIyllHM

O){HO

ypaBHeHHe

HyJIeBOro

npH6JIHJKeHHH

I d .

-

, ,

-d [fEo (f)J

=-

jO>o!J.oHo

(4)

H

cHcTeMy

ypaBHeHHH

nepBoro

npH6nHJKeHHSI

rot

e =

jO>o!J.oh,

rot

h = 0,

dive

= 0,

divh=O.

(4a)

Ha

H){eaJIbHO

np080){SI~eH

nOBepxHocTH

IT

){OnJKHO BbIIIOJIHHTbCSI

yCJIOBHe

paBeHCTBa

Hy

nlO

yr

nOBOH KOMnOHeHTbl

BeKTopa

.9JIeKTpH-

tIeCKOH HanpSlJKeHHOCTH:

E(jllrr=E

o

(ro+ep

(z,

t»+e(jl

(fo+ep(z,

t), z,

t)=O.

Jis

IIOCne){HerO

paBeHcTBa

nyTeM

rpynnHpoBKH

BenHlJHH

no

HX

no-

PH){Ky

ManOCTH

nOJIyllaeM

.l{Ba

rpaHHlJHblX

yCJIOBHH

){JISI

COcTaBnHIO-

~HX

Eo

H-e(jl:

Eo

(f

o

)

= 0,

(5)

e(jl

(r

o

,

z,

t)

= -

E~p

(z,

t),

r){c

(6)

E~

=

d~

Eo

(r)

Ir=r..

(7)

JiHTerpHpoBaHHe

ypaBHeHHH

(2) C YlIeTOM

ero

rpaHHIIHoro

yCJIOBHH

VPABHEHHSI

non51

H

PABHOBECHblA

PE)KHM

339

,

/O>ollo

H ( , r0 )

E

o

(

f

)

=

--2-

ofo

r;;-,

,

fo~f<f1'

(8)

.LJ:JlSI

Bhl60pa

paClIeTHblx

napaMeTpoB

CHCTeMbI

He06xo){HMO

no-

}IYllHTh

SlBHylO saBHCHMOCTb

Eo

(f)

OT

aMnJIHTy){bI.

no

3aKoHy

KHpX-

. "ro4>a M01KHO

aanHcaTh,

lJTO

TOK J I

Is

Bbl6paHHOH

aaMKHyToH

UenH,

CocToSlrqeH

H3

O.l{HOH

KaTylllKH

C HMne){aHCOM Z (jo>o) H

aKTHBHoro

'conpOTHBJleHHSI R,

paBeH

J =

e2

u+

2w

I

n1Eo

('1)

I

R+z

(/0>0)

r){e

n

1

-nOJlHOellHCJIO

BHTKOB O){HOH

KaTylllKH,

paBHoe

5

1

1'5

1

n

1

=T

I

Y(~-~1)d~=T

y(x)dx

-I -I

H <l>aKTHlJeCKH

He

3aBHcSlrqee

OT

~1'

TaK

K8K

nOJlHblH TOK

lIepe3

npO){OnhHOe

CelJeHHe

O.l{HOH

KaTylllKH

B

aMnep-BHTKax

B n

1

/e

paa

60Jlhllle,

a

06rqee

lJHCnO

KaTyllleK

paBHO

2llefJ,

TO

nOJlHaSi

nJlOT-

,HOCTb

TaKa

Ha

nOBepXHOCTH

S1

paBHa

J =

n1

U+

2w

1

n

1

E

o

('1)

o

II

R+z

(/0>0)

Ho

CKalJOK HanpSl1KeHHOCTH

MarHHTHoro

nonSi

B CHCTeMe

eM

lIHCneHHO

paBeH

nJlOTHOCTH

TOKa

Ha

nOBepXHOCTH

pa3pbIBa'MarHHT-

Horo

IIOJlSl. CJIe.l{OBaTeJlhHO,

/0

= H

0,

TaK

KaK

H0 = °

npH

f>

r

l'

OTCIO.l{a C YlJeTOM (8)

nOJlYlJaeM

H

o

":"

6

U

-

n1

(R

+z

(/ro

o

» +

/O>OllOnl1t

(,f

-

r~)

.LJ:anee, TOK

J 0

lJepe3

HCTOqHHK nHTaHHSI B

2l/efJ

paa

60nbllle

TaKa

J

l'

nO.9TOMY

J

=e..E.!..U+21tr

1

n

1

E

o

('1)=e:l...H

=

o 6

R+z

(10)0)

n1

0

21

U

= e -

..-------=-

n1

-

6

(R

+

('"

10>0llon1

1t

(2

Z

10>0

»

+'

/

'1

-'0

2)

n1

'

OrcIO.l{a

BH){HO, qTO nOJlHOe 3J1eKTpHqeCKOe

conpOTHBneHHe

HarpysKH

Ha

HCTOqHHK nHTaHHSI

paBHO

Z(.) 2 (

,,)

t

[6

(R

+

(.

'»)..L

/O>ollonf

('

2

2)]

0)0>0

=8%0

)0>0

=8

21

'

10>,'r

21

3t f

1

-f

o

•

341

I

-\

c\

-c>

ynPABnEHHE

nnA3MoA

340

lIneH,

COOTBeTcTBylOrn.RH caMORHJJ;yKURR

060JIUtIKR,

OTCIOJJ;a

B~pa3RTb

no

¢opMYJIe

II,

n

2

\'

8-

2

L

o

=

8-

2

r~/

11:

(r~-r~).

TaKRM

06pa30M,

Hy:>KHble

p~CtIeTHble

cooniOWeHR5I

npRHRMalOT

BRn

u

ft

n

l

u

J

o

=8-(-.

-)'

J

l

=8-

H

O

=-2l-(-'

-);

Zo

/000

2l

J

o

,

Zo

/00

0

Tenepb

¢opMyJIa

(3)

npHHRMaeT

BRJJ;

E (r) = _

joool.to

n

i

_u_

r .

(-!.-

_!.J..)

ro~r

<r

l

•

o ,

4l

Zo

(jooo)

0 , 0 "

JJ.R¢¢epeHuRpy5I

nOCJIeJJ;Hee

paBeHcTBo

no

r

npR

r = r

o

, HaXOJJ;RM

E

'

()

joool.tO

n

l U •

H

oro

= -

-2-l-

-,(-.

-)

= -

Jrool"'o

o'

(9)

, Zo /00

0

§

4.

3aKoH

OMa

JJ;Ji~'pacnpeJJ;eJJeHHoH

CTaOHJIH3HPYIOrn.eH

OOOJJOtIKH

JJ.JI5I

'Toro

tIT06bI

npRHTR

K

KOHeIJItoH

ueJIR

-BbIBOJJ;y

onepaTopa

06paTHoH

CB5I3R, -

HaM

npR,netC5I

pewaTb

KpaeBylO

3aJJ;atIy

JJ;JIH

onpeJJ;eneHR5I

3JIeKTpRtIeCKOrO

JIon5I.

B 3TOH CB5I31i

nOTpe6yeTC5I

3aJJ;aHRe

rpaHWlHoro

·yCJIOBR5I

Ha

cTa6RJIR31ipylOrn.eH

060JIOtIKe

Sl'

K BbIBOJJ;y

KOToporo

MbI

Tenepb

If

nepeHJJ;eM.

3TR

ycnOBR5I,

no

CYTR

JJ;eJJa,

nOJJytIalOTC5I R3

3aKOHa

OMa:

j=ae,

(I)

CB5I3bIBalOrn.ero

nnOTHOCTb

TOKa

j C Hanp5I:>KeHHOCTblO

3JIeKTpRtIeCKOrO

nOJI5I e

tIepe3

npOCTpaHCTBeHHO-BpeMeHHOH

TeH30p

npOBOJJ;RMOCTIf a.

YtIIfTbIBa5I

COOT,HOWe'HR5I (1)

If

(3) npeJJ;bIJJ;yrn.ero

naparpa¢a,

JIefKO

BIfJJ;eTb, tITO

TeH30p

a

RMeeT

BRJJ;

Ql

0)

{1 = (

a

a°. (2)

UHpKyJI5IUH5I

BeKTopa

e B

KaTywKe

L~

BJJ;OJIb

3TOH

KaTywKR

paBHa

1

2:?

Se

Ql

(r

l

,

~,

jro)Y(~-~l)d~.

-I

3Ha5I

TOtIHOe

conpOTRBJIeHReKaTywKR,

BbIBeJJ;eHHoe

paHee,

HaXOJJ;RM

TOK:

1

-

23t'l S

Ql

. 1 e r .

J~1(jro)=82

-I

(1.

~./OO)Y(~-~l)d~

(3) .

9:;

~~.

3AKOH

OMA

JJ;J151

OB0J10tIKH

, tITO JJ;aJIbHeHWRe BbIKJIaJJ;KR 6yJJ;yT BeCTHCb B

TepMHHaX

rrpe06pa30BaHH5I

ypbec

YIJeTOM

TOro,

tITO

npRMeHeHRe

3TOrO

,

npe06pa30BaHH5I

K

KBa3HrapMOHRtIeCKRM

KOJIeOaHR5IM JJ;aeT,

HanpH-

Mep,

CJIeJJ;ylOrn.ee

COOTHoweHRe:

(4)

F

[y

(t) e

iC1lol

] = y [j

(ro-ro

o

)),

. (JJ;e y

(jro)

-npe06pa30BaHHeYPbe

<pyHKURH

y

(t).

naJIee,

3aMeTRM,

.~}'O

B03Myrn.eHHe

nJIOTHOCTH

TOKa

B MaJIOH OKpeCTHOCTR TOIJKH Z

Ha

';/VIHHe

RHTepBaJIa

d~l

np5IMO

nponopUROHaJIbHO

wary

eB

pacnpeJJ;e-

Jr

HH5I

06MOTOK.

TI03TOMY

nOJIHOe

B03Myrn.eHRe

nJIOTHOCTR

TOKa

e

onpeJJ;eJIRTC5I

cyMMapHbIM

JJ;eilcTBHeM

Bcex

napaJIJIeJIbHO

BKJIlOtIeHHbIX

idlTyweK:

jQl

(z,

jro)

=

1 1

23t'l 1 1

S

eQl(r,~,

jro)d~x

+5

Y(Z+Sl)Y(~l)d~,

=

-f>-

R+z

(j

(00+00

0

»1

_I

-I

(5.)

.

jiJIR

"

1

jQl

(z,

jro)

= +5s

(z-~,

jro).e

Ql

(r

l

,

~,

jro)

d~,

(6)

-I

.

r:n.e

.

23t'l

g (z) ,

(7)

S (z,

Jro)

= T R

+z

(j

(00+000»

"-

(8)

g(z)=+

5

I

y(z+~)y(~)d~.

-I

TIOJIyIJeHHoe

Bblpa:>KeHRe

JJ;JI5I

nJIOTHOCTR

TOKa

jQl

(z,jro) R

eCTb

B~pa:>KeHHe

3aKOHa

OMa

JJ;JI5I

B03Myrn.eHH5I

TOKa

B

060JIOtIKe

C

npo-

CTpaHCTBeHHO-BpeMeHHOH

JJ;RcnepCReH.

yHKUR5I s

(z--:"~,

jro)

npeJJ;CTaBJI5IeT.

C060H

5IJJ;po

RCKoMoro

one-

paTopa

npOBOJJ;RMOCTR.

PaCCMOTpRM

OJJ;RH

tIaCTHbIH

CJIytIaH,

KorJJ;a POJIb

06MOTKIi

HrpaeT

CnJIOWHOH

npOBOJJ;~mRH

KO:>KyX,KOTOpbIH,

eCJIR OTBJIetIbC5I

OT

He-

cymecTBeHHoi%,

npOBOJJ;HMOCTR

era

BJJ;OJIb

OCIi Z,

MO:>KHO

pacCMaTpR-

BaTb

KaK

Ha60p

napaJIJIeJIbHO

BKJIlOtIeHH~X

KaTyweK,

Ka:>KJJ;a5I

R3

KOTOpbIX

Ha

HHTepBaJIe

(-1,

l)

RMeeT

JlliWb

bJJ;RH

BRTOK,

B 3TOM

CJIyIJae

>

.)

2n;'l

l

J: ( )

(9)

Y

(z)

-:-lB (z),

g (z) =

lB

(z),

S

(

z,

Jro

=

Rf>

U z .

'"

,

••••

>

,y

r,','

~:~.>

C

343;

nOCTAHOIlI<A

II

PEIIIEHHE

I<PAEBOA 3AJI.Al.IH

ynPABJIEHHE

nJIA3MOP!:

342

3,u.eCb H

){aJlee

0003HatIeHHe

h

Z

'

(r

o

,

z, -

}ro)

nOHHMaeTCSI

cJIe){YIOw.

HM

epaKTHqeCKHHe

3aBHCSlT

,OT

ro.

3aKoH

OMa

TOr){a

npHHHMaeT

60bIqHbIH

BH){:

'qJ

(

.)

-

qJ

( . )

J z,

Jro

-

aa

e

r

i,

z,

Jro

,

f){e

aa

= 2nr

1/

R6

-

3KBHBaJIeHTHaSl

npOBO){HMOCTb.

§

5.

Cpe){Hee

MarHHTHoe

){aBJIeHHe

KaK

rOBOpHJIOCb

BbIIIIe,

CTaOHJIH3HpYIO~aSl

060JIOqKa,

){OJI)f{Ha

COCTOHTb

H3

){BYX

CHCTeM OOMOTOK

co

B3aHMHO

0pTOrOHaJIbHbIM

HanpaBJIeHHeM

BHTKOB,

npHqeM

B

paBHoBecHoM

pelKHMe

TOK B

HHX

){OJIlKeH

HMeTb

BpeMeHHoH

C){BHr

epa3 n/2.

3TO

n03BOJISleT

BMecTO

,

nOJIHOrO

MarHHTHoro

){aBJIeHHH

OT

){BYX

CHCTeM

paccMaTpHBaTb'

CyMMy

Cpe){HHX

3a

nepHO){

MarHHTHbIX

){aBJIeHHH,

C03){aBaeMbIX

KalK){OH

cHcTeMoH

OOMOTOK B OT){eJIbHOCTH.

ECJIH

CHCTeMbIO){HHaKOBbI,

TO ){JISI

Me){JIeHHblx(no

cpaBHeHHIO

C OCHOBHbIM

nepHO){OM)

B03MY-

w.eHHH

nOJIHOe

MarHHTHoe

){aBJIeHHe

oy){eT

paBHO

y){BOeHHoMy

Cpe){HeMy

3a

nepHO){

MarHHTHoMy

){aBJIeHHIO

OT O){HOH CHCTeMbI.

BbIpa3HM

B03MYllJ,emle

p

(z,

t)

nOJIHOrO

MarHHTHoro

){aBJIeHHH

P

(z,

t)

Ha

nOBepXHOCTH

So

qepe3

B03MYllJ,eHHe

eqJ

(r

o

,

z,

jro)

9JIeKTpH-

qeCKOrO

nOJIH.

Y

Cpe){HeHHbIe

3a

nepHO){

2n/ro

o

BeJIHqHHbI

0603Ha

i

mM

npHMoH

qepTOH

cBepxy.

BOJIHHCTOH

qepTOH

cBepxy

oy){eM

0003HaqaTb

Cpe){HHe

BeJIHqHHbI

){JI5I

){B~X

0pTOrOHaJlbHbIX

CHCTeM.

vlTaK,

HMeeM

P(z,

t)=P

o

+8p(Z,

t)+Od8)=2Po+28~+02(8)=

'{

1

'1

} 2

=

fLo

2"

[H

o

+

8h

z

(r

o

,

z,

t)]

e

fro

•

I

+"2

[H

o

+8h

Z

(r

o

,

i,

t)]e-

/ro

•

1

+

82fLo

{

Re

[h

Z

(r

0,

z,

t)]

e

iro

•

I

}2.

Crp,ynnHpyeM

qJIeHbI

no

nOpSI){Ky

MaJlOCTH,

yqHTbIBaSl,

qTO

roo

=

= V

o

/8

H

qTO,

CJIe){OBaTeJIbHO,

Cpe){HHe

3a

nepHO){

,3HaqeHHSI

BeJIH,-

qHH,

KOJIe6JIIOIlJ,HXC5I C qaCTOTOH

2ro

o

,

HMelOT

nOpSl){OK

8.

TIOJIyqHM

2

Po

=

2"

1

fLo

I H

o

1

,

(1)

P(z, t) =

-}

fLo

[Hoh

z

'

(r

o

,

z,

t)

+

H~hz

(r

o

,

z,

t)].

(2)

BbIIlOJI~HSI

npe06pa30BaHHe

ypbe

Ha){

nOCJIe){HHM

paBeHcTBoM,

nOJIyqHM

p(z,

jro)=;

fLo

[Hohz'(r

o

, z,

jro)+H~hZ(ro,

z,

jcu)].

(3)

OTCIO){a

BmeKaeT

CBSI3b

MelK){y

OTHOCHTeJIbHbIM

H3MeHeHHeM

MarHHT-

,

l'

.

Horo

nOJ:ISI

H;; h

z

H

){aBJIei:lHeM

n (z,

}ro):

. )

__

1_-(

.

)_hZ(,o,z,/w)+

hZ'(,o',z,/w)

(4)

n

(

z,

Jro

- - p z,

Jro

- H

•.

Po

0 H

o

o6pa30M:

h"

,(r

..

z.

-

jro)

~

[I

h'

(r

..

z.

t)

,'.'

dt

r-

GO

=

~

hZ'(r

o

,

z, t)e-iroldt.

(5)

-CD

, _

Hpe06pa3yeM

no

ypbe

rpaHHtIHOe

yCJIOBHe

(6) § 3

,noH

rJIaBbI:

eqJ

(r

,

z,

jro)

= -

E~p

(z,

jro).

(6)

o

Pa3){eJIHB

nOtIJIeHHO

nepBoe

H3

ypaBHeHHH

(4a)

§ 3

::noH

rJIaBbIHa

pilBeHcTBo

(9) § 3

::noH

lKe

rJIaBbi

H

ytIHTbIBaSl

nOCJIe){Hee

paBeHcTBo,

nOJIyQHM

-H'

1h

Z

(

ro,z,

Jro

• )

=--p

1 +

-a

a

[1

-,e

qJ

('

")J

'

(7)

r,t,Jro

•

o

'0'

Eo

r=

r.

.

TIo.n.CTaBHB TOJIJ>KO tITO

nOJIyqeHHOe

BbIpalKeHHe

B (4),

nOJIyqH~

HcKOMoe

BbIpalKeHHe

OTHOCHTeJIbHOrO H3MeHeHHSI

MarHHTHOrO

){~lBJIe~

gHSI

Qepe3

OTHOCHTeJIbHoe

H3MeHeHHe

3JIeKTpHQeCKOrO

nOJlSI:

.

)=_~(

.

)+~[eqJ(,;z,fCiJ)+,eqJ*("z,-/(o)}]

(8)

n

(

z,

1ro

P

z,

1

ro

a ' (

')*

.

'0

'

En

Eo

r=r.

§ 6.

nOCTaHOBKa

H

peweHHe

KpaeBoH

3a){aqH

CKaQOK

KaCaTeJIbHOH

COCTaBJISIlOllJ,eH

HanpHlKeHHocTH

MarHHTHoro

nOJIH

Ha

nOBepxHocTH

Si

B

cHCTeMe

e){HHHU

CI1

qHCJIeHHO

paBeH

nOBepXHOCTHOH

nJlOTHOCTH

TOKa.

TI03TOMY,

ytIHTbIBaSl

aaKOH

OMa

,u.JISI

OOOJIOqKH (10) § 4,

MomHO

3anHcaTb

1

hz

(r

1

-0,

z,

jro)

_h

z

(r

i + 0, z,

jro)

= +SS

(z-'

6,

jro)

eqJ

(r

i'

;"

jro)d~.,

-I

(1)

YMHomaSl

ooe

tIaCTH

SToro

paBeHCTBa

Ha

jro

o

'

yqTeM

BbIpameHH8

,,u.JISI

(rot

e)Z

B

UH./IHH){pHtIeCKHX

Koop,u.HHaTax:

(rot

e)Z

=.!.

,

eqJ

(r, z,

jro)

+

aa

,

eqJ

(r,

t,

jro),

(2)

a

TaKme

HenpepbIBHOCTb

no

r B

TOqKe

'1

epyHKUHH

eqJ

(r,

z,

l

ro

).

.~\"';.

..

0:,

~~>

~mPABJIEHHE

nJIA3Moti

344

B

HTore

nOJIy'lHM

-iJ

OQl(

e r, z,

JW

')1

--iJ

iJ

e

Ql

(

r, z,

JW

.)/

=

,

''''''.+

0 ,

'='.-

0

1

.= +SjwoUoS(Z-

S,

jw) e

Ql

(r

I'

£,

jw)

dS·

-I

DOJIy'leHHoe COOTHoweHHe

(3)

H

HBJIHeTCH

Hl;KOMbIM rpaHH'IHhIM

yCJIOBHeM

Ha nOBepxHocTH

Sl'

Ha

nOBepxHocTH

So

)!.omKHO BhlIIOJI-

HHTbCH

yCJIOBHe(6)

§ 5

9TOH

t'JIaBhI.

Ha

6eCKOHe'lHOCTH <PyHKUHH;

eQl(r,

Z,

jw)

)!.OJI2KHa

ObITh

OrpaHH'IeHHo~.

I1cI(JIIO'IaH OOhI'IHhIM npHeMOM BeKTOp h

H3

ypaBHeHHH

(3a)

§ 3

)!.JIH

OOJIaCTeH

BaKyyMa

V

o

H VI' nOJIy'laeM BeKTopHoe ypaBHeHHe

JIanJIaca

)!.JI5I

BeKTopa e:

~e=O.

(4)

B

HaweM

CJIy'lae OTJIH'IHa

OT

HyJIH

TOJIhKO

npoeKUHH

9Toro

BeK-

Topa

Ha

XQl'

D09TOMY

OCTaeTCH

JIHWh CKaJISlpHOe ypaBHeHHe

)!.JIH

Ql

KOMnOHeHTbI e HanpH2KeHHOCTH 9JIeKTpH'IeCKOrO nOJIH:

1

~eQl-2eQl

=0.

(5)

,

I1TaK, TpeoyeTcH 'peWHTh

CJIe)!.yIOIUYIO

KpaeByIO 3a)!.a'ly: HaHTH

Ql

<PYHKUHIO

e

(r,

z,

jW),

y)!.oBJIeTBOpHIOIUYIO ypaBHeHHIO

(5)

B

OOJIaCTH

Vo,

VI' KpaeBhIM

yCJIOBHHM

(6)

§

5,

(3)

H OrpaHH'IeHHOCTH

Haoec-

KOHe'lHCCTH.

DepeH)!.eM K peweHHIO

9TOH

3a)!.a'lH.

D.JIH

KpaTKOCTH

BBe)!.eM

HOBbIe

0003HaqeHHH:

. Q

(r,

z)

= e

Ql

(r,

Z,

jW)

npH

r

o

~

r

~

rl>

(6)

o

Q

1

(r,.

z)

=e

Ql

(r,

Z,

jw)

npH

r

1

~

r <

00,

(7)

f

(z)

=

eQl(r

0'

Z,

jw),

(8)

S

(z)

=

jWO/U1S

(z, jw). (9)

Tenepb

3a)!.aQy

M02KHO

nepe<popMyJlHpOBaTh

CJIe)!.yIOIUHM

oopa30M:

HaHTH

)!.Be

<PyHKUHH' Q

o

(r, z), QI(r, z), onpe)!.eJIeHHhle B

CBOHX

OOJIaCTHX,

y)!.oBJIeTB:;pHIOIUHe rpaHHQHhIM

yCJIOBHHM

Qo(r

o

,

z)=f(z),

(10)

Q1

(rl>

z)

=Qo

(r

1

,

z),

(11)

iJ

2

1

r

1

i

(r

1

,

Z)-rliJa~O(rl>

z)=+

5

S(Z-S)Qd

r

l,S)ds,

(12)

,'-I

Q

1

(00,

z)

<

00.

(13)

Qo(r,

z+2l)=Q.(r,

t).

(14)

Qdr,

z+21)=Qdr,

1)

(15)

<-

c·

(2.;nocTAHOB'KA

Ii

PElIIEHHE

KPAEBOIl

~A.u.AqH

345

H~

ypaBHeflHIO

-!-~(riJQi)

iJ2Qi_.l.-

.-0

,

a,

iJ,

+

az2

,2

Q,-

,

(16)

r)!.e i =

1,2.

OOIUHe

peweHIHI

ypaBHeHHH

(16)

)!.JIH

OOJIaCTeH

VoH

VI

,cqOTBeTcTBeHHO C

YCJIOBH5IMH

nepHo)!.H'IHOCTH

(14),

(15)

no

KOOp)!.H-

'HaTe

Z

HMeIOT

BH)!.

aD

Z

Qo

(r,

z)

=

~o

+bor +

L.

[(

a

k

cos

k~Z

+b

k

sin

k7

)/

1

(kx) +

k='

.

knz

,knz

) ]

+

ckcos-1-+dksm-

l

-

K

1

(kx) ,

(17)

(

•

aD

Ql (r,

z)

=

~o+Bi

+

t:,

[(

Akco~

k7'

+B

k

sin

k7)

11

(kx) +

+(G

k

cos

k7

z

+ D

k

sin

k;Z

)K

1 (kX)] , (18)

r)!.e x=:rtrll.

113

yCJIOBHH

(13)

OrpaHHQeHHOCTH peweHHH Ha

6ecKo-

He'lHOCTH

cpa3Y

CJIe)!.yeT,

QTO

B

o

A

k

= Bk = 0

l:!PH

k = 1,

2,

•

..

~

aD

I

Qdr,

z)=

~o

+

L.

(GkCOs

k7

z

+Dksin

k7

z

)K

1

(kx). (19)

k=l

Pa3io2KHM

B

pH)!.

¢yphe

<PYHKUHIO

f (z):

aD

a.o

,-.

(.

knz

. knZ)

(20)

f(z)=2"+

~I

akcos-l-+~k

sm-

1

- ,

r)!.e

1

I

S .

knz

~k

= +Sf

(z)

sin

k7

z

dz.

(21)

a

k

=[

1

f(z)cos-1-dz,

-I

113

yCJIOBHH

(10) nOJIyQaeM

ao

+b

a.o

(22)

-

oro

=-2

'

'0

a

k

/

1

(kx

o

)

+

CkK

1 (kx

o

) = ak'

k=

1,

2, ,

(23)

b

k

/

1

(kx

o

) +d

k

K

l

(kx

o

)

=

~k'

k=

1,

2, ,

(24)

r)!.e X

o

=:rtrl/l.

I1cnoJIh3yH

yCJIOBH5I

HenpephIBHocTH

(

11),

nOJIy'lHM

a

'1

. ..J!.+b

O

r

l

= A

o

-,

(25)

'1.

' .

k =

1,2,

.••

,

(26)

akldkx1)+CkKi

(kx

l

) = G

k

K

l

(kx

1

),

bill1 (kx

1

)

+d/!K

1

(kx

1

)

=

D/!K

1 (kx

1

),

k=1,2,

...

,

(27)

VnPABJIEHHE

nJIG'loA

,r:

346

rJ{e

Xi

=

n'il

1

.

HCnOJIb3yH

H3BeCTHble COOTHOIlIeHHH,

Me>KJ{yMo;llH.,:

TIPH

HeH3BeCTHbIX

HHTepecoBaTb

(30)

. (29)

pa3JIa-

<PHI..{HpOBaHHbIMH

<PYHKI..{HHMH

BeCCeJIH

OT

MHHMoro

apryMeHTa

d 1 '

dx

Ii

(X)

= 1

0

(X)-x-

11

(X),

dx

d

Kdx)

=

-K

o

,

(x)--

x

1

Kdx),

npeJ{CTaBHM B BHJ{e pHJ{a YPbe JIeByIO qaCTb yCJIOBHH

(12):

aQI

( )

aQo

( )

(ao-A

o

) b

'1

-()

,

i'

Z

-,

i

-a

'

i'

Z =

----'-

0'1

+

l'

,

'I

...

+

~

{(Ck-C

k

) [kx

1

K

o

(kx

1

)

+K

1

(kX

i

)]-

k=1

k~

-ak

[

kx

1

/

0

(kx

1

)-/

i

(kx

1

)]

}

GOS

-1-+

...

+

~

{(dk-Dk)

[kx

1

K

o

(kx

1

)

+K

1

(kx

1

)]-

k=1

-b

k

[kx

1

/

0

(kx

1

)

-11

(kx

1

)]}

sin

~z.

TaK

KaK

<PYHKI..{HH

S (z),

KaK

H g (z),

qeTHaH,

TO

OHa

raeTCH B pHJ{

Ha

(-1,

1)

TOJIbKO

no

KocHHycaM:

...

S(z)=So+

~ S

cos

Imz

2

~

k

Z.

(31)

k=1

rJ{e

I ,

1 S

k~

2Wi

Iro

o

/1oTi

g

Sk=T

S(z)cos-z-dz=-l}-,

R+z[j(ro+ro

)]

k'

(32)

o

-I

HCnOJIb3yH

pa3JIO>KeHHe (31), npeJ{cTaBHM B BHJ{e pHJ{a YPbe

npaByIO

qaCTb YCJlOBHH (12r.

i-

S

I

S

(z-~)

Qd'i,

6)

d6

=

-I

...

1

-~

(

Imz

k1tZ)

=SoA

o

-+

~

Sk Ckcos

-Z-'

+D

k

sin

-Z-

K

1

(

kx

i)'

(33)

1'{

k=1

IlpupaBIHfBaH

B~pa>KeHHH

(30) U (33), nOJlyqUM

nyTeM

nOqJIfHHOro

~

CA10CTAHOBI(A

H

PEIIIEHHE

I<PAEBOA 3AAA-4H

341

PSlJ{OB

CJIeJ{YIOIUHe eOOTHomeHHSI:

'.

J . 1

(a.-A

o

)

'i-bO'i

.....

SoA

o

ft.

(34)

(c.-C.)

[kx

1

K

o

(kx

1

)+K

1

(kxl)]-a.

[kx

1

/

0

(kx

1

)-/

1

(kXi)] =

=SkCkKi

(kx

1

),

(35)

(d/c·":"D.)

[kxiK

o

(kXi)

+K

1

(kx

1

)]-b.

[kx

1

/

0

(kXl)-/1

(kx

I

)]

=

-S,P

k

K

I

(kx

1

)·

'(36)

113

ypaBHeHHFt (22).

(23)H

(34)

nOJIyqaeM

1

1+

2

S

o

'

0

a

0==2

----,--....",...=--.,....-.....,...-_=:_

I+~

(37)

So

[I_(~:r]ao

.

b So'o 1

o

(38)

0=--4-'1-

2

1+

~

So

[1-

(~~

YJ

a

,

A

(39)

0=2

'o

'I

[

1

(

1'0

)1]

ao·

1+2'

So

1-

r;

.

I

JII060M k

~

1 HMeeM

cHcTeMy

IlIeCTH JIHHeAHbIx

aJIre6paHqeCKHX

ypaBHeHHH

(23), (24),

(26). (27),

(35),(36)

J{JIH

onpeJ{eJIeHHH IlIeCTH

K09<p<pHI..{HeHTOB

ak, b

k

,

ck,d

k

,

C,;,

D

k

•

Hac

6yJ{yT

TOJIbKO.

K03<p<p

H

I..{HeHTbI

ak' b

k

•

ck.dk.

Heo6xoJ{HMble

J{JIH

onpeJ{eJIeHHH

MarHHTHoro

J{aBJIeHHH

Ha

cTa6HJIH3HpyeMoA

nOBepxHocTH.

TI09TOMY CHaqaJIa

HCKJIIOQHM

H3 9THX

ypaBHeHHH

C.

H

V",

a

3&TeMpa3peIlIHM

9TycHcTeMy

OTHOCHTeJIbHO HCKOMbIX BeJIH-

qHH.

TIOJIb3YHCb

ypaBHeHHHMH

(26),

(27),

onpeJ{eJIHM C

k

"

D

k

qepes

ak. b••

Ck'

dk:

h(

kx

l)

'

+'

D

Ii

(

kX

l)

b 'd

C

k =

Kdkxi)

ak

c

•• k = K

1

(

kX

l)

k+ k'

nOJ{CTaBJIHR 9TH BbIpa>KeHHH B

ypaBHeHHH

(35)H

(36).

nOJIyqHM

J{Ba Ha3aBHCHMbIX COOTHOIlIeHHH. H3 KOTOpbIX

BbIpa3HM

Ck

qepe3

a.

H d

k

qepe3

b

k

COOTBeTCTBeHHO:

c.

= - ( Nk+

~~

)ak'

(40)

(41)

d.=-

(N

k +

~:

)b

••

rJ{e

N =

Id

k

"I) , P

=kx

10(kxl)Kdkxl)+ldkxl)Ko(Ul)

•

k K

1

(kx

1

) • k

i

K~

(

kX

l)

tIO~CTaBJ1RH

reneph

C/c

H d

k

•

onpeJ{eJIReMble

<pop~YJIaMH

(40) H (41).

B

(23)

H

(24).

nOJIyqaeM

onHTb

J{Ba He3aBHCHMhIX COOTHOIlIeHHS.

~,

~

I.

£?",

ynPABJIEHHE

nJC~Of[

348

H3

KOTOpWX

Onpe~eJIHeM,

qTO

~

b

=_

Sk

R

k

l..<'.IAA.t-'k'

rJle

L-

11

(

kX

l)

K

CkX

o

)-/

1

(kxO)

K

1

(kx

1

)

k - K

1

(

kX

l)

M

=kx

K(kx)

10(kxl)K(kxl)-,-11(kxl)Ko(kxl)

k

1

lOKi

(

kX

l)

BeJIHqHHbI

L

k

,

M

k

,

N

k'P

k

3aBHCHT

TOJIbKO

oT'reoMeTpHH

CHCTeMbI,

B

~aHHOM

CJIyqae

UHJIHH~pHqeCKOR

H

onpe~eJIHIOTHeHsMeHHeMylO

qaCTb

CHCTeMbI

aBTOMaTHqeCKOrO

ynpaBJIeHHH.

3TH

BeJIHqHHbI

He

HesaBHCHMbI

H CB5JSaHbI

COOTHOlIleHHHMH

L

k

= N

kK

1

(kx

o

)

-11

(kx

o

)'

M

k

= P

k

K

1

(kx

o

)'

J1sMeHHeMaH

qaCTb

CHCTeMbI

aBTOMaTHqeCKOrO

ynpaBJIeHHH,

nO~JIe

)KamaH

cHHTesy,

onpe~eJIHeTCH

Ha6opOM

K03<p<pHUHeHTOB

Sk'

AHaJIO-

rHqHO

Haxo~HM,

qTO

_

NkSk+P

k

d'

NkSk+P

k

A

C

k

=

LkSk+M

k

ak'

k =

LkSk+M

k

t-'k'

nOCJIe

nO~CTaHOBKH

HaR~eHHbIX

KOS<p<pHUHeHTOB

Uk'

b

k

,

C

k

H d

k

B

<pOpMyJIy

(17)

o6mero

pelIleHHH

Sa~alJH

~JIH

06JIaCTH

V

o

H

rpyn-

nHpoBKH

no~o6HbIX

qJIeHOB

HaXO~HM

HCKOMoe

qaCTHOe

pelIleHlfe:

'0

( 1 ) , I (

'0

)2

- I+-S

o

---so

-

, )

I,

2

'0

2

'1

U

u

(f'Z=2

1 r ('0\2

1

a~+

1+-So'l

1- -

2

'1

..

+

L:

(akcos

k7

z

+Pksin

kltZ)

(N

k

S

k

+P

k

)Kl(kX)-Sk

/

l(kX») ,

(42)

k=l 1

Lk

S

k

+

M

k

.

JI.JIH

no~cTaHoBKH

B

iPOpMyJIy

(8) § 5,

~alOmyro

HCKOMoe

H3Me-

:HeHHe

MarHHTHoro

~aBJIeHHH,

OCTaJIOCb

HaRTH

TOJIbKO

BeJlHqHHY

au

(,.

Z>'I

~

r=r.'

KOTopaH

JIerKO

onpe~eJIHeTCH

~H<p<pepeHUHpOBaHHeM

nOCJIe~HerO

paBeHCTBa:

au

o

1

1+{-So[I+(

~

YJ

-a;:-(f

o

'

z) =

2'0

1 [ ('0

)'J

a~+

1+-S

o

1-

-

2

'1

GO

1

~

T

k

S

1t

+R/t

(

k1tt

R •

knZ)

+,

~

L S

+M

akCOS-r

+

t-'k

sm-

1

- ,

(43)

- 0 k=l k k k

\'

I

-

npO@HCTBEHHOoBP'EMEHHOft

onEPATOP

OBPATHOf[,CB5ISH

349

1

j

T

=kx

1

0

(kxo)K

1

(kx])+/d

kx

l)K(kx

o

)

I

k 0

K~

(kx

1

)

I

R

k

==

[kxoK

o

(kx

o

)

+K]

(kxun

P k'

'I

I

j

§ 7.

npOcTpaHcTBeHHo-BpeMeHHoH

onepaTop

o6paTHoH

CBD3R

nJIH

nOJIyqeHH5lI

oKOHqaTeJIbHOrO

pesyJIbTaTa

Tenepb

OCTaJIOCb

1

.ycTaHoBHTb

CB5ISb

Me)K~y

K03<p<PHUHeHTaMH

<l>ypbe

B

paSJIO)KeHHHX

B

PH~bI

BosMymeHHH

cTa6HJIHSHpyeMoR

nOBepxHocTH

H

MarHHTHoro

~aBJIeHH5I_

ltlTaK,

nycTb

\

-"1

""

'-'1

j

'I'J

(z,

jro)

=~

p

(z,

jro)

=

ao

(2

O)

+

L.

[Uk

(jro)

cos

knz +

't

k

(jro)

sin

knZJ

'

,'0

k=l I I ,

(1)

1

r~e

I

1

j

.'

1 5p (z

/00)

knz

'tk

(J'ro)

=.!. 5p

(z,

/0)

sin

kltZ

d

I

Uk

(jro)

= 1

-----f;:-

cos

-1-

dz,

1

'0

1

z.

I

-I

nYCTb

TaKme

n (z,

jro)=';'"

P(z,

iro)

=

Po

""

Po

(j0)

~

[

(.)

krrz

+

(.).

kltZ]

=

-2-

+

~

Pk

Jro

cos

-1-

qk

Jro

sm

-1-

,

(2)

k=l

r~e

Pk

(jro)

=-4-

S

1

P

(::

jO)

cos

k1f2

dz,

-I

Po

I

z

qk

(jro)

~+

S

P(z..:

jO)

sin

k7 dz.

-I

Po

YlIHTbIBaH,

lITO

K03<p<PHUHeHTbI

PH~OB

<l>ypbe

~JIH

<pyHKUHH

f

(z)

=

elp

(r

0'

z,

jro)

H S (z) =

jroof.toS

(z,

jro)

saBHCHT

OT

jw:

ak=ak(jw),

~k=~k(jW),

S

s-

(.)

21f'1

jO)o/-l

0'1

(3)

k=

k

JW

=-6-R+z

[j(0)+0)0)] qk'

HS

cooTHoIlleHHH

(6) §

5,

(20) § 6 H

(1)

HaXO~HM

a

k

(jw)

= - f

uE~

Uk

(jw);

~k

(jw) =

---::-

foE;

'tk

(jro).

(4)

"350

ynPABJlEHHE

nJle

:~oA

,naJlee.

H3

CooTHolUeHHH

(8)

§ 5.

(6)

§ 6, (43) § 6.

(3)

H.

(4)

no~

':npH

nOJIYllHTb

Jly'laeM

n (z.

jro)

=

~2'1l

(z.

jro)

+

1

1+

2

I [

1+

('0

r;

)2J'

So(jro)

1+

2

1 [

1

+('0

t;

)2J.

So

(-/ro)

}

.

+2

{

1 , 2 +

1·

,

2.

00 (JCJ)+

1+

2

[1-(,:)] So(jro)

1+

2

[1-(,:)

]So(-/ro) .

+

i:

[TkS

k

(jro)+Rk

+TkSk(-jro)+Rk] X

k=1

LkSk(jro)+Mk LkSk(-/ro)+Mk

X

[Ok(jro)

cos

k7

Z

+

'tk

(jro)

sin

k7

z

J. (5)

nOJIaraSl

B nOCJIe,nHeM

paBeHCTBe

Sdjro).

0

,nJIH

Bcex

k =

O.

I.

2,

....•

nOJIy'lHM

BeJIH'lHHy

no

(z,

jro)

OTHOCHTeJIbHOrO

MarHHTHoro

,naBJIeHHSI B

oTcyTcTBHe

cTa6HJIH3HpyIOIU.eii

060JIOllKH:

no(z, jro)=

GO

.K

(kW.)

2

~

kn,o

0

-,-

[

(.)

k1tZ+

(.).

knzJ

2 (

.)

(6)

=

~

-'-,

K ( kn;o)

Ok

Jro

cos

-,

.

'tk

Jro

Sill

-1-

-

'Il

z.

Jro

•

1

nO,nCTaBJISISI

Tenepb

B (5)· 3HaQeHHSI

BeJIH'lHH

L

k

•

M

k

•

R

k

H T

k

H Bbl,neJISlSI lIJIeH

no

(z.

jro).

nOJIyllaeM

OKOH'laTeJIbHOe Bblpa:>KeHHe:

n (z,

jro)

=

no

(z.

jro)

+

~

F

o

(jro)

00

(jro)

+

GO

+

{;:

F

k

(jro)

[Ok

(jro)

COS

k7

1Z

+

'tk

(jro)

sin

k~ZJ

'

(7)'

r,ne

F

k

(jro)

~<Dkgk

[H

k

(jro)

+H:

(jro)]

,

(8)

H

('

)

jrooL

(9)

k Jro = R+z

[j(ro+roo)]+jroOL'P'kgk'

L

2n'1

2

=!-to-ll-'

(10)

<D

=(~)2

'l'

=1-

[I_(~)I]

(

11)

o

'1'

0 2

'1'

'0

[Kl(~)]l/o(9)KO(~)+/l(~)KO(~)

<Dk=r; K

1

(k';'0)

10(kn'I)Kl(k';'I)+/l(k~'I)Ko(k~'I)'

(12)

'l'k

=_!_K

1

(~)10(~

)K

1

(~)-/

l

(~)Ko

(~)

.

n

(

13)

kWI

K

1

(k~'o)

/0

C7

1

)

K

1

e

;l)

+/i

(k';'l)

K

o

C~'I)

.

np£DAHCTBEHHO~B~EMEHHOA

onEPATOP

,OBPATHOA CB5I3H

351

'1-+00

K03<p<pHnHeHTbI

<D

k

H Fk(jro) CTpeMSlTCSI K HyJIIO H

.n

(z.

jro)

-+

no

(z.

jro),

liTO

COOTBeTcTByeT yCTpaHeHHIO

cTa6HJIH3HPYIO-

.III.eH

060JIO'lKH.

npH

60JIblUHX

k

,nJISI

K03<p<pHnHeHTOB

<D

k

H 'l'k

MO:>KHO

CJIe):{yIOIII.He aCHMnTOTHlIeCKHe Bblpa:>KeHHSI:

'0

[2kn

]

<D

..

"",-exp

--1-('1-'0)

,

(14)

,'1

.

'l'

"",_1_

'/'0

(15)

k

2nk'1

r

'1'

OrClO,na BH,nHO.

'lTO

psi,n B

npaBOH

qaCTH (5)

CXO,nHTCSI.

,UaJIee.

BH,nRO.

1JTO·

eCJIH

Tpe6yeTcSI.

lITo6bI

K03<p<pHnHeHTbI yCHJieHHH F

k

(jro)

,nJISI

pSl,na

rapMOHHK

He

SaBHceJIH

OT

k.

TO

Heo6xo,nHMO.

'lTo6b1

K03<p<pH-

nHeHTbI

cI>ypbe

Ylc

xapaKTepHCTHlIeCKOH

<PYHKllHH HaMOTKH pOCJIH

3KcnOHeHnHaJIbHO

~

nOKa3aTeJIeM

n<rt~'o)

.

n03TOMY

,nJISI

KOPOTKHX

.

~

BOJIH.

A.<n('I...:....'O).

CHJIbHO

B03paCTaIOT

Tpy,llHOCTH

peaJIH3aUHH

•

li3

nOJIylleHHoro

pelUeHHSI

HeTpy,nHO

HaHTH

pelUeHHe

,llJISI nJIOC-

Koro

CJIYllaSl.

KOr,lla,

So H

SI-napaJIJIeJIbHble

nJIOCKOCTH.

HaXO.D.SI-

III.HeCSI

Ha

paCCTOSlHHH h

,npyr

OT

,npyra.

nOJIaraSl

'1

='0

+h H

ne:

pexo,nSl

K

npe,neJIy

npH

'0

-+

O.

nOJIy'lHM.

liTO

B

STOM

CJIy'lae

<D

k

=exp(....,.-

2k

t)

.

'¥k=O.

k=0.I,2,

•••

(16)

,naJIee,

3aMeTHM,

'lTO

eCJlH

BeJIH'lHHa

R+z

(jro

o

) +

jrooL'l'

kgk

=1=

0,

TO

F

k

(jro)

aHaJIHTH'lHa

B

HeKOTopOH

OKpeCTHOCTH HyJISI

H,

CJIe,llOBa-

TeJIbHO,

MO:>KeT

6bITb

pa3JIO:>KeHa B CTeneHHOH pSl,n:

1

. F

k

(jro)

=

1'1

+jroFk+ 2,

(jro)1

Fk

+ ...

(17)

3TOT

PSl,n

CXO,nHTCSI

B

Kpyre,

HMeIOIII.eM

pa,nHyc,

paBHblH

paCCTOSlHHIO

OT

TOllKH

jro

o

,no 6JIH:>KaHUlerO HyJISI

<pyHKnHH

R+z

(jro)

+

jrooL

'l'

kgk'

,

TaK

liTO

pa,llHyc

CXO,llHMOCTH HMeeT nOpSl,llOK roo' B

3ToMKpyre

cnpaBe,nJIHBO

TaK:>Ke pa3JIO:>KeHHe

-}

[H

k

(jro)

+

Hi.

(jro)]

~

(jro)2m

a2m

(fro)

1m

+1

aim

+1

GO

. =

L.

(2m)

I at2mReHk(t)

I

+L (2m+l)rat2th+lImHk(t).

I

.•

m=.O

t.o

m"O

t=o

(18)

lis

<pOpMyJIbI (9)

MO:>KHO

YCTaHOUHTi),

'lTO

eCJIH R>0 H Z

[(fro

+roo)]

eCTb lIaCTOTHaSl

xapaKTepHcTHKa

<pH3H'leCKH

peaJIH3yeMoij

CHCreMPI,

1'9

F~

(jro)

TaK)Ke

<l>H3HlI~C~H

pea.~H3~eMa,

"'r,1',~

353

(>

ynPABJIEHHE

nJI~,jM~f:I

352

HaH.n.eM.

nepBble

TpH

lJJIeH~

pa3JIoiKeHHH

F

k

(jro)

B

pH.n.nO

CTe·

neHHM

jro.

3TH

lJJIeHbI

HrpaIOT

OCHOBHyio

POJIb

npH

HCCJIe.n.OBaHHH

MeJi:JIeHHbIX

(no

cpaBHeHHIO

C lJaCTOTOH roo)

KOJIe6aHHH

cTa6HJIH3&

pyeMoro

06beKTa:

F~=Fk(0)=2<DkgkReR+

C

~W+L.

VI'

'

(19.)

Z

JWo

.

TWo

kgk

F'

-

8F

k (jw) " = _

2<1>

Re jwoLz',(jw

o

) (20)

k-

8(jw) III= 0 kgk

[R+z(jw

o

)+jw

o

LlJ'kgk]2' .

2

r = 8 Fk

(jw)

I .

k 8

(jw)2

III

=0

.

2jwoL<1.Jkgk

[z'(jw

o

)]2

jwoUDkgkZ' (jw

o

) }

=2<1>~kRe

{

[R+z(jwo)+jw

o

LlJ'kgk]3

[R+z(jwo)+iw

o

L<Il

k

g

k

J2

•

DepBbIH

lJJIeH pa3JIO>KeHHH (17)

cooTBeTcTByeT

pacnpe.n.eJIeHHoiF

B.n.OJIb

cTa6HJIH3HpyeMoH

nOBepxHocTH

npHcoe.n.I-lHeHHOH >KeCTKOCTH,

BTOpOH

-npHcoe.n.HHeHHoH

BH3KOCTH,

TpeTHH

-npHcoe.n.HHeHHOH

/.

Macce.C60THolIIeHHH

(19), (20) H (21)

n03BOJIHldT

onpe.n.eJIHTb KOH-

cTpyKTHBHble

napaMeTpbI

060JIOlJKH

no

3a.n.aHH

..

IM CTaTHlJeCKHM H

.n.H-

HaMHlJeCKHM

xapaKTepHcTHKaM

CHCTeMbI, lJTO' 6y.n.eT c.n.eJIaHO B CJIe-

.n.YIOIIJ.eM

naparpa<p

e

.

§ 8.

PaCqeT

napaMeTpoB

CHCTeMW

PaCCMOTpHMnpOCTeHlIIHH

CJIylJaH,

Kor.n.a

conpOTHBJIeHHe

npe.n.cTaBJIHeTC060H

KOH.n.eHcaTop

nocToHHHOH

z(iro) =

i~C

.

OTcIO.n.a

no

<popMyJIaM

(19), (20)

npe.n.bI.n.ymerp

naparpa<pa

nOJIylJHM

• _ ro.L

(""VI'

"'-d.c)

]'

Fk-2<1>~k

( I

)2'

R2+

woL

lJ'i.gk

-

woC

t

1

(1)

,

LR

woL

IJ'

kgk -

ro;;c

F

k

= -4<1>kgk

woR

[(

1

)2J

2'

R2+

woLlJ'kgk-

woC

)

B

9TOM

CJIylJae

3HaK

K09<p<pHUlfeHTa .n.HCCHnaUHH

Fi.

npOTHBono-

JJOmeH

3HaKy

K03<p<pHUHeHTa >KeCTKOCTH

Fg,

lJTO

npH

CJIa60H

.n.HC-

CHnaU-HH B CTaTHlJeCKH HeYCTOHlJHBOM

06beKTe

MO>KeT

6bITb

HemeJIa-

TeJIbHbIM.

DPH

.n.oCTaTOlJHO

60JIblIIOH

.n.HCCHnaUH:H B CTaTHlJeCKH

.HeycTOHlJHBOM

06beKTe

HJIH

npH

OTpHUaTeJIbHOH

.n.HccHnaUHH

B CTa-

THlJeCKH YCTOHlJHBOM

06beI<Te

9TOT CJIylJaH MO>KeT rrpe.n.cTaBJIHTb

npaKTHlJeCKHH

HHTepec.

fl09T0My

KpaTKo

OCTafl9BHMcH

aa

ero

liCCJIe-

~08aHijH.

8~'

~~

,

PAClJET

nAPAMETPOB

CU<i:TIlMbI

.

Mai{CfiMYM

BeJIHlJJoIHbl

FE

no

C nOJIylJa.eTcH

npn

C =

[ro~L'1'

kgk

-w

o

R]-1

(2)

H

paBeH

F

tmax

=

CDkg/tQ,

r.n.e Q=

cooL/

R.

BeJIHlJHHa

Q HMeeT CMbICJI

'.n.

06

POTHOCTH

,KoJIe6aTeJIbHOrO

KOHTypa

C

pe30HaHcHoH

lJaCToToHroii)

HH.n.YKTHBHOCTbIO L H

conpOTHBJleHHeM

R.

DPR

.n.OCTaTOlJHO

60Jlb-

1II0M Q K03<p<PHuHeHT

ycfijjeHHHMO>KeT

CTaTb .n.OCTaTOlJHbIM

.n.JlH

_y,cTpaHeHHSl CTaTHlJeCKOH Hey,cTOHlJH'BOCTH

06beKTa.

.

,..

QnTHMaJIbHa51

eMKOCTb,

onpe.n.eJIHeMaH

<pOpMyJlOH

(2)',

Bo06m6

rOBOp5l,

3aBHCHT

OT

HOMepa

rapMoHHKR

k.

4T06bI

yCTp~HHT!?

3TY

3aBHcHMoCTb,

- OlJeBH.n.HO, Ha.n.o

nOJlOmRTb

'I'

kg/t

':"''I'~"

=

const.

Tor.n.a

BeJIHlJHHa

.

<Ilk'

F~

=

IJ'

k

'¥

ogoQ

OKa>KeTCH

nponopUHoHa.iIbHoH

ofHOlIIeHHlO

/t/'I'/t,

KOTopoe

C pOCTOM k

CHalJaJIa

B03paCTaeT,

a

3aTeM

CTpeMHTCH K

HyJlIO.

PaCCMOTpHM

cJiylJaH

H.n.eaJIbHO

npOBO.n.Hmero

KO>Kyxa.

Tor.n.a

Ha.n.o rrOJlO>KHTb R . 0, z

(jro)

=

OH,

CJIe.n.oBaTeJIbHO, .

,.,

FO

-'-

2

lDk

.

k-~'

BbIHrpbIlli

B K09<p<PHuHeHTe

yCHJleHH5I,

.n.ocTHraeMbIH

3a

ClJeT

npHMe-

HeHHH

cTa6HJlH3H

pYIOLUeH

060JIOlJKH,

paBeH

m = ;

qr

oI1

oQ =

~

[ 1-(

~:'

rJ

niQ

= [

1-{

~:

YJ

~

Q

i,"

w L

nt~ni

r.n.e Q] =

°R

1

,.

L] =

f.to

-w-

-COOTBeTCTBeHHO

Jl06pOTHOCTb

HHHn:yK-

THBHOCTb O.n.HHOlJHOH

KaTyllIKH.

OpR

Q]=

5,

1/6

= 20, ro/r

I.

= 1/2 HMeeM m

=75.

K09<p<pHUl1eHT

YCHJleHH5I

k

y

JlJIH k

-=

0 onpeJleJl5lE.'TCH B aTOM

CJlYlJae

TaK:

k

y

=

F~

= 4

~

(

~:

rQ1 =

100.

BeJlHlJHHa

K03<p<pHUHeHTa D-HCCHnaI.(HH

F;

npH

roo

= 10

7

l/ceK

paBHa

r

.)

2l Q

{.

I·

r (,.

).2J'

}

F~=-4

(

':

.6'

w~2(SQll1-\,:

.

-r

=-1,49·1O-

3

ceK.

DOCTOH,HHyIO

BpeMeHH

06pilTHOH

CBH3H onpe.n.eJIHM,

HCXO.n.fJ

-H3_t~e.

JlYIOllI,HX

co06pa)l{eUHH.

ECJiH

Il'epe.n.aTOlJHyfO epyHKU,HIO

anepHp,l.{ih

lJeCKOrO

3BeIia

- ,

W

-~

(p)

-pT+l

1

~

A.

r.

5yrKoBCKJJII

"

":::>:.

C

;fl,HM

. Ui,HK

HllSl

.

K03<P<PHUHeHT

yCHJIeHH51

6bIJI60JIbWHM,

Yo.n.OJl1KHO 6bITb

He3HallH~

TeJIbHO

60JIbWe

e,ll.HHHUbI. llJI51 nOJIy"eHH51 B03MOJKHO

60JIbWerO

nOJIOJKHTeJIbHOrb K03<P<PHUHeUTa ,ll.HCCHnaUHH JKeJIaTeJIbHO,

lIT06bI

6bIJIOX

l

> 0,

Yl

<

O.

\

;'CoBMew.a51YKa3aHHbie

Tpe60BaHH51, 6y.zteMHC}(O,ll.HTb

H3'

CJIe.z:r:ylO-

yCJIOBHH: 0 < X

o

=

Yo:"'"

1,

Xl>

0;

fit

<:

O.

npH

3TOM BbIpaJKe-

354

.

VHJOABJIEHHE

nJIA3MOA

Ir

JI;

,6'

pa3JIO»UITb B

pSl;n

no

C'reneHSlM

Tp,

TO

npH

MaJIbIX

Tp

nOJIyllHM

"

. k

y

.

W

(P)=Tpf-I

~ky(1-Tp)=Fo+FtP.

QTcIO.n.a k

y

=F", T =

~

F

1/

F

o

•

B

.n.aHHOM

CJIyllae

_ ' _

1,49.10-

3

-

-5

ky -

100

H T -

100

,-

1,49

·10

ceK.

n:

TaKHM

06pa30M,

cTa6HJIH3HpYlOlllaSl

060JIot/Ka

C

npocTpaHcTBeHH(j;~

BpeMeHHoH .n.HcnepcHeH

.n.aJKe

B

npocTeHweM

cJIyllae

MOJKeT

HMeTb

BbICOKHR K03<p<pHuHeHT YCHJIeHHSI' H MaJIyIO nocToSlHHYIO BpeMeHH.·

if

H(

t

--I

C

f

PHC; 5.8.1.

1I.aJIee

paccMoTpHM

CJIyllaR,

Kor,ll.a· z

(iro)

peaJIH3yeTcSI C nOMOIII.blO

cxeMbI,

H306paJKeHHOR Ha pHC. 5.8.1. llJI51 ynpOIII.eHHSI

3anHceH

BBe.n.eM

cJIe.n.YIOIII.He

0603HalleHHSI:

x(ro) = I +

~

Rez

(jro),

Y (ro) = Q'I'kgk +

~

Irnz

(iro),

x(roo}=x

o

'

y(roo)=Yo,

x'(roo)=x

lt

y'(roO)=Yl>--

_ R H (0) R _ a

1m

H

k

(t) I '

ex.- e k ,

1"-

at

1=0'

Tor,ll.a COfJIaCHO <popMyJIaM (8) § 7 H (9) § 7

nOJIyllaeM

F

k

(jro)

=

2<D

k

g

k

[ReH

k

(0) +

jro

a

Ima~k

(t)

11=0

+ ...

J.

B npHHSlTbIX

Bblwe

0603HalleHHSlx

ex.

-

Yo

Q

(3)

-

x~+Y~

,

P

_xl(y~-xg)-2xoYOYl

Q

(4)

=

(X:+y~)2

. •

Ha

Tp~60BaHHA

. <pH3HlIecKOR peaJIH3yeMocTH COnpOTHBJIeHHSI z

(jco)

,nOJIJKHO

BbIDOJIHSlTbCSI

ycJioBMe X

o

>0,

'a

,ll.mI' nOJIOJKHTe.nbHocTH

K03<p<pHUHeHTa yCHJIeHH$I H, CJIe.n.OBaTeJIbHO,

npaBoH

lIaCTH paBeHcTBa

(3) He06xo,ll.HMO, '1T06bI 6bIJIO.

Yo

>

O.

MaKCHMaJIbHOe

3HalleHIfe

K03<p-

'!>uUHeHTa yCHJIeHHSI,

QlIe~HAHoi

AOCT1iraeTC;SI

npH

x

p

=

lip'

LJT099J

..

,

" "

C

"

.'

PACtJET

nAPAMETP013CHctllMbI

355

(3) H (4) npH,HHMalOT

:BH,ll.

ex.

=.!d:...~

=

-Q

.It!.

=

~a

l!!.

2xo'

.

'2xg

xo·

YCJIOBHSlM

BnOJIHe

MOJKHO

y ,ll.OBJIe:rBopHTb,

ripH~eHSl51

cxeMy,

yKa3aHHylO

Ha

pHC. 5.8.1.

0603HallHB

pe30HaHcHylOlIacToTy

lIepe3

ro

p

;

Jl06pOTHOCTb

lIepe3

Q0

npHI

ro-ro

p

I

~rop,

MOJKHO

BOCnOJIb30BaTbCSI

H3BecTHoR

<pOpMyJIOH

,ll.JISI

conpOTHBJIeHHSlnapaJIJIeJIbHOrO

KOHTypa

Hnpe.n.cTaBHTb

z(jro)

B

BH.n.e

(

.)

.1

+

Ri

z }ro

=-.-

lroC

1+'

2

Q

o

( )

·loo

ro

-

rop

p

OTCIO.n.a

JIerKO HaRTH

R 2Q

o

(ro-ro

)

R

l

1 Oe

ro

p

,

'-,

p

;~

-Rez(}ro)=

1+

4~~(ro_rop)2

1m

z

(iro)

= -

roC-

I +

4~~

(ro-ro

)2

p

ro

p

rop

2

Q

o

(ro

_'co

)

1

R

l

,

nr

-

R

l

ro

'Q2

p

o

p

2

,

X = 1+

if

4Q~

(ro-ro

)2

YO=QTkgk-

R

I+~(roo-rop)

o

1+-

p

rop rop

2

4Q2

2

I--T(roo-ro

p

)

o

l

1 2Q R

rop

J2'

(5)

Yl

=

rog

RC

-,wp

R

[1

+

4~~

(ro

O

-ro

p

)2

rop

Y'Ka3aHHbIM BbIUIe

YCJIOBHSlM

HaH60JIee

BblrO,ll.HOrO peJKHMa

JIerlle

Bcero

y.n.OBJIeTBOpHTb, nOJIaraSl

ro

p

= roo' Tor,ll.a

1

Yl

=

_1

__

2Qo

Rl

X

o

= I +

~l,

Yo

=

Q'I'kgk-

rooRC'

ro~RC

(00

7['

113

yCJIOBH51

X

o

=

Yo

Haxo.n.HM

1 Q'I' 1 ' Roc

rooRC=

kgk-

-1[".

12*

356

ynPABJIEHHE

nC:;lOA

TIO.llCTa13HB 3TO

13bIpameHHe

B

4>oP~YJIY

(5),

H;iH/{eM

Yl

=

~o

[Q'l'

k8"k-

~

(1

+2Qo)-I

] •

TaKKaK

meJIaTeJIbHO,

HT06bI

.lh

~

0,

nOTpe6yeM,

tIT06f,J

Qo

MaKCHMaJIbHbIM.

I-h

yCJIOBHSl x

o

"""

1 ,CJIe,n:yeT, tITO

Roc""

R.

B

H

6bI.ITO .

yMeHb-

napa-

/R=1O

,

nOJIy-

TaKH~

06pa30M,

KoppeKTHpYIO~HH

KOHTyp

,n:OJImeH

HMeTb

BbICpKylO

4

06

PClT-

HOCTb

npH

OTHOCHTeJIbHO

He60JIbiliOM

pe30HaHCHOM

conppTHBJIeHHH.

3T0

,n:OCTHraeTCSl

3a

CtIeT

npHMeHeHHSlKOH.lleHCaTOpa

C ,n:OCTaTOtIHO

1

60JIbiliOH

eMKocTH

npH

MaJIOH

BeJIHtIHHe

HH,n:yKTHBHOCTH L H

IIIeHHSl

aKTHBHbIX

nOTepb

B

KOHType.

1

TIpoBe,n:eM tIHCJIOBylO

O~eH:KY

peaJIbHO

ocy~eCTBJIHeMbIX

MeTpOB.

TIyCTb,

KaK

n

13

npe,n:bI,n:y~eM

IipHMepe,

Ql

~5,

[/8=20,

'0/'1=1/2,

Ul

o

=IO-7

IlcelC.

Tor.n:a

npH

n=O,

Qo=20,

R

oc

HMeeM··

.

xo=Yo=I+

~1

=

11,

(Oo~c=2

[1-(;:

YJ

~

QJ-(

1+

~1

)-:-139,

Yl

=

~o

((Ookc-

2Qo

~1

) =

-2,61.10-

5

celC,

F

k

(jUl)

= 4

(~)2

~

Ql

-l

(1-

jUl.!tL

+...)

~

10[1

+

jUl,

2,6I.'IO-

o

f

rl

u

~ ~

TIpHBe,n:eHHbIe

'p

e3

YJIbTaTbI

CBH.lleTeJIbCTBYIOT 0

B03MomHOCTH

tIeHHH

nOJIOmf!TeJIbHOrO

K034>epH~HeHTa

,n:HCCHna~HH

npH

,n:OCTaTOtIHO

BbICOKOM

K03epepH~HeHTe

yCHJIeHHSl.

3a

CtIeTBbI60pa

K03epepH~HeH

TaB

11k

MomHO

06eCrretIHTb

npaKTHtIeCKH

o,n:HHaKOBbIe

K03epepH~HeHTbI

meCTKOCTH

H

,n:HCCH.na~HH

O.llHOBpeMeHHO

,n:JISl

3a,n:aHHoro

KOHetIHOro

tIHCJIa

rapMOHHK.

"

C}

fJl(lBA

6

ynPABJIEHHE

BOJIHOBbIMH

PACnPE~EJIEHHbIMH

fiPOlJ,ECCAMH HA

nPHMEPE

nOPIllHEBbIX

.

p.

KOMnPECCOPHbIX YCTAHOBOK

.-

r;

§

J.

npOOJIeMa

YMeHbweHHSI

nyJlbcao,H"

)l.aBJIeHUSI

H paCXO)l.a B

Tpy6onpOBO)l.HbIX

JIHHHSlX

COBpeMeHHoe

COCTOHHHe H

rrepcrreKTHBbI

pa3BHTHH

HeepTerrepepa-

~aTbIBalO~eH

H

HeepTexHMHtIeCKOH,

XHMHtIeCKOH

H

ra30BOH

rrpoMbI-

IUJleHHOCTH BbI3bIB31OT

He06xo.llHMOCTb

rrpHMe,HeHHH

13

Hapo,n:fioM

X03HHcr'Be

OOJIhiliOrO

tIHCJIa

MO~HhIX

KOMrrpeccopHhIX

MaIIIHH':

HaCTOH~ee

BpeMH

Tpy,n:HO

HaHnl

xHM.HtIecI\HfI

rrpo~ecc,

ocy~e

CTBJleHHe

KOToporo

He

Tpe6oBaJIO

OhI

BbIcOKoro

,n:aBJIeHHH, r

JIy6oKoro

OXJlam,n:eHHH HJlH

TpaHcrropnipOBKH

OOJIbiliOrO

KOJI,HtIeCTBa

ra30B.

Bce

3TO

,n:OCTHraeTCHC

rrOMOllJ,blOnpHMeHeHHH

HMeHHO

MO~HhiX

KOMnpeccopOB.

.'

3HatIeHHe

KOMrrpeccopHoro

X03HikTBa

Ha

npe,n:rrpHHTHHX

XHMdH

HeepTeXHMHH Tpy,n:HO

rrepeo~eHHTb.

CeoeCToHMoCTh

KOMrrpeMHpOBa-

HHH

ra3a

COCTaBJIHeT

cy~eCTBeHHylO

tIaCThCe6ecTOHMoCTH

OC:HOBHbIX

npo,n:yKTOB

COBpeMeHHOH

XHMHH.

KOMrrpeccopHbIe

M?IIIHH~I

HBJIHIOTCH

OCHOBHbIMli rrOTpeOHTeJIHMI.f

9J1CI<TpO:lHeprHH

Ha

HeepTeXHMHtIeCKHX

IIpe)l.IIpHHTHHX

•.

B

raSOBOH

fJPOMbIIIIJIeHHOCrH

KOMilpeccopHbIe

Ma·

liJHHbI

'fIBJlHlOTCH OCHOBHbIM

T.eXHOJIOrHqeqpIM

oOopy,n:OBaHHeM.

Bce

::lTO

npe,n:'bHBJlHeT

qpe3Bbl

U

aHl-:IO BbICOKHe

Tpe60BaHHH

K 3epepeI\:-

THBHOCTH

3KCIIJIyaTau,HH

KOMilpeccopHbIX

MaIIIHH.

Mem4Y

TeM

~OPOiliO

H3BeCTHbI

cJlytIaH

KpyIIHbIX

.aaapHH

B

KOMrrpeccopHhIX

u,exax

C

THme-

JlbIMH

rrOCJIe,n:CTBHHMH,

qe~OBeqeCKHMH

m.epTBaMH.

B

HaCTOSl~ee

BpeMH

KOl\mpeCCOpHbIe

MaIIIHHhI

'lame

Bc~ro

9Kcn.JJyaTHpYIOTCH

C

60JIbIIIHM

rrepepacxo,n:OM

9J1eKTp03HeprHH,

qeMy

B HeMaJIOH CTerreHH

crroco6cTBylOT

Bpe,n:HbIe .

IIYJIbCa~HH

,n:aBJIeHHH H

pacxo,n:a,

HMelO~f!e

MeCTO

Ha

BbIxo,n:e H3

KOMilpeccopa

H

3aTeM

IIepe,n:alO~HecH

B JIHHHIO.

KpoMe

Toro,

3.0

HaJIH4He

Bpe)J.HbIX

IIYJIbCa~I1H

3HatIHTeJIhHO

GHf!-

m.aeT

K.

II.

)I..

,KOMilpeccopHblX

yCTaHoBOK,

OHH C03,n:aIOT

rpoMaiHble

Hey)J.o6cTBa

IIpH

HaJIa,n:Ke H

9KcIIJlyaTarJ,HH

yCT3HOBOK,

a rro,n:tIac

HBJIHIOTCH

rrpHtIHHOH

KpyrrHbIx

aBapHH

H

KaTaCTpoep.

YtIHTbIBaH

BbIIIIeCKa3aHHoe,

MomHO,

yTBepm,n:aTb,

<ITO

HCCJIe,n:o-

BaHHH,

HailpaBJleHHh,Ie

Ha

IIOBhIllIeHHe

Ha,n:e.>KHOCTH

H

9KOHOMHtIeCK9H

3epepeKTHBHOCTH

3KCrrJIyaTau,HH

IIopIIIHeBhlX

KOMilpeccopOB,

HMeIOT

60JIhiliOe

3HatIeHHe

,n:JIH

HeqJTerrepepa6aThlBalO~eH

H

HeepTeXHMHqe-

CKOH,

XHMHtIeCKOH

H

raaOBOH

IIpOMbiIIIJleHHOCTH

HMoryT

paCCMaTpH-

BaTbCH

KaK

pe3epB

rrOBbIIIIeHHH

peHTa6eJIhHOCTH

Bcera'IIpOH3Bo,n:CTBa.

Бутковский А.Г. Методы управления системами с распределенными параметрами

Подождите немного. Документ загружается.