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lrJI.

11

360

TaK

KaI{

COrJIaCHO

(45.4)

HMeeM:

gIl

g1211'[

0]

=

[g12]

=

[1

].

II

g21

g22

1

g22

0

IIpH

aTOM

g2l

=1=

0 B

CHJIy

Toro,

lITO

MaTpH~a

G

no

npep;nOJIOmeHHIO

HeocoOafl.

IIepeHp;eM

K

pemeHHIO

aa.n;alIH.

)J;JIfl

BhInOJIHeHHfl

x

(t)

===

0

npH

'I'J'

+ e

~

t

~

2'1'J'

+ e

Heooxo.n;HMO

H .n;OCTaTOlIHO

(B

COOTBeTCTBHH

.C

ypaBHeHHeM

(45.1»,

lITOOhI

BhInOJIHJIJIHCh

YCJIOBHfl:

gIl

Xl

(t -

'I'J')

+ X

2

(t -

'I'J')

+ u

(t)

= 0

('I'J'

+ e

~

t

~

2'1'J'

+ e),

(45.5)

g21Xl

(t

-

'I'J')

= 0

('I'J'

+e

~

t

~

2'1'J'

+e),

(45.6)

X

('I'J'

+e) =

O.

(45.7)

PaBeHCTBO

(45.6)

paBHOCHJIhHO

TOMy,

lITO

Xl

(t)

= 0

npH

e

~

t

~

'I'J'

+

e.

(45.8)

Ho

Tor.n;a

Ha

(45.5)

CJIe.n;yeT,

lITO

Ha

npoMemyTKe

'I'J'

+

e·~

t

~

2'1'J'

+e

if

~

f.

yrrpaBJIeHHe

u (t)

orrpe.n;eJIfleTCJI

paBeHCTBOM

!!'

!;i

u (t) = - X

2

(t -

'I'J').

(45.9)

,

II'

TaKHMoopaaOM,

.n;JIfl

pemeHHflaa,ll;alIH

45.1

Ha,ll;JIemHT

y,ll;oBJIeTBOpIITh

J"

II·

~:

;

Terreph

,ll;ByM

YCJIOBIIJIM:

yCJIOBIIIO

(45.8) H

yCJIOBIIIO

X

2

('I'J'

+

e)

=

O.

".

OOCy.n;IIM

CHalIaJIa

YCJIOBlIe

(45.8).

BhI)IOJIHIITh

ero

MomHO

Torp;a

:11

1

i;'il'

H

TOJIhKO

Torp;a,

Kor.n;a,

C O,ll;HOH

CTOpOHhI,

BhIOOPOM

u

(t)

rrplI

0

~

t < e

"tUI

y,ll;aCTCfl

rrpHBeCTH

OO'heKT

B

COCTOflHHe

I:

oIti

.":'

Xl

(e)

= 0, (45.10)

a C

,ll;pyroH,

- y.n;aC'1'Cfl

aa

ClIeT

BhIoopa

yrrpaBJIeHHfl

u

(t)

rrpH

e

~

t <

< e +

'I'J'

yp;epmlIBaTh

OO'heKT

B

COCTOflHHH

Xl

(t) . 0

BrrJIOTh

,ll;0

MOMeHTa

t = e +

'I'J'.

Ho

rrOCJIep;Hee

TpeOOBaHlIe

oaHalIaeT,

lITO

CJIep;yeT

BhIrrOJIHHTh

yCJIOBHe

~l

(t)

= a

12

x

2

(t) +

gIl

Xl

(t

....,.-

'I'J')

+

g12X2

(t

-

'I'J')

+

u,

(t)

==0,

(45.11)

KOTopoe

H

orrpe,ll;eJIfleT

yrrpaBJIeHHe

u

(t)

Ha

OTpe3Ke

e

~

t <e

+-tt.

ECJIII

(45.8)

BhIrrOJIHfleTCJI,

TO .n;JIfl

Koop,ll;HHaThI

X

2

(t)

IIMeeM

rrpH

e

~

t

~

e +

-tt

ypaBHeHlIe

X

2

(t) = a

22

x

2

(t)

+

g2l

Xl

(t -

-tt).

(45.12)

RaK

BII.n;HO

lIa

rrplIBe.n;eHHhIX

paccym,ll;eHIiIH,

MhI

.n;OJImHhI

Terreph

rro-

CTapaThCfl

TaK

paCrrOpJIp;IIThCJI

yrrpaBJIeHHeM

u (t)

rrpH

0

~

t <

e,

lITOOhI

BhIIIOJIHIiITh

yCJIOBHJI

(45.7)

II

(45.10).

OOCYP;IIM

HX.

C

aTOH

c:

§ 45]

aA}l;A

lIA

OB

YCIIOIWEHlIl1

CUCTEM

C

nOCJIE}l;E1WTBIIEM

361

~eJIhIO

aarrHmeM

BhIpameHlIe

,ll;JIJI

Xl

(e)

H X

2

(-tt

+

e),

rrOJIhaYflCh

tPop-

MyJIOH

ROIlIH.

lICXO,ll;fl

HS

(45.1),

Haxop;IIM,

lITO

Xl

(e)

eCTh

rrepBaJI

KOOp,ll;IIHaTa

BeKTopa

X

(e),

r.n;e

e

. X

(e)

= X

[e,

0]

XO

(0) +

~

X

[e,

't']

Gxo

('I:'

-

'I'J')

d't'

+

o

e

+~X[e,'tlbu('t')dr.

(45.13)

o

3.n;ech

X

[t,

't] -

tPYH,ll;aMeHTaJIhHafl

MaTpH~a

CHCTeMhI

X =

Ax.

BeJIHlIlIHa

X

2

(-tt

+

e)

BhItIliICJIJIeTCJI

CJIe,ll;YIO~IiIM

rryTeM.

lIa

(45.12)

HaXO)l;lIM:

e

X2

(e

+

'I'J')

= e

al2e

X2

('I'J')

+

~

e

a22

(e+&-l;)

g2lXl

(~

-

-tt)

dt;,.

(45.14)

{}

l1a

Toro

me

COOTHOmeHIIJI

(45.12)

IIMeeM

CJIep;YIO~ee

paBeHCTBO

,lJ;JIJI

X

2

('I'J'):

-&

X2

(-tt)

=

X2

(e)

ea,,({}-e)

+

~

ea,,(~-l:)

g2lXO

(t;,

-

'I'J')d~.

(45.15)

e

HaKOHe~,

HCXO.n;JI

lIa

(45.1),

rrOJIylIaeM:

X2

(e)

= X21

(e,

0)

x~

(0) + X

22

(e,

0)

x~

(0) +

e

+\

[X2l

(e,

~)

g11X~

(~-

'I'J')

+

v

'0

•

+:li22

(e,

t;,)

g2lX~

(~-

'I'J')]

dt;,

+

~

X2l

(e,

t;,)

u

(t;,)

dt;,

(45.16)

o

U

Xl

(t) =

Xu

(t,

0)

X~

(0) +X12

(t,

0)

X~

(0) +

t

+

~

[Xll

(t,

t;,)

g11X~

(0)

+X12 (t,

~)

g2lX~

(0).1

dt;,.+

o

+

Xll

(t,

t;,)

u

(~)

ds

(O~t~e).

(45.17)

.J

o

IIop;cTaBJIJIfl

B (45.14)

BeJIliIlIlIHhI

X

2

(it) (45.15)

II

Xl

(t)

(45.17), a

TaK-

me

YlIUThIBaJI

(45.16),

orrpep;eJIHeM

nOJIHOe

BhIIJameHlIe

P;JIH X

2

(e

+

-tt).

352

I

, .

~".;".'

".

."

'.

.

...

":"

,.;.'

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::

.

:.,

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..

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C?

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C

OCOBEHHOCTflMI1

[rJl.

11

lloJIarafl

Teneph,

qTO

Xl

(e)

= 0, X

2

(1't

+B) = 0, B

pl:laYJIbTaTe

He-

CJIOiKHbIX

npe06pa<lOBaHI1H

nOJIyqI1M

CJIe,n;yroDJ;He

,n;Ba

ypaBHeHHB

,n;.IJl

<PYHRJJ;HI1

u (t):

E

Ci

= \

Mi)

fe,

-r]

u

(-r)

a-r

(i

=

1,2),

(45.18)

15

r,n;e

h(l)

fe,

-r]

03HaqaeT

nepByIO

Roop,D;I1HaTy

BeRTopa

X

[e,

-r]

b,

a <pYHIr-

E

IJ;IH! h(2)

fe,

-r]

= e

a

,'&

X

2

l

(e,

-r)

+

~

eadE-l:;)

g21

X

n

(~,

-r)

a~;

Cl

OaHaqaeT

"t

nepBylo

Roop,n;HHaTy

BeRTopa

I

-

[X

[e,

0] X

O

(0) +

~

X

[e,~]

Gxo(~-1't)a~J,

o

a qI1CJIO

C2 = e

a

.,(}

[X2l

(e,

0)

x~

(0) +

X22

(e,

0)

x~

(0)] +

j~i

•

'lli

'

+e

a

,,(}

~

[X21

(e,

~)

gllX~

(~-1't)

+

X22

(e,

~)

g2lX~

(~-1't)]

a~

+

ml

o

~I

'j;!j

.&

,

'~"

,

!'"

~i,

+

~

ea,,('&-tE-l:;)

g21X~

(~-

'fr)

a~

+

•

'I~

I

:i

'::::

E

I

,1~

I

~~,

+

~

ea,,(E-t)

[xn

(~,

0)

x~

(0) +

X12

(~,

0)

x~

(0)]

g2la~+

..

~II

\

()

"III

I

l:;

.~t,

,

+

~

ea,,(E-l::)

g2l

{~

[xn

(~,

A)

(gnx~

(A.

-1't)

+,.

° 0

+

x~

(A

-1't»

+

X12

(~,

A)

g21X~

(A

-1't)]

all,}

a~.

IhaR,

npI1Xo,n;HM

R CJIe,n;yIODJ;eMy BbIBO,n;y:

npI1

,n;aHHOM

HaqaJIbHOM

COCTOflHHI1 X

O

(t)

(-

1't

<:

t

-<

0)

aa,n;aqa

paapemHMa

Tor,n;a H TOJIbRO

Tor,n;a,

Ror,n;a

paapemHMa

np06JIeMa

MOMeHTOB (45.18).

llpH

3TOM

yn-

paBJIeHHe

u

(t)

Ha

oTpeaRe

0

-<

t < e onpe,n;eJIJIeTCH

Ha

ypaBHeHHH

(45.18), a

npH

e

-<

t

-<

e +

21't

<PYHRIJ;HfI u

(t)

Haxo,n;HTCH ¥Ia

yCJIOBHH

(45.9), (45.11).

TaRHM

o6paaoM,

Bonpoc

0

pemeHHH

3a,n;aqn

45.1

CBeJICfI R

paapemeHHIO

ypaBHeHHH

(45.18).

RaR

HaM

H3BeCTHO,

y,n;OB-

JIeTBOpHTb

TaRHM

ypaBHeHHHM

npH

JII06bIX

qHCJIaX

ClM

C

2

MOiKHO B TOM

H TOJIbRO TOM

CJIyqae,

Ror,n;a

<PYHRJJ;HliI

hill

fe,

1:]

H h(2)

fe,

1:]

JIHHeHHO

HeaaBHCHMhI.

MOiKHonpoBepHTb,

qTO

,n;JIH

TaROH

JIHHeHHOH

HeaaBHCH-

MOCTH <PYHRJJ;HH

h(l)

fe,

1:]

H h(2)

fe,

-r]

BO BCHROM

CJIyllae

,n;OCTaTOqHO,

qTo6bI

BhIllOJIHflJIOCb

YCJIOBIH~

g21-a21&',&

=f=

o.

Ha

npOBepRe

3Toro

yTBep,H,n;eHHfI

MhI

OCTaHaBJIHBaTbCa

lie

6YlW

M

•

Q\

~::.~.~;

145]

3A.D:A

LIA

OD

YCrrOI\OEHHII

CI1CTEM C

nOCJIE.D:EHC'fBHEM

363

II

P

1I

M e ,'{ a H

II

e 45.1.

06CYAHM

em;e

paa

OIIHcaHHoe

TOJIblW

'ITO

perneHHe

31lIl;a'IlI,

ll;eJIaJI

aK~eHT

Ha

ero

reOMeTpH'IeCKOM

CMblCJIe.

PaCCMOTpHM

TpeXMepnoe

IIpOCTpaHCTBO

{t,

Xl'

x~}.

HaM

Tpe6oBaJIOCb,

'ITo6b1

IIpH

'It

+e

~

t

~

2'lt

+e

KpHBaJI

X (t)

mJIa

IIO

OCll

t.

IIPll

OTCyTCTBHll

yIIpaBJIeHllJI

(T.

e.

rrpll

U (t)

=0

0)

TO'IKa

X

(t)

c6paCblBaJIaCb

6bl

C OCII t B

narrpaBJIeHHll

BeKTopa

CKOpOCTII

xu=O

(t)=

=

Gx

(t

-

'fr).

iha

CI<OpOCTb

p;OJImna

raCHTbCJI

BeKTopOM

bu

(t).

Ho

aTO

BoaMom-

HO

Torll;a

II

TOJIbKO

Torll;a,

Kor,n:a

BeKTopbI

Gx

(t

-

'fr)

II

b

KOJIJIllHeapHbI.

IIocJIell;-

Hee

06CTOJITeJIbCTBO

Tpe6yeT

B CBOIO

Oqepep;b,

'lTo6b1

KpllBaJI

x

(t

-

'fr)

('fr

+

+e < t <

2'fr

+e)

JIemaJIlI,

B

IIJIOClWCTH:Tt,

IIpOXOIl;~eii

'Iepea

OCb t H

BeKTop

e

.z;

:C//20{t)",Am(t) +6'.2'(t

-#)

,

I

i

t

I

j

I

t

!Juft)

1

~

PHC.

45.1.

TaKOH,

'ITO

Ge

= b

(pHC.

45.1).

CoxpaHeHHJI

me

KpHBOii:

X

(t)

IIpll

e < t <

'fr

+e

B

IIJIOCROCTll

:Tt

BO

BCJIKOM

CJIy'Iae

MomHO

Ao611TbCJI,

eCJIll

BeKTopbl

b

II

e

He

KOJIJIHHeapHbl.

TaKHM

06paaOM,

IIpH

'fr

+-

e

~

t <

2'1'l:

+e H e

~

t

~

'fr

+

+ e

yIIpaBJIeHlle

u (t)

OIIpell;eJIJIeTCJI

113

YCJIOBIlJI

coxpaHeHHJI

KpHBOH

X

(t)'

na

OCH

t H B

IIJIOCKOCTH

:Tt

COOTBeTCTBeHHO.

)J;aJIee

Il;JIJI

pemeHllJI

aall;a'IH

OCTaeTCJI

JIllmb

paCIIOpJIp;HTbCJI

yIIpaBJIeHlleM

U (t)

IIpH

0

~

t

~

e

TaR,

'ITo6b1

06eCIIe'IHTb

IIpH-

Bell;eHHe

KpHBOH

X (t) K

MOMenTy

t = e

na

IIJIOCI<OCTb:Tt

II

l,

MOMeHTy

t =

'fr

+

e-

B

TO'IKy

x =

O.

IIocJIell;HHe

YCJIOBllJI

BbIpamaIOTCJI

paBenCTBaMH

(45.18).

AHaJIOrHqHOe

IIOCJIell;OBaTeJIbHOe

coxpaHenlle

KpllBOii:

X (t) B

JIllHeHnblx

IIOAIIpO-

CTpaHCTBax,

IIOll;06Hb1x

IIJIOckOCTH

:Tt,

6yp;eT

corrpOBO;Rll;aTb

pemeHHe

aall;a'IH

06

ynpaBJIeHHH

CHCT9MOH

C

IIOeJIep;eHCTBHeM

H B

CJIy'IaJIX

60JIee

BblCO}(OH

(HemeJIH

:!)

paaMepHOCTH

Bel,Topa

x (t).

IIpH

aTOM

MomeT

IIOTpe6oBaTbCJI

yBeJIH'IeHHe

Il;JIHHbl

y'IaCTKa

t{3

na

COOTBeTcTByrom;ee

'lHCJIO

llHTepBaJIOB

'fr.

.

3aRJIIOqaH

naparpa<p.

OTMeTHM,

qTO

paCCMOTpeHHaH

BliIWe

aa,n;aqa

45.1

CTaBHT,

no

CYTH

,u;eJIa,

Bonpoc

0

BoaMOiKROCTH

ycnoRolITb

CHCTO-

My

C

aanaa,n;hIBaHHeM.

OTBeT

Ra

3'1'0'1'

Bonpoc

ORaaaJICfI

B

npHRJJ;Hne

IlOJIOiKHTeJIbRbIM,

o,n;HaKo

Ha

npHBe,n;eRHbIX

HaMH

paccy,H,n;eHHH

BH,n;RO,

f':'

\_.':'.'i

....:...:....:

JIUHEfiHbIE

CHCTEMbI

C

OCOBEHHOCTHMH

lrJI.

11

364

qTO

3ap;aqa

AonycRaeT

HeOAH03HaqHOe

perneHliIe.

BbIAeJIeHJile

RaROH-JIliI-

00

OAHOH

<PYHRIJ;liIli1 U

O

(t)

1iI3MHOmeCTBa

perneHliIH3aAaqlil

45.1

MomHO

ocym;ecTBliITb,

HaRJIaAbIBaH

Ha

<PYHRIJ;liIIO U (t) eIIJ;e

TpeOOBaHJile

onTliI-

MaJIbHOCTliI

B

CMbICJIe,

HanplilMep,

MliIHJilMyMa

RaROH-JIliIOO

HOpMbI

<PYHR-

IJ;1iI1i1

U (t). B

pernemm

TaROH

HOBOH

3aAaqlil

CJIeAyeT

onlilpaTbCH

Ha

pe3YJIbTaTbI

npeAbIAym;liIx

rJIaB.

IIplil

aTOM,

RaR

JiI

B

APyrlilx

nOAOOHbIX

CJIyqaHX,

HalilMeHbmlile

TPYA-

HOCTJiI B

B~liICJIeHliIHX

B03HliIRaIOT

TorAa,

RorAa

MliIHliIMJil3Ji1pyeMaH

Be-

JIJilqlilHa

%~lJ.+E

[U]

BbIOlilpaeTCH

B

BliIAe

2&+.

%~&+dU

('t)] =

~

II

U ('t)

li

2

d't =

min.

(45.19)

°

l

)J;JIH

cBeAeHliIH

3aAaqJil

(45.18), (45.19) R

npooJIeMe

MOMeHTOB

I

(45.18) C

AOnOJIHliITeJIbHbIM

YCJIOBJileM B

nplilB~HoH

AJIH

Hac

<popMe,

J

,

I

HaAJIemlilT

TorAa

CBeCTJiI YCJIOBJile

MJilHliIMYMa

BeJIliIqlilHbI

(45.19) R

yCJIO-

BliIIO

MliIHliIMYMa

HeRoTopoH

BeJIliIqlilHbI

"Ii

%.

[u ('t)] =

min,

(45.20)

,

-:w

!

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TeMII

paccyJI<,a;eHIIHMH,

ROTOpbIe

yn\0

BCTpelJaJIIICb

HaM

B § 33

(CM.

CTp.

280).

Ho

Tor,a;a,

RaR

H B § 33,

()

MbI

rrplI,a;eM

R BL1BO,a;y,

lIT

0

nCROMaH

onepan;nH

<P

=

CPi,

DbIIIOJIHeUHaH

Ha

H,a;eaJIbHOM

ClIrHaJIe

y

('t'),

,a;OJIJ-RHa

,a;aBaTb

mO'lHOe

3HalJeHIIe

Xi

(t).

lIHalJe

rOBopH,

,a;JIH

3TOH

onepan;HII

,a;OJImHO

6bITb

W

x

=

O.

TOJIbRO

Te

:

..

;'::.: .....

::~:

......

:.:.;;

'lP"

.•

>:;,-

..

.,

t'

"

",",'

••

~.'..

-.:::

:,,:w.

'R:'-,,::;

",:,:

;.:"

.,

••

0 t

J

"

0,

r

',",_"-i

, '"

o

368

JIMHEJ!1HhIE

CMCTEMbI

C

OCOBEHHOCTHMIi

[rJI,11

onepaD;1i1i

<p,

ROTOpbIe

YP;OBJIeTBOpHIOT

aToMy

yCJIOBliIO,

Ii

paCCMaTpll-

BaIOTCH

p;aJIbIIIe

B

aTOM

naparpa.pe.

Ho

Torp;a

OIIIlIORa

W

oyp;eT

CJIy-

u

...

... ... ...

(I)

'lallHOll

BeJIlI'lllHOll,

COCTaBJIeHHOll

1I3

ee

peaJI1I3aD;1I11

W~,

P;JIH

ROTO-

pbIX

MomHO

nOp;C'lllTaTb

pacnpep;eJIeHlle

BepOHTHOCTeH

(no

RpaHHeH

Mepe

B

npllHD;lIne),

lICXOP;H

1I3

3ap;aHHoro

pacnpep;eJIeHllH

BepOHTHO-

CTeH

P;JIH

nOMeXlI

L\

(1:).

Op;HaRO

npOCTeHIIIeH

MepOH,

ROTopaH

yP;OB-

JIeTBOpllTeJIbHO

OD;eHlIBaeT

BeJInqllHY

nop;OOHOH

CJIyqaHHoH

OIIIlIORlI

W,

HBJIHeTCH

MaTeMaTlI'leCROe

omllp;aHlIe

ee

RBap;pa;ra

M{w

2

}.

TaRHM

oopa30M,

npllXop;lIM

R

CJIep;yIOID;eH

3ap;aqe.

3a

p;

aq

a 46.1.

Haumu

OnmUMa.lf,bliYlO onepa/{ulO

<p~[t,

z

(1:)],

pa6omalOUfYlO

lia

UOea.lf,bliOM

cumaM

Y

(1:)

6ea

oUlU6Jl;u u

6occmaliaMU-

6alOUfYlO

~oopouliamy

Xi (t)

no

pea.lf,bliOlrtY CU8lia.lf,y z

(1:)

Xi (t) =

<P~

[t, Z

(1:)]

+W

C

liaUMeliblUeu

608MO:JICliOU

noepelUliocmblO

M {w

2

}

=

min.

il

IIepeHp;eM

R

peIIIeHlIIO

aToH

3ap;aqll.

3anllIIIeM

CHaqaJIa

HBHoe

BbI-

pameHlIe

P;JIH

ClIrHaJIa

Z

(1:),

nOJIaraH,

RaR

OOblqHO,

1I3BeCTHbIMlI

P;JIH

ClICTeMbI

(46.2)

.pYHp;aMeHTaJIbHYIO

MaTpllD;y

Z

[t,

1:1

1I

lIMnyJIbcHyIO

nepexoP;HYIO

.pYHRD;lIIO

H

[t,

1:]

no

B03p;eHcTBlIIO

X

(1:).

B

paCCMaTpll-

t.,

BaeMOM

CRaJIHpHOM

CJIyqae

ClIrHaJIa

Z

(1:)

aTO

OYP;YT

COOTBeTCTBeHHO

CRaJIHpHaH

nepeMeHHaH

Z (t,

1:)

1I

BeRTop-cTpoRa

hi

[t,

1:]

=

,

f:

'ii

.'

III

..

'

=

{hI [t,

1:],

...

, h

n

[t,

1:]}

= Z (t,

1:)

g'

(1:).

lhaR,

3anllCbIBaeM

no

:0'"

mopMyJIe

ROlllI:

.w'

t

JO

"

f

o

.,

I

z('t')=z('r,t-'I'})zo+

~

h'('t',~)x(~)d~+

-"1-

'IM

t-&

'~II

"

+

~

z(t,~)

L\(~)d~.

(46.7)

t-&

B

COOTBeTCTBMlI

C

HalleH

p;oroBopeHHoCTbIO

aTO 1I

p;pyrMe

BbIpameHMH,

cop;epmaID;Me

c.If,yttaUlible

.pYHRD;lIM,'

YCJIOBlIMCH

nOHlIMaTb

RaR

COOT-

HOlleHMH,

CBH3blBaIOID;Me

cooTBeTcTByIOID;Me

P;Pyr

p;pyry

peaJIM3aD;MlI

!;

ClTlIX

.pyHRD;MH.

MbI,

op;HaRO,

He

nOp;qepRlIBaeM

B

p;aJIbHeHIIIeM

yRa3aH-

t

Hoe

OOCTOHTeJlbCTBO

lIHp;eRCOM

<il,

qTOObI

coxpaHllTb

P;JIH

paCCMaTpH-

I

BaeMbIX

COOTHOIIIeHlIH

CMbICJI

paBeHCTB,

BepHbIX

P;JIH

JIIOOOH

peaJI1I3a-

~'

L'

D;HlI.

B

p;aJI);,HeHIIIeM

Hac

OYP;YT

lIHTepeCOBaTb

JIlIIIIb

p;Ba

nocJIep;HlIx

'~

.

t·

CJIaraeMbIX

B

npaBOH

qaCTlI

paBeHCTBa

(46.7).

,n;eHcTBMTeJIbHO,

oqe-

,,"

•

..

BMP;HO,

p;OCTaTOqHO

HaHTlI

onepaD;MIO

<po

[q

(1:)],

ROTopaH

OnTlIMaJIbHO

BbIp;eJIHeT

BeJIMqMHY

Xi

(t}1I3

ClIrHaJIa

q

(1:)

= Z

(1:)

- Z

(1:,

t -

'I'})

zO,

1I00

Tor,lla

MbI

MomeM

npOCTO

nOJIOmlITb:

I

<P~

[t, z

(1:)]

=

<p°[z

(1:)

- z

(1:,

t -

'I'})

ZO]

(46.8)

§

q6J

HABJIIO,D;EHlIE

B CJIY<IAJ.il:HbIX

OBCTOHTEJIbCTBAX

369

(CM.

aHaJIOrnqHYIO

clITyaD;lIIO,

paCCMoTpeHHyIO

B § 34

(CTp.

290)

npliI

peIIIeHlIliI

3ap;aqlI

0

HaOJIIOp;eHHlI).

IlTaR,

HaM

Hap;o

HaHTliI

onepaD;liIIO

<pO

[q

(1:)],

ROTopaH

Ha

CliIrHaJIe

" "

q('t') =

~

h'[t,~]x(~)d~+

~

z('t',~)L\(~)d~

(46.9)

t

----&

t----&

p;aeT

OIIIliIORY

W = Xi (t) -

<po

[q

(1:)]

C

HaMMeHbIIIlIM

B03MomHbIM

cpep;HHM

3HaqeHlIeM

RBap;paTa

M{w

2

}

ClTOH

OIIIlIORlI.

OTCIOp;a

BbITeRaeT,

'ITO

onepaD;lIH

<po

p;OJImHa

yp;OBJIeTBOpHTb

CJIep;YIOID;lIM

p;ByM

YCJJOBlIHM:

"

<po[

~

h'[t,~lX(~)d~]=xdt)

(46.10)

t-&

liI

"

M{w

2

}=M{[qJo[

~

z(t,~)L\(~)d~JJ2}=min,

(46.11)

t-~

npliIqeM

nepBoe

1I3

aTliIX

YCJIOBlIH

03HaqaeT,

'ITO,

RaR

MbI

yme

yCJIOBlI-

JIliICb,

onepaD;HH

<pOi

(46.8) B

OTCyTcTBHe

nOMeXH

L\

(1:)

p;aeT

mOttliOe

3Ha-

qeHlle

Xi (t).

HeM3BeCTHYIO

onepaD;HIO

<pO

[q

1

oyp;eM

lICRaTb

B

BlIp;e

t

<pO

[q('t')] =

~

q(t)dV('t'),

(46.12)

t-&

rp;e

V

(1:)

-

HCROMaH

.pYHRD;liIH

C oepaliUttelilibl.M--UiJlrtelie1iUeM.

3p;ecb,

npaBp;a,

B03Hl1RaeT

HeOOJIbIIIOe

Hep;opa3yMeHHe:

peaJI1I3aD;1I1I

q"'(

1:)

CMrHaJIa

q

(1:)

(46Jl)

HBJIHIOTCH

.pYHRD;liIHMliI

Jl;ycO'lliO-lienpepbl6liblMU,

nOCROJIbRY

peaJI1I3aD;HlI

nOMeXliI

L\'"

(1:)

HBJIHIOTCH

.pYHRD;liIHMlI

liIM-

rrYJIbCHbIMlI.

Ho

onepaD;MH

(46.12),

CTporo

rOBopH,

onpep;eJIeHa

.If,UlUb

a.lf,R,

lienpepbUJliblX

.pYHRD;liIH

q ('t'),

liI

B

CJIyqae

COBnap;eHliIH

TOqeR

pa3-

pbIBa

y q

(1:)

H V

(1:)

nOJIyqaIOTCH

OCJIOmHeHMH.

Op;HaRo

MbI

MOi!{eM

npeHeOpeqb

3THM

OCJIOmHeHHeM

no

CJIep;yIOID;eH

npHqHHe.

B

pe3yJIb-

TaTe

peIIIeHlIH

3ap;a~

MbI

nOJIyqlIM

.pYHRD;MIO

V

(1:)

Bcero

C OP;HOH

TOq-

ROH

pa3pbIBa.

Torp;a

MomHO

nOp;CqllTaTb,

'ITO

BepOllTHOCTb

COBnap;eHHH

TOqeR

pa3pbIBa

y V

(1:)

H

q'"

(1:)

paBHa

HyJIIO.

RpoMe

Toro,

BCJIep;-

cTBHe

MaJIOCTlI

e

p;ame

H

COBnap;eHHe

pa3pbIBOB

V

(1:)

H q

(-t)

He

npliI-

Bep;eT

R

OOJIbllMM

HenplIHTHOCTHM,

TaR

RaR

Torp;a

B

RaqeC)'Be

q

(1:)

MomHO

BbIOMpaTb

oe3

cyrn;ecTBeHHoro

BJIHHHHH

Ha

pe3YJIbTaT

H q

(1:

+0)

H q

(1:

- 0).

OrpaHHqMMCH

3p;eCb

TOJIbRO

TaRHM

He

BnOJIae

CTporHM,

HO .

npaBp;onop;OOHbIM

paccymp;eHHeM,

OOOCHOllbIBaIOID;HM

B03MOi!{-

HOCTb

BbIoopa

onepaD;MH

<po

B

BliIp;e

(46.12).

3aMeTHM

em;e,

'ITO

qaCTO

BMeCTO

nOMeXH

L\

(1:)

(46.3)

paCCMaTpHBaIOT

<<HenpepbIBHyIO»

nOMexy,

HBJIHIOm;YIOCH

npep;eJIbHbIM

CJIyqaeM

P;JIlI

(46.3)

nplI

e

-+

0

liI

').,

-+

00,

2

'taR

'ITO

BeJIHqHHa

e

').,

OCTaeTCH

ROHetIHOH

(CM.,

HanpHMep,

[228]).

"·":'7:

..

:......-..

:.:

">,

-:"

••.•..•

):,.:.,J.

:\':

370

n,'

V':',::i

'",,,..'

JIIIHE1tHbIE

CMCTEMbI

G

OCOBEHHOCTflMH

[rJI.

11

IIoJIy'IaIOnweCH

Torp;a

peaJIHSa~HII

ZOO

(,;)

MomHO

C'IIITaTb

HenpepbIB-

HbIMII,

HO

060CHOBaHlIe

pa60TbI

C

TaKOB

«npep;eJIbHOH»

nOMeXOB

t1

(,;)

Tpe6yeT

saTO

p;OCTaTO'IHO

CJIOmHOrO

MaTeMaTII'IeCKoro

annapaTa,

II

n03TOMY

OT

TaKoro

nyTII

MbI

ap;eCb

OTKaSaJIIICb.

IhaK,

6yp;eM

IICKaTb

onepa~IIIO

cpo

B

BIIp;e

(46.12).

Torp;a

yCJIOBlIe

(46.10)

nplIMeT

BlIP;

t

't

~

[

~

h'

[t,

~]

x

(~)

dt]

dV

('r)

=

xd

t

).

(46.1?)

t-& t-&

¥qIITbIBaH,

'ITO

X

(~)

= X

[~,

t]

X (t),

h'

[';,

~]

= z

(,;,

~)g'W

II

MeHHH

nopHp;oK

IIHTerplIpoBaHlIH

B

JIeBOB

qaCTlI

paBeHCTBa

(46.13),

nOJIyqlIM

lIS

3Toro

paBeHCTBa

CJIep;YIO~ee

ypaBHeHlIe

P;JIH

IICKOMOB

4JYHK~1I1I

V (,;):

t t

[

~

g'

(~)

X

[~,

t]

[~z

(';,

~)

dV ('t)]

d~

] x (t) = xd

t

),

t-&

l;

I

KOTopoe

p;OJImHO

yp;OBJIeTBOpHTbCH

nplI

JII060M

SHaqeH1I1I

X (t).

OTcIOp;a

SaKJIIOqaeM,

'ITO

4JYHK~IIH

V (,;)

p;OJImHa

yP;OBJIeTBOpHTb

CJIep;YIO~eB

CIICTeMe

ypaBHeHHB:

..

, t t

::1

~:

~

hU)(s)

[~z

(t,

s)dV(1:)

Jd~

=

CJ

(j

= 1,

...

,

n),

(46.14)

...

t-&

l;

..

".,

rp;e

Ct

= 1 H

cJ

= °

nplI

j

=F

i, CHMBOJI

hU)

(s)

OSHaqaeT

j-IO

KOMno-

,,",

HeHTy

BeKTopa-cTpoKII

g'

(s) X

[~,

t].

OM

Tenepb

npeoopasyeM

BbIpameHlIe

(46.11). ,IJ;JIH

3Toro

HaM

Hap;o

If!

BbI'IHCJIHTb

MaTeMaTlIqeCKOe

oamp;aHHe

OT

KBap;paTa

HHTerpaJIa

t

't

1=

~

[~

z(t,

~)~(~)d~]dV(1:).

t-&

1-&

IIpe06pasyeM

3TOT

MHTerpaJI,

nOMeHHB

CHOBa

nopHp;OK

HHTerpH-

pOBaHHH.

IIOJIyqHM:

t t

1=

~

t1(~)[~z(t,

~)dV(t)]d~.

1-&

l;

,IJ;JIH

p;aJIbHeHmero

yp;06HO

BBeCTH

HOByIO

4JYHK~HIO

U ('t),

onpe-

p;eJIeHHyIO

MS

yCJIOBHH

I

u (s) =

~

z

(1:,~)

dV

(1:).

(46.15)

l;

,-~~,:,.",:,,,:.~_-,:--,,,,:,,,_._,,,,

't'·

,

,.~,'

'>.:.'

. .

...

<,':,

,·:·::::·::::·:<::,;:·.·:~:;n!~

C)

§ 46]

HABJIIO.D:EIUlE

n

CJIY'IAitHbIX

OBCTOHTEJIbCTBAX

371

Torp;a

npHp;eM

K

sap;aqe

0

BblqHCJIeHHH

BeJIH'IHHbI

MaTeMaTHqecKoro

I

O,RHp;aHHH

\

t

M {[

~

(s)

l.l

(~)

d~

]2}.

t-&

iho

BbIqlICJIeHHe

npOHSBOP;HTCH

no

IISBecTHbIM,

cTporo

060cHoBaH-

HbIM

npaBHJIaM

TeopHM

BepoHTHocTeB.

MbI,

op;HaRO,

BbIJIOJIHHM

3TO

BblqlICJIeHHe,

IICXOP;H

HS

HarJIHp;HbIX,

He

BnOJIHe

CTporHX,

HO

npaBp;o-

f

nop;06HbIX

co06pameHHH.

,IJ;JIH

noB

~eJIH

rrpep;cTaBHM

HHTerpaJI

I

B

BHp;e

eYMMM

HHTerpaJIOB,

BbIqI1CJIeHHbIX

no

OTpeSKaM

[~k'

~h+l)

(k = 0,

...

,

In

- 2), [Sm-l,

~m],

COCTaBJIHIO~HM

BeCb

OTpeSOR

[t

- it,tl.

IIoJIY'IHM:

m-ll;/.+1

I =

~

(~)

u.

(s)

d~.

I,

,;=0 l;k

B

pesyJIbTaTe

nora

HHTerpaJI

I

npep;CTaBJIHeTCH

KaR

cyMMa

CJIy-

qaBHbIX

BeJIHqHII

l;1'+1

llk

=

~

t1(~)

u (s) ds.

l;k

06CYP;HM

BeJIHqHHbI

llk'

113

paBeHCTBa

(46.4)

HeTpyp;HO

BbIBeCTII,

'ITO

npH

MaJIOM

SHaqeHHH

paaHOCTH

~1'+1

-

~,

..

BepOHTHOCTb

Toro,

'ITO

Ha

HHTepBaJIe

[SI"

~h+l)

ocy~ecTBHTCH

B

nOMexe

t1

(,;)

60JIbme

op;Horo

HMnyJIbCa,

eCTb

BeJIHqHHa

BbIcmero

nopHp;Ka

MaJIOCTH

no

cpaBHeHHIO

C

P;JIIIHOH

noro

HHTepBaJIa.

Ho

Torp;a,

npeHe6peraH

TaKHMH

BeJIHqH-

HaMIiI

BbIcmero

nopHp;Ka

MaJIOCTH

H BbIBOP;H

lIS

(46.4),

'ITO

BepOHTHOCTb

nOHBJIeHIIH

o8noeo

HMnyJibca

Ha

HHTepBaJIe

[~I.,

~"+l)

eCTb

PI

[[~k'

Sl.+l)];::;:;;

(~k+l

-

~k

)A"

MbI

Moa<eM

nplIHHTb,

'ITO

BeJIHqHHa

llk

nplIHII-

MaeT

JIHJIIb

TpH

SHaqeHHH:

ll~l);::;:;;

e u

(~),

1l1~2):::::::

- e u

(~),

1l!(3)

= 0

cooTBeTCTBeHHO

C

BepOHTHoCTHMH

1

p~;l):::::::"2

A,

(~k+l

-

~k)'

1

p(2)

:::::::

_

A,

(~k+1

-

~k)'

p(3)

=

1-

pO)

_

p(2).

k 2

I, I,

J.

OTCIOp;a,

Memp;y

npO'IHM,

cpasy

BbIBOP;HTCH,

'ITO

MaTeMaTH'IeCKOe

omH-

p;aHlre

M

{lll,}Ramp;oH

CJIyqaHHOH

BeJIlIqHHbIllk

eCTb

HyJIb,

a

SHaqHT,

I

6yp;eT

HyJIeM

H

MaTeMaTH'IeCKOe

omHp;aHHe

HHTerpaJIa

M {I},

TaR

KaR

H3BecTHO,

'ITO

MaTeMaTH'IeCROe

omlIp;aHHe

CYMMbI

CJIyqaBHbIXBeJIH-

q1IH

paBHO

cyMMe

IIX

MaTeMaTHqecKHX

omlIp;aHIIR.

IhaK,

HaM

HymHo

Tenepb.

COCqHTaTb

MaTeMaTII'IeCKOe

GrRHp;aHHe

M {I2} OT

Imap;paTa

I

CYMMbI

I

CJIyqaBHbIX

BeJIH'IIIH

llh'

Kamp;aH

lIS

KOTOpbIX

HMeeT

HyJIeBOe

t

1

,

...

,"

'-'

....

~

..

>

..

,:

:",

;'

..

,

..

;:'_:~<:

..

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Красовский Н.Н. Теория управления движением

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