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11

IiHHrH

cop;epmHT

aap;a'IH,

OTHOCHm;HeCH

,

:'1

K CHCTeM.aM,

orrHChIBaeMbIM

ypaBHeHHHMH,

OTJIH'IHbIMII

01'

06hIK-

HOBeHHbIX

p;HlfllflepeHIJ;:iIaJIbHhIX

ypaBHeHHH,

COCTaBJI-HIOm;HX

oc-

HOBy

MaTeMaTH'IeCKOrO

arrrrapaTa

BO

Bcex

rrpep;blp;ym;HX

paap;eJIax.

MMeHHo,

B

§§

43,44

paCCMaTpHBaIOTCH

cucmeMbL

c

nOC,l/,ef)eucmeueM,

KO-

TophIe

OIIHCbIBalOTCH

TaK

HaahIBaeMhIMH

ypaBHeHHHMH

C

aarraap;hIBa-

:.

HHHMH

BpeMeHH.

3p;ecb

rrOKaaaHO,

'ITO

rrop;06HbIe

ypaBHeHHH

MOmHO

"

TpaKTOBaTI,

KaK

ypaBHeHHH,

OnHChIBaIOm;He

H3MeHeHHe

BO

BpeMeHH

KOOpp;HHaT

Yj

(t),

KOTOphIe

lflHrypHpOBaJIH

B

III

'IaCTH.

KHHrH

rrpH

HSy'IeHHH

sap;a'IH

()

Ha6JIIop;eHHH

H COCTaBJIHJIH

TaM

lIa6JIIop;aeMbIH

CHrHaJI

Y (,;).

81'0

rrOSBOJIHeT

BhIHCHHTI,

CBHaI,

sap;a'I

0

Ha6JIIOp;eHHH

06hIKHoBeHHhIx

JIHHeHHhIX

p;HHaMH'IeCRHX

CHCTeM C

TeopHeH

JIHHeH-

HhIX

ypaBHeHHH

C

aarrasp;bIBaHHHMH

BpeMeHIL

liIay'IeHHe

:nOH

CBHaH

H COCTaBJIHeT OCHOBHOH

MaTepHaJI

§§

43, 44. B § 45

paCCMaTpHBaeTCH

op;Ha

sap;a'Ia

06

yrrpaBJIeHHH

JIHHeHHOH

CHCTeMOH C

rrocJIep;eiiCTBHeM.

IIocJIep;HHH

§

46

rrOCBHm;eH

HSy'IeHHIO

JIHHeHHhIX

CHCTeM,

rrop;Bep-

meHHhIX

CJIy'IaHHhIM

Boap;eHcTBHHM.

PaccMaTpHBaeTcH~o,n;Ha

aa,n;a'Ia

0

Ha6JIIO,n;eHHH

lflaaoBhIx

Roopp;HHaT

JIHHeHHOH

CHCTeMbI

nplI

yCJIOBHlI,

'ITO

CHrHaJI

Z

(';):BhIpa6aThIBaeTCH

HaMeplITeJII,HhIM

YCTPOHCTBOM,

rrop;-

BepmeHHhIM

CJIyqa:RHOH

lIMnYJII,CHOH

nOMexe.

IIORaahIBaeTCH,

'ITO

3Ta

sa,n;a'Ia

BKJIIO'IaeTCH

eCTeCTBeHHhIM

06pasoM

B

Kpyr

np06JIeM

Ha6JIIop;e-

HlIH,

HSy'IeHHhIX

B

nlaBe

8;

CJIe,n;yeT

JIHillI,

COOTBeTCTBYIOIIJ;lIM

06pasoM

BIJ6paTI,

yCJIOBHe

OrrTHMaJlI,HOCTH

P;JIH

paSpeillaIOIIJ;eH

onepan;HlI.

Ct

I1TPOBAH

3A,IJ;A

qA

0

BCTPE'LJE

,IJ;BHJKEHHll

r.lt a 8 a 10

§ 40.

IIocTaHoBl,a

saJ(a'IH

0

rrpeCJIeJ(OBaHHIl

3ap;a'Ia

0

npeCJIep;OBaHlIlI,

ROTopaH

CJIymHT

npep;MeTOM

HaCTOHIIJ;eH

rJIaBhI,

yme

ynoMHHaJIaCI,

BhIme,

B § 38,

rrplI

oocymp;eHHlI

CBHSII

Teo-

plIlI

Hrp

C

np06JIeMaMH

ynpaBJIeHHH

p;BlImeHlIeM.

TenepI,

MhI

06cy-

P;HM 31'01'

Bonpoc

no,n;pOOHee.

Ha'IHeM

C

nOCTaHOBRII

Sap;a'IH.

liITaK,

B

COOTBeTCTBlIH C

MaTeplIaJIOM

IIa

§ 38,

paccMoTplIM

sap;a'Iy

0

npeCJIep;o-

BaHlI1I

RaR

IIrpy

,n;BYX

JIlIn;.

B

3TOH

IIrpe

POJII,

nepBoro

lIrpORa

oyp;eT

lICrrOJIHHTI,

rrpeCJIep;yeMbm

06'1>eKT,

POJII,

BToporo

HrpORa

-

npeCJIe-

,n;OBaTeJII,.

CTpaTerHHMII

sp;ecI,

HBJIHIOTCH

saKOHhI

peryJIlIpOBaHHH,

KOTophIe

orrpe,n;eJIHIOT CHJIhI,

ynpaBJIHIOIIJ;lIe

OO'l>eRTaMH.

IIoRaaaTeJIeM

HrphI

6y,n;eT

CJIymHTI,

BpeMH

t =

'I't,

Rorp;a

OO'l>eRThI

BCTpe'IaIOTCH.

ECTeCTBeHHO,

'ITO

npeCJIep;OBaTeJII,

CTpeMHTCH

y'MeHI,illHTI,

'I't,

a

npe-

CJIep;yeMhIH

saHHTepeCOBaH

B

npe,n;eJII,HOM

yBeJIH'IeHHH

'I't

.KaR

II

Bcer-

p;a

B

lIrpe,

MhI

p;OJImHhI

npe,n;rrOJIaraTI"

'ITO

npH

BhIoope

CBoeH

CTpa-

TerlIH

Ramp;IJH

lIS

HrpOKOB

He

IIHlflopMHpOBaH

0 TOM,

RaKyIO

CTpa-

TerHIO

IISOepeT

p;pyroH

rrapTHep.

YTO'IHHM

MaTeMaTlI'IecRyIO

nOCTaHoBKy

sap;a'IlI.

IIYCTI,

TeRyIIJ;IIe

COCTOHHHH

npeCJIep;yeMoro

06'1>eRTa

onHChIBaIOTCH

n-MepHhIM

BeRTO-

pOM

Z (t)

ero

lflasoBhIx

Roop,n;HHaT:

Z (t) =

{Zl

(t),

...

,

Zn

(tn,

a CO-

CTOHHHH

OO'l>eKTa-npeCJIep;OBaTeJIH-BeRTopOM

Y (t) =

{YI

(t), .

..

,

Yn

(tn.

Byp;eM

npep;noJIaraTb,

KaK

OOhI'IHO,

'ITO

1I3MeHeHHe

BeKTopOB

Y (t)

1I

z(~

onlIChIBaeTCH

ypaBHeHHHMII

tJ

(t) = A (t) Y (t) +B

(t)

u,

(40.1)

i (t) = P (t) z (t) + Q (t) v,

(40.2)

rp;e

u =

{u

l

,

•••

, u

r

}

H V =

{VI'

••.

, v

r

} -

BeRTophI

ynpaBJIHIOIIJ;HX

CHJI,

A,

B,

P, Q -

MaTplIIJ;hI,

1'.apaKTepHsyroIIJ;He

p;HHaMIIRy

OO'l>eKToB.

MhI

OrpaHH'IHBaeMCH

JIHHeHHbIMH

ypaBHeHHHMII

(40.1)

II

(40.2),

TaK

RaKIToJIbRo

B

aTlIX

npocTeHillrrx

CJIyqaHx

oyp;eM

pemaTI,

B

p;aHHoii

KHlIre

3ap;a'Iy

0

rrpeCJIep;OBaHHH.

Boap;eHcTBHH

u

1I

V

CTeCHIIM

orpaHII-

'IeHIIHMlI

x [u (,;), ,; >

t]

<;

1!~(t),

(40.3)

xl

[V

(,;), ,;

:>

t]

<."

(t),

(40.4)

:

'.'

'.

.

__

.~'~,

~,_.

'_'_;_'

;."

:

...

-...:.~~:..~,'

.;

..

'

:",.:::.'

....

,',:: .

:._;..'",.'

",".

::'

..

::.

':::

..

:

:':'.;

:",

.:'

'::':

..

'.'

~

...

:_:

:::_._:~

:···:r~:~:I·

I

,',·

.

-11

•

Ii!

~

,

11

:1'1

"1,

I,

ii

"

I~;

r~

~

. '

~

[rJI.

10

HrpOBAff

3AAAqA

0

BCTPE~E

ABHlliEHHfi

326

rAe

x

[U],

x [vl _

HHTeHCHBHOCTH,

O~eHHBaIOm;He

pecypchI

yrrpaB-

JIeHHH

U

(T)

H V

(T),

KOTOphIe

MoryT

TpaTHTbCff

rrpH

T ;:;> t

(CM.

§ 26).

HarrpHMep,

MoryT

6hITb

oroBopeHhI

orpaHHlIeHHff

(40.5)

II

U

(T)

II

-<

11,

11

v

(T)

II

-<

v

(11

=

const,

v =

const)

HJIH

orpaHHlIeHHff

00

2

(40.6)

~

1\

v

(t)

1]2

dt

-<

v

(t),

\

Ilu

(T)11

2

dt-<

11

2

(t),

t

HJIH,

HaKOHe~,

orpaHHlIeHHff

00

(40.7)

j

II

dU

(T)

II

-<

11

(t),

~

l\dV

(t)

II

-<

v (t).

t

B

rrOCJIeAHeM

CJIylIae'

pellb

HAeT,

eCTeCTBeHHO,

06

0606m;eHHhIx

yrrpaBJIeHHffX

H,

CJIeAOBaTeJIbHO,

0

CHCTeMax,

orrHCbIBaeMbIX

ypaB-

HeHHffMH

(40.8)

dy =

Aydt

+

BdU,

dz =

Pzdt

+ QdV.

HarrOMHHM,

lITO

BeJIHlIHHhI

11

(t)

H v

(t)

3aAaIOTCff

TOJIbKO

B

HalIaJIb-

HhIH

MOMeHT

BpeMeHH

t =

to'

B

AaJIbHeHmeM

IIpH

t;:;>

to

HX

3HalIeHHJI

orrpeAeJIffIOTCff

pecypcaMH

yrrpaBJIeBHH,

OCTaIom;HMHClI,

BalIHBaff

C

KaiRAOro

MOMeBTa

t, B

rrpo~ecce

peaJIH3a~HH

BhI6paBBhIX

3aKOBOB

yrrpaBJIeBHH

(CM.

§ 29).

IIpeArrOJIOiRHM,

lITO

rro

YCJIOBHffM

aaAalIH

OTMelIeBbI

BarrepeA

HeKO-

TophIe

KoOPAHHaThI

Yi' H

Zi'

(j =

1,

...

, m

-<

n),

cOBrraAeHHe

KOTO-

J J

phIX

COCTaBJIHeT

~eJIb

rrpeCJIeAOBaHHff.

AJIH

yrrpo~eHHff

3arrHCH

MhI

rrpHMeM,

'ITO

BhI6paHa

TaKaH

HYMepa~Hff,

rrpH

KOTOpOH

81'0

6YAYT

rrep-

BhIe

m

KooPAHHaT

Yi

H

Zi

(i

= 1,

...

, m).

Ha60phI

3THX

KooPAHBaT

6YAeM

paCCMaTpHBaTb,

AaJIee,

KaK

HeKOTophIe

m-MepHhIe

BeKTophI,

060-

3HalIaH

lIX

ClIMBOJIaMH

{Y}m

H

{z}m·

MeHT

PaccMoTpHM

ABlIiReHlIff

(40.1)

11

(40.2) B

HeKOTOpI:.I:ii

M9

t =

tao

;:;>

to'

TorAa

MOM

e H

'1'

0 M B

CT

P e

1I1I

ABlIiI,eHHH

(40.1)

11

(40.2)

Ha30BeM

TaKoH

MoMeHT

BpeMeHlI

t =

{}

=

ta.

+ T,

KorAa

BrrepBhIe

COBrraAyT

Bee

OTMelIeHHhIe

KooPAHHaThI

Yi

H

Zi

(i

= 1,

...

, m),

T.

e.

KorAa

BrrepBhIe

{y

(ta.

+ T)}m = {z

(ta.

+ T)}m'

OTpeaoK

BpeMeHH

T;:;>O

OT t=ta. AO

MOMeHTa

BCTpelIH

t={}=ta.

+T

6YAeM

Ha3hIBaTb

epeMeneM

ao

ecmpe1.f.u.

PaccMaTplIBaeMan

HaMH

lIrpa

6YAeT

COCTOHTb B

orrpeAeJIea

lIH

aJIrOplITMOB,

BhIlJlICJIHIOm;lIX

yClIJIlIH

U

11

v,

<··)

':.-,.:>

C

40j

TIOCTAHOBKA3AAAQH

0

TIpgCJIEAOBAHMH

321

KOTophIe

06eCTIelIHBaIOT

min

max

T.

0AHaKo

}J.JIH

aaBepmeBHH

rrOCTa-

u v

HOBKH

3aAalIH

MhI

AOJIiRBbI

em;e

'IeTKO

orOBopHTb,

'ITO

rrOBHMaeTCH

3AeCb

rroA

CTpaTerHHMH

rrepBoro

H

BToporo

lIrpOROB.

CAeJIaeM

3'1'0.

XapaRTep

paCCMaTplIBaeMOH

HrphI

3aBHCHT

0'1'

Toro,

RaROBa

HH-

<popMa~HH

0

ABHiReHHHX

Y

(t)

H Z (t),

rrocTyrraIOm;aH

B

opraBhI,

rAe

<pOpMHpyIOTCH

yrrpaBJIHIOm;He

yCHJIHH.

Eha

HH<popMa~HH

AOCTaBJIHeT

Te

AaHBHe,

Ha

ROTOphIX

6a3HpYIOTCH

aJIrOpHTMbI,

BhIlIHCJIffIOm;He

TeRy-

m;He

3HalIeBHH

U

(t)

H V (t).

OrpaBHlIHMcH

HaH60JIee

rrpocThIM

CJIyl:J.a-

eM,

RorAa

B

RaiRAhIH

MOMeHT

t B

060HX

opraBax

yrrpaBJIeHlIH

H3BeCTHO

ece

0

pea.tlU30eaeUJ,UXc.fl.

1'j,

amOMY

MOMenmy

ABHiReHlIHX

Y (t) H Z (t).

:.3'1'0

03Ha'IaeT,

'ITO

B

MOMeBT

t

060HM

rrapTHepaM

H3BeCTHbI

Bce

ROOPAH-

HaThI

Yi(t)

H

zi

(t)

H

H3BeCTHhI

o~eHRH

11

(t)

H v

(t)

(40.3), (40.4)

OCTaB-

mHXCH

pecypcoB

yrrpaBJIeBHH,

ROTophIe

MoryT

6hITb

lICrrOJIbaOBaHhI

rrpH

T;:;>

t.

IIOA'IepRHeM,

OABaRO,

em;e

paa,

'ITO

6YAym;ee

ABHiReHHe

rrpOTHBHIIRa

rrplI

T > t

RaiRAOMY

H3

rrapTBepOB

HeU3eecmno

(rrOMIIMO

IIH<popMa~IIII

06

IIMeIOm;HXCH

pecypcax

11

(t)

H v (t)). B

TaROM

CJIy'Iae

MOiRHO

BeCTlI

yrrpaBJIeBHe

no npuH,Tfuny

o6pamnoii

Ce.fl.3U

H

HCRaTb

aaROBhI

peryJIIIpOBaHIIH,

orrpeAeJIHIOm;IIe

BeJIII'IHHhI

U ttl H v ttl B

MO-

MeHT

t, B

<popMe

1)1

U ttl = U

[t,

Y (t), Z (t),

11

(t), v (t)l, (40.9)

v

ttl = v

[t,

Y (t), Z (t),

11

(t), v (t)l.

(40.10)

3aROBhI

perYJIIIpOBaHIIH

(40.9), (40.10)

HaaOBeM

AorrycTHMhIMH,

ec-

JIH

rrpH

IIX

HCrrOJIbaOBaHIIII

B

rrpo~ecce

yrrpaBJIeHIIH

Be

BapymaIOTcH

OrpaHII'IeHHJI

(40.3), (40.4).

RpoMe

Toro,

HaM

CJIeAOBaJIO

6hI

em;e

oro-

BOpIITb,

'ITO

AorrYCTIIMhI

JIIImb

TaRIIe

RJIaCChI

<PYHR~IIH

(40.9) H (40.10),

rrpII

rrOACTaHOBRe

ROTOphIX

BMeCTO

U

11

v B (40.1) H (40.2)

rrOJIy'IaeTCJI

CHCTeMa

ypaBHeHlIH,

IIMeIOm;aJI

CTporIIe

MaTeMaTlIlIeCRlIe

pemeBHH

Y (t), Z (t), U [tl, v ttl,

COBMeCTHhIe

C

yCJIOBHHMII

(40.3), (40.4) H

orr-

peAeJIeHBhIe

rro

RpaHHeH

Mepe

AJIH

Bcex

t,

rrpeAmecTByIOm;IIX

MOMeBTy

BCTpe'IH.

0ABaRo

MbI

HaMepeHHO

He

aR~eHTIIpyeM

aAeCb

BHIIMaHIIe

Ha

3TOM

Borrpoce,

TaR

RaR

06m;IIH

MaTeMaTH'IeCRIIH

aHaJIHa

CRJIaAhIBaIO-

m;eHCH

TyT

cIITya~IIII

CJIOiReH

H

yBeJI

6hI

Hac

AaJIeRO

0'1'

rJIaBHOH

3aAa-

'Ill

-

0'1'

rrOIICRa

crroc060B

ROBRpeTHoro

BhIlIHCJIeHHH

yrrpaBJIeHHH

U

H

V.

II03TOMY

Ha

Borrpoce

0

RJIaCCaX

<PYHR~HH

(40.9) H (40.10)

MhI

6y-

AeM

OCTaHaBJIlIBaTbCH

RpaTRo

rro

XOAY

AeJIa

rrpII

paCCMOTpeHHII

BCTpe-

lIaIOIIJ;IIXCH

HIIiRe

'IaCTHhIX

CJIylIaeB.

IITaR,

rroA

CTpaTerlIHMH

lIrpOROB

MbI

6YAeM

rrOHIIMaTb

Terrepb

AO-

rrycTHMbIe

aaROHbI

peryJIHpOBaBHH

U

II

v (40.9)

11

(40.10).

TorAa

AJIH

RamAoro

BOaMOiRHOrO

Ha'IaJIbHOrO

rrOJIOiReHHJI

t =

ta.,

Y

(ta.)

=

ya.,

1)

ITo

IIOBOAY

0603HaQeHlIH

u (t), v (t)

11

u

[t],

v

[t]

eM.

B § 28.

328

,..

a-u.s'.,

a .'

'x'

.tGi"...,....,

.

...:...

.

..:...~b.,.i:

''P'"

•

.~

I

"<

:.",:::

C

[rJI,

10

MrpOBAH

SAAAqA

0

BCTPEqE

ABMlliEHMft

z (t(X) =

z(X,

f..I.

(t"J

=

f..I.(X,

'\I(t(X) =

'\I(X

BpeMH

T

P;O

BCTpel:IM

p;BMllieHMH

y (t) M z (t)

6YAeT

<pyHKIJ;MeH

(TOlIHee,

<pyHKIJ;MOHaJIOM) T

u

•

v

0'1'

<pyHR-

IJ;MH U

[t,

y,

z,

f..I.'

'\I]

M V

[t,

y,

z,

f..I.,

'\I].

(ECJIM

npM

BbI6paHHhIX

V M U

BCTpelIa

Bo06m;e

He

HacTynaeT,

TO,

lIT06bI

He

HapymaTb

sanMCb

Tu,v'

nOJIaraeM

yC.I£oeno

T

u.v

=

{oo},

rp;e

{oo}

-

Hecym;ecTByIOm;ee

'IMCJIO,

KOTopoe

6yp;eM

C'IMTaTb

60JIbmMM

JIIo60ro

p;eHCTBMTeJIbHOrO

lIHCJIa

T.

II

pH

aTOM

MbI

HaMepeHHO

He

nMmeM

npOCTO

T

'U.

V =

00,

lIT06bI

He

coa-

p;aBaTb

omM601IHOH

HJIJIIOSMH,

'ITO

MMeeTCH B

BHP;y

C6JIHIIWHHe

P;BM-

meHW

y (t) H z (t)

npH

t

-+

00.)

Tenepb

3ap;alIy

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HCrrOJIbaOBaTb

co06pameHHJI, paaBHTLIe BLIrne B

,rraHHOH

MOHOI'palj):~IH

.

IIpem,rre qeM

H,rrTH

,rraJIbrne, paCCMOTpllM

O,rrHY

reOMeTpWleCI{yIO

llHTeprrpeTa:U;I2lIO

,rrJIJI

MOMeHTa

nOrJIOm;eHHJI

itO.

IIyc1'b

B HeROTopLIH

MOMeHT

t = t

a

peaJIl2laOBaJIHCb

BeJIl2lql2lHLI

y(t

a

),

z

(t

a

),

f-t

(ta), v (ta).

BbI6epeM 'IllCJIO

it

> t

a

H paCCMOTpllM B

m-MepHOM

npOCTpaHCTBe

q = {q}m 06JIacTb

G(l),

COCTOJlm;yIO

lla

Bcex

Tex

TOqeK

q.,

P;JIJI

Kam-

,rrOR

Ha

ROTOpLIX

MomHO nop;o6paTb

ynpaBJIeHlle

U.

(e)

(ta

<;

1:

<; it),

cTeCHeHHoe

yCJIOBHeM

x

[U.(1:)

,

1:

> t

a

]

<;

f..t

(ta)

H nepeBo,rrJlm;ee

06'I>eRT

(40.1)

Ha

COCTOJlHllJl

Y (ta) B

nOJIOmeHl2le

{y

(it)}m = q•.

06JIaCTb

G(l)

onpe,rreJIJleTCJI BeJIH'IHHaMll t

a

,

y

(t

a

),

f-t

(t

),

it, rr03TOMY

a

060aHaqHM

ee

CHMBOJIOM

G(l)

[t

a

,

y (ta),

f..t

(t

a

),

it].

AHaJIorllqHYlo

o6JIacTb

,rrJIJI

BToporo

06'I>-eKTa

060aHaqHM

CHMBOJIOMG(2)

[ta, Z (t

),

a

v

(t

a

),

it]. 8TH 06JIaCTH

CYTb

06JIaCTH

OOcmU;)fCuJ.wcmu

,rrJIJI

,rrBll-

meHHH

(40.1)

H

(40.2).

BamHo

OTMeTllTb, 'ITO B CJIyqaJlX, Rorp;a BeJIH'IH-

Ha x B

YCJIOBHJlX

(40.3)

I2l

(40.4)

HMeeT

CMLICJI

HOPMbL

lj)YHR:U;llR

U

(1:)

H

V

(e),

06JIaCTH

G(l)

I2l

G(2)

JlBJIJlIOTCJI

BLIrryRJILIMl2l, aaMlmyTLIMIf MHomeCTBa-

MIf B npOCTpaHCTBe

{q}m' COBeprneHHo

O'IeBHp;HO,

'ITO

MOMeHT

rrorJIom;e-

HHJI

t =

itO

-

3TO

raROR nepBLIH

MOMeHT

BpeMeHH t = it, Kor,rra

<:

:,··:,·::~~._,2'.:.~~~·~'~~·

..

:'.

:,:

....

:.;

..

:~:

-'

'",

.....

.:,.'

..

;'

....

-:: .'

.....

:

::.

<:'

.

':.::',~:~.~~,~~~~.

o

332

ltrpO.BMi:

3AnA

etA

0

13CTPE

q:El

AmuItEHI1:l!I

Ll'.iI.

i 0

06JIaCTh

G(2)

OKa31i1BaeTCH

JIeJRam;eM

B

06JIaCTlI

G(I).

THnHlJHa

cHTya-

n;IUJ,

KorAa

C

1I3MeHeHlIeM

{t

3aMKHYTble

OOJT

aCTn

G(1)

n

G(2)

Ae-

q>OpMnpyloTcH

HenpeplilBHO

n

noaToMy

OK

a31i1BaeTCrr , lJTO B

MOMeHT

{t =

{to

HX

rpaHHn;1iI

conpUr;,aCa1Omest

(pIIC.

41.1).

Teneph

Mlil

MomeM

Cq>OpMyJIlIpOBaTh

npaBlIJIO

;rJ;JIfl

BliIlJIICJ1eHUrr

ynp

aBJIeHHn

U U V C n;eJThIO

ocymeCTBJIOHHfl

BCTpOlJU

B

MOMOHT

t = {to.

CKaJReM,

lJTO

ynpaBJIeHUe

U

[tCl

J B

MOMeHT

t =

tCl

nalfe.ltueaem

p;BUJRe-

Hue

(40.1) B

TOlJKY

{y

('t'})}m

= q.,

eCJIU

cymecTByeT

ynpaBJIeHHe

u. (r)

(HenpeplilBHoe

cnpaBa),

CTe-

fz

CHeHHoe

yCJIOBHeM

(40.3) H

npHBOp;Hm;ee

cHcTeMy

(40.1)

B

nOJIOJReHHe

{y

({t)}m = q.,

npUlJeMU

[tot]

= u.(tot).AHa-

JIOrHlJHO

onpep;eJIHeTCH

yn-

paBJIeHHe

v

[tCl),

KOTO-

poe

Han;eJIHBaeT

B

TOlJKY

{z

({t)}m =

q.

p;BHJReHHe

(40.2).

llYCTh

q.

-

HeKo~

.

~';

,1\1

I

~p

I

o

•

~I!I

:~I;

~'q

'lIlt

~~I!

ynpaBJIeHHe

B

,~

TOpaHTOlJKa

H3

G(I)

[tCl,

Y

(tot),

,....

(t

a

),

{t).

Torp;a

no

CMIiICJIY

06JIaCTlI

G(I)

MOJRHO

.

BIiI6paTb

ynpaBJIeHHe

u =

--

If,

_0-

= u

[tCl

l,

Han;eJIHBaIOIIJ;ee

PIlC.

41.1.

ABHJReHHe

(40.1) B

TOlJKY

{y

({t)}m

=Ifl..

llyCTb,

Ta-

KHM

06pa30M,

BIiI6HpaJIOCb

TelJeHHe

HHTepBaJIa

BpeMeHH

[tel,

tel

+

dt]

(dt>

0).

PaccMoTpHM

06JIaCTb

G(l}

[t

a

+ dt, y (t

a

+ dt),

,....

(tCl

+ dt),

{t).

I;""

HCHo,

lJTO

3Ta

oOJIacTb

JIeJRHT

BHyTpH

06JIaCTlI

G(I)

[tot,

y (t«),

,....

(ta), {t] H

IIO-IIpeJRHeMy

COp;epJRHT

TOlJKY

q•.

AHaJIOrHlJHaH

cHTya-

n;HH

IIMeOT

MeCTO

H B

OTHomeHHH

p;BHJReHUH

z

(t)

(40.2).

IIcKoMoe

HaMH

npaBHJIO

6yp;eT

COCTOHTb B

npHU;eJIHBaHHH

p;BHJReHUM

B

Te

TO'1KH

q.

=

qO,

rp;e

B

MOMeHT

{to

KacaIOTCH

rpaHHn;1iI

OOJIaCTen

G(I)

H

G(2).

JIraK,

nyCTb

'B

MOMeHT

t = t

a

peaJIH30BaJIUCb

BeJIHlJIIHIiI

y (t«),

z (ta),

,....

(ta), v

(tot)

H

IIyCTh

{to

[tot,

y

(tCl),

z

(t

a

),

,....

(tilt),

v (t

a

)]

-

MOMeHT

nOrJIOIIJ;eHUH

npon;ecca

(40.2)

npOn;eCCOM

(40.1).

Torp;a

co-

rJIaCHO

npep;liIp;yIIJ;eMy

6yp;eM

npep;nOJIaraTh,

lJTO

06JIaCTh

G(2)

[ta,

Z (t

a

),

v(t

a

),

{to]

JIeJRl'I'f

u;eJIUKOM

B06JIaCTH

G(l)

[tilt,

Y (t

a

),

,....

(ta), {to] U

rpaHHn;1iI

aTHXa06JIaCTen

KacaIOTCH

B

TOlJKe

qO

(CM.

pHC.

41.1).

Orpa-

HHlJUMCH

CJIylJaeM,KOrp;a

aTO

KacaHue

ocyIIJ;eCTBJIHeTCH

B

O,llHOn-ep;HH-

CTBeHHon

TOlJKe

qO.

B

IIpOTHBHOM

CJIylJae

p;aJIbHenmHe

paccymp;eHHH

HaTaJIKHBaIOTCH

Ha

Cepne3H1i1e

TPYP;HOCTH.

B

KalJeCTBe

ynpaBJIeHHH

u =

UO

[ta,

y (t

a

), z

(tel),

,....

(t«), v (ta)]

BIiI6epeM

ynpaBJIeHHe

U =

= u

(tot),

Han;eJIIIBaIOm;ee

ABUJReHUe

y

(t)

(40.1) B

TOlJKY

{y

({t°)}m =

=

qO.

ECJIH

B

KaJRp;IiIM

MOMeHT

1:

= t

a

;>

to

BbI6HpaTb

ynpaBJIe-

c>

~,_:~.

§

41]

8BPltd'fllqlWHIIE

coog

P

MHEHlfH

tIue

u

[tCl]

MMeHHO

TaKIIM

06pa30M,

TO,

IWI{

OJ>I'

HM

P;OIIYCTIIMoeJ

ynpaBJIeHlIe

v

(1:),

BCTpelJa

P;BHJROHUM

333

BhIOlIpaJIOCh

y

(1:)

H Z

(1:)

npOH30Hp;eT

He

n03JRe,

lJeM

B

MOMeHT

~.O[to,

Y

(to),

z

(to),

,....

(to),

v

(to»).

lloKaJReM

aTO

1).

JIraK,

paCCMOTpUM

cHTyan;HIO,

KOTopaH

CKJIap;IiIBaeTCH

npH

yCJIOBHU,

lJTO

p;BH}KeHHe

y

(t)

Han;eJIH-

BaeTCH

Bce

BpeMH

Ha

TOlJKY

qO.

llYCTb

P;BUJROHHe

HalJaJIOCh

npu

t =

=

to'

,I!;onycTIIM

CHalJaJIa,

lJTO

ynpaBJIeHUe

v

(t)

BIiIOHpaeTCH

B

I{aJRp;IiIM

MOMeHT

t ;>

to

Bce

BpeMH

TaKJRe

H3

YCJIOBlIH

npHn;eJIIIBaHIIH

B

Ty

JRe

TOlJKY

qO.

Torp;a

B

IIpOn;ecce

ABHJReHIIH

TOlJRa

qO

U

MOMeHT

{to

H3Me-

HHTbCH

He

6YP;yT,

a

06JIaCTH

G(1)

ft, y (t),

,....

(t), {to] H

G(2)

[t, z (t),

v (t), {to]

npu

npH6JIuJReHHH

t K

{to

6YP;yT

CTHrHBaTbCH

K

HeIIOp;BIIJR-

HOM

TOlJKe

qO.

IIpH

aTOM

B

MOMOHT

t =

{to

[to,

y

(to),

z

(to),

,....

(to),

v

(to)]

nOJIylJUM

COBnap;e-

.

HHe

TOlJeK

{y

({t°)}m =

-

{Z

({t°)}m'

TaKUM

06pa-

30M,

eCJIH

06a

ynpaBJIe-

HHH

u

[t]

H v

[t]

6YP;yT

BIiI6HpaThCH

no

yKa3aH-

HOMy

3aKoHy,

TO,

Bo06IIJ;e

rOBopH,

BCTpelJa

COCTOHT-

CH B

MOMeHT

t =

{to

[to,

Y (to)' z

(to),

,....

(to),

'V

(to»).

(CJIylJaHHO

BCTpelJa

MO-

JReT

oCYIIJ;eCTBHTbCH

U

paHbme.)

IIpep;IIOJIOJRUM

v

fJl

Tenepb,

lJTO

Ha

HeKOTO-

PIlC.

41.2.

pOM

UHTepBaJIe

(t, t +dt)

IIpH

BIiI60pe

yIIpaBJleHHH

v

(1:)

BnepBlile

61i1JIO

p;onym;eHO

OTKJIOHeHUe

01'

yKasaHHoro

IIpaBHJIa.

IIMeHHo,

npep;nOJIOJRHM,

lJTO

Ha

aTOM

IIHTOpBaJIe

pa60TaJIO

ynpaBJIe-

HHe,

ROTopoe

Han;eJIUBaJIO

p;BHiHOHlIe

z

(1:)

B

HeKoTopyIO

TOlJKY

{Z}m

=

q.

=1=

gO.

llYCTh

npu

3TOM

yIIpaBJIOHHO

U

[1:

J

no-npenmeMY

Han;eJIU-

BaeT

p;BHJReHUe

y

(1:)

B

TOlJKY

{y({t°)}m

=

qO.

Torp;a

sa

BpeMH

t

<:

1:

<

<t + dt

06JIaCTH

G(1)

II

G(2)

HeCKOJIbKO

p;eq>opMupYIOTCH,

CTHrHBaHCh

BOKpyr

TOlJeR

qO

U

q.

COOTBeTCTBeHHO

(pHC.

41.2).

IIpH

aTOM,

Bo06IIJ;e

rOBopH,

06JIaCTb

G(2)

[t

+ dt, z (t + dt),

'V

(t + dt), {to]

OKa1I{eTCH

yJRe

cTporo

BHYTPII

OOJIaCTH

G(I)[t + dt, Y (t + dt),

,....

(t + dt), {to].

lloaToMy

cTapoe

3HalJeHHe

t =

{to

[t,

Y (t), z (t),,.... (t), v (t)]

yJRe

He

6YP;Wf

P;JIH

peaJIH30BaBmHXCH

BeJIHlJIIH

y (t + dt), z (t + dt),

,....

(t + dt),

'V

(t + dt)

MOMeHTOM

rrOrJIOm;eHHH

rrpon;ecca

(40.2)

1)

CJIe~yeT

Bce

BpeMH

liMeTb

n

BII~y,

'ITO

Mbl

lirHopHpyeM

oco6uii

CJIyqaH,

KorAa

B

rrpoIi;ecce

~Bnil\eHHII

B

OT~eJIblffiJ.e

MOMeHTbl

t = t

rpaHHI:(bl

06JIaCTeH

a

G(l)

II

C<2)

6YAyT

HMeTb

60JIbIJIe

''!eM

OAHY

TO'lKy

HacarrliH

qO

HJIH

Hor~a

rparrHI:(bI

flTHX

06JIaCTen:

corrpUlmcaIOTcH

C,llHJUlWM

meclto U

Hepeey.MtpHO

B03JIe

TOQHH

qO.

:-;'~;"':;:;"'----'~-".

'.'

'.

.

....

.

...

>." .

:.>::

...

:::;

..

'.' ..'

"'):.:

...

:<>.;

.....

:.,:.::.::.:.:

..

;'~~~

C

'C.

334

H1'POBAJI

3A)l;A"rA

0

BCTPE<IE

,tiBHrREHHft

[rJi.

10

npo:u;eCCOM

(40.1).

,D;JIJI

8THX

HOBLIX

HCXO)l;HLIX

p;aHHLIX

MOMeHT

norJIO-

m;eHHJI

l}0

[t

+ dt, y (t + dt), z (t + dt),

f.t

(t

+ dt),

'V

(t + dt)]

HacTynHT

Torp;a

paHbme.

OTCIOp;a CJ.£ep;yeT,

lITO

B

CJIylIaJIX,

Rorp;a

p;BHiReHHe

z (r)

Ha

HeROTopLIX

HHTepBaJIaX

BpeMeHH

npH:u;emIBaeTCH

B

TOlIRH

q.,

OTJIHlIHbIe

OT

TOlIRH

qO,

rp;e

RacaIOTCH

rpaHH:U;LI

G(l)

H

GJ.2),

BeJIHlIHHa

l}0

MOiReT

H3MeHHTbCH,

HO

He

B03paCTaH.

OTCIOp;a CJI.e-

p;yeT,

lITO

H

Torp;a

BCTpelJa

p;BHiReHHH

{z

(r)}m

H

{y

(r)}m

npOH30Hp;eT

He

n03iRe,

lJeM

B

MOMeHT

l}0

(t)

(H

p;aiRe

B

MOMeHT

l}0

(t+

dt) <

l}O).

Ta-

RHM

06pa30M,

eCJIH B

rrpo:u;ecce

npeCJIep;OBaHHH

He

ecmpemiLmCiL

ynoMH-

HyTLIe

BLIme

HenpUiLmHOffmu,

npH:u;eJIHBaHHe

p;BHiReHHH

y

(t)

Bce

BpeMH

B

TOlJRY

{y

(l}°)}m

=

qO

06eCnelJHBaeT

BCTpelJy

He

n03iRe,

lJeM

B

MOMeHT

t =

l}0

(t

lZ

) ,

ROTOpLIH

JIH60

OCTaeTCH

nocToHHHLIM,

JIH60

y6LIBaeT

C

pOCTOM

t

lZ

•

C4>opMyJIHpOBaHHoe

npaBHJIO

P;JIH

BLIlJHCJIeHHH

U = uo, v =

VO

6yp;eM

B

p;aJIbHeHmeM

Ha3LIBaTb

npaBHJIOM

f)Yicmpe-

ltta/l,bHOeO npu/fe/l,UeaHUiL, a

ynpaBJIeHHH

UO

H vo,

rrocTpoeHHLIe

no

8T0-

My

npaBHJIy,

- i)YiCmpeMa/l,bHbutU

ynpaBJIeHHHMH.

OTHMH

He

BnOJI-

He

CTporHMH

paccyiRp;eHHHMH

MLI

H

OrpaHHlJHMCH

nORa

rrpH

06CyiR-

p;eHHH

06m;eH

np06JIeMLI

0

rrpeCJIep;OBaHHH.

B CJIep;yIOm;eM

naparpa4>e

MLI

nOCMOTpHM,

RaR

OHH

nOp;TBepiRp;aIOTCH

B OP;HOM

npocTeHmeM

ROH-

I

RpeTHOM

CJIylJae.

I

§

42.

3aAa'Ia

0 BCTpe1Je

0AHOTHOHbIX

OO'heRTOB

I

06m;He

co06paiReHHH,

BLICRa3aHHLIe

B

npep;LIp;ym;eM

naparpa4>e,

I

MLI

OpOHJIJIIOCTpHpyeM

Tenepb

Ha

3ap;alJe

0

opeCJIep;OBaHHH

B

CJIy-

l

lJae,

Rorp;a

06a

06'heRTa

OOHCLIBaIOTCH

ypaBHeHHHMH

op;Horo

H

Toro

.,-

iRe

BHp;a.

B

8TOMCJIYlJae

rrp06JIeMa

BCTpelJH

ORa3LIBaeTCH

HaH60JIee

t

npocToH

H

npoBep;eHHe

paccyiRp;eHHH

II3

§

41

BCTpelJaeT

HaHMeHbmlIe

J

TPYP;HOCTH.

IhaR,

paCCMOTplIM

3ap;alJy

40.1

0

MHHHMaRce

BpeMeHH

T

'I

p;o

BCTpelJlI

npeCJIep;YIOm;ero

(y

(t»

II

npeCJIep;yeMoro

(z(t»

p;BlIiReHHH,

ROTopLIe

onlICLIBaIOTCH

COOTBeTCTBeHHO

ypaBHeHHHMII

tJ

=

Ay

+

Bu,

(42.1)

-1

z =

Az

+

Bv,

(42.2)

rp;e,

RaR

Bcerp;a,

y,

z -

n-MepHbIe

4>a30BLIe

BeRTopLI;

U,

v -

r-

MepHLIe

BeRTOpLI

yrrpaBJIeHIIH;

A,

B -

MaTplIn;LI,

xapaRTepH3YIOm;He

p;IIHaMHRy

06'heRToB.

IIpHMeM

P;JIH

ynpom;eHHH,

lJTO MaTpH:U;LI A

II

B

CYTb

nOCTOHHHLIe.

B

COOTBeTCTBHII

C

3ap;alJeH

40.1

HaM

Hap;JIeiRIIT

YlJHTLIBaTb

OrpaHHlJeHIIH

(40.3)

II

(40.4).

,D;JIH

onpep;eJIeHHOCTII

nplI-

MeM,

lJTO

pelJb

IIp;eT

06

OrpaHH'leHIIII

(40.6),

XOTH

paccyiRp;eHHH

pac-

npOCTpaHHIOTCH

II

Ha

p;pyrlIe

CJIylJaH

HepaBeHCTB

(40.3)

H

(40.4),

RO-

TopLIe

p;onycRaIOT

HenpepLIBHLIe

H3MeHeHIIH

y

(t)

II

z

(t)

B

rrpo:u;ecce

o

~~,:'j

§ 42J

3A)l;AqA

0

BCTPEQE

O)l;HOTHnHbIX

OE'bERTOB

335

npeCJI8,L10BaHHH.

HanOMHIIM,

RCTaTII,

lJTO B

CJIylJae

OrpaHHlJeHIIH

(40.6)

H3MeHeHlIe

BeJIlIlJlIH

f.t

(t)

II

'V

(t)

onlICLIBaeTCH

ypaBHeHIIHMII

(CM.

BLIme,

§

29)

lIuW

v=

_"viP

'fl.

= -

2~

I

2v '

(42.3)

ROTopLIe

nOJIYlJaIOTCH

p;1I4>4>epeH:U;HpoBaHlIeM

HHTerpaJIOB

(40.6)

rrQ

HlIiRHeMY

npep;eJIy

t.

Byp;eM

paCCMaTplIBaTb

3ap;alJy

0

MIIHHMaRce

T

nplI

YCJIOB~II,

lJTO :u;eJIb

npeCJIep;OBaHIIH

3aRJIIOlJaeTCH

B

COBnap;eHIIII

ecex

Roopp;IIHaT

Yi

(l})

C

Zi

(l})

B

MOMeHT

BCTpelJlI

t'} =

tlZ

+ T.

MHa-

lJe

rOBopH,

BeRTopbI

{Y}m

H

{z}m,

BBep;eHHbIe

B §

40,

npOCTO

COBna-

p;aIOT

Tenepb

C

4>a30BLIMII

BeRTopaMII

Y

II

z.

IIplIMeM

TaRiRe,

lJTO

CH-

CTeMLI

(42.1)

II

(42.2)HBJIHIOTCH

enO/l,He ynpaB/tiLelttbutU

(CM.

§

19).

OTO

npep;nOJIOiReHHe

He

OrpaHlIlJlIBaeT

06m;HOCTII.

,D;eHcTBIITeJIbHO,

1I3

Ma-

TeplIaJIa

§

19

JIerRO

BLIBOP;IITCH, lJTO B

CJIylJae

OTCyTCTBIIH

nOJIHOH

yn-

paBJIHeMOCTII

pelJb

0

BCTpelJe

p;BlIiReHIIH

y (t)

II

z (t)

Bo06m;e

MOiReT

1IP;-

TII

TOJIbRO

Torp;a,

Rorp;a

pa3HocTb

x (t) = Y

(t)

- z

(t)

BeRTopoB

y

(t)

II

z

(t)

npIIH3.p;JIeiRIIT

nop;npocTpaHcTBy

X

m

,

rrOpOiRp;eHHoMy

BeR-

TopaMII-CTOJI6:u;aMII

MaTplIIJ;LI

{B,

AB,

... , A

n-l

B}.

MHalJe

HIIRaRIIM

nop;60poM

ynpaBJIeHIIH

U (t)

IIV

(t)

ocym;ecTBIITb

BCTpelJy

p;BlIiReHIIH

y

(t)

II

z

(t)

no

BceM

Roopp;IIHaTaM

HeJIb3H.

Op;HaRO,

eCJIH

BeRTOp

x

(t)

COp;epiRIITCH

B

nop;npOCTpaHCTBe

X

m'

TO

MOiRHO

BbIIIOJIHIITb

npe06pa-

30BaHlIe

ypaBHeHIIH

(42.1)

II

(42.2)

R

RaHOHlIlJeCROMY

BIIP;y

(19.3),

(19.4)

H B

p;aJIbHeHmeM

paCCMaTplIBaTb

JIlImb

Te

lJaCTII

CIICTeM,

ROTO-

pLIe

BnOJIHe

ynpaBJIHeMLI.

OCTaJIbHbIe

(HeyrrpaBJIHeMLIe)

Roopp;IIHa-

TLI

p;BlIiReHHH

(42.1)

II

(42.2)

'ML!

MOiReM

Torp;a

rrpOCTO

IIrHOplIpOBaTb,

TaR

RaR

npHHap;JIeiRHOCTb

x

(t)

R

:JC

m

B

HeROTopLIH

MOMeHT

t

03HalJaeT,

lJTO B 8TOT

MOMeHT

Bce

HeynpaBJIHeMLIe

Roopp;IIHaTLI

BeRTopa

x

(t)

=

=y

(t)

- z

(t)

CYTb

HyJIII,

II

OHII

6YP;yT

OC'I'aBaTbCH

TaRIIMH

nplI

Bcex

't'

>

tBCJIep;CTBHe

IIHBaplIaHTHOCTII

npOCTpaHCTBaX

m'

Ho

8TO

II

03HalJaeT,

lJTO

TOrp;a

P;JIH

OCym;eCTBJIeHIIH

BCTpelJlI

p;OCTaTOlJHO

P;O-

6lITbCH

JIlImb

COBnap;eHIIH

ynpaBJIHeMbIX

ROOpp;IIHaT

p;BlIiReHIIH

y (t)

II

Z

('t').

Terrepb

B COOTBeTCTBlI1I C

rrJIaHOM,

HaMe'IeHHbIM

B §

41,

HaM

Hap;-

JIeiRlIT

CHalJaJIa

1I3YlJlITb

c4>opMyJIlIpOBaHHoe

B §

41

npaBIIJIO

i)Yicmpe-

ltta/l,bHOeO

npU/fMueaHUiL,

a

3aTeM

IICCJIep;OBaTb

nepexop;

R IICXOP;HOH

np06JIeMe

40.1.

MTaR,

nYCTb

B

MOMeHT

t =

tlZ

peaJIlI30BaJIlICb

BeJIH-

lJlIHbI

y

(t

lZ

), Z (t

lZ

),

f.t

(t1Z)'

'V

(t

lZ

),

nplIlJeM

nycTb'1-1

(t1Z)

>

'\t

(t1Z)'

P

a3-

HOCTb x (t) = Y (t) - z (t)

yp;OBJIeTBOpHeT

ypaBHeHIIIO

jJ =

Ax

+Bw, (42.4)

ROTopoe

nOJIylJaeTCH

BLIlJHTaHlIeM

(42.2)

1I3

(42.1).

3p;ecb

w = U -

V.

,D;JiH

Toro

lJT06LI

B

MOMeHT

l}

=

tlZ

+ T

ocym;ecTBIIJIaCb

BCTpelJa

P;BII-

IKeHIIH

y

(t)

H z (t),

He06xop;HMO

II

p;OCTaTOlJHO, CJIep;OBaTe,JIbHO,

QT06LI

ynpaBJIeHlIe

w

(t)

= U

(t)

- v

(t)

nepeBOp;IIJIO

clIcTeMY

(42.4)

1I3

'~.

,';'

>.....

'.":,',

:.:...:.:.:"

...

'

..

,\;.

..

,;

....

'e,,',

',':.'

.....

=~~"=-'"='''-=-,.,._

....

_--

n

~~.;

D

~

..

':"

HrpOBAIT 3A~AqA 0

BCTPEQE

~BHffiEHHa

[rn.

10

336

nOJIOfROHHH

X (t,,) = y

(tilt)

- z

(tat)

B

COCTOHHHe

X

(i})

=

O.

YlIH-

TbIBaH orpaHHlIOHHH

(40.3)

H

(40.4),

MOfRHO

cp;eJIaTb

TaRoe npep;rro-

JIOiReHHe.

ConocmaeuJft

aaoa'1,e

40.1

c.lteoylOutYlO

aaoa'1,y

0

npeoe/tbllOJft

6b~cm

pooeucmeuu.

3 a

p;

a

II

a

42.1.

Tpe6yemcJl,

llaumu

OnmUJIW.ltbllOe

ynpa8.ltellue

w\1.

(-r),

cmeCllelllloe

yc.ltoeueJft

X [w(-r),

-r>t,,]<"~(tet)

(~(toc)=

~(tet)-v(tet»

(42.5)

u

nepeeooJl,Ufee

o6'be/'im

(42.4)

ua

COCmMllUJl,

x

(t,,:)

e

nO.ltO:JfCel-f,Ue

x

(tat

+ 1') = 0

aa

llaUJftellbUlee

e03JftO:JfCllOe

epeJftJl,

T°

=

min

T.

w

l'

oeoa

JftOJfteum

n08.lt0UfelIUJI,

i}0=

tat

+

T~

U

imCmpeJfta.ltbllb~e

yn-

pae.ltellUJI,

u~1lt

[t"

1,

v~"

[tat],

npu~fe/tuea70Ufue

OeU:JfCelIUJI,

y

(t)

U z

(t)

e

mo'1,/'iy

qO

(t

et

),

onpeoe/tJl,lOmCJl,

peUlel-lUeJft

aaoa'1,U

42.1,

npUtteJft

T~

= T°,

un

[ ] =

fJ.

(tet)

0

()

0 [ ]

-'--

v (t,,)

w;

(-r)

(l.

2

6)

t<1.

-r

s

(Tilt)

w

iet

-r,

V

tllt

-r

- s

(toc)

t,,'

01

.

IIoRamoM,

lITO

3TO

rrpep;rrOJIOiReHHe

crrpaBep;JIHBO. B

caMOM

p;OJIe,

p;onycTHM CHa'IaJIa,

lITO

sap;alIa 42.1

HMeeT

perneHHe.

Torp;a,

RaRoe

6J)I

yrrpaBJIeHHO

V.

(.)

(toc

-<

1:

<"

tilt

+

T°),

CTeCHeHHoe

YCJIQBHeM

X [V. (-r),

-r;:;;;.t

et

] <"v(toc),

(42.7)

MbI

HH

BSHJIH,

yrrpaBJIeHHe

u.

(1:)

=

V.

(1:)

+

w~1lt

(1:)

(t

oc

<"

1:

<"

tilt

+

T°),

Bo-rrepBbIX, 6yp;eT

yp;OBJIeTBOpHTb

orpaHHlIeHHIO

X [u" (-r),

-r

> t

oc

] <

~

(l&), (42.8)

"

r'

I~",

H60 rro

CBOHCTBy

HOpMbI

X

[u.

(-r)]

<"

X [v.

(1:)]

+X

[wf,,(1:)]

<"

V

(toc)

+ S

(tet)

=

It

(tet),

a BO-BTOpbIX, yrrpaBJIeHHe u.

(1:)

06ec-

rrelIHT

paBeHCTBO

x

(toc

+ T°) =

0,

T.

e. B

MOMeHT

i}

=

tilt

+

T°

6yp;eT

06ecrrolIeHa BCTpelIa

p;BHiReHHH

y (-r) H z (-r).

OTcIOp;a

CJIep;yeT,

lITO

rrpH

i}

= t

oc

+

T°

06JIaCTb

G(2)

[tilt,

z (t,,), V

(tilt),

i}]

COp;epiRHTCH

n;e-

JIHI{OM

B 06JIaCTH

GW

[tilt,

y (t,,), v

(tat)

,i}].

CJIep;OBaTeJIbHO

i}0

<"

i},

HJIH, HHalIe rOBopH,

T~

<" T°. Op;HaRo

cTporoe

HepaBeHCTBO

sp;eCb

He-

BOSMOiRHO.

B

caMOM

p;eJIe,

rrpep;rrOJIOiRHM

Terrepb,

lITO

cyrn;ecTByeT

MOMeHT

rrOWIorn;e~HH

t =

i}0

=

tat

+

T~,

1I

BbIBep;e~

HepaBeHcTBo

T° <"

~,

(42.9)

rrpOTlIBOrrOJIO}HHOe

rrpep;bIp;yrn;eMy. PaccMoTplIM C

3TOH

n;eJIbIO

sap;alIy

13.1

06

OrrTlIMaJIbHOM yrrpaBJIeHlIH cHcTeMoH

(42.4)

HS

TOlIHH

x =

=

x

(tilt)

B TOlIRY x

(t

et

+

T~)

= 0 rrpH

yCJIOBHH

MHHHMyMa HHTeH-

ClIBHOCTH

f{

[w

(-r)]

=

~o

=

miJ1

(tilt

<"

1:

~

t" +

T~).

(42.10)

§

42]

SA~AlIA

0

BCTPEQE

O~HOTHl1HbIX

OE'DERTOB

337

CHCTeMa

(42.4)

BrrOJIHe

ynpaBJIHeMa,

rrOC>TOMY

rrOCJIep;HHH

sap;a'Ia

06HSaTeJIbHO

HMeeT

pemeHHe

(CM.

cTp.138).

ECJIH ORameTCH rrpH

3TOM,

lITO

B (42.10)

SO

<" s (l,,),

TO

31'0

YiRe 6yp;eT OSHalIaTb

BbIrrOJIHeHHO

He06xop;HMoro HaM HepaBeHcTBa

(42.9).

II03TOMY rrpep;rrOJIOmHM

01'

rrpOTHBHoro,

lITO

SO

>s

(let).

(42.11)

'I'

IlTaR,

rrYCTb

Wt"

(1:)0

- OrrTlIMaJIbHOe ynpaBJIOHHe, paspomaIOrn;eo

sap;alIy

06

yrrpaBJIeHHH

ClICTeMOH

(42.4)

rrpH

yCJIOBHHX

(42.10), (42.11).

I

Torp;a

BbIrrOJIHHeTCH

paBeHCTBO

x(toc

+

T~)

= X

(let

+

T~,

tilt]

x(t,,) +

o

tet+T.

+

X

ftet

+

T~,

-r]

BW

IIlt

(-r)O

dT

= 0,

(42.12)

k

let

:;

rp;e X - <pYHp;aMeHl'aJIbHaJI MaTpHn;a

CHC1'OMLI

(42.4)

npM W =:

O.

BhI6epeM yIIpaBJIOHlIO v.

(1:)

= v

(tet)

w

iet

(1:)0:

so.

81'0 yrrpaBJIOHHe

yp;OBJIeTBOpHeT

orpamrlIoHMIO(42.7)

M

rrpHBop;M1'

P;BHiR0HHe

z

(t)

B

.J

HeRoTopoe

COCTOHHHO

.

f

Z

(tet

+

T~)

= X

(let

+

T~,

t

et

] z

(let)

+

't

+T

o

et

•

v

(tet)

+

-s-o-

:)

X

[ta.

+

1'~,

-r]

Bw,a.

(-r)O

dt.

(42.13)

ta.

IIo

CMbICJIY

BeJIHlIHHbI

T~

P;OJIiRHO

cyrn;OCTBoBaTb yrrpaBJIeHHe

u.

(1:),

ROTopoe

YP;OBJI0TBOpHeT

orpaHHlIeHHIO

(42.8)

H

rrpHBop;J!I'l'

CH-

:i

•

,

cTeMy

(42.1)

B

TO

me

caMoe

rrOJIOiReHHe

y

(tilt

+

T~)

= z

(ta.

+

T~).

:r

I

lfHalIe

rOBopH, cyrn;ecTByeT yrrpaBJIeHHe u. (-r), ROTopoe

COrJIaCHO

(42.13)

yP;OBJIe1'BOpHeT

yCJIOBHIO

X[ta.+T~,

ta.](y(t,,)-z(la.»

+

t

+To

Ilt •

[V

(t 1 ]

+ X (la. +

T~J

-r]

B u.

(-r)

- T w

iet

(-r)O

d-r

= 0

(42.14)

Ilt

H HepaBeHcTBy

(42.8).

lfs

(42.12)

H

(42.14)

saJ\JlIOlIaeM Terrep!>, lITO

yrrpaBJIeHHe

U.

(1:)

yP;OBJIeTBOpHeT

paBeHcTBy

O

I

+T

a.

•

.\

X

[tilt

+

T~,

-r]

Bu.

(-r)

d-r

+

fer,

V

(tilt»)

Tn,

'.

+

(

1 +r X

[ta.

+ 1'.,

l",]

x (t",) =

O.

:·,:,:·:·::

.•

:~:.:.;£~m~

-------------~~-~,~,

,~

..

__

_'

..

_'_'>o_",'.;..:,'

~.~~,_~~~__,___

"

'~""_

'..:....'

~...:..:.......:~~

._ ...

-,,-

,',

.....

--;~._:~~

o

~-.:>~

[rJI,

10

llTPOBAfl

3A;O:AqA

0

BCTPEqE

;O:BMlREHllJl

338

r

Ho

B

TaKOM

CJIy'lae

yrrpaBJIeHlIe

w.

('t) =

U.

('t) :

(1

+v

(tex)!~O)

!

6y;o:eT

y;O:OBJIeTBOpHTb

HepaBeHcTBy

f.t

(tocH

O

<ro

f

X

[

W.

(

't

)]

~

~o

+"

(t

oc

)

...

(TaK

KaK

rro

rrpe;o:rrOJIOlReHIIIO

~o

+ V

(t

oc

) >

"'"

(t(1.

)).

KpOMe

Toro,

l!>YHK:U;IIH

w.

('t)

06eCrre'llIBaeT

pa:BelICTBO

t

ex

+1'0

•

~

X

[tex

+

T~,

't] Bw. ('t) d't + X

[t

ex

+

T~,

tocl

x

(tex)

= 0,

lex

KOTopoe

03Ha'laeT,

'ITO

yrrpaBJIeHlIe

w.

(1:)

rrepeBO;O:lIT

CIiCTeMY

(42.4)

1I3rrOJIOlReHlIH

x = X (t

ex

) B

COCTOHHlIe

x

(t

ex

+

T~)

= 0.

O;O:HaKo

TaKoro

yrrpaBJIeHIIH

w.

(t)

C

HOpMOH

x [w. {'t)]

MeHbmeH,

'1eM

orrTII-

MaJIbHaH

IIHTeHCIIBHOCTb

~o,

6bITb

He

MomeT.

IIoJIyqeHHoe

rrpoTIiBope-

'1lIe

;o:oKa3bIBaeT

HepaBeHcTBo

(42.9).

'Y''1IITbIBaH

06a

;o:oKa3aHHbIx

He-

paBeHcTBa

~ <

TO

II

TO

<

T~,

y6em;o:aeMcH

B

3KBliBaJIeHTHOCTII

Be-

JIH'IHH

T~

II

TO.

Terrepb

OCTaJIOCb

rrpOBepHTb,

'ITO

rrpaBHJIO

(42;6)

03Ha'laeT

KaK

pa3

rrplI:u;eJIHBaHHe

B

TO'lKy

qO.

He

rrplIBop;H

rro;o:p06Horo

;o:oKa3aTeJIbCTBa,

rrOHCHHM

JIMmb

COBceM

KpaTKo,

rro'leMy

aTO

TaK.

JIerKo

BIIp;eTb,

'ITO

yrrpaBJIeHIIH

U~(1.

['t]

II

vt

[1:](42.6)

HBJIHIOTCH

'~~.

onmUJ,WAbHbUtU

yrrpaBJIeHHHMH

(B

CMbICJIe

3ap;aQH

13,1),

rrepeBop;HIIJ;H-

.~

~II

MM

COOTBeTCTBeHHO

CHCTeMbI

(42.1) H (42.2)

H3

COC1'OHHIIH Y

(t

ex

) H

"'~r.

Z

(t

) B

O;O:HO

II

TO me

COCToHHlIe

q = y

(t

ex

+

TO)

= z

(tex

+

TO).

"

~,

ex

OTcIOp;a

BbIBO;O:HTCH,

'ITO

TO'lKa

q =

qO

JIemliT

Ha

rpaHH:u;ax

06JIaCTeH

1',

G(l)

H

G(2),

'ITO

II

Tpe60BaJIOCb

rrpOBepliTb.

liTaK,

MbI

BII;O:HM,

'ITO

rrpH-

'1""1

:u;eJIHBaHHe

BTO'lKy

qO

CBO;o:HTCH K

pemeHIIIO

yme

1I3BeCTHOH

HaM

3ap;a-

'1H

42.1 0

rrpe;o:eJIbHOM

6blcTpop;eHcTBIIII.

O

Terrepb

rrOKameM,

'ITO

iJ1i,CmpeMaJl,bHbLe

ynpaBJI8HHfl

UO, V , ;o:eH-

CTBHTeJIbHO,

06eCne'lHBaIOT

MHHHMaKC

TO

=

min

max

T

BpeMeHH

p;o

'H.

'lJ.

BCTpe'lH.

'Y

CJIOBIiMCH

CHa'laJIa

06

O;O:HOM

0603Ha~IeHIIII.

IIMeHHo

1'y

06JIaCTb

B

npOCTpaHCTBe

{x,

~},

r;o:e

3a;o:a'la

42.1

HMeeT

pemeHHe,

6y-

;o:eM

0603Ha'laTb

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(42.6),

a CJIe;O:OBa-

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II

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{x

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ex

) = Y

(tex)

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oc

), 9

(tex)

=

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(t

ex

) -

_ v

(t

)}

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B r.

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t =

to

ex

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BbI6lIpaH

Bce

BpeMH

no

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(42.6)

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It, y (t), z (t),

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=

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(42.15)

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§

41,

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o

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0

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~39

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1}0]

no

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P;O

Tex

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y (t) C z (t).

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(40.6)

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y

(1}),

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t =

1},

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.&

Y

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= X

[1},

t]

Y

(t)

+

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[1},

't] U

(1:)

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t

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~

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t

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C

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U

(1:)

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n

U ('t) =

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lihli]

[1},

't] =

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[1},

't]l,

i=l

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13.1.

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3

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,

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.&

y

(1})

= X

[1},

t]

Y (t) +

~

H

[1},

't]

H'

[1},

't]ld't,

,t

t

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li

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.&

~

l'H

[1},

't]

H'

[1},

't']ld't' -

CD

(l17

...

,

In)

<

"",2

(t),

t

r;o:e

<D

-

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li'

A aTO H

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H

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ero

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G(2).)

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t = t.

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lReTCH,

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rpaHHIJ;bI

G(l)

[t., y (t.),

"'"

(t.),

1}0]

H

G(2)

[t.,

z (t.),

v (t.),

1}0]

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npo-

CTO

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CBOHMH

TO'lKaMIi.

Ho

aTO

B03MomHO,

O'leBII;O:HO,

JIlImb

rrpH

yCJIOBHH,

'ITO

"'"

(t.) = v (t.)

H x (t.) = Y (t.) - z (t.) = 0,

T.

e.

'ITO

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yme

rrp01I30-

mJIa

nplI

t = t. <

1}0.

IITaK,

rrpHXO;O:lIM

K

BbIBo;o:y,

'ITO

yrrpaBJIeHHe

Красовский Н.Н. Теория управления движением

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