Красовский Н.Н. Теория управления движением

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CBOJlIC'!'BA

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220

ConOCTaBKM

KCXOAHOH

aaAalJe

06

ynpaBJIeHKH

cKcTeMoH

(24.37)

dx

=

Axdt

+

BdU

npK

yCJIOBKlI

t/3

(24.38)

~

IldU

('t)11

~

mJn

'"

aaAa'ly

06

ynpaBJIeHKII

clIcTeMoH

(24.39)

X=[A+81

G

]x+Bu

nplI

yCJIOBKK

t/3

(24.40)

x

E

[u]

=

max

[8

2

m"axllu

('t)

II,

~

Ilu('t)lld't] =

min

t U

'"

(K

npK

Tex

iRe

KpaeBbIX

AaHHbIX

x"',

x.B).

3Aecb

nOCTOHHHbIe

8i

> 0 -

MaJIbIenapaMeTpbI,

a

MaTpKD;a

G

nop;611paeTCH

TaKIIM

06paaoM,

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WYHKIJ;KH

hE

('r),

nOJIYlJaIO-

I

m;aHCH

npK

pemeHKK

aap;alJK

(24.39), (24.40)

no

npaBKJIY

MUl-lU-

MaK-Ca

17.1,

OKaaaJIaCb

OTJIK'lHOH

01'

TOiRp;eCTBeHHoH

nocToHHHOH

Ha

JII060M

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1I3

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t",

-<

't'

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tfJ.

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K

nplITOM

6eClJHCJIeHHbIM

KOJIKlJeCTBOM

cnoc060B.

3ap;a'la

06

ynpaBJIeHHK

(24.39), (24.40)

aHaJIOrKlJHa

aap;alJe

06

ynpaBJI8HlIK

nplI

!'

yCJIOBKH

MHHKMyMa

BeJIKlJHHbI

x

[u]

(24.19) . .Il:JIH

nepexop;a

01'

BeJIK-

'lKHbI

x"

[u]

BIIp;a

(24.19) K

BeJIKlJlIHe

XE

[u]

(24.40) p;OCTaTOlJHO

YMHOiRKTb

(24.19)

Ha

1/A

K'nOJIOiRlITb

82=

1/A.

OTCIOp;a

3aKJIIOlJaeM,

lJTO

OnTIIMaJIbHOe

ynpaBJIeHKe

u

E

(t)'P;JIH

BcnOMOraTeJIbHOH

3ap;alJK

(24.39), (24.40)

OTJIlIlJHO

01'

HyJIH

JIKmb

B

oKpecTHocTHX

Tex

TOlJeK

t

==

't'j,

rp;e

WYHKD;KH

W(E)

('t')

=

II

hE

('t')

II

p;ocTHraeT

CBOKX

Ha1I60JIb-

mKX

3HalJeHKH.

CyMMapHaH

p;JIKHa

3TlIX

oKpecTHocTeH

eCTb

8

2

,

MII-

HHMaJIbHaH

WYHKD;KH

h!

('t')

= B's!.('t')

onpep;eJIHeTCH

npK

3TOM

K3

yCJIOBKH

/3

1

\"

m:x

e;

.)

IIB'sE

('t)

II

d't

=

m:n

= p!, (24.41)

ll",

rp;e

s!

('t')

-

p;BKiReHKe

CIICTeMbI,

COnpHiReHHOH

K

CKCTeMe

(24.39),

CTeCHeHHoe

YCJIOBlIeM

SE'

(t/3)c = 1.

3p;ecr;.

!1

-

CIICTeMa

IIHTepBaJIOB

U3

oTpe3Ka

[tilt, t/3],

IIMeIOm;aH

cyMMa

pHyIO

P;JIUHy

(J

(/1)

= 8

2

,

CaMO

OnTUMaJIbHOe

ynpaBJIeHUe

U!('t')

TaM,

rp;e

OHO

OTJIK'lHO

01'

HyJIH,

yp;OBJIeTBOpHeT

paBeHcTBy

II

U!

(t)

II

=

_1

.'

(24.42)

82p!

c~·

~~~-:

' .

§ 25]

YITPABJIEHHE

MEXAHH~ECHOJII

CHCTEMOJII

221

.Il:aJIee

npOBepHeTCH,

lJTO

npu

8i

--+ 0

ynpaBJIeHKe

u

E

(t) CXOP;UTCH

K

HMnyJIbCHoMy

OnTUMaJIbHOMY

ynpaBJIeHKIO

HCXOP;HOH

aap;alJH.

Ta-

HHM

.

nyTeM,

paCCMaTpUBaH

npep;eJIbHbIH

nepexop;

lim

'tj = 'tj

]I

lim

hE

('t')

=,

hO

('t')

npu

8i

--+ 0,

onpep;eJIHeM

MOMeHTbI

't'j

npKJIOiReHKH

KMnYJIbCHbIX

B03p;eHCTBliH

U

UX

BeJIU'lHHy

P;JIH

3ap;alJU

(24.37), (24.38).

1I3YlJeHUe

H

060cHoBaHHe.

3Toro

npep;eJIbHOrO

nepexop;a

B

06m;eM

CJIy-

'lae

H

BO

Bcex

p;eTaJIHX

BbIXOP;HT

3a

paMKK

3TOH

KHKrK.

TaKoe

H3YlJe-

Hue

MOiReT

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npep;MeT

nOJIe3HOrO

caMOCTOHTeJIbHOrO

UCCJIep;o·

BaHlIH

P;JIH

KHTepeCYIOm;erOCH

lJIITaTeJIH.

llpH

3TOM

u3y'leHHe

p;aHHoro

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MOiRHO

HalJaTb

C

KOHKpeTHoro

MaTepKaJIa,

pacnpOCTpaHKB

a

ap;alJy

,

paCCMoTpeHHyIO

B

npHMepe

23.3,

p;o

HOpMbI

x!

[u]

BHp;a

(24.40).

§

25.

YnpaBJIeHHe

MeXaHUlJeCKOH

CUCTeMOH

B

OKpeCTHOCTH

nOJIOmeHHH

paBHOBeCHH

B

npep;bIp;ym;ux

naparpawax

MbI

1I3YlJaJIlI

aap;alJy

06

OnTIIMaJIb-

HOM

ynpaBJIeHlI1I

K

YCTaHOBIIJIU

aaKOHOMepHOCTH,

xapaKTeplI3yrom;lIe

ee

pemeHlIe,

paCCMaTplIBaH

ynpaBJIHeMbIe

CIICTeMbI

npOlI3BOJIbHOH

npHpop;bI,

nOBep;eHlIe

KOTOpbIX

MOiReT

6bITb

onHcaHO

B

JIKHeHHOM

nplI6JIKiReHlIH

p;uwwepeHD;lIaJIbHbIMH

ypaBHeHHHMH

(2.4) , (6.11).

B

HaCTOHm;eM

naparpawe

MbI

06CYP;IIM,

KaKOH

BlIP;

npHHHMaIOT

3TlI

aa-

KOHOMepHOCTII

B

KOHKpeTHOM

CJIylJae

ynpaBJIHeMbIX

MeXaHKlJeCKHX

CIICTeM,

a

TaKiRe

BbIHCHKM

BJIKHHKe

P;OIIOJIHIITeJIbHbIX

CHJI

pa3JIK'lHOH

nplIpop;bI

Ha

CBOHCTBO

ynpaBJIHeMOCTH

3TlIX

CIICTeM.

llPII

3TOM

orpa-

HH'lIIMCH

paCCMOTpeHlIeM

p;BHiKeHHH

rOJIOlIOMHbIX

KOHcepBaTKBHbIX

CIICTeM

[22*] B

OKpeCTHOCTII

nOJIOiKeHHH

paBHOBeCIIH.

IITaK,

nYCTb

3ap;aHa

rOJIOHOMHaH

KOHcepBaTIIBHaH

MeXaHlIlJeCKaH

CIICTeMa

C n

CTeneHHMH

cB060P;bI,

nop;BepiKeHHaH

BHemHIIM

ynpaBJIH-

IOm;IIM

B03p;eHcTBIIHM

U

1

(t),

...

,U

r

(t).

COCTOHHHH

TaKOH

CIICTeMbI

xapaKTeplI3yIOTcH

ee

0606m;eHHbIMH

Koopp;HHaTa~m

qi

(i

=

1,

... ,

n).

llpep;nOJIOiRKM,

lJTO

nplI

Uj

= 0

(j

,=

1, , r)

CKCTeMa

06JIap;aeT

no-

JIOiKeHlIeM

paBHoBecHH

qi

= 0

(i

= 1, , n).

YpaBHeHIIH

JIarpaHiRa

([22*],

CTp.

331),

onlICbIBaIOm;Ke

ee

p;BHiKeHIIH

B

OKpeCTHOCTH

TOlJKH

qi = 0, qi = 0

(i

= 1,..., n), B

nepBOM'

npH6JIlIiKeHlIU

KMeIOT

BlIP;

(CM.

§ 3)

d

(aTO

\)

aTO

+

ano

_

'"

r

b?

,

-

-.-

-

--

- LJ

tJU

J

(i=1,

...

,n).

(25.1)

dt

'aqo

aqi'

aqi

.

~

J=1

3p;eCb

n

1

'"

0

••

TO

="2

aijqiqj,

LJ

°

i,

i=1

t,I-1

LJ

no

="2

1

'"

'Ciiqjqj

(25.2)

222

._--~--~._

c

CBOnCTBA

OllTHMAnhHOrO YllPABnEHHH

lrJI. 6

-

cOBoKynHocTH

qneHOB

HaHHHamerO

(BTOpOrO)

nOpHp;Ka

B

paaJIO-

meHHHX

KHHeTHqeCKOH

H

nOTe~HanhHOH

aHeprHH

B pHP;

no

qi

H

qi

B

OKpeCTHOCTH

nOJIOmeHHH

paBHOBeCHH;

aiOj,

bf

j

,

Ci}

-

nOCTOHHHMe.

06cymp;eHHe

aap;aqH

B

JIHHeHHOM

npH6JIHmeHHH

(25.1)

IJ;eJIeco06-

paaHO

no

TOH

nplfqHHe,

'ITO

pemeHHe

JIHHeapHaHpOBaHHOH

aap;aqH

pacnpOCTpaHHeTCH,

KaK

MM

BIIp;eJIII

B § 22,

Ha

aHaJIOrlfqHYIO

aap;aqy

P;JIH HCXOP;HOH

HeJIHHeHHOH

CIICTeMM

d ( aT ) aT

all

dt

aqi

- oqi + aqi = Bi (qto

...

,

qn;

Ul, ,

U,.)

(i

-1,

, n). (25.3)

n

00

3p;ech

T =

~

aij

(qh

...

, qn) qiqj,

TI

=

~

n(k)

(qb

•..

,

qn)-

i,

j=1

/,=2

COOTBeTCTBeHHO

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H

nOTe~lIaJIhHaH

aHeprHH

CIICTeMM,

npH

aTOM tPYHKIJ;HH

aij

-

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OT

0606m;eHHMx

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qi

CIICTeMM,

a tPYHKIJ;1I1I

TI(k)

npep;CTaBJIHIOT

C060H

OP;Hopop;HMe

tPOPMM

k-ro

nopHp;Ka

OT

aTIIX

Koopp;IIHaT;

B

i

=

00

•

~

b(i" ... ,

in;

""

"',

/',.)

i,

.

in

Ii,

/',. 6

6'

LJ i .

ql

...

qn

,Ul

...

u,.

- 0 0 -

i,+.

..

+in+k,+.

..

+k,.=l

m;eHHMe

CHJIM,

nopomp;aeMMe

ynpaBJIHIOm;HMH

BoaJ~eHCTBHHMH

·b(i"

....

,

in;

k"

....

, Ii,.) _ .

b(ih

... ,

in;

0,

...

,0)

-

0)

3 .

U

1

,

.••

, U

r

(

i -

const,

- .

aMerHM,

'ITO

B (25.2), (25.1)

a?;

= aij (0,

...

, 0),

n°

=

n(2)

(ql,'

.

..

, qn),

? _

b~o,

.... 0;

0,

...

, "'j'

...

,0)

b

.,

- . .

3anlImeM

ypaBHeHHH

(25.1) B

r;,aH01f,U'lecr;,ux

r;,oop8uJ-lamax([25*l.

CTp.

97-99).

liIMeeM:

,.

iii

-:-

"'iYi +

L}

di;uj

(i

= 1,

...

,

n).

(25.4)

j=1

3p;ecb

nepeMeHHMe

Yi

(KaHOHII'IeCKHe

Koopp;IIHaTM)

CBHaaHM

C

0606-

m;eHHbIMH

Koopp;IIHaTaMII

qi

paBeHCTBaMH

qi =

~ilYI

+...+

~inYn'

(25.5)

,l];eHcTBIITeJIbHMe

qlICJIa

Ai

YP;OBJIeTBOpHIOT

ypaBHeHlIIO

I

aY

j

",

+

c?j

If,

j=1

= 0, . (25.6)

a

p;eHCTBHTeJIbHMe

qlICJIa

diJ

onpep;eJIHIOTCH

cooTHomeHHHMH:

n

d

tj

=

~

~ktbgj,

k=1

(25.7)

n

l;

(agj"'k +

C~j)~ki

=

O.

}j.=J

r.·.·

'(-,,>-~~~

§ 25]

YITPABJIEHME

MEXAHHqECKOn

CHCTEMOfi

223

TIYCTb

aap;aHa

IIHTeHCIIBHOCTb

x[uJ

(HJIH

x [dU)),

oIJ;eHHBaIOm;aJl

pecypcM,

aaTpaqHBaeMMe

Ha

ynpaBJIeHHe

CHCTeMOH (25.4)

Ha

OTpea-

Ke

BpeMeHH

t

a

-<

't'

-<

tf3.

B COOTBeTCTBHH C

MaTepHaJIOM

§§

12-14

MM

CqHTaeM,

'ITO

BeJIH'IlIHa

x

[uJ

(IIJIH

x[dU))

IIMeeT

CMMCJI

KaKOH-

HlI6YP;b

HOpMM

p*

[uJ

(p*

[UJ) tPYHKIJ;HII u('t')

(HJIH

tPYHKIJ;HHU('t'».

01'paHHqlIMCJI

JIlImb

aap;aqeH

06

ycnoKoeHlI1I

CIICTeMM

(25.4),

Korp;a

Tpe6yeTcJI

nepeBeCTII

ee

Ha

COCTOJlHIIJI

p;BHmeHHH

{Yi

(t

),

iii

(t

)}

B

a

nOJIOmeHlIe

paBHOBeCIIJI

{Yi

(tf3)

= 0,

tli

(til) =

O}

C

MHHIIMaJIbHOH

aaTpaTOH

ynpaBJIHIOIIJ;HX

yCIIJIHH,

T.

e.

npH

YCJIOBIIH

x

[uJ

=

min

(x[dUJ =

min).

YCJIOBHH

paapemHMOCTH

:3TOH

aap;aqH

IIpH

JII06MX

HCXOP;HMX

p;aH-

HMX

{Yi (t

a

),

Yi

(ta)}

yKaaaHM

BMme,

B § 17, a B §

19

OHH

CtPOPMYJIH-

pOBaHM

B

atPtPeI{TIIBHOH

tPopMe.

BMJlCHHM,

'ITO

OaHaqaIOT

3TH

yCJIOBHJI

B

HameM

KOHKpeTHOM

CJIyqae

ynpaBJIHeMOH

MeXaHHqeCKOH

CHCTeMM

(25.4), (25.1). B COOTBeTCTBHH C

MaTepHaJIOM

§

19

COCTaBHM

MaTpHIJ;y

K =

(B,

AB,

...

,An-lB)

(25.8)

P;JIJI

CHCTeMM

(25.4). C

aTOH

IJ;eJIbIO

aanHmeM

cHcTeMy

(25.4) B

HOp-

MaJIbHOH

tPopMe,

BBep;H

060aHaqeHlIH

X2i-l

=

Yi'

X

2

i =

Yi

(i

= 1,... ,

n)

•

TIOJIyqaeM:

.

X2i-l

= x2

i,

X

2

i =

Ai

X

2i-l +

l;

dijuj

(i

= 1,

...

, n). (25.9)

j=1

Torp;a

....,

o 1

..

a

011

110

...

0

Al

0 :

..

0 0

Id

ll

'"

d

Ir

o 0

...

0 0

A =11·····

"'1/

B=

I

j

I/O

...

0

o 0

...

0

1/1

: d

nI

.,.

dn,.

o 0

...

"'n

0

H

c

0

...

0

d

",1.:

d

AI;

11 1 .

..

1,. 1

/.

I'

d

ll

",;

'"

dll'Ai

0

" .

0

.

..

.

..

A.

2k

B

II'"

....

=

.........

,

A

21

""IB =

.....

"

...

::: ::::::

II.

(25.10)

0

...

0

d ",Ii d

","

1ft n •

..

nr

11.

,.

I'

d111A,,~

'"

dn,.'}.,~

0

...

0

W~.'~~.·_

224

~~~=~~~~.'

.'.

;.,:'

.....

,....

:.'.~.;;',":.'';'"~;

CBOn~TBA

OnTMMAnhHOrO

YITPABnEHMH

[l'JI.6

3aAaqa

06

OTITMManbUOM

ynpaBneUJIlI

CMCTeMOH

(25.9)

MMeeT

peme-

UMe

TorAa M TonbRO TorAa,

RorAa

paul'

MaTpMn;bI

K

(25.8)paBeu

2n

(CM.

§

19).

IIpoBepRa

;}TOrO

ycnoBMH

HBJIHeTCH,

Bo06~e

rOBopH, aa-

TpYAHMTenhuoH.

B uameM CJIyqae

MeXaUMqeCROH

CMCTeMbI

(25.4)

YC-

JIOBMH

paape~OCTM

aaAaqH 06

OnTBMaJIbUOM

ynpaBJIeUJIlI

Momuo

CcPOpMyJIHpOBaTb

TaRme B

APyroH

cPopMe,

ROTopaH

HuorAa

YA06uee.

C

;}TOH

n;eJIbIO

paCCMOTpHM

MaTpMD;y

K

1

=

II~

:11,

rAe

d

n

d

1r

,

..

"

dllA.~-l

...

dlrA.~-l

,

...

,

N=

,

"0'

(25.11)

,

..

0'

dnI

dnr~

. .

0'

dn1A~-1

...

dnrA~-l

HeTpYAUO y6eAHThcH B

TOM,

qTO

B

CHJIy

paBeUCTB

(25.10)

MaTpHD;a

K

1

nOJIyqaeTCH

JIa

MaTpMD;bI

K

(25.8)

npocToH rrepecTauoBRoH

CTpOR.

CneAOBaTeJIbUO,

paul'

MaTpMD;bI

K

(25.8)

paBeu

2n

TorAa M

TOJIbRO

TorAa,

RorAa

paul'

MaTpMD;bI

N

(25.11)

paBeu n.

Ho

MaTpMD;a

N

COB-

rraAaeT C

MaTpMn;eH

K,

COCTaBJIeUUOH

AJIH

CMCTeMbI

r

Yi

=

AiYi+

2j

dijuj

(i

= 1,

...

, n). (25.12)

j=l

TaRHM 06paaoM,

MbI

rrpHxOAHM R aaRJIIOqeUnIO,

qTO

MeXaHMqeCRaH

CHCTeMa

(25.4)

BrrOJIUe

yrrpaBJIHeMa TorAa H

TOJIbRO

TorAa, RorAa

BnOJIUe

ynpaBJIHeMa

CHCTeMa

(25.12),

rrOJIyqaIOID;aHcH

Ha

uee aa-

MeUOH

yCRopeuMH jii

ua

cooTBeTcTByIOID;Me

CROpOCTH

Yi'

)J;aJIee, rrpeArrOJIOmMM,

qTO

cpeAH

qMCeJI

AI"

..

,An

6YAeT k paaJIMq-

UbIX,

HMeIOID;HX

COOTBeTCTBeuuo

RpaTUOCTM

i

1

,

...

,i

k

•

He

uapyIIiaH

06ID;UOCTH,

yrropHp;oqMM Bce

qHCJIa

Ai

TaR:

Al

=

...

=

Ai"

Ai,+!

=

...

=

Ai.+i"

...

,

Ai,+

...

+ik_1+

1

=

...

=

Ai,+

...

+i

k

;

Ai,+

...

+i

j

<

Ai.+

...

+ij+t

(j

= 1,

...

,

k-1).

CocTaBMM

MaTpMD;LI

d

n

...

d

1r

II II

di,+.

..

+ili_l+1.

1

...

di,+...

·Hk_1+1.

r

,1.1=

,

'"

,1.

-

k-

dill

dill'

d

i

,+

...

+i

k

,l

...

d

i

.+.

..

+

ik

, r

(25.13)

I

C

··'

.'.'.:~>i

§

25]

yrrPABJIEH

ME

MEXAHM'IECIWi1

CMCTEMOi1

225

lIMeeT

MeCTO

CJIeAYIOID;ee

yTBepmAeUMe.

Teo

p e M a

25.1.

JI:/L.fL

m080

"tmo6b~

:MeXallUtteC1W.fL

cucme:Ma

(25.4)

«25.1»

6b~a

enOlLlle ynpaM.fLe:MOU, Jieo6xo8u:MO U

8ocmamo"tllo,

f.lmo6u

pall8U

:Ma.mpu/,f

.1

1

,,,,

,Ll,.

(25.13)

6b~U

paeJib~

coomeemcmeellJiO

i

1

,

..

, i

k

• .

MbI

ue

6YAeM

OCTaUaBJIMBaTbCH

ua

rroAPo6HOM

AORaaaTeJIbCTBe

;}TOrO

yTBepmAeUMH,

OTCbIJIaH

qMTaTeJIH

R

CrreD;MaJIbUOH

JIMTepaType

[51,

160aJ.

OTMeTMM

JIMIDb,

qTO

cnpaBeAJIMBOCTb

ero

BbITeRaeT

M3

CTpyRTypbI

MaTpMD;bI

N

(25.11),

ROTopaH C

rrOMO~bIO

;}JIeMeHTapHbIX

rrpe06paaOBaUMH

([3*],

CTp.

110)

MomeT 6bITb rrpMBeAeua K

BMAY

[160a]

.1

1

0

o

M

21

.1

2

o o

o

M

I

•

l

M

I

•

2

• . •

d

k

0

o

1I3

TeOpeMbI

25.1 BbITeRaeT, B

qaCTUOCTM,

qTO

MeXaHMqeCRaH

CMCTeMa

(25.1)

«(25.4»,

rrOABepmeHHaH

B03AeHcTBMIO

bO

u, rAe u -

CRaJIHp, Brronue yrrpaBnHeMa

3TMM

B03AeHcTBMeM

TorAa H TonbRO

TorAa, RorAa

Ai=FAj npM

i=Fj

M

dli=F0

(i,j=1,

...

,n).

(25.14)

IIOCJIeAHMe

COOTuomeHHH

HMeIOT

rrpocToH

MeXaHMqeCRMH

CMbIcn,

ROTOpbIH

MomHO

yHCHMTb

M3

cneAYIOID;ero npMMepa.

ITpoMep

25.1.

PaccMoTpHM

p;BHJKeHHe TRJKeJIOH

MaTepHaJIbHOH

TO'IRH

m

no

rJIap;ROH

nOBepxHocTO

Z = f

(x,

y).

ITpep;nOJIaraeM,

'ITO

COCTeMa

Koopp;HHaT

(x,

y,

z)

opTOrOHaJIbHaR,

OCH

x H y

HanpaBJIeHhI

no

rJIaBHhIM

HanpaBJIeHHRM

(HanpaBJIeHHR

:lKCTpeMaJIbHOH

KPHBH3HbI)

:lTOU

nOBepxHocTIl

([22*],

CTp.

202)

B

TO'IKe

{O,

0,

O},

KOTOpyIO

C'IHTaeM

CTaI(II:OHapHOH

TO'IKOH

lJ!YHRIJ;HlI Z =

=

I(x,

y),

TaK

KaK

nOJIaraeM

COCTORHHe x = 0, y = °

TO'IKOH

paBHoBecHR;

OCb z

HanpaBJIeHa

BepTHKaJIbHO

BBepx.

ITYCTb

paccMaTpHBaeMaR

MaTepIlaJIbHaR

TO'I-

Ka

nO)J;BepJKeHa

ynpaBJIRIOm;eMY

B03p;eucTBHIO

U,

HMeIOm;eMY

HeH3MeHHoe

ropH30R-

laJIbHOe

HanpaBJIeHHe.

OTKJIOHeHHe

TO'IKH

OT

nOJIOJKeHHR

paBHOBeCHJI

{O,

0,

O}

H CKOpOCTb

ee

p;BHJKeHHR

C'IHTaeM

MaJIl,IMH

BeJIH'IIlHaMH.

BbI6epeM

3a

o606m;eHHblc

Koopp;HHaThI

p;BHJKym;eHcR

TO'I:I\H

ee

p;eKapToBbI

Koopp;HHaThI

x H y.

Torp;a

BeJIH-

'IHHbI

TO,

ITo

(25.2)

onpep;eJIRTCJI

paBeHcTBaMIl:

TO

=

_1_

m

(:t~

+

iJ2),

2

ITO =

_1

mg

[(

(PI)

x~

+

(fl)

Y2J

2

ax~

x=O ay2

x=o

'

y=O y=O

H

ypaBHeHHR

p;BIDl.{eHHR

TO'lKH

B

nepBoM

npH6JIHJKeHHIl

(25.4)

HMeIOT,

CJIep;OBa-

TeJIbHO,

BIlP;

x =

1..

1

X + d

l

U,

fj

=

1..

2

Y + d

2

U.

(25.15)

8 H. H

..

RpacoBcRHH

G'

l....:.

:.'

[rJl.

G

CBOaCTDA

OITTHMAnbHoro

YITPADnEHHfl

226

3AeCb

Al

= g

(a

2

j/ax

2

)=u=O'

A

2

= g

(a

2

j/a

y

2)x=y=o; d

1

=

a)m,

d

2

=

aim

-

OTHomeHHC

HanpaUJIaIOm;1IX

KocHnycou

ax,

au

CRJIbI U

I{

Macce

TO'iKU.

YCJIOUHII

(25.14)

nOJIHOH

ynpaUJIaeMOCTR

CHCTeMbI (25.15)

03Ha'IarOT,

eJIe)l;OUaTeJIbHO,

'ITO

nOJIOIKeHne

paunOBeCHJI

MaTepnaJIbHOH

'T01IKR

{O,

0,

O}

ne

aBJIJIeTCJI map080U

TO'IKOH

(AI

=1=

A

)

nOBepxHocTH

Z = j

(x,

y)

H

ynpaUJIJIIOm;eC

u03ABHcTunc

HC'

2

nanpaUJICHO

no

rJIaBHbIM

HanpaBJIenaaM

nouepxHocTR

B

TO'lKe

{O,

0,

O}

(dt

4=

=1=

0, d

=1=

0).

HanpHMep,

eCJIH

MaTepliaJIbHaJI

T01IKa

m

ABHiKCTCll

no

3JIJIHnTH-

2

'leCKoMY

napa60JIOHAY

(pHC.

25.1)

llJIH

mnep60JIll'IeCKOMY

napa60JIORAY

,(pHC.

25.2),

TO

yCJIOBRe

Al

=1=

A

2

a.n;eCb BbIUOJIHJleTCJI

,

H,

TaKRM

06pa30M,

YCJIOBllJI (25.14)

3

z

PHC.

25.2.

PRC.

25.1.

nOJIHOH

ynpaBJIJIeMOCTll

03naQaIOT,

'lTO

B03.n;eHcTBRc

U

ne

.n;omRHO 6!>lTb

narrpaB-

JIeno

napaJIJIeJIbHO

nJIOClWCTllM

CllMMeTpHll

3THX

nOBepxHocTeR

B

TO'lKe

{O,

0,

O}.

Ecml

lRe

TO'lKa

In

.n;BHIKeTCJI

no

napa60JIOH.n;y

npam;CHHll

(D OHpeCTHOCTR

ero

uepwnlibl),

TO yCJIODRJI (25.14)

3~\eCb

He

DhlllOJIHeHbI,

TaR

Imr,

A'l

= A2·

MTaH,

B

CJIy'rae

pHC.

25.1,

25.2

MbI

ucer.n;a

MOIKeM

pacnopllARThCJI

CHJIOR U

Tau,

'IT06bI

nepeueCTH

MaTepHaJIhnylo

TO'IKy

113

JIl060ro

na'laJIbHOrO

nOJIOIKeHllll

{x

(ta)'

y

(t,J}

n

nOJIOllWIIHe

paBnoBecHlI

{x

(t~)

= 0, y

(t~)

=

O}

R

3aqlHKcll-

ponaTb

ee

Ta~1

C

nyJIeBOH

CIWpOCTbIO.

B

CJJy'lae

IKe

Al

= A

2

,

HanpoTFlB,

AJIa

60JIbllIRHCTna

na'raJIbIlhlX

OTKJIOnenaR

3Toro

c.n;eJIaTb

neJIb3ll,

KaK

6bI

MbI

I1H

Bbl6HpaJIH

ynpaBmnom;ee

n03AeRcTBHe

U,

llMerom;ee

HE'1I3MeHIIOe

ropH30IlTaJIbHOe

HanpanJIeIDIe.

BepHeMCH

K

06m;eH

aap;a'Ie.

I1pep;rrOJIOlliHM

Terrepb,

'.ITO

yCJIOBHH

TeOpeM:bI

25.1 B:bIrrOJIHeH:bI,

H,

CJIep;OBaTeJIbHO,

aap;a'Ia

06

OrrTHMaJIbHOM

yrrpaBJIeHHH

CHCTeMoH

(25.1)

HMeeT

peIIIeHHe.

PaccMoTpHM,

KaK

TpaH-

C<popMHpyeTcH

B

CJIy'Iae

MeXaHH'IeCKOH

CHCTeM:bI (2.5.1)

rrpaBHJIO

MHHHMaKca

17.1.

COrJIaCHO

3TOMy

rrpaBHJIy

P;JIJI

orrpep;eJIeHHH

orr-

THMaJIbHOrO

yrrpaBJIHIOm;erO

Boap;eHcTBHH

U

O

(t)

CJIep;yeT,

rrpellip;e

Bce-

ro,

HaRTH

MHHHMaJIbHOe

p;BHllieHHe

SO

(t)

CHCTeM:bI,

conpHllieHHOH

K

HCXOP;HOH CHCTeMe

(25.9)

(rrpH

U =0).

::ha

conpHllieHHaH

CHCTeMa

HMe8T

BHP;

(25.16)

82i-l

= -

"'i

S

2i,

8

2

i

=

-S2i-l'

TaK

KaK

CHCTeMa

(25.1)

CTaIJ,HOHapHaH,

TO,

He

HapyiliaH

06m;HOCTH,

',

,:""

,

C

§ 25]

ynPABJIEHHE

MEXAHH'IECRotl:

CHCTEMOtl:

227

MOlliHO

C'IHTaTb,

'ITO

t«

~O.

060aHalJHM

t(&

=

'6'.

MCKOMoe

MHHH-

MaJIbHOe

p;BHllieHHe

SO

(t)

M:bI P;OJIlliH:bI

HCKaTb

cpep;lI

p;BlIllieHIilH

S (t)

ClICTeM:bI (25.16), CTeCHeHH:bIX

KpaeB:bIM

YCJIOBlIeM

(CM.

CTp.

114)

S'

(0)

·x

(0)

=

-1.

(25.17)

Cp;eJIaeM

B (25.16)

aaMeHy

BpeMeHlI

t =

{}

-

l'

(0

~

l'

<

(}).

'YpaB'

HeHHH

(25.16)

rrpHHHMaIoT

BlIP;

dS2i_1(~

- 't') =

A.i

S

2i

({)

_

1'),

dS

2i

({) - 't') _

S.

({)

_

1').

(25.18)

--;---

-

2t-1

BYAeM

060aHa'IaTb

'Iepea

S 2n-MepH:bIH

BeKTOp,

nOJIy'IaIOm;HHCH

lIa

BeKTopa

S = {sr'

S2""

,S2n-r,

S2n}

nepecTaHoBKoH

KOMIrOR8RT

C

'IeTH:bIMB

H

HelJeTH:bIMlI

HOMepaMH:

S=

{sr',

S2""'S2n'

S2n-r}.

Torp;a

nOCJIe

BBe·

p;eHHH

060aHalJeHHH

z('t)

=

;(1't-'t)

(25.19)

M:bI

MOllieM

aarrHcaTb

cHcTeMy

ypaBHeHHH

(25.18)

B BHp;e

dZ

2

i-!

('t') =

Z2i

(1'),

dZ

2i

('t') =

"'hi-!

(1'),

(25.20)

a

I<p,aeBoe

yCJIOBHe

(25.17)

B

BHp;e

Z'

({})

.x

(0) = - 1.

(25.21)

Ho

ypaBHeHHH

(25.20)

cOBnap;aIOT

C OCHOBHOH

lJaCTbIO

CHCTeM:bI (25.9).

TaKHM

06paaoM,

aap;a'Ia

onpep;eJIeHHH

MU1f,UMMbfWeo

p;BHllieHHH

SO

(t)

CBeJIaCb

K

HaXOllip;eHHIO'

p;BHllieHHH

~o

(1')

UCXOOHOU

CHCTeM:bI (25.9)

(npH

U 0),

yp;OBJIeTBOpHIOm;ero

KpaeBoMy

yCJIOBHIO

(25.21) H

HMeIO-

=

m;ero

HaHMeHbmylO

BoaMOlliHYIO

HOpMy

pO

= p

[t>

(1')

1

=min

p

[z

(1')]

z

(CM. § 17).

I10cJIe

Toro

KaK

:no

p;BHllieHHe

ZO

(1')

HaHp;eHo,

OrrTlIMaJIb-

Hoe

ynpaBJIeHHe

CJIep;yeT

onpep;eJIHTb

Ha

yCJIOBHH

MaKcHMyMa

(17.1),

KOTopoe

npHHHMaeT

ap;eCb BHP;

~ ~

) B';'o

({)

_

t)

U

O

(t)

dt =

m:x

~

B';'o

({)

-

t)

u (t) dt

(25.22)

o 0

rrpH

p*

[ul

= 1/po.

TaKHM

06paaoM,

M:bI

rrplIxop;HM

K

CJIep;ylOm;eMy

B:bIBOP;Y.

Teo

p e M a

25.2.

OnmUMaltbHOe

ynpa8ltILIO/1.fee

80soeucm8ue

U

O

(t),

peUJ,alO/1.fee

saoa'ly 06 ycnoJli,oenuu

eOltOlwlt£/WU

Jli,OnCep8amu8nou

~eX4Hl,(,'let;JIi,9u

C~Cl1JeMb{.

(~5.1),

onpeOe4MmCi£

npa8u40~~

~~U1f,UMaJli,Ca

8-

·,'

" "

'

..

-~.':.'

C

CBOnCTBA

OllTRMAnbHOrO

YITPABnEHRH

[rJI.

6

228

17.1,

e8e

8a

MUHUMa.ltbHOe

8eu:neeHue

c.ltef)yem

e8R.mb

8eu:neeHue

SO

(t) =

=

ZO

({}-t),

npU'le:M

ZO

(t) - amo

8eu:neeHue

mou

:nee

caMOU

cucme:M1it

(25.9)

npu

U =

0,

y806.ltemeopR.10utee

YipaeeoMy

yc.ltoeulO

(25.21) U

UMelOutee

MUHUMa.ltbHYlO

HOpMy

P

[SO

(I:)

l,

npu

amOM

CaMO

OnmUMa.ltb-

Hoe

ynpae.lteHue

UO

(t)

onpe8e.1tR.emcR.

yc.ltoeueM

MaYiCuMyMa

(25.22).

06CYAHM

Tenepb,

RaRoe

B~HHHHe

Ha

ynpaB~HeMOCTb

ROHcepBa-

THBHOH

MeXaHHqeCROH

CHCTeMLI

ORa3LIBaIOT

AonO~HHTe~bHLIe

CH~LI,

ROTopLIe

MoryT

Ha~araTbCH

Ha

:ny

cHcTeMy,

nOMHMO

ynpaB~mOm;HX

B03AeHcTBHH.

PaccMoTpHM

npH

aTOM

AHCCHnaTHBHLIe

H

rHpOCROnHqe-

CRHe

CH~LI.

OrpaHHqHMCH

c~yqaeM

CRa~HpHoro

ynpaB~HIOm;ero

B03-

AeHCTBHH.

'YpaBHeHHH

(25.4)

HMeIOT,

c~eAOBaTe~bHO,

BHA

ih=

'AiYi

-+

diu.

(25.23)

IIpeAno~omHM,

qTO

CHCTeMa (25.23)

He

BnO~He

ynpaB~HeMa

B03-

AeHCTBHeM

u.

81'0'

03HaqaeT,

qTO

He

BLInO~HeHLI

YC~OBHH

(25.14).

B03HHRaeT

Bonpoc:

cym;eCTBYIOT

~H

TaRHe

AHCCHnaTHBHLIe

H~H

rH-

pOCROnHqeCRHe

CH~LI

([25* l,

CTp.

102, 106),

nOMe

npH~omeHHH

ROTOpLIX

He

BnO~He

ynpaB~HeMaH

CH~OH

U CHCTeMa (25.23) CTaHOBHTCH

BnO~He

ynpaB~HeMoH?

IIpH

aTOM B COOTBeTCTBHH C

06m;enpHHH-

TOH

TepMHHo~orHeH

oYAeM

Ha3LIBaTb

eupOCYionU'I{eCYiUMU

CH~aMH

0606-

m;eHHLIe

CHJT'hI

Qi

BHAa

",

n

ar

r =

~

Yijqiqj

(Yij

= -

YH,

Yii =

const),

Qi

=

aqi

'

i,

j=l

a 8uccunamueliblJrtU -

0606m;eHHhIe

CH~hI

n

aR

1

~

..

Qi=-

aqi'

R

="2

LJ

aiiqiqj

(aii

=

aii,

aii

=

const),

i,

j=l

q>opMa

R -

3HaRono~omHTe~bHa.

IIpemAe

Bcero,

BLIHCHHM

He06xoAHMLIe

YC~OBHH

no~omHTe~bHoro

pemeHHH

nora

Bonpoca.

CnpaBeA~HBo

yTBepmAeHHe.

Ec.ltu cpe8u

'{UCe.lt

'Ai

no

YipaUHeU

Mepe

8ea

paelibt

liy.ltlO

U.ltU

ec.ltu

xomR.

6bt

e 08-

liOM

ypaeHeHUU

cucmeMbt

(25.23) d

i

= 0,

'Ai

= 0,

mo

cucmeMa

(25.23)

He

},w:neem

cmamb

8nO.ltHe

ynpae.fl,R.eMOU

e080eucmeueM

U

Yia-

Yioelit

61it

HU

6bt.ltU

80nO.ltHUme.ltblibte

8uccunamu8libte

U.ltU

eupocYionu'{e-

CYiue

CU.ltbt,

npU.lto:neeHHbte

Yi

amou

cucmeMe.

B

caMOM

Ae~e,

B

YRa3aHHLIx

c~yqaHx

XOTH

6LI

OAHO

H3

ypaBHeHHH

CHCTeMLI (25.20)

MomHO

npHBeCTH

R

BHAY

,

iii

=

O.

(25.24)

IIycTb

Ha

CHCT8MY

AeacTByroT

AOIIOJIHln'eJIbHO

RaRHe-JIH6o

AHC-

canaTHBI:lble

IlJIH

rHpOCROnHqeCRHe

CHJILI.

TorAa

([25*],

CTp.

t06)

'~--'-'-'-"':""~'

.',

....

-

'"."

.:.

"::,

:-,

".

.:"

,"

;,', :

:":,:..

-"

..

",

,',

',,,

>'-'>.';::-:,':.".

''':'''-:::':':;~~

c·

§ 25J

YllPABJIEHHE

MEXAHHqECIWJ;t

CHCTEMOn

229

ypaBHeHHe

(25.24)

nplIMeT

BIIA

iii =

alYI

+... +

anYn

(25.25)

H,

c~eAOBaTe~bHO,

Be~HqHHa

fJj

-

alYI

-

...

-

anYn

=

const

(25.26)

HBJIHeTCH

nep81itM

UHmeepa.ltOM

CHCTeMLI (25.23)

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na~HH

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01'

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u.

HoJ

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(25.26),

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a

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(25.23)

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(25.23)

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1,

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(25.27)

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...

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d

t

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0

(i

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(j=1,

...

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k+1,

...

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l'lpH

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(25.27) CTaHOBHTCH

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He

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[50,51].

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'A

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O.

CHcTeMy

(25.27)

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q>opMe

(25.9):

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i-1

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'A

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+diU,

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= X

2

i-l

(i

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...

, n),

(25.29)

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no

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H3

qHCe~

dj=l=O.

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102),

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aii

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(25.30)

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(i=1,

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(2,5,0'30)

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~

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(25.32)

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19.1,

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(x,

y),

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0,

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e.

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He

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(25.33)

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2 H

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(25.15)

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=.

J.,2

= A CTaHOBlITCH BlIOJIHe

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1

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const.'

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113

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det

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p*

(U

('t'),

ta.

< 't' <

0']

=

p*

[U

('t'),

ta.

< 't' < '\'t] P;JIH

Bcex

<pYHKn;Hii

U

('t'),

yp;OBJIeTBOpHIO-

m;HX

yCJIOBIIIO

U

('t')=

0

npH

't'

;;>

'\'t.

3p;eCb

'\'t

II

0' -

JII06Me

'IHCJIa,

CBHaaHHMe

HepaBeHCTBOM

'\'t <

0'.

B

p;aJIbHeiimeM

MM

BCTpeTHMCH

em;e

CO

CJIyqaHMH,

Korp;a

MOMeHT

t =

ta.

6yp;eT

HaMeHHTbCH.

Torp;a

CIIMBOJIbI

p*

[U

('t'),

T;;>

t

..

]

6YP;yT

060aH3.'IaTb

BeJIH'IHHM

p*

[U

('t'),

ta.

"'t'

< '\'t]

npH

JII060M

tt,

KaKoe

TOJIbKO

MOiKeT

OKaaaTbCH

MOMeHTOM

OKOH'IaHHH

npon;ecca.

B

'IaCTHOCTH,

OrpaHH'IeHHe

p*

[u

('t'),

1:'

:>

ta.]

<

It

(26.6)

6yp;eT

OaHa'IaTb,

'ITO

B

npon;ecce

ynpaBJIeHHH

MoryT

HCnOJIb30BaTbCH

JII06Me

<PYHKrt;HH U

(T)

('t';;>

ta.),

yp;OBJIeTBOpHIOm;He

yCJIOBHIO

p*

[u

(1:'),

ta.

<

1:'

< tt] <

It

(npH

JII060M'\'t

>

ta.).

MMeHHo

B

aTOM

CMM-

CJIe

CJIep;yeT

nOHHMaTb

TaKme

H

aanHCb

OrpaHH'IeHHH

(26.6) B

KaKoii-

JIH60

HBHOii

KOHKpeTHoii

<popMe,

HanpHMep

B BHp;e

00

I,

r\

Ilu(T)112

d

't']

'< [1.

(26.7)

ta.

1)

TaRHM rryTeM

MomHO

OXlJaTHTL

6oJIl:.moe

'lHCJIO

peaJILHbIX OrpaHH'leHlm.

IIop;o6IUJii Borrpoc

yme

o6cymp;aJICH B aHaJIOrH'lHOM CJIY'iae

rrplI

o6cymp;eHHlI

aap;a'lH

13.1. )];o6aBHM

ap;eCb

e~e

OP;HO

HOBoe

coo6pameHHe.

qacTo

B aap;a'lax

rrpep;eJIbHOrO

6hlcTpop;eHcTBHH OrpaHH'leHHH

Ha

U

(t)

IIMeIOT

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HeClIM-

MeTplI'lIUJii

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aj

<:;;;

Uj

<:;;;

bj,

rrpH'leM,

CJIep;OBaTeJIbHO,

aj

=1=

-

bi'

Op;HaKo

II

B

aTOM

CJIy'lae

aaMeHOH

rrepeMeHHhlX

Uj

Ha

Vj

P;eJIO

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CBecrn R OrpaHlI'leHHIO

BHp;a

p*

[v]

<Jt.

)];JIH

aToro

p;OCTaTO'lHO

BbIIIOJIHHTb,

HarrpHMep, rrpeo6paao-

BaHHe

. _

(.

ai

+

2

b

j

)

2!Jo

v,

-

U,

-

b

j

-

aj

H

rrOJIO}KHTh

p*

[v]

=

m~x

sup

{ IVj

('I:)

n.

, '

.

,.'

,

··:··i..:....:....:....:...:_..:...:.:...~~...:.·'.'

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...

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~/

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C

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r

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235

§,271

.PEmEHHE3A,D,A'I1I

0

BbICTPoAE:BCTBIIII

IlPOEJIEMA

IIPE,D,EJIbHOrO

BbICTPO,D,EflCTBl1H

[rJI,

7

?3''!

CIIMBOJI

00

BJiIOllpaeTCH

npll

aTOM B

KalfeCTBe

BepXHero

npep;eJIa

B

HHTerpaJIe

(26.7)

KaK

pas

C

TOR

:n;eJIbIO, lfTOOJiI

nO;:i;l£epKHYTb,

'ITO

MOMeHT

t =

tfJ

>

trx.,

Korp;a

OKOH'IIITCH

npo:n;ecc

ynpaBJIeHIIH,

HaM

MomeT

OJiITb

HellSBeCTeH

II,MJiI

Ha

Bce

oYP;yIIJ;ee

BpeMH

t;;>

ta.

OOJIap;aeM

peCYPCOM

ynpaBJIeHIIH,

paBHJiIM

fL

(t",).

AHaJIOrll'IHOe

SaMe'IaHlIe

'CJIep;yeT

IIMeTb

B

BIIP;y

B

Tex

CJIyqaHx,

Korp;a

P;JIH

sanllclI

o:n;eHKII

yn-

paBJIeHIIH

U (t)

npll

t;;>

t«

oyp;eT

IICnOJIbSOBaTbCH

OOIIJ;IIR CIIMBOJI

lIHTeHCIIBHOCTII

'X

[U

('r:),

't;;>

ta.]. .

3aMeTlIM

eIIJ;e,

'ITO

sap;a'Ia

0

npep;eJII>HOM

OJiICTpOp;enCTBIIII

MomeT

OJiITb

UOCTaBJIeHa

II

P;JIH '1'01'0

CJIy'IaH,

Korp;a

CIICTeMY

(26.1)

HJIII

(26.5)

'TpeoyeTcH

nepeBeCTIIB

KpaT'IaRmHR

CpOK

liS

sap;aHHoro

COCTOHHIIH

X

(ta.)

=

xa.

He

B

onpejJ;eJIeHHyIO

sapaHee

TO'IKy

xf'

<paSOBoro

npOCT-

paHCTBa,

a

Ha

HeKOTopoe

MHorooopaSHe

@

KOHe'IHJiIX

COCTOHHIIR xII.

CJIep;yeT

TaKme

IIMeTb

B

BHP;y,

'ITO

BeKTOp

xll

no

YCJIOBHHM

sap;a'IlI

MO-

meT

saBlIceTb

0'1' til.

~ha

cHTya:n;IIH

BOSHIIKaeT

B

CJIyqae,

Korp;a

Tpe-

oyeTcH

npllBeCTII

p;BHJUeHHe

x'

(t)

He

B

Henop;BlImHyIO

TO'IKy

x =

xI',

HO

TpeoyeTcH

BJiIBeCTII

ero

Ha

sap;aHHoe

p;BHmeHHe

x =

xii

(t).

tIH-

TaTeJIb,

yqllTJiIBaH

C<POPMYJIlIpOBaHliJiIe

yme

Sap;a'IH

13.2

II

26.1,

MomeT

p;aTb

COOTBeTcTByIOIIJ;He

<pOpMyJIHpOBKII

caMOCTOHTeJIbHO.

§ 27.

PemeHlIe

3a;:i;a'IlI 0

npep;en:bHOM

ObIcTpoAeiicTBlIlI

'"

OnllmeM

pemeHHe

Sap;a'IH

0

npep;eJIbHOM

OJiICTpOp;eRcTBHH,

onH-

,paIOIIJ;eeCH

Ha

pesyJIbTaTJiI

HS

npep;JiIp;yIIJ;HX

rJIaB.

B COOTBeTCTBHH C

''''It

)l!,:

nOCTaHOBKOR

Sap;a'IH

26.1

HaM

TpeoyeTcH

HaRTH

ynpaBJIeHHe

UO

(~,

nepeBOp;HIIJ;ee

cHcTeMy

(26.1)

HS

COCTOHHHH x

(ta.)

=

xa.

B COCTOHHlIe

p:'

x

(tfJ)

=

xii

sa

KpaT'IaRmee

BpeMH

TO

=

t~

-

ta.

npH

sap;aHHOM

OrpaHH'IeHHH

'X

[u]

<'

fL-

(27.1)

II

pHHHMaH

BO

BHHMaHHe

npaBHJIO

MHHHMaKca

17.1,

yCTaHOBJIeHHOe

B §.17,

MomHO

yKasaTb

cJIeP;YIOIIJ;IIR

nYTb

pemeHIIH

Sap;a'IH

0

npep;eJIb-

HOM'

OJiICTpOp;eRCTBHH. •

IIYCTb

til

>

ta.

-

HeKOTOpJiIR

<pHKCHpOBaHHJiIR

MOMeHT

~peMeHH.

Torp;a

MOmeT

OJiITb

nOCTaBJIeHa

H

pemeHa

sap;a'Ia

13.1

00

OnTHMaJIb-

HOM

Bosp;eRCTBHH

UT

(t),nepeBOp;HIIJ;eM

cHcTeMy

(26.1)

H3

COCTOHHHH

X

(ta.)

=

xa.

BnOJIOmeHHe

x

(tfJ)

= xII

sa

BpeMH

T =

til

-

ta.

H

. HMeIOIIJ;eM

npH

aTOM

uaHMeubmyIO

BosMomuyIO

UOpMy

p*

CUT

] =

=

'X

rUT] =

min.

ECJIH

Tenepb

MJiI

oyp;eM

HSMeUHTb

MOMeUT

BpeMeHII

tfJ'

TO,

nOJIaraH

sap;a'Iy

13.1

Kamp;JiIR

pas

paspemHMoif

1),

nOJIyqIlM,

'ITO

BCJIKOMy

tfJ >

ta.,

a

SUa'IHT,

BCJIKOMy

'IHCJIy

T > 0, OYP;YT CO-

OTBeTCTBOBaTb

onpep;eJIellUOeOnTHMaJIbHOe

ynpaBJIeliHe

UT

(t)

H

on-

1)

l1mI,

DO

BCRKOM

CJIyqae, paapemllMoii: IIpll

aHaqeHllRX

tfJ

llalieKOToporo

MHOmeCTBa

@t.

l1aMeHeHHJr

.

uOTpe6HhIe

Tor~a

B

paccym,n:eHllJIX,

OqeBll~HhI.

pep;eJIeUHaH

ero

IIUTeUCHBHOCTb

'X

CUT],

KOTopaH·

HBJIHei'CH;CJI-e;;(OBa-

T,eJIbUO, <pyHK:n;HeR 0'1'

T.

OOOSHa'IHM'X

CUT]

=

'XT

HpaCCMOTpllM

He-

paBCHCTBO

(27.2)

'XT<'fL·

O'IeBHp;HO

Torp;a,

'ITO

ORTHMaJIbHIiIM

BpeMeHeM

nepexop;Horo

npo:n;ec.:

ca

TO

npH

OrpaHH'IeHIIII

(27.1)

oyp;eT

HaHMeHb.mee

liS

nOJIOmHTeJIbHJiIX

TJ:JIceJI

T,

:yp;OBJIeTBopHIOIIJ;HX yCJIOBHIO (27;2).

ConOCTaBJIHH

3'1'0

06:-

CTOHTCJIbCTBO C

npaBHJIOM

MHHHMaKca

17.1,

nOJIY'IaeM,

CJIep;yIOIIJ;HR

pesyJIbTaT.

Teo

p

eM

a 27.1.

IIycmb

h~

('t)

(ta.

<'

't <'

tfJ

=

ta.

+

T)

-

MUl-UM£MbUaJL

(/Jyur;,LfUJL,

uau8euuaJL

COMacuo

npaeu.f1,y

MUUUMar;,ca

17.1,

mar;,

'limo

p[h~

('t')] =

minp

[B'ST('t')]

=

PT,

(27.3)

ST

z8e

ST(t)

-

8eu:JfCeuue

conpJL:JfCeuuou

CUCmeMbL

(17.3),

y8oMemeOpJLlO:"

Ufee

r;,pae8oMy

YC.f1,QeUlO

S'T

(tfJ)

C

(T)

= 1. Toz8a uaUMeubutee us 'luceiL

T

=

tfJ

- t«,

y8oe.f1,emeOpJLlOUfUX

coomuOuteUUlO

1

pX;;>!1'

(27.4)

6y8em

onmUMMbUbLM

epeMeUeM

nepexo8lloeo npoLfecca

TO,

a coomeemcm-

QYlOUfee

amoMy

epeMeUU

QnmUMa.f1,bUOe

ynpaeJLeuue

UTQ

(t) =

'uo

(t)

(ta.

<'

t

<'

tg

=

ta.

+

TO)

JLeumCJL

UCr;,OMbLM,

OnmUMa.f1,bllbLM

no

6bLcm-

po8eucmeUlo ynpaMeuueM.

IIpu

amOM,

r;,ar;,

c.f1,e8yem

us

npaeU.f1,a

JI£U-

UUMar;,ca,

OnmUJl£MbUOe

no 6bLcmpo8eucmeUlo

ynpaMellue

6y8em

UMemb

UOpMy

p*

[UO

('t)] =

lIpro

U 6y8etn

ebL8MJLmbCJL

cpe8u ecex

8onycr;,aeMbLX

ynpaMe7-l.UU

U ('t)

(ta.

<' 't

<'

tg)

c

llOpMOU

p*

[u

('t)] =

=

1Ipro

ceoucmeoM'Mar;,cUMYMa (17.1)

lla

MU7-l.UJlta.f1,bllOU

(/JYUr;,LfUU

h~.

('t).

,.

3p;ecb

c (T) -

BeKTOp

(15.2).

CtpopMyJIHpOBaHHaH

TeopeMa

YKaSJiIBaeT

crrocoo

pomCHlIH

S3.p;a'IH

o

npep;eJIbHOM

OJiICTpOp;eHCTBJIH,

KOTOpJiIR

SaKJIIO'IaeTcJi

B

rroCTpOemIJI

TOR

HJIH

HHOH

BJiI'IHCJIJITeJIbHOR

npo:n;ep;ypJiI,

nOSBOJIHIOIIJ;eR

orrpep;e-

JIllT'b

HallMeHbmee

HS

nOJIOmHTeJIb.HJiIX'IJICeJI

T,

yp;OBJIeTBopHIOIIJ;HX

yCJIOBIIIO (27.4)

HJIH,

lfTO

TO

me

caMoe,

yCJIOBHIO (27.2).

IIpJI

aTOM

.rrpo:n;ecc

OTJiICKaHHH

caMoro

OnTHMaJIbHOrO

no

OJiICTpOp;eHCTBHIO

yrr-

paBJIeHIIH

CJIep;yeT

HsyqeHHoMy

yme

pyCJIy,'

KOTopoe

onpep;eJIeHO

~BORCTBOM

MaKclIMyMa

(17.1),

cop;epmaIIJ;HMCH

B

rrpaBHJIe

MHHJIMaK-

ca

17.1.

ECJIH

BeJIH'IIIHa

pr

HBJIHeTCH

HenpepJiIBHOH

<PYHKIJ;HCH T

(a

~TO,

p;eHCTBHTeJIb.HO,

HMeeT

MeCTO B

mllpOKOM

KJIaCCe

CJIY'IaeB),

to

P;JIH OTJiICKaHHH

BeJIHlfllHJiI

TO

BMeCTO

YCJIOBlIJI

(27.4) yP;OOHO

·,

! '

c

'

.....

····,'

J

. ,:.,'

236

llPOBJIEMA

llPEAEJIhHOrO

BMCTPOAEfiCTBllH

[rJI.

7

nOJIbaOBaTbCH

ypaBHeHlleM

PT

=

min

[B'

('t)ST

('t')]

=

_1

rrpll

s~

(t{3)c(T)

= 1.

(27.5)

ST

!A.

HallMeHbIIIllH

nOJIOmllTeJIbHLIH

RopeHb

aToro

ypaBHeHllH

H

p;acT

on-

THMaJIbHOe

.BpeMH

nepexop;Horo

npoIJ;eCca

TO.

II

p

II

M e q a H

II

e 27.1.

yqlIThIBaJI,

qTO

BhIpalRemIe

p

[B'

sT

(1")]

O,n:HO-

pO,n:HO

IIO

BeKTopy

l = s

(t{3)'

HepaBeHCTBO

(27.4)

IIJIII

ypaBHeHlIe

(27.5)

BMecTe

C

YCJIOBlIeM

s~

(t{3)

C (T) = 1

MOlRHO

saMeHIITb

HepaBeHcTBoM

min

{-

s~

(t{3)

C

(T)

+

!A.P

[B'

(1")

ST

(1"))}

;> °

II

s

T

II=l

HJIH

COOTBeTCTBeHHO

ypaBHeHHeM

min

{!A.P

[B'

(1")

ST

(1")]

-

s~

(t{3),

c

(Tn

= 0,

II

s

T

II=l

qTO

IIo,n:qaC

MOlReT

OKaSaUCJI

60JIee

y,n:06HhIM

,n:JIJI

pa60ThI

C

llTHMlI

COOTHOmeHIIJI-

MH.

~aJIee,

yqlIThIBaJI

paBeHCTBO

(12.19),

ypaBHeHlIe

(27.5)

MOlRHO

em;e

saMe-

HlITb

COOTHomeHlIeM,

He

co,n:eplRam;HM

HBHO

p [h

(1")],

t{3

'4i~1

min

[max

~

B'

(1")

ST

(1")

U

(1")

d1"-

s~

(t{3)

C

(T)

] = 0.

II

STII=l

x [u('t")]<::1h

ta.

~I~

!f'h

TpYP;HOCTH

peaJIHaaIJ;HH

npep;JIOmeHHOro

cnoco6a

B

BHp;e

Bbl'IHC-

JIHTeJIbHOrOaJIrOpHTMa

He

HOCHT

npHHIJ;HnHaJIbHOrO

xapaRTepa

H

[l'"

RacaIOTCH

TaRHX

CTaHp;apTHLIX

np06JIeM

lJHCJIeHHLlX

MeTop;OB,

RaR,

HanpIIMep,

Bonpoc

0

BbI60pe

HalJaJIbHOrO

npH6JIHmeHHH

llJIll

Bonpoc

o

CROpOCTH

CXOP;HMOCTH

BLIlJHCJIllTeJIbHOrO

npOIJ;ecca.

IIpH

paCCMOT-

peHHll

onpep;eJIeHHLIX

OrpaHHlJeHHH

(27.1), a

TeM

60JIee

rrpH

peIIIeHHH

ROHRpeTHLIX

aap;alJ

aTH

06m;He

TPYP;HOCTH

MorYT

6LITb

TaR

HJIH

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npeOp;OJIeHLI.

I

,IJ;orrYCTHM,

HanpHMep,

lJTO

Tpe6yeTcH

peIIIHTb

aap;alJy

26.1

npH

i

j

OrpaHlllJeHHH

Ha

aHeprHIO

ynpaBJIHIOm;erO

Boap;eHcTBHH,

T.

e.

npH

!

I

yCJIOBUll

00

'/

x[u]

=

[~

Ilu('t')11

2

d't'J

'<~.

(27.6)

tlX.

B

aTOM

CJIylJae

aap;alJa

0

npep;eJIbHOM

6LIcTpop;eHcTBHH

peIIIaeTCH

C

HallMeHbIIIHMH

TPYP;HOCTHMH.

,IJ;eJIo B

TOM,

lJTO

OnTHMaJIbHOe

ynpaB-

JIeHlle

npH

yCJIOBHH

MHHHMyMa

aHeprHH

Haxop;HTCH

B

aaMRHyToH

iPopMe

(CM.

§ 18) H

n03TOMY

yp;aeTCH,

HCRJIIOlJHB

l~

(T)

Ha

COOTHOIIIeHHH

(18.3),

aanHcaTb

HHTeHcllBHoCTb

X

T

OnTHMaJIbHoro·

ynpaBJIeHHH,

a

cJIep;OBaTeJIbHO,

H

BeJIHlJHHY

PT

= 1/xT, B

BHp;e

HBHOH

iPYHKIJ;HH

'-

..

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

"'''',' :

<,

..

:.:-..

",

..

'

·:::<-:·::·

:·:·

~~;·;;~;,)i;·:

",'

.'

.\

,,·.·

••

1

c

237

§ 27]

PEmEHIIE

aA):l;A

11M

()

BhICTPO,UEfiCTBHH

BpeMeHH

T.

,IJ;eHcTBHTeJibHO,

npHHHMaH

BO

BHHMaHHe

(18.11),

nOJIy-

lJaeM,

lJTO

n

.t{3

II

UO

('t')

11

2

d't't

=

[i,f.L;iCkDikf,

(27.7)

XT

=

[~a.

rp;e

c = c (T) =

{Ci

(T)} -

BeRTOp

(15.2), a CMLICJI

0603HalJeHHH

D H D

ik

rrOHCHeH

Ha

CTp.

126.

HeTpyp;Ho

HenOCpep;CTBeHHO

yCTaHoBllTb,

lJTO X7' (27.7)

HBJIHeTCH

HenpepLIBHoH

iPyHRIJ;HeH

T,

eCJIH

TOJIbRO

D

-+

O.

Ho

rrOCJIep;Hee

yCJIOBlle

BLIrrOJIBJIeTCH

BO

BCHROM

CJIylJae,

eCJIH

CllCTeMa

(26.1)

8nO.!U-ie

ynpaMReJ,ta.

Torp;a

BemJ'illHa

PT

=

lIx

T

6yp;eT

TaRme

iPyHRIJ;HeH

HenpepLIBHoH

H P;JIH

onpep;eJIeHHH

TO

MomHO

BOCnOJIbaOBaTbCH

ypaBHeHHeM

(27.5),

ROTopoe

npllMeT

BHP;

n

~

Ci (T) Ck (T)

Due

(T) -

~2D

(T) =

O.

(27.8)

i,

k=l

TaRHM

06paaOM,

B

paCCMaTpllBaeMOM

CJIylJae

3ap;alJa

0

npep;eJIbHOM

6LICTpop;eHCTBllH

CBOP;HTCH R

Haxomp;eHHIO

HaHMeHbIIIero

nOJIOmH-

TeJIbHOrORopHH

TO

ypaBHeHllH

(27.8).

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me

OnTHMaJIbHOe

no

6LI-

cTpop;eHcTBllIO

ynpaBJleHlle

UO

(t)

onpep;eJIHeTCH

no

iPopMyJIaM

(18.10), B

ROTOpLIX

BMeCTO

t{3

nOp;CTaBJIeHO

tg

=

tlX.

+

TO.

IIpHBep;eM

rrpocToH

HJIJIIOCTpHpyIOm;HH

npHMep.

.

I1puMep

27.1.

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MaTeplIaJIbayIO

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OIIIIChIBa-

eTCH

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xopomo

HaM

SHaKOMO:H CHCTeMO:H ,n:1I<p<pepeHIJ;lIaJIbHhIX

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(18.11)

(CM.

riplIMep

18.1),

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IIepeBeCTH

HS

JIOJImKeHHJI

xa.

=

{-1,

0,

1,

o}

IIplI

tlX.

= °B

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xf3

=

{a,

0, 0,

o}

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06paSOM.

IIpli

llTOM

,n:OJIlRHO y,n:OBJIeTBOpHTbCJI

HepaBeHCTBO

00

2

(27.9)

~

cui

(1")

+

U~

(1")]

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<

!A.

•

tlX.

~

COrJIaCHO

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BhIille

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pemeHIIJI

sa,n:aqll

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<PYHK-

IJ;IIIO

PT'

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Ha

paBeHcTBo

(27.7),

IIOJI,yqIlM:

1

[.

24J-2

P

=-=

g-T+

- (27.10)

T

x

T

T3

II,

CJIe,n:oBaTeJIbHO,

ypaBHeHlIe

(27.8)

saIIHChIBaeTCR

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g2

T4

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2

T3

+ 24 =

O.

(27.11)

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pRC.

27.1

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rpa<pHK

<PYHKIJ;II:1i

PT

(27.10).

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BH)l;HO

liS

llToro

rpa<p

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,

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Toro

qTo6hI

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(27.11)

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B

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B

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238

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=

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0,

0)

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rrpll

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T,

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96 g

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TO

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(27.11)

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-

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I

/

I

IT

=(

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I

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T

+

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3

J

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\

(

T

o

PliC.

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'It:l

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(18.10)

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no

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U

O

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=

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l

[TO]S

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u~

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(0

~

t

~

TO),

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-

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(27.5).

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paa06paHHoM

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TO

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(CM.

plIC.

27.1),

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/A.

> 0 ypaBHeHHe

(27.11)

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p;OCTaBJlHeT

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B

p;OCTaTO'IHO

06m;eM CJlY-

'Iae

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(27.1)

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CTBHH.

A HMeHHO, IIyCTb B ypaBHeHHH

(26.1)

w

(t)=

0 H

x/3

=

0,

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PT HBJlHeTCH

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He-

[rJl. t

('-::'>:

~~>

239

y6MBarom;eH <pYHKn;lIeH T. )J;JlH p;OKaaaTeJlbCTBa

3Toro

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§ 27]

PEIJIEHlIS aA,D;A'lfi 0

BbICTPO,D;E:f[CTB.lHI

HHM, 'ITO BeJlH'IlIHY

PT

MomHO IICKaTl.

Ha

YCJIOBIIH

(CM.

IIpaBIIJlO

17.3 B §

17)

PT

=

min

P

[B'

("t')

S

('r)

1=

min

P [B'

("t')

S

["t',

tex

]l]

(27.12)

I

IIplI

s'

(tex)

'x

ex

=

l'

x

ex

=

-1.

)J;onYCTHM, 'ITO

P;JlH

Kamp;oro

l'

>0

uaHp;eHM 'IlICJla

l~

(1')

=

s~

(t

ex

)

lIa

YCJlOBIIH

(27.12).

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npH-

HHMaH

BO

BHHMaHHe

aTO

yCJlOBHe, IIOJlyqHM

O'IeBHp;HOe

HepaBeH-

CTBO

PT+D.T = P

[B'

("t')

S

["t',

tex]lO

(T

+

LlT),

tex

<"t'

<

tex

+ T +

LlT]

;>

P

[B'

("t')

S

["t',

tex]lO

(T

+ LlT),

tex

<"t'<

tex

+

T]

;>

min

P

IB'

("t')

S ['t, tex]l,

tex

< 't <

tex

+

T]

= pT, (LlT

>0).

I

(27.13)

TaKHM

06paaOM, MM BIIP;IIM, 'ITO, p;eHCTBHTeJlbHO, nepeMeHHaH PT

HBJlHeTCHMOHOTOHHOH, He y6MBarom;eH <PYHKn;lIeH BeJllI'IHHbI

1'.

Bam-

HO

nOp;'IepKHYTb, 'ITO

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cJlyqaHx

BM60pa

IIHTeHCHBHOCTII x

[l.4]

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YCJlOBIIII IIOJlHOH ynpaBJlHeMOCTII

CIICTeMM

(26.1)

B COOTHomeHlI1I

(27.13):6yp;eT

BMIIOJlHHTbCH

CTporoe

HepaBeHcTBo, T. e.

<PYHKn;HH

PT

6yp;eT

cmpoao

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<pyHKIJ;HeH

T.

TaK,

HanplI-

Mep,

B paccMoTpeHHoM

BMille

cJlyqae

OrpaHlI'IeHHH

Ha

aHeprlIlo

yn-

paBJIeHIIH

(27.6)

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(18.2)

II

(27.13)

6yp;eT:

t/3+D.T

2

p}+D.

T -

p}

~

~J~

II

B'

("t')

S ['t,

tex]lO

(T

+LlT)

11

d't.

(27.14)

<.

t/3

IIpaBaH

'IaCTb HepaBeHcTBa

(27.14)

cTporo

IIOJlOnmTeJlhHa,

eCJIH

TOJibKO

CIICTeMa

(26.1)

BnOJlHe

ynpaBJlHeMa,

nOCKOJlbKY Torp;a BeKTOp-

<PYHKIJ;1I1I

h[i]

('r) -

CTOJl6IJ;M

MaTpHn;M

B'

('r) S ['t, t

ex

] - JlIIHeHHO

HeaaBHCHMbI.

TIPII

OrpaHH'IeHIIH

Ha

MaKclIMyM

ynpaBJlHrom;eH

CHJlM

(21.1)

HMeeM:

t{3+D.T

PT+D.T -

PT ;> J r

{B'

('t) S

["t',

tex]lo

(T

+

LlT)}

d"t',

//3

H

npaBaH

'IaCTb

aToro

HepaBeHCTBa

npH

YCJlOBHH

nDJIHOH

..

y..npallJIll-

ellfQCTII CIICTeMbI

(26.1)

CHOBa

6yp;eT

CTporo

nOJlOl-KHTeJlbHoH.

Ho,

HanpHMep,B

cJly'Iae

OrpaHH'IeHHH

Ha

HMIIyJibC

ynpaB-

JlH:IO.m;ero

Boap;eHcTBHH

(23.5)

BeJlH'IHHa paaHOCTH

PT+:>T

- PT

BCJlep;CTBHe

(23.10)

HBJIHeTCH

JlHillb

HeoTpHn;aTeJIblIOH

II

noaToMy

Красовский Н.Н. Теория управления движением

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