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MHomeCTBO

KOHeq-

Hl:>IX

COCTOHHHH CHCTeMl:>I (13.1).

8'1'0

MnO:JICeemeo

HBJIJIeTCH

rJilnepllJ10-

CKOCTI>IO B

4JaaoBoM

npOCTpaHCTBe

{~,

•••

, x

n

},

onpeAeJIeHHOH

CH-

CTeMOH

ypaBHeIDlH

Xi)

=

Xi~

(j

= 1,

...

, k). MCXO,r(H Jila

TaKOH

TOqKJiI

apeHlIH

MOiKHO

o606m;JilTI>

aaAaqy

13.1,

CqHTaH,

'ITO

aaAaHO

HeKOTopoe

Mnoaoo6paaue @

KOHeqHhIX

COCTOHHliIH

xf3

iPaaoBoro

BeKTopa

X (t).

Ta-

KHM

MHoro06paaHeM

@

MoryT

OKaaaTI>CH

HeKOTopaH

AOCTaTOqHO

MaJIaH

,

....

"

....

"::.

:: .

~:"

':

' : . : ' ..' ...

'.'

...

,-::

.....

::""::":,"~':,'<

.,

.....

:.:.::>

";:.::',

t:>'~::-~<~4';

c·

~~';.'

[rJI.

4

nOCTAHOBKA

3AAATfM

OB

OllTMMAnI.HOM

YITPABnEHMM

84

OKpeCTHOCTI.

HRqaJIa

KOOpp;MHaT

X = 0,

HeKOTOpaH

rMnepnJIOCKOCTI.

B

npOCTpaHCTBe

{Xl'

...

, X

n

},

HeKOTOpaH

KpMBaH

l'!

T.

p;. B aTOM

CJIy-

lIae

sap;alIa

06

OnTMMaJII.HOM

ynpaBJIeHlIM

cMcTeMoH

(13.1)

<pOpMyJIM-

pyeTcH

CJIep;yIOm;MM

06pasoM.

3 a

p;

a 1I a 13.2.

lIycmb

ynpa6.tuleMaR cucmeJrta

OnUCb16aemcR

AU-

neufUilJrf,

ypa6l-leHueM (13.1).

3aoaUbl

ompeaon

tao

< t <

t~,

nal{aAbnOe

cocmoRnue

xa.

U Mno3006paaue @ nOnel{UblX cocmORnuu

x(3

o6u1i:ma.

Bbl6pana

UT-lmenCU6nocmb 'X

[ul

ynpaMeHUR

U

(t),

nomopaR

MO:HCem

6bzmb

UCmOJl,n06ana

nan

nopMa

p*

[ul

rj'JynnUruu U

(t)

6

npocmpancm6e

5iJ*{u}.

Tpe6yemcR

cpeou

oonycnaeMblxynpa6Mnuu

U

(t)

naiimu

onmu-

JrtaAbl-We

ynpaMenue

Uo

(t),

nepe600RUfee cucmeMy (13.1) ua cocmORnUR

X (ta.)

=

xa.

6 nanYlO-nu6yob mOl{ny X

(tfJ)

=

x(3

us

saoann030 Mn03006-

pasuR

@ u

UMelOUfee

npu

amo},t naUMenbUlYlO

60SMO:HCnylO

nopMY

p*

ruG].

IIOJleSHO

saMeTHTI.

CJIep;yIOm;ee.

RaK

M

06hllIHO

B

BapMan;MOHHOM

HClIMCJIeHMM,

pemeHMe

sap;alIM

13.2

CBHsaHO

C

pemeHHeM

sap;alIM

13.1.

MMeHHO,

eCJIM

MMeeTCH

pemeHHe

sap;alIM

13.1

npM

<pHKcMpoBaHHoM

X

(t~)

=

x(3,

npHlIeM

<popMa

pemeHlIH

nOSBOJIHeT

npOCJIep;HTI.

saBHCH-

MOCTI.

aToro

pemeHHH

13.1

0'1'

x(3,

TO

nepexop;

K

pemeHHIO

Mp;alIH

13.2

CBOp;HTCH

JIHmI.

K

p;OnOJIHHTeJII.HoMy

aHaJIHsy

saBHCHMOCTH

pemeHHH

sap;alIH

13.1

OT

x(3,

Korp;a

x(3

np06eraeT

MHQ)l{eCTBO

@.

TaKHM

06pa-

SOM,

THnHlIHa

CHTyan;HH,

Korp;a

CBOHCTBa

pemeHHH

sap;alIlI

13.1

COCTaBJIHIOT OCHOBHOH

aJIeMeHT

pemeHlIH

sap;alIlI

13.2 (CM. § 38).

IIo

yKasaHHoH

npl!l1IlIHe

B

p;aHHoH

KHHre

BHHMaHHe

yp;eJIeHO,

npeiRp;e

Bcero,

lIMeHHO

sap;alIe

13.1,

pemeHHe

KOTOpOH

onpep;eJIHeT

OCHOBHhle

aJIeMeHTI.I

pemeHHH

sap;alIlI

13.2

II

aHaJIOrl!l1IHhlX

eH

sap;alI

C

6t)JIee

CJIOiRHLIMlI

KpaeBhlMlI

yCJIOBlIHMlI

P;JIH

p;BHiReHHH X (t).

B

SaKJIIOlIeHlIe

nOJIeSHO

em;e

pas

06paTHTI.

BHlIMaHlIe

lIHTaTeJIH

Ha

n;eHTpaJII.HhlH

nyHKT

Hamero

nop;xop;a

K

np06JIeMe

06

onTlIMaJII.HOM

ynpaBJIeHHl!l.

IIpep;nOJIaraeTCH,

lITO

B

TOH

lIJIlI

lIHOH

clITyan;lIlI

lICXOP;H

lIS

<pHslIlIeCKHX

C006paiReHHH

nOJIylIeHa

sap;alIa

4.2.

IIop;rOTOBJIHH

aTy

sap;alIY

P;JIH

pemeHHH,

MI.I

nepeBop;lIM

ee

B

<pop~y

sap;alIlI

13.1,

yp;o6-

HyIO

P;JIH

MaTeMaTl!l1IeCKOrO

lICCJIep;OBaHHH.

TaKlIM

06pasoM,

sap;alIy

13.1

Hap;o

paCCMaTplIBaTI.

He

KaK

lISHalIaJII.HYIO

np06JIeMy,

HO

EaR

nepBhlH

mar,

HanpaBJIeHHI.IH

K

pemeHHIO

lICXOP;HOH

sap;alIlI

4.2.

06

aTOM

CJIep;yeT

nOMHHTI.

II

B

Tex

CJIy'HiHX,

Korp;a

MI.I

6yp;eM

lICXOP;lITI.

cpasy

lIS

sap;alIlI

13.1.

§ 14.

OOOOm;eHHall

sap;a'la

TenepI.

HaM

Hap;JIeiRHT

pacnpOCTpaHlITI.

sap;a1Il!l

1:3.1

H 13.2

Ha

CJIy-qaH

0606m;eHHhlx

ynpaBJIeHlIH

BlIp;a

dU

(t).

IIycTb,

CJIep;OBaTeJII.-

HO,

nOBep;eHlIe.

ynpaBJIHeMOH

ClICTeMhl

onlIChlBaeTCH

ypaBHeHlIeM

(6.11),

T.

e.

dx

= A

(t)

xdt

+ B

(t)

dU

+

dW.

(14.1)

,"

:,.:-:,:.;.

C

§ 14]

OBoBmEHHAH

SAAAQA

85

IIop;06HO

TOMy

KaK

aTO 6bIJIO Cp;eJIaHO B

npep;hlp;ym;eM

naparpa<pe,

nplI-

MeM

II

sp;ecI.,

lITO

BeJIlIlIlIHa

'X

[dUl,

On;eHlIBaIOm;aH

pecypChl,

saTpa-

'1lIBaeMhle

Ha

ynpaBJIeHlIe

Ha

OTpeSKe

tao

< t <

t~,

MOiReT

TpaKTO-

BaTI.CH

KaK

HopMa

p*

[U

1

<pYHRn;HlI

U (t).

ll.JIH

TOro

tIT06hl

nOHCHlITI.

aTO

06m;ee

npep;nOJIOiReHlIe

Ha

KOHRpeTHOM

MaTeplIaJIe,

nplIBep;eM

CJIep;YIOm;lIH

nplIMep.

PaccMoTplIM

sap;alIy

06

ynpaBJIeHlIlI

p;BlIiReHHeM

MaTeplIaJII.HOH

TOlIKlI

C

nepeMeHHoH

MaCCOHno

roplISOHTaJII.HOH

OClI

1;.

Byp;eM

rOBO-

plITI.,

lITO

peaKTlIBHaH

Macca

pacxop;yeTcH

HenpephlBHO,

eCJIlI<PYHK-

IJ;lIH m

(t)

p;mp<pepeHIJ;HpyeMa.

Torp;a

p;BlIiReHlIe

TOtIKlI

onlICbIBaeTCH

ypaBHeHlIeM

Mem;epcKoro

(3.8),

KOTopoe

B

p;aHHOM

op;HOMepHOM

CJIy-

lIae

nplIHHMaeT

BHP;

m

(t)

~

- a

(t)

m(t) = 0, (14.2)

rp;e

m (t) = m

o

+ m

l

(t) -

Macca

TOlIKlI,

m

o

-

HelISMeHHaH

lIaCTI.

Macchl,

m

l

(t)

-

sanac

peaKTHBHOH

MaCChI,

a

(t)

-

OTHOClITeJII.HaH

CKOpOCTb OTp;eJIHIOm;lIXCH

lIaCTlIIJ;.

CKOpOCTI.

a

(t)

6yp;eM

ClIlITaTb

no-

CTOHHHOH

no

a6COJIIOTHOH

BeJIlIlIlIHe

II

nplIHHMaIOm;eH

sHalIeHlIH

+ a

lIJIlI

- a B

saBlIClIMOCTlI

0'1'

HanpaBJIeHlIH

oT6paChlBaHlIH

lIaCTlIn;.

YpaBHeHHe

(14.2)

,sanHmeM

B

CJIep;ylom;eH

<popMe:

f = (t)

dm

= ±

dIn

m

(t)

'"

a

mdt

a

dt

lIJIH

B

BlIp;e

ClICTeMhl

dX

I

= x

2

dt,

dX

2

±

ad

In m (t),

(14.3)

rp;e

Xl

=

1;,

X

2

=

~.

:fhaK,

nycTb

CHalIaJIa

qaCTlIIJ;hl

peaKTlIBHOH

MaCChl

Bce

BpeMH

Bhl6paChlBaIOTCH

HenpepbIBHO,

TaK

lITO

cym;ecTByeT

npOlIsBop;HaH

dIn

m

(t)/dt.

BBep;eM

<PYHKIJ;HIO U

(t),

CBHsaHHyIO

C <pYHKIJ;lIeH In

m(t)

COOTHomeHlIeM

dIn

m (t)

npH

a (t) = a,

(14.4)

dU

(t) =

{

_

dIn

m (t)

nplI

a (t) = - a.

Torp;a

ClICTeMa

(14.3) MOJ-I{eT

6hlTI.

sanlIcaHa

B

BlIp;e

dX

I

= x

2

dt,

dX

2

=

adU(t).

(14.5)

ll.OnyCTlIM

Tenepb,

lITO

HapHp;y

C nenpepbl6l-lbLM

pacxop;OM

MaCChl

m

l

(t)

B OTp;eJII.Hhle

MOMeHThl

BpeMeHH

t =

t.

MoryT

MrHOBeHHO

oT6pa-

ClilBaTbCH

KOHelIHhle

KOJIlIlIeCTBa

peaKTlIBHOH

MaCChI

L1

mI'

8'1'0

BhlShl-

BaeT

lIMnYJIbCHhle

YClIJIlIH

Ha

p;BlIiRym;YIOCH

TOlIKy.

Byp;eMTpaKToBaTI.

p;aHHoe

HBJIeHHe

KaK

IIp;eaJIlIsaIJ;lIIO

HenpephlBHQro

oT6pacbIBaHHH

aeMaJIOH

MaCChl,

HO

sa

lIpesBhlllaHHo

MaJIhlH

npOMeiRyToK

BpeMeHH.

MHalIe

rOBopH,

paCCMOTplIM

npep;eJIbHhlH

CJIylIaH

ynpaBJIeHHH

BHp;a

"

....

~

.........

'.... .

....

;

......

;.:.

.:;,

,'.

.",'

~~

',""

':',

::'\

(~."';

~

••

>

C

>

86

nOC'l'AHOBI<A 3A,D:A'Ill OB

OJI'l'HMAJlbHOM

~'In)

ABJIEHI1M

[rJI.

4

I

§

14]

OEOEIUEHHAII

3A)J;A'lA

!

87

(14.4)

Ha

oTpeime

(t.,

t.

+

f!

t)

rrpM

yCJIOI;lMM,

lITO

f!

t

~

0,

HO

rrpII

HHII

KOJIHlIeCTBa

ABumeHHH

CHCTeMhI

([13*],

CTp.

102),

MhI

nOJIYlIHM:

t.+t.t t.+t.t

aTOM

IIHTerpaJI

~

I

dU

I=

~

Ia

(t)

dIn

m

(t)

1=

0>0

eCTbrrOCTOHH-

f!~

= ± a

Llml

•

t. t.

m.

HaH.

BCJIep;CTBIIe

MaJIOCTII

OTpeaKa

f!

t

rrOJIOmIIM

em;e,

lITO

rrpII

t. <

<

t <

t.

+ f! t

BeJIIIlIIIHa

a (t)

OCTaeTCH

rrOCTOHHHOH.

Torp;a,

rrpHBJIe-

OlIeBHp;HO,

lITO

liml/m.

=F

f!

In

m.

3'1'0

u

nop;TBepmp;aeT

BhICKa-

KaH

rrOHHTMe

UMnY,/f,bCHOU 6-¢ym;/fUU

II

rrOHHTIIe

o606UfeHH080

yrrpaB-

aaHHYIO

MbICJIb.

JIeHIIH

dU

(t),

BBep;eHHble

B §

6,

MhI

MomeM

II

B

OKpeCTHOCTH

TaKIIX

BepHeMcHK

ypaBHeHHIO

(14.5).

Orrpep;eJIHM

aaKOH

p;BHmeHHH

TOlIKH

aHalIeHMH

t = t.

aarrHcaTb

ypaBHeHHH

p;BHmeHMH

CHOBa

B

BHAe

B

COOTBeTCTBHH

C

OOOOm;eHHoH

<pOpMyJIOH

ROIIIlI

(6.6).

IIplI

t =

tfl

(14.5),

rrOJIaraH,

lITorrpH

rrepeXOAe

~Iepea

TOlIKY

t = t.

HMeeM:

oyp;eM

lIMeTb:

tf)

liU

i

__

{

Ii

In

m (t.)

UpH

a (t.) = a,

(14.6)

Xl

(t{3)

=

X~

+

(t{3

-

t",)

X:

+

~

h(l)

[t{3,

,;]

dU

(';),

( .)

-:-

-

Ii

In m

(tJ

rrplI

a (t.) = - a,

t",

rAe

f!

In

m

(tJ

=

In

m (t. + 0) -

In

m (t.).

t{3

X2

(t{3)

=

X:

+

~

h(2)

[t{3.,

,;]

dU

('r).

I-baK,

MI.I

BHP;lIM,

lITO

aAecI.

B

Ka~IeCTBe

OOOOm;eHHoro

yrrpaBJIHIO-

tot

m;ero

BoaAeHCTBlIH

dU

(t)

eCTeCTBeHHO

rrplIHHTb

yrrpaBJIeHlIe,

rropom-

AeHHoe

<pYHKIJ;lIeH

U (t),

CBHaaHHOH

C

<pYHKIJ;lIeH

In

m (t)

COOTHOIIIe-

3p;eCb,

KaK

H

BhIIIIe

(CM.

CTp.

70), <PYHKIJ;HlI

HIIHMM

(14.4)

II

(14.6).

::ha

<PYHKIJ;lIH,

C OP;HOH

CTOpOHbI,

yP;OOHO

orrHChIBaeT

yrrpaBJIHIOm;lIe

YClIJIlIH,

a C

APyroH

CTOpOHhI,

OHa

HBHO

xa-

h(l)

(,;)

=

h(l)

[t{3,

,;] = a .

(t{3

- ';),

h(2)

(1:)

=

h(2)

[t{3.

,;] = a

paKTepHayeT

pacxop;

peaKTlIBHOH

MaCChI

BO

BpeMeHlI.

B

caMOM

AeJIe,

rrOJIHOe

IIaMeHeHHe

<PYHKIJ;lIlI

U (t)

Ha

OTpeaI\e

[t,,, tfl]

BhIpamaeTCH

pa-

BeHCTBOM

(CM.

[23'],

T.

3,

CTp.

78, a

TaKme

BhIIIIe

B

aToH

KHHre

CM.

I

CYTb

aJIeMeHThI

lIMrrYJIbCHOH

rrepexop;HoH

MaTplIIJ;hI

H [tfl,

T]

CHCTeMhI

(14.5).

paBeHcTBo

(4.4».

I

IIYCTb

TpeoyeTcH

rrepeBeCTH

TOlIKY

ua

aap;aHHoro

HalIaJIbHOrO

nOJIO-

meHlIH

X

(t",)

=

xet.

B

aap;aHHoe

KOHelIHoe

rrOJIOmeHUe

X (tl') =

xf3

C

MlI-

1{3

. m m (l",)

HlIMaJIbHhIM

pacxop;OM

peaKTHBHOH

MaCChI

mi

(t",).

IIpH

aTOM,

KaK

var[U

(,;)]:0

J IU (,;) Id,; + h I

liU

(tIl')

I=

In

m

'(t{3)

,

(14.7)

.

'"

OObIlJ:HO,

OYAeM

rrpep;rrOJIaraTb,

lITO

7(1,1

(tfl) = 0,

T.

e.

'ITO

aarrac

peaK-

tet.

I.

=1

TlIBHOH

MaCChI

B

rrpOIJ;ecce

yrrpaBJIeHlIH

nOJIHOCTbIO

HCrrOJIbayeTCH.

TorAa

B

COOTBeTCTBIIH

C (14.7) B

KalIeCTBe

MHHHMlIaHpyeMoH

lIHTeHCHB-

rp;e

tic

-

TOlIKa

paapI.IBa

<PYHKIJ;lIlI

U (t).

HOCTlI

x

[dU]

yrrpaBJIeHHH

dU

(T)

CJIeAyeT

BblopaTb

BeJIHlIlIHY

(CM.

OTMeTlIM

ap;ecb

em;e

CJIeAYIOm;ee

OOCTOHTeJIbCTBo.

MhI

TpaKTyeM

TaKme

§ 7,

CTp.

52)

MrHoBeHHhIH

BI.IOPOC

KOHelIHOH

MaCChI

KaK

lIAeaJIIIaaIJ;lIIO

H-8UpephIB-

tf3

I

Horo

BhIopoca

ee'

aa

lIC~Ie3aIOm;e

MaJIOe

BpeMH

II

IICXOP;HM

rrplI

3TOM

lIa

ypaBHeHlIH

Mem;epcKoro

(14.2),

KOTopoe

BhIBeAeHO

KaK

paa

B

rrpep;-

I'

x[dU]

=

~

IdU(1:)/=var[U(t)]::

(14.8)

t

tot

UOJIOmeHlIlI,

'ITO

peaKTHBHaH

Macca

lIaMeHHeTCH

HerrpephIBHO

(CM.

[13*],

CTp.

110). B

COOTBeTCTBIUI

C

aTHM

rrpep;eJIbHhIH

rrepexop;

rrpHBeJI

t

Hac

K

.cOOTHOIIIeHHIO

(14.6).

TaKaH

HAeaJIbHaH

Kap'l'lma

HBJIeHHH,

paBHyIO

rrOJIHoMy

uaMeHeHlIIO

var

[U

(t)lt~

<PYHKIJ;HlI U

(t)

Ha

oTpeaKe

BHP;HMO,

YAOBJIeTBOpHTeJIbHO

OTBelJ:aeT

ero

peaJIbHoMy

cOAepmaHlIIO.

t",

-<

t < tfl·

11:TaK,

TpeooBaHlIe

ocym;ecTBlITb

rrepeBoA

TOlIKlI

lIa

Ha-

ECJIH

me,

OAHaKO,

ClJ:HTaTb,

lITO

KOHelIHaH

Macca

OTopaChIBaeTCH

MrHO-

lIaJIbHOrO

COCTOHHlIH

X'"

B

KOHelIHoe

COCTOHHHe

xf3

C

MUHHMaJIbHhIM

BeHHO,

KaK

eAlIHoe

IJ;eJIoe,

TO

ypaBHeHlIe

(14.5) C

<pyHKIJ;HeH

U (t),

paCXOAOM

peaKTuBHOH

MaCChI

m

l

(t)

(npII

aaAaHHOH

HeuaMeHHOH

lIac-

orrpe)l;eJIeHHOH

COOTHOIIIeHHeM

(14.6),

yme

He

rrOJIYlIHTCH.

~eHCTBlI

TII

m

o

)

aKBlIBaJIeHTHO

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x

[dU]

TeJIbHO,

rrpeArrOJIOmHM,

lITO

B

HeKOTophIH

MOMeHT

BpeMeHH

t = t* '1'0'1-

(14.8).

Ka

IIMeJIa

Maccy

m =

m.

+

lim

l

H B aTOT

MOMeHT

BpeMeHH

MhI

OTOpO-

ECJIH

Tenepb

paQcMaTplIBaTb

<PYHI<IJ;HM

hW

(T)

H h(2)

(T)

KaK

CHJIlI

cpaay

Maccy

lim

l

=F

0.

Torp;a,

rrOJIbaYHCb

TeopeMOH

00

HaMeH€-

aJIeMeHThI

<PYHKIJ;lIOHaJIbHOrO

npoCTpaHCTBa

~{h

(T)}

CKaJIHpHIiIX

88

-------~.

--~.~~-_

..

~.

"-'-~~~,

..

",'

:.'

.,

','

',',

:

..

:,'

;::

",

~::'-"

.

·~··:~··:~~1~:.:.:

···:

....

:."

c.

"-.:.

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OB

OIITHMAJTbHOM

YIIPABJIEHHH

[rn.

4

HerrpephIBHhIX

4>YHR~HH

h ('t)

(ta.

< 't <

t~),

rAe

HopMa·

p [h ('t)]

orrpeAeJIeHa

paBeHCTBOM

(CM.

§§

56,59)

P [h {'t}] =

max

Ih

(1:')

I,

ta.

-<

1:'

<ti>],

(14.9)

'"

TO

BeJIHlIHHhI

10

I,q

~

h(l) [t/3'

1:']

dU

(1:'),

~

h(2)

[til'

1:']

dU

(1:')

la

MomHO

paCCMaTpHBaTI>

RaR

3HalIeHHH

JIHHeHHOH

orrepa~HH

CPu

[h]

Ha

::meMeHTax

kil)('t) H

h(2)

('t). B

caMOM

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4>YHR~HH

U ('t),

orrHCbIBaIOm;aH

paCXOA

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rrpeArrOJIaraeTCH

HerrpephIBHO

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t.,

rAe

copacbIBaloTcH

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MaCC:bI

ml

(t).

CJIeAOBaTeJIb-

HO,

M:bI MOJI<eM

rrpeArrOJIaraTb

TeM

OOJIee,

lITO

U

HBJIHeTCH

4>YHR~HeH

C

orpaHHlIeHHhIM

H3MeHeHHeM.

Ho

H3BeCTHO

(CM.

§ 59),

lITO

JIIOOaH

4>YHR~HH

U ('t) C

orpaHHlIeHHhIM

H3MeHeHHeM

Ha

OTpeSRe

Itlt'

t~]

RaR

pas

H

rropomAaeT

B

rrpOCTpaHCTBe

re{h ('t); t

a

< 't

-<

t~}

JIHHeHHYIO

orrepa~HIO

II'>

CPu

[h

(1:')]

~

h('t)dU(1:'),

(14.10)

la

rrpHlIeM

113

p*rcpu]=p'rU]=

~

IdU(1:')I=varrU(1:')]:~.

(14.11)

Ia.

8TO

H

OOOCHOBhIBaeT

Hallie

yTBepmAeHHe.

l1:TaR,

B

AaHHOM

rrpHMepe

MhI

BHAHM,

lITO

MOJImO

SarrHCaTI,:

Xl

(ti»

=

X~

+

(ti>

- ta)

X:

+

CPu

[h(l)

(T)],

X2

(ti»

=

X:

+

CPu

[h

(2)

(1:')].

IlpH

<ITOM

3aMelIaeM

TaRme,

lITO

BhIOpaHHaH

HS

4>HsHlIecRHX

COOO-

pameHHH

HHTeHCHBHOCTb

x

rdU]

(14.8) B

AaHHOM

CJIylIae

HMeeT

RaR

pas

CMbICJI

nopMb~

p*

[CPu]

(14.11)

orrepa~HHcpu.

PaccMoTpHM

TerrepbOOJIee

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rrpHMep.

IlycTb

B

ypaBHeHHH

(14.1) U -

CRaJIHp

H,CJIeAOBaTeJIbHO,

B (t) -

BeRTOp

b (t). B

RalIeCT-

Be

HHTeHCHBHOCTH

x

rdU]

BhIoepeM

orrHTb

BeJIHlIHHY

rrOJIHOrO

HMrryJIb-

ca

(14.8)

yrrpaBJIHIOm;eH

CHJIhI.

HaxoAH

CHOBa

saROH

ABHmeHlm

i

I

f

!

t

I

!

I

f

i

!

§

14

1

CHCTeMhI

(14.1)

rro

C

O"

',':

,~>

OEOEll(EHHAH

3A,[(ATIA

89

4>opMyJIe

KOlliH

(6.6),

rrOJIylIHM

rrpH

t =

t~:

113

1/3

Xi

(ti»

=

x[i]'

(tf], ta) X (ta) +

~

XCi]'

(t/3'

1:')

dW

(1:')

+

~

h(i) Iti>' 1:']dU(1:')

Ia.

la

(i

= 1,

..

'J

n). (14.12)

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Taro

lITOOhI

HHTeHCIIBHOCTb

x

rdU]

(14.8)

MomHO

ObIJIO

IIC-

TOJIROBaTb

RaR

HOpMy

P*

[CPu]

orrepa~IIH

CPu,

CJIeAyeT

paCCMaTplIBaTI>

4>YHR~IIII

h(l) ('t) =

h(l)rt~,

'tl,

...

,

h(n)

('t) = kin)

[t~,

't] OrrHTI>

RaR

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re{h ('r)}

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h ('t) (t

a

< 't <

t~)

C

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(14.11).

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RaR

II

B

rrpeAbIAY-

m;eM

lIaCTHOM

rrplIMepe,

rrOCJIeAHlIe

CJIaraeMbIe

D

paBeHCTBax

(14.12)

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TpaRTOBaTb

RaR

SHalIeHIIH

JIIIHeHHOH

orrepa~IIII

CPu

[h),

rropom-

AeHHOH

4>YHR~HeH

U ('t)

II

BhIrrOJIHeHHOH

HaA

<IJIeMeHTaMII

h(i) ('t)

(i

~

1,

...

, n),

T.

e.

1/3

Xi

(tf»

= X

[i]'

(tfj, ta) X (ta) +

~

xCi]' (til'

1:')

dW

(t)

+

CPu

[h

1i

) (t)]

.

la

(i=1,

...

,n),

rrplIlIeM

OrrHTI>

BhIrrOJIHHeTCH

(14.11).

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0

~eJIeCOOOpaSHOCTII

paCllilIpeHIIH

RJIaCCa

e03-

MO;)fCl-lb~X

yrrpaBJIeHIIH

dU

('r)

AO

yrrpaBJIeHIIH

AorrycRaeM:bIX,

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AeHHhIe

B § 12

AJIH

CJIylIaH

CHCTeMhI

(12.8), CJIeAYElT

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II

SAeCb

B

CJIylIae

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(14.1).

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me

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II

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AYIOm;ee

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rdU]

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meT

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HopMa

p*

[CPu]

JIIIHeHHOH

orrepa~IIII

(14.10)

Ha

HeRoTopoM

4>YHR~IIOHaJIbHOM

rrpocTpaHcTBe5<1{h

('t)}.

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CRameM,

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('t)},

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R

5<1

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('t)} ,

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00

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OhITb

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BaHa

Terrepb

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oopasoM.

3 a A a q a

14.1.

II

ycmb ynpaMJle.MaJl

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onUCb~eaemCJl

0606-

Ufel-ll-lbW

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ta.

<'

t <

t~

U

l-la'l(,a/1,bl-lOe

x

a

U

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x·

9

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l-la

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x

rdU]

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p*

r

U]

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dU

(t)

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dUO

(t),

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(14.1)

U3

co-

a

cmOJll-lUJl

X (ta) = x e

cocmOJll-lue

X

(t,q)

=

xf3

U

UMelOUfee

npu

amOM

l-lau-

Mel-lbUtylO

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C

90

IIOCTAHOBHA

3A,I:J:AtIH

on

OfITHMAJILHOM

YIlPABJIEIlJUI

(fn.

4

II P M M e q a H H e 14.1. B

AaHHOM

CJIyqae

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CtPOpMyJIlIpOBaTL TaRlRe

H 3aAaQy,

COOTBeTCTBYIOIl..\YIO

3aAaqe 13.2.

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op;HaKO,

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tP0PMYJIR-

pOBKy npMBop;UTh

He

6YAeM,

TaR

KaK

OHa

nOJIyqaeTCR

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3aAa'lH 13.2.

B

npe,n:hI,n:y~HX

npHMepax

<PYHR~HH

U

(t)

6hIJIH

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H B

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(t). B COOTBeTCTBHH C

3THM

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H

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sa,n:alJH

14.1 CJIe,n:yeT OTHOCHTb R

06~eMY

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dU

(t).

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H B § 12,

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14.1 C

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('t)}

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p*

[UJ

lIMeeT

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L

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B

npHJlOiKCHHHX.

T a 6

JI

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IT

a 14.1

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[dU]

= p •

[U]

I

p

[h]

I

U ('t)-cRaJIJlp h (t')-cRaJIlIp .

l{-l

~

IdU

(t')[

=var

[U

('t)l/:

lex

U

('t)

=

{U

I

(t'),

•...

, U

I'

('t)}-I!eI{'rop

I;,

,.

~

(2J

dUr

(t')f'

lex

i=1

max

Ih

(t)!

lex""

't""

l{-l

II

h

('t)=

{h] (T),

...

, hr(t')}-BeK-

TOp

I'

max

(2J

hr(t')f'

lex

""

't

""

If]

;=1

.

III

If]

,.

~

2}ldU

i

(T)I

lex

;=1

l{-l

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mfx

jdU

i

(T)I

lex

max

(max

Ih

i

(t)1)

t(L~'t<tp

i

,.

lex

~a'tx<;;

If]

(~jhi(-t>l)

;=1

IV

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nOJIe

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sa

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t~

- t

ex

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14]

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91

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m

l

(t

ex

).

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ca

m

l

(t

ex

) B

npo~ecce

ynpaBJIeHHH

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lICnOJlbsyeTcH,

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lJTO

mi

(tfJ)

=

O.

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C

sap;aHHoH

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BHP; (CM.

n.

5° § 3)

..

V x

2

~

r =

-7

+

---,:a

+

a,.

(t)

In'

x=

ra<l>

(t)

~,

m

rp;e

m = m

o

+ m

l

(t), r - ,n:JIHHa

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Z2'

Za

-

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01'

HX

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lJeHHH

Ha

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e.

Zl

= r

~

ro,

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=

f,

za

= X -

Xo'

3p;ecb

r

o

-

pap;Hyc

sap;aHHoH

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-

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X

npH

p;BlIlKeHlIH

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no

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a

Zl =

Xl,

Z2

=

-V

:

X2'

Za

= ; V

1':

Xa,

t =

_0

t'

r

~

v '

o

sanHmeM

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p;BHIKeHHH

TOlJRlI

B

irepBoM

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(CM.

§ 3,

ypaBHeHHH

(3.23)):

dXI =

X2

dt

',

dX

2

= -

xldt'

+:radt' +

..

~_

a

r

(t,)m

dt',

r v m

d

21'0

(

')

~

d '

xa

=

..

r-

a<l>

t -

t.

r

v

m

Bes,n:e B

p;aJIbHeHmeM

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t 6y,n:eM

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ml

(t)

BI>I6paChIBaeTCH

C

OTHOCHTeJIbHOHCROPOCTbIO a,

nOCTOHHHOHno

Mop;yJlIO

(la/ =

Va~

+a~

=

const),

HO

nepeMeHHOH

no

HanpaBJleHHIO.

TaR

IKe

RaR

1I

BhI-

me

(CM.

CTp.

85),

6Y,I:J:eM

npe,n:nOJIaraTb,

lJTO

BhI6poc

MaCChI

npollCXO-

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HenpephIBHO

(TaR,

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cy~ecTByeT

npOlISBop;HaH

dm (t)ldt),

sa

lIC-

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BpeMeHlI

t

k

(k = 1, ..., l)

llS

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[tot:,

tfJ]'

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7n

1

(t)

MOIKeT

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llpep;eJIbIlhIil:

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92

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(rJl.

q

ee

aa

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npoMemyI'oR,

npHTIeM

B

I'eTIeHHe

:noro

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a OI'.n;eJIHIOrn;HXCH TIaCTHIJ;

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H

HanpaBJIeHHIO.

Tor.n;a

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RaTIeCTBe

0606rn;eHHhIx

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dU

I

(t)

H

dU

z

(t),

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C

MaCCOH

m

(t)

COOI'HorneHHHMH:

dU I = a

r

(t)

!!!....

dt

= a

r

(t) d In m (t),

m

dU

z

=

a",

(t)

!!!....dt

=

a",

(t)

dIn

m (t),

m

Ror.n;a

cyrn;ecTByeI'

rrpOH3Bo.n;HaH

m(t), H

COOTHorneHHHMH:

I:1U

I

= a

r

(tIJ

1:1

In

m (tk),

I:1U

z =

a",

(t

k

)

1:1

In

m (tk)'

eCJIH

B

MOMeHT

t

k

Macca

H3MeHHeTCH

CRaTIR006pa3Ho.

IITaR,

OROHTIaI'eJIbHO

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.n;BHmeHHH

TOTIRH B

nepBOM

rrpH6JIHmeHHH:

.

dXI=Xzdt,

I

.

ro

dxz =

-xldt

+

xsdt

+

yv

dU

I

,

(14.13)

dxs =

~~

dU

z

·

IIYCI'b

Tpe6yeTcH

rrepeBecTH

TOTIRy

H3

3a.n;aHHoro

ml.TIaJIbHOrO

rro-

JIOmeHHH

x

(t,,;)

=

xa.

B

3a.n;aHHoe

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rrOJIOmeHHe

x

(t~)

= x

13

=O

C

MHHHMaJIbHhIM

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m

l

(ta.).

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B

Ra-

TIeCTBe

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%

[dU]

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dU

(t)

= {dU

I

(t),

dU

z

(t)} CJIe.n;yeT

BbI6paTb

BeJIHTIHHy

(CM.

§ 60, r.n;e .n;aHO TOTIHOe

orr-

pe.n;eJIeHHe

CMhICJIa

IIHTerpaJIa

(14.14»

t{3

% [dU] =

~

IldU

(t)I]·

(14.14)

ta.

,[J;eHCI'BHTeJIbHO,

rrOBTopHH

C

He3HaTIHTeJIbHhIMH

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.n;o-

BO.n;hI,

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CTp.

86,

MomHO

CHOBa

y6e,ll.HTbCH

B

crrpaBe.n;-

m

o

+m

l

(ta.)

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paBeHCTBa

% [d

U]

=

II

a

II

In . . A aTO H

03HaTIaeT

I

mo

TITO,

MHHHMH3HpyH

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%

[dU]

(14.14),

MhI

ocyrn;ecI'llHM

rrepeBO.n; TOTIRH

Ha

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C

MHHHMaJIhHhIM

pacxo.n;oM

ua-

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3anaca

peaRTHBHOU

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ml

(tot).

!

',

.:",'

..

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C

§ J4]

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aA,ll;ATIA

93

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Toro

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rrpaBO

CIJIOpMyJIHpOBaTb

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B

1JI

0

pMe

3a.n;aTIH 14.1,

HaM

He06xo.n;HM06hITb

yBepeHHhIMH

B

B03-

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no.n;o6paTb

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fiiJ

{h ('rn

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o6pa30M,

TII'o6M

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% (dU] (14.14)

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IJIYHRIJ;HH U

('I:)

n

rrpo-

CTpaHCTBe

fiiJ*{

U

(T)},

corrpHmeHHoM

R

fiiJ{h

(T)}.

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aTOH

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R

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fiiJ

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(T)}

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npocI'paHcTBo

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h(T)

= {hI

(T),

h

(T)}

C

HOpMOH

z

p [h

(t)]

=

max"

h

(t)

II.

(14.15)

't"

TaRHM

06pa30M,

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(14.13)

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3a.n;aTIY

06

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(t)

={dU

I

(t),

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(tn,

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t

(t)

rropom.n;eHhI

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U

i

(t) C

orpa-

HHTIeHHhIM

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(t),

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(14.13)

II3

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x

(ta.)

=

xa.

=

{x~,

x~,

x~}

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x

(t~)

--:-

x!3

=

{O,

0,

O}

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aTOM

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p*

[U0].

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(T)

(i = 1, 2, 3),

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orrepa-

IJ;HH

CPu

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(14.12) .n;JIH

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(t).

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IJ;a H

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(14.13)

HMeIOT

BH.n;

!

cos (t -

ta.)

sin

(t

-

tot)

1-

cos (t -

ta.)

X (t,

ta.]

= -

sin

(t -

ta.)

cos (t ~

ta.)

sin

(t -

ta.)

II'.

o 0 1

H[t,

t]

=X[t,

t].B(T)=

cos

(t-t)

sin(t-t)

1-cos(t-t)

o

-

ro

0

_

,,-

,sin (t -

t)

cos

(t

---,-

t)

sin

(t

-

t)

Yv

,,-

o 2ro

o

1

Yv

ro

2ro

yv

sin

(t -

t)

Vv(1 - cos (t -

t»

ro

2ro .

yv

cos(t-t)

yv

SID

(t

-

t)

2ro

o

yv

'

:-:...c~:2-~L..l.~~--'--"'-"'.;",:,,"~~':""

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eo"

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lrJI. 4

IlOCTAHOBHA SA,D;AqH

on

OIITHMAJIbHOM YIIPABJIEHI1H

94

CJIep;OBaTeJIbHO, COrJIaCHO (14.12) BeHTOp

x(t(3)

OnIIC:bIBaeTCH

paBeH-

CTBaMII:

Xl

(t(3)

=

x~

cos

(t(3

- t

rx

) +

x~

sin

(tfl-

ta.)

+

x~

(1

-

cos

(t(3

-

ta.»

+

t(3

1(3

+

.v"

~

sin

(tl>

-

1:)

au

d't')

+

~~

..

~.

(1-

cos

(tl>

-

1:»

au

2

(1:),

ta. ta.

X2

(t(3)

-:-

-

x~

sin

(t(3

-

ta.)

+

x~

cos

(tl>

-

ta.)

+

x;

sin

(tl>

-

tr1.)

+

t(3

II>

+

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~

cos

(t(3

-

1:)

au

I

(1:)

+

~

sin

(t(3

-

't')

au

2 ('t'),

ta.

Ia.

tl>

X3

(t(3)

=

x~

+

~

\

aU

2

('t').

ta.

lIs

:3TIIX paBeHCTB CJIep;yeT, lITO

JIIIHenHaH

onepaIJ;IIH

t(3

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('t')] =

~

h'

(1:)

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('t'),

ta.

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Xi

(t~),

Ha

BeKTop-<pYHKIJ;HHX

2

r

o

MIl

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';0"

sin

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-

1:),

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-

cos

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-

1:»}

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. }

IPl

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=

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sm

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--r)

,

3

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={O,

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=

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h

2

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-<

t

-<

t~)

C HOpMOH

(14.15).

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~'.':,'-'

r.llaBa

5

OBIIJ;EE

PEillEHIfE

3A,[J;A

qlf

OB

YllP

ABJIEHIfIf

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3ap;aQa

06

ynpaBJIeHHII

RaR

np06JIeMa

MOMeHTOB

Hama

6JIlImanmaH

IJ;eJIb -

p;aTb

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sap;alI

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II

14.1.

IIpep;JIaraeMoe

pemeHI:Ie

6yp;eT

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Ha

MaTeMaTHlIecRyro

TeopHIO,

CBHsaHHyIO C

TaR

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npo6/1-eJJwu

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IIO:3TO-

MY,

npemp;e

lIeM

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pemeHlIro

sap;alI

06

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CHa'IaJIa

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ROTopaH

cy-

m;ecTByeT

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H

ynoMHHyToH

np06JIeMon.

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paCCMOTpHM

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X

(trx)

=

xrx

II

X

(tf»

=

xi'

B

paBeHCTBa

(12.10),

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p;BlIme-

HHe x

(t)

ynpaBJIHeMOrO

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t,')

,

(

Ill

i

]'

[t

p

,

-r]

U

(-r)

a't

=

Ci

(i~1,

...

,n),

(15.1)

t"a.

B KOTOpbIX <PYHI\IJ;HH h[iJ

[t~,

't]

= h ('t) H lIHCJIa

Ci

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co-

rJIaCHO (6.1), (8.4)

II

(12.10) cJIep;yrom;HMH COOTHomeHHHMH:

n

Il'J

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=

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l

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=

{2J

Xii,

[t(3,

-r]

bl,j

(-r)},

/.=1

tl>

(15.2)

Ci

= X r-

apr

[tl>,

t

rx

]

xa.

-

~

X[iI'

[t[3,

-r]

w ('t)

a-r

ta.

(i=1,

...

,n).

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Ha

TO, lITO BeRTop-<pYHKIJ;HH

h[i]

[t~,'t

1

II

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no

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a

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rx

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(T),

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96

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p*

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TO

sa,n;alJa

pemaeTCH

cpasy,

TaR

RaK

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,n;OCTaTOlJHO

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lJTO

U ('t) :=

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n

meM

Mol

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O.

i=l

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(15.1),

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12, 13, HMeIOT

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BH,n;e

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p 0 6

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e M a

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8 npocmpalicm8e

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r

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15]

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e.

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(15.4).

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pellIe-

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B COCTOHHHH

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ee

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nopom,n;aeTcH

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yrrpaBJIeHHeM

U ('t). B <popMe

HHTerpaJIa.

HHalJe

rOBopH,

He06xo,n;HMO, lJTo6bI

JII06aH

JIHHeHHarr

orrepan;HH

<p

[hl,

HMeIOIIJ;aH CMliICJI

Ha

5iJ{h

('tn,

MOrJIa

6LITb

sanHcaHa

B

BH,n;e

t~

q>

[h] =

~

h'

(or)

u

(or)

d't', (15.5 )

tl1l.

rAe

<pyIi~HH

U ('t)

co8ep:JICumc5t

cpe,n;H

paccMaTpHBaeMLIx

HaMH

ynpaB-

JIeHHH U ('t).

,[(0

CIIX

rrop

MIiI

He

aR~eHTlIpOBaJIH

BHIIMaHHe

Ha

aTOM

06CTOHTeJIbCTBe.

RaR

Mor

saMeTHTb

l:JHTaTeJIh, B § 12, r,n;e

BrrepBbIe

6LIJIO HCnOJIbSOBaHO

nOHHTHe

JIIIHeHHOH

orrepa~HH

B CBHSII C

ABH-

meHHeM

x (t),

MbI

OnHpaJIHCb

Ha

aTy

CBH8b Mem,n;y

yrrpaBJIeHHeM

U ('t)

H

orrepa~HeH

<Pu

[h],

HCnOJIbsyH

ee"

JIHillb

B

O,n;HOM

HarrpaBJIeHHII

-

0'1'

U ('t) R

<Pu

[hI.

IIpH

aTOM 6LIJIO OTMelJeHO,

lJTO

COOTBeTcTByIOIIJ;He

'lJIe-

HbI

B

sanHCH,

onpep;eJIHIOIIJ;eH ,n;BHmeHHe x (t),

MomHO

HCTOJIROBaTb

KaR

JIHHeHHyro

orrepa~IIIO

<Pu

[hJ,

rropomp;eHHylo

803MO:JIClibLM

ynpaBJIe-

HHeM

U ('t).

Tenepb

me,

RaR

MLI

BH,n;HM,

,n;JIH

Hac

llameH

H

06paTHbIH

nepexo,n;.

IIp06JIeMa

MOMeHTOB :3RBHBaJIeHTHa SaAalJe 13.1 Tor,n;a H

TOJIbRO Tor,n;a, Ror,n;a 'JIIIHeHHLIe

orrepa~HH

<p

[h], HMeIOllI,He

CMbICJI

Ha

fJiJ{h

('tn,

Haxo,n;HTCH

BO

BsaHMHO 0AHOSHalJHOM COOTBeTCTBHH C

pac-

CMaTpHBaeMLIMH

ynpaB-!IeHHHMH]

u(t).

ECJIH

6bI

MLI

OrpaHH'lHJIHCh

JIHillb

KpyroM

803MO:JIClibLX

ynpaBJIeHHH

U (t),

TO

TaRoro

COOTBeTCTBHH

nOJIyqHTb

6LI

He

YAaJIOCb H

Ham

nepexoA

R

np06JIeMe

MOMeHTOB 6bIJI

6bI

HeSaROHHbIM.

HMeHHo,

B rrOCJIeAHeM CJIylJae

He

P;JIH

BCHROH

onepa-

~IIH

<p

[h]

MomHO

6LIJIO

6bI

nop;o6paTh

803MO:JICH,Oe

ynpaBJIeHHe

U ('t),

ROTopoe

onpe,n;eJIHJIO

6bI

ee

B

BIIAe

(15.5).

OAHaRo

MLI

yme

yCJIOBH-

JIHCb B

§ 12

pacmHpHTb

Rpyr

BOSMomHbIX

ynpaBJIeHHH

AO

MHomeCTBa

8onyc1'i.aeJt1,bl,x

ynpae.n.e1-luu.

8'1'0

RaR

pas

H 6bIJIO CAeJIaHO C TOR

~eJIbIO,

lJT06LI

06eCne'lIlTb

HymHoe COOTBeTCTBlIe

MemAY

U ('t) H

<Pu

[h ('t)

1:

MHomeCTBO

AorrycKaeMMX

ynpaBJIeHHH

MLI

orrpeAeJIHJIH

HMeHHO

TaR,

lJTO

JII06aH

JIHHeHHa8

onepaU;HH

<pu[hj,

onpeAeJIeHHaH

Ha

npOCTpaHCT-

Be

fJiJ{h

('tn,

ORaSaJIaCb

npeACTaBHMOH

B BHAe (15.5),

rAe

<PYHK~HH

U ('t) -

8onyC1'i.aeMOe

ynpa8.11.eliue.

l1TaR,

eCJIH MLI HIIJ;eM

ynpaBJIeHHe

U ('t) cpe,n;H

,n;onycKaeMliIx

ynpaBJIeHHR,

TO

saAa'la

13.1 nOJIHOCThIO

aK-

BHBaJIeHTHa

np06JIeMe

MOMeHTOB. CJIeAOBaTeJIbHO, AJIH

Toro

'lT06hI

HaHTH

OnTHMaJIbHOe

.ynpaBJIeHHe

UO

('t), AOCTaTOlJHO

pemHTb

np06JIe-

i

My

MOMeHTOB.

PemeHHe

aTOM

rrp06JIeMbI

orrHcaHo

B

CJIeAYIO~eM

napa-

i

rpa<p

e.

,I

\

4

H.

HpacoBClUfH

i

98

.,:

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'::".

::.::

..

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.

',::'

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...

.'

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,:'

..

:

..

~-

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.....

:~~.,:

:.,::.-:

r::~

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OBJJ~EE

PEJUEHHE

3AAALUI OB

YIIPABJIEHHM

lrJI. 5

3aAa

t

Ia

14.1

06

ynpaBJIeHHlI

B tPopMe

dU

(t)

TaRme

MomeT

6MTb

CBe.n;eHa

I{

np06JIeMe

MOMeHTOB.

,IJ;JIJI

SToro

HymHo

JIlImb

no.n;06paTb

COOTBeTCTBYIOII~HM

06paaoM

npOCTpaHCTBO

fJiJ{h

('t);

tat

<"

't

<"

tl3}.

BM-

CRaaaHHoe

yTBeplI{.n;eHHe nOJICHHM,

HanpHMep,

.n;JIJI

CJIyqaJI CRaJIJIpHO-

ro

ynpaBJIJIIOm;ero

Boa.n;encTBlIJI, Ror.n;a MaTpHIJ;a

B(t)

B

ypaBHeHHJIX

(14.1) eCTb npOCTO BeRTOp bet).

IIHTeHcHBHoCTb

%

[dU]

ynpaBJIeHlIJI

dU

(t)

B:bI6epeM B BII.n;e (14.8). IIo.n;cTaBJIJIJI

RpaeBMe

YCJIOBIIJI x

(ta.)

=

xa.

II

x(tl3)

=

xf3

BpaBeHCTBO

(14.12),

nOJIyqaeM

CIICTeMY

ypaB-

HeHlIn

t(3

~

h(i)

[t(3,

'tJ

dU

('t) =

Ci

(i=1,

...

,n),

(15.6)

tat

r.n;e

Ci

onpe.n;eJIJIIOTCJI CHOBa

BMpameHlIeM

(15.2)

(caaMeHon

w ('t) d't

Ha

dW).

CHcTeMy

(15.6) CJIe.n;yeT

Tenep:bpaapemlITb

nplI

yCJIOBHlI, qTO

p*

[U]

=

min.

B .n;aHHOM

CJIyqae

tPYHRIJ;HII

h(i)

[tl3,

't] Ha.n;JIemlIT

paccMaTplIBaTb

RaR

SJIeMeHTM tPYHRIJ;lIOHaJIbHOrO

npocTpaHcTBa

7iS{h

('t);

tat

<"

't

<"

tl3}

HenpepMBH:bIX

Ha

[ta.,

tl3J

tPYHRIJ;lIn h ('t),

r.n;e

HopMa

p

[h

('t)] =

max

Ih ('t)

I.

l]pocTpaHcTBoM

fJiJ*

{U

('t);

tat

<"

't

<"

~

.

<"

tl3},

onpe.n;eJIJIIOm;IIM·

.n;onycRaeMMe

(ynpaBJIeHIIJI,

6y.n;eT Tor.n;a

npocTpaHcTBo

tPYHRIJ;lIn U ('t) C

OrpaHlIqeHHMM

lIaMeHeHlIeM

Ha

[tat

,tl3],

C

HopMon

p*

[U],

paBHon

BeJIrrqlIHe

%

[dU]

(14.8).

COrJIaCHO

MaTeplIaJIy

§

14

JIeByIO

tIaCTb

(15.6)

MomHO

paCCMaTpH-

BaTb

RaR

peayJIbTaT

JIHHenHOn

OnepaIJ;HH,

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tPYHRIJ;lIen

U ('t)

Ha.n;SJIeMeHTaMlI

h(i)

[t13,

't]

lIa

7iS

{h ('t);

tat

<"

't

<"

tl3}'

TaRIIM

06paaoM,

HaxO.n;lIM, qTO

paCCMaTpHBaeMaJI

aa.n;aqa

CBO.n;lITCJI

R

np06JIe-

Me

MOMeHTOB B tPYHRIJ;IIOHaJIbHOM npOCTpaHCTBe

7iS[ta.,

t(3]'

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CJIyqaeM,

no.n;qepRHeM, qTO H

a.n;eCb

aa.n;aqa 14.1 ORaaMBaeTCJI

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JIeHTHOn

np06JIeMe

MOMeHTOB

no

Ton

nprrqlIHe,

qTO

06m;lIn

BlI.n;

JIHHen-

Hon

OnepaIJ;HlI

Cpu

[h]

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3JIeMeHTax

h ('t)

lIa

7iS[tat,

t,s] onpe.n;eJIJIeTCJI

paBeHCTBOM

(14.10),

r.n;e

dU

(t)

-

HeRoTopoe

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yn-

paBJIeHlIe.

TIOJIeaHO

aaMeTlITb,

qTO

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aa.n;aqH

13.1,

ROTOpaJI CBo.n;IIJIaCb R

paBeHCTBaM

(15.1), tPYHRIJ;lIlI

Mi)

[tl3,

't] HeJIbaJI 6MJIO

paccMaTplIBaTb

B RatIeCTBe 3JIeMeHTOB Ha

7iS[t

at

,

t13J'

B caMOM .n;eJIe,

06m;lIn

BlIll

JIIIHenHOn

onepaIJ;HH

cp[h]

B

7iS[ta.,tI3J

aa.n;aeTCJI

tPopMyJIOn

(14.10), a

He

tPOpMy-

JIon

(15.5). TIOSTOMy OnepaIJ;IIJI

cp

[h],

pemaIOm;aJI

np06JIeMy

MOMeHTOB,

COOTBeTCTBYIOm;yIO aa.n;atIe

13.1 (eCJIII

YRaaaHHyIO

np06JIeMy

paCCMaT-

plIBaTb

B npOCTpaHCTBe tPyHRIJ;Hn

7iS[t""

t,9l),

MomeT

ORaaaTbCJI BOBce

He

npe.n;CTaBlIMon

B BlI.n;e (15.5).

TaRaH

onepaIJ;lIH

He

6y.n;eT, OqeBlI.n;HO,

.n;aBaTb

pemeHHe

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aa.n;atIH 13.1.

,IJ;JIJ-I

CUCTeMbI C

BeRTopHMM

ynpaBJIeHUeM

dU

(t) aa.n;aqa 14.1

BO

MHorux

CJIytIaJIX

TaIOI,e

MOJI\01'

6bITb

cne.n;eHa R

np06JIeMe

MOMeHTOB.

o

b'

§ 15]

aAAAtIA

OB

ynPABJIEHHII

RAR

npOBJIEMA

MOMEHTOB

99

MbI

He OCTaHaBJIlIBaeMCJI

a.n;eCb

Ha

3TOM

Bonpoce,

TaR

RaR

BceBoaMom-

HbIe

CJIyqalI,

CBo.n;IIMbIe

TaRlIM

nyTeM

I{

np06JIeMe

MOMeHToB, Tpy.n;HO

060apUMM.

,IJ;JIJI

pJI.n;a ROHRpeTIU,IX

CJIyqaeB

yRaaaHHoe

cBe.n;eHue 6y.n;eT

lICnOJIbaOBaHO

HlIme,

B § 23,

no

Xo.n;y

pemeHUJI

aa.n;atI.

IIp06JIeMa

MOMeHTOB, R

ROTopon

MM

BMme

CBeJIU aa.n;aqu

06

yn-

paBJIeHlIII,-

31'0 BeCbMa

06m;aJI

MaTeMaTlIqeCRaJI

np06JIeMa,

BOaHlI-

RaIOm;aJI

lIa

npIII{JIa.n;HMX aa.n;aq 0 pacnpe.n;eJIeHHU

Macc,

YCIIJIlIn,

BepOJITHOCTen

II

T.

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06eCneqlIBaIOm;eM

meJIaeMbIe

BeJIUtIlIHM

COOTBeT-

CTBYIOm;IIX MOMeHTOB (MOMeHTOB ClIJI, MOMeHTOB

lIHepIJ;lIlI,

BepOJITHOCT-

HMX

MOMeHTOB

II

T.

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B RaqeCTBe

lIJIJIIOCTpaIJ;lIlI~

I{

CRaaaHHOMy

U

meJIaJI

pacmlIplITb

npe.n;CTaBJIeHlIe 0

co.n;epmaHlIlI

np06JIeMbI

MOMeH-

TOB,

paCCMOTplIM

CJIe.n;YIOm;lIn

nplIMep

tPlIarrqeCRon

aa.n;aqlI, OTJIUqHOn

01' aa.n;atIlI

06

ynpaBJIeHIIII,

HO

npuBO.n;JIm;en

TaRme

R

np06JIeMe

MO-

MeHTOB.

llpoMep 15.1.

IIycTb

Ha

nJIOCKOCTII

g,

1')}

saAaHa

KpIIDaH,

OUIICbIDaeMaH

rrapaMeTpHqecKIIMlI

ypaDHeHIIHMlI

S

=>

S(-r),

1')

=

1')

(-r)

(t"

~

-r

~

tfJ).

(15.7)

IIpeArroJIaraeTcH,

qTO

rrJIOCKOCTb {s,

1')}

HaXOAIITCJ;l D OAHOPOAHOM

llJIeKTpOCTa-

THqeCKOM

rrOJIe

II

pacrrOJIOmeHa

rreprreHAlIKYJIHpHO

K

DeKTopy

HarrpHmeHHOCT.lI

llToro

rrOJIH.

3aAaqa

COCTOIIT D

TaKOM

pacrrpeAeJIeHlIII

sapHAoD

DAOJIb

KplIDon

(15.7),

rrpH

KOTOpOM

MOMeHTbI

8l; H 8

11

CIIJI

llJIeKTpOCTaTlIqeCKOrO

DsalIMoAencTDIIH

OTHOCIITeJIbHO

ocen

S = 0

II

1')

= 0

nplIHlIMaJIII

6bI

saAaHHbIe

SHaqeHIIH

8l; = c

,

I

8

11

= C2'

IIPII

llTOM

6YAeM

CqlITaTb,

qTO

paCIIpeAeJIeHlIe

SapJIAOD

DAOJIb

KplIDon

xapaKTepHsyeTcH

JIlIHenHOn

IIJIOTHOCTbIO u

(-r)

STIIX

sapHAoD.

PaCCMoTplIM

llJIe-

MeHT

d-r

KPlIDOii

(15.7) D

oKpecTHocTH

TOql{H

S(-r),

1')

(-r).

STOT

llJIeMeHT

HeceT

SapJ;lA ds = u ('t')

d-r.

CocTaDlID

DbIpameH:lre

AJIH

CTaTMqeCKIIX

MOMeHTOD 8l; M

8

11

,

IIOJIyqnM

p;na

ypaBHeHlIJ;l

AJIJ;l

HeHsDecTHon

IIJIOTHOCTII

u ('t'):

tl3

8l; = A

~

u (-r)

1')

(-r)

d-r

=

CI,

tar.

__

tl3

= A

~

u (-r)!; (-r)

d-r

= C2,8

11

tar.

rAe

A-

l{olliPw~eHT

npOIIOpqlIOHaJIbHOCTII,

06eCIIeqHBaIO~IIii:

COBIIaAeHlUI

pasMepnocTeii.

IIpeAIIOJIOmIIM,

qTO

IIO

YCJIODIIJ;lM

SaAaqH

BeJIMqlIrra

max

lu

('t')1

B.n;OJIb

RplIBon

.n;OJImHa

6bITb

IIO

BOSMomrrOCTH

naHMenI,meiL

BBe.n;J;l

o60SnaqeHIIH

1..1')

('t')

=-

h(l)

(-r),

As

('t') =

h(2)

(-r),

IIOJIyqnJ,l

CJIe.n;Y!OllQ'lo

sa.n;aqy:

HanTH

WYHK~IIIO

u (-r),

y.n;oBJIeTBOpHIo~yIO

ypaB-

EleHlIHM

tl3

.

~

h(i)

(-r)

u

(-r)

d-r

=

Ci

(i =

1,

2),

(15.8)

tar.

TanyIO,

qTO

max

(I

u ('t')j

rrplI

tar.

~

-r

~

t =

min.

(15.9)

13

)

4·

Красовский Н.Н. Теория управления движением

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