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2007

2

- 28 2007 ., 6

-

. 2- 3- .

: 010200 (010501) –

3

4 §1. . 5 §2. 7 §3. .

12 §4. 15 §5. 18 §6. 20 §7. 23 §8. 24 §9. 27 §10. 29 §11. 35 §12. [2] 48 §13. 49

53

4

- 010501 (010200) « »,

, . .03.1 «».

2 2 . ,

, 68 (72 / ).-

– . –

.

.

.

5

§1. .

:

Fwm .

m – ;w – ;F – .

,, .

,.

/ ,.

I. xOyz .,

,

,

r xi yj zk

v xi y j zk

w xi yj zke

tzztyytxx

.

kFjFiFF zyx zyx FFF ,, – OzOyOx ,, .

,,

.II. zr ,, .

)()()(

tzzttrr

.

,o o o o odr d dzo

v r r z r r r z zdt dt dt

( , , , , , , ),( , , , , , , ), (1)

( , , , , , , ).

x

y

z

mx F t x y z x y zmy F t x y z x y zmz F t x y z x y z

6

oooz

dtzd

dtd

dtdr

dtdrr

dtdr

dtrdw 2

2

2

22

2

2

2 .

zr FFF ,, – ooo

zr ,,.

F – , , .22

2

2

2

2

2

, , , , , , ,

2 , , , , , , ,

, , , , , , .

r

z

d r d dr d dzm r F t r zdt dt dt dt dt

d dr d dr d dzm r F t r zdt dt dt dt dt dt

d z dr d dzm F t r zdt dt dt dt

(2)

III. bn,, .)(tSS – .

dtdSv ,

bnvdt

Sdw 02

2

2

.

bn FFF ,, – .

2

2 , , ,

1 , , ,

0 , , .

n

b

d S dSm F t Sdt dtdS dSm F t Sdt dt

dSF t Sdt

.

7

§2.

zJ N

zN

kkkz rmJ

1

2.

.

kr – - z .0zJ . 2J z

, 2zJ .

( )

)(

2

)(

2

)(

2 ),,(VVM

z dvrVM

VMconstdvrzyxdmrJ ,

M – ,V – ,dv – ,

VM

– .

( ):2

1MdJJ

czz .

M – ,cz – , ,

1z – cz ,d – .

R z – ,: 2MRJ z .

N () ,

:N

NNC mmm

rmrmrmr

21

2211 .

8

1. M .z ,

.3 3 3 2

2 2 2 ,0( ) ( ) 3 3 3 30

x M MJ r dv r dv x dxz V V

. .Mconst – (

). xr ; dxdv )(V .

,.

1238382

2

3

23332

2

2 MMMdxxJCz

/ :

124323

222222 MMMMMMdJJ

AC zz .

33

22 RR .

3212

22 RR .

, D.

4813

483

4816

43

22222 MMMMMJ z .

9

2. baM . .

dxdydSdm .

abM

SMconst –

.yr – x .

dxdydSdv – .

SV – ~ .a b a b

SSVMx dxdyy

abMdydxy

abMdSydSydvrdmrJ

0 0 0 0

22

)(

2

)(

2

)(

2

)(

2

dxbabMdx

by

abM a a

0 0

33

30

3 330

33

222

0

3 Mbaa

Mba

xbaMdxb

abM a

–

Ox ,.

3

2MaJ y .

oJ – O ( , ).

2 2 2

( ) 0 0

( )a b

oV

J r dv x y dydx

)(3

)( 22

0 0

22 baMJJdydxyxabM

yx

a b

,

r – dv .

cyx JJJ ,,11

,

C .

10

3. R M .zJ –

Cz ,

dl dmi.Ri=R,

n

iiincz dmRJJ

1

2lim MRdmRn

iin

2

1

2 lim .

( z ).M

R . ( / ) R M .

zJ – Cz

.

r dr rdrdS 2 ,

rdrdm 2 , 2RM

–

.

, MRJ z2 ,

drrdmrdJ c32 2 ,

222

24

0

3 MRRdrrJJR

zc .

rdrddxdydS , ,,

.2 2

2( ) ( )

2 22 3

2 20 0 0 0

23

20

2 .2

zV S

R R

R

MJ r dv r dsR

M Mr rd dr r d drR R

M MRr drR

11

4.zyx ,, r ,

a , . M –-.

)2()1(xxx JJJ ,

)1(xJ –

,)2(

xJ – .

ZY JJ , .

2

2)1()1( rMJ x , 6

2)2()2( aMJ x .

)2()1( , MM M :

2(1) (2) (1) (1)

2

2(1)

2(1 ),

aM M M M Mr

aMr

Mar

rM 22

2)1(

,

Mar

ar

aMM 22

2

2

2)1()2(

,

MararJ x )(6

322

44

.

:

yx JJ ,R

M ..

yx JJ , ab M ,

11, yx JJ . –

.

12

MR . z

. 23,0 MRJ z .

M R z.

24,0 MRJ z .

§3. .

1. = 98 ,

, . ,, 1

, 14,4 .: . –

.0 .

: – ,, F =cd,

. : 0FP0cdP , d= / – , –

.

13

:

0 00 : , 0Pt x x x xc ,

cDF , F – ,(

); D – ,

. . xdMMDx 0 ., )( xdcFx . (1)

:

xFPxm , (2) (1), (2):

cxcdPxgP

. (3)

(3) 02 xkx , (4)

Pcgk – ( ).

(4) 022 k .ki2,1 .

(4) )sin()cos( 21 ktcktcx . (5)

,)cos()sin( 21 ktkcktkcx . (6)

(5) (6) 0 00 , 0Pt x x x xc .

0, 201 ccPx .

)cos()cos( tPcg

cPkt

cPx .

cP

Pcgk 8,6,12 1 .

a = 6,8 c ; = – /2;

k = 12 -1.

14

kT 52.02

.

2. .

S. ,

, v0.. = 1 T/ 3.

.;

– .

d.: –

, R –

,.

R V , ,

. . dSVR , ,

0dSP . (7)

0,0:0 vxxt . (8) - v0

. , ,)( xdS . , R

)( xdSRx . (9)

xx RPxm .

= , (1) (3), SxxgP

.

:02 xkx , (10)

kP

Sg – .

15

(4) )sin(ktax , (11)

2

202

0 kx

xa – ,0

0

xkx

arctg – .

(2), :

gSPv

kv

a 00

, 0 . (12)

(6),(7) (5),

tP

gSgSPvx sin0 . (13)

gsP

kT 22

.

§4.

1. = 98 H, ,. =10 / .

:R= , =1,6 / . ,

4 0=4 / .

.

,.

:0

0

0, 4 ,4 / .

t x xx x c

,

D = d + x. ,

,)( xdcFx . (1)

:xx RFPxm .

Fx Rx:

16

xcxcdPxgP

. (2)

: – , ,

F =cd, . ,0cdP . (3)

(2) 02 2 xkxnx , (4)

Pgn

Pcgkxx 2

,, , k = 10 c-1, n = 8 -1,

, n < k. (4):

.,02 22222,1

22 nkinknnkn,

))sin()cos(( 222

221 tnkctnkcex nt . (5)

,

22

00201 ,

nk

nxxcxc .

,

cos,sin22

000 A

nk

nxxAx . (6)

)sin( 22 tnkAex nt . (7) ( . . t x 0)

22 nkkc . ,=7,2 , =0,59 , kc=6 -1.

,tex t )59,06sin(2,7 8 . (8)

kT

cc 05,12

.

2. , R= , =5,2 H / .

4 ,0=240 / .

. ,.

17

0

0

40 .

240 /x x

tx x

,

02 kxxnx , .2

,,Pgn

Pcgkxx

, k=10 c-1, n=26 -1,, n>k ( ).

,02 22 kn., 22

222

1 nknknnn > k, 1 2

.tt ecex 21

21 . (9) , :

21

0012

21

0021 ,

xxc

xxc .

(1) :tt exxexxx 12 )()(1

00200121

. (10)

1 2, (2)

tknchknxknshnxxkn

exnt

22220

220022)( . (11)

, . . t x 0.

(3) ,

)529(61 242426 ttt eeex

)24172412(31 26 tshtchex t

,:

0)529(61 242426 ttt eee . , t1=0,037 , t2= .

18

t1, t2 . ,

.

§5.

1.

.=196

,

. ,

D,

.

S = H·sin(pt), = 1,6 , = 60 -1.

= 2 / .

,.

:1)

;2)

,

R= , =25,6 / .

. ,, .

: 0,00 xxt ..

D = . : – , N – , S,

F.

:

xx FSxm xPcgpt

PHgx sin ,

,sin2 ptkxkx (1)

Phgh

Pcgk , . k=100 -1, h=8000 / -2.

(1),

19

)sin()cos( 211 ktcktcx . (2) (1)

)cos()sin(2 ptBptAx ,

)sin(222 ptpk

hx . (3)

(1), (2) (3):

)sin()sin()cos( 2221 ptpk

hktcktcx . (4)

,:

2221 ,0pk

hkpcc .

,

)sin()sin( 2222 ptpk

hktpk

hkpx . (5)

,cttx ))60sin(25,1)100sin(75,0( . (6)

. ,,

R, , .

xxx RFSxm xPgx

Pcgpt

PHgx sin ,

,sin2 2 ptkxkxnx (7)

Phgh

Pgn

Pcgk ,

2, . k = 100 -1, h = 80 -2,

n = 64 -1, = 60 -1. , n < k p < k.

8,04)(

8,7622222

22

pnpkhank 87,02

22 pknparctg .

, )sin(2 ptax ,

).87,060sin(8,0)8,76sin8,76cos( 2164 ttctcex t (8)

,: 1 = 0,62; 2 = 0,12. (9):

).87,060sin(8,0)8,76sin12,08,76cos62,0(64 tttex t (9) : 0,62=bsin ; 0,12=bcos , b=0,63, =1,74.

, :.)87,060sin(8,0)74,18,76sin(63,0 64 ttex t (10)

20

, e-64t.

.§6.

1.

=196 , F

S. : Fx = –cx,

Sx = H·sin(pt), = 2 / , = 1,6 , = 101 -1.

.

.

.xx SFxm ptHcxxm sin

pthxkx sin2 , (1) 2 1 1 210000 , 101 , 8000 / .c Hk p h

m m .

( p=k)

(p>k).,

ptpk

hktpk

hkpx sinsin 2222 . (2)

,,

k .. . k p,

.22,1 pkkpkp

(3)

(3), (2),

)sin(sin))((

ktptpkpk

hx .

(3), :

)sin(sin)(2

ktptpkk

hx . (4)

,pttax cos)( , (5)

21

tkppkk

hta2

sin)(

)( . k p a(t)

kpTa

4. (6)

(5), ,

pT 2

. (7)

k p, >> T, kpp

TTa 2

.

,

( ) cos(101 ), ( ) 80sin 12,56 , 0,063 .2tx a t t a t

2.

,

.,

D ,

),3cos()cos( 310 ptHptHHS, 0, 1, 3 –

.

.

, –

. D .

.,

,.

.,

0 F . d

0. F = d.: 00 cdH .

22

, F :

)( xdccdF xx . (8) , : – , N – , S – .

D: xx FSxm .

(1) S,

.)3cos()cos( 310 cxcdptHptHHxgP

,

)3cos()cos( 312 pthpthxkx , (9)

.,, 33

11 P

gHh

PgHh

Pcgk

(2) )3cos()3sin()cos()sin( 22112 ptBptAptBptAx . (10)

1, 1, 2, 2 ,2 (2),

,.

.9

,0,,0 223

33221

11 pkhBA

pkhBA

(3), :

).3cos()cos( 223

221

2 ptpk

hptpk

hx

k = p . k = 3p .

Pcgk ,

, k p k 3p. .

23

§7.

1. . .«

», 2, , 1975.

l

1 2

.

. 1 2.

.n

k

ekFwM

1c =

n

k

ekF=R

1CwM .

l/2 ,,

22

21OCwn , ,

R2

2gPlMR C .

1 2.

2. . . «», 2, ,1975.

F,

.,

. f. x, y

.

24

. : – , F – , R – , F – ,

, F.:

c FFxM , PRyM c (1) constryC . 0C ,

R = P. , fRF .

F (1), PfPFgxC .

1CtP

fPFgxC .

t = 0, 0 () 1=0. 1

: tP

fPFgxC .

F > fP.

§8.

1.

0 ,

0u .

.:

. M – ,m – .

.n

i

eiFQ

dtd

1

)(.

« », :

25

0 1

"0" ,0

"1" .d Q Q const Q Qdt

.

0 0 00

1

( ),

,

Q M m u

Q M m

0 0 0

0 0 00 0

( ) ,

( ) .

M m u M m

M m u m um M m M

2. . . «», , ,1999.

, 90 / .

,

0,12 ,

?.

1. , ( ),

.2. ,

, : G,R F.

V0 = 90 / = 25 / , V = 0. S t,.

3.

:0

01 )( dttFQQ .

( ), ,G nR , F

, : 0V, –Ft = –mV0.

26

4. t: t = mV0/F.F = 0, 12G = 0,12mg,

gVt 2,21

81,912,025

12,00

.

3. . .«

», ,,1999.

( ,),

V0 = 60 / 5

.

.1. ,

. . ,.

2. ; G – , ,

iR , fRi , ., ,

, ,nR fR ,

G , nR , fR .

3. G nR. ,GffRR nf .

4. ,,

t

fdtGVmVm0

01 .

27

t =5 , GftmV0 ,

fGtVgG

0 , , , V0 = 60 / = 16,7 / ,

34,0581,9

7,160

gtVf .

§9. 1. . . «

», 2, ,1975. 1 1

r z

, 2 2,

.,

m ,

f. .

. z,

. . )(1

ek

n

kZ

Z Fmomdt

dL.

: m – ; 1 – , 2 – , R1

'1R –

, R2 – , F – . , 22 PR , F = fP2,

1, R1'

1R , z, :rfPmFmom e

kZ 2)( . (1)LZ

z 2212

21

22)2()1(

22

2)( r

gPPr

gPr

grPvmmomJLLL ZZZZZ .

LZ , ,

28

221

22 rg

PPdt

dL Z. (2)

(1) (2) )(1

ek

n

kZ

Z Fmomdt

dL

, :

)()2(

222

21

rfPmrPP

g.

2. . . «», 2, ,1975.

,.

.

.. ,

.. .

.

, z

. : – , r –

.

,

: – , 1 2 –, R x, R y, R z,

RBx, R y – .

z. n

k

ekZ

Z Fmomdt

dL1

)( .

29

,z, z :

0)(1

n

k

ekZ Fmom . 0

dtdLZ , , 21

ZZ LL . ,

. ZZ IL ,:1

11ZZ IL , 2

22ZZ IL , 21 , ZZ II –

. 21ZZ LL 1

11ZZ IL ,

222ZZ IL , 2

21

1ZZ II , 12

1

2Z

Z

II

.

z. ,.

z 2

103 r

gP

.

,

22

21

4516

342

103 r

gPr

gPr

gPIZ ,

2222

54

42

103 r

gPr

gPr

gPIZ .

, 12 94

.

§10.

Tm vk k

k

n 2

1 2 .

, ,

TMv

TC2

2 ,

T –

. ( )., ,

30

TMvC

2

2 ,

M – ;Cv – ( ).

, ,

2

2ZJT ,

ZJ – , – .

()

TMv JC C

2 2

2 2 ,

JC – , (

); – .

, ,

A F dr Fds F F dx F dy F dzx y zcos( , ) .

A Fdr F ds F dx F dy F dzL L

x y zL

( ) .

sA Fs F scos( , ) .

A P P z zC C12 1 2( ) ( ) ,

P – ;zC1

zC2 –

.F cxx

Ac

x x12 12

22

2 ,

x1 x2, ,

,A R v M dtO O ,

31

R M0 – ; – .

, , ,

;dMA Z

2

1

12 dMA Z ,

MZ – Oz..

, ,

NA

dtF v F v F x F y F zx y z .

, ,OO MvRN .

, ,,

ZMN .

L, . ,, ,

xF

yF yx , x

Fz

F zx , yF

zF zy

,

.dA ; A12= 1 – 2,

(x, y, z) – ;1 2 – (x, y, z).

constPZc ., Ph , h –

,« » ,

. ( )

2

2c,

:c – , ,

;

32

– ;

:c – , ,

1 ; – .

,, , . .

T2 – T1 = A12.,

)(1212

eATT , )(12

eA – .

, , . . NdtdT

.

,,

, , . . + = const.

1. m

,,

,.

S ,

,.

f, .tgf

. : P , F ,N F , F .

,: )(1212 FATT . ,021 vv 021 TT ,

, ,012A . . 0)()()()( 1212121212 NAFAFAPAA .cosfPF , cxF , .mgP

33

;sin)(12 PSPA ;cos)(12 fPSSFFAS cScxdxFA0

2

12 2)( ; 0)(12 NA ,

.

, ,02

cossin2cSPSfPS

.)cos(sin2c

fmgS

2. . . «», , , 1999.

, 90 / .

, ,

0,12 ,

?. 1. S

)(1001 FATT . V1=0,GSFSFSFA 12.0)cos()( ( F

180 ), G nR( ),

FsmV

2

20

.

2. S: gV

FmVS

12,022

20

20

(F =0,12G=0,12mg).

225 2652 0,12 9,81

S .

3. . . «», , , 1999.

34

500 , 2 ,, 12 / , .

,, ,

106 , 19600 ..

1. ,

)(1001 FATT ,

,S = 500 .2.

,

.

.

G 1G 2G ., , :

2G , R F .

3. (F+R+G2),,

10 = –(F+R+G2)S.4. 10 ,

V1=0, 2

20

2mVSGRF .

F:

2

20

2GR

SmVF .

5. G2:

ShGGG )sin(2 .

35

G2 F,

ShGR

SmVF2

20

.

F, , G=mg,

SmghR

SmVF 85100

500281,91019600

50021210

2

62620

.

§11.

[1], [5]

jjj

QqT

qT

dtd

(j = 1, 2, ... , s),

q1, q2,..., qS – ;s – ;

,1q ,2q …, Sq – ; – ;

Q1, Q2, ..., QS – ., ,

0jj q

LqL

dtd

,

L = T – – .:

;, ,

,, , ,

, . .);,...,,;,...,,( 2121 tqqqqqqTT SS ( t

T);

36

jqT

;

jqT

dtd

, ,jq

T,

t;;

.

1..

B, ,. D

m. 2m, 3m, ,

, 30 . .

. ,

. :xq1 – D;

37

yq2 – ;zq3 – .

,.

.DBA TTTT ,

TA – ;TB – ;TD – .

zv3 , ,

.2

32

22 zmzmT AA

D : () xv1 ( )

zv3 .,

, . . 31 vvvD , , .2223

21

2 zxvvvD

, )(22

222

zxmvmT DDD .

.

: 22

20

20 BB

BJvm

T ,

v0 – ;B – ;

J0 – .

, . .,32 vvvO

yv2 – , ,cos2cos2 22

3223

22

2 zyzyvvvvvO .

B . ,B

.: xvr

E ; .yvrO

38

, .EKv

OKv r

ErO

B ,

,R

xyEKOKvv r

ErO

B R – .

,2

22

mRRmJ BO

, .)(2

)cos2( 222 yxmzyzymTB

.)(2

)cos2()(22

3 222222 yxmzyzymzxmzmT

:

)( yxmxmxT

;

)()cos(2 yxmzymyT

;

)cos(23 yzmzmzmzT

.

x, y, z

T, ,0zT

yT

xT

;2)( ymxmyxmxmxT

dtd

;cos23)()cos(2 zmymxmyxmzymyT

dtd

.6cos2)cos(23 zmymyzmzmzmzT

dtd

.

QX ,1

xAQX

A1 – ,,0;0 zyx , ,1 xmgxPA D .mgQX

39

yAQY

2 , A2 -

.0;0 zxy ,,sin2sin2 ymgyPA B .sin2mgQY

,0;0 zyx A3 0, , , .0ZQ

.

.j

j qQ P x P y constD B sin ,

,

.0

;sin2sin

;

zQ

mgPy

Q

mgPx

Q

Z

BY

DX

,2

3 2 22 6 0

mx my mgmx my mz mg

my mz

;cos sin ;

cos ., 30 ,

2

3 3

;

;

x y g

x y z g3 6 0.y z

. 2 ,

5 2 3 .y z g,

.24

3;41;

83 gzgygx

,

.321 243;

41;

83 CgtzCgtyCgtx

1, 2, 3 ,.x y z 0

40

,:

.48

3;81;

163 222 gtzgtygtx

2. , . 2

1. 2 ,

m0. , 2 – 2, r2 – 2, r1 – 1. 2 , –

. ..

,.

.

QTTdtd

.

41

: – , 2 – 2, m0 – , . ,

, .

, . . .Q:

coscos 20 OAPOCPmA .

OA=OP-AP=r1-r2, OCOA r r2 2

1 2,

A m P P r r12

2 20 2 1 2 cos . (1)

, A Q , ,

: Q m P P r r12

2 20 2 1 2 cos . (2)

, 2 (

1 ), . . T T T( ) ( )1 2 . (3) ,

, ,

T IO( )1 21

2 , IPg

OAPg

r rO

13

13

21 2

2–

. , TPg

r r( ) ( )11 2

2 216 . (4)

2,

, TPg

v IA A( )2 2 2

221

212 . (5)

, :v OA r rA ( )1 2 . (6)

, 2, P : 22rvA . (7)

(6) (7), : 21 2

2

r rr . (8)

2

IP r

gA2 2

2

2 . (9)

42

v A , 2 I A (6),

(8) (9) (5) TPg

r r( ) ( )2 21 2

2 234 . (10)

(3), (4) (10), :

TP P

gr r

2 912

21 2

2 2( ) . (11)

: 221

2 )(6

92 rrg

PPT

: 221

2 )(12

92 rrg

PPTdtd

. (12)

, , (11), , :

0T. (13)

(2), (12), (13)

:2 9

612

2 221 2

2

0 2 1 2

P Pg

r r m P P r r cos ,

:

cos3

2 22 9

0 2 1 2

2g

m P P r rP P . (14)

: m P P r r0 2 1 2

12

2 cos .

3. 1 2. D

. 5 L 6; 3 – D, 4 – , 5 –

. ,.

, L. .

.

43

..s1 s2,

.s1 s2:

111

SQsT

sT

dtd

,

222

SQsT

sT

dtd

. (1)

: 1 – , 2 – , 3 – D, 4 – , 5 – , 6 – L.

, , , ( ,

, ).

1SQ2SQ

s1 s2 ,

s1 s2.

44

1SQs1 , s2 , . .

s s1 20 0; . ( , s1 s2

.), ,

N . s1 ,,

rM , s1 . ,N ,

rO5 ( . ),

rM , . .

221

5srr M

O . (2)

,s1 :

56511 )(sin OrPPsPA .

(2),

1651 )(21sin sPPPA . (3)

2 , s2 0 ,3 4 ,

.1SQ ,

(3), . .

)(21sin 6511

PPPQS . (4)

2SQs2 , s1 :

s s2 10 0; .

, , D. s2

, , NNr ,

s2 . ,, ( . )

45

222

5srr N

O . (5)

,s2 :

56522 )(sin OrPPsPA .

(5), :

2652 )(21sin sPPPA . (6)

1 , s1 0 ,3 4 ,

.2SQ ,

(6), . .

)(21sin 6522

PPPQS . (7)

, : , L D, :

T T T T T T T( ) ( ) ( ) ( ) ( ) ( )1 2 3 4 5 6 . (8) v A vB ,

.s1 s2 s1 s2 .

D, r3, r4 r5. D:

31

3

sr , 4

2

4

sr . (9)

v M

v A , . . v sMx 1 . v sNx 2 ., . ), 5 ,

:

2221

5ssvvv NxMx

xO . (10)

46

5

, N: v v vMx Nx MN x( ) , . .( )v v v s sMN x Mx Nx 1 2 .

( )v MN rMN x 5 5 52 , 2 5 5 1 2r s s ,

: 51 2

52s s

r . (11)

,,

TPg

s( )1 1121

2 , TPg

s( )2 2221

2 . (12)

D ,:

235

)3(

21

OIT ,244

)3(

21

OIT .

grPI

grPI OO 2

,2

244

4

233

5 (9),

TP s

g( )3 3 1

2

4 , TP s

g( )4 4 2

2

4 . (13)

, ,255

25

5)5(

21

21

OO IvgPT .

grPIO 2

255

5

(10) (11), :

TPg

s sPg

s s( )5 512

22 5

1 2

316

18 . (14)

L, ,

TPg

vO( )6 6 21

2 5 .

(10),

47

TPg

s s( )6 61 2

218 . (15)

(8) (12)–(15). :

2 2 2 231 2 41 2 1 2

2 2 25 5 61 2 1 2 1 2

1 1 1 12 2 4 4

3 1 1( ) ( ) ,16 8 8

PP P PT s s s sg g g g

P P Ps s s s s sg g g

. .

TP P P P

gs

P P P Pg

sP P

gs s

8 4 3 216

8 4 3 216

28

1 3 5 612 2 4 5 6

22 5 6

1 2 . (16)

s1 s2

165

26542

2

265

16531

1

82

82348

,8

28

2348

sg

PPs

gPPPP

sT

sg

PPsg

PPPPsT

165

26542

2

265

16531

1

82

82348

82

82348

sg

PPs

gPPPP

sT

dtd

sg

PPs

gPPPP

sT

dtd

. (17)

, , (16) s1 s2 ,

01s

T, 0

2sT

. (18)

(4), (7), (17) (18) (1) s1 s2 :

48

8 4 3 28

28

12

8 4 3 28

28

12

1 3 5 61

5 62 1 5 6

2 4 5 62

5 61 2 5 6

P P P Pg

sP P

gs P P P

P P P Pg

sP P

gs P P P

sin ,

sin .

, :( ) sin sin

s gD C B C P B P

AB C12 1

28 ,

( ) sin sins g

D C A A P C PAB C2

2 128 ,

212

5sin)(sin)()2(4

CABPCBPCABACDgwO ,

8 4 3 21 3 5 6P P P P A ,

8 4 3 22 4 5 6P P P P B ,

P P C5 62P P

D5 6

2 .

§12. [2]

26.1, 26.5, 26.6, 26.9.

27.1, 27.3, 27.12, 27.13, 27.34, 27.42, 27.53, 27.56.

28.1, 28.6, 28.7, 28.15, 28.17, 28.21.

29.7, 29.14.

30.4, 30.6, 30.11, 30.12, 30.15, 30.19, 30.22.

32.1, 32.2, 32.5, 32.16, 32.24, 32.26, 32.37, 32.53, 32.55, 32.57, 32.68, 32.70, 32.78, 32.79, 32.86, 32.93, 32.94, 32.96, 32.98, 32.99.

34.2, 34.3, 34.9, 34.11, 34.12, 34.15, 34.21.

35.3, 35.4, 35.10, 35.13, 35.14, 35.19.

49

36.3, 36.6, 36.9.

37.1, 37.3, 37.9, 37.14, 37.34, 37.39, 37.43, 37.46, 37.53, 37.56.

38.2, 38.4, 38.9, 38.20, 38.24, 38.27, 38.30, 38.40, 38.42, 38.44, 38.46, 38.47, 38.50.

48.5, 48.6, 48.12, 48.13, 48.28, 48.29, 48.30, 48.35, 48.44.

§13.

:

Fwm ,

m – ,w – ,F – .

dtdz

dtd

dtdrzrtF

dtzdm

dtdz

dtd

dtdrzrtF

dtd

dtdr

dtdrm

dtdz

dtd

dtdrzrtF

dtdr

dtrdm

z

r

,,,,,,

,,,,,,2

,,,,,,

2

2

2

2

2

2

2

.

dtdSStF

dtdSStF

dtdSm

dtdSStF

dtSdm

n ,,0

,,1

,,2

2

.

( , , , , , , )( , , , , , , ) .

( , , , , , , )

x

y

z

mx F t x y z x y zmy F t x y z x y zmz F t x y z x y z

50

N

kkkz rmJ

1

2.

)(

2

)(

2

)(

2 ),,(VVM

z dvrVMdvrzyxdmrJ .

21 MdJJ ZcZ .

2MRJ z .

n

k

ekFwM

1c = .

n

i

eiFQ

dtd

1

)( ;

001 )( dttFQQ .

)(1

ek

n

kZ

Z Fmomdt

dL.

)(1001 FATT .

Tm vk k

k

n 2

1 2 .

TMvC

2

2 , M – ; Cv – .

2

2ZJT , ZJ – ,

– .

TMv JC C

2 2

2 2 .

A F dr Fds F F dx F dy F dzx y zcos( , ) .

A Fdr F ds F dx F dy F dzL L

x y zL

( ) .

A Fs F scos( , ) .A P P z zC C12 1 2

( ) ( ) .

51

F cxx ; Ac

x x12 12

22

2 .

A R v M dtO O ;

;dMA Z

2

1

12 dMA Z .

.

NA

dtF v F v F x F y F zx y z .

OO MvRN .

ZMN .

xF

yF yx , x

Fz

F zx , yF

zF zy

– .

dA ; A12= 1– 2 , (x, y, z) – ;Ph , h – ,

2

2c.

,, , . . T2 – T1 = A12.

, , . . NdtdT

.

,,

, , . . + = const.

jjj

QqT

qT

dtd

(j = 1, 2, ... , s),

q1, q2,..., qS – ;s – ;

,1q ,2q …, Sq – ; – ;

Q1, Q2, ..., QS – .

0jj q

LqL

dtd

, L = T – – .

:

52

;, ,

,, , ,

, . . );,...,,;,...,,( 2121 tqqqqqqTT SS (t T);

jqT

;

jqT

dtd

, ,jq

T,

t;;

.

53

1. . . : .. / . . . – 12- ., . – . : . ., 2002. –

416 .2. . . : .

. , . . / . . ; . . . , . . . – . : ,

2004. – 447 .3. . . : .

. , . . / . . ,. . . – 8- ., . – . : , 2001. – 763 .

4. : . . / . . [ .]. –

. : - , 2004. – 382 .5. . . : .

. : 3 . / . . , . . ,. . . – . : , 1990. – . 1 : . – 670 .

6. .. . . / . . [ .]. – . : . ., 1996. – 431 .

7. . . : .. - / . . ; . . . . –

. : , 1972. – . 1 : , ,. – 467 .

54

. 3.

-

:,

15.08.07. 60×84/16. . . . 3,1. 100 . 1678.

-.

394000, . , . . , 10. . 208-298, 598-026 ( )http://www.ppc.vsu.ru; e-mail: [email protected]

-.

394000, . , . , 3. . 204-133.

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