D2_metodichka_obschie_teoremi_dinamiki

' (1.15) x ,

Q =10m V + 2m V − 2m πr cos α + m V − m π l sin πt ,
x	4		1	4	1					4			4			1		4 2
, (1.14),
								π(2r cos α +						l	sin πt)
				V1 =									2				.	(1.16)
											15
	$ (1.16),
				V1	= 0,314 + 0,2sin πt ( / ).
	& ,
(1.16),			V =	dS1		.
			1	dt
									t	2πr cos α					t	πl sin πt
	dS1 = V1dt ,			S1 = ∫									dt + ∫					dt ;
										15
					0								0			30
		2πr t cos α +					l		(1 − cos πt)									(1.17)
	S1 =				2								.
					15
!											S1 = 0,314t + 0,2(1 − cos πt) ( ).
	(			∑Fkxe = 0 ,

S1 (t) V1 (t) , -

' (1.13):

xC = xC 0 = const .

! $ (1.9)

xC0 =	m x0	+ m	x0	+ m x0	+ m	x0
	1 1	2	2	3 3	4	4	,
				M
						(1.18)

xC = m1x1 + m2 x2 + m3 x3 + m4 x4 . M

11

t					= 0 ' x
: x0	, x0	, x0	, x0	,	x , x , x , x – -
1	2	3	4		1	2	3	4

ϕ . )

S1 , 2, 3 4

. - 2, 3 4 -

, , S1 . )

-

x .

x = x0		+ S ,	x = x0		+ S ,
1	1	1	2	2		1
x	= x0	+ S − ϕr cos α,				x	= x0	+ S −	l	(1 − cos πt).
3	3	1				4	4	1	2

) $ « ϕr cos α » « l (1 − cos πt) » – - 2

' 3 4 -

x .

$ (1.13)

(1.18) S1 ,
		2πr t cos α +			l		(1 − cos πt)
S1	=	2							.	(1.19)
		15
! V1 , $$' (1.19)
:
			π(2r cos α +		l		sin πt)
V1	=	2						.		(1.20)

15 ! (1.19) (1.17) (1.20) (1.16),

.

12

	3.1.2. ' 1 ('() * + )*)
! 1, m1 ,						$ -
A. % 2 m2 ,							' C2
OC2 = l . 0-
ϕ = ϕ(t) .
' $ .
:	m =100 ,	m = 20 ,		ϕ =	πt 2		, l =10 .
	1	2				2
		y
		φ
	2	C2
			φ
		O		V2			x
		P2
		C1
	1
	Y /	P					Y //
	A			A			A
	A	1
	X A/						X A//

0. 2

. 0 , :

1 2. ) ,

r r

, P1 , P2 ' $,

13

X				A	= X '	+ X '
					A	A
Y	A	= Y '	+Y ' .
		A	A

,

. 2.

- ' (1.6):

dQx

=

∑Fkxe ,

dQy

=∑Fkye .

dt

dt

' x y

∑

F e = X

A

,

∑

F e

= Y − P − P ,

kx

ky

A

1

2

dQ

= X

,

dQy

= Y − P − P .

(1.21)

x

A

dt

dt

A 1

2

2 (1.21) '

X A , YA , Qx Qy

. ! $ (1.2)

Q = Q1 + Q2 . (

,

Q1 = 0

,

-

r

r

, Q = Q2 . ! (1.3) Q2

= m2V2 , V2

–

' . , ' :

Qx = m2V2 x ,

Qy = m2V2 y ,

V

= V cos ϕ,

V

= ω OC

ω =

dϕ

= πt ,

2

2 x

2

2

dt

V

πt 2

= πl t cos

.

2 x

2

3, V

πt

2

= −V sin ϕ = −πl t sin

.

2 y

2

2

( Q = m πl t cos

πt2

Q

πt 2

,

y

= −m πl t sin .

x

2

2

2

2

14

			dQx			= m πl(cos πt 2		− t 2πsin πt 2 ) ,
			dt			2	2		2
			dQy			= −m πl(sin πt 2			+ t 2πcos πt 2 ) ,
			dt			2	2			2
$ (1.21)
X	A	= m πl(cos πt 2					− t 2πsin πt 2 ),				(1.22)
			2			2			2
Y							πt 2		πt	2	(1.23)
	= P + P − m πl(sin							+ t 2πcos		).
A		1	2			2	2		2

, -

' - ' (1.11),

&&

e

,

&&

e

M xC = ∑Fkx

M yC

= ∑Fky .

! . 2

∑

F e

= X

A

,

F e

= Y

A

− P − P ,

kx

∑

ky

1 2

X

&&

&&

+ P + P .

(1.24)

A

= M x , Y

A

= M y

C

C

1

2

xC

yC

' -

$' . ! $ (1.9)

MxC = m1x1 + m2 x2 , MyC = m1 y1 + m2 y2 .

) $ x1, y1 x2 , y2 – '

, x1 = y1 = 0 ( -

). ( MxC = m2 x2 , MyC = m2 y2 .
) $ x = l sin	πt2	, y		πt	2
			2	= l cos .
2	2			2

$$' -

$ (1.24), :

X		= m πl(cos		πt	2	2		πt	2		(1.25)
	A				− t		πsin		),
			2	2				2
Y	= P + P − m πl(sin					πt 2			πt	2	(1.26)
								+ t 2πcos		).
A		1	2	2			2		2

15

(1.25) (1.22) (1.26) (1.23), -

, .

,

X A	= 6,28(cos πt 2	− πt 2 sin πt		2	) ( ) ;
	2		2
Y =1176 + 6,28(sin πt 2			+ πt 2	cos πt		2	) ( ).
A		2			2

3.2. 2

, –

-

.

.

( -

) !

- -

" !

r	r	r r
l0 = r	× mV	= M 0 (mV ).	(2.1)

l0 -

: l0

r

, # r mV ,

- r -

.

$ :

r r
l0 = mV r sin(r ; mV ) = mVh ,	(2.2)
	r

h – mV

! O.

16

! (2.1) ,

lx = M x (mV ), ly = M y (mV ), lz = M z (mV ). (2.3)

-

- , ,

-

z -

! mV -

, z , " ! mV

z
lz = ±mV h.	(2.4)
lz > 0,

lz < 0

.

% -

! L0 , -

( ) "

!			r	r	r
r
L0	= ∑	M	0 (mkVk ) = ∑rk		× mkVk .	(2.5)
% -

, ! (5)

:

r	r	r
Lx = ∑M x (mkVk ), Ly	= ∑M y (mkVk ), Lz	= ∑M z (mkVk ). (2.6)

& # z

ω, -

"
Lz = J zω,	(2.7)

J z – ! .

' #

. #

17

, – !-

.

(

- ! :

-

:

	dL0	r	r
		= ∑M	0 (Fke ) .	(2.8)
	dt

rr

) ∑M 0 (Fke ) – ,

.

! (2.8) ,

-

" :

	dL		r	dLy	r	dL		r
	x	= ∑M x (Fke ),			= ∑M y (Fke ),	z	= ∑M z (Fke ). (2.9)
	dt			dt		dt
	'

- z

" , . .

r
∑M z (Fke ) = 0 (2.9) -
Lz = const .	(2.10)

' ! -

z zC , !

(! ), #

*– +

	J z	= J zC + M d 2 ,	(2.11)
J zC	– ! ! ;
M –	;
d –	.
	!

!.

18

	3.2.1. 2 ( )
,:	m = m = 2 ,		m = 5 ,	m = 40 ,
	1	2	3	4
	a = 8 , d =10 , M C = 2ω (H ), t1 = 0,1 c.
, . 3, #-
# AB , 1 , 2
3,	d . %

# 4.

# M C . 5 .

, . - ω t = t1 .

1

1

x1

P

RA

1

3

RB

5

1,5a

ω

Mc

d

x

5

4/3a

2

x2

P3

C2

4

a

P2

P4

2a

.. 3

. , -

# x . $ (2.9)

19

	dLx	r
		= ∑M x (Fkxe ) ,	(2.12)
	dt

r

∑M x (Fkxe ) – -

x .

) " $ . .

, # . / - r r r r

: P1 , P2 , P3 , P4 – , M C –

rr

# RA , RB – ! .

( ! #

r r

(2a) P P ,
1	2

" #

( - r r r

). P3 , RA , RB x

, " .

rr 1

P4 : M x (P4 ) = 2 P4d . M C -

# , " !-

: MC = −2ω. -

r
∑M x (Fk ) =	1	m4 gd − 2ω .	(2.13)
2
. (2.12).
(2.6) Lx			-
x
Lx = L1x + L2 x + L3x + L4x .			(2.14)

% # # 1, 2, 3

$ (2.7)

L1x = J1xω, L2 x = J2 xω, L3x = J3xω.

! J1x , J2 x ,

*– + (2.11):

20