Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.pereplet.ru/nauka/Soros/pdf/9809_096.pdf Äàòà èçìåíåíèÿ: Sun Jun 3 12:10:20 2001 Äàòà èíäåêñèðîâàíèÿ: Tue May 27 13:37:56 2008 Êîäèðîâêà: Windows-1251

ON DYNAMICS OF A ROLLING BODY AND SOME CURIOUS PROPERTIES OF A ROTATING TOP
A. P. MARKEEV Problems of dynamics of bodies being contiguous to a rigid surface are discussed briefly. A theoretical basis is given for curious dynamical effects being observed in experiments, namely the changes in the direction of rotation of the celtic stone about the vertical without any external action, and the origin of rotation in any direction due to oscillations about the horizontal axis. ä ÚÍÓ ÓžÒÛÊÚÒfl Ô ÓžÎÂÏœ ËÌÏËÍË ÚÂÎ, ÒÓÔ ËÍÒËiÒfl Ò Ú, ÓÈ ÔÓ, iÌÓÒÚå. ÑÂÚÒfl ÚÂÓ ÂÚË~ÂÒÍÓ ӞÓÒÌÓ,ÌË ÍÛ åÂÁÌœi ËÌÏË~ÂÒÍËi ÙÙÂÍÚÓ,, ÌžÎÂÏœi , ÍÒÔ ËÏÂÌÚi: ËÁÏÂÌÂÌË ÌÔ ,ÎÂÌËfl , ÂÌËfl ÍÂÎåÚÒÍÓ,,Ó ÍÏÌfl ,ÓÍ Û,, , ÚËÍÎË žÂÁ ÍÚË,ÌÓ,,Ó ,ÌÂ?ÌÂ,,Ó ,ÓÁÂÈÒÚ,Ëfl Ë ,ÓÁÌËÍÌÓ,ÂÌË , ÂÌËfl , ÚÓÏ ËÎË ËÌÓÏ ÌÔ ,ÎÂÌËË Á Ò~ÂÚ ÍÓΞÌËÈ ,ÓÍ Û,, ,,Ó ËÁÓÌÚÎåÌÓÈ ÓÒË.


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, . . , , . . [1, 2]. , . .
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, , , . . : , , , . , , [4], [3]. , , . , (. 1), , . A C G G , . [3] . , .

, . , . G . M0 , : , M0 , M0G; , M0G, ; G, B. , r1 r2 M0 , M0 . , , A C G G . , , ( ) . , , , , ( . celtis - ). , (Celtae - ), I .. [3]. . , .

. 1.

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ìëíéâóàÇéëíú ëíÄñàéçÄêçéÉé ÇêÄôÖçàü ÇéäêìÉ ÇÖêíàäÄãà

.. , .. .. (. [5]), , ( , ). . , . . Oxyz O z = 0 (. 2). Oz . G, , . Gxyz , Oxyz. , , : - , - , - . . 2. Gxyz G : Gz ( Gx , GN, ), G.
z

. , .. , , , . . , ( ) = -- , 2 = 0, = = const. (1)

G, . M0 (. 3). . > 0 , < 0 - ( G). = 0 M0 . . , , . , = -- + x 1 , 2 | xi | = x2 , = + x3 , (2)

1, i = 1, 2, 3.

(1)





G

N M

y

z O y x

x

. 2. . M -

98

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(B ) O
2

r1 (. 3), G , G , (1). .. , .. .. ,
(A )

O G h (C ) M
0

1

(R - P2)2 - S > 0,

S > 0,

(4) (5)

(A - C )(r2 - r1) sin cos > 0

r

2





r

1

. 3. . M0 - , M0rj - , rj , j = 1, 2; M0O1 = r1 , M0O2 = r2 ; G G M0 M0; A, B C - G, G G

(2) xi . , (P 4 + Q3 + R 2 + Q3 + S ) = 0, P = (A + mh2)(C + mh2), Q = mlh(A - C ), (3)

(1) , , , , , . (4), (5) (1) . (4) , , (5) ( ). , . ( r1 < r2 C < A, . 3 .) (4), (5) , , , , G, . Q = 0, || (1) , || (4) .
é åÄãõï äéãÖÅÄçàüï Ç éäêÖëíçéëíà ëíÄñàéçÄêçéÉé ÇêÄôÖçàü

R = [(A + C - B + 2mh2)2 - (A + C - B + 2mh2)mh(r1 + + r2) + m2h2r1r2]2 - (A + mh2)[(A - B )2 + m(h - l1)(g + + 2h)] - (C + mh2)[(C - B )2 + m(h - l2)(g + 2h)], S = (A - B )(C - B )4 + m(g + 2h)2[A(h - l2) + + C(h - l1) - B(2h - r1 - r2)] + m2(g + 2h)2(h - r1)(h - r2), l = (r2 - r1) sin cos , l2 = r1 cos2 + r2 sin2. l1 = r1 sin2 + r2 cos2, m - , g - , A, B, C - , h - G ( M0G ), r1 r2 - M0 , - G , -

() (1). , A - C, r2 - r1 , sin 2 . Q = 0 (5) . (3) - , . + i1 , + i2 , 1 > > 2 > 0. (1) , 1 2 . (3) Q. ,

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Q + ij , j = 1, 2, j , j = 1, 2: Q ( - 1 ) 1 = ------------------------------- , 2D
2 2 2

1 , 2 . (1 > 2 > 0) (A + mh2)(C + mh2)4 - mg[(A + mh2)(l1 - h) + + (C + mh2)(l2 - h)]2 + m2g2(r1 - h)(r2 - h) = 0. (8) , x1 , y1 , y2 , y3 , x2 , x3 x1 = u11y1 + u12y2 , x2 = u21y1 + u22y2 , x3 = y3 , u1j = kjmgl,
2

Q ( 2 - ) 2 = ------------------------------- , 2D
2 2 2 2 2

(6)

D = ( A + m h ) ( C + m h ) ( 1 - 2 ) . , , Q > 0, (5) . (6) , 2 2 2 2 < < 1 , j > 0, j = 1, 2, , (1), 2 2 ; 0 < < 2 , 1 < 0, 2 > 0 ( 1), ( 2) ; 2 2 > 1 , , , , . Q < 0 .
èêàÅãàÜÖççõÖ ìêÄÇçÖçàü çÖãàçÖâçõï äéãÖÅÄçàâ ÇÅãàáà èéãéÜÖçàü êÄÇçéÇÖëàü

u2j = kj[(A + mh2) j + mg(h - l2)], j = 1, 2,
2 2 2 2 2

k j = { ( A + m h ) ( mgl ) + ( C + m h ) [ ( A + m h ) j + + mg ( h - l 2 ) ] }
2 -1 / 2

.

(8) y1 + 1 y1 = Y 1 ,
2

y2 + 2 y2 = Y 2 ,
2

(9)

(1) = 0, M0 . , .. , r1 > h, r2 > h, (7)

By 3 = Y 3 , Yi , i = 1, 2, 3, y 1, y 1 , y 2, y 2, y 3 . (9) Yi , , y1 = 1 sin (1t + 1), y2 = 2 sin (2t + 2), y3 = 3 , 1 , 2 , 3 , 1 , 2 - . 1 2 ( 1) ( 2) . , j 0, j = 1, 2. (9) Yi , (10), 1 , 2 , 3 , 1 , 2 , . , , , : 1 = 0,
2 1 = - 1 1 3 ,

, G O1 O2 M0 (. 3). (7) , . , ( A + m h ) x 1 = mg ( h - l 2 ) x 1 - mgl x 2 + X 1 ,
2

(10)

( C + m h ) x 2 = - mgl x 1 + mg ( h - l 1 ) x 2 + X 2 ,
2

(8)

Bx 3 = X 3 , Xi , i = 1, 2, 3, x j, x j , j = 1, 2, 3. (8) , , , . -

2 = 0,
2 2 = 2 2 3 ,

(11) (12)

42 42 B 3 = ( 1 1 - 2 2 ) ,

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( A - C ) mh ( r 2 - r 1 ) sin cos = ----------------------------------------------------------------------------- . 2 2 2 2 2 ( A + m h ) ( C + m h ) ( 1 - 2 ) (11) , (12) - k , k = 1, 2, 3. (11) (12) . j(t) = j(0) = const. (12) 1 2 - 3 . .
çÖãàçÖâçõÖ äéãÖÅÄçàü äÖãúíëäéÉé äÄåçü

, (13) (14). ÷ 0 < < * =
-1

B÷ ---------------------2 ( 1 + ) 1
2

1+ -----------2

.

, (12) 1 1 + 2 2 + B 3 = B ÷ ,
2 2 2 2 2 2

> * . 2 2 . 4 1 1 = 2 2 , (12). 3 = 0. . . 4 > 0; < 0 . (12) . P1 = (0, 0, ÷), 0 0 P2 = (0, 0, -÷), P3 = ( 1, 2, 0 ) . 4 (12). P1 P2 ÷ -÷. , . 4 ( 0) (1) ||. P3 1 2 . = * , ÷ B 0 1 = ----- ---------- , 1 1 + ÷ B 0 2 = ----- ----------------- . 1 ( 1 + )

÷
2 -2

0,

(13) (14)

1 2 = ,

= 2 1 ,

÷ - , . (12) . 4 1 , 2 , 3 . 1 0, 2 0

3 P
1

÷

, : (3 0).

A2

2

(12) , 1 2 : 1 = 0, ÷B 2 ( t ) = ----------- sch [ 1 ( t + e 1 ) ] , 2 3 ( t ) = ÷ th [ 1 ( t + e 1 ) ] , 1 = - ÷ 2 ,
2

1

A

P
1

3

(15)

3 ( 0 ) 1 e 1 = ---- Arth ------------ ; ÷ 1 2 = 0,

P2 -÷
. 4.
2

÷B 1 ( t ) = ----------- sch [ 2 ( t + e 2 ) ] , 1 3 ( t ) = ÷ th [ 2 ( t + e 2 ) ] , 2 = ÷ 1 ,
2

(16)

.
-1 2 2

A j = ÷ B j , j = 1, 2; 1 1 = 2

3 ( 0 ) 1 e 2 = ---- Arth ------------ . ÷ 2

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. 4 , P1 P2 . (15) , , , 2 . 3(0) 0, , , 2 ( > 0, . 4) 2(0) , . -÷. 3(0) > 0, , , 3(0) 0, . 0 < t < t* = - e1 2 , . t = t* , 2 A2 . t > t* , . , 3(0) > 0 . (16) , , , . . ÷. , t = - e2 . 1 A1 . (12), P1 , P2 , P3 (15), (16). (13), (14) 2 = 1 ,
-

1(t), 2(t) 3(t) (17). 1(t) . (12). , , (12) 3 . (. . 4). t > 0 (3 ). 1 , 2 . , (12) . . 4 , 2 2 1 1 = 2 2 . 3 ( ). 1 - , 2 . , 3 . 1 2 , 1 , 2 - , (3 < 0) . 2 1 , (12) ( . 4 2 2 1 1 = 2 2 , 3). . 1 , 2 - . , 3 , 1 2 , . . . 4 . (18). [3, 4] .

3 = + f ( 1 ) ,

(17)

- 2 + B ÷ 1 - 1 1 f ( 1 ) = ------------------------------------------------------------------------- . B 1
2 2 2 2 2 2( + 1)

3 (17) (12) , d 1 ------------------ = - 2 dt . -+1 1 f ( 1 ) (18)

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áÄäãûóÖçàÖ

ãàíÖêÄíìêÄ
1. .. . .: , 1989. . 1: . 576 . 2. . . .; .: . . .-. . ., 1935. 92 . 3. Walker G.T. On a Curious Dynamical Property of Celts // Proc. Cambridge Phil. Soc. 1895. Vol. 8, Pt. 5. P. 305-306. 4. Walker J. The Mysterious "Rattleback": A Stone That Spins in One Direction and Then Reverses // Sci. Amer. 1979. Vol. 241, 4. P. 144-149. 5. .., .. // . . .: . 1983. . 6. 131 . 6. .., .. . .: , 1967. 519 . 7. .. , . .: , 1992. 335 .

. , . XIX - XX . , ( ) , . . , , , . , , , , . - , . , , .

*** , , , . : , , . 100 .

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103