Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/64e/PMM0011_rus.pdf Äàòà èçìåíåíèÿ: Wed Feb 24 16:55:42 2016 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:57:40 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 17

. . 1, 2016

80.

. 1, 2016 11

531.44 © 2016 . . . , . .

­ , , ­, , . , . . 1. . ­ , . ,

F = ­ NV / V
F ­ , V, ­ , N ­ . , ( ) , . [1]. [2], , ( < 1, ). ( , , h) [3]. h 0 , , . [2, 3] . , , ­ , , , , , . , , (, , [4], [5], , [2, 6], ­ [7] .). ( [1] XX ) ,


12

.. , ..

( ., , [8, 9] ). , . , , , , (). , , , [4]. , . ­ , ( , . , (, IPOS [10], [11] ..), . , ( ) , () [12­14]. , , , . , , , , . , . ( [4]) ­ : . , , . 2. . ­ ( . . 1) , M1 Ox . ­ l, M1 M2 m. : ­ xM1M2, x1 ­ M1, (x2, y2) ­ M2. M1 : P = mg, R, N F Ox. M2 P2 = mg, ­ R. F Ox ( ), , , , . F ,


. . 1, 2016

13

N O M1 P F R x

y
. 1

M2

. (, M1 ) · · : ) ( x 1 = 0), ) ( x 1 0). . , l,

l / g mg. · 1 = 0). . M1 ( x

, ( )

·· = cos

(2.1)

M1, F + N (. . 1) Ox ():

N > F

(2.2)

, (2.2), M1 M2

·· ·· x 1 = F + R cos , y 1 = 1 + N + R sin ·· ·· x 2 = ­ R cos , y 2 = 1 ­ R sin x 2 = x 1 + cos , y 2 = sin

(2.3)

, ,

y1 = 0 ,

(2.4)

(2.4), (2.3)

N = ­ ( 1 + R sin ) ,

F = ­ R cos ,

·2 R = + sin

(2.5)

(2.3) , (2.2) ( )
2 ·2 ( sin - cos ) > ­ ( 1 + sin ) ± sin cos +

(2.6)


14

.. , ..

2 .

= 0.3

=1

0

/2





0

/2





. 2

. 2 ( ) (, ) = 0.3 () = 1 (). = 0 . 2 . 4, 5, 7, 8 9 0. . M1 · Ox x 1 0. P, R, N F Ox, · x 1 , ,

·

·

·

F = ­ N ;

· = sign x

1

(2.7)

(2.3). C. N , . (2.4) C;

N + N + = 0


(2.8)

= 1 + cos , N = N;

2

= sin cos ,

·2 = 2 + sin

(2.9)



· = sign ( x 1 N )

(2.10)



N = ­ / f ,

f = 1 + cos + sin cos

2

(2.11)

> 0 [0, ], N f , , . , , N . , , .


. . 1, 2016

15

N N : f [0, ],

< ( 1 + cos ) / sin cos
(2.12) ,

2

(2.12)

< * = 2 2

(2.13)

. (2.12) [3] (. [15], . 91). , , , , .

: · x 1 N ( )

· = sign( x 1 N), , f, , N. , (2.12) (2.13), : 1) * M1 f1 f­1 (f±1 > 0); , C [0, ]; 2) > * f1 f­1 (. 3); , ( C ), ( C ). · , * N < 0, ( x 1 > 0) · = ­1, ( x 1 < 0) ­ = 1.
­ · > * ( x 1 < 0) 1 < < ­ ­ ­ 2

( 1 2 ­ f­1, . . 3), f1 > 0 ( N < 0) f­1 < 0 ( N > 0). , · ( x 1 > 0) f1 f­1 (f1 > 0, f­1 < 0), .. C . 1 < < 2 ( 1 2 ­ f1, . . 3)
+ + + +

· · x 1 < 0 x 1 > 0.
. N ·· , y 1 [16], (2.8) (2.9). , N || < , (2.12). , : < ­ N = /( ­ ), > .

­1 1

N>0 = 1, f1 < 0 = ­1, f­1 > 0

N<0 = ­1, f­1 > 0 = 1, f1 > 0


16 f

.. , ..



=1 2

= ­1

­ ­ 2 1

+ 1 /2
. 3

+ 2

0





(2.11) C, , M1:

·· ·2 = [ sin ( cos + sin ) + 2 cos + 2 sin ] / f 2 ·2 ·· x 1 = [ ( cos + sin ) + sin cos + ( 2 ­ cos ) ] / f

(2.14)


(2.14) , · · x 1 x 2 M1 M2, , (. [4]) (, ).

·

· x 1 , N, .
3. . (2.5)

· (2.14). , (0), (0), x 1 (0) (2.5), (2.14), () · (, , x 1 ), (, ) . . , , , ­ . · ­ (, ) ( x 1 = 0), . ­ · , x 1 = 0 .
. . , * N < 0 [0, ], · · · x 1 = 0, = 1 x 1 > 0 = ­1 x 1 < 0. 4 , ­

·

·

·

·


. . 1, 2016

17

2 . /2

. x1 = 0

= 0.3 . = ­1, x1 > 0

. = 1, x1 < 0

0

0

/2
. 4





/2





= 0.3 =1 = ­1

2 .

2 .

0

/2





0

/2





. 5

(, ) = ­1 = 1 < * ( = 0.3). . , , .. . (2.14) ·· · x 1 x 1 = 0.

·

·· · . 5 , x 1 x 1 = 0 = ±1 < * ( = 0.3). , ·· · x 1 x 1 = 0.
, * . , (, ) (2.6). , . 5 · (2.6): > /2 ( x 1 < 0, = 1)

·

· < /2 ( x 1 > 0, = ­1), · ·· x 1 (, ).
, (


18 2 .

.. , ..

4 5 n=0 0 7 3 6 1 2

n 0 1 2 3 4 5 6 7

°cn tn 0 0.47 0.91 2.02 4.40 5.54 7.36 8.83

­2 . x1

0 n=0

/2





0.2

12 0

3

4

5

6

7

.. x2, y2

. y2

. x2

0

­1

0

5

t

10

. 6


. . 1, 2016

19

. 2

a

. x1 > 0

= ­1



. x1 < 0

=1

0 . 2 =1 = ­1

0



­ 1

­ 2

/2

+ + 1 2


. 7

0

­ 1

­ 2

/2

+ + 1 2





. x1 > 0 . 2 a = ­1

. x1 < 0 =1

0 . 2 =1 = ­1

0



­ 1

­ 2

/2

+ + 1 2




. 8

0

­ 1

­ 2

/2

+ + 1 2





. 2

0

/2
. 9






20

.. , ..

· ) . , x 1 = 0 ( ). , * . . . 6 , = 0.3 , (0) = 3/4, · (0) = 0, · x 1 ( 0 ) = 0.3
(3.1)

· (). , x 1 = 0 (2.6) . cn tn (n = = 0, 1, ..., 7), . 6. , , · cn, x 1 · · M1 ( . 6) x 2 y 2 M2 ( ) , (3.1). · x 1 (0) c0 t1 = 0.47 ( c1), , , ( ). M1 t2 = 0.91 c2, , , · (2.3), x 1 = 0 (2.6) c3 t3 = 2.02. (2.14) . , , . , (2.5) ( ), . , · (, ) x 1 = 0 (. . 2). , · x 1 = 0.
. , > * f (. . 3). , , . 2, , . (2.14) (, ). , f .

·

·

·

· x 1 ; , (. . 7 , , , ):
­ ­ · ) x 1 > 0, = ­1 N < 0, . . f­ > 0, (0, 1 ) ( 2 , ),

· ) x 1 < 0, = 1 N < 0, . . f+ > 0, (0, 1 ) ( 2 , ), · ) x 1 > 0, = 1 N > 0, . . f+ < 0, ( 1 , 2 ),
+ +

+

+


. . 1, 2016

21
­

· ) x 1 < 0, = ­1 N > 0, . . f­ < 0, ( 1 , 2 ). · , , x 1 > 0 = ±1 (. 7, , ),
, ( 1 , 2 ) : ,
+ +

­

· 1 . , x 1 < 0 = ±1 (. 7, , ), N, ,
( 1 , 2 ) : , 1 . N, . , , . , , , . , · < *, , x 1 = 0 (, ) ( . 8) = ±1 > * ( = 3). , , , . , , (. . 7). , > * . (, ) .
­ ­ ­

+

·

·

·

· (2.3) x 1 = 0 . 9. , = 3, . ( ), < *, , · . 8, ( x 1 < 0, = 1) > /2 · = ( x 1 > 0, = ­1) < /2 = 0.
m1 m2. ­ M1 M2. * = 2 2 . , , > *, , * . ­ m1 m2. , ­ m1 m2 M1 M2, l. ·· ·· ( C) x 2 y 2 A = = m2/m1,


22 * 1.5 1.0 0.5 0

.. , ..

5

10
. 10

15

A

N=­

2 2 2 ·2 A cos + sin + A sin cos

1 + A cos + A sin cos

2

(3.2)

[0, ], , N (3.2) . , , , (N < 0) [0, ],

< * = 2 1 + A/A

(3.3)

*(A) (3.3) . 10, , (* < 1) A > 5 (m2 > 5m1). A (m2 m1) * . , M1 , .. ­ , , (. [17], . 1, § 5). 4. . , ­, , , * = 2 2 . , , . , < * , , ( ) . , . > * , · N x 1 . , N, , . (, ) , , . , .

·


. . 1, 2016

23

, , > * , , . , , , , . , , [18] . (3.3) . .. , .. .. . (14 50 00005) . .. .
1. PainlevÈ P. LeÃons sur le frottement. Paris: Hermann, 1895. 2. . . . 2. . . .: . . . . , 1960. 488 . 3. De Sparre. Sur le frottenent de glissement // Comptes Rendus. 1905. V. 141. P. 310. 4. Mamaev I.S., Ivanova T.B. The dynamics of a rigid body with a sharp edge in contact with an in clined surface in the presence of dry friction // Regul. Chaotic Dyn. 2014. V. 19. 1. P. 116­139. 5. .. . , . : , 1980. 64 . 6. Thiry R. ètude d'un problÕme particulier ou intervient le frottement de glissement // Nouv. Ann. Math. 1922. SÈr. 5. V. 1. P. 208­216. 7. Klein F. Zur Painleves Kritik der Coulombschen Reibungsgesetze // Ztschr. Math. Phys. 1909. B. 58. S. 186. 8. Zhao Z., Liu C., Chen B., Brogliato B. Asymptotic analysis of Painleve's paradox // Multibody System Dynamics. 2015. 21 p. 9. Stewart D.E. Dynamics with Inequalities: Impacts and Hard Constraints. Philadelphia, PA: SIAM, 2011. 387 p. 10. Or Y., Rimon E. Investigation of Painleve's paradox and dynamic jamming during mechanism sliding motion // Nonlinear Dyn. 2012. V. 67. 2. P. 1647­1668. 11. Or Y. Painleve's paradox and dynamic jamming in simple models of passive dynamic waling // Regul. Chaotic Dyn. 2014. V. 19. 1. P. 64­80. 12. Butler J.P. Hopping hoops don't hop // Amer. Math. Monthly. 1999. V. 106. 6. P. 565­568. 13. Theron W.F.D. The rolling motion of an accentrically loaded wheel // Am. J. Phys. 2000. V. 68. 9. P. 812­820. 14. Tokieda T. Roll models // Amer. Math. Monthly. 2013. V. 120. 3. P. 265­282. 15. .. . .; : , 2011. 302 . 16. .. // . 2012. . 76. 2. . 197­ 213. 17. .., .. . . 5 . .: , 2012. 224 . 18. Pfeifer F. Zur Frage der sogenannten Coulombshen Reibungsgesetze // Ztschr. Math. Phys. 1909. B. 58. S. 273. . .. , e mail: tbesp@rcd.ru mamaev@rcd.ru 28.IV.2015