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539.3

. .
, * E-mail: heepper@gmail.com . .
* E-mail: strelkiny@gmail.com * . , 66, . , , 69600
1.
, . , , - . , . , , .
2.
, [1, 2] [3-10].
, -, , [1]. -, , [4, 9] .
[3-8] [3-5], , , [6, 7]. , [9, 10], ( ) . , . , [3-8].
3|................................................................................................

3.
, , .
, , :
- , ;
- ;
- .
4.
, - , . , , xyz ^) , t, (. 1). , - (^ t 0 > 0 . , , . , , , .
- , - V; ^) = (1( ()^, ()^, ()), - ( 1-3 z, , 1 0, 1, 2,.., ). - [11]:
11 ()> ∞; ^ ()> 0; ()- ^ () = 0;
/+ ();
^2, (^ +[^31 00]2 21 (^^2, () = 0; (1)
^2, (^ +[^31 (3)]2 , (3)+^- ()'1^"3, (0 = ∞; ; , = 1, ,
V, () = ('2, (),'3, ()) - . (1) , . (1) .
, , () V, () L2 () [9] -, L2() [9]. , ^ ^2 ()^ L2 (), (,] = 1,3), , () V, ()
^() = (], ) - fk1 () = Fk (,, , ) ^ = 1,3,
(2)
]=1
. 1.
[0, ]
0 = ^ < t1 < t2 <... < ^ = . , ^,^,^,...^. , - , () = (1, (),2, (),3, ()) , , -
- , () = (4 (),^, (),3, ())
L2 (), ^ () = -80 ()-1,, 2, () = -2,, 4, () = -3,. ( 2,, 3, ) = ^, ) = , 80 () (0 () L2 ()).
(1) V 2, (),'1 3, () - :
V ()= '(8>-" ; , = ; = 2,3;
^ ' 1-1
, = ^ - ti-l.
,00=---^(Pi,,) , ,=1,n, =2,3;
(3)
(s) = Ak1 - (-1- -1 )s (1 )s (2). (3) = 1 ,
10-20-30 (s),P20 (^ (s). , , t = 0 ^) . ,
10 =20 =30 = 0 10 (s),20 (s), (s) .
(3), (1) :
110# ∞; (l, I ^0; 11 (s) (l, 1 )s=∞;
< (s);
21 (s)+^ 1 ()-(l, 2 )s = 0;
00+ 11 ()- , (l, )=0;
; 1 = 1, .
,
- -
, (),2, (),3, ()
L2 (), -
(4) -
,. (3),
, (),2, (),3, () (4)
,
, ,..., (,-1) - - (),2 () ,..., ,- () . 1 (),2 (),3 () , (4) , = .
(4) , - , (s)L2 () , , - ( ),
1 = ( -((1)-^)),
(5)
,
?, (3) = (51, 0), 0\)4 (3))=(-8 (s)-li, -2, (s), -3, ())
:L32 () ^ L32 ()
=(1,2,), =(1,2, ) ();
= (4 = ](]) =1,3
=1
(6)
(4)
[|,), ^ :

;)={ 1 1 (l, )=0},
^ = { 01) 1 (l, 21 ) + (l, ) = 0} ,
=
^ = {3 01) 1 22 (l, 21) + 2 (l, 1 ) > 0} ,
1 (1, 11 ^, 2 (1, 21) , (1, 1) (2). 3
5.
- 1 (8)2 () (4) [9]
G(l () ^ L32 ()
=(1,2,), =(1,2, )L32 (); = ^(
1 ()= (1 (4
2 00=q(x2 (^ (s), (1))
()=q ( (4 2 (s), (1 (s))), 3 0;

() = |, 0;
1 ' }0, < 0;
, ^/2 + 2 < z;
(x,,z) =
, ^2 + 2 > z.
(6) ] :2 () ^ L2 ()
() = 1 ;0,4 x(s,)ds,, (, = 13) ,
(7)
(5) 11 0),21 0),31 () L2 () .
(5), [9, 10], , . , , '' , ',' - , . 1 12
3
4,2,...,[2 , . , : (s),P2i (s),P3i (8) * 3-2,3-1,3 , (5) -
, (), (),()
4,2,...,|2, 312 :
-2 (3-2 ( 1
312
-=q(xgk-l - (1
1
3-2' (-2));
'3-1 ' - 3-1),
(_ 1


( =
3,-2 3,-2 1
3-3
-IX
=1
>+4
-I
=3-1
'() + ()
3-2 3-2
3,-1 3,-1
3.-2 312
-1 3-1 ' -1 3;_1 ' =1 3
-()
3-1
, (11)
()
3 3,
3,-1
-I
3 '
(+1)
-I
' () + ()
3 + -
=3+1
3 , 1 -
32
''( 3, ,1 ',
32
I 3-1, ' ,
=1
[9] (8) ^^
, (8) (10)(11), -
, -
312
, (8). . ,
[10] , .
3-2 =-80 )- 3-1 =-2, I 3 = - 3 ) ^ =1,12;
.( =
,(0=.
6.

3-2 3-1 = 3-2 3 = 3-1 3-2 = 3-1 3 = 3 3-2 = 3 3-*

)
. ,
)
-: 3 = +{ 3-; 3-;0^ ^ = 0,2; ^2;
(9) = 0 = 2' -
1
0,3 . >0 , ] = 20
3- 3^ = ^) ' 3- 3-g ^) = ^ ,8 = 0,2
>0 , ] (]) ].
]- (8) A2i,Ai ,
:
Ej, Ej :


^3 -2
1
- -
3'.-1
1
3,-2 3,-2

-, 3',=
V, 1,12.
3,-1 3,-1
3, 3,
] - (8) xj
,302 )32;
(<)) ,
?= ((),(), (3)2;
1>= ((),(),(3-2)) , ] = 1,12; = 0,1,2,...,
(10) ,20),
(+1) 3,-2
(+1) 3, q
(+1) 3, q
(10)
^ = -0,0000035 1, 2i = 3i = 0, 1 = 1,20.
.
1, 2 1,v2 : 1 = 3105 , 2 = 105 , v1 = v2 = 0,2. = 0,12375. , [12]
1 (1-22)(1 + ^2)-2'(1 -2)(1+1) 2 (1 -2 ) + 2 (1 -V?)

0,1875. = 0,66.


31 312- , - _____________________________^ _________________________
((0),20),...,302| , 4141=1681
(8), (-1)- -


3
3-
3
3

3-
3-2
(1 = 41, (]) = 6,2510-8 2). ^ (9), ^ (,'), (7), -, :
= 10 1 J K11 (sj,s)ds .
(10)(11).
, , ', , [12] / = 0,66. . 1 / ( ) ' / ( [12], - ). , - .
1

x' /a 0,00 0,11 0,22 0,33 0,44 0,55 0,66 0,77 0,88 0,99
1 0,00 0,18 0,29 0,37 0,43 0,47 0,48 0,43 0,33 0,09
2 0,00 0,21 0,30 0,39 0,45 0,49 0,48 0,41 0,31 0,09
, , , .
7.
- .
, , , . .

1. , . . . , [] / . . // . . . - 2009. - . 73, 3. - . 492-502.
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3. , . . [] / . . // . . . - 1981. - 5. - . 61-67.
4. , . . - [] / . . // . . . - 1985. -. 49, . 5. - . 827-835.
5. , . . [] / . . , J. J. Kalker, . . // . . . - 1999. - 4. - . 51-55.
6. , . . [] / . . , . . // . . . - 2004. - . 68, . 3. - . 516-527.
7. , . . [] / . . // . . . -2002. - 6. - . 59-68.
8. Sundaram, Naravan Mechanics of advancing pin-loaded contact with friction [Text] / Naravan Sundaram, T. N. Farris // Journal of the Mechanics and Physics of Solids. - 2010. - Vol. 58, Issue 11. - P. 1819-1833.
9. , . . [] / . . // i i-ii . - 2013. - .56, 3. - . 29-42.
10. , . . [] / . . // . . - 2012. - 11. -. 47-52.
11. Kalker, J. J. A survey of the mechanics of contact between solid bodies [Text] / J. J. Kalker // ZAMM. - 1977. -. 57, 5. - P. T3-T17.
12. , . [] / . . . . , . . . . . . - .: , 1989. - 510 .