_________
XXXI 2000
34
519.63:512.643 539.3

. . , . .
(). -- , , . .
, , . . , .
- , . , , , , , .
1. . . , , ,
, , 0x^x23, . \, 2, 3. , , [],
[ ] = [ 123 ]
[] ,
, '
' = [ ] , = [ ]'. (1.1)
, , .
[], :
" 0 - 3 2
[ ] = 3 0 - 1
2 1 0
,
1
axb = [ a]b,
. . .
. . . , , , , .
, 123 . , = ( = 1, 2, 3) [],
( ).
. , ,
, . , (. ),
2^3,
0[23. , Rq , [].
. [] , , . . , . : , , , . . . , , , . Cij (i, j = 1, 2, 3), (1.2),
.
, .
[ C ]=[ c ]
(1.2)
R(t) = R o(t) +p (t)+u (t).
(1.3)
, , . [1]
+ 2 ^ 2 2 (]2 3 ) 2 (]3 2 )
2 (]2 + 3 ) 2 + "2 2 2 (23 ]) .
2 (] 3 2 ) 2 (23 + ]) ^ 2 "2 + 2

2 ]] 22 33 ,
2] ] + 32 23 ,
22 2 + ] ] ,
2 + 2] ]2 .
. . (1.3),
) (0+[ ]('+'(?)). (1.6)
, , ' , , [].
' + ',
' '. (1.6)
+ [](' + ') + [] . (1.7)
, . , (1.2),
+ [][ ] (' + ' ) + [] . (1.8)

(1.8)
+ [ ][]( ' + ') + [ ][]( ' + ') + [ ][]' + [] ' + [] ' (1.9)
(1.2)
+ [] ' + [ ][]( ' + ' ) + [ ] [ ][]( ' + ') + 2[ ][]'.
[] = [] :
+ + [ + ] + [][ ]( + ) + 2 [ ] (1.10)
-
.
, , .
[ ]
, 5 5 5
5 5. (1.11)
, (1.6)
5 5 + [5](' + ') + []5'. (1.12)
[] , 5'
5' 5 + [] [5] (' + ') + 5 '. (1.13)
; (/ = 1, 2, 3), [],
, , . .
[] [5] = ] [] 5; [50 ' ], (1.14)
/
^ [ ]
[ ] ------5; ,
;
5 '. [] [] 5;
;
. ; [ ] [ ]
[][] [] []+[] \] =0
; ; ;
.
1[ ]
;
[ ]
[]
;
= []
[]
;
. , , (1.14) 5', (1.13)
5' 5 + [ 50' ] (' + ') + 5'

5 5 + [5 ]( + ) + 5 . (1.15)

[5]( + ) = [ + ]5,
, , ,
( \
5 5
+
+ 5
|[ + ] Pd + |[ + ] ^

/
+
+ ^5<^+ |"5^ .
(1.16)

,
,
52 = -|5Pdd-|5pdd
(1.17)


, , , , . .
= - >
( ). (1.16),


53 = -5 | ^ - 50 |[+ ] -15 ^,



dm (X], 2, 3)d .
53 (1.10), :
+ | - |[ + ^ + [][] |( + ) dm + 2[] | Udm>
53 =-5^ \ + I - | | + ^ + ||| ( + )dm + 2

- 5 \ | [ + ] dmR + |[ + ] - |[ + ] [ + ] dm +



| [ + ][][]( + )dm + 21[ + ][] ;
+ ^ +


15dmR + 15 -15 [ + ]dm +



+ 15 [] [] ( + ) dm + 215 [] I.
(1.18)

,
5 = 5^1 + 5^2 + 53 = 5ad
(1.19)

<
.
(1.19) 5, 50,
5 5. .
(1.19) 5, 50 . 5, 50 5.
(1.19) (1.16)(1.18) 5,
| <^ +1 - - | + | [ + ]

+ ) - 2 [] / = 0.

(1.20)

, (1.19) 50 | [ + ] ^ +1 [ + ] ^ - | [ + ] - | [ + ] +



+
| [ + ] [ + ] - |[ + ] [][] ( + ) -

- 2 |[ + ] [] = 0.
(1.21)

(1.20), (1.21) (1.19)
15< = 15 ^. +| 5 ~ .
(1.22)



= - + >
- - 1
-
∞, ] + =
= -( ,} + ), = ] >
(1.23)
, , , ^ .


. . (1.23), , I, 8 (/,] = 1, 2, 3). , . .
I , . (1.23),
8 =
- [(5-) + (5). ],
2

5 = 5.
(1.24) (1.22) ,
15 ( + ) -| 5 ( + ) = 0.

5 (1.25)
+ = 0,
^ - = 0 .
(1.24)
(1.25)
(1.26)
, .
(1.23). , /1 / , (1.23) ( ) ,
-
\} ( + )+ = 0

- ( , + , ) =

= ,
= ~ .
2
, , (1.19),
' - ' -(*6 + ['] ' + [']['] ')- ([ї '] ' + [']['] ')-
- 2[' ] - ' = '. (1.27)
(1.20), (1.21) (1.27), , ї . , , .
. .
[2], [3],
, . , . , , , , , . , - . . ( ї )
= [] [' ]q' , (1.28)
[ ' ] , q' , . ,
se = [] [ ' , (1.29)
[ ' e ] .

5 = [] [e]'e
>
5 = [] [ ']^' ,,
(1.30)
. . .
, , [' ]
' = [] 8 ' .
, :
8 =(5q'e) |['] ['] [']4 (1.31)


^')[']q' , (1.32)
[ ' ] .
(1.32) , . , , . .
, . , , (= 1, 2, ..., ). (1.20) (1.21), (1.28) (1.30), q :

2 } +| +1 - 0 - [] | [N'] q ' +
=1
+| [] - [] | [ '][]^' ] q' - [][] | -

[][][]|[N'] q'-2[][]|[N'] q' = 0; (1.33)

-2[ ](+ [ ] 4) -|[](+ [][ N' ] q') } =1
/ ]( + [] [ N ] q') + [01|( + [] [N'] q') -

- | [] [N'] ^ ' -1 [] +| [ + ][ + ] +

+ [] |[ + ] [ + ] 2^[ + ] [] 0. (1.34)

(1.27) (1.19). (1.19) 5. , [ N ]. (1.28)(1.32),

+ ' I ^^
2 } + | [N'] +1 [N'] -1 [N'] ' -
}1
-1 [N'] -1 [N'] (0 + [ '] ' + [ '][ '] ') -


- | [N'] ([ '] ^' ] + [ ']['' ])' - 21N'] [ ']^'] -

- I[N'][N'] ' = [] [&][]^'.

(1.35)
(1.35) , .
, (1.33), (1.34), (1.35) , (0> (, (0 ;() ( = 0, 1, 2, 3). , , .
2. . . , 0*^2^3. 0x^23,
[]. . . , . , . , ,
(0 = (*)+ (-
Q:
Q =
, ,
dQ ()

, . , = [] ,
-[] + [][] = . (2.1)
. , , , , ,
= |[ + ] ( + ) = [] + []| + |[] + |[] . (2.2)

, , ,
| [ ] = 0. (2.3)

, (2.1) , :
= [ ] + | [ ] .

(2.4)

^ = * + *1 ] , (2-5)


, , . (2.5),
(
[ ] + [ ]
- I 1+| [ ] + I [ = [*/ ] - (2-6)
V /
(2.6)
. [] = 0,
.
|[] = 2[(, ] . (2.7)

(2.7),
I = | [*][]* + I [][][] =

= -|[][] - [] |[][] . (2.8)

,
[ ^ ] = -| [*][*] ,

[/] , (2.7) -

1 +
[/] + [][/] = [*, ] . (2.9)
[*] :
[/ * ]= -1 [ + ][ + ] = - [][] - [] | [] -

| [] [] -1 [][] = [ / ] - [][]. (2.10)

(2.10) (2.9), (2.1)
[] +[/*] = 2 [4 ] + [] - [] [/*] . (2.11)
(2.1), (2.11) . (1.20) (2.1).
(1.20) . (2.1).
(1.20) , . (2.1) (1.20), .
(1.20) (2.1). , (1.20) (2.1) , , ,
, . , ( (1.21), , , (1.21), . (1.33),
(1.20) , . (1.34), (1.21), . , , ,
ї.
. (1.35) , ( 0() (0).
(1.35)
: ( ), , .
, , (/)
(/).
. , . , , , , , .
, . , , (0 :



(1.20)
, . (1.21) (2.11). ,

(2.12)
=1


/
(0 = ^'] ] [] ' = [] ',
/
[' ] . .
0( , 0x^x2 3
0 () = | [N'] (0 + [ '] ' + [ ']['] ' ) .

, (-
).
} ()
' () = | [ N ' ] + ([ ' ]^'] + [ ' ][ ' ]^ ' ])' + 21 [N ' ] [ ' ][N' ] ' +

+ | [ N' ] [ N' ] '.

\ ()
} (0 = [ '+ ] '+[' ] 4'+[ ' ] 4'. (2.13)
, #(0, ['] [<2'], . .
[' ]
[ ' ] = | [ N' ] [ ' ]^ ' ] .,


[ '] =| [N'] [ '][N'] . = | [N'] [ ']^ '] . = [ '].

[<2' ]
['] = | [N'] [ '][ ''] .

, [<2' ] ( ['] = [ ' ]). [' ],
[ ' ] = 21 [ N ] [ ' ]^' ] .,

. () </' (), , . , } () . ['] . [' ].
[' ], (2.13),
['] = | [ N'] [ N ' ] .

'(), (2.12),

'() = |['] [ '][ ']' = [']4().

[' ] [ ' ] .
0(), } () , .
, , (2.12)
[' ] 4' + [ ' + ' ] 4 ' + [ ' + ' + ] 4 ' = ' () (). (2.14)
(2.14)
4 1/=0 = ", 4 |/= = 0. (2.15)
(2.14), (2.15) , . , , (2.14), (2.15) , .
, (1.33), (1.34),
(2.14), (2.15) -^(), (), () (5 =
= 0, ... , 3), '(0. ,

.
. , , (1.33), (1.34) (2.1), (2.11), . . .
(2.14),
(2.15).
, (2.14), (2.15).
1) (' = 0) [' ]
[' ] 4'+[ ' + ' ] 4' + [ ' + ] 4 = ' ()0 ().
. [' ] [ ' + Q' ]
, [' + ' ] . ([' ] = 0), ([ ] ).
2)
[' ] 4' + [ ] 4' + [ ' ] 4' = ' ()0 ()
['], [' ] [']. [3], [10].
3) (2.14), (2.15) , ,
[' ]4' + [' ] 4 ' = ' ()
'( ( , ).
3. . , . . () , ( ) , . , . , , [9], , . . , . . , , , .
( ) . , . , . , .
[4], [5]: , , , . .
. , [4]. , , . , . .
(, . .), . , . ,
, (2.14), (2.15). , , . . , . ,
. , . . . 0x2 , 0\2 . , 02 , . . , (2.14), (2,15).
.
, . ї. , . , -
, , (1.33)(1.35).
, , [6] . , . , . .
, [7] . ,

[6].
(3.3) . , (1.33)(1.35) .
(3.3) [7].
. (3.4) , . , . [8].
98 -01-
= 0, * (, = 1, 2, 3),
(3.1)
(3.2)
(3.3)


(3.4)



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2. . . .: . 1975.
3. ., . -
. .: . 1982.
4. . . . .: . 1961.
5. Canavin J. R., Likins P. W. Flotating reference frames for flexible spacecraft//! Spacecraft and
Rockets. 1977. V. 14, N 12.
6. . . . .: . 1987.
7. . . // . 1989. . 27. . 2.
8. . ., . ., . ., -
. ., . ., . . . .: - ї. 1997.
9. . . . .: . 1964.
10. Argyris J., Mlejnek H. P. Dynamics of Structures. North Hol-
land. 1991.
27/IX1999 .