К ЗАДАЧЕ СТАБИЛИЗАЦИИ ДВИЖЕНИЙ МЕХАНИЧЕСКИХ СИСТЕМ, СТЕСНЕННЫХ ГЕОМЕТРИЧЕСКИМИ И КИНЕМАТИЧЕСКИМИ СЕРВОСВЯЗЯМИ

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<ul><li><p> 4 (12), 2009 - . </p><p> 27 </p><p> 531.31+62-50 . . </p><p> , </p><p> . , . , , . : , ()-, -, , , , , , , . Abstract. In work the equations of motion of the mechanical systems constrained by geometrical and kinematical constraints of the first and second sort are deduced. The obvious kind of forces of reactions of servo constraints is received, and also ques-tions of stability of system under the relation of the variety defined by servo con-straints are considered. Keywords: servo constraint, (A)-moving, parametrical clearing, compulsions of re-actions, high-speed parameters, clearing parameters, quasicoordinate, quasispeed, the stability, not indignant movement. </p><p> - . [1]. , . , - . [2], . [3], . . - [4], . . [5], . . [6, 7], . . [8], . . [9, 10] . -, , . - , - , , , , [1, 2, 9, 10]. </p><p> -, - , . . . [11]. - [12] [13], . . , - , - [9, 10]. [14], - . . . </p><p>. . [9, 10]. -</p></li><li><p> . </p><p> 28 </p><p> , -. </p><p> , , - . </p><p> , q1, , qn, - </p><p> 1</p><p>0n</p><p>i ii</p><p>b q b </p><p> ( = 1, , b), (1) </p><p> [1] </p><p> 1 , , ..., 0nt q q ( = 1, , a); </p><p> 1 1, , ..., , , ..., 0n nt q q q q ( = 1, , c), (a + c = ). (2) , , -</p><p> (1), , </p><p> 2 11</p><p>, ,..., 0n</p><p>i n ii</p><p>a t q q q</p><p> (2 = 1, , ), (3) </p><p> [6]. , </p><p> [9, 10], </p><p> * 1 1 2, , ..., , , , ..., 0n a t q q , * 1 1 1, , ..., , , ..., , , ..., 0n n ct q q q q (4) ( = 1, , ; = 1, , ; + = ), </p><p> , , - (2). - , ,p (2) - . , , (2), - [14]. </p><p> Np P , - , , , </p><p>p pN (p = 1, , a); </p><p> P ( = 1, , c). (5) </p><p> , </p><p>1 1,</p><p>a c</p><p>p pp</p><p>N P </p></li><li><p> 4 (12), 2009 - . </p><p> 29 </p><p> 2</p><p>1 1 1</p><p>n a c</p><p>i p pii p</p><p>A R q N P </p><p> . (6) </p><p> , ()- [6] , . (6) - </p><p>p = 0; = 0 (p = 1, , a; = 1, , c). </p><p> (2), (4) - q1, , qa 1, , a. , (1) (4), [15] ev: </p><p>1</p><p>n</p><p>i iv v iv b</p><p>Z d e d </p><p> (i = 1, , n), </p><p> i iZ (i = 1, , a), i iZ q (i = a + 1, , n), + = - ev , . (3) [15] </p><p> 2 2</p><p>1 10</p><p>n n</p><p>i iv v v vi v b</p><p>a d a </p><p> (2 = 1, , ), (7) </p><p> v , ev. [15] </p><p>10</p><p>n</p><p>vvv b</p><p>Se </p><p> ()- (7) - </p><p> 2 2</p><p>2 1v v</p><p>v</p><p>S Q ae</p><p> (v = b + 1, , n), (8) </p><p> S ; 2 . </p><p>p px , = xa + , p = xk + , </p><p>Np = pU , cP U </p><p> (5) </p><p> x Ax BU , (9) </p></li><li><p> . </p><p> 30 </p><p>a</p><p> O O</p><p> O O</p><p>E O O</p><p>, </p><p>a</p><p>c</p><p>E O</p><p>E</p><p>O O</p><p>, </p><p>1</p><p>1</p><p>c</p><p>xx</p><p>x</p><p> , 1U</p><p>UU</p><p>, </p><p> Ea , c1 = 2a + c. (9) [16, 17], -</p><p> . - , , [9, 10] </p><p>3U K x , </p><p> K3 (k1), - = 0 </p><p> 3x A BK x . (1), (2) . </p><p> 1 1</p><p>12</p><p>n n</p><p>ij i ji j</p><p>T A q q </p><p> , (10) </p><p> ev (10) - </p><p>1 111 1</p><p>12</p><p>n n</p><p>vv v vv b v b</p><p>T A e e </p><p> , </p><p> 1 1</p><p>1 1</p><p>n n</p><p>vv ij jv vii j</p><p>A A C C </p><p> , </p><p> S [15] </p><p> 1 1 1 21 1 2</p><p>1 21 1 1 1 1</p><p>1 , ,2</p><p>n n n n n</p><p>vv v v v v vv b v b v b v b v b</p><p>S A e e v v v e e e </p><p> . </p><p> (8) [16]: </p><p>1) 11</p><p>, , 11 1 1 1 1</p><p>, ,a c n b a a</p><p>p p a pp p</p><p>A A A e p</p><p> 11</p><p>11 1 1 1</p><p>, , , ,a c a n b</p><p>p pp p</p><p>p a p e</p></li><li><p> 4 (12), 2009 - . </p><p> 31 </p><p> 1 1 1 1</p><p>, , , ,c c c n b</p><p>a a a e</p><p> 1 2 21 2</p><p>11 1 1</p><p>, ,n b n b</p><p>e e Q a </p><p> ( = 1, , ); </p><p>2) 1 1</p><p>1</p><p>, , ,1 1 1</p><p>a c n b</p><p>p p a a ap</p><p>A A A e</p><p> 11</p><p>11 1 1 1</p><p>, , , ,a a a c</p><p>pp</p><p>p a a a </p><p> 11</p><p>11 1 1 1</p><p>, , , ,a c a n b</p><p>a a a a e</p><p> 1 1</p><p>, ,c n b</p><p>a a e</p><p> 111 1</p><p>, ,n b n b</p><p>a e e </p><p>2 22</p><p>,1</p><p>a aQ a</p><p> ( = 1, , ); </p><p>3) , 1 11</p><p>, ,1 1 1 1 1</p><p>, ,p</p><p>a c n b a a</p><p>p ap p</p><p>A A A e p</p><p> 11</p><p>11 1 1 1</p><p>, , , ,a c a n b</p><p>p pp p</p><p>p a p e</p><p> 11</p><p>11 1 1 1</p><p>, , , ,c c c n b</p><p>a a a e</p><p> 1 21 2</p><p>2 11 1</p><p>, ,n b n b</p><p>e e </p><p> 2 2</p><p>2</p><p>,1</p><p>Q a</p><p> ( = 1, , n b). (11) </p><p>, , (1), , , -, (3) . </p></li><li><p> . </p><p> 32 </p><p> (11) 2 22 , , aR R R </p><p> 11</p><p>21</p><p>1 1, ,</p><p>n b n bo oR e e Q</p><p> ' " ,1 1</p><p>a c</p><p>p p p p ap</p><p> ( = 1, , a); </p><p> 1</p><p>21</p><p>1 1, ,</p><p>n b n bo o</p><p>aR a Q </p><p> ' ", , ,1 1</p><p>a a</p><p>a p p a p p a ap</p><p> ( = 1, , c); </p><p> 1 21 2</p><p>21 2</p><p>1 1, ,</p><p>n b n bo oR e e Q</p><p> ( = 1, , n b), </p><p> (11) : </p><p>1) 111 1</p><p>1 1</p><p>, ,1 1 1 1</p><p>a a c ao o</p><p>p p p a ap</p><p>A A A A </p><p> 11 1</p><p>01 1</p><p>1 1 1 1, , , ,</p><p>a n b c n bo</p><p>pp</p><p>p e a </p><p> 1 111 1 2</p><p>1 2,1 1 1 1</p><p>, ,a n b n b n b</p><p>ppa</p><p>pe Q A e</p><p> 1 2 ' 2 2 2, , 0p p p pe e Q X ( = 1, , a); </p><p>2) , , ,,1 1 1 1</p><p>c a a apo o</p><p>a a p p a p a p pa ap p</p><p>A A A A </p><p> 11 11 1 1</p><p>1 1 11 1 1 1</p><p>, , , ,c n b a n b</p><p>o oaa</p><p>pa a e Q p a</p><p> 11 1 2</p><p>1 2,1 1 1 1</p><p>, ,a n b n b n b</p><p>ppp a a</p><p>pe A e a</p></li><li><p> 4 (12), 2009 - . </p><p> 33 </p><p> 1 2 ' 2 2 2, , 0p p a p pae e Q X ( = 1, , c); </p><p>3) , , , ,1 1 1 1</p><p>c a c ao p o</p><p>p p p a ap p</p><p>A A A A </p><p> 11 1</p><p>1 1,1 1 1 1</p><p>, , ,a n b c n b o</p><p>pp</p><p>p e a </p><p>1 1 11 1 11</p><p>, ,1 1</p><p>n b ap pa pe Q A e A e Q</p><p> 1 21 2</p><p>1 21 1</p><p>, ,n b n b</p><p>ppe e</p><p> ' 2 2 2, , 0p pX ( = 1,, n b), (12) ' 2 2 2 ' 2 2 2 ', , , , , ,p p a p pX X X - 2 2 2, ,p p . </p><p> c n b (12), e , (12), </p><p>, ,1 1 1 1</p><p>a c a c</p><p>p p a p p ap p</p><p>A B C D </p><p> 2 2 21</p><p>, , 0a</p><p>p p p pp</p><p>E </p><p> ( = 1, , a); </p><p>, , , ,1 1 1 1</p><p>a c a c</p><p>p p a a a p p a ap p</p><p>A B C D </p><p> ,1</p><p>0a</p><p>a p p ap</p><p>E </p><p> ( = 1, , c), (13) </p><p> 1</p><p>11 1</p><p>1</p><p>,1 1 1</p><p>,</p><p>1a n b n bo op p p To</p><p>A A A AA</p></li><li><p> . </p><p> 34 </p><p>1212 2</p><p>1</p><p>,, ,</p><p>1</p><p>ao</p><p>p pA A A </p><p> ; </p><p> 21, , ,,</p><p>1 1 1,</p><p>1a n b n bpo oa a pa Top</p><p>B A A AA</p><p>22</p><p>,,,</p><p>1</p><p>apo</p><p>paap</p><p>A A A </p><p> ; </p><p>pC pC 1 2 11</p><p>1 ,1 1 1</p><p>,</p><p>1, ,n b n b n b</p><p>o oTo</p><p>p e AA</p><p> 2 11</p><p>,1 2</p><p>1, ,</p><p>n bA p e</p><p> ; </p><p> 11 1</p><p>, 11</p><p>,</p><p>1, ,n b</p><p>o aa To</p><p>D a e QA</p><p> 2 3 21 3</p><p>,, 3 2</p><p>1 1 1, ,</p><p>n b n b n bo aA A a e Q</p><p> ; </p><p> 111 1 2</p><p>1 2,1 1 1</p><p>, ,n b n b n b</p><p>ppp A e</p><p> 1 2 1,</p><p>1 1, 1</p><p>1 n b n bp oTo</p><p>e e Q AA</p><p>231 3 2</p><p>3</p><p>,,</p><p>1</p><p>n bp pA A e Q</p><p> 3 43 4</p><p>3 41 1</p><p>, ,n b n b</p><p>p e e </p><p> ; </p><p> 21, , , ,</p><p>1 1 1,</p><p>1a n b n bo oa p a p a p aTo</p><p>A A A AA</p></li><li><p> 4 (12), 2009 - . </p><p> 35 </p><p>22 2</p><p>,, ,</p><p>1</p><p>ao</p><p>p pA A A </p><p> . </p><p> = , p a px , = + (p = 1, , a), ( = 1, , c) </p><p> (13) </p><p> 2 2 2 2 2 22 1 1 , 1... , , ...,i i i n n i ndx P t x P t x t x xdt (i2 = 1, , n2 = 2a + c), (14) </p><p>1 1 11 1 1</p><p>1 1</p><p>1 1</p><p>1 1</p><p>1 1</p><p>, ,,</p><p>1 1 1</p><p>, ,</p><p>, ,, ,</p><p>,</p><p>1 1</p><p>, ,</p><p>, ,, ,</p><p>p</p><p>a c ap p p</p><p>a</p><p>p pa a</p><p>a a a aa p a p</p><p>a ca</p><p>p</p><p>p pa a</p><p>a a a aa p a p</p><p>E O O</p><p>A C A D A E</p><p>A AB BB BA AP</p><p>A C A</p><p>A AB BB BA A</p><p>1</p><p>1</p><p>1</p><p>,</p><p>,,</p><p>,,</p><p>1</p><p>,</p><p>,,</p><p>p a</p><p>a aa p</p><p>aa a</p><p>a</p><p>p a</p><p>a aa p</p><p>A BBA</p><p>A E</p><p>A BBA</p><p> , (14) - . , (2) , - [17]. </p><p> 22 2 2 21 1 ...</p><p>ii i n n</p><p>dxd x d x</p><p>dt (i2 = 1, , n2 = 2a + c), (15) </p><p> 2 2 21, ...,i i nd d (i2 = 1, , n2 = 2a + c) -</p><p> [17]. , 2i -</p><p>2</p><p>2</p><p>2,2 2 2</p><p>1,1211</p><p>2221 2,</p><p>2,1 ,</p><p>.....</p><p>.....0................... ..... ......</p><p>.....</p><p>n</p><p>n</p><p>nn n n</p><p>ddddd d</p><p>dd d</p></li><li><p> . </p><p> 36 </p><p> (15) </p><p>1 1e iR (1 &gt; 0; i1 = 1, , 2a). </p><p> (15) , . </p><p> 2 2 2 22 2</p><p>2 2</p><p>1 1</p><p>a c a c</p><p>i j i ji j</p><p>w x C x x </p><p> 2 2 2 22 2</p><p>2 2</p><p>1 1</p><p>a c a c</p><p>i j i ji j</p><p>x a x x </p><p> , </p><p> 15</p><p>d w xdt </p><p> . </p><p> 15</p><p>ddt </p><p> , </p><p>(15), 2 2i ja 2 2 2, 1, ...,i j n </p><p> 2 2 2 22 2</p><p>2 2</p><p>1 1</p><p>a c a c</p><p>i j i ji j</p><p>x a x x </p><p> 2 2 2 7 7 2 2 7 7 27</p><p>2</p><p>1</p><p>a c</p><p>i j i j j iK</p><p>C a d a d</p><p> 2 2i ja 2 2 2, 1, ...,i j n . , -</p><p> 2 2</p><p>,i ja .. </p><p>2</p><p>2 2 2 2</p><p>1,11 12</p><p>1 2 ,</p><p>......... ..... .... .....</p><p>....</p><p>n</p><p>n n n n</p><p>dd d</p><p>d d d</p><p> (n2 =2a + c) </p><p> . </p><p> 14</p><p>ddt </p><p>: </p><p> 2 2 2 7 2 72 2 7</p><p>2 2 2</p><p>14 1 1 1</p><p>a c a c a c</p><p>i j j ji j</p><p>d w x a p t ddt</p><p> 7 2 2 2 2 2 2 7 2 2 2 22 2</p><p>2 2</p><p>1 1,</p><p>a c a c</p><p>j j i j i i i j j ii j</p><p>a p t d x x a x t x </p><p> . </p></li><li><p> 4 (12), 2009 - . </p><p> 37 </p><p> [17], 2 ,i t x </p><p> 2 2 7 77</p><p>2</p><p>1, ,</p><p>a c</p><p>i it x h t x x</p><p> . </p><p> 2 7 2 2 2 7 2 72 2 7</p><p>2 2 2</p><p>14 1 1 1</p><p>a c a c a c</p><p>i i j j ji j</p><p>d C a p t ddt</p><p> 7 2 2 2 2 2 2 2 2 7j j i j i i j ja p t d a h 7 2 2 2 2 7j j i ia h x x . (16) = 0 (16) -</p><p>, 2ix (16) [18]. </p><p> , </p><p> 2 2 72 7</p><p>2 2</p><p>71 1</p><p>/a c a c</p><p>i k i i </p><p>l x x d dt </p><p> [18]: </p><p> 1 1 1 10, 0, ..., 0n n (n1 = 2a). (17) </p><p> 2 7i kl , -</p><p> (d/dt), , (17). - </p><p>21, ..., nx x , .. - </p><p> 2 2 ,n 2 1 2 1 2, ...,n (2 &gt; 0; n2 = 2a + c). (18) </p><p> (17) (18) (14). </p><p> 1. , . / . . . : , 1967. </p><p>192 . 2. Appel , P. Sur les une forme generale des equations de la dynamique (memorial des </p><p>Sciences Mathematique, fasccule 1) / P. Appel. Paris, 1925. . 150. 3. Przeborsri , A. Die allgemeinsten Gleichunden der Klassischen Dunamik / A. Prze-</p><p>borsri // Math., teitschift. T. 36. 2. . 184194. 4. , . . -</p><p> / . . // . . 1967. 217. . 5083. </p><p>5. , . . / . . // - . . 4. , 1958. 183 . </p></li><li><p> . </p><p> 38 </p><p>6. , . . / . . // . 1961. . 6166. (. . .). </p><p>7. , . . / . . - // . 1976. . 40. . 5. . 771781. </p><p>8. , . . () / . . // . 1967. . 31. . 3. . 433446. </p><p>9. , . . / . . // . . 476. , 1975. . 6775. </p><p>10. , . . : / . . . , 1980. . 23. </p><p>11. , . . / . . - // . 1953. . 27. . 2635. </p><p>12. , . . / . . // . . . . . : , 1962. . 311316. </p><p>13. , . . / . . // . . - . . . : , 1962. . 329334. </p><p>14. , . . / . . -, . . , . . , . . . . : , 1971. 352 . </p><p>15. , . . / . . . . : , 1961. 824 . </p><p>16. , . . , / . . // . 2005. 1. . 37. </p><p>17. , . . / . . . . : , 1959. 211 . </p><p>18. , . . / . . . . : , 1987. 304 . </p><p> , , ( , . ) </p><p>Teshaev Mukhsin Khudayberdievich Associate professor, sub-department of higher mathematics, Bukhara Technological University of Food and Light Industry (Republic of Uzbekistan, Bukhara) </p><p>E-mail: Teshayev_Muhsin@rambler.ru </p><p> 531.31+62-50 </p><p>, . . , </p><p> / . . // . . -- . 2009. 4 (12). . 2738. </p></li></ul>

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