﻿ 有多余坐标完整系统的自由运动
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 力学学报  2016, Vol. 48 Issue (4): 972-975  DOI: 10.6052/0459-1879-15-392 0

### 引用本文 [复制中英文]

[复制中文]
Chen Ju , Wu Huibin , Mei Fengxiang . FREE MOTION OF HOLONOMIC SYSTEM WITH REDUNDANT COORDINATES[J]. Chinese Journal of Ship Research, 2016, 48(4): 972-975. DOI: 10.6052/0459-1879-15-392.
[复制英文]

### 文章历史

2015-10-27 收稿
2016-06-01 录用
2016-06-06网络版发表

1. 北京理工大学宇航学院, 北京 100081 ;
2. 北京理工大学数学学院, 北京 100081

0 引言

1 系统的运动微分方程

 $f_\beta \left( {q_s ,t} \right) = 0 \quad \left( {\beta = 1,2,\cdots ,g ; s = 1,2,\cdots,n} \right)$ (1)

d'Alembert-Lagrange原理有形式

 $\left( {\dfrac {d }{d }\dfrac {\partial T}{\partial \dot {q}_s } - \dfrac {\partial T}{\partial q_s } - Q_s } \right)\delta q_s = 0 \quad \left( {s = 1,2,\cdots ,n} \right)$ (2)

 $\dfrac {\partial f_\beta }{\partial q_s }\delta q_s = 0$ (3)

 $\dfrac {d }{d }\dfrac {\partial T}{\partial \dot {q}_s } - \dfrac {\partial T}{\partial q_s } = Q_s + \lambda _\beta \dfrac {\partial f_\beta }{\partial q_s } \\ \left( {\beta = 1,2,\cdots ,g ; \ s = 1,2,\cdots,n} \right)$ (4)

2 约束力的确定

 $T = T_2 + T_1 + T_0$ (5)

 $T_2 = \dfrac {1}{2}A_{sk} \dot {q}_s \dot {q}_k ,T_1 = B_s \dot {q}_s$ (6)

 $\dfrac {d }{d t }\dfrac {\partial T_2 }{\partial \dot {q}_s } - \dfrac {\partial T_2 }{\partial q_s } = \\ A_{sk} \ddot {q}_k + \left( {\dfrac {\partial A_{sk} }{\partial q_m } - \dfrac {1}{2}\dfrac {\partial A_{km} }{\partial q_s }} \right)\dot {q}_k \dot {q}_m + \dfrac {\partial A_{ks} }{\partial t}\dot {q}_k =\\ A_{sk} \ddot {q}_k + \left[{k,m;s} \right] \dot {q}_k \dot {q}_m + \dfrac {\partial A_{ks} }{\partial t}\dot {q}_k$ (7)

 $\left[{k,m;s} \right] = \dfrac {1}{2}\left( {\dfrac {\partial A_{ks} }{\partial q_m } + \dfrac {\partial A_{ms} }{\partial q_k } - \dfrac {\partial A_{km} }{\partial q_s }} \right)$ (8)

 $\dfrac {d }{d t }\dfrac {\partial T_1 }{\partial \dot {q}_s } - \dfrac {\partial T_1 }{\partial q_s } = \dfrac {\partial B_s }{\partial t} - \left( {\dfrac {\partial B_k }{\partial q_s } - \dfrac {\partial B_s }{\partial q_k }} \right)\dot {q}_k$ (9)
 $\dfrac {d }{d t }\dfrac {\partial T_0 }{\partial \dot {q}_s } - \dfrac {\partial T_0 }{\partial q_s } = - \dfrac {\partial T_0 }{\partial q_s }$ (10)

 $A_{sk} \ddot {q}_k + \left[{k,m; s} \right] \dot {q}_k \dot {q}_m = \\ \left( {\dfrac {\partial B_k }{\partial q_s } - \dfrac {\partial B_s }{\partial q_k }} \right)\dot {q}_k + Q_s - \dfrac {\partial B_s }{\partial t} + \dfrac {\partial T_0 }{\partial q_s } -\\ \dfrac {\partial A_{sk} }{\partial t}\dot {q}_k + \lambda _\beta \dfrac {\partial f_\beta }{\partial q_s } \quad \left( {s = 1,2,\cdots ,n} \right)$ (11)

${\rm{det}}\left( {{A_{sk}}} \right) \ne 0$，故可由式(11)求出所有广义加速度

$\ddot {q}_l = A^{ls}\left[{ - \left[{k,m; s} \right] \dot {q}_k \dot {q}_m + \left( {\dfrac {\partial B_k }{\partial q_s } - \dfrac {\partial B_s }{\partial q_k }} \right)\dot {q}_k + Q_s } \right. -$

 $\left. { \dfrac {\partial B_s }{\partial t} + \dfrac {\partial T_0 }{\partial q_s } - \dfrac {\partial A_{sk} }{\partial t}\dot {q}_k + \lambda _\beta \dfrac {\partial f_\beta }{\partial q_s }} \right]$ (12)

 $A^{ls}A_{sk} = \delta _k^l$ (13)

 $\dfrac {\partial ^2f_\beta }{\partial t^2} + 2\dfrac {\partial ^2f_\beta }{\partial t\partial q_s }\dot {q}_s + \dfrac {\partial ^2f_\beta }{\partial q_s \partial q_k }\dot {q}_s \dot {q}_k + \dfrac {\partial f_\beta }{\partial q_s }\ddot {q}_s = 0$ (14)

 \eqalign{ & {{{\partial ^2}{f_\beta }} \over {\partial {t^2}}} + 2{{{\partial ^2}{f_\beta }} \over {\partial t\partial {q_s}}}{{\dot q}_s} + {{{\partial ^2}{f_\beta }} \over {\partial {q_s}\partial {q_k}}}{{\dot q}_s}{{\dot q}_k} + {{\partial {f_\beta }} \over {\partial {q_l}}}{A^{ls}} \cr & \left[ { - \left[ {k,m;s} \right]{{\dot q}_k}{{\dot q}_m} + \left( {{{\partial {B_k}} \over {\partial {q_s}}} - {{\partial {B_s}} \over {\partial {q_k}}}} \right){{\dot q}_k} + {Q_s}} \right. - \left. {{{\partial {B_s}} \over {\partial t}} + {{\partial {T_0}} \over {\partial {q_s}}} - {{\partial {A_{sk}}} \over {\partial t}}{{\dot q}_k} + {\lambda _\gamma }{{\partial {f_\gamma }} \over {\partial {q_s}}}} \right] \cr & = 0\left( {\beta = 1,2, \cdots ,g} \right) \cr} (15)

 ${\rm{det}}\left( {{{\partial {f_\beta }} \over {\partial {q_l}}}{A^{ls}}{{\partial {f_\gamma }} \over {\partial {q_s}}}} \right) \ne 0$ (16)

 $\varLambda _s = \lambda _\gamma \dfrac {\partial f_\gamma }{\partial q_s } \quad \left( {\gamma = 1,2,\cdots ,g;s = 1,2,\cdots ,n} \right)$ (17)
3 自由运动的条件

 $\dfrac {\partial ^2f_\beta }{\partial t^2} + 2\dfrac {\partial ^2f_\beta }{\partial t\partial q_s }\dot {q}_s + \dfrac {\partial ^2f_\beta }{\partial q_k \partial q_s }\dot {q}_k \dot {q}_s +\\ \dfrac {\partial f_\beta }{\partial q_l }A^{ls}\Bigg [- \left[{k,m;s} \right] \dot {q}_k \dot {q}_m + Q_s - \\ \dfrac {\partial B_s }{\partial t} + \dfrac {\partial T_0 }{\partial q_s } - \dfrac {\partial A_{sk} }{\partial t}\dot {q}_k \Bigg] = 0 \\ \left( {\beta = 1,2,\cdots ,g} \right)$ (18)

 $\left.\!\!\begin{array}{l} B_s = 0 \left( {s = 1,2,\cdots ,n} \right) \\ T_0 = 0 \\ \dfrac {\partial A_{ks} }{\partial t} = 0 \left( {k,s = 1,2,\cdots ,n} \right) \end{array} \right\}$ (19)

 $\dfrac {\partial ^2f_\beta }{\partial t^2} + 2\dfrac {\partial ^2f_\beta }{\partial t\partial q_s }\dot {q}_s + \dfrac {\partial ^2f_\beta }{\partial q_s \partial q_k }\dot {q}_s \dot {q}_k + \\ \dfrac {\partial f_\beta }{\partial q_l }A^{ls}\left( { - \left[{k,m; s} \right] \dot {q}_k \dot {q}_m + Q_s } \right) = 0$ (20)

 $\dfrac {\partial ^2f_\beta }{\partial t^2} + 2\dfrac {\partial ^2f_\beta }{\partial t\partial q_s }\dot {q}_s + \dfrac {\partial ^2f_\beta }{\partial q_s \partial q_k }\dot {q}_s \dot {q}_k + \\ \dfrac {\partial f_\beta }{\partial q_l }A^{ls}Q_s = 0 \quad \left( {\beta = 1,2,\cdots ,g} \right)$ (21)

 $\dfrac {d }{d }\dfrac {\partial T}{\partial \dot {q}_s } - \dfrac {\partial T}{\partial q_s } = Q_s \quad \left( {s = 1,2,\cdots ,n} \right)$ (22)
4 算例

 $T = \dfrac {1}{2}\left( {\dot {q}_1^2 + \dot {q}_2^2 + \dot {q}_3^2 } \right) ,f = tq_1 + 2q_2 - 3q_3 = 0$ (23)

 $2\dot {q}_1 + tQ_1 + 2Q_2 - 3Q_3 = 0$ (24)

 $\left. {\matrix{ {T = {1 \over 2}\left[ {\dot q_1^2\left( {2 + \sin t} \right) + \dot q_2^2 + \dot q_3^2} \right]} \cr {f = {t^2}{q_1} - {q_2} - {q_3} = 0} \cr } } \right\}$ (25)

 $A_{11} = 2 + \sin t ,A_{22} = 1 ,A_{33} = 1$

 $A_{11} = \dfrac {1}{2 + \sin t} ,A_{22} = 1 ,A_{33} = 1$

 $\left[{k,m;s} \right] = 0 \quad \left( {k,m,s = 1,2} \right)$

 $2q_1 + 4t\dot {q}_1 + \dfrac {t^2}{2 + \sin t}\left( {Q_1 - \dot {q}_1 \cos t} \right) - \\ Q_2 - Q_3 = 0$ (26)

5 结论

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FREE MOTION OF HOLONOMIC SYSTEM WITH REDUNDANT COORDINATES
Chen Ju1, Wu Huibin2, Mei Fengxiang1
1. School of Aerospace, Beijing Institute of Technology, Beijing 100081, China ;
2. School of Mathematics, Beijing Institute of Technology, Beijing 100081, China
Abstract: If the parameters are not completely independent for holonomic systems, it is called holonomic systems with redundant coordinates. In order to study the forces of constraints for holonomic systems, we use the Lagrange equations with multiplicators of redundant coordinates or the first kind of Lagrange equations. Because there are no forces of constraints in the second kind of Lagrange equations. In some mechanical problems, the forces of constraints should not be equal to zero. In other conditions, the forces of constraints are very tiny. However, if the forces of constraints are all equal to zero, we called the free motion of constraints mechanical systems. This paper presents the free motion of holonomic system with redundant coordinates. At first, the differential equations of motion of the system are established according to d'Alembert-Lagrange principle. Secondly, the form of forces of constraints is determined by using the equations of constraints and the equations of motion. Finally, the condition under which the system has a free motion is obtained. The number of this conditions is equal to the constraints equation's, its depend on the kinetic energy, generalized forces and constraints equations. If the two arbitrary conditions are given, the third one should be obtained when the system becomes free motion. At the end, some examples are given to illustrate the application of the methods and results.
Key words: redundant coordinate    holonomic system    force of constraints    free motion