有多余坐标完整系统的自由运动

发布时间：2018-11-04 11:14

【摘要】：对于完整力学系统,若选取的参数不是完全独立的,则称为有多余坐标的完整系统.由于完整力学系统的第二类Lagrange方程中没有约束力,故为研究完整力学系统的约束力,需采用有多余坐标的带乘子的Lagrange方程或第一类Lagrange方程.一些动力学问题要求约束力不能为零,而另一些问题要求约束力很小.如果约束力为零,则称为系统的自由运动问题.本文提出并研究了有多余坐标完整系统的自由运动问题.为研究系统的自由运动,首先,由d’Alembert--Lagrange原理,利用Lagrange乘子法建立有多余坐标完整系统的运动微分方程;其次,由多余坐标完整系统的运动方程和约束方程建立乘子满足的代数方程并得到约束力的表达式;最后,由约束系统自由运动的定义,令所有乘子为零,得到系统实现自由运动的条件.这些条件的个数等于约束方程的个数,它们依赖于系统的动能、广义力和约束方程,给出其中任意两个条件,均可以得到实现自由运动时对另一个条件的限制.即当给定动能和约束方程,这些条件会给出实现自由运动时广义力之间的关系.当给定动能和广义力,这些条件会给出实现自由运动时对约束方程的限制.当给定广义力和约束方程,这些条件会给出实现自由运动时对动能的限制.文末,举例并说明方法和结果的应用.
[Abstract]:For holonomic mechanical systems, if the selected parameters are not completely independent, they are called holonomic systems with redundant coordinates. Because there is no binding force in the Lagrange equation of the second kind of holonomic mechanical system, in order to study the binding force of holonomic mechanical system, the Lagrange equation with superfluous coordinates or the Lagrange equation of the first kind should be adopted. Some dynamic problems require no binding force, while others require little binding force. If the binding force is zero, it is called the free motion of the system. In this paper, the problem of free motion of a system with redundant coordinate holonomic system is proposed and studied. In order to study the free motion of the system, the differential equations of motion of the system with redundant coordinates are established by using the d'Alembert--Lagrange principle and the Lagrange multiplier method. Secondly, the algebraic equations satisfied by the multipliers are established by the equations of motion and constraint equations of the holonomic system with redundant coordinates and the binding expressions are obtained. Finally, by the definition of free motion of constrained system, all multipliers are zero, and the condition of realizing free motion of system is obtained. The number of these conditions is equal to the number of constraint equations. They depend on the kinetic energy, generalized forces and constraint equations of the system. That is, given kinetic energy and constraint equation, these conditions will give the relationship between generalized forces when realizing free motion. Given the kinetic energy and the generalized force, these conditions will give the constraints on the constraint equation when the free motion is realized. Given the generalized force and the constraint equation, these conditions will give the restriction of kinetic energy when the free motion is realized. At the end of the paper, examples are given to illustrate the application of the method and the results.
【作者单位】：北京理工大学宇航学院;北京理工大学数学学院;
【基金】：国家自然科学基金资助项目(10932002,11272050,11572034)
【分类号】：O316

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