微振动力学系统模型构造微分方程系数矩阵方法

摘要

对于多自由度定常约束微振力学系统的数学建模有许多方法，但这些方法不是受到使用条件限制就是计算繁琐，并且不适合于计算机建模。根据系统的动能、势能、耗散能量与运动方程决定的动能、势能、耗散能量分别相等的原理，提出了构造微分方程系数矩阵方法(下称能量偏导法)。论证了直接法［1］、视察法［2,7］是能量偏导法的特例。能量偏导法的优点是：在建立运动方程的过程中，设有对时间求导的过程，也没有对方程整理和化简的过程。该方法既适用于单自由度系数又适合于多自由度系统，既适用于手工建模又方便于计算机建模。在该方法的基础上，本文又提出了运动方程解耦的数学条件，为广义坐标的选取指明了方向。文中也给出了能量偏导法的应用和结论。%Although there are many methods of building mathematical model for permanently constrained discrete system,the methods are usually toilsome and not convenient for computer modeling.With increasing of degrees of frecdom of the system,the task of reduction and arrangement of equations becomes arduous.This paper presents a simple and direct method of mathematical modeling for micro-vibration system.The method,called as the method of partial derivation of energy,is based on the supposition that the kinetic energy,potential energy and dissipation power of the system should be equal to those derived from the motion equtions.The method has more advantages than others:no derivative with repect to time in the process of modeling;no any process of reduction and arrangement of equations and no any constrained force appeared during modeling.It is pointed out that the Direct Method in［1］or the Watch Method in［2,7］is a special case of the method of partial derivation of energy.Mathematical conditions of equation de-coupling,applications of the method and some conclusions are given in the paper.