一般力学初值问题三类变量的广义变分原理

Generalized variational principles with three kinds of variables of initial value problems in general mechanics

一般力学初值问题的广义变分原理的研究,是一个相当重要的研究领域.它不仅在有限元素法和其他近似计算方法中得到广泛应用,而且可以方便地求得一般力学初值问题的精确解.首先,明确了一般力学初值问题的基本方程,应用Laplace变换将基本方程变换到像空间,按照广义力和广义位移之间的对应关系,将各基本方程乘上相应的虚量,代数相加,进而建立了一般力学初值问题的像空间中的三类变量的广义变分原理.然后,应用Laplace逆变换将像空间中的广义变分原理反演到原空间,进而建立了一般力学初值问题的原空间中的三类变量的广义变分原理.并且,将三类变量的广义变分原理进行退化,得到像空间和原空间中的几个两类变量的广义变分原理和经典变分原理.最后,以弹性动力学为例,说明了像空间和原空间中的势能函数和余能函数的丰富的内涵.

Research of generalized variational principles of initial value problems in general mechanics is vital because they are widely applied to finite element methods （FEM） and other calculation methods, as well as accurately solving initial value problems in general mechanics. First of all, the basic equations of initial value problems in general mechanics were transformed into the image space by Laplace transformation, then according to corresponding relations between general forces and general displacements, the basic equations were multiplied by corresponding virtual quantities, and added algebraically, the generalized variational principles with three kinds of variables of initial value problems in the image space were established. Afterwards, the generalized variational principles in the image space were inversed into the original space by Laplace inverse transformation, subsequently, the generalized variational principles with three kinds of variables of initial value problems in the original space were acquired for general mechanics. Furthermore, several generalized variational principles with two kinds of variables and classical variational principles were obtained by reducing generalized variational principles with three kinds of variables. Finally, elasto-dynamics was taken as an example to explain the abundant meaning of potential energy and com- plementary energy functions.