结构动力方程一种新的级数形式的解析解

New series form of analytical solution for the structural dynamic equations

将结构的位移及速度响应作为状态变量,采用Lyapunov(李雅普诺夫)人工小参数法求解状态方程,导出状态方程的一个新的级数形式的解析解,该解析解还可以推广到非线性动力方程的计算。将秦九韶算法引入级数解的计算,提高了计算的效率和稳定性,同时给出了算法的计算格式和步骤。该算法无需对转换矩阵H求逆,仅使用矩阵向量相乘,计算稳定,精度仅由收敛项数控制,很容易达到任意精度要求,而且适合并行计算及压缩存储。最后通过算例进一步证实了该算法的精度和效率。

The displacement and velocity responses of the structure were taken as state variables and in solving the state equations,the Lypaunov artificial small parameter method was used,in which a new series form of analytical solution was presented.The series solution can also be used to solve nonlinear dynamic equations.Hornor's scheme was applied to the calculation of series solution to increase,the computation efficiency and stability.At the same time,the corresponding computation formats and steps for linear dynamic equations were established.The algorithm needs only repetitive matrix-vector multiplication instead of inversion of H matrix,so good computation stability is achieved.The precision is only controlled by the series convergence number,so theoretically,the algorithm can easily reach the accuracy of arbitrary-order,and is fit for parallel calculation and compression storage.A numerical example was given to demonstrate the validity and efficiency of the method.