结构动力微分方程的一种高精度摄动解

A HIGH-PRECISION PERTURBATION SOLUTION FOR STRUCTURAL DYNAMIC DIFFERENTIAL EQUATIONS

将结构的位移及速度响应作为状态变量,把结构动力方程转化为状态方程,采用摄动方法求解状态方程,推出一种级数形式的摄动解,同时给出了该文算法的迭代格式和计算步骤。该算法无需对转换矩阵H求逆,也无须作指数矩阵eH运算,仅做矩阵向量相乘及向量求和运算,计算稳定而且效率高,收敛速度快,解的级项数及精度可由允许误差参数直接控制,很容易达到任意精度要求,该方法兼具线性加速法的高效率和精细积分法的高精度,可应用于结构大型稀疏线性动力方程组的求解。最后通过典型算例进一步验证了该文算法的精度和效率。

The structural dynamics equations were converted to the state equations in which the displacement and velocity response were taken as the state variables. In order to solve the state equations, the perturbation method was used, and a new series form of analytical solutions was presented. At the same time, the corresponding iterated computation formats and steps for the dynamics equations were established. The algorithm needs only repetitious matrix-vector multiplication and vector summation without inversion of H matrix and calculation of exponential matrix e/~. Thusly, the computation stability and efficiency are very high. The items of the series solution and the accuracy of the algorithm can be directly controlled by the tolerance parameter e, and theoretically, the algorithm can easily achieve arbitrary-order accuracy, and be suitable for parallel computing and compression storage. Generally, the algorithm combines the high-efficiency of the linear acceleration methods and the high-precision of precise integration methods. This method can be used for calculating the large sparse linear dynamic equations of engineering structures. At last, a model numerical example was given to demonstrate the validity and efficiency of the method.