基于三次样条插值的精细积分法  

A HIGH PRECISE DIRECT INTEGRATION BASED ON CUBIC SPLINE INTERPOLATION

作　　者：高小科[1] 邓子辰[1] 黄永安[1] 

机构地区：[1]西北工业大学工程力学系,西安710072

出　　处：《振动与冲击》2007年第9期75-77,82,共4页Journal of Vibration and Shock

基　　金：国家自然科学基金(10372084和10572119);教育部新世纪优秀人才支持计划(NCET-04-0958);大连理工大学工业装备结构分析国家重点实验室开放基金资助项目

摘　　要：结合指数矩阵的精细算法,提出了一类基于三次样条插值的精细积分方法。针对结构动力学方程一般解中的积分项,考虑在一个时间步长内激励为线性和正余弦两种变化形式,通过对积分项中的指数矩阵进行三次样条插值函数模拟,得到一组新的被积函数,最后通过多次分部积分,构造了一类新的高精度计算格式。在三次样条插值函数构造过程中引入了指数矩阵的精细算法,有效避免了中间过程中有效数字的丢失,同时还有效解决了HPD-F算法中涉及的矩阵求逆问题,大大增加了算法的数值稳定性。数值算例显示了该方法的有效性。The analysis of dynamic response plays an important role in dynamic system analysis. Generally, there are two ways to solve the differential equations of dynamic systems: direct integration method and mode superposition method. Here, an improved direct integration method based on cubic spline interpolation is given to solve the inhomogeneous dynamic equations. With this method, when the applied loading is variable in sinusoidal form or linear form, a new integrand is obtained by simulating the exponential matrix in integral items with cubic spline. After several times of integration by parts, a new precise time step integration method is proposed. The calculation technique of matrix exponential function is used in the construction of cubic spline interpolation function to avoid the loss of effective digits. In addition, this new method avoids the computation of the inverse matrixes from which the HPD-F algorithm suffers, so the stability of this algorithm is improved greatly. Numerical example is given to demonstrate the validity and efficiency of the algorithm proposed.

关 键 词：结构动力方程 时程积分 样条插值 指数矩阵 精细积分法 

分 类 号：O327[理学—一般力学与力学基础]