非齐次动力方程Duhamel项的精细积分

提出了不需要矩阵求逆运算的求解Duhamel积分项的精细积分方法.通过将精细积分法的关键思想--加法定理和增量存储--直接应用于Duhamel积分响应矩阵的求解,可给出当非齐次项分别为多项式、正弦/余弦以及指数函数等基本形式时Duhamel积分在计算机上的精确解.特别的,该算法不依赖于系统矩阵(或相关矩阵)的形态.当系统矩阵奇异或接近奇异时,其优越性更为显著.算例验证了该算法的有效性.

PRECISE INTEGRATION METHOD FOR DUHAMEL TERMS ARISING FROM NON-HOMOGENOUS DYNAMIC SYSTEMS

With the precise integration method (PIM) proposed for linear time-invariant systems, one can obtain precise numerical results approaching the exact solution at the integration points. However, it is more or less difficult to use the algorithm in the Duhamel's integration arising from the non-homogenous dynamic systems due to the inverse matrix calculations. So the precise integration method for Duhamel terms without the inverse matrix calculations is proposed. By applying the techniques of addition theorem and increment storage, which are the key ideas of PIM, directly to the Duhamel integration terms, it can also give precise numerical results can be obtained close to the computer precision when the non-homogenous terms are polynomial, sinusoidal, exponential or their combinations. In particular, this method is not affected by the quality of the system matrix (or the relative matrix). If the system matrix is singular or nearly singular, the advantages of the method will be more remarkable. Numerical examples are given to demonstrate the validity and efficiency of the method.