具有重频特性系统的动力响应分析

Dynamic response analysis of the linear system with repeated frequencies

一般有阻尼线性系统出现重特征值时,基于振型正交性的复振型分解法将不再适用。本文综合运用高等数学、线性代数和复变函数理论,对具有重频特性的一般有阻尼线性多自由度系统给出了系统动力响应在时域中的计算方法。该方法充分利用复振型分解法和留数矩阵解耦法的优点,不仅概念清晰,而且易于理解和掌握,适合于大型复杂系统的动力响应分析。此外,本文给出了双自由度体系产生重特征值的条件,对典型实例进行了地震响应分析,并通过与Newmark-β法计算结果的对比,论证了文中所给计算公式的正确性。本文提出的分析方法具有普适性,对线性结构、机电和控制系统也都是适用的。

For the generally damped linear systems with repeated eigenvalues,a hybrid approach based on the complex modal superposition method and residue matrix decomposition method is presented. The hybrid approach incorporates the merits of the modal superposition method and residue matrix decompo- sition method, and has clear physical concept and is easily to be understood and mastered by engineering designers to analyze the large structures. Besides,the conditions producing repeated eigenvalues for the double degrees of freedom system are deduced, and the implementation procedure of the proposed hybrid approach in the paper is illustrated by analyzing simple numerical examples. Finally, correctness and effectiveness of the formula are judged by comparing the results obtained from Newmark-β methods. It pointed out that the method derived in this paper is also suitable for linear system, electro-mechanical and control system.