基于频响函数矩阵计算阻尼系统动力响应的新方法

A new method for computing dynamic response of a damped linear system based on frequency response function matrix

首先引入模态坐标变换及矩阵函数变换,将阻尼系统转化为无阻尼系统,再通过分析无阻尼系统的模态参数的特点,推导出计算其频率响应矩阵的公式,然后使用矩阵函数逆变换和模态坐标反变换,推导出原阻尼系统响应的精确表达式,方法应用于经典阻尼系统时,取得了与传统的振型迭加法等价的解析表达式,同时对于无法直接使用振型迭加法处理的非经典阻尼系统,也获得了解析解。数值算例表明了该方法的正确性及有效性。

Firstly,by introducing modal coordinate transformation and matrix function transformation,an original damped system was transformed into a non-damped one.Secondly,by analyzing the properties of modal parameters of the non-damped system, its frequency response function matrix was derived. Furthermore, by using inverse matrix transformation and inverse modal coordinate transformation,the precise expression for the response of the original damped system was derived.When the presented method was applied to a classical damped system,the analytical expression equivalent to the traditional modal superposition method was obtained.And then,the analytical expression for a non-classical damped system which could not be solved with the modal superposition method was obtained as well.Finally,the correctness and effectiveness of the proposed method were verified with two examples.