复阻尼多自由度系统动力分析的模态叠加法

Two mode-superposition methods for dynamic analysis of MDOF systems with complex damping characteristics

工程实际中,复阻尼多自由度系统的瞬态响应过去一直是通过频域方法求解的。频域方法的一个突出问题是求传递函数矩阵的计算工作量过大。本文给出一种有效而实用的时域解法———实模态叠加法,此外还介绍了复模态叠加法。实模态叠加法是基于一个事实,即n维复向量空间中的复向量可以在n维实向量空间中的一组线性无关的实向量构成的基下表出;另外,就此方法还讨论了复阻尼多自由度系统初始运动条件的给出和转换问题。复模态叠加法则是通过变量替换的方法,变其中的复特征值问题为形式上的一个实特征值问题来解决的。

In engineering practice, the vibrational response computations for the systems with complex damping are fulfilled mainly in frequency domain. A conspicuous defect of the method in frequency domain is that the computation of the transfer function matrix is in large quantities. In this paper, an effective and practical procedure in time domain,the real mode-superposition method,is proposed. Besides, a complex mode-superposition method is also depicted.The real mode method is based on the fact that a complex vector in n-dimensional complex vector space can be represented on the basis of a set of linearly independent real vectors in n-dimensional real vector space. In addition, the transformation of initial conditions of motion of an MDOF system with complex damping characteristics is discussed. And the complex mode method is based on two variable replacements which get the complex eigen value problem changed formally into a real one.