基于奇异值分解的复模态矩阵摄动法

对非自伴随系统的振动重分析问题，提出了一种简单的通用方法。从子空间缩聚出发，基于复矩阵的奇异值分解定理，推导了同时适用于孤立 特征值，相重特征值和相近特征值三种复特征值情况的一阶和二阶摄动公式。算例表明，该方法通用性好，且具有足够的精度。

A general matrix perturbation method for complex modes using singular value decomposition

A good understanding of the dynamic behavior of a structure under vibration and the ability to perform dynamic analysis is of vital importance for a designer nowadays. To satisfy a variety of requirements of structural dynamics, quite often in the dynamic design of structural systems, a so-called iterative design or reanalysis is generally carried out until a satisfactory outcome is achieved. A frequently encountered scenario in the dynamic reanalysis is determining the changes of the dynamic characteristics as a result of the changes in the system parameters. For general damped gyroscopic systems, the dynamic analysis is significantly difficult and most complicated. In such systems, there frequently co-exist distinct, repeated and closely spaced complex eigenvalues. In the existing techniques dealing with the three cases of eigenvalues, complete modal expansion is usually used. For large complicated systems, however, quite often it is difficult to obtain all modes. Therefore, using complete modal expansion probably gives rise to errors. In some cases these errors maybe too large to be acceptable.To this end, in this paper a simple and general matrix perturbation method is proposed for the dynamic reanalysis of systems with complex modes. This method is developed by performing a sub-space condensation and by using the singular value decomposition of complex matrix. The first and second perturbation formulas are derived. The present method can be universally applicable to systems with all the three cases of complex eigenvalues: distinct, repeated and closely spaced complex eigenvalues but complete modal expansion is not used. Illustrative examples are given to demonstrate the validity of this proposed technique. Three examples corresponding to the three cases of complex eigenvalues have shown that the present method is very effective.