特征值重分析的递推算法和若干技术比较

A progressive algorithm and comparison of some techniques for eigenvalue reanalysis

对于利用虚材料结构的模态试验数据提取(辨识)真实材料结构的试验模态参数(简称虚材料结构试验模态)的问题,作者曾建立过一种基于动柔度的广义伽略金法(GG)和基向量组合法(DF)。为了特征值重分析的"快速性"要求,作者又得到广义伽略金法的一种退化解(DS),其"快速性"非常好,但精度比广义伽略金法差。为了全面了解,还对若干相关方法,如GG、DF、DS和CA(combined approximation)等,在满足修改结构特征方程的近似性方面作了探讨,同时还就它们的数值结果进行了比较。其探讨表明,GG法的精度最好,而其余方法的精度是相同或相当的、且基本满意。这些方法在修改结构之特征方程Ku=λMu的右边惯性项中都不同程度引入了Kirsch的近似假设,故而会引入不同程度的误差。现在要问:右边惯性载荷项的误差对方程解之精度的影响倒底有多大?为此,首次提出了一种递推算法。该递推算法的第一级递推解恰好是广义伽略金法的退化解(DS法)。递推算法的概念适用于上述各种方法。为此,作者又首次建立了CA法的递推公式。最后,该递推算法从数值上得出一个重要结论:导致右边惯性载荷项误差的"Kirsch近似假设"对特征值重分析结果的影响不严重。同时,该递推算法还能逐级减小由Kirsch的近似假设给重分析结果带来的误差。

Reanalysis procedures of eigenvalue play an important part in many fields. For pseudo-material structural modal testing, the authors proposed a generalized Galerkin（GG） method and dynamic flexibility method（DF） that are based on the structural dynamic flexibility matrix. To satisfy the requirement of fastness of eigen reanalysis, a degenerated solution（DS） of the GG method were obtained. The fastness of the degenerated solution is very good. To compared we discuss accuracy of CA（combined approximation）,dynamic flexibility（DF）, GG methods and degenerated solution based on the approximation that eigenequation is approximately satisfied. The comparison shows that accuracy of GG method is better than that of CA, DF and DS methods, and accuracy of CA, DF methods and degenerated solution is the same. Theses methods all employ in varying degrees Kirsch＇s approximate assumption in right-hand side inertial term of eigenequation Ku=2Mu of modified structure, such that the varying degrees error is introduce into various method. In order to investigate the influence of the inertial term （corresponding to load term） error on accuracy of the equation solution as well as to decrease these errors, the authors ftrst develop a progressive algorithm. The first-level progressive solution of the present progressive algorithm is just the degenerated solution （DS method）. Idea of the progressive algorithm can suit other some methods. For this the authors first propose progressive CA （PCA） method. The numerical results of the progressive algorithm find an important conclusion： the influence of Kirsch＇s assumption on eigen reanalysis is not serious.