频变系统特征问题及其灵敏度分析的扩阶法

Extended Order Method for Eigenproblem and Its Sensitivity of Frequency Dependent Systems

当振动系统的物理矩阵（刚度矩阵和／或质量矩阵）是频率的函数时，该特征问题就变为非线性特征问题．由于振型的非正交性，使得非线性特征问题及其灵敏度分析存在困难．针对可展成幂级数形式的非线性特征问题，提出了一种通过初始系统矩阵的扩阶来使原非线性问题线性化的扩阶方法．该方法的优点是：由于在线性化过程中未引入任何误差，因而是精确的；可适用于强非线性特征问题；可求得系统的所有特征值及其灵敏度．数值示例表明，该方法虽很简单，但十分有效．

The eigen problem is defined as a nonlinear one when the physical matrices (stiffness and/or mass matrices) of a vibration system are frequency dependent.It is difficult to solve the nonlinear eigenproblem and its sensitivity due to the non orthogonality of its eigenvectors.In this paper,an extended order method (EOM) is derived for solving the nonlinear problem which can be expanded in a power series.The nonlinear eigenproblem is linearized by extending the order of the system matrices.As an accurate method,the approach can be used to solve strong nonlinear eigenproblems.All the eigenvalues and their sensitivities of the system can be obtained by using the proposed method.The numerical example shows that the method is simple but very efficient.