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Calculation of complex modes of non-proportionally damped systems using real modes

DOI：10.7511/jslx20230105001

 作者 单位 E-mail 魏向天 广东工业大学 机电工程学院, 广州 510006 吴柏生 广东工业大学 机电工程学院, 广州 510006 wubs@gdut.edu.cn 钟慧湘 广东工业大学 机电工程学院, 广州 510006

提出一种利用实模态计算非比例阻尼系统复模态的新方法。首先,引入一个嵌入参数将无阻尼系统与阻尼系统联系起来,然后构造模态规范化条件。将非比例系统特征值和特征向量在无阻尼系统基础上对该参数实施摄动展开,得到展开系数的控制方程。每个特征向量展开系数可借助Nelson法求解具有同一个对称实系数矩阵的方程组来求得。该方法仅涉及感兴趣的模态,无模态截断问题,不要求广义逆,也不需要将矩阵扩维。本文以两个数值算例来说明本文方法的有效性。

In this paper,a new method for calculating complex modes of non-proportionally damped systems using real modes is proposed.First,an embedded parameter is introduced to directly connect the undamped system with the damped system,and then the modal normalization condition is constructed.The eigenvalues and eigenvectors of the non-proportionally damped system are expanded as the power series of this parameter to obtain the governing equations of the expansion coefficients.The eigenvector expansion coefficients can be determined by solving systems of equations with the same symmetric real-coefficient matrix by Nelson’s method for each eigenpair.The proposed approach involves only the modes of interest,no modal truncation,but no generalized inversion,and no matrix expansion.Two numerical examples are given to illustrate the effectiveness of the proposed method.