基于随机摄动方法的广义随机复特征值问题研究

Research on Generalized Stochastic Complex Eigenvalue Problem Based on Stochastic Perturbation Method

将结构振动特征值和特征向量的一阶摄动展式与概率理论相结合,结构随机特征值的方差就可以由刚度矩阵的摄动项、质量矩阵的摄动项和原始特征向量直接推导出来。上述方法在对实模态结构随机特征值变化范围的计算中,不涉及结构参数的相关系数矩阵,在工程实际中是易于实现的。具体操作上,只需要知道随机结构参数本身的方差(或标准差),而不是结构参数之间的相关系数矩阵,这使得计算随机结构特征值的统计学性质变得更为便利。实模态结构随机摄动方法被进一步改进并推广到复模态特征值领域。将改进的复模态结构随机摄动方法与概率理论相结合,推导出了随机复特征值方差关于随机结构参数的显式表达式。根据本文提出的方法,随机复特征值的方差能够由随机结构参数的统计量直接求出,那么相应的复模态结构的动力学重分析和不确定性分析也能够很方便地完成。通过两个数值算例表明了所提方法的计算效率和计算精度,所提方法对随机摄动方法在工程中复特征值问题的应用起到了推动作用。

Combining the first-order perturbation expansion of structural vibration eigenvalues and eigenvectors with probability theory,the variance of structural stochastic eigenvalues can be directly derived from the perturbation term of the stiffness matrix,the perturbation term of the mass matrix and the original eigenvector.The above method does not involve the correlation coefficient matrix of structural parameters in the calculation of the random eigenvalue variation range of the real modal structure,which is easy to implement in engineering practice.In terms of specific operation,only the variance(or standard deviation)of the random structure parameters itself is required,not the correlation coefficient matrix between the structure parameters,which makes it more convenient to calculate the statistical properties of the stochastic structure eigenvalues.The stochastic perturbation method of real modal structure is further improved and extended to the field of complex modal eigenvalues.Combining the improved stochastic perturbation method of complex modal structure with probability theory,the explicit expression of stochastic complex eigenvalue variance with respect to stochastic structure parameters is derived.According to the method proposed in this section,the variance of stochastic complex eigenvalues can be directly obtained from the statistics of stochastic structure parameters,so the dynamic reanalysis and uncertainty analysis of the corresponding complex modal structure can also be easily completed.Two numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method.Additionally,this method will make conspicuous contributions to the development and application of the stochastic perturbation method for complex eigenvalue problems in engineering.