摄动力方法在模态扩展中的应用讨论

Application discussion of perturbed force approach in modal expansion

摄动力方法是一种模态振型扩展方法。首先,从摄动方程出发重新推导了模态扩展矩阵。在测试模态频率与有限元模态频率相等时会有矩阵奇异问题,这可以通过一个较小的移轴量消除奇异,从而简化扩展矩阵的表达。其次,实际应用中为了避免计算满阵,可以展开伪逆,得到一个更实用的模态扩展矩阵。接着,根据伪逆展开结果,给出了这种方法和SEREP方法等效的条件。最后,针对有限元模态基选择问题,讨论了如何挑选低阶振型以及如何确定扩展基振型数目,并以一个具体的测试案例演示了整个分析流程。

The perturbed force approach is a modal shape expansion method,it can also be derived from the perturbed eigenvalue equation.Here,the simplification of this method was discussed when tested eigenvalues were very much close to FE model eigenvalues to cause a singularity problem of eigen-matrix.Again using perturbation analysis,an arbitrary small shift was introduced to eliminate the matrix singularity and simplify the modal expansion matrix expressing.To avoid computation of a full matrix in actual application,the expansion of the generalized inverse of a matrix could be used to obtain a more efficient modal expansion scheme.Through implementing the generalized inverse expansion,the necessary conditions for the perturbed force approach being equivalent to SEREP method were gained.Finally,some criteria for FE modal base selection were proposed to choose lower order modal shapes and determine the number of modal expansion bases,a notched beam modal test demo was conducted to validate the discussion.