A simplified method is presented for the computation of eigenvalue and eigenvector derivatives of damped systems with repeated eigenvalues. In the proposed method, adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation whose order is (n+m)x(n+rn), where n is the number of coordinates and m is the number of multiplicity of repeated eigenvalues. One algebraic equation developed can be computed eigenvalue and eigenvector derivatives simultaneously.
Since the coefficient matrix of the proposed equation is symmetric and based on N-space, this method is very efficient compared to previous methods. Moreover the numerical stability of the method is guaranteed because the coefficient matrix of the proposed equation is non-singular.
The more large structure, the more limits are present in approximation dynamic behavior of structure using only first sensitivities of eigenpair. Therefore the second sensitivities of eigenpair become important and the proposed method is so expanded as to obtain the second derivatives of eigenvalue and eigenvector of damped systems with repeated eigenvalues.
This method can be consistently applied to both structural systems with structural design parameters and mechanical systems with lumped design parameters. To verify the effectiveness of the proposed method, the finite element model of the cantilever beam and a 5-DOF mechanical system in the case of a non-proportionally damped system are considered as numerical examples. The design parameter of the cantilever beam is its width, and that of the 5-DOF mechanical system is a spring.
