Eigensensitivity Analysis Of Damped Systems

The sensitivity analysis of eigenvalue problems arises in a remarkable variety of applications, such as damage detection, system identification, structural optimization, model updating and so on. Studies on the sensitivity analysis have important theoretical significance and application value. This dissertation deals with the sensitivity analysis of generalized eigenvalue problems, quadratic eigenvalue problems and nonviscously damped systems. The main contributions are as follows.Using the implicit function theorem, the mean of the defective eigenvalues and corresponding left and right eigenvector matrices of the generalized eigenvalue problems and the quadratic eigenvalue problems are proved to be analytic. The first-order partial derivatives of the left and right eigenvector matrices are derived.The governing equations for computing the derivatives of the multiple eigenpairs of symmetric quadratic eigenvalue problems are derived. The eigenvector derivatives are expressed as the sum of a particular solution to the governing equations and the general solution of the corresponding homogeneous equations. The particular solution to the governing equations is computed by using the “elastic” flexibility matrix, the generalized inverse matrix and constructing extended linear systems, respectively. Three numerical methods for computing the multiple eigenpair derivatives of the symmetric quadratic eigenvalue problems are proposed. Nnumerical results show that the proposed methods are efficient.In order to guarantee the uniqueness of the eigenvectors associated with semisimple eigenvalues of asymmetric quadratic eigenvalue problems, a new normalization condition is presented. The governing equations for computing the derivatives of the left and right eigenvectors of asymmetric quadratic eigenvalue problems are derived. The particular solution to the governing equations is constructed by using the generalized inverse matrix and solving linear equations with nonsingular coefficient matrix, respectively. Two numerical methods for computing the semisimple eigenpair derivatives of the asymmetric quadratic eigenvalue problems are provided. Nnumerical results show that the proposed methods are efficient.The governing equation for computing the derivatives of the multiple eigenpairs of symmetric nonviscously damped systems is derived. The particular solution to the governing equation is found by solving a linear equation with nonsingular coefficient matrix. A novel method is derived for the computation of the eigenpair derivatives of symmetric nonviscously damped systems with repeated eigenvalues. A numerical example is provided to illustrate the effectiveness of the proposed method.