Natural Frequency and Mode Shape Sensitivities of Damped System, Part II: Multiple Natural Frequencies

Abstract

An efficient algorithm with proven numerical stability is derived for computation of eigenvalue and eigenvector derivatives of damped vibratory systems with multiple eigenvalues. In the proposed method, adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation whose order is (n + m) x (n + m), where n is the number of coordinates and m the number of multiplicity of a multiple natural frequency. The mode shape derivatives of the damped systems can be obtained by solving the algebraic equation. The method can be consistently applied to both structural systems with structural design parameters and mechanical systems with lumped design parameters. As an example of a structural system to demonstrate the theory of the proposed method and its possibilities in the case of multiple eigenvalues, the finite element model of the cantilever beam is considered, and also a 5-DOF mechanical system in the case of a nonproportionally damped system. The design parameter of the cantilever beam is its height, and that of the 5-DOF mechanical system is a spring. (C) 1999 Academic Press.