NATURAL FREQUENCY AND MODE SHAPE SENSITIVITIES OF DAMPED SYSTEMS: 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 condition are used to compose an algebraic equation whose order is (n+m)×(n+m), wherenis the number of co-ordinates andmthe 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 methods 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 non-proportionally damped system. The design parameter of the cantilever beam is its height, and that of the 5-DOF mechanical system is a spring.