Natural frequency and mode shape sensitivities of damped systems: Part II,Multiple natural frequencies

An efficient algorithm with proven numerical stability is derived for compu tation of eigenvalue and eigenvector derivatives of damped vibratory system s 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 deri vatives of the damped systems can be obtained by solving the algebraic equa tion. The method can be consistently applied to both structural systems wit h structural design parameters and mechanical systems with lumped design pa rameters. As an example of a structural system to demonstrate the theory of the proposed method and its possibilities in the case of multiple eigenval ues, the finite element model of the cantilever beam is considered, and als o a 5-DOF mechanical system in the case of a nonproportionally damped syste m. The design parameter of the cantilever beam is its height, and that of t he 5-DOF mechanical system is a spring. (C) 1999 Academic Press.