Bounding properties for eigenvalues of a transcendental dynamic stiffness matrix by using a quadratic matrix pencil

Abstract

An approximate representation of a transcendental dynamic stiffness matrix K(rho) by a simple quadratic matrix pencil A-rho B-rho(2)C is studied in this paper. The matrix pencil is formed by expressing the elements of K as parabolic functions based on choosing three fixed values of the eigenparameter rho. General bounds on the exact eigenvalues of the transcendental eigenvalue problem provided by the quadratic matrix pencil are shown to exist, provided that the three fixed values are below the lowest pole of the transcendental stiffness matrix considered and that the three coefficient matrices are positive definite. Numerical examples illustrate and confirm these bounding properties. Furthermore, the bounding properties are extended to constrained dynamic stiffness matrices. e.g., matrices formed by using Lagrangian multipliers to couple individual stiffness matrices of several different responses when a single response does not satisfy the desired boundary conditions.