Reduction of Dynamic Cable Stiffness to Linear Matrix Polynomial

Abstract

For the dynamic stiffness of a sagging cable subject to harmonic boundary displacements, frequency‐dependent closed‐form analytic functions can be derived from the corresponding continuum equations. When considering such functions in stiffness matrices of composed structures, however, these matrices become frequency dependent, too—a troublesome fact, especially in regards to the eigenvalue problem, which becomes nonlinear. In this paper, a method for avoiding such difficulties is described whereby an analytic dynamic stiffness function is reduced to a linear matrix polynomial; the matrices of this polynomial are of any desired order. The reduction corresponds to a mathematically performed transition from a continuum to a discrete‐coordinate vibrating system. In structural dynamic applications (dynamic cable stiffness), the two resultant matrices correspond to a static stiffness matrix and a mass matrix. Beyond the particular problem focused on, the method may be applied to all kinds of analytic impedance functions. In every case, the resultant matrices can easily be considered within the scope of a linear matrix‐eigenvalue problem.