REDUCTION OF ANALYTIC IMPEDANCE FUNCTION TO LINEAR MATRIX POLYNOMIAL - EXEMPLIFIED FOR DYNAMIC CABLE STIFFNESS

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. By consideration of such functions in stiffness matrices of composed systems, however, these matrices become frequency-dependent, too - a troublesome fact, especially with regard to the eigenvalue problem which becomes nonlinear. In this paper a method for avoiding such difficulties is described: A complex analytic impedance function is reduced to two constant matrices of any desired dimension. This reduction corresponds to a mathematically performed transition from a continuum to a discrete-coordinate vibrating system. In structural dynamics applications such as for dynamic cable stiffness the two resultant matrices correspond to a static stiffness matrix and a mass matrix. In every case, these matrices can easily be considered within the scope of a linear eigenvalue problem.