Eigenproblem formulation, solution and interpretation for non-proportionally damped continuous beams

Abstract

A rationalized approach is described for formulating the eigenproblems associated with continuous beams where damping non-proportionalities are introduced through the use lumped dampers, partial damping layers or arbitrary complex impedances. An efficient algorithm is then presented for determining an exact solution to the resulting complex eigenvalue equations, and for computing the coefficients associated with the complex eigenfunctions. A normalization procedure is developed which embodies all pertinent modal information, yet presents the complex modes in a manner which approximates the instantaneous displacement patterns associated with the natural frequencies. This scheme also allows an analyst to assess qualitatively and quantitatively the extent of non-proportionality in the damping distribution and determine which parts of the system it most affects. The formulation and normalization techniques are illustrated with two example cases.