Modal analysis of the steady state response of a driven periodic linear system

Abstract

Abstract A modal analysis method is developed that predicts the steady state response of discrete linear systems that are governed by systems of ordinary differential equations with periodic coefficients. The systems are excited both parametrically by periodic coefficients and directly by inhomogeneous forcing terms that have the same period. In the method, the solution for the steady state vibration is expressed as a linear combination of the Floquet eigenvectors, which are orthogonal with respect to solutions of the associated adjoint problem. The approach is applicable either when the Floquet eigensolutions have been obtained numerically through the transition matrix, or when they are found analytically through perturbation methods. For this reason, implementation of the present modal analysis solution can be computationally efficient, even if the dimension of the system is large. The approach is demonstrated by an application to Mathieu’s equation.