An efficient computational scheme for the analysis of periodic systems

Abstract

In this paper a new efficient numerical scheme for the stability analysis of linear systems with periodic parameters is suggested. The approach is based on the idea that the state vector and the periodic matrix of the system can be expanded in terms of Chebyshev polynomials over the principal period. Such an expansion reduces the original problem to a set of linear algebraic equations from which the solution in the interval of one period can be obtained. Furthermore, the technique is combined with Floquet theory to yield the transition matrix at the end of one period and provide the stability conditions via an eigen-analysis procedure. Two formulations are presented. The first formulation is suitable for a set of equations written in state space form, while the second can be applied directly to a set of second order equations. The application is demonstrated through practical illustrative examples including the parametric excitation problem of a fixed-fixed column. The numerical results thus obtained are compared with those obtained from DVERK, a Runge-Kutta code available in the IMSL library. An error-bound analysis is also included. It is concluded that the suggested schemes not only provide accurate results but are also computationally very efficient. In particular, the second formulation is found to be several times faster than the standard Runge-Kutta type codes.