On The Response of Linear Time-Periodic Systems Subjected to Deterministic and Stochastic Excitations

Abstract

In many situations, engineering systems modeled by a set of linear, second-order differential equations, with periodic damping and stiffness matrices, are subjected to external excitations. It has been shown that the fundamental solution matrix for such systems can be efficiently computed using a Chebyshev polynomial series solution technique. Further, it is shown that the Liapunov-Floquet transformation matrix associated with the system can be computed, and the original time-periodic system can be put into a time- invariant form. In this paper, these techniques are applied in finding the transient response of periodic systems subjected to deterministic and stochastic forces. Two formulations are presented. In the first formulation, the response of the original system is computed directly. In the second formulation, first the original system is transformed to a time-invariant form, and then the response is found by determining the response of the time-invariant system. Both formulations use the convolution integral to form an expression for the response. This expression can be evaluated numerically, symbolically, or through a Chebyshev polynomial expansion technique. Results for some time-invariant and periodic systems are included, as illustrative examples.