Free Response Approach for the Second Order Partial Differential System with Time Periodic Coefficient

Abstract In this paper, a new method to approach the free response of the second order partial differential system with time periodic coefficient is presented with a special form of a trigonometric series with mode functions, and it can be expressed in a closed-form solution. As a result that the partial differential equation with time periodic coefficient can be transformed into a linear algebra equations, from which a characteristic equation can be obtained. Then, complex oscillation frequency and all harmonic coefficients can be computed. All arbitrary constants in a general solution can be obtained with initial conditions. Investigations show that the proposed approach not only computes free vibration with multiple modes with higher accuracy, but also it is suitable to analyze vibration instability. So this study is valuable for vibration analysis in an elastic continuum with time periodic coefficient.