Analytical Evaluation of Dynamic Responses of Time-varying Systems

Abstract

An analytical approach for forced vibration of single-degree-of-freedom (SDOF) systems with arbitrary time-dependent parameters is presented. Unlike most of the previous studies on this topic, the function for describing the variation of mass of a SDOF system with time is an arbitrary continuous or piecewise real-valued function, and the variation of stiffness with time is expressed as a functional relation with the variation of mass and vice versa. Using appropriate functional transformation, the closed-form homogeneous solutions of the governing differential equations for forced vibrations of such a SDOF system with continuous time variable parameters (mass and stiffness) are derived for 10 important cases. The particular solutions of the non-homogeneous differential equations are determined based on the Lagrange method. Furthermore, the proposed exact method and solutions are developed to analyze the forced vibration of SDOF systems with multi-step (piecewise) non-periodical variable parameters. A numerical example illustrating the application of the proposed method shows that the proposed exact procedure is an efficient method.