A Computational Technique for Free Vibration Response of a 2-DOF Parametric System

Abstract

In this paper, a closed-form solution is presented for the free vibration response of a 2-DOF parametric system by a technique with special trigonometric series. The technique extends the concept of modulation feedback to the problem of multiple degrees of freedom (M-DOF) parametric system using an equivalent dynamic system. The characteristic equation is derived by the harmonic balance in a parametric equation with the principal oscillation frequencies computed. The computation of modes and splitting modes is carried out by operation on a mixed matrix. The arbitrary constants in the general solution are determined by the initial conditions. A real-life example of a coupled inverted-pendulum system is provided to illustrate the applicability of the proposed approach. An approaching error is adopted to assess the computational errors both using the proposed technique and the Runge–Kutta algorithm. Of interest is that the time-periodic coefficient results in the changeability of the mode in the 2-DOF parametric system, while exciting the splitting mode in the harmonic combination resonance. The proposed technique is numerically demonstrated to possess not only high approaching accuracy, but is also capable of revealing some new phenomena in the parametric system, such as characteristics of mode and slitting mode, etc. It proves to be useful for future research and engineering applications of parametric systems.