Application of Differential Transformation Method to Detect Chaotic Motion in Nonlinear Forced Vibration Systems

摘要

This paper employs the differential transformation method to solve Duffing's equation for a forced vibration system with a time-dependent external force. Initially, the basic rules, of differential transformation are applied to the nonlinear vibration differential equation and its initial conditions. The initial values are then used to obtain the solution of the next time step. Applying an iterative procedure in which the solutions of one time step are used as the initial values of the next step, the differential equation is solved over the entire time domain. The solution of Duffing's equation is then obtained by applying inverse differential transformation. Time histories, phase portraits and Poincare maps are used to detect the existence of chaotic behavior. It is found that chaotic motion occurs at certain values of the excitation force amplitude. The accuracy and versatility of the proposed method is demonstrated by solving several examples of Duffing's equation with different parameter values. The results are found to be in good agreement with those obtained from the Runge-Kutta-Huta scheme.