A Simple Collocation Scheme for Obtaining the Periodic Solutions of the Duffing Equation, and its Equivalence to the High Dimensional Harmonic Balance Method: Subharmonic Oscillations

Abstract

In this study, the harmonic and 1/3 subharmonic oscillations of a single degree of freedom Duffing oscillator with large nonlinearity and large damping are investigated by using a simple point collocation method applied in the time domain over a period of the periodic solution. The relationship between the proposed collocation method and the high dimensional harmonic balance method (HDHB), proposed earlier by Thomas, Dowell, and Hall (2002), is explored. We demonstrate that the HDHB is not a kind of "harmonic balance method" but essentially a cumbersome version of the collocation method. In using the collocation method, the collocation-resulting nonlinear algebraic equations (NAEs) are solved by the Newton-Raphson method. To start the Newton iterative process, initial values for the N harmonics approximation are provided by solving the corresponding low order harmonic approximation with the aid of Mathematica. We also introduce a generating frequency (ω_g), where by the response curves are effectively obtained. Amplitude-frequency response curves for various values of damping, nonlinearity, and force amplitude are obtained and compared to show the effect of each parameter. In addition, the time Galerkin method [the Harmonic-Balance method] is applied and compared with the presently proposed collocation method. Numerical examples confirm the simplicity and effectiveness of the present collocation scheme in the time domain.