@Article{cmes.2012.084.459,
AUTHOR = {Hong-Hua  Dai, Matt  Schnoor, Satya N.  Atluri},
TITLE = {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},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {84},
YEAR = {2012},
NUMBER = {5},
PAGES = {459--498},
URL = {http://www.techscience.com/CMES/v84n5/25825},
ISSN = {1526-1506},
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. <i>We demonstrate that the HDHB is not a kind of "harmonic balance method" but essentially a cumbersome version of the collocation method</i>. 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 <i>N</i> harmonics approximation are provided by solving the corresponding low order harmonic approximation with the aid of Mathematica. We also introduce a generating frequency (ω<sub>g</sub>), 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.},
DOI = {10.3970/cmes.2012.084.459}
}