A multiple scale time domain collocation method for solving non-linear dynamical system

作者：Dai, Honghua^*; Yue, Xiaokui; Liu, Cheinshan

来源：International Journal of Non-Linear Mechanics, 2014, 67: 342-351.

DOI：10.1016/j.ijnonlinmec.2014.10.001

• Multiple scale time domain collocation method
• Duffing equation
• Ill-posedness
• Multiple scale differential transformation method

Recently, a simple time domain collocation (TDC) method was proposed by researchers including the first author of the present paper, and has been successfully applied to obtain the harmonic, subharmonic, and superharmonic responses of the nonlinear Duffing oscillator. The TDC method is based on the point-collocation method performing within an appropriate period of the periodic solution, wherein the approximate solution is assumed as a Fourier series. Upon using the TDC method, the ordinary differential equation is transformed into a system of non-linear algebraic equations (NAEs), which can be readily solved by an NAE solver. In this study, using the Duffing oscillator as the prototype, we develop a multiple scale time domain collocation (MSTDC) method, by introducing a series of optimal multiple scales to the Fourier series of the approximate solution, to alleviate the ill-posedness of the system of collocation-resulting NAEs due to the inclusion of very high order harmonics in the approximate solution. Besides the MSFDC method, a multiple scale differential transformation (MSDT) method is proposed by introducing the multiple scales to the classical differential transformation method. Finally, numerical experiments are carried out to verify the accuracy and efficiency of the MSTDC and the MSDT methods.

• 出版日期2014-12
• 单位西北工业大学