A modified two-variable expansion method for a nonlinear jerk equation

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Journal of Vibration and Shock ›› 2012, Vol. 31 ›› Issue (23) : 118-122.

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A modified two-variable expansion method for a nonlinear jerk equation

ZHENG Min-yi， ZHANG Nong, SUN Guan-yong

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Abstract

A Modified two-variable expansion method is applied to determine approximate periods and analytical approximate periodic solutions of a third-order differential equation with cubic nonlinearities. This method combining the Lindstedt-Poincare techniques and the two-variable expansion method is not only valid for weakly nonlinear oscillations but also valid for strongly nonlinear oscillations. In this paper, a nonlinear jerk equation excluding linear part of velocity term as an example is calculated. The second-order approximate period and the second-order analytical approximate periodic solution are obtained. A comparison of the first and second order analytical approximate periodic solutions with the numerically exact solutions shows that the second order analytical approximate periodic solution is much more accurate than the first one. The result shows that the modified two-variable expansion method could be suitable for the calculation of the nonlinear jerk equation. Moreover, when the jerk equation doesn’t have the linear part of velocity term this method still works.

Key words

Nonlinear jerk equation / approximate periodic solution / Perturbation / Multiple Scales method / two-variable expansion method

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ZHENG Min-yi;ZHANG Nong; SUN Guan-yong. A modified two-variable expansion method for a nonlinear jerk equation[J]. Journal of Vibration and Shock, 2012, 31(23): 118-122