一类非线性jerk方程的改进两变量展开法

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振动与冲击 ›› 2012, Vol. 31 ›› Issue (23) : 118-122.

论文

一类非线性jerk方程的改进两变量展开法

郑敏毅，张农，孙光永

作者信息 +

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

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

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文章历史 +

摘要

应用改进的两变量展开法求解非线性含有三次非线性项的三阶微分方程的近似频率和近似解析周期解。该方法结合了Lindstedt-Poincare方法与两变量展开法不仅可以适用于弱非线性振动问题的求解而且还可以适用于强非线性振动问题的求解。文中以一个不含速度线性项的非线性jerk方程作为例子分析并得到二阶近似周期和二阶近似解析周期解，与数值方法给出的“精确”周期解比较，二阶近似解析周期解比一阶近似解析周期解要精确得多。结果表明，改进的两变量展开法能够适用于求解非线性jerk方程。而且在jerk方程不含速度线性项时该方法仍然有效。

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.

导出引用

郑敏毅;张农;孙光永. 一类非线性jerk方程的改进两变量展开法[J]. 振动与冲击, 2012, 31(23): 118-122

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