带三次恢复力项频率依赖于速度的非线性振子研究

带三次恢复力项频率依赖于速度（Velocity-Dependent-Frequency, VDF）的非线性振子 的周期及其性质目前没有文献讨论，且使用传统的摄动法或谐波平衡法求解这类振子一阶近似解时往往失效。特别的，其频率在有限的幅值范围内奇异。首先求得了该振子周期的积分表达式，基于积分表达式可积性条件采用谐波平衡法得到了该振子一类初始条件下的精确解；研究了该振子的周期性质，给出了由第一类完全椭圆积分表示的周期-振幅的近似解析表达式，分析了振子的方波现象及产生原因。研究表明，振子周期最终随着幅值的增大衰减到0；振子方波现象产生原因是由于系统参数 ，随着幅值的增大，方波现象更明显。最后提出使用一种Hermite插值法求解该振子的周期解，该方法将时间变量转换为新的谐振时间变量，其频率为振子频率的一半，对应的控制微分方程转变为适合于Hermite插值分析的形式，其解与数值解的对比证明了该方法的有效性。

A study of a velocity-dependent-frequency nonlinear oscillator with cubic restoring force

The period and its property of a Velocity-Dependent-Frequency nonlinear oscillator with cubic restoring force has not been discussed in literatures, what’s more, the traditional standard methods, i.e. perturbation methods or harmonic balance methods for determining approximations to the periodic solutions for this kind of oscillators may breakdown in their corresponding first-order calculations. In particular, the frequencies become singular for finite values of the amplitude. This paper firstly gives the exact period expression of this oscillator; based on the integrability condition of period integral expression the exact solution of special class of initial conditions for the oscillator is obtained by using harmonic balance method. The property of the period of the oscillator is studied, and the approximate analytical expression of the period is given by the complete elliptic integral of the first kind, and then the phenomenon and reason of somewhat like the square waves of the oscillation are analyzed. It shows that the period of the oscillator decays to zero finally when the amplitude increases to infinity, and the reason of square waves is the parameter of the system , the square waves will be squarer when the amplitude increases. At last, a Hermite interpolation method is presented for the periodic solution of the oscillator, the time variable is transformed into a new harmonically oscillating time of which the frequency is one-half the one of the oscillator, with the corresponding governing differential equation transforms into a form suitable for Hermite interpolation analysis. The solutions are compared with the numerical solution and results show good agreement.