非线性保守系统周期运动的Hermite插值解法

摘要

Based on the Hermite interpolation,a highly accurate method was proposed for the periodic motion of nonlinear conservative systems.It is shown that a Hermite interpolation solution for a dynamical system can be obtained by transforming the independent time variable to a vibration time.The corresponding transformed differ-ential equation becomes well-conditioned for a solution by the Hermite interpolation method (HIM).The conver-gence and accuracy of the proposed HIMis superior to the traditional Qaisi’s power series method for Hermite in-terpolation using the information of two points instead of one point.By a way of illustration,the approximate gen-eral solutions of a class of nonlinear oscillators were derived by the HIM.The results show that the approximate general solutions can be used in the analysis on the vibration characteristics of oscillators with high accuracy.%提出了非线性保守系统周期运动的Hermite插值解法。该方法首先将时间转换为周期运动时间，由此系统的微分方程变为适用于Hermite插值的形式。与Qaisi提出的传统幂级数法不同，采用两点Hermite插值函数代替一点幂级数展开，保证了求解的收敛性及精度。使用Hermite插值解法给出了一类非线性振子的近似通解。研究表明，该近似通解不但可用于进一步分析振子的振动特性，且具有较高精度。