用线化和校正法近似求解一种非谐振动

A non-harmonic oscillator treated by linearization and correction method

利用拉格朗日方程建立了单质点弹簧振子非线性振动方程,作出了回复力、势能随坐标的变化曲线以及相图.应用第一类完全椭圆积分求出了非线性振动周期的精确解.应用线化和校正法对单质点弹簧振子的周期和近似解进行了求解.利用Maple计算机绘图,分别作出了它们的周期近似解与周期精确解随振幅的变化曲线以及近似解与数值解的变化曲线.利用线化和校正法所求得的近似解与数值解比较,具有简单实用、精度高、相对误差低等优点,在求解非线性振动中具有一定的实用价值.

Through the Lagrange＇ s equation, the nonlinear vibration equation of a particle non-harmonic oscillator is established; the curves of the restoring force and potential energy varying with coordinates and phase plot are drawn. The exact solution of the period is got by using the first kind of complete elliptic integral. Using the linearization and correction methods, the approximate solutions on period and the displacement of the oscillator are solved, And the curves of the approximate solutions and exact solutions of the period varying with amplitude and the approximate solutions and the numerical solution are drawn by using the Maple 9.5. The approximate solutions have advantages of simplicity, practicality, and higher accuracy, which is of practical value in calculating nonlinear oscillations.