从摆的非线性振动方程的派生性质求它的运动

EVALUATION OF THE NONLINEAR VIBRATION MOTION FOR A PENDULUM FROM ITS DERIVATIVE PROPERTIES

探讨了摆的非线性振动方程的新解法.由此方程和初始条件着手,可推导出一系列派生性质,它们包括:最大位移,最大速度,初始加速度和相平面上的相轨线.把近似运动表成Fourier级数的形式,其中圆周频率也是待定的.令近似运动满足这些派生性质,便可以定出待定的Fourier系数和圆周频率.文中提出了4参数法和5参数法,即:4个或5个待定的Fourier系数和圆周频率.分析计算表明,4参数法己有较高的精度,5参数法的结果己和精确解相差甚微.

This paper suggested a novel solution for the non-linear vibration equation of a pendulum. From the relevant differential equation and the initial condition for the problem, there are some derivative properties, which include the maximum displacement, the maximum velocity, the initial acceleration and the trajectory on the phase plane. The studied approximation motion for a pendulum was expressed in the form of Fourier series, in which the circular frequency was also an undetermined value. Let the approximate motion to be close to those derivative properties, the involved Fourier coefficients as well as the circular frequency can be evaluated, in which the four-parameter method and five-parameter method are used. It is found that the results obtained from the four-parameter method have a high accuracy, and that the results obtained from five-parameter method has a very high accuracy.