【摘要】 应用微扰分析法,对大角度单摆的非线性方程进行求解,得到单摆的微扰解,应用Matlab软件,模拟非线性方程的微扰解,画出任意摆角下的微扰系数曲线,结果表明,当摆动角小于5°,运动方程的解只需近似到基波项,当摆动角小于30°大于5°,基波和三次谐波可以精确表达单摆运动,当摆动角小于90°大于30°,单摆的微扰解包含基波、三次谐波和五次谐波项。最后将二阶微扰方法得到的理论值与用相平面法、线性内插法、叠代法等多个方法得到的解析解进行比较,结果表明,二阶微扰法在任意摆角下所得的偏差最小,其与精确解的最小二乘法偏差为2. 724×10^-4,该值相比其他文献结果小很多。更多还原

【Abstract】 The perturbation method is used to find the solution for the nonlinear equation of the simple pendulum moving back and forth at a large angle. The software Matlab is adopted to simulate the perturbation solution of the nonlinear equation and to draw the curve line of the perturbation coefficient at all of the possible angles. The results show that the solution can be approximated up to the fundamental wave item when the angle is less than 5°. When the angle is greater than 5° and less than 30°,the third harmonic wave should also be used. The fifth harmonic wave should also be involved when the angle is greater than 30° and less than 90°. Finally,the solutions obtained by the proposed method are compared to the results obtained by phase plane method,linear interpolation method,iterative method,etc. The least square deviation obtained by using the second order perturbation method compared with the exactly solution is 2. 724 × 10^-4,which is nearly the smallest one among the solutions of other papers.更多还原