单摆大振幅振动的解析逼近解

构造了单摆大振幅振动的高精度解析逼近周期和周期解.首先,利用Maclaurin展开和Chebyshev多项式加速技术将单摆振动方程中的正弦型恢复力用三次多项式近似代替,得到一个Duffing型方程;然后,将牛顿法与谐波平衡法结合起来建立Duffing方程的解析逼近周期及周期解,从而给出单摆振动的解析逼近解.因此,在求解过程中避免了关于参数的非线性代数方程组的出现,只需解线性代数方程组就能建立单摆振动的解析逼近周期及周期解.几乎在振幅(初始摆角)的全部取值范围内,与数值方法给出"精确"周期及周期解比较,得到的解析逼近解都有很高的逼近精度.

Analytical Approximate Solutions to Large Amplitude Oscillation of a Simple Pendulum

Analytical approximate periods and periodic solutions to large amplitude oscillation of a simple pendulum are constructed using a new method.Firstly,the function sinx that appears in the simple pendulum equation is replaced with a polynomial of degree 3 using Maclaurin series expansion and Chebyshev polynomial approximation.Subsequently,the resulting equation which is a Duffing type equation is approximately solved with combing Newton's method and the harmonic balance method.It yields simple linear algebraic equations instead of nonlinear algebraic equations withno analytical solution.The new analytical approximate periods and periodic solutions show excellent agreement with the numerically exact solutions,and they are valid over almost the whole allowable range of oscillation amplitudes.