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.