Analytical approximations for the period of a nonlinear pendulum

This paper deals with the non-linear oscillation of a simple pendulum and presents an approach for solving the non-linear differential equation that governs its movement by using the harmonic balance method. With this technique it is possible to easily obtain analytical approximate formulas for the period of the pendulum. As we shall see, these formulas show excellent agreement with the exact period calculated with the use of elliptical integrals, and they are valid for both small and large amplitudes of oscillation. The most significant feature of the treatment presented is its simplicity because for the level of approximation considered in this paper the required work can be done “by hand”.