Approximate periodic solution for the large-amplitude oscillations of a simple pendulum

Abstract

This paper presents approximate periodic solutions to the anharmonic (i.e. not harmonic or non-sinusoidal) response of a simple pendulum undergoing moderate- to large-amplitude oscillations. The approximate solutions were derived by using a modified continuous piecewise linearization method that enabled very accurate solutions to the pendulum oscillations for the entire range of possible amplitudes i.e. $0<{\theta }_0<180°$ . The present solution method is very simple and can be used to obtain amplitude-frequency solutions as well as the displacement and velocity histories of the simple pendulum without the need for a complementary method. The purpose of this paper is to present simple and accurate approximate analytical solutions to the large-amplitude oscillations of the simple pendulum that can be applied by undergraduates.