Differential Analysis on A Simple Pendulum System

Abstract

Given a system of a simple pendulum, we analyze the solution that is obtained from the nonlinear second-order differential equation that characterizes the pendulum and compare it to the corresponding small-signal linear approximation of the nonlinear system. In addition, the properties that are present for the simple pendulum system like the exact period of the pendulum system is derived and compared to that of the small-signal linearization approximate system. The time per unit angular displacement for the pendulum system is proposed, derived, and discussed. During this process, an approach is presented to compute the incomplete elliptic integral of the first kind. Finally, the concept of the daily average periodicity of the pendulum is proposed and several considerations are provided in regulating the daily average periodicity.