Simple but accurate periodic solutions for the nonlinear pendulum equation

Abstract

Despite its elementary structure, the simple pendulum oscillations are described by a nonlinear differential equation whose exact solution for the angular displacement from vertical as a function of time cannot be expressed in terms of an elementary function, so either a numerical treatment or some analytical approximation is ultimately demanded. Such solutions have been thoroughly investigated due to the abundance of distinct pendular systems in nature and, more recently, due to the availability of automatic data acquisition systems in undergraduate laboratories. However, it is well-known that numerical solutions to differential equations usually loose accuracy (due to accumulation of roundoff errors) and polynomial approximations diverge after long time intervals. In this work, I take a few terms of the Fourier series expansion of the elliptic function sn(u; k) as a source of accurate periodic solutions for the pendulum equation. Interestingly, these approximations remain accurate for arbitrarily long time intervals, even for large amplitudes, which shows its adequacy for the analysis of experimental data gathered in classical mechanics classes.

Periodic Solution of Nonlinear Conservative Systems

“…(13) is expressed in terms of elliptic functions. The displacement and natural frequency are given as [58]:…”

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

“…However, the elliptic integral of the first kind and the Jacobi sn function have no closed-form solution in terms of elementary functions and are normally evaluated numerically or in series form. The series form equivalent of equations (2) has been derived as 6…”

“…The simple pendulum is very useful for introducing nonlinear concepts (i.e. phenomena, modelling and solution) to undergraduate students and has been used to investigate the response of mechanical systems such as pendulum clock, acoustic vibrations, 2 optically torque nanorods 2 and swings on children playground. One of the main applications of simple pendulum in undergraduate laboratory is to measure the local acceleration due to gravity.…”

“…Exact analytical solutions to equation (1) are not available in terms of elementary functions but only in terms of elliptic functions. The exact time period can be derived in terms of the complete elliptic integral of the first kind as 2 where T0=2π/ω0,, k=sin (θ0/2) and K(k) is the complete elliptic integral of the first kind given as …”

“…7 Consequently, numerous studies have been carried out to formulate approximate analytical solutions to the undamped oscillations of the simple pendulum (see refs. 1,2,6–13 and many of the cited articles in these references). Most studies concentrate on finding approximate analytical expressions for the time period in terms of the oscillation amplitude, 6–8,10,11 while a few other studies 1,2,9,12,13 have attempted to find approximate solutions for the oscillation histories as well.…”