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Using Elliptic Integrals And Functions To Study Large Amplitude Oscillations Of A Pendulum

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2007 Annual Conference & Exposition


Honolulu, Hawaii

Publication Date

June 24, 2007

Start Date

June 24, 2007

End Date

June 27, 2007



Conference Session

Applied Mathematics

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Page Count


Page Numbers

12.1543.1 - 12.1543.13



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Paper Authors


Josue Njock-Libii Indiana University-Purdue University-Fort Wayne

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Josué Njock Libii is Associate Professor of Mechanical Engineering at Indiana University-Purdue University Fort Wayne, Fort Wayne, Indiana, USA. He earned a B.S.E in Civil Engineering, an M.S.E. in Applied Mechanics, and a Ph.D. in Applied Mechanics (Fluid Mechanics) from the University of Michigan, Ann Arbor, Michigan. His areas of interests are in mechanics, particularly fluid mechanics, applied mathematics, and their applications in engineering, science, and education.

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NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

Using elliptic integrals and functions to study large-amplitude oscillations of a pendulum


The solution to the oscillations of a pendulum that includes large amplitudes is presented for the purpose of comparing it to that for small amplitudes. Such a comparison allows for the determination of the limits of applicability of the linearized equation. It is shown that, in both cases, the angle of swing is a periodic function of time but that the nature of the functions involved varies with the amplitude of motion. For small angular displacements, the period of oscillation is a constant and the ensuing angle of swing can be represented accurately by means of circular functions. However, for large amplitudes, the period is represented by Jacobi’s complete elliptic integral of the first kind and varies with the initial amplitude, while the corresponding angle of swing is represented by elliptic functions of Jacobi. It is shown that the period of the linearized motion is always smaller than, or equal to, that from the nonlinear motion. The errors induced by the linearization process are determined analytically and represented graphically. It is demonstrated that those in the magnitude and phase of swing vary with time and the initial amplitude of the pendulum. Consequently, as a general rule, it is inaccurate to use the error in the angle as an estimate of the accuracy of how well the linearized solution approximates the actual motion.

1. Introduction

The motion of a pendulum is studied in the first college physics course; and its governing differential equation is amongst the first ones that are solved in an introductory course on ordinary differential equations. This equation is encountered again and again in courses such as dynamics, controls, vibrations, and acoustics. In all these cases, however, it is linearized by assuming that the amplitude of oscillation is small. As a consequence, students do not see what happens to the oscillation of a pendulum when the amplitudes are large and the restoring force becomes nonlinear. More importantly, they do not know the limits of applicability of the linearized solution they have studied.

In this article, we present the solution to the oscillations of a pendulum that includes large amplitudes and compare the general solution to that which is valid only for small

Njock-Libii, J. (2007, June), Using Elliptic Integrals And Functions To Study Large Amplitude Oscillations Of A Pendulum Paper presented at 2007 Annual Conference & Exposition, Honolulu, Hawaii. 10.18260/1-2--1561

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