Take-Home Problem 8--A Non-Linear Pendulum Problem

Using the non-linear model for the oscillating pendulum, suppose there is an external force acting on the weight at the end of the rod that is always in the direction tangent to the motion of the weight and that the magnitude of this force is given by

F = ml(2sin(cos(t))-cos(t)) where m is the mass at the end of the rod and l is the length of the rod.  We are neglecting the mass of the rod and damping.  Using a gravitational constant of g = 32 and a rod of length l = 16 feet our governing differential equation becomes (with t = time in seconds):

Suppose the mass at the end of the rod is pulled back one radian in the positive (counterclockwise) direction and released (no initial velocity).  In this case the governing differential equation with initial conditions would be

Your task is to find the first 6 terms in a series solution to this differential equation and use this to approximate the value of theta when t = 2 seconds.  Graph your six term series solution over the interval [ 0 , 2pi ].  You may use any means at your disposal including Maple.  Here is a link to the development of the homogeneous non-linear pendulum model.  Here is a link to a Maple Worksheet investigating a series solution to a non-linear second order differential equation and here is a link to a second Maple Worksheet (whose equation could be interpreted as modeling a non-linear pendulum problem with an external force and damping).

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats