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# Pendulum with a moving pivot

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Pendulum with a moving pivot

Nova Scotia Agricultural College

Truro, N.S.  B2N 5E3

Introduction

The problem being considered is a non-linear pendulum where the point of suspension is moving.  Damping is ignored but can easily be included.

If  X represents the horizontal and Y the vertical components of the motion of the pivot then by resloving the accelerations along the pendulum we have

where g is the acceleration due to gravity.

the pendulum's

Setting F = ma and equating

elliminating T from these two equations we have

simplifying

We will assume L =1 and consider three types of motion for the pivot      a)  horizontally

b)  vertically       X = 0

c)  circular

In the examples below XX(t) and YY(t) are used to define the pivot

Example 1    Horizontal motion of the pivot

Plot of the positon wrt time

animation of the motion for this pendulum

Example 2     Vertical motion

Plots of the position with respect to time and also of the phase plane for this example

animation of the motion of this pendulum.

Example 3    Circular motion

plots of the position with respect to time and of the phase plane

animation of the motion for this pendulum

A Procedure for drawing the pendulums

The procedure  is called drad and has the following  imputs

L = length ,

angl = inital displacement angle,

vel = initial velocity

a =  the x component of the path for the pivot

b =  the y component of the pivot's path

n = the number of iterations used to draw the annimations

examples   for horizontal motion try  a =

for vertical motion   try    a = 0    and b =

for circular motion   try    a = and b = sin(3t - π/2)

experiment with other paths  such as  a = sin(2.25t - π/2)  b = cos(2t + π/4)  etc  try varying the lenght and the initial conditions.  Have Fun with it!!!

Example 1.  pivot moving along a curve

Example 1   horizontal motion of the pivot

Example 2 vertical motion of the pivot

Example 3 cicular motion of the pivot

Drawing the pendulums

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