Pendulum with a moving pivot 

Carl Madigan 

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 

 

 

 

Additional examples 

 

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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