The Spring Pendulum  

Prof. C. Madigan
Nova Scotia Agricultural College
Truro, N.S.    B2N 5E3
cmadigan@nsac.ca 

The problem being considered is a  pendulum attached to a spring.  (Damping is ignored but can easily be included.) 

Initialization 

 

Theory 

Part 1.     A  pendulum of mass m having a variable length 

The problem will be broken up into two parts.   

We begin by modelling a pendulum of mass m having a variable length 

 

                                             

Taking the origin at the point of suspension we have the following: 

  

               Pendulum's position is OP = [ Lsin(θ), -Lcos(θ) ] 

                                  velocity = OP' = [Lcos(θ)θ'+  L'sin(θ), Lsin(θ)θ'-L'cos(θ) ] 

                            acceleration = OP'' = [                                                                                                                                    

                                   Also  F = [ -Tsin(θ), Tcos(θ)-mg ] 

Setting F=ma and equating  we obtain 

                 

 

               (1) 

                         (2)  

Elliminating T from these equations and simplifying we obtain the following result 

                                                                                                                       (3) 

One can add a damping term if one wishes.  Since the variation in L will be oscillatory we assume that 

                                 Where      

 

Also the natural frequency is ω = the phase angle δ controls where in the swing the maximum length occurs. 

Example 1. 

 

 

 

(2.1.1.1)

 

 

(2.1.1.2)

 

 

(2.1.1.3)

 

 

 

 

 

Part 2.    The spring pendulum 

 

 

We now consider the Spring Pendulum  In this case the mass m is at one end of a spring and the other is attached to a fixed point of suspension.  The spring remains straight, and the tension in it obeys Hooke's law; the spring constant is k.  The unstretched spring has a length  and its length at time t is L(t).    

 

                                         

Again resolving the forces perendicular to and parallel to the pendulum we obtain the following system of equations 

 

                                                           (1) 

                                      (2)                    

For the second equation you need to determine the acceleration along the spring.  ( Recall: accel vector dot unit position vector)  

 

Example 1   Set  k/m = 4, have as initial values L(0) = 3 ,D(L)(0) = 1,  θ(0) = 0.1 and D(θ)(0) = 0.1 

 

 

 

 

(2.2.1.1)

 

 

(2.2.1.2)

 

 

(2.2.1.3)

 

 

(2.2.1.4)

 

 

(2.2.1.5)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Procedure to draw the spring pendulum having the following as inputs 

 

                    the spring-mass ratio  (k/m)        R 

                    unstretched spring length             L1 

                    Initial length of the spring     L(0)  =  L 

                    Initial velocity for the spring  D(L)(0) = dL 

                    Initial angle of the pendulum    θ(0)  = a 

                    Initial velocity of the pendulum  D(θ)(0) = b  

                    number of frames to use in the animation   n 

 

NOTE Since the procedure, as written, uses several of the same labels as used above it is best to restart before executing the procedure. 

 

Enjoy playing with the spring pendulum!!! 

 

 

 

 

 

 

 

 

procedure to draw the phase planes. 

 

The first procedure draws the phase planes for both the pendulum aand the spring.  The second one does an animation of these drawings and takes some time to do the plots. 

 

 

 

 

 

 

 

 

 

 

 

 

 

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