The second order D. E. approach

Math-Model (Nonlinear Pendulum)  A simple pendulum consists of a point mass  m  attached to a rod of negligible weight.   The torque    is  

(1)        ,
        
where    denotes the the angle of the rod measured downward from a vertical axis.  The moment of inertia for the point mass is    where  l  is the length of the rod.  The torque can also be expressed as  ,  where    is the angular acceleration, using Newton's second law, and the second derivative, this can be written as  

(2)        .

Equating  (1) and (2) results in the nonlinear D. E.

(3)        .

Proof Pendulum Models  Pendulum Models    

Math-Model (Linear Pendulum)  Introductory courses discuss the pendulum with small oscillations as an example of a simple harmonic oscillator.  If the angle of oscillation   is small, use the approximation    in equation (3) and obtain the familiar linear D. E. for simple harmonic motion:  

(4)        ,

Using the substitution  , the solution to (4)  is known to be

(5)        ,

which has period  .  When the solution (5) is written with a phase shift, it becomes

(6)        .

Proof Pendulum Models  Pendulum Models    

Computer Programs Pendulum Models  Pendulum Models    

Mathematica Subroutine (Runge-Kutta Method for second order D.E.'s)  To compute a numerical approximation for the solution of the initial value problem  

      with  , ,  

over the interval    at a discrete set of points.  

Example 1.  Solve the second order  I.V.P.  
      with  ,   .    
Use the Runge-Kutta method to compute the solution over the interval .  
Solution 1.

Example 2.  Find the analytic solution to the second order  I.V.P.  
      with  ,   .    
Plot the solution over the interval .  
Solution 2.

Example 3.  Solve the second order  I.V.P.  
      with  ,   .    
Solution 3.

Example 4.  Compare the graphs of the solutions to  
(i)          with  ,   ,    
    and
(ii)          with  ,   .    
Solution 4.

The systems of  D. E.'s approach

    The pendulum can also be explored using a phase curve in the phase plane.  This requires a method to solve systems of D. E.'s, and our choice will be the Runge-Kutta method.  Plotting several curves will enable us to make a phase portrait which help understand some of the subtle features of the non-linear pendulum.

Mathematica Subroutine (Runge-Kutta Method in 2D space)  To compute a numerical approximation for the solution of the initial value problem  

      with  ,  
      with  ,  

over the interval    at a discrete set of points.

Note.  The Runge-Kutta method in 2D is a "vector form" of the one-dimensional method, here the function f is replaced with F.

Example 5.   Solve   ,  by using the Runge-Kutta method to solve the system of D. E.'s  
         ,  
and
        .    
Use  g = 32  and  l = 32.
Solution 5.

Example 6.   Solve   ,  by using the Runge-Kutta method to solve the system of D. E.'s  
         ,  
and
        .    
Solution 6.

Example 7.   Solve  ,  by using the Runge-Kutta method to solve the system of D. E.'s  
         ,  
and
        .    
Solution 7.

Research Experience for Undergraduates

The Pendulum  The Pendulum  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook The Pendulum

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(c) John H. Mathews 2004