THE VIRTUAL CHAOTIC PENDULUM

To find the equation of motion for a rotational system you just sum the torques and set them equal the the moment of interia (I) times the angular acceleration ( tex2html_wrap_inline38 ).

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For a simple pendulum this is:
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where the gravity(g) provides the only torque on the lenght(l) of the pendulum. To make a pendulum chaotic, we need to add a couple of terms. We can add an external driving force which varys with a frequency. The equations of motion would be found by summing all the torques as above:

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where tex2html_wrap_inline46

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After rearanging terms the equation of motion will look like this:

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For out purposes, we will call the coefficient of friction tex2html_wrap_inline54 and the force constant tex2html_wrap_inline56 , both of these will be input by the user. Other constants, g is set to 9.8, l is set to 9.8, m is set to 1.


Here are some of the more interesting cases
"Results" has plots and animations to view.

Period-1 limit cycle, normal pendulum.(b=0,a=0,w=0,T=100,t=0.1,xi=0.8,vi=0)Results

Period-1 limit cycle, complicated pendulum.(b=0.52,a=0.2,w=0.689,T=100,t=0.1,xi=-0.8,vi=0.1234)Results

Period-3 limit cycle, complicated pendulum.(b=0.52,a=0.2,w=0.694,T=100,t=0.1,xi=0.8,vi=0.8)Results

Period-5 limit cycle, complicated pendulum.(b=0.52,a=0.2,w=0.689,T=100,t=0.1,xi=0.8,vi=0.8)Results

Chaotic pendulum.(b=0.9,a=0.2,w=0.54,T=100,t=0.1,xi=-0.8,vi=0.1234)Results

Here you can use the parameters of your choice
Just input some parameters and click on "Create Animation". Animation will be created and played right away. Depending on the selected time and time intervals, it may take some time to create a long movie. Please be patient. We have to solve a 2nd order differential equation, create a GIF file for each time interval, combine the gifs into a animated GIF(and/or make a FLI or MPEG movie), and send the result to you.
Driving Force b:
Drag Coefficient a:
Driving Frequency w:
Time T:
Time step length t:
Initial Position xi:
Initial Velocity vi:

Select the type of animation you would prefer to view:
Gif89a Format (Animation will be displayed right away. May be slow over a slow network)
MPEG Format (A MPEG movie will be created for you to download)
FLI format (A FLI movie will be created for you to download)
NOTE: You will need your own local animation player to view a MPEG or FLI movie
(Please be patient)

Mail to Jon Maestri (Web Implementor).