ANN: Simple Pendulum Simulation

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ANN: Simple Pendulum Simulation

RogGuay
A simple simulation of a simple pendulum . . .


                                 revOnlin -> User Spaces -> RogerG or  
Education.



Cheers, Roger
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Re: ANN: Simple Pendulum Simulation

jim hurley-3
>
>A simple simulation of a simple pendulum . . .
>
>
>                                  revOnlin -> User Spaces -> RogerG or
>Education.
>
>
>Cheers, Roger
>

Roger,

Nice job! Good interface.

If you have the inclination, you might want to tackle the large
amplitude pendulum. There is no nice analytic solution but you could
numerically integrate the equation of motion. Something like this:

Let A represent the angle. Then you  would do a numerical integration with

repeat loop
   set the location of the pendulum to R,A --using radial coordinates
   add c *  sine(A) to the angular velocity -- where c depends on the
mass, L and  g
   --The angular acceleration is proportional to  the torque which is
proportional to sine(A)
   --For small amplitudes sine(A) = A, in radial coordinates
   add the angular velocity to A
end repeat loop

Where I have assumed the time interval between loops is one second,
so that dt =1

It would be interesting to show how the period (determined by the
number of loops between changes in sign of the angular velocity)
depends on the amplitude. Show that the clock slows down as it runs
down, i.e. the period decreases with decreasing amplitude--albeit
slowly; it is a second order effect in the amplitude. That's why
pendulum clocks work so well.

Jim


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Re: ANN: Simple Pendulum Simulation

RogGuay
In reply to this post by RogGuay
Thanks, Jim.  I do indeed have this inclination.  In fact my original  
intent was to use the simple pendulum to learn and apply the Runge-
Kutta Method.  I just haven't gotten around to it yet.  Might your  
suggestion be a variation of this?

Cheers, Roger


On Oct 31, 2005, at 2:32 AM, [hidden email]  
wrote:

> If you have the inclination, you might want to tackle the large
> amplitude pendulum. There is no nice analytic solution but you could
> numerically integrate the equation of motion. Something like this:
>
> Let A represent the angle. Then you  would do a numerical  
> integration with
>
> repeat loop
>    set the location of the pendulum to R,A --using radial coordinates
>    add c *  sine(A) to the angular velocity -- where c depends on the
> mass, L and  g
>    --The angular acceleration is proportional to  the torque which is
> proportional to sine(A)
>    --For small amplitudes sine(A) = A, in radial coordinates
>    add the angular velocity to A
> end repeat loop
>
> Where I have assumed the time interval between loops is one second,
> so that dt =1
>
> It would be interesting to show how the period (determined by the
> number of loops between changes in sign of the angular velocity)
> depends on the amplitude. Show that the clock slows down as it runs
> down, i.e. the period decreases with decreasing amplitude--albeit
> slowly; it is a second order effect in the amplitude. That's why
> pendulum clocks work so well.
>
> Jim

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Re: ANN: Simple Pendulum Simulation

jim hurley-3
In reply to this post by RogGuay
>
>Thanks, Jim.  I do indeed have this inclination.  In fact my original
>intent was to use the simple pendulum to learn and apply the Runge-
>Kutta Method.  I just haven't gotten around to it yet.  Might your
>suggestion be a variation of this?
>
>Cheers, Roger
>


Roger,

Actually I was thinking of something even simpler than the
Runge-Kutta approximation.

Using the Euler approximation, the repeat loop to generate the
pendulum motion is really simple and looks like this:

repeat until the mouseClick
     setRA r,270+psi -- Polar coordinates; 270 so that the pendulum hands DOWN
     add -c*psi to angVel --Add angular acceleration to the angular velocity
     add angVel to psi --Add angular velocity to the angle
end repeat

where psi is the angular displacement of the pendulum.

I am using Turtle Graphics, but I think you get the idea. To see this
in action, put this in the message box:

go stack url "http://home.infostations.net/jhurley/ControlGraphics.rev"

and go to the last card.

Control graphics is a variation on TG. It allows you to identify any
control as a Turtle which not only responds to Transcript, but also
to TG. So you can create a circle graphic and call it "pendulum" and
then talk to the circle like it was a turtle, i.e. forward 10, right
90, setXY 20,30, setRA 200,35 etc.

Polar coordinates are particularly  useful in the pendulum problem

I tried to show the dependence of the period on the amplitude but no
luck so far. Maybe Runga-Kutta is required.

The period depends on the amplitude (to second order in the
amplitude) in this way:

T = T(0) (1 + A^2/16)

where A is the angular amplitude in radians.

Jim
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