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How to simulate a spherical pendulum in matlab?

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Hi,
I want to simulate a spherical pendulum in matlab. So far, I have found the equation of motion of the spherical pendulum:
But I haven't a clue how to simulate this. Any help whil be useful.
Kind regards, Bas

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Accepted Answer

Roger Stafford
Roger Stafford on 11 Feb 2015
Edited: Roger Stafford on 11 Feb 2015
Bas, you have a sign error in the gravity term. It should be:
-g*sin(phi)/L
Use one of matlab's 'ode' solvers to numerically solve these equations. Read about how to use them at:
http://www.mathworks.com/help/matlab/math/ordinary-differential-equations.html
Note that you would have numerical accuracy difficulty if the angle phi approaches near zero because of the division by sin(phi) in the first equation. At that point theta would change very rapidly. That is inherent in the physical situation as measured by the two angles.

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Bas Siebers
Bas Siebers on 25 Feb 2015
I was able to simulate the equation of motion, but as you mention I experience a rapidly changing teta when the phi approaches zero. Is there a way to get rid off this numerical accuracy difficulty?
Roger Stafford
Roger Stafford on 26 Feb 2015
Yes, there is a way. If you (carefully!) transform your coordinate system to cartesian coordinates, this problem should not occur. That will require three dependent variables, x, y, and z. A third equation will express the constraint that the distance from the support point is the value 'L'.
As I mentioned, the problem is caused by the nature of the definition of the two angles - when phi is near zero, a small motion can change theta by large amounts. That is an artificial difficulty.
Bas Siebers
Bas Siebers on 22 Apr 2015
Hi Roger,
I started again with solving this problem. Do you known how to transform my coordinates to cartesian coordinates?

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