How to simulate a spherical pendulum in matlab?

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

Answer 1

[Roger Stafford]

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.

[Bas Siebers]

Thank you for your answer. I have one question, why should you use a ordinary differential equation (ODE) solver when you have a differential equation that depends on two angles that are changing in time? Because of that I thought it was a partial differential equations.

[Roger Stafford]

No, there is still only one independent variable, namely t, the time, so these are ordinary differential equations. The fact that you have more than one dependent variable does not alter that fact. You have partial differential equations when there is more than one independent variable and hence you have partial derivatives.

[Bas Siebers]

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]

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]

Hi Roger,

I started again with solving this problem. Do you known how to transform my coordinates to cartesian coordinates?

Answer 2

[Youssef Khmou]

You can look for solution in this link :

http://scicomp.stackexchange.com/questions/8595/numerical-instability-of-spherical-pendulum