Help getting started with a numerical solution

[davidwinth]

Hello,

I am trying to refresh my knowledge, and so I dug out my copy of Marion and Thornton to look through. I came across an example problem, 10.2, that involves a hockey puck sliding on a flat, frictionless and rotating surface. The example problem shows some solution plots for the puck position as a function of time that were found numerically, and so I tried to code this up to see if I could get the same answers but I am stuck on how to actually code this. The puck starts out at a certain location on the rotating merry-go-round and is given an initial velocity. The idea is to find the subsequent motion.

In the example, the authors give the effective acceleration as measured by an observer on the surface as,

$a_{eff} = -\omega \times (\omega \times r) - 2\omega \times v_r$

where $v_r$ is the velocity as measured in the rotating frame and $\omega$ is constant. Performing the cross product and breaking this into Cartesian components, I get

$a_{eff} =\langle 2 \omega v_{r_y} + r \omega^2cos(\theta),-2 \omega v_{r_x} + r \omega^2sin(\theta),0 \rangle$

Can this be solved with a 2D Euler method? I have never used such a method, so before I try to figure that out I want to see if anyone knows a simpler numerical scheme to solve this, or if I am over complicating things. I think the $v_{r_x}$ term is going to be complicated to evaluate because $\theta$ is a function of time too.

Thanks for any input!

 

[mfb]

There is an analytic solution - quite easy to find if you start in a non-rotating frame (assuming the total rotation angle until time t has an analytic expression).

You can solve it numerically, of course. Everything will work, the accuracy depends on the method and step size.

 

[davidwinth]

Thanks, mfb. I don't see how there is an analytic solution possible, but I will take your word for it. Any insight into which numerical scheme I should look into for this?

 

[mfb]

Find an analytic solution in Cartesian coordinates (trivial without friction), convert it to non-rotating polar coordinates, convert it to the rotating polar coordinates. Or do I miss something?

Any insight into which numerical scheme I should look into for this?

That depends on what you want to do with the result.

 

[davidwinth]

I was looking to refresh my skills. It has been a long while since I wrote numerical solutions to problems like these and I thought this would be relatively simple. I am starting to think otherwise!

If anyone else has any input, I would like to hear that too. The way I see it, the task is to find the position as a function of time starting with the above acceleration as a function of position and velocity (it is not a constant).

Thanks.

 

[davidwinth]

After a little more work on this, I came up with the following coupled 2nd order equations.

$\frac{d^2x}{dt^2} = \omega^2 x + 2\omega\frac{dy}{dt}$
$\frac{d^2y}{dt^2} = \omega^2 x - 2\omega\frac{dx}{dt}$

Do I need to break this into 4 coupled 1st order equations, or is there another approach? Don't be shy! If you have a suggestion, please make it.

 

[mfb]

My suggestion was in post #4 above the quote.

You can work in the rotating coordinate frame (2 higher order equations or 4 1st order equations will equally work), it just makes everything much more complicated.