Trying to find the equation of motion of a model

[Broseph]

Homework Statement

I need to derive an equation of motion for the system modeled in this java app:
http://web.whittier.edu/physweb/lagan/250/ejs_ehu_oscillations_cylinders.jar

I was told to treat the weight force as if it was acting only at the center of mass of the rod, and that normal forces and friction forces are crucial to the problem. I have been playing around with this forever though!

μ is the coefficient of kinetic friction, a is half the distance between the circles.

Homework Equations

F=ma, ƩF_x=0, Ʃτ=0

The Attempt at a Solution

Pages and pages of manipulations :p
I tried to mainpulate things to get it to look the way the mass on a spring system looks, so I could say F=C*x, and then just use the fact that f=ma=mx''.
It didn't seem to want to work out that way. I am having trouble finding an equation the relates the normal forces to the position of the center of mass at time t. ( the x(t) ). I now that when the displacement of the center of mass from the resting position is zero, then the normal forces are equal. i think that the normal force on the left side = mg/2 * (a-x(t))/a and the right side = mg/2 * (a+x(t))/a. (remember a is half the distance between the cylinders in the app)

 

[Spinnor]

You should be able to write down the net force on the rod as a function of the displacement of the center of mass from equilibrium. With that expression it should be down hill from there.

A first guess is the force on the rod is proportional to the displacement? Complications might arise if the rotating cylinders were rotating slow enough to the point that friction force on one cylinder changed direction?