Solving a Complicated 2D Rotation Problem: Seeking Answers

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This is similar to finding the net force of a moon around and planet and the planet around the sun but in perfect circular orbits in 2d without consideration for gravitational effects.
Does anyone have a solution? If not there's a complicated paper that could be simplified from NASA in the 1930s out there. Or some parabolic equations with conic sections and calculus and vectors could be used.

The exact details for this is that a shaft has a fixed center on one end and the other end is a center of rotation for another shaft which rotates at the same speed but relative to the inner shaft. So at theta=0, they are both pointing the same direction with maximum radius. At theta=pi the farther rotation will be pointing to the center and will have the shortest radius. The mass is at the end of the outer shaft, assuming no weight for the shafts.

I'm not sure, but I think there are 2 sources for acceleration to find force. One is the normal force from rotation and the other is the relative acceleration caused by the outer shaft.

Any help would be appreciated, thanks.

 

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So I found the equations for a parametric form equation of the motion of the system:
x=cos t + cos t/2
y=sin t + sin t/2

the 2nd derivative of each gives

x''=-sin(t)-sin(t/2)/4
y''=-cos(t)-cos(t/2)/4

the curve by inspection seems to have symmetry and average out on all angles so no net force should be caused by the system.

Did I do this right? and does this make sense or intuitive?