Thanks! What I'm trying to do is develop an equation that will give me the torque the beam exerts on object to which it is mounted in the cantilever setup.
It's easy enough to make this a time dependent function by making P a sinusoid, and thus, the acceleration isn't hard to get after that. This function will then give me the acceleration at any position along the beam for a given time t. From there I need inertia, which could be modeled as a point mass in orbit around a central location (in this case the mounting point of the beam to the rigid structure).
My problem is that this equation gives the deflection in rectangular coordinates, I need to convert to polar so that I can eventually arrive at torque provided by each infinitely small portion of the beam. You're saying that Hooke's law keeps the beam the same length, I can see and agree with that, but will the individual infinitely small portions of the beam not change their distance from the mounting position if the beam deflects and remains the same length? If the beam curls, it seems like the tip would get closer to the mounting position.
If they don't, I suppose at an instance in time the relative motion of the individual particles of the beam could still be modeled in rectangular coordinates, since the subsequent torque provided by each is due to the tangential motion of that infinitely small portion (along the length axis since I keep mentioning it).
End game is to get the torque provided by the beam on it's mounting position, if you could imagine an axis coming out of the page where beam meets wall/rigid body.