Mathematical modeling of the deflection of a rod

A rod is radially mounted on a motor shaft and provided with a flat end plate. By turning movement, the end plate will touch a pointed object, and the angle turned is measured.By the torque of the motor during acceleration and deceleration, a vibration and deflection happens on the rod. The model shows the deflection depending on temporal torque and angle.

This deflection will affect the angle measurement . So the idea is to find the mathematical formulae for the deflection and vibration, account it as the error and minimize it using controller(Simulink model or so).
This system is a higher order system(parameters like acceleration, length, torque, angle, etc) and for me it looks a very complicated task but interesting.

Can someone guide me to start the modeling. Can an FEM analysis help me with understanding the behaviour of vibration and deflection?
Even after FEM how do/ where do I start the modeling task?
Can anyone suggest me/ tip me some similar mathematical modeling examples or even a suitable literature so that I can learn more?