I am a ME student trying to understand this system pictured below. I have a mass sliding on a surface with a coefficient of friction [u], attached by inextensible massless rope to a tension spring of constant [k], and also attached to a flywheel with a moment of inertia [I] which spins as the mass slides x+ and acts as a damper. The forcing function F(t) is of the form Ax^2. The mass is confined to only slide between two stops and the spring is already preloaded under some tension (i will define this as it is stretched by [s]). Essentially a small burst pushes the mass x+ for some distance, damped by friction and the flywheel and then returns to x(0). I would like to assume it is a slightly underdamped system, where it impacts the end stops with some non-zero velocity, and the stops sort of truncate the natural travel. I would imagine that this problem is best solved by using a work-energy balance, but it looks like it fits a second order DE system well. Could someone provide me some guidance on how to solve for the travel function x(t) for this?
I would like to add that I came up with this problem and it is not a homework assignment I just want to cheat on, I drew the diagrams and would just like to get a better understanding of my basic physics.
Below is the block diagram and a graph showing the assumed behavior: