$1.$ A prototype $2-$stroke free piston engine is shown in FIGURE Q$1.$ The coefficient of friction of the linear support bearing is $0\mathrm{.}05$ and the total mass of the mover unit is $6$ kg $.$ The total stroke of the engine between bottom dead centre $($BDC$)$ and top dead centre $($TDC$)$ is $60$ mm $,$ diameter of the combustion chamber is $50$ mm and the maximum mean effective pressure $($MEP$)$ of the engine is $12$ bar. in order to finalize the prototype, a starting model of the engine is required.

a$.$ Assuming that there is no combustion for the first few cycles during engine starting process, derive the equation of motion and the transfer function for the displacement of the mover unit.

b$.$ If the engine is required to operate at idling frequency of $10$ Hz $,$ determine the required stiffness and dimensions of the gas spring chamber.

c$.$ In normal operation, the engine delivers $2$ kW electrical output and the steady state operation is maintained by complete control of the combustion pressure. Using a sketch, suggest a viable multiple feedbacks closed loop control system that can maintain this steady state operation.