A dynamic system can be modeled as a mass, spring, damper system as shown in the schematic diagram. The system vibrates under a Sine wave force.

Where $F(t)=180sin(5t)N,M1=50kg,C1=2N\frac{s}{m},K1=25\frac{N}{m}$

$(1)$ You are requested to simulate the vibration of the system using Simulink, Note the largest displacement, and the steady state displacement.

$(2)$ Add a PID controller to command a linear actuator to provide actuation force to the system in the purpose of reducing vibration. Simulate the control system using Simulink. Compare the results with that in $(1).$