The suspension system of a car is modelled by the following differential equation:

Question

M*((d^2*y)/(d*t^2) + D*(dy/dt) + Ky( t ) = x( t )

where y(t) is the vertical position of the car body, x(t) is the external force coming from the ground acting on the car body, M is the mass of the car body, and D and K are the damping coefficient and spring coefficient of the suspension system, respectively. For the given mass M = 1000kgs, we need to design the suspension system by choosing D and K such that the car is responsive but comfortable. Being responsive means that the step response should be reasonably quick, and being comfortable requires the oscillating frequency not to be too fast. More specifically, we are required to find D and K such that the step response is slightly oscillatory with oscillating period of 0.5 second and overshoot of 10%. Overshoot is defined to be the ratio of the maximum peak and the steady-state value of the step response.

1.1 Determine the values of D and K and plot the resulting step response. Verify the oscillating period and overshoot. (Hint: Express the transfer function of the system in the form of

H(s)= k/(s^2 +2ςω s+ω^2)

and use the fact that the oscillating frequency and overshoot are controlled by
the natural frequency and damping ratio. You may tune D and K while simulating the step response until you get the desired results.)