For designing an automobile suspension, a two-mass system can be ·used for modeling as shown in the following diagram.

This is called a quarter-car model because it comprises one of the four wheel suspensions. The car and wheel positions are denoted by y(t) and x(t) respectively. These displacements are from static equilibrium which corresponds to no inputs except gravity.

(a) Draw the free-body diagram of this system, assuming one-dimensional vert'ical motion of the mass above wheel.

(b) Write down the equations of motion for the au,tomobile.

(c) Ex:p1·css these equations in a state-variable matrix form ( A,B, C,D)
using the follow'ing state-variable vector, x(t) = [x x y y]T, and justify this choice of state variables. Note that the car and wheel positions, y(t) and x(t), are the two outputs of the car system while the input is the unit step bump r ( t).

(d) Plot the position of the car and the wheel after the car hits a "unit bump" (i.e. r(t) is a unit step) using MATLAB. Assume m1 = lOkg, m2 = 250kg, k111 = 500, OOON/m, ks = 10, OOON jm. Find the value of b that you would prefer if you were a passenger in the car.