A representation of a car's suspension suitable for modeling the bounce and pitch motions is shown in Figure, which is a side view of the vehicle's body showing the front and rear suspensions. Assume that the car's motion is constrained to a vertical translation x of the mass center and rotation θ about a single axis which is perpendicular to the page. The body's mass is m and its moment of inertia about the mass center is IG. As usual, x and θ are the displacements from the equilibrium position corresponding to y1 = y2 = 0. The displacements y1 (t) and y2(t) can be found knowing the vehicle's speed and the road surface profile.
a. Assume that x and θ are small, and derive the equations of motion for the bounce motion x and pitch motion θ.
b. For the values k1 = 1100 lb/ft, k2 = 1525 lb/ft. c1 = c2 = 4 lb-sec/ft, L1, = 4.8 ft, L2 = 3.6 ft, m = 50 slugs, and IG = 1000 slug-ft2, use MATLAB to obtain a state-variable model in standard form.