3.38. (Automobile suspension system) [M.L. James, G.M. Smith, and J.C. Wolford, Applied Numerical Methods for Digital Computation, Harper and Row, 1985, p. 667.] Consider the spring mass system in Figure 3.4, which describes part of the suspension system of an automobile. The data for this system are given as m = x (mass of the automobile) = 375 kg, mass of one wheel = 30 kg. m = k= spring constant = 1500 N/m, k = linear spring constant of tire = 6500 N/m, c = damping constant of dashpot = 0, 375, 750, and 1125 N sec/m, #1 = displacement of automobile body from equilibrium position m. 23 displacement of wheel from equilibrium position m. v = velocity of car = 9, 18, 27, or 36 m/sec. A linear model = Az+ Bu for this system is given by #1 # #3 A m 0 m k 1791 0 ki ma 1 I C 1761 0 C 7122 0 k 771 0 ki+k 11/2 0 C 771 1 11/2 where u(t)=sin 20 describes the profile of the roadway. (a) Determine the eigenvalues of A for all of the above cases. (b) Plot the states for t20 when the input u(t) = sin 2 and (0) = [0,0,0,0] for all the above cases. Comment on your results. = 20 +x 21 I +x TA + 1712 2kv 20 Figure 3.4. Model of an automobile suspension system. 4. Problem 3.38 of the textbook. Use MATLAB. Replace part (b) by the following two parts. (b) Plot the body to wheel displacement - 3 and the body acceleration 2 when x(0) [0 0 0 0] and u(t) = sin 2ut for three cases of the damping constant = 20 C = 375, 750, and 1125 sec/m when the car velocity is v= 27 m/sec. For each 23 and 2, plot the three responses in the same graph. Comment on your of x1 results. 1 - 20 (c) Plot the body to wheel displacement - 3 and the body acceleration 2 when x(0)= [0 0 0 0] and u(t)=sin 2t for the four given values of the car velocity v when the damping constant is c = 375 sec/m. For each of - 3 and #2, plot the four responses in the same graph. Comment on your results.