DP3.1 A spring-mass-damper system, as shown in Figure 3.3, is used as a shock absorber for a large high-performance motorcycle. The original parameters selected are m = 1 kg, b = 9 N s/m, and k = 20 N/m.

(a) Determine the system matrix, the characteristic roots, and the transition matrix (t). The harsh initial conditions are assumed to be y(0) = 1 and dy/dt |t=0= 2.

(b) Plot the response of y(t) and y(t)

for the first two seconds.

(c) Redesign the shock absorber by changing the spring constant and the damping constant in order to reduce the effect of a high rate of acceleration force y(t) on the rider. The mass must remain constant at 1 kg.