Chapter 5, Problem 55P

A car active suspension system adds an active hydraulic actuator in parallel with the passive damper and spring to create a dynamic impedance that responds to road variations. The block diagram of Figure P5.36 depicts such an actuator with closed-loop control.

In the ?gure, K_tis the spring constant of the tire, M_USis the wheel mass, r is the road disturbance, x₁is the vertical car displacement, x₃is the wheel vertical displacement, ${\omega }_0^2=\frac{K_t}{M_{US}}$ is the natural frequency of the unsprung system and $\epsilon$ is a ?ltering parameter to be judiciously chosen (Lin, 1997). Find the two transfer functions of interest:

1. $\frac{X_3\left(s\right)}{R\left(s\right)}$

$\frac{X_1\left(s\right)}{R\left(s\right)}$