Question: 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

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.

K, M E X3 m, FIGURE P5.364

In the figure, K_t is the spring constant of the tire, MUS is the wheel mass, r is the road disturbance, x₁ is the vertical car displacement, x₃ is the wheel vertical displacement, ω²₀= K_t/M_USis the natural frequency of the unsprung system and ε is a filtering parameter to be judiciously chosen (Lin, 1997). Find the two transfer functions of interest:

a. X₃(s)/R(s)

b. X₁(s)/R(s)

K, M E X3 m, FIGURE P5.364

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