4. Consider the hydraulic system shown in Fig. 6, which consists of a hydraulic cylinder, a tank and a valve. The hydraulic cylinder is a gyrator governed by v = Q/A and f = Ap, where v(t) and f(t) are the velocity and the force of the piston, A is the area of the cylinder, and p(t) and Q(t) are the pressure and flow rate at the inlet of the tank. Also, the tank has fluid capacitance C and the valve has fluid resistance R. Moreover, Fig. 7 shows the linear graph of the hydraulic system. Answer the following questions. (a) Determine the driving point impedance Z(s). (b) Define H(s) as the transfer function with the piston velocity v(t) as the input and the inlet pressure p(t) as the output. Derive the relationship between the transfer function H(s) and the driving point impedance Z(s). v(t) f(t) A C R p(t) q(t) f(t) 12 Figure 6: A hydraulic cylinder driv- ing a tank and a valve Figure 7: System linear graph R