The system shown in Figure 5.42 simulates a machine supported by rubbers, which are approximated as four identical spring-damper units. The input is the force \(f\) and the output is the displacement \(x\) of the mass. The parameter values are \(m=500 \mathrm{~kg}, b=250 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}\), and \(k=200,000 \mathrm{~N} / \mathrm{m}\).

a. Draw the necessary free-body diagram and derive the differential equation of motion.

b. Determine the transfer function. Assume zero initial conditions.

c. Determine the state-space representation.

d. Find the transfer function from the state-space form and compare with the result obtained in Part (b).

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FIGURE 5.42 Problem 6. m