An accelerometer attached to an object can be modeled as a mass-damper-spring system, as shown in Figure 5.122. Denote the displacement of the mass relative to the object, the absolute displacement of the mass, and the absolute displacement of the object as \(x(t), y(t)\), and \(z(t)\), respectively, where \(x(t)=y(t)-z(t)\) and \(x(t)\) is measured electronically.

a. Draw the necessary free-body diagram and derive the differential equation in terms of \(x(t)\).

b. Using the differential equation obtained in Part (a), determine the transfer function \(X(s) / Z(s)\). Assume that the initial conditions are \(x(0)=0\) and \(\dot{x}(0)=0\).

c. Using the differential equation obtained in Part (a), determine the state-space representation. The input is the absolute displacement of the object \(z(t)\) and the output is the displacement of the mass relative to the object \(x(t)\).

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