**3.1. (a)** For the mechanical translational system illustrated in Figure P3.1, write the differential equation relating the position *y*(*t*) and the applied force *f*(*t*).

**(b)** Determine the transfer function *Y*(*s*)/*F*(*s*).

**(c)** Determine the phase-variable canonical vector form of this system.

**3.2. (a)** For the mechanical translational system illustrated in Figure P3.2, write the differential equation relating the position *y*(*t*) and the applied force *f*(*t*).

**(b)** Determine the transfer function *Y*(*s*)/*F*(*s*).

**(c)** Determine the phase-variable canonical vector form of this system.

**3.3. (a)** For the mechanical rotational system illustrated in Figure P3.3, write the differential equation relating *T*(*t*) and *θ*(*t*).

**(b)** Determine the transfer function *θ*(*s*)/*T*(*s*).

**3.4.** Figure P3.4 represents the diagram of a gyroscope which is used quite frequently in autopilots, stabilized fire control systems, and so on. Assume that the rotor speed is constant, that the total developed torque about the output axis is given by

where *K*′ is a constant, and that the inner gimbal’s moment ...

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