3. Consider a rack and pinion system driving hydraulic components as shown in Fig. 4. The pinion has radius r and negligible mass. Moreover, the pinion is subjected to a prescribed torque Ts as shown. The rack has mass m, and is connected to a hydraulic cylinder with area A. An elbow turn with fluid resistance R connects the cylinder outlet to a tank with capacitance C. (a) Draw the linear graph and the corresponding normal tree (b) Derive the state equations in the matrix form using the normal tree (c) The output variables of interest are the pressure loss pR at the elbow and the forcef transmitted to the hydraulic cylinder. Derive the output equations in the matrix form. 4. Consider an electrical circuit containing an unknown transducer (a black box); see Fig. 5. After some initial testing, it is found that the voltages are related by v2-kvi, where k is a constant determined by the tests (a) Is this unknown transducer a trasnformer or a gyrator? Why? Black R Box L R, Figure 5: Electric circuit involving a Figure 4: A rack and pinion system driv- black box ing hydraulic components 3. Consider a rack and pinion system driving hydraulic components as shown in Fig. 4. The pinion has radius r and negligible mass. Moreover, the pinion is subjected to a prescribed torque Ts as shown. The rack has mass m, and is connected to a hydraulic cylinder with area A. An elbow turn with fluid resistance R connects the cylinder outlet to a tank with capacitance C. (a) Draw the linear graph and the corresponding normal tree (b) Derive the state equations in the matrix form using the normal tree (c) The output variables of interest are the pressure loss pR at the elbow and the forcef transmitted to the hydraulic cylinder. Derive the output equations in the matrix form. 4. Consider an electrical circuit containing an unknown transducer (a black box); see Fig. 5. After some initial testing, it is found that the voltages are related by v2-kvi, where k is a constant determined by the tests (a) Is this unknown transducer a trasnformer or a gyrator? Why? Black R Box L R, Figure 5: Electric circuit involving a Figure 4: A rack and pinion system driv- black box ing hydraulic components