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Problem 1, Consider the system with two masses shown in the figure below. Each mass is on a surface with a friction coefficient. X1 (t) x2(t) fv3 f(t) K1 K3 0000 M1 K2 M2 0000 fv, fv2 (a. ) Derive the differential equations that describes the motion of the system. (b. . ) Express the system in state space form with the displacement and velocity of both masses as the system outputs. Problem 2 Consider the pitch control system of a space shuttle in the figure below. Commanded Commanded pitch pitch rate acceleration Shuttle Commanded Controller dynamics + + pitch + + + Pitch K1 K2 G1(s) G2(s) Pitch Pitch Pitch error rate acceleration error error Accelerometer 32 Rate gyro 5 Inertial measuring unit 1 (a. ) Find the closed loop transfer function relating pitch to commanded pitch. (b. . ) Find the closed loop transfer function relating pitch rate to commanded pitch rate. (c. Find the closed loop transfer function relating pitch acceleration to commanded pitch acceleration. Problem 3 Consider the block diagram shown below. R(s) + 1 + 50 + C(s) S S + 1 - 2 2 s (a. ) Draw the signal flow graph, and find the transfer function G(s) = C(s) / R(s) for the system. Problem 4 Consider the approximate model of a tensile tester shown in the figure below. Current Load, frame, specimen, Desired load Controller amplifier load cell Actual load F¿(s) + E((s) 0.06 15 x106 F(s) 1666.67 s+720 33+8232+4X 106s + 2x 107 - E(s) 1 (a.) Model the system in state space. (b.) Simulate the unit step response using MATLAB. Is the response predominately first order or second order? (c.) Plot the controller output ei(t) for the unit step response in (b.) . Problem 5 Consider the two stage Operational Amplifier circuit shown below. 100 k§2 4 uF 2 F v1(t) V V1(t) vi(t) vo(t) + 500 kQ2 2F + 400 k§2 4 F - - (a. ) Find the transfer function relating the output voltage Vo(t) to the input voltage VI(t). .

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