Problem 1) Consider the system with two masses shown in the figure below. Each mass is on a surface with a friction coefficient.

(a) Derive the differential equations that describe 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.

(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.

(a) Draw the signal flow graph, and find the transfer function for the system.

Problem 4) Consider the approximate model of a tensile tester shown in the figure below.

(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.

(a) Find the transfer function relating the output voltage vo(t) to the input voltage vi(t).