AOE 3034 Final Exam Fall 1998

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1(30 pts) Consider the system shown in the figure. Assume that the bar is rigid and that it has no mass. Assume that (t) is small (less than five degrees).

Figure 1

a)(5 pts) Derive the linear equation of motion.

b)(5 pts) Derive the transfer function of the system from the applied moment M(t) to the angle of rotation (t).

c)(10 pts) A moment M(t)=sin(t) is applied. Derive an expression for the steady state response, (t).

d)(10 pts) Consider that l1=1 m, l2=2 m, m1=1 kg, m2=2 kg, g=9.81 m/sec2. A moment M(t)=sin(t). The response is (t)=-0.01sin(t). Calculate the frequency .

  1. (35 pts) The unicyclist in exam 2 decided to try her luck on a road that has a step profile as shown in Fig. 2.

a)(8 points) Derive the differential equation of motion of the approximate model shown.

b)(7 points) Derive the transfer function from the road elevation, z(t), to the displacement, x(t), and the sinusoidal transfer function.

c)(20 points) Find the displacement of the unicyclist, x(t). Assume that the wheel will hit the step at t=0. Assume that the wheel always maintains contact with the road.

3) (35 pts) Do not justify your answers to true false questions (marked T-F). However, you can explain your answer if you think that a question is ambiguous. Answers to the remaining questions must be short (about 50 words).

A)An engine is mounted to a base using mounts. An excitation force is applied to the engine due to an eccentric mass that rotates with constant angular frequency. The engine is modeled as a single degree of freedom system consisting by a mass supported by a spring and a damper. Transmissibility is the ratio of the amplitude of the force transmitted to ground to the excitation force applied to the engine in steady state (T-F).

B)Transmissibility always decreases as the damping ratio, , increases (T-F).

C)The block diagram shown below is equivalent to the single block shown (T-F).

D)The transfer function of a linear system may depend on the excitation force (T-F).

E)The inverse Laplace transform of the transfer function of a linear system is equal to a unit impulse (T-F).

F)The following force is applied to a linear system:

The system is at rest at t=0. The response has the form: , where G(j) is the sinusoidal transfer function of the system, and  is the phase angle of the sinusoidal transfer function (T-F).

For questions G-J, consider the steady state response of a linear single degree of freedom system subjected to harmonic (sinusoidal) excitation as shown in the figure below. b is non zero.

G. The frequency for which the amplitude of the response becomes maximum is less that the natural frequency of the system. (T-F)

H.The response does not oscillate in time if the system is over damped. (T-F)

  1. For excitation frequencies that are practically zero compared to the system natural frequency, it is reasonable to assume that the response amplitude is zero (T-F).
  1. For excitation frequencies that are much larger than the natural frequency of the system, the response amplitude is practically zero. Explain why using an intuitive argument (do not use equations).
  1. The block diagram shown is equivalent to the single block shown below (T-F).
  1. The steady state error of a system with integral control subjected to a step input is zero. (T-F).
  1. The response of a feedback control system with proportional control to a step input involves oscillations (T-F).
  1. It is not a good idea to use derivative control alone. Why?
  1. What is the main advantage of a feedback control system compared to an open loop system?

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