AOE 3034 Midterm 1 Fall 1998

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1) (40 points) Answer the following questions using less than 50 words. You do not need to justify your answers to truefalse questions (these questions are marked by TF). However, you can write an explanation if you feel that a true false question is ambiguous.

1. Give one example of the equation of motion of a) a linear system, and b) a nonlinear system.

2. The superposition property holds for both linear and nonlinear systems (T-F)

3. The Laplace transform of the derivative of a function with respect to time, df(t)/dt, is always sF(s), where F(s) is the Laplace transform of f(t). (T-F)

4. The inverse Laplace transform of the sum of two functions of s is equal to the sum of the inverse Laplace transforms:

where L-1(.) is the Laplace transform operator and F1(s) and F2(s) are the functions. (T-F)

5. What is the use of Laplace Transforms?

6. Consider two springs connected in parallel and a third connected in series to the first two springs. These springs can be replaced by a single equivalent spring whose stiffness is . (T-F)

7. The radius of gyration of a body is the mass moment of inertia divided by the mass. (T-F)

8. Consider the free vibration of an undamped (=0) single degree of freedom system. The frequency of vibration depends on the initial conditions. (T-F)

9. Consider the free vibration of an under damped single degree of freedom system. The amplitude decays to zero faster as the damping ratio increases. (T-F)

10. The inverse Laplace transform of the transfer function of a linear system is the impulse response function. (T-F)

11. The transfer function of a linear single degree of freedom system does not depend on the excitation. (T-F)

12. If we know the impulse response of a single degree of freedom system, then we can find its response to any arbitrary excitation. (T-F)

2) (40 points)

a)(20 points) Derive the linear equation of motion of the system shown. The wheel cannot slip relative to the ground.

b)(10 points) The natural frequency of the system was measured. It was found equal to n. Find the distance from the point where the spring is connected to the wheel to the center of the wheel a.

c) (10 points) Derive the transfer function G(s) of the system.

In all questions, assume that the rotation angle, (t), is small.

3) (20 pts) The free vibration response of single degree of freedom system is shown below. T is the period of oscillation. Find the damping ratio 

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