AOE 3034 Example problems Fall 1999

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1) Answer to 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.

A. Define the term “linear system”.

B. The superposition property holds for all time invariant systems (T-F)

C. State the initial value theorem.

D. The Laplace transform of the sum of two functions of time is equal to the sum of the Laplace transforms:

where L(.) is the Laplace transform operator and f1(t) and f2(t) are the functions. (T-F)

E. The equation of motion of a pendulum whose mass is concentrated at a point and is suspended by a weightless non extensible cord from the ceiling, is:

where l is the cord length and g is the acceleration of gravity. This equation is nonlinear. What is the linearized equation and under what conditions can it be used?

F. Consider two springs in parallel. These springs can be replaced by a single equivalent spring whose stiffness is the sum of the stiffnesses of the two springs. (T-F)

G. When applying Newton’s law for rotation, it is better to use as reference point the center of gravity or the fixed point of rotation (if there is such a point). Why?

H. The terms “mass moment of inertia” and “area moment of inertia” mean the same thing. (T-F)

I. Consider the signals:

Sketch these signals and state which signal is leading relative to the other.

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

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

L. The transfer function of a linear single degree of freedom system is not equal to the ratio of the Laplace transform of the response over the Laplace transform of the excitation. (T-F).

Answer

  1. A linear system is a system whose e.o.m. is linear w.r.t. the unknown displacement and its derivatives.
  2. F
  3. If f(t) and its derivative are Laplace transformable, and is the limit of sF(s) as s tends to infinity exists, then
  1. T
  2. . It holds when the angle is small (i.e. less than 10 degrees).
  3. T
  4. Because then: , where A is the center of mass or a fixed center of rotation (if there is one). The equation expressing N.L. w.r.t. an arbitrary point is far more complicated.
  5. F

x1(t) leads x2(t).

  1. F
  2. T.
  3. F

2. a) Derive the linear equation of motion of the system shown. The bar is weightless and rigid. Mass, m, is concentrated at a single point. An external moment, M(t), is applied to the system as shown.

b) Derive the transfer function G(s) of the system.

c) Write the equation for the free vibration response of the system (in free vibration the external moment, M(t), is zero).

Assume that the initial conditions are known.

In all three questions, assume that the rotation angle, (t), is small so that the vertical distance of the lower end of the bar, A, from point O is still l2 when the bar rotates by (t).

Solution

NL:

b)Initial conditions are zero by definition of the transfer function.

We need to find the Laplace transform of the impulse response function:

The Laplace transform of this equation is:

c)The free vibration response of an undamped single-degree-of-freedom system is:

(see class notes or p. 242 of the text)

where

3. The free vibration response of single degree of freedom system is shown below. T is the period of oscillation.

a) Find the damping ratio 

b) Find the ratio of the displacements at two points in time that are separated by four periods.

Solution

a)

b)

1