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The problems and solutions for examination iii of the em406 course, focusing on vibrations and systems. The problems involve determining fundamental frequencies, odd/even functions, mode shapes, initial displacement patterns, time responses, and using lagrange's equations. Students are expected to find the solutions using maple, matlab, and hand calculations.
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Name 25 pts EM406 Examination III Problem 1 October 26, 2006
Problem 1. A second order system is forced with a periodic input as shown below (only the portion of the input displacement for t>0 is shown – the dotted line is the beginning of the next cycle).
Determine a) What is the fundamental frequency of the input? (4 pts)
b) Is the function odd, even or neither? What is the implication of this when you look at your Maple worksheet results? (4 pts)
c) What is a 0 for this function? (4 pts)
Problem 1.2 What is the mass moment of inertia of the system shown below about its center of gravity? Assume each of the masses is a point mass (4 pts)
0.5 (^) 1.0 Time (s)
y (cm)
m (^) 2m m
a (^) a
Name 35 pts EM406 Examination III Problem 2 October 26, 2006
A 2-DOF linear dynamic system has the mass and stiffness matrices given below.
and
a) Write down the characteristic polynomial of the system. Do not find the roots of this polynomial, and you do not need to simplify the polynomial in any way. (7 pts)
b) The natural frequencies of the dynamic system are ω 1 = 3.0 rad/sec and ω 2 = 5.0 rad/sec.
The mode shape associated with the first natural frequency is
. Find the mode shape
associated with the frequency ω 2 = 5.0. (8 pts)
c) Specify an initial displacement pattern, 1 2
, which would produce a free vibration having
the single frequency ω 1 = 3.0 rad/sec. You may assume zero initial velocity. (5 pts)
d) Suppose the initial displacement pattern for a free vibration was
. Assume zero initial
velocity and determine the time responses for x 1 and x 2. (15 pts)
b) Assuming the equations of motion are found to be:
determine the steady-state response of mass 2 (neglect the homogeneous solution).
c) Determine the values of k 3 and m 1 so that mass 1 acts like vibration absorber for mass 2 and mass 1 has a displacement less than 0.05.
( )
( )
4cos t
2
1
3 3
3 3
2
1 1
x
k k
k k
x
m x
& &
