Advanced Vibration
Ex. 3
Submit Wed. March 23, 2011
1.
Consider the six d.o.f. shown below,
(a)
Explain how you can find one natural frequency of the system by inspection.
(b)
Determine the six natural frequencies and mode shape of the system by solving
three eigenvalue problems of dimensions 1, 2 and 3.
(c)
Plot the modeshapes and observe their special features.
2.
Consider the axial vibrations of a rod with
E
=
=
ρ
1,
Ax
e
x
()
=
, fixed at
x
=
0 and
free to oscillate at the other end
x
=
1.
(a)
Evaluate and plot the first 10 natural frequencies and mode shapes of the rod.
(b)
By using finite differences with 10 equal elements approximate the values
obtained in (a).
3.
Consider the axially vibrating rod with
E
=
=
⎩
⎨
⎧
<
<
<
<
=
1
5
.
0
,
2
5
.
0
0
,
1
)
(
x
x
x
A
, fixed at
x
=
0 and free to oscillate at the other end
x
=
(a)
Evaluate and plot the first 10 natural frequencies and mode shapes of the rod.
(b)
By using discrete model with 10 equal elements approximate the values obtained
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 Fall '08
 Ram,Y
 Natural Frequency, natural frequencies

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