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22
Chapter 2
Fourier Series
9.
(a) Natural frequency of the spring is
ω
0
=
±
k
μ
=
√
10
.
1
≈
3
.
164
.
(b) The normal modes have the same frequency as the corresponding components
of driving force, in the following sense. Write the driving force as a Fourier series
F
(
t
)=
a
0
+
∑
∞
n
=1
f
n
(
t
) (see (5). The normal mode,
y
n
(
t
), is the steadystate
response of the system to
f
n
(
t
). The normal mode
y
n
has the same frequency as
f
n
. In our case,
F
is 2
π
periodic, and the frequencies of the normal modes are
computed in Example 2. We have
ω
2
m
+1
=2
m
+ 1 (the
n
even, the normal mode
is 0). Hence the frequencies of the ±rst six nonzero normal modes are 1, 3, 5, 7, 9,
and 11. The closest one to the natural frequency of the spring is
ω
3
= 3. Hence, it
is expected that
y
3
will dominate the steadystate motion of the spring.
13.
According to the result of Exercise 11, we have to compute
y
3
(
t
) and for this
purpose, we apply Theorem 1. Recall that
y
3
is the response to
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.
 Fall '11
 StuartChalk

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