January 2014
Intermediate to advanced
144 pages
4h 2m
English
Content preview from Mathematical Formulas for Industrial and Mechanical Engineering
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a
0
520:78539753
a
k
5
2
T
ð
π
2π
f ðt Þcosðkw
0
tÞdt
where
w
0
5
2π
T
5
2π
2π
5 1
Hence,
a
k
5
2
T
ð
π
2π
f ðt ÞcosðktÞdt
or
a
k
5
2
2π
ð
2π=2
2π
2
π
2
cosðktÞdt 1
ð
π=2
2π=2
ð2tÞcosðktÞdt 1
ð
π
π=2
2
π
2
cosðktÞdt
()
Similarly
b
k
5
2
T
ð
π
2π
f ðt Þsinðkw
0
tÞdt 5
2
T
ð
π
2π
f ðt ÞsinðktÞdt
or
b
k
5
2
2π
ð
2π=2
2π
2
π
2
sinðktÞdt 1
ð
π=2
2π=2
ð2tÞsinðktÞdt 1
ð
π
π=2
2
π
2
sinðktÞdt
()
The “integration by part” formula can be utilized to compute the second integral on
the right-hand side of the above equations for a
k
and b
k
.
For k 5 1; 2; ...; 8, the Fourier coefficients a
k
and b
k
can be computed and summarized
as in
Table 5.1.
The periodic function (shown in Example 1) can be approximated by Fourier series as
f ðt Þ5 a
0
1
X
N
k51
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