Engineering Mathematics III

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Engineering Mathematics III
Fourier Series
FOURIER SERIES
Dr.A.T.Eswara Professor and Head Dept of
Mathematics P.E.S.College of Engineering Mandya-57
1401
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Engineering Mathematics III
Fourier Series
.
HALF-RANGE FOURIER SERIES The Fourier expansion
of the periodic function f(x) of period 2l may
contain both sine and cosine terms. Many a time
it is required to obtain the Fourier expansion
of f(x) in the interval (0,l) which is regarded
as half interval or half range. The definition
can be extended to the other half range in such
a manner that the function becomes even or odd.
This will result in cosine series or sine series
only.


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Engineering Mathematics III
Fourier Series
. Sine series (Fourier half range sine
series) Suppose f(x) ?(x) is given in the
interval (0,l). Then we define f(x) -?(-x) in
(-l,0). Hence f(x) becomes an odd function in
(-l , l). The Fourier series then is

where

The series (11) is called half-range sine series
over (0,l).
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Engineering Mathematics III
Fourier Series
Putting l? in (11), we obtain the half-range
sine series of f(x) over (0,?) given by
Cosine series (Fourier half range Cosine series)

Suppose f(x) ?(x) is given in the interval
(0,l). Then, we define f(x) ?(-x) in (-l,0).
Hence f(x) becomes an even function in (-l , l).

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Engineering Mathematics III
Fourier Series
Then the Fourier series of f(x) is given by
The series (12) is called half-range cosine
series over(0,l).Putting l ? in (12), we get


n1,2,3,..
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Engineering Mathematics III
Fourier Series

• Examples
• Expand f(x) x(?-x) as half-range sine series
  over the interval (0,?).
• We have,

Integrating by parts, we get

The sine series of f(x) is
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Engineering Mathematics III
Fourier Series
2. Obtain the cosine series of
Here

Performing integration by parts and simplifying,
we get

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Engineering Mathematics III
Fourier Series
Thus, the Fourier cosine series is f(x)
3. Obtain the half-range cosine series of f(x)
c-x in 0ltxltc Here
Integrating by parts and simplifying we get,

The cosine series is given by f(x)

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Engineering Mathematics III
Fourier Series
Obtain the half-range sine series of the
following functions over the specified intervals
13. f(x) cosx over (0,?) 14. f(x) sin3x
over (0,?) 15. f(x) lx-x2 over (0 , l) Obtain
the half-range cosine series of the following
functions over the specified intervals 16. f(x)
x2 over (0,?) 17. f(x) xsinx over
(0,?) 18. f(x) (x-1)2 over (0,1)

19. f(x)
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Engineering Mathematics III
Fourier Series
HARMONIC ANALYSIS The Fourier series of a known
function f(x) in a given interval may be found
by finding the Fourier coefficients. The method
described cannot be employed when f(x) is not
known explicitly, but defined through the values
of the function at some equidistant points. In
such a case, the integrals in Eulers formulae
cannot be evaluated. Harmonic analysis is the
process of finding the Fourier coefficients
numerically.

.
To derive the relevant formulae for Fourier
coefficients in Harmonic analysis, we employ the
following result
The mean value of a continuous function f(x) over
the interval (a,b) denoted by f(x) is defined
as
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Engineering Mathematics III
Fourier Series
The Fourier coefficients defined through Eulers
formulae, (1), (2), (3) may be redefined as

With these values of , we obtain
the Fourier series of f(x). The term
a1cosxb1sinx is called the first harmonic or
fundamental harmonic, the term a2cos2xb2sin2x is
called the second harmonic and so on. The
amplitude of the first harmonic is and
that of second harmonic is

and so on.
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Engineering Mathematics III
Fourier Series

• Examples
• Find the first two harmonics of the
• Fourier series of f(x) given the following table
  

Note that the values of y f(x) are spread over
the interval 0? x ? 2? and f(0) f(2?) 1.0.
Hence the function is periodic and so we omit the
last value f(2?) 0.We prepare the following
table to compute thefirst two harmonics.
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Engineering Mathematics III
Fourier Series

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Engineering Mathematics III
Fourier Series
We have
as the length of interval 2l 2? or l ?
Putting, n 1,2, we get



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Engineering Mathematics III
Fourier Series
The first two harmonics are a1cosxb1sinx and
a2cos2xb2sin2x. That is (-0.367cosx 1.0392
sinx) and(-0.1cos2x 0.0577sin2x)

• Express y as a Fourier series upto the third
  harmonic
• given the following values

The values of y at x 0,1,2,3,4,5 are given and
hence theinterval of x should be 0 ? x lt 6. The
length of the interval 6-0 6, so that 2l 6
or l 3.The Fourier series upto the third
harmonic is

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Engineering Mathematics III
Fourier Series
, then
Put


We prepare the following table using the given
values
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Engineering Mathematics III
Fourier Series
?


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Engineering Mathematics III
Fourier Series


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Engineering Mathematics III
Fourier Series
Using these in (1), we get
This is the required Fourier series upto the
third harmonic.
3. The following table gives the variations of a
periodic current A over a period T

Show that there is a constant part of 0.75amp. in
the current A and obtain the amplitude of the
first harmonic.
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Engineering Mathematics III
Fourier Series
Note that the values of A at t 0 and t T are
the same. Hence A(t) is a periodic function of
period T. Let us denote
. We have


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Engineering Mathematics III
Fourier Series


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Engineering Mathematics III
Fourier Series
Using the values of the table in (1), we get

The Fourier expansion upto the first harmonic is


The expression shows that A has a constant part
0.75 in it. Also the amplitude of the first
harmonic is
1.0717.
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Engineering Mathematics III
Fourier Series

• Exercises
• The displacement y of a part of a mechanism
• is tabulated with corresponding angular movement
• x0 of the crank. Express y as a Fourier series
  upto
• the third harmonic.

2. Obtain the Fourier series of y upto the second
harmonic using the following table

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Engineering Mathematics III
Fourier Series
3. Obtain the constant term and the coefficients
of thefirst sine andcosine terms in the Fourier
expansion of y as given in the following table

4. Find the Fourier series of y upto the second
harmonic from the following table

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Engineering Mathematics III
Fourier Series
5. Obtain the first three coefficients in the
Fourier cosine series for y, where y is given in
the following table
6. The turning moment T is given for a series of
values of the crank angle ?0 750 .


Obtain the first four terms in a series of sines
to represent T and calculate T at ? 750 .