UNIT – I FOURIER SERIES PROBLEM 1

[Pages:63]UNIT ? I FOURIER SERIES

PROBLEM 1:

The turning moment T lb feet on the crankshaft of steam engine is given for a series of values of the crank angle degrees

0

T0

30 60

90

120 150 180

5224 8097 7850 5499 2626 0

Expand T in a series of sines.

Solution:

Let T = b1 sin + b2 sin2 + b3 sin3 + ........., Since the first and last values of T are repeated neglect last one

T

T sin T sin2

0

0

30 5224

60 8097

90 7850

120 5499

150 2624

Total

0 2612 7012.2 7850 4762.27 1313 23549.47

0 4524.11 7012.2

0 - 4762.27 - 2274.18 4499.86

T sin3

0 5224

0 -7850

0 2626

0

b1 = 2 [mean value of T sin ] = 2[ 23549.47 ] = 7849.8 6

b2 = 2 [mean value of T sin2 ] = 2[ 4499.86 ] = 1499.95 6

b3 = 2 [mean value of T sin2 ] = 0

f ( x ) = 7849.8 sin + 1499.95 sin2

PROBLEM 2:

Analyze harmonically the given below and express y in Fourier series upto the third harmonic.

x0

2

4 5

2

3

3

3

3

y 1.0 1.4 1.9 1.7 1.5 1.2

1

Solution:

Since the last value of y is a repetition of the first, only the six values will be used. The length of the interval is 2.

Let y =

a0 2

+ (a1 cosx + b1 sinx) + (a2 cos2x + b2 sin2x) + (a3 cos3x + b3 sin3x) + ...

(1)

x

y

cosx

0 1.0

1

sinx cos2x sin2x cos3x sin3x

0

1

0

1

0

1.4

0.5

0.866 - 0.5 0.866 -1

0

3

2 1.9

- 0.5

0.866 - 0.5 - 0.866 1

0

3

1.7

-1

0

1

0

-1

0

4 1.5

- 0.5

- 0.866 - 0.5 0.866 1

0

3

5 1.2

0.5

- 0.866 - 0.5 - 0.866 -1

0

3

a0 = 2 [mean value of y] = 2 (8.7) = 2.9 6

a1 = 2 [mean value of y cosx] = 2 [1 (1.0) + 0.5 (1.4) ? 0.5 (1.9) ? 1 (1.7) ? 0.5 (1.5) + 0.5 (1.2)] 6 = - 0.37

b1 = 2 [mean value of y sinx] = 2 [0.866 (1.4 + 1.9 -1.5 ? 1.2)] 6 = 0.17

a2 = 2 [mean value of y cos2x]

= 2 [1(1.0 + 1.7) ? 0.5 (1.4 + 1.9 + 1.5 + 1.2)] 6

= - 0.1 b2 = 2 [mean value of y sin2x]

= 2 [0.866(1.4 ? 1.9 + 1.5 ? 1.2)] 6

= - 0.06 a3 = 2 [mean value of y cos3x]

= 2 [1.0 ? 1.4 + 1.9 ? 1.7 + 1.5 ? 1.2] 6

= 0.03 b3 = 2 [mean value of y sin3x]

= 0

Hence y = 1.45 + (-0.37 cosx + 0.17 sinx) ? (0.1 cos2x + 0.06 sin2x) + 0.03 cos3x.

PROBLEM 3:

Find the Fourier series expansion for the function f(x) = x sinx in 0 < x < 2 and deduce

that 1 1 1 ......... 2

1.3 3.5 5.7

4

SOLUTION:

The Fourier series is

f (x)

a0 2

(an

n 1

cos nx

bn

sin

nx)..................(1)

Given f(x) = x sinx

a0

=

1

2

f (x)dx

0

=

1

2

x sin xdx

0

=

1

[x (- cosx) - (1)(- sinx)]0 2

= 1 (-2) = -2

an

=

1

2

f (x) cos nxdx

0

=

1

2

x sin x cos nxdx

0

=

1 2

2

[x sin(1 n)x sin(1 n)x]dx

0

[ 2SinACosB Sin(A B) Sin(A B)]

1 2

x(

cos(1

1 n

n)x

cos(1 n)x 1 n

)

(

sin(1 n)x ( 1 n)2

sin(1 n)x (1 n)2

2 ) 0

= 1 [2( cos 2(1 n) cos 2(1 n))] (

2

1 n

1 n

[ 1 1 ] 1 n 1 n

[1

n 1 1-n2

n

]

=

an

2 n2 1

where n 1

a1

=

1

2

f (x) cos xdx

0

=

1

2

x sin x cos xdx

0

=

1 2

2

x sin 2xdx

0

=

1 2

x(

cos 2

2x

)

(1)(

sin 2x 4

)

0

2

= 1 [2 (- cos 4 ) = - 1

2

2

2

1 [2( cos 4)

2

2

bn

=

1

2

0

f

(x)

sin

nxdx

=

1

2

0

x

sin

x

sin

nxdx

=

1 2

2

0

x[cos(1

n)x

cos(1

n

)x]dx

[ 2SinASinB Cos(A B) Cos(A B)]

=

1 2

x(

sin(1 n)x 1 n

sin(1 n)x 1 n

)

(

cos(1 (1

n)x n)2

cos(1 (1

n

n)x )2

)

2 0

=

1 cos 2(1 n)

2

(1 n)2

cos 2(1 n) (1 n)2

cos 0 (1 n)2

cos 0

(1

n)2

=

1 1

2

(1

n)2

1 (1 n)2

1 (1 n)2

1

(1

n

)2

= 0 where n 1

b1

=

1

2

0

f

(x)

sin

xdx

=

1

2

0

x

sin

x

sin

xdx

=

1

2

0

x

sin

2

xdx

=

1

2

0

x(1

cos 2

2x

)dx

=

1 2

2

0

x(1

cos

2x)dx

1 2

x(x

sin 2x ) 2

(1)( x2 2

cos 2x 4

) 0

2

= 1 [22 1 1]

2

44

=

Substitute in (1), we get

f

(x)

=

1

1 2

cos x

n2

n

2 2

1

cos

nx

sin

x

.......(2) +

...................(2)

Deduction Part:

Put x [x is a point of continuity] [x= is a point of continuity]

2

2

f ( ) = f ( ) sin =

2

2 2 22 2

(sin = 1) 2

(2)

(2)

2

1

0

n2

2 n2 1

cos

n 2

= -1-0+

2

1

2

n2

1 n2

1

cos

n 2

-

+1

=

- 1 2

1 cos n +1 =

2

n2 (n 1)(n 1) 2

- 1

1

cos n =

4 2 n2 (n 1)(n 1) 2

=

1

cos n 2

n2 (n 1)(n 1) 2

4

1 (1) 1 (0) 1 (1) ........ 2

1.3

2.4 3.5

4

(1)[ 1 1 1 ........] [ 2]

1.3 3.5 5.7

4

1 1 1 ........ [ 2]

1.3 3.5 5.7

4

Problem:4

Find the Fourier series for f(x) = sin x in - < x <

Solution:

Given f(x) = sin x

This is an even function. bn = 0

f(x) =

a0 2

a nCosnx.....................................(1) n 1

a0

=

2

0

f

(x)dx

=

2

0

Sinx

dx

=

2

0

sin

xdx

=

2

[

cos

x]0

= - 2 (cos cos 0)

= - 2 [11] [-1-1]

= 4

an

=

2

0

f

(x)

cos

nxdx

=

2

0

Sinx

cos nxdx

=

2

0

sin

x

cos

nxdx

=

2

1 2

0

[sin(n

1)x

sin(n

1)x]dx

=

1 [ cos(n 1)x n 1

cos(n 1)x n 1

]0

= 1 [ cos(n 1) cos(n 1) 1 1 ]

n 1

n 1 n 1 n 1

= 1 [ (1)n1 (1)n1 1 1 ] n 1 n 1 1 n n 1

= 1 [(1)n (1)n 1 1 ] n 1 n 1 n 1 n 1

= 1 [(1)n 1 (1)n 1]

n 1

n 1

= (1)n 1[ 1 1 ] n 1 n 1

=

(1)n

1[n

1 n n2 1

1]

=

(1)n

1[

2 n2

] 1

=

2[(1)n 1] (n 2 1)

if n 1.

a1

=

2

0

f

(x)

cos

xdx

=

2

0

Sinx

cos xdx

=

2

0

sin

x

cos

xdx

=

2

1 2

0

sin

2xdx

=

1

[

cos 2x 2

]0

= 1 [ 1 1] = 0 22

Substitute in equation (1), we get

f(x) =

2

0

n2

2[(1)n (n 2 1)

1]

cos

nx

 ................
................

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