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3 Pages

### Fourier Series Pt 3

Course: CIVE 353, Spring 2010
School: W. Alabama
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Word Count: 595

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of University Waterloo CivE 331 Determine the Fourier Series representation of the following function using the half-range cosine expansion. Graph the expansion for n = 10. 6.00 5.00 4.00 3.00 2.00 1.00 0.00 0 1 2 3 4 5 6 7 8 9 10 f (x ) = 5e - k ( x - 4 ) ; k = 0.26824 f (x ) = 1 + x By inspection, the domain length (L) is 10. In the period of interest (0 &lt; x &lt; 10): ; 0&lt; x&lt;4 x +1...

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of University Waterloo CivE 331 Determine the Fourier Series representation of the following function using the half-range cosine expansion. Graph the expansion for n = 10. 6.00 5.00 4.00 3.00 2.00 1.00 0.00 0 1 2 3 4 5 6 7 8 9 10 f (x ) = 5e - k ( x - 4 ) ; k = 0.26824 f (x ) = 1 + x By inspection, the domain length (L) is 10. In the period of interest (0 < x < 10): ; 0< x<4 x +1 f (x ) = -k ( x-4 ) = 5e 4 k e -kx = 14.62 e -kx ; 4 < x < 10 5e Substitute L = 10 into the HRC expression: n x f ( x ) = a0 + an cos ; 0 < x < 10 10 n =1 Substitute L = 10 and f(x) into the coefficient expressions: 10 4 10 1 1 a0 = f ( x )dx = (1 + x )dx + 14.62e - kx dx 10 0 10 0 4 an = 10 4 10 2 2 n x n x n x - kx f ( x ) cos dx = (1 + x ) cos dx + 14.62e cos dx 10 0 10 0 10 10 10 4 Now evaluate a0 and an. The following relationships will be useful (and would be provided in an exam or quiz context): cos(ax ) x sin (ax ) + a2 a ax e [a cos(bx ) + b sin (bx )] ax e cos(bx )dx = a2 + b2 cos(2 A) = 1 - 2 sin 2 ( A) x cos(ax )dx = University of Waterloo CivE 331 Evaluate a0: 1 x 2 14.62 -kx a0 = x + + - e 10 2 0 k 4 4 10 1 14.62 -4 k = 12 + e - e -10k = 2.691 k 10 ( ) Evaluate first integral of an: n x n x n x (1 + x )cos 10 dx = cos 10 dx + x cos 10 dx 0 0 0 4 4 4 10 100 n x n x 10 n x = sin x sin + 2 2 cos + n 10 0 n 10 0 n 10 0 4 4 4 = = 10 2 100 sin n + 2 2 n 5 n 2 40 2 cos n 5 - 1 + n sin n 5 - 1 50 2 100 2 sin n + 2 2 cos n n 5 n 5 50 2 200 2 = sin n - 2 2 sin n n 5 n 5 Evaluate second integral of an: 10 14.62e 4 - kx 1462e - kx n x cos dx = 100k 2 + n 2 2 10 n x n n x - k cos 10 + 10 sin 10 4 10 = = = = -10 k n n 1462 2 2 e sin (n ) - k cos(n ) - e -4 k sin n - k cos n 2 2 2 100k + n 10 5 5 10 -10 k n 1462 2 2 n e - k (- 1) - e -4 k sin n - k cos n 2 2 2 100k + n 5 5 10 -4k 1462 2 n 2 e k cos n sin n - 2 2 2 100k + n 5 10 5 [ ] -10k n - e k (- 1) n - 0.018347(- 1) 1462 2 2 0.091737 cos n - 0.107441 n sin n 2 2 7.1953 + n 5 5 2 2 n 134.12 cos n - 157.08 n sin n - 26.824(- 1) 5 5 = 2 2 7.1953 + n Substitute integration results into expression for an: 2 2 n 26.824 cos n - 31.416 n sin n - 5.3648(- 1) 10 2 40 5 5 2 an = sin n - 2 2 sin n + 2 2 n 7.1953 + n 5 n 5 University Waterloo CivE of 331 Substitute a0 and an into the HRC expansion: 2 2 n 26.824 cos n - 31.416 n sin n - 5.3648(- 1) 10 2 40 5 5 2 cos n x f ( x ) = 2.691 + sin n - 2 2 sin n + 2 2 10 7.1953 + n 5 n 5 n =1 n We can use a spreadsheet to plot the expansion for n=10. Set up the spreadsheet as shown below, with a series of columns for each value of n and an and a series of rows for each x and f(x) that is to be plotted. C 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 D f(x) 1.2059 1.2155 1.2438 1.2901 1.3533 1.4317 1.5233 1.6258 1.7367 1.8535 1.9737 2.0948 2.2146 2.3313 2.4435 2.5503 2.6512 2.7463 2.8364 2.9224 3.0059 E F G 1 2 3 a1 a2 a3 0.6763 -1.3521 -0.6242 0.6760 -1.3495 -0.6214 0.6750 -1.3415 -0.6131 0.6733 -1.3282 -0.5994 0.6710 -1.3097 -0.5803 =2.691+SUM(E6:N6) 0.6680 -1.2860 -0.5561 0.6643 -1.2572 -0.5270 0.6600 -1.2235 -0.4932 0.6551 -1.1849 -0.4550 0.6495 -1.1416 -0.4128 0.6432 -1.0939 -0.3669 0.6363 -1.0418 -0.3177 0.6288 -0.9857 -0.2658 0.6207 -0.9256 -0.2114 0.6120 -0.8619 -0.1552 0.6026 -0.7948 -0.0976 0.5927 -0.7245 -0.0392 0.5821 -0.6514 0.0196 0.5710 -0.5757 0.0782 0.5594 -0.4978 0.1362 0.5472 -0.4178 0.1929 H 4 a4 -0.1028 -0.1020 -0.0996 -0.0956 -0.0901 -0.0832 -0.0749 -0.0655 -0.0551 -0.0438 -0.0318 -0.0193 -0.0065 0.0065 0.0193 0.0318 0.0438 0.0551 0.0655 0.0749 0.0832 I 5 a5 0.1268 0.1252 0.1206 0.1129 0.1026 0.0896 0.0745 0.0575 0.0392 0.0198 0.0000 -0.0198 -0.0392 -0.0575 -0.0745 -0.0896 -0.1026 -0.1129 -0.1206 -0.1252 -0.1268 J 6 a6 -0.0208 -0.0204 -0.0194 -0.0176 -0.0152 -0.0122 -0.0089 -0.0052 -0.0013 0.0026 0.0064 0.0100 0.0133 0.0160 0.0182 0.0198 0.0207 0.0208 0.0202 0.0188 0.0168 K 7 a7 -0.1042 -0.1017 -0.0943 -0.0823 -0.0664 -0.0473 -0.0259 -0.0033 0.0195 0.0414 0.0612 0.0781 0.0913 0.1000 0.1040 0.1029 0.0969 0.0862 0.0713 0.0530 0.0322 L 8 a8 -0.1023 -0.0991 -0.0896 -0.0746 -0.0548 -0.0316 -0.0064 0.0192 0.0435 0.0652 0.0827 0.0951 0.1015 0.1015 0.0951 0.0827 0.0652 0.0435 0.0192 -0.0064 -0.0316 M 9 a9 -0.0034 -0.0032 -0.0028 -0.0022 -0.0014 -0.0005 0.0004 0.0013 0.0021 0.0028 0.0032 0.0034 0.0033 0.0029 0.0023 0.0015 0.0006 -0.0003 -0.0012 -0.0021 -0.0027 N 10 a10 0.0216 0.0205 0.0175 0.0127 0.0067 0.0000 -0.0067 -0.0127 -0.0175 -0.0205 -0.0216 -0.0205 -0.0175 -0.0127 -0.0067 0.0000 0.0067 0.0127 0.0175 0.0205 0.0216 f(x) 5.00 4.00 3.00 2.00 1.00 0.00 0 1 2 3 4 5 6 7 8 9 10 HRC, n=10 ... 6.00
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