This preview has intentionally blurred parts. Sign up to view the full document

View Full Document

Unformatted Document Excerpt

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 ... View Full Document

End of Preview

Sign up now to access the rest of the document