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TAMU PETE 301 - Numerical Methods for Engineers Ch. 19 Solutions

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1 CHAPTER 19 19 1 In the following equations 0 2 T sin t dt cos t T 0 0 0 T T 0 0 T cos t dt sin t T 0 0 0 T T 0 0 T 0 cos 2 cos 0 0 T 0 T t sin 2 0 t 4 0 0 2 T T 0 sin 2 0 t dt T T t sin 2 0 t 4 0 0 2 T T 0 cos 2 0 t dt T cos 0 t sin 0 t dt T 0 T T sin 2 sin 0 0 T sin 4 0 0 2 4 0 1 T 2 T sin 4 0 0 2 4 0 1 T 2 sin 2 0 t sin 2 2 0 0 2 0 T 2T 0 0 T 19 2 The angular frequency can be computed as 0 2 360 0 017453 Because the data are equispaced the coefficients can be determined with Eqs 19 14 19 16 The various summations required to set up the model can be determined as t 15 45 75 105 135 165 195 225 255 285 315 345 sum Radiation 144 188 245 311 351 359 308 287 260 211 159 131 2954 cos 0t 0 96593 0 70711 0 25882 0 25882 0 70711 0 96593 0 96593 0 70711 0 25882 0 25882 0 70711 0 96593 sin 0t 0 25882 0 70711 0 96593 0 96593 0 70711 0 25882 0 25882 0 70711 0 96593 0 96593 0 70711 0 25882 ycos 0t 139 093 132 936 63 411 80 493 248 194 346 767 297 505 202 940 67 293 54 611 112 430 126 536 614 175 ysin 0t 37 270 132 936 236 652 300 403 248 194 92 916 79 716 202 940 251 141 203 810 112 430 33 905 164 429 The coefficients can be determined as y 2954 246 1667 N 12 2 2 A1 y cos 0 t 614 175 102 363 N 12 2 2 B1 y sin 0 t 164 429 27 4048 12 N A0 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 2 Therefore the best fit sinusoid is R 246 1667 102 363 cos 0 017453t 27 4048 sin 0 017453t The data and the model can be plotted as 400 300 200 100 0 0 60 120 180 240 300 360 The value for mid August can be computed as R 246 1667 102 363 cos 0 017453 225 27 4048 sin 0 017453 225 299 1698 19 3 The angular frequency can be computed as 0 2 24 0 261799 The various summations required for the normal equations can be set up as t y cos 0t sin 0t sin 0t cos 0t cos2 0t 0 2 4 5 7 9 12 15 20 22 24 sum 7 6 7 7 1 6 5 7 4 7 2 8 9 8 8 8 9 7 9 7 84 3 1 00000 0 86603 0 50000 0 25882 0 25882 0 70711 1 00000 0 70711 0 50000 0 86603 1 00000 2 31784 0 00000 0 50000 0 86603 0 96593 0 96593 0 70711 0 00000 0 70711 0 86603 0 50000 0 00000 1 93185 0 00000 0 43301 0 43301 0 25000 0 25000 0 50000 0 00000 0 50000 0 43301 0 43301 0 00000 0 00000 1 00000 0 75000 0 25000 0 06699 0 06699 0 50000 1 00000 0 50000 0 25000 0 75000 1 00000 6 13397 sin2 0t 0 00000 0 25000 0 75000 0 93301 0 93301 0 50000 0 00000 0 50000 0 75000 0 25000 0 00000 4 86603 ycos 0t ysin 0t 7 60000 6 06218 3 55000 1 68232 1 91526 5 09117 8 90000 6 22254 4 45000 6 84160 7 00000 15 05713 0 00000 3 50000 6 14878 6 27852 7 14785 5 09117 0 00000 6 22254 7 70763 3 95000 0 00000 10 28615 The normal equations can be assembled as 2 317837 1 931852 A0 84 3 11 0 2 3178 6 133975 A1 15 0571 0 4 866025 B1 10 2862 1 9319 This system can be solved for A0 7 96464 A1 0 55488 and B1 1 04815 Therefore the best fit sinusoid is pH 7 96464 0 55488cos 0t 1 04815sin 0t The result can also be expressed in the alternate form of Eq 19 2 by computing the amplitude PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 3 C1 0 55488 2 1 04815 2 1 185965 and the phase shift 1 04815 2 05767 0 55488 arctan Therefore the fit can also be expressed as pH 7 96464 1 185965cos 0t 2 05767 Consequently the mean is 7 96464 and the amplitude is 1 185965 To determine the time of the maximum inspection of Fig 19 3 indicates that a positive phase shift represents the time prior to midnight that the peak occurs Therefore the time of the maximum can be computed as tmax 24 2 05767 24 hr 16 14026 hrs 2 This is equal to about 16 08 25 or 4 08 25 PM The data and the model can be plotted as 10 9 8 7 6 0 6 12 18 24 19 4 a0 0 ak 2 T 2 T T 2 T 2 2t cos k 0 t dt T 2 4 1 t cos k 0 t sin k 0 t 2 T k 0 k 0 T 2 bk 4 T T 2 T 2 2t sin k 0 t dt T 2 1 t sin k t cos k t 0 0 2 k 0 k 0 T 2 On the basis of these all a s 0 For k odd bk 2 k PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 4 For k even bk 2 k Therefore the series is f t 2 sin 0 t 1 sin 2 0 t 2 1 sin …


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