# Tema19 - CHAPTER 19 19.1 The normal equations can be...

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CHAPTER 19 19.1 The normal equations can be derived as 11 2 416183 2 018098 2 416183 6 004565 0 017037 2 018098 0 017037 4 995435 83 9 1543934 1081054 0 1 1 . . . . . . . . . . . = A A B which can be solved for A 0 = 7.957538 A 1 = -0.6278 B 1 = -1.04853 The mean is 7.958 and the amplitude and the phase shift can be computed as C 1 2 2 1 0 6278 104853 1222 104853 0 6278 2 11 8 06 = - + - = = - - + = × = - ( . ) ( . ) . tan . . . . θ π π radians 12 hrs hr Thus, the final model is f t t ( ) . . cos ( . ) = + + 7 958 1222 2 24 8 06 π The data and the fit are displayed below: 6 8 10 0 12 24 Note that the peak occurs at 24 - 8.06 = 15.96 hrs. 19.2 The normal equations can be derived as

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1890 127 279 568187 0 5 1 0 1 5 350 265 381864 156 281 0 1 1 . . , . . - - - = - A A B which can be solved for A 0 = 195.2491 A 1 = -73.0433 B 1 = 16.64745 The mean is 195.25 and the amplitude and the phase shift can be computed as C 1 2 2 1 73 0433 16 6475 74 916 16 6475 730433 3366 192 8 = - + = = - - + = × = - ( . ) ( . ) . tan . . . . θ π π radians 180 d d Thus, the final model is f t t ( ) . . cos ( . ) = + + 195 25 74 916 2 360 192 8 π The data and the fit are displayed below: 0 100 200 300 0 90 180 270 360 19.3 In the following equations, ϖ 0 = 2 π / T ( 29 ( 29 [ ] ( 29 sin cos cos cos ϖ ϖ ϖ ϖ π 0 0 0 0 0 0 2 0 0 t dt T t T T T T = - = - - = ( 29 ( 29 [ ] ( 29 cos sin sin sin ϖ ϖ ϖ ϖ π 0 0 0 0 0 0 2 0 0 t dt T t T T T T = = - =
( 29 ( 29 sin sin sin 2 0 0 0 0 0 0 2 2 4 2 4 4 0 0 1 2 ϖ ϖ ϖ π ϖ t dt T t t T T T T T = - = - - + = ( 29 ( 29 cos sin sin 2 0 0 0 0 0 0 2 2 4 2 4 4 0 0 1 2 ϖ ϖ ϖ π ϖ t dt T t t T T T T T = + = + - - = ( 29 ( 29 ( 29 cos sin sin sin ϖ ϖ ϖ ϖ π ϖ 0 0 0 2 0 0 0 2 0 2 2 2 0 0 t t dt T t T T T T = = - = 19.4 a 0 = 0 ( 29 ( 29 ( 29 ( 29 a T t k t dt T k k t t k k t k T T T T = - = - + - 2 2 1 0 2 2 0 2 0 0 0 2 2 cos cos sin / / / / ϖ ϖ ϖ ϖ ϖ ( 29 ( 29 ( 29 ( 29 b T t k t dt T k k t t k k t k T T T T = - = - - - 2 2 1 0 2 2 0 2 0 0 0 2 2 sin sin cos / / / / ϖ ϖ ϖ ϖ ϖ On the basis of these, all a ’s = 0. For k = odd, b k k = 2 π For k = even, b k k = - 2 π Therefore, the series is ( 29 ( 29 ( 29 ( 29 f t t t t t ( ) sin sin sin sin = - + - + + ⋅⋅⋅ 2 1 2 2 3 3 1 2 4 0 0 0 0 π ϖ π ϖ π ϖ π ϖ The first 4 terms are plotted below along with the summation:

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-1 0 1 -2 0 2 19.5 a 0 = 0.5 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 a t k t dt t k t dt k t k t k t k k t k t k t k k = - + = - - + + - - 2 2 1 1 0 0 1 2 1 0 2 0 1 cos cos cos sin cos
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## This note was uploaded on 03/13/2012 for the course AEROSPACE 301 taught by Professor Pfchang during the Spring '12 term at Shandong University.

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Tema19 - CHAPTER 19 19.1 The normal equations can be...

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