SolSec3.3 - 3- 21 Problems and Solutions Section 3.3...

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3- 21 Problems and Solutions Section 3.3 (problems 3.22-3.28) 3.22 Derive equations (3.24). (3.25) and (3.26) and hence verify the equations for the Fourier coefficient given by equations (3.21), (3.22) and (3.23). Solution: For n m , integration yields: 0 0 22 2 2 2 2 2 T TT T n t m tdt nm t nm T T T T = () + + = + + = sin sin sin sin sin sin sin ωω ππ π ( ) [] + ( ) + = 2 2 2 0 sin Since m and n are integers, the sine terms are 0, so this is equal to 0. Equation (3.24), for m = n : 0 2 0 1 2 1 4 2 28 2 2 4 2 T T T T T n tdt t n nt nT T n n T =− = sin sin sin sin ω Since n is an integer, the sine term is 0, so this is equal to T /2. So, 0 0 2 T n t m tdt mn Tm n = = sin sin / Equation (3.25), for 0 0 2 2 2 2 2 T T n t m tdt m T T T T = + + = + + = cos cos sin sin sin sin sin η + ( ) + = 2 2 2 0 sin Since m and n are integers, the sine terms are 0, so this is equal to 0.
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3- 22 Equation (3.25), for m = n becomes: cos sin sin sin 2 0 0 1 2 1 4 2 28 2 2 4 2 T T T T T n tdt t n nt TT n n T T n n T =+ () [] = ω π Since n is an integer, the sine term is 0, so this is equal to T /2. So, 0 0 2 T n t m tdt mn Tm n = = cos cos / ωω Equation (3.26), for : 0 0 22 2 2 2 2 1 2 T T n t m tdt nm t nm T T T T = + + = + + + + cos sin cos cos cos cos ππ 1 2 2 2 2 2 1 2 1 2 0 = ( ) + ( ) + + + = cos cos Since n is an integer, the sine term is 0, so this is equal to 0. So, 0 0 T n t m tdt = cos sin Equation (3.26) for n = m becomes: 0 2 0 2 1 24 20 T T T T nt nt d t n T n n = == cos sin sin sin Thus 0 0 T d t = cos sin
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3- 23 3.23 Calculate b n from Example 3.3.1 and show that b n = 0, n = 1,2,…, for the triangular force of Figure P3.23. Also verify the expression a n by completing the integration indicated. ( Hint : Change the variable of integration from t to x = 2 π nt/T .) Solution : From Equation (3.23), b T F t n tdt n T T = () 2 0 sin ω . Computing the integral yields: b TT t n tdt T t T n tdt b t n tdt n tdt n tdt T t n tdt n T T T T T n T T T T T T T T T T =− +− + ∫∫ 24 11 4 2 3 4 0 2 2 0 2 0 2 22 / / // sin sin sin sin sin sin ωω
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This note was uploaded on 02/11/2010 for the course MECHANICAL ms316 taught by Professor Abduljaba during the Spring '09 term at Kalamazoo.

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SolSec3.3 - 3- 21 Problems and Solutions Section 3.3...

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