Explanation Definition used: Fourier series with period 2 L : The Fourier series of a function with period 2 L is obtained as f ( x ) = a 0 + a 1 cos ( π x L ) + a 2 cos ( 2 π x L ) + ⋅ ⋅ ⋅ + a n cos ( n π x L ) + ⋅ ⋅ ⋅ + b 1 sin ( π x L ) + b 2 sin ( 2 π x L ) + ⋅ ⋅ ⋅ + b n sin ( n π x L ) + ⋅ ⋅ ⋅ The n th coefficients are obtained as a 0 = 1 2 L ∫ − L L f ( x ) d x a n = 1 L ∫ − L L f ( x ) cos ( n π x L ) d x b n = 1 L ∫ − L L f ( x ) sin ( n π x L ) d x Result used: For an even function, the Fourier series contains only cosine terms and possibly constant term. For an odd function, the Fourier series contains only sine terms and no constant term. Calculation: It is given that the pulsating force function is F ( t ) = { 0 − 2 ≤ t < 0 , 1 ≤ t < 2 8 0 ≤ t < 1 . This function is neither even nor odd; hence all the coefficients are to be calculated. The period of the function is 2 L = 4 ⇒ L = 2 . Obtain the value of a 0 as follows. a 0 = 1 2 ( 2 ) ∫ − 2 0 0 d x + 1 2 ( 2 ) ∫ 0 1 8 d x + 1 2 ( 2 ) ∫ 1 2 0 d x = 2 [ x ] 0 1 = 2 [ 1 − 0 ] = 2 Obtain the value of a n as follows. a n = 1 ( 2 ) ∫ − 2 0 0 cos n π x 2 d x + 1 ( 2 ) ∫ 0 1 8 cos n π x 2 d x + 1 ( 2 ) ∫ 1 2 0 cos n π x 2 d x = 4 [ 2 n π sin n π x 2 ] 0 1 = 8 n π [ sin n π 2 − 0 ] = 8 n π sin n π 2 However, for all even values of n , the values of a n are 0. Thus, for n = 1 , the value of b 1 becomes 8 ( 1 ) π sin ( 1 ) π 2 = 8 π ; for n = 3 , the value of b 3 becomes 8 ( 3 ) π sin ( 3 ) π 2 = − 8 3 π

Basic Technical Mathematics with Calculus (11th Edition)

11th Edition

ISBN: 9780134437736

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

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Chapter 30.7, Problem 27E

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To expand: The given function in the form of sine series.

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Chapter 30.7, Problem 26E

Chapter 30.7, Problem 28E

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Chapter 30 Solutions

Basic Technical Mathematics with Calculus (11th Edition)

Ch. 30.1 - Prob. 1PE Ch. 30.1 - Prob. 2PE Ch. 30.1 - Prob. 1E Ch. 30.1 - Prob. 2E Ch. 30.1 - Prob. 3E Ch. 30.1 - Prob. 4E Ch. 30.1 - Prob. 5E Ch. 30.1 - Prob. 6E Ch. 30.1 - Prob. 7E Ch. 30.1 - Prob. 8E

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