To determine To find: The Fourier series of the function that represents the force. Explanation Definition used: Fourier series with period 2 π : The Fourier series of the function with period 2 π is obtained as f ( x ) = a 0 + a 1 cos x + a 2 cos 2 x + ⋅ ⋅ ⋅ + a n cos n x + ⋅ ⋅ ⋅ + b 1 sin x + b 2 sin 2 x + ⋅ ⋅ ⋅ + b n sin n x + ⋅ ⋅ ⋅ The n th coefficient is obtained as a 0 = 1 2 π ∫ − π π f ( x ) d x a n = 1 π ∫ − π π f ( x ) cos n x d x b n = 1 π ∫ − π π f ( x ) sin n x d x Calculation: It is given that the periodic force function is f ( t ) = { 0 − π ≤ t < 0 t 2 + t 0 < t < π . Obtain the value of a 0 as follows. a 0 = 1 2 π ∫ − π 0 ( 0 ) d t + 1 2 π ∫ 0 π ( t 2 + t ) d t = 1 2 π [ t 3 3 + t 2 2 ] 0 π = 1 2 π [ π 3 3 + π 2 2 − 0 3 3 − 0 2 2 ] = 1 2 π [ 2 π 3 + 3 π 2 6 ] That is, a 0 = π 12 ( 2 π + 3 ) . Obtain the value of a 1 as follows. a 1 = 1 π ∫ − π 0 ( 0 ) cos t d t + 1 π ∫ 0 π ( t 2 + t ) cos t d t = 0 + 1 π ∫ 0 π t 2 cos t d t + 1 π ∫ 0 π t cos t d t = 1 π [ t 2 sin t + 2 t cos t − 2 sin t ] 0 π + 1 π [ t sin t + cos t ] 0 π = 1 π [ 2 ( π ) cos ( π ) ] + 1 π [ cos π − cos 0 ] That is, a 1 = 1 π [ 2 π ( − 1 ) ] + 1 π [ − 1 − 1 ] = − 2 − 2 π . Obtain the value of a 2 as follows. a 2 = 1 π ∫ − π 0 ( 0 ) cos n t d t + 1 π ∫ 0 π ( t 2 + t ) cos 2 t d t = 0 + 1 π ∫ 0 π t 2 cos 2 t d t + 1 π ∫ 0 π t cos 2 t d t = 1 π [ t 2 sin 2 t 2 + t cos 2 t 2 − sin 2 t 4 ] 0 π + 1 π [ t sin 2 t 2 − cos 2 t 4 ] 0 π = 1 π [ ( π ) cos 2 ( π ) 2 ] + 1 4 π [ − cos 2 π + cos 0 ] That is, a 2 = 1 π [ π ( 1 ) 2 ] + 1 4 π [ 0 ] = 1 2

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

11th Edition

ISBN: 9780137554843

Author: Allyn Washington, Richard Evans

Publisher: PEARSON+

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Chapter 30.6, Problem 21E

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To find: The Fourier series of the function that represents the force.

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Chapter 30.6, Problem 20E

Chapter 30.6, Problem 22E

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Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

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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