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+

See similar textbooks

Videos

Question

Chapter 30.6, Problem 21E

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 30.6, Problem 20E

Chapter 30.6, Problem 22E

Students have asked these similar questions

To: [Boss's Name] From: Nathaniel D Sain Date: 4/5/2025 Subject: Decision Analysis for Business Scenario Introduction to the Business Scenario Our delivery services business has been experiencing steady growth, leading to an increased demand for faster and more efficient deliveries. To meet this demand, we must decide on the best strategy to expand our fleet. The three possible alternatives under consideration are purchasing new delivery vehicles, leasing vehicles, or partnering with third-party drivers. The decision must account for various external factors, including fuel price fluctuations, demand stability, and competition growth, which we categorize as the states of nature. Each alternative presents unique advantages and challenges, and our goal is to select the most viable option using a structured decision-making approach. Alternatives and States of Nature The three alternatives for fleet expansion were chosen based on their cost implications, operational efficiency, and…

Golden Ratio search Method f(x) = 2x^3 - 3x^2 - 12x + 1 Golden ratio search rules 1.If f(x) < f(x2): 1. Eliminate all x values less than x2 2. X2 becomes the new a 3. x, becomes the new x2 4. no change in b If f(x) > f(x2): 1. Eliminate all x values greater than x 2. x, becomes the new b 3. x2 becomes the new x 4. no change in aquesion=Narrow the interval in which the minimizer of the function f is located using the golden search method, starting with the initial interval (0,6], until its width is less than 2. Then, accept the midpoint of this interval as an approximate value of the minimizer of the function fand determine it. (ф=0.62)According to the question above, fill in the table below using the algorithm until the appropriate place.please write every step by step in a verry comprehensive way

Business

Chapter 30 Solutions

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

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.