Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Express the periodic loading shown using Fourier's series expansion. F(t) F(t)= F sin 171 ) o

Express the periodic loading shown using Fourier's series expansion. F(t) F(t)= F sin 171 ) o

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 78E
See similar textbooks
icon
Related questions
Question
--- **Title: Fourier Series Expansion of Periodic Loading Function** --- **Fourier Series Expansion** In this example, we are tasked with expressing a given periodic loading function using Fourier's series expansion. ### Given Function The periodic loading function \( F(t) \) is defined piecewise as shown below: \[ F(t) = \begin{cases} F_0 \sin \left( \frac{3 \pi}{T_n} t \right) & 0 < t < \frac{2 T_n}{3} \\ 0 & \frac{2 T_n}{3} < t < T_n \end{cases} \] ### Graph of \( F(t) \) The graph provided shows a piecewise periodic function \( F(t) \). The specific intervals are defined by \( \frac{T_n}{3} \), \( \frac{2 T_n}{3} \), and \( T_n \), aligned with the piecewise definition above: 1. From \( t = 0 \) to \( t = \frac{2 T_n}{3} \): \( F(t) \) is a sinusoidal function given by \( F_0 \sin \left( \frac{3 \pi}{T_n} t \right) \). 2. From \( t = \frac{2 T_n}{3} \) to \( t = T_n \): \( F(t) \) is 0. The graph clearly illustrates these intervals, showing the sinusoidal wave in the first section and a flat line at zero in the second section. Each cycle repeats after \( t = T_n \). ### Fourier Series Representation The goal is to represent this function, \( F(t) \), using a Fourier series, which is a sum of sinusoidal functions (sines and cosines). A Fourier series expansion takes the form: \[ F(t) = A_0 + \sum_{n=1}^{\infty} \left[ A_n \cos \left( \frac{2 n \pi}{T_n} t \right) + B_n \sin \left( \frac{2 n \pi}{T_n} t \right) \right] \] Where: - \( A_0 \) is the average value of the function over one period. - \( A_n \) and \( B_n \) are the Fourier coefficients and represent the
Transcribed Image Text:--- **Title: Fourier Series Expansion of Periodic Loading Function** --- **Fourier Series Expansion** In this example, we are tasked with expressing a given periodic loading function using Fourier's series expansion. ### Given Function The periodic loading function \( F(t) \) is defined piecewise as shown below: \[ F(t) = \begin{cases} F_0 \sin \left( \frac{3 \pi}{T_n} t \right) & 0 < t < \frac{2 T_n}{3} \\ 0 & \frac{2 T_n}{3} < t < T_n \end{cases} \] ### Graph of \( F(t) \) The graph provided shows a piecewise periodic function \( F(t) \). The specific intervals are defined by \( \frac{T_n}{3} \), \( \frac{2 T_n}{3} \), and \( T_n \), aligned with the piecewise definition above: 1. From \( t = 0 \) to \( t = \frac{2 T_n}{3} \): \( F(t) \) is a sinusoidal function given by \( F_0 \sin \left( \frac{3 \pi}{T_n} t \right) \). 2. From \( t = \frac{2 T_n}{3} \) to \( t = T_n \): \( F(t) \) is 0. The graph clearly illustrates these intervals, showing the sinusoidal wave in the first section and a flat line at zero in the second section. Each cycle repeats after \( t = T_n \). ### Fourier Series Representation The goal is to represent this function, \( F(t) \), using a Fourier series, which is a sum of sinusoidal functions (sines and cosines). A Fourier series expansion takes the form: \[ F(t) = A_0 + \sum_{n=1}^{\infty} \left[ A_n \cos \left( \frac{2 n \pi}{T_n} t \right) + B_n \sin \left( \frac{2 n \pi}{T_n} t \right) \right] \] Where: - \( A_0 \) is the average value of the function over one period. - \( A_n \) and \( B_n \) are the Fourier coefficients and represent the
Expert Solution
steps

Step by step

Solved in 9 steps with 9 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer