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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00c227c7-5e7d-4c97-86f7-329995c469cc%2Febe94036-6ada-4299-a02f-b27887713dae%2F9yysoft_processed.png&w=3840&q=75)
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
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