Express the periodic loading shown using Fourier's series expansion. F(t) F(t)= (1) Oct 25 +++ 0 2T
Express the periodic loading shown using Fourier's series expansion. F(t) F(t)= (1) Oct 25 +++ 0 2T
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Fourier Series
The Fourier Series is a way to represent a function as the sum of simple sine waves. It decomposes periodic functions into sums of simpler, sinusoidal components.
#### Formula Breakdown
1. **Angular Frequency:**
\[
\Omega = \frac{2\pi}{T_n}
\]
2. **Piecewise Function Definition:**
\[
F(t) =
\begin{cases}
F_0 \sin \frac{3\pi}{T_n} t, & 0 < t < \frac{2\pi}{3} \\
0, & \frac{2\pi}{3} < t < T_n
\end{cases}
\]
3. **Fourier Series Representation:**
\[
F(t) = \frac{a_0}{2} + \sum a_n \cos n\Omega t + \sum b_n \sin n\Omega t
\]
Or, equivalently:
\[
F(t) = \frac{a_0}{2} + \sum a_n \cos \frac{2\pi n}{T_n} t + \sum b_n \sin \frac{2\pi n}{T_n} t
\]
4. **Coefficient Definitions:**
- \(a_0\)
\[
a_0 = \frac{2}{T_n} \int_0^{T_n} F(t) dt
\]
- \(a_n\)
\[
a_n = \frac{2}{T_n} \int_0^{T_n} F(t) \cos n \Omega t dt
\]
- \(b_n\)
\[
b_n = \frac{2}{T_n} \int_0^{T_n} F(t) \sin n \Omega t dt
\]
5. **Example Expansion:**
\[
F(t) = \frac{6F_0}{\pi} \left( \frac{1}{5} \cos \frac{\pi t}{T_n} - \frac{1}{7} \cos \frac{3 \pi t}{T_n} + \frac{2}{35} \cos \frac{5 \pi t}{T_n} - \frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00c227c7-5e7d-4c97-86f7-329995c469cc%2F60f59b3b-8b2e-46d9-a05d-d09903bd6dc9%2F5bmsnqk_processed.png&w=3840&q=75)
Transcribed Image Text:### Fourier Series
The Fourier Series is a way to represent a function as the sum of simple sine waves. It decomposes periodic functions into sums of simpler, sinusoidal components.
#### Formula Breakdown
1. **Angular Frequency:**
\[
\Omega = \frac{2\pi}{T_n}
\]
2. **Piecewise Function Definition:**
\[
F(t) =
\begin{cases}
F_0 \sin \frac{3\pi}{T_n} t, & 0 < t < \frac{2\pi}{3} \\
0, & \frac{2\pi}{3} < t < T_n
\end{cases}
\]
3. **Fourier Series Representation:**
\[
F(t) = \frac{a_0}{2} + \sum a_n \cos n\Omega t + \sum b_n \sin n\Omega t
\]
Or, equivalently:
\[
F(t) = \frac{a_0}{2} + \sum a_n \cos \frac{2\pi n}{T_n} t + \sum b_n \sin \frac{2\pi n}{T_n} t
\]
4. **Coefficient Definitions:**
- \(a_0\)
\[
a_0 = \frac{2}{T_n} \int_0^{T_n} F(t) dt
\]
- \(a_n\)
\[
a_n = \frac{2}{T_n} \int_0^{T_n} F(t) \cos n \Omega t dt
\]
- \(b_n\)
\[
b_n = \frac{2}{T_n} \int_0^{T_n} F(t) \sin n \Omega t dt
\]
5. **Example Expansion:**
\[
F(t) = \frac{6F_0}{\pi} \left( \frac{1}{5} \cos \frac{\pi t}{T_n} - \frac{1}{7} \cos \frac{3 \pi t}{T_n} + \frac{2}{35} \cos \frac{5 \pi t}{T_n} - \frac
![### Fourier Series Expansion of a Periodic Loading Function
**Objective:** Express the periodic loading function shown using Fourier's series expansion.
#### Mathematical Definition
The periodic loading function \( F(t) \) is defined as follows:
\[
F(t) =
\begin{cases}
F_0 \sin \left( \frac{3\pi}{T_n} t \right) & 0 < t < \frac{2T_n}{3} \\
0 & \frac{2T_n}{3} < t < T_n
\end{cases}
\]
Where:
- \( F_0 \) is the amplitude.
- \( T_n \) is the period.
#### Graphical Representation
The function \( F(t) \) is graphically represented in a diagram with respect to time \( t \), showing the periodic nature of the function within one period \( T_n \). Key features of the graph include:
- The function \( F(t) \) exhibits a sinusoidal waveform from \( t = 0 \) to \( t = \frac{2T_n}{3} \).
- From \( t = \frac{2T_n}{3} \) to \( t = T_n \), the function remains at zero value \( F(t) = 0 \).
#### Detailed Breakdown of the Graph
1. **Interval \( 0 < t < \frac{2T_n}{3} \):**
- The function \( F(t) \) follows a sinusoidal path, represented by \( F_0 \sin \left( \frac{3\pi}{T_n} t \right) \).
- The sine wave rises to its peak and then descends back to zero within this interval.
2. **Interval \( \frac{2T_n}{3} < t < T_n \):**
- The function \( F(t) \) is zero in this interval, indicating no loading.
The periodic nature of the function implies that this pattern repeats for subsequent periods \( T_n \), making it suitable for expansion using Fourier series techniques.
### Fourier Series Expansion
To find the Fourier series expansion of \( F(t) \), you would typically calculate the coefficients \( a_n \) and \( b_n \) using integrals over one period of the function. The periodic function can then be expressed as an infinite sum of sines and cosines:
\[
F(t](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00c227c7-5e7d-4c97-86f7-329995c469cc%2F60f59b3b-8b2e-46d9-a05d-d09903bd6dc9%2Fxfh9ryg_processed.png&w=3840&q=75)
Transcribed Image Text:### Fourier Series Expansion of a Periodic Loading Function
**Objective:** Express the periodic loading function shown using Fourier's series expansion.
#### Mathematical Definition
The periodic loading function \( F(t) \) is defined as follows:
\[
F(t) =
\begin{cases}
F_0 \sin \left( \frac{3\pi}{T_n} t \right) & 0 < t < \frac{2T_n}{3} \\
0 & \frac{2T_n}{3} < t < T_n
\end{cases}
\]
Where:
- \( F_0 \) is the amplitude.
- \( T_n \) is the period.
#### Graphical Representation
The function \( F(t) \) is graphically represented in a diagram with respect to time \( t \), showing the periodic nature of the function within one period \( T_n \). Key features of the graph include:
- The function \( F(t) \) exhibits a sinusoidal waveform from \( t = 0 \) to \( t = \frac{2T_n}{3} \).
- From \( t = \frac{2T_n}{3} \) to \( t = T_n \), the function remains at zero value \( F(t) = 0 \).
#### Detailed Breakdown of the Graph
1. **Interval \( 0 < t < \frac{2T_n}{3} \):**
- The function \( F(t) \) follows a sinusoidal path, represented by \( F_0 \sin \left( \frac{3\pi}{T_n} t \right) \).
- The sine wave rises to its peak and then descends back to zero within this interval.
2. **Interval \( \frac{2T_n}{3} < t < T_n \):**
- The function \( F(t) \) is zero in this interval, indicating no loading.
The periodic nature of the function implies that this pattern repeats for subsequent periods \( T_n \), making it suitable for expansion using Fourier series techniques.
### Fourier Series Expansion
To find the Fourier series expansion of \( F(t) \), you would typically calculate the coefficients \( a_n \) and \( b_n \) using integrals over one period of the function. The periodic function can then be expressed as an infinite sum of sines and cosines:
\[
F(t
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