6. Consider a spring-mass system with small damping and driven by a cosine force: x" +0.01x' + 4x = cos 2t, x(0) = 0, x'(0) = 0. Find the solution and plot the result.
6. Consider a spring-mass system with small damping and driven by a cosine force: x" +0.01x' + 4x = cos 2t, x(0) = 0, x'(0) = 0. Find the solution and plot the result.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please do number 6
![### Transcription of Educational Content
**Figure Description:**
- **Figure 2.9** shows a graph of the solution \( x(t) = 10^{-3} (\cos(45t) - \cos(55t)) \) illustrating the phenomenon of beats. The plot displays oscillations over a time interval from \( t = 0 \) to \( t = 4 \).
**Exercises:**
4. **LC Circuit Resonance:**
Consider a general LC circuit with input voltage \( V_0 \sin \beta t \). If \( \beta \) and the capacitance \( C \) are known, what value of the inductance \( L \) causes pure resonance?
5. **Undamped Spring-Mass System:**
An undamped spring-mass system with mass \( m = 4 \) and stiffness \( k \) is forced by a sinusoidal function 412 sin 5t. What value of \( k \) causes pure resonance?
6. **Spring-Mass System with Damping:**
Consider a spring-mass system with small damping and driven by a cosine force:
\[
x'' + 0.01x' + 4x = \cos 2t, \quad x(0) = 0, \quad x'(0) = 0.
\]
Find the solution and plot the result.
7. **Equation Solution with Initial Conditions:**
Consider the equation
\[
x'' + \omega^2 x = \cos \beta t.
\]
a) Find the solution when the initial conditions are \( x(0) = x'(0) = 0 \) when \( \omega \neq \beta \).
b) Use the trigonometric identity \( 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \) to write the solution as a product of sines.
c) Take \( \omega = 55 \) and \( \beta = 45 \) and plot the solution from part (b) on the time interval \([0, 4]\). The solution can be interpreted as a high frequency response contained in a low frequency amplitude envelope. We see the phenomenon known as beats.
### Understanding the Graph
**Graph Explanation:**
The graph in Figure 2.9 is a visual representation of wave](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc10cce41-ef51-4545-a724-9736e236d3b7%2F5f7dca60-da48-4cba-b2a9-c720bd742474%2Feou0k4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription of Educational Content
**Figure Description:**
- **Figure 2.9** shows a graph of the solution \( x(t) = 10^{-3} (\cos(45t) - \cos(55t)) \) illustrating the phenomenon of beats. The plot displays oscillations over a time interval from \( t = 0 \) to \( t = 4 \).
**Exercises:**
4. **LC Circuit Resonance:**
Consider a general LC circuit with input voltage \( V_0 \sin \beta t \). If \( \beta \) and the capacitance \( C \) are known, what value of the inductance \( L \) causes pure resonance?
5. **Undamped Spring-Mass System:**
An undamped spring-mass system with mass \( m = 4 \) and stiffness \( k \) is forced by a sinusoidal function 412 sin 5t. What value of \( k \) causes pure resonance?
6. **Spring-Mass System with Damping:**
Consider a spring-mass system with small damping and driven by a cosine force:
\[
x'' + 0.01x' + 4x = \cos 2t, \quad x(0) = 0, \quad x'(0) = 0.
\]
Find the solution and plot the result.
7. **Equation Solution with Initial Conditions:**
Consider the equation
\[
x'' + \omega^2 x = \cos \beta t.
\]
a) Find the solution when the initial conditions are \( x(0) = x'(0) = 0 \) when \( \omega \neq \beta \).
b) Use the trigonometric identity \( 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \) to write the solution as a product of sines.
c) Take \( \omega = 55 \) and \( \beta = 45 \) and plot the solution from part (b) on the time interval \([0, 4]\). The solution can be interpreted as a high frequency response contained in a low frequency amplitude envelope. We see the phenomenon known as beats.
### Understanding the Graph
**Graph Explanation:**
The graph in Figure 2.9 is a visual representation of wave
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