We said above that G(w) does not depend on the number of people or their mass, and that it increases with smaller B; why are these state- ments true? Which terms in Equation 5 persist for large times? For large times, what happens to the dependence of the solution on the initial conditions? What is the amplitude of the equilibrium oscillation in Equation 5? How does the gain function relate the amplitude of the equilibrium oscillation to the amplitude of a forcing term in the form of Equation 3? In particular, what is the maximum displacement of the bridge given a gain function G(w), a walking frequency w, a number of people N, and the lateral force per person W?
We said above that G(w) does not depend on the number of people or their mass, and that it increases with smaller B; why are these state- ments true? Which terms in Equation 5 persist for large times? For large times, what happens to the dependence of the solution on the initial conditions? What is the amplitude of the equilibrium oscillation in Equation 5? How does the gain function relate the amplitude of the equilibrium oscillation to the amplitude of a forcing term in the form of Equation 3? In particular, what is the maximum displacement of the bridge given a gain function G(w), a walking frequency w, a number of people N, and the lateral force per person W?
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Please show all work for Exercise 2 please!
![The general solution to Equation 3, unless \( \omega = \sqrt{\frac{K}{M}} \) and \( B = 0 \), is
\[
X(t) = c_1 e^{-\frac{Bt}{2M}} \cos \left( t \sqrt{\Omega_0^2 - \left( \frac{B}{2M} \right)^2} \right) + c_2 e^{-\frac{Bt}{2M}} \sin \left( t \sqrt{\Omega_0^2 - \left( \frac{B}{2M} \right)^2} \right)
\]
\[
+ \frac{NW}{\sqrt{M^2 (\Omega_0^2 - \omega^2)^2 + B^2 \omega^2}} \cos \left( \omega t - \text{atan} \left( \frac{B\omega}{M(\omega^2 - \Omega_0^2)} \right) \right)
\]
where \( \Omega_0^2 = \frac{K}{M} \) and \(\text{atan}(x) = \text{arctan}(x)\), the inverse tangent of \( x \). It is useful to have a measure of how a certain system (for this lab, the bridge), amplifies or reduces forces (resulting from walking humans) applied to it. The ratio of the amplitude of the input (i.e., the right-hand side of the equation) to the long-term amplitude of the output (i.e., the solution to the differential equation) as a function of input frequency is called gain (we explore this in class in §4.6 of Brannan & Boyce). The gain function for Equation 3 is
\[
G(\omega) = \frac{1}{\sqrt{M^2(\Omega_0^2 - \omega^2)^2 + B^2 \omega^2}}
\]
There are a number of important observations to make about \( G(\omega) \): (1) because it's a ratio of output to input amplitudes, it doesn't depend on the input amplitude (for our bridge, the number of people or their mass); (2) it gets larger as the damping \( B \) decreases; and (3) in general, it gets larger as the forcing frequency \( \omega](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ea01508-f9b6-4a19-b038-56e12c298daf%2Fd2dfe426-248a-49e8-9c65-f92214e64774%2Fek8h27r_processed.png&w=3840&q=75)
Transcribed Image Text:The general solution to Equation 3, unless \( \omega = \sqrt{\frac{K}{M}} \) and \( B = 0 \), is
\[
X(t) = c_1 e^{-\frac{Bt}{2M}} \cos \left( t \sqrt{\Omega_0^2 - \left( \frac{B}{2M} \right)^2} \right) + c_2 e^{-\frac{Bt}{2M}} \sin \left( t \sqrt{\Omega_0^2 - \left( \frac{B}{2M} \right)^2} \right)
\]
\[
+ \frac{NW}{\sqrt{M^2 (\Omega_0^2 - \omega^2)^2 + B^2 \omega^2}} \cos \left( \omega t - \text{atan} \left( \frac{B\omega}{M(\omega^2 - \Omega_0^2)} \right) \right)
\]
where \( \Omega_0^2 = \frac{K}{M} \) and \(\text{atan}(x) = \text{arctan}(x)\), the inverse tangent of \( x \). It is useful to have a measure of how a certain system (for this lab, the bridge), amplifies or reduces forces (resulting from walking humans) applied to it. The ratio of the amplitude of the input (i.e., the right-hand side of the equation) to the long-term amplitude of the output (i.e., the solution to the differential equation) as a function of input frequency is called gain (we explore this in class in §4.6 of Brannan & Boyce). The gain function for Equation 3 is
\[
G(\omega) = \frac{1}{\sqrt{M^2(\Omega_0^2 - \omega^2)^2 + B^2 \omega^2}}
\]
There are a number of important observations to make about \( G(\omega) \): (1) because it's a ratio of output to input amplitudes, it doesn't depend on the input amplitude (for our bridge, the number of people or their mass); (2) it gets larger as the damping \( B \) decreases; and (3) in general, it gets larger as the forcing frequency \( \omega
Transcribed Image Text:Certainly! Below is the transcription and explanation for an educational context:
---
### Transcription of the Text:
1. **(1)** We said above that \( G(\omega) \) does not depend on the number of people or their mass, and that it increases with smaller \( B \); why are these statements true?
2. **(2)** Which terms in Equation 5 persist for large times? For large times, what happens to the dependence of the solution on the initial conditions?
3. **(3)** What is the amplitude of the equilibrium oscillation in Equation 5? How does the gain function relate the amplitude of the equilibrium oscillation to the amplitude of a forcing term in the form of Equation 3? In particular, what is the maximum displacement of the bridge given a gain function \( G(\omega) \), a walking frequency \( \omega \), a number of people \( N \), and the lateral force per person \( W \)?
### Explanation of the Questions:
1. **Understanding \( G(\omega) \) and \( B \):**
- The function \( G(\omega) \) is discussed in the context of its independence from the number of people or their mass. The question prompts a deeper investigation into why this independence occurs and how the function is influenced by the parameter \( B \), suggesting that \( G(\omega) \) increases as \( B \) decreases.
2. **Persistence of Terms in Long Time Scales:**
- This question focuses on the analysis of Equation 5 over large time periods. It asks which terms remain significant and whether the solution's reliance on initial conditions diminishes over time, a typical inquiry in the study of dynamic or differential equations.
3. **Amplitude of Equilibrium Oscillation:**
- The inquiry revolves around understanding the effect of parameters on bridge oscillations. The emphasis is on determining how the gain function \( G(\omega) \) connects the amplitude of the equilibrium oscillation to the force applied by walking people (given by Equation 3). It also seeks to calculate the maximum displacement of a bridge, considering variables like walking frequency \( \omega \), number of people \( N \), and lateral force \( W \).
---
This text is written to guide students or readers through an analysis of dynamic systems, particularly in physics or engineering scenarios involving oscillations and forcing functions.
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