For small positive damping, c, in the sinusoidally forced mass-spring system mz" + cz' + kr = Ggcos (wt) there is a value of the forcing frequency w for which the amplitude of the response is greatest - a resonance peak. very light damping maderate daming very heay dampng tecing freeency Derive a formula for the location of the peak in terms of the parameters in the differential equation (m, c, k, Go). So the location of the peak depends on the damping - Where does the peak move to as the damping constant cis reduced to zero? Does the above frequency have a name? ptude of remponse
For small positive damping, c, in the sinusoidally forced mass-spring system mz" + cz' + kr = Ggcos (wt) there is a value of the forcing frequency w for which the amplitude of the response is greatest - a resonance peak. very light damping maderate daming very heay dampng tecing freeency Derive a formula for the location of the peak in terms of the parameters in the differential equation (m, c, k, Go). So the location of the peak depends on the damping - Where does the peak move to as the damping constant cis reduced to zero? Does the above frequency have a name? ptude of remponse
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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![For small positive damping, \( c \), in the sinusoidally forced mass-spring system
\[ m x'' + cx' + kx = G_0 \cos(\omega t) \]
there is a value of the forcing frequency \( \omega \) for which the amplitude of the response is greatest - a resonance peak.
### Graph:
The graph illustrates the relationship between the forcing frequency \(\omega/\omega_n\) (along the x-axis) and the amplitude of response (along the y-axis) for various damping conditions: very light damping (blue), moderate damping (green), and very heavy damping (red). The y-axis shows a peak for very light damping, indicating a resonance at a certain frequency. As damping increases, the peak becomes less pronounced and broadens, illustrating how damping affects the system's response to forcing frequencies.
### Questions to Explore:
1. **Derive a formula for the location of the peak in terms of the parameters in the differential equation \((m, c, k, G_0)\).**
\(\omega = \)
[Input formula]
2. **So the location of the peak depends on the damping…**
3. **Where does the peak move to as the damping constant \( c \) is reduced to zero?**
\(\omega = \)
[Input formula]
4. **Does the above frequency have a name?**
[Input name]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb493ecdd-cbdc-400d-a05a-de2010eb2d52%2F552658e7-4745-4092-942d-57c3862481a4%2F7pnk6aq_processed.png&w=3840&q=75)
Transcribed Image Text:For small positive damping, \( c \), in the sinusoidally forced mass-spring system
\[ m x'' + cx' + kx = G_0 \cos(\omega t) \]
there is a value of the forcing frequency \( \omega \) for which the amplitude of the response is greatest - a resonance peak.
### Graph:
The graph illustrates the relationship between the forcing frequency \(\omega/\omega_n\) (along the x-axis) and the amplitude of response (along the y-axis) for various damping conditions: very light damping (blue), moderate damping (green), and very heavy damping (red). The y-axis shows a peak for very light damping, indicating a resonance at a certain frequency. As damping increases, the peak becomes less pronounced and broadens, illustrating how damping affects the system's response to forcing frequencies.
### Questions to Explore:
1. **Derive a formula for the location of the peak in terms of the parameters in the differential equation \((m, c, k, G_0)\).**
\(\omega = \)
[Input formula]
2. **So the location of the peak depends on the damping…**
3. **Where does the peak move to as the damping constant \( c \) is reduced to zero?**
\(\omega = \)
[Input formula]
4. **Does the above frequency have a name?**
[Input name]
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