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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
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]
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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