Finding Omega and Zeta from a Magnitude and Phase Plot

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Discussion Overview

The discussion revolves around estimating the natural frequency (ωn) and damping ratio (ζ) from the magnitude and phase plot of an atomic force microscope (AFM). Participants explore methods for deriving these values, including the use of -3dB bandwidth and the characteristics of underdamped systems.

Discussion Character

  • Homework-related
  • Technical explanation
  • Exploratory
  • Debate/contested

Main Points Raised

  • One participant mentions that the -3dB bandwidth can provide the quality factor (Q), which is related to the damping ratio (ζ).
  • Another participant seeks clarification on the origin of the -3dB bandwidth and its relationship to Q and ζ.
  • A participant reports successfully calculating the damping ratio but struggles to find the natural frequency, asking for suggestions.
  • It is proposed that for an underdamped system, the natural frequency is approximately equal to the frequency at which the response peaks, with a possible correlation to the 90° phase crossing.
  • One participant confirms the assumption that the AFM cantilever can be modeled as a second-order system and estimates the natural frequency to be around 72.7 kHz.
  • Another participant suggests that there is an equation linking the peak frequency and ζ back to ωn for more precision, noting that the phase behavior indicates the presence of higher-order terms in the system.

Areas of Agreement / Disagreement

Participants generally agree on the approach to estimate ωn and ζ from the phase and magnitude plots, but there are differing views on the specifics of the calculations and the implications of the phase behavior, indicating that the discussion remains unresolved.

Contextual Notes

Some assumptions about the system's behavior, such as its underdamped nature and the modeling as a second-order system, are present but not universally accepted. The relationship between Q, ζ, and the -3dB bandwidth is discussed but not fully clarified.

Loppyfoot
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Homework Statement


I've been asked to estimate ωn and ζ from the magnitude and phase plot of an atomic force microscope. The magnitude and phase plot is attached.

Does anyone know how to solve for these values?
W0nWUxM.png
 
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The -3dB bandwidth will give you Q, and Q is related to ζ
 
NascentOxygen said:
The -3dB bandwidth will give you Q, and Q is related to ζ

Could you elaborate? Where does this -3dB bandwidth come from? Is Q inversely related to the damping ratio, ζ?

Thanks for your help.
 
I suggest that you try a google search for the details.
 
I was able to calculate the value for the damping ratio; but I am having trouble finding the value of the natural frequency. Any ideas on how to solve for the natural frequency?
 
Loppyfoot said:
I was able to calculate the value for the damping ratio; but I am having trouble finding the value of the natural frequency. Any ideas on how to solve for the natural frequency?
Because the system is so underdamped, the natural frequency is practically equal to the frequency where the response peaks. (Which looks suspiciously co-incident with that 90° crossing on the phase plot, though off-hand I can't say that's right.) I suppose you are approximating this to a second-order system?
 
NascentOxygen said:
Because the system is so underdamped, the natural frequency is practically equal to the frequency where the response peaks. (Which looks suspiciously co-incident with that 90° crossing on the phase plot, though off-hand I can't say that's right.) I suppose you are approximating this to a second-order system?

Yes, I am assuming that the AFM cantilever is being modeled as a second order system. With the cantilever tune, the phase curve is set to 90° of the resonant frequency. So would I be close enough in estimating that the natural frequency is 72.7 kHz?
 
Loppyfoot said:
Yes, I am assuming that the AFM cantilever is being modeled as a second order system. With the cantilever tune, the phase curve is set to 90° of the resonant frequency. So would I be close enough in estimating that the natural frequency is 72.7 kHz?
Most likely. Though there is an equation relating frequency at the peak and ζ back to Ѡn if you really wanted to be precise.

The phase in a true second order system approaches 180° at infinity, so your system seems to have some higher order term because it apparently goes beyond 220°.
 

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