# Finding Omega and Zeta from a Magnitude and Phase Plot

1. Mar 1, 2013

### Loppyfoot

1. The problem statement, all variables and given/known data
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?

2. Mar 2, 2013

### Staff: Mentor

The -3dB bandwidth will give you Q, and Q is related to ζ

3. Mar 2, 2013

### Loppyfoot

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

4. Mar 2, 2013

### Staff: Mentor

I suggest that you try a google search for the details.

5. Mar 2, 2013

### Loppyfoot

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?

6. Mar 2, 2013

### Staff: Mentor

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?

7. Mar 3, 2013

### Loppyfoot

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?

8. Mar 3, 2013

### Staff: Mentor

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°.