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Message ID: 319     Entry time: Fri Feb 24 14:44:22 2017
Author: Gabriele 
Type: General 
Category: General 
Subject: Disk model 

It's possible to build an analytical model of the resonant frequencies of a simple thin disk. For example, see J. Sound and Vibrations 188, 685 (1995), section 2

The solutions are given in term of Bessel functions:

W_{mn}(r, \theta)=\left[ J_m(\lambda_{mn} r/a) + C_{mn} I_m(\lambda_{mn} r/a)\right] \cos m \theta

where J is a Bessel function of the first kind, and I a modified Bessel function of the first kind, a is the disk radius.

The coefficient Cmn and the eigenvalue can be found as solution of the following two equations

Then the eigenfrequencies are given by

f_{mn} = \frac{\lambda_{mn}^2}{2\pi a^2}\sqrt{\frac{D}{\rho h}}

where rho is the material density, h the disk thickness and D the flexural rigidity

D =\frac{E h^3}{12(1-\nu^2)}

where E is the material Young's modulus and nu the Poisson's ratio.

From all those results we can conclude that the frequency scaling with respect to disk radius and thickness are very simple:

  • the frequencies scale linearly with the thickness
  • the frequencies scale inversely proportionally to the square of the radius

Also, the frequencies scales as sqrt(E/rho)

f_{nm} \propto \sqrt{\frac{E}{\rho}} \frac{h}{a^2}

The dependency on the Poisson's ratio is more complex since nu is involved in the eigenfrequency equation shown above.

Unfortunately the thin disk model does not exactly match the COMSOL results: deviations of few tens of Hz are present, probaly due to the thin disk approximation. The COMSOL model is more accurate to match the experimental frequencies.

However, I checked that the eigenfrequencies predicted by COMSOL also scales as predicted with thickness and radius.

Using the measurements on the six samplex we got from Mark Optics, after annealing, I was able to tune the COMSOL model to fit all measured frequencies within 6 Hz. I chose to change the disk thickness (since diameter and Young'r modulus are degenerate) and the Poisson's ratio. 

Here is an example of the difference between the measured and modeled frequencies:

The table below summarizes the best fit for each of the disks

SN Thickness [mm] Poisson's ratio
S1600519 1.0194 0.1669
S1600520 1.0186 0.1663
S1600521 1.0217 0.1669
S1600522 1.0223 0.1635
S1600523 1.0227 0.1655
S1600524 1.0209 0.1654

Since the material is the same, I would expect the Poisson's ratio to be constant. So for future modeling I'm using the average of the values above: 0.166

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