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