Finally, here are the simulation results of dude500's previous phones (large earcup, thin pads). One important thing is that I realized how critical the pads construction / configuration / size affects the results, in particular the fundamental resonance of the diaphragm (it is heavily influenced by the effective mass and stiffness of the earpad cavity which in turns depends on the earpad to some extent).
Since I only had one dimension with work with (old earpad inner surface area), everything else is approximate base of the pics posted by dude500 and of coarse I did not have any of the materials properties so I don't expect things to line up with tests. However, it was very good learning experience to find out about the importance of the pads.
First, a simulation of the response of the headphone in free air for mics along the vertical axis between the diaphragm and base of the earcup. A couple of notes: this type of simulation is based on "Boundary Element Method" in which you only mesh the boundaries of the domain, compute the pressure and particle velocity at every nodes of the surface(s) and can then post-calculate the SPL at any point in the domain. The nice thing is that this enables calculation in free field / over large volumes:
Below is a contour plot of the diaphragm velocity and SPL around the headphone at a single frequency (~100Hz, the free air resonance of the diaphragm):
And here is the 3rd resonance of the diaphragm (+/-/+ across the length) at about 300Hz.
Note that the 2nd resonance (+/- across the length) isn't visible in the SPL response because this "mode" is a very poor "radiator" (+/- cancellation over the surface).
Next step consisted in comparison the results from "BEM" simulation above to those of more traditional "Finite Element Method" where the whole acoustic volume is meshed. In that case, "rigid walled" acoustic modes are first computed and combined to "in-vaccuo" structural modes of the diaphragm during a coupled structural/acoustic solution. The main limitation is that you cannot resolve the response outside the meshed domain and computations get expensive with increasing domain size. Special care needs to be taken to make sure you don't get reflections from the boundary of the exterior domain, it then returns reasonably close results to the BEM approach (red vs. black curve). Note the drastic shift in the diaphragm resonance frequency compared to free air response:
Contour plot around 3.5kHz where the SPL dips at the center of the earcup (this is an acoustic resonance in the earcup). First the BEM model, next the FEM model:
Then comes the question why bother with FEM instead of using BEM? The reason is to be able to include more complicated earpad dynamics. In particular, in the BEM and FEM simulations above, the earpad effect was modeled as just a surface impedance (using infinite size trim model discussed in previous post) with every node acting independently of the other. It turns out this simplified earpad model totally misses the physics at low frequency, in particular because the earpad has such large influence on the first diaphragm resonance frequency. Here's a comparison with a "FEM/PEM" coupled model with the earpad (PU foam + leather) modeled in 3D using "Porous Finite Elements". The pad is assumed bounded to on the diaphragm side and "sliding" on the base (the surface in contact with the skin). You can see the diaphragm resonance (above 25Hz) shifted back to about 100Hz and you barely see the blip (I now understand why Stax headphones measure like they do below 100Hz: this is actually a coupled diaphragm / earcup cavity / earpad resonance):
FEM model (rigid earpad) at ~25Hz (note the SPL response inside / outside is quite boring, this is because we're at very low frequency):
FEM/PEM (3D earpad visible in grey color below) at ~100Hz:
So, this brought me to the next stage. Now, it was clear I couldn't cut corners and had to include a full 3D model of the earpad. I did the rest of the simulations using this coupled FEM/PEM approach and investigated the sensitivity of the response to the earpad. I first changed the foam (PU foam) stiffness drastically. The disappointing thing is that I could not reproduce the huge 1kHz peak in dude500's response (it's rather easy to imagine though since I probably used the wrong material properties, foam dimensions, leather properties/dimensions and especially mounting configuration...). But the interesting thing was how sensitive the low frequency response was to the foam used. Below I am showing the SPL in the middle of the earcup, diaphragm velocity response and contour plot at 60Hz (soft foam only):
At last, I investigated the effect of removing the leather and having a porous ("unsealed") or smooth foam surface ("sealed", impervious...). The motivation was that I did not know the actual leather properties / thickness and noticed that dude500 was only partly covering the foam with leather. Below are the SPL and diaphragm velocity, effect is pretty drastic:
Conclusion: well, it's no surprise to some of you maybe but indeed the earpad matters a lot more than I thought! This actually has taught me a thing or two and I will be able to revisit my 009 model next...