Chladni experiment verification

Question

I refurbished my post in order to be more understandable.

After computing simulations of Chladni patterns with Mathematica (see my previous topics), I finally went to practice. I realized my own experience. So, I have compared my results with the theory. And the obtained patterns are not really matching the expected ones. My hypothesis is that the boundary conditions I used in Mathematica are not the good ones. I think all simply-supported edges bc must be replaced by all fully free edges bc, because, in original Chladni experiment, the plate is “clamped” in the middle and excited from an edge. So, all edges are free. And it is more true with modern variant, because the plate is excited from the center with a Melde Vibrator, and the plate is no more clamped.

So my first question is: how to introduce the fully free edges boundary condition within the eigenvalues equation: Dirichlet? Neumann? I am a little bit lost.

My second question is: what is the formula to get f(m,n,a,b) - f=natural frequency - for a fully free ends plate (m and n are the mode coeff. and [a,b] the plate dimensions)?

You will find hereafter my code to illustrate all my questions:

(* My code to illustrate my questions *)
a = 0.18;
b = 0.18; h = 0.001;(*length,witdh,thickness in m*)
Ey = 2.1 10^11;(*N/m^2*)(*Young modulus*)
\[Rho] = 7800;(*kg/m^2*)(*density*)
\[Nu] = 0.3;(*Poisson coeff.*)
Df = (Ey h^3)/(12 (1 - \[Nu]^2))(*flexural rigidity*)
d = Sqrt[Df/(\[Rho] h)] (*coeff.correponding to the plate's \
mechanical behavior to introduce within Double Laplacian equations*)
eqnr = {-(d) Laplacian[u[x, y], {x, y}] + 
    v[x, y], -(d) Laplacian[
     v[x, y], {x, y}]};(bi-harmonic eigenvalue system)
(bi-harmonic eigenvalue system)
bcsr = DirichletCondition[u[x, y] == 0, True];
(BC used with SS plate/what is the equivalent for Fully Free Edges 

plate?)
{valr, funr} = 
   NDEigensystem[{eqnr, bcsr}, {u, v}, {x, 0, a}, {y, 0, b}, 
    80]; // Quiet
f = valr/(2 [Pi]) (to get all modal frequencies and functions)
Table[ContourPlot[Re[funr[[i, 1]][x, y]] == 0, {x, 0, a}, {y, 0, b}, 
  PlotRange -> All, PlotLabel -> Re[valr[[i]]/(2 Pi)] "Hz", 
  AspectRatio -> Automatic, ImageSize -> Tiny, 
  FrameTicks -> None], {i, 1, Length[valr]}]
(to get the nodal lines patterns which should match the results of 

my experiment-see photos below)
(below is the formula to get natural freqencies for a simply-supported plate)
fss[m_, n_] := [Pi]/2 Sqrt[
   Df/([Rho] h)] (m^2/a^2 + n^2/b^2);(Hz)TMss = 
 Table[fss[m, n], {m, 1, 9}, {n, 1, 9}];
(* Natural frequencies table (in Hz) computed with the well-known 

formula: *)
TableForm[TMss, 
 TableHeadings -> {{"m1", "m2", "m3", "m4", "m5", "m6", "m7", "m8", 
    "m9"}, {"n1", "n2", "n3", "n4", "n5", "n6", "n7", "n8", "n9"}}]
(what is the equivalent formula for a fully free edges plate?)
(* Animation of the plates vibrations: *)
ListAnimate[
 Table[Plot3D[Re[funr[[i, 1]][x, y]], {x, 0, a}, {y, 0, b}, 
   PlotRange -> All, PlotLabel -> Re[valr[[i]]/(2 Pi)] "Hz", 
   ColorFunction -> "Rainbow", AspectRatio -> Automatic], {i, 1, 
   Length[valr]}], AnimationRepetitions -> 1, DefaultDuration -> 20]

You will see the frequency values are not matching the values computed with my Mathematica code. Thank you in advance for your answers. You will find hereafter the diagram showing the details of the assembly.

---

In order to clarify this topic, you will find additional elements :

1/ The formula to compute the natural frequencies (or pulsations) I have found in the litterature is the following one :

fss[m_, n_] := \[Pi]/2 Sqrt[Df/(\[Rho] h)] (m^2/a^2 + n^2/b^2)

The table computed with the formula

There are frequency discrepencies between the table and the patterns computed with MMA code (see below).

This table applies for a Simply Supported plate. What is the good one for Fully free edges plate?

2/ The BC I used is the following one, but I think it s not correct for my purpose :

bcsr = DirichletCondition[u[x, y] == 0, True]

3/ Some pictures to improve the understanding

Under the Monacor exciter:

And the correponding assembly diagram:

The nodal patterns computed with the MMA code (there a lot of "intermediate" modes):

My own patterns (not all, because the big difficulty I met was to get a very accurate tuning, according the fact my wife was getting angry because of the high-pitched whistling generated by the devices):