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 the same with modern variant, because the plate is excited from the center with a Melde Vibrator.

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 my code below which shows that something is wrong. Maybe is it my scientific thought?

``(*---------------------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*)  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] ``

Below is depicted my own Chladni experiment (my first one, not very cute but I will do better the next time). 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. You will see the frequency values are not matching the values computed with my Mathematica code. Thank you in advance for your answers.

Posted on Categories cheapest proxies