Hi Eric
I repeated my run with a dedicated mesh (before I had modified the inputs of an aspect ratio test). Same results to 4 significant figures. I suspect we are looking at differences in meshing with all our runs, because I doubt that 4mm is thick enough to show any deficiency in thin plate theory.
I specified a 5mm grid in the neighborhood of all points. I don't know how that translates to a triangular grid, but the triangle seems to be ~20mm on each side.
I doubt we will ever get to a point where the higher accuracy of the thick plate theory will matter. I'm mainly looking for trends and indications of improvement.
I repeated my run with a dedicated mesh (before I had modified the inputs of an aspect ratio test). Same results to 4 significant figures. I suspect we are looking at differences in meshing with all our runs, because I doubt that 4mm is thick enough to show any deficiency in thin plate theory.
I specified a 5mm grid in the neighborhood of all points. I don't know how that translates to a triangular grid, but the triangle seems to be ~20mm on each side.
I doubt we will ever get to a point where the higher accuracy of the thick plate theory will matter. I'm mainly looking for trends and indications of improvement.
Here's another run with a 'true' 5mm grid. I found an option that was setting a hard limit on the grid size, explaining why my 5mm gid was actually ~20mm.
Cant tell if its any better just browsing a few values... Looks like there are more modes, so maybe in-plate modes? Dunno.
6.554225947209E+001 8.086218961498E+001 1.064642682221E+002 1.423747305539E+002 1.885945210975E+002 2.451182170946E+002 2.468906831794E+002 2.621745856817E+002 2.876821363936E+002 3.119417256610E+002 3.234495808750E+002 3.695043426655E+002 3.890598467747E+002 4.258638949652E+002 4.764696327141E+002 4.925395475418E+002 5.491697454286E+002 5.644408555318E+002 5.695305292589E+002 5.741704960837E+002 5.899110757467E+002 6.256058875852E+002 6.568343008481E+002 6.715501051795E+002 6.821599859572E+002 7.277674398621E+002 7.544508862144E+002 7.942810210743E+002 8.004402279101E+002 8.623729809290E+002 8.711012605289E+002 9.290080572819E+002 9.582351393185E+002 9.724180774173E+002 9.806059581032E+002 9.876794931535E+002 1.013126150167E+003 1.048777251983E+003 1.055691385887E+003 1.067864138528E+003 1.094651587263E+003 1.109138359968E+003 1.150770980257E+003 1.163465520040E+003 1.217010951662E+003 1.217159932444E+003 1.247970424852E+003 1.281570822592E+003 1.293833276388E+003 1.376446142470E+003 1.380804207814E+003 1.397108807456E+003 1.409991244641E+003 1.478089034204E+003 1.516692135563E+003 1.531943766485E+003
Cant tell if its any better just browsing a few values... Looks like there are more modes, so maybe in-plate modes? Dunno.
6.554225947209E+001 8.086218961498E+001 1.064642682221E+002 1.423747305539E+002 1.885945210975E+002 2.451182170946E+002 2.468906831794E+002 2.621745856817E+002 2.876821363936E+002 3.119417256610E+002 3.234495808750E+002 3.695043426655E+002 3.890598467747E+002 4.258638949652E+002 4.764696327141E+002 4.925395475418E+002 5.491697454286E+002 5.644408555318E+002 5.695305292589E+002 5.741704960837E+002 5.899110757467E+002 6.256058875852E+002 6.568343008481E+002 6.715501051795E+002 6.821599859572E+002 7.277674398621E+002 7.544508862144E+002 7.942810210743E+002 8.004402279101E+002 8.623729809290E+002 8.711012605289E+002 9.290080572819E+002 9.582351393185E+002 9.724180774173E+002 9.806059581032E+002 9.876794931535E+002 1.013126150167E+003 1.048777251983E+003 1.055691385887E+003 1.067864138528E+003 1.094651587263E+003 1.109138359968E+003 1.150770980257E+003 1.163465520040E+003 1.217010951662E+003 1.217159932444E+003 1.247970424852E+003 1.281570822592E+003 1.293833276388E+003 1.376446142470E+003 1.380804207814E+003 1.397108807456E+003 1.409991244641E+003 1.478089034204E+003 1.516692135563E+003 1.531943766485E+003
Hello Eric
Thank you for this feedback. I agree the exercise was very good for me too.
- Up to now as my script was running without external comparison, I remember decided not take care of the Poisson's coeff as you mentioned due to the errors that exists in other terms. Correction done.
- No problem with the dimensions. It show me an other improvement not too difficult to implement to allow rectangular (not too rectangular!) tiles.
Very good!
I wonder also why the 2 Elmer simulations are slightly different. Probably the mesh. I don't know where to read this information in Elmer. 36x121 is the number of points in the FDM. As the points are outside the elements, it is a simple 35x120 10mm square cell grid. FDM works at simple points.
Paul,
I'll add these tonight, if I can get my wife to dictate the numbers for me again!
Eric
Could be, but I still wonder.
I originally got LISA when I was building wind chimes and adding weights to change the timbre. Kind of like a one dimensional impact driven DML! Anyway, I was really surprised to see how poorly the Euler beam equations predicted anything beyond the 2nd natural frequency. Even for chimes that were pretty long, like 60 L/D.
But I agree, no doubt the trends will be there with or without thick beam theory.
Eric
I can access the data from the vtu file output by Elmer.
So now, looking again at the equations from Putra/Thompson paper (another good one is Xie and Thompson) it looks like we should be able to do a simple numerical integral to estimate far field sound at a point.
The Rayleigh integral adds up the contributions from each point on the panel. The 1/r term accounts for distance (why not 1/r^2 ? ). The complex exponential accounts for the phase effect of each point on the panel with regard to its distance from the observation point. In any case, we can do the calcs directly in complex numbers.
The velocity term is given by the mode shape (which we have) times a constant Umn
Umn is given by this expression below, which also makes sense. The j makes sense to me because it represents a 90 degree phase shift between maximum displacement and velocity. F is the applied force, which we would assume constant for our exciter in the first instance. Mmn is also an integral over the shape function which we can calculate. The phi(x0, y0) represents the forcing location - if its at an antinode, it causes stronger excitation as we know.
The term in square brackets represents the resonance peak, which would cause Umn to go to infinity without damping. Omegamn is the modal frequency (which we have) Eta is the damping loss factor, which we can estimate. Seems like there is an absolute value missing? I don't think we would want Umn to be negative?
Anyhow seems to me if we get a few details straight, we can use these directly to estimate power at a given point in the far field from any mode due to point excitation on the panel. Should be useful for comparative purposes, particularly for strange mode shapes to tell how productive they are.
So now, looking again at the equations from Putra/Thompson paper (another good one is Xie and Thompson) it looks like we should be able to do a simple numerical integral to estimate far field sound at a point.
The Rayleigh integral adds up the contributions from each point on the panel. The 1/r term accounts for distance (why not 1/r^2 ? ). The complex exponential accounts for the phase effect of each point on the panel with regard to its distance from the observation point. In any case, we can do the calcs directly in complex numbers.
The velocity term is given by the mode shape (which we have) times a constant Umn
Umn is given by this expression below, which also makes sense. The j makes sense to me because it represents a 90 degree phase shift between maximum displacement and velocity. F is the applied force, which we would assume constant for our exciter in the first instance. Mmn is also an integral over the shape function which we can calculate. The phi(x0, y0) represents the forcing location - if its at an antinode, it causes stronger excitation as we know.
The term in square brackets represents the resonance peak, which would cause Umn to go to infinity without damping. Omegamn is the modal frequency (which we have) Eta is the damping loss factor, which we can estimate. Seems like there is an absolute value missing? I don't think we would want Umn to be negative?
Anyhow seems to me if we get a few details straight, we can use these directly to estimate power at a given point in the far field from any mode due to point excitation on the panel. Should be useful for comparative purposes, particularly for strange mode shapes to tell how productive they are.
A trick? Copy paste the frequencies in a text editor (wordpad?) Save as a text file (.txt, .cvs...). Open the file with Excel or Calc with the import options from a text file (I don't have all the details in mind for Excel) with a "space" as separator. Probably some adaptions needed but should be closed.
Christian
Paul,
Yep, it's better. Nearly perfect it appears, at least compared to the exact. It's the blue line right labelled Elmer 5 mm. Nice.
I can't do a mesh that fine with LISA. Not sure if it's my laptop of LISA but something runs out of something....
Eric
Paul,
I am interested in the way you get the data from the vtu file. Which Python library? vtk?
For the procedure above, it is what I have found also.
There is a missing element for me which is the link between the mode shape from elmer and the mode shape in this procedure. The papers about this procedure are based if I remember well on an "exact" calculation". Is it the same from Elmer (the question is valid for FDM... I should be able to check it!).
I don't think there is a problem in Umn being negative as it carries a phase idea too
Christian
The 'exact' solution just happens to be from a formula, but how closely does it model reality compared with Mindlin?
Good test that shows a possible improvement behind each deviation from the theoretical values. For the FDM script I am used to see increasing error with the mode number when the mesh is not fine enough. Before seeing those graphs, I haven't noticed it is almost a straight line. The FDM in orange is a 10mm mesh. Is a correction function from that particular case might be applied to other boundary condition? No interest for FEM as far as the computation time is acceptable, more interesting for the FDM. The interest I see for FDM and I should test next (except if an other solution comes from our exchanges) is the possibility of local "devices" like spring, damper, mass attached to each point (starting from the exciter location). I still wonder how the modes are modified mainly in light membrane because of the exciter moving mass and spring (spider).
My thoughts are this: Of course the exact solution applies strictly only to an imaginary plate, ie one that is vanishingly thin. So I would hope and expect the Mindlin model to compare better with a real plate than the exact solution would. But also I expect that the Mindlin model would converge to the exact solution when used to model a very very thin plate, as long as the model has a sufficiently fine mesh. And I think that’s exactly what our combined results are showing us. Your 5 mm mesh is sufficiently fine, while all the other shell/plate FEM results are close but not quite there. Today I was able to run a Lisa model with about 3000 elements ( my earlier model was 2000 elements) and the results fell closer to yours than the coarser model, but still it didn’t converge to the exact solution. It’s the best I can do though with LISA for now.
Today I also tried modeling thicker plates which should reveal the effect of shear deformation better than the 4 mm thick plate we have been working with. I doubled the thickness in a series from 4 to 8 then 16 and finally 32 mm. For each doubling of the thickness I divided E by a factor of four, to give the same D in every case. The result was that for every step in thickness the new frequencies fell farther below the previous thinner case. Also the separation between the curves increased as frequency increased. This is exactly (qualitatively at least) what I was expecting/hoping to see from the beginning with these tests. I’m feeling more and more confident in our models myself.
Eric
Meshio seems to work, and there is an example on the computational acoustics site. I have not investigated closely yet though.
My guess is that the eigenmodes of more complex shapes have the same relationship to these formulas as does the simple rectangular sin/sin case. Normally however there is another step of harmonic analysis that takes as input the modes calculated during modal analysis. But I feel that as a first cut, if we can get a realistic numerical proxy for ‘productive’ modes, that will tell us whether these more exotic shapes hold promise for improvement of frequency response.
Here are the modeling results I mentioned in my previous post. Otherwise same dimensions and properties as previous results.
I used these parameters:
thickness, E
4 mm , 8e9 Pa
8 mm, 2e9 Pa
16 mm, 5e8 Pa
32 mm, 1.25e8 Pa
I believe that the results for the 4 and 8 mm thickness panels land slightly above the "Exact" solution only because my mesh isn't quite fine enough. I expect if it was fine enough, the 4 mm results would land ever so slightly below the Exact solution, and the 8 mm result just below the 4 mm (as it is now). But other than that, the results look like I would expect. As the plate gets thicker, the farther the result lands below the exact solution.
Plywood plates are typically around 4 mm thick or less, so it looks like that pretty much qualifies as a "thin" plate, at least up to 1000 Hz. But PS foam plates tend to be thicker, closer to the 16 mm lines, and hence are probably best considered "thick-ish" plates for which the Exact solution isn't so good, especially at high frequencies.
Oh yeah, I only plotted four frequencies this time. One reason is that fewer points were easier to plot! But the other is that I noticed that increasing thickness had stronger effect on some types of modes than others, so the "order" of the modes changed. In this case I plotted the 1,1 2,2 3,3 and 4,4 modes only.
Eric
That will be interesting to see. I have noticed in my tap test that with 4 mm plywood panels there is virtually no change to the natural frequencies after adding an exciter. But with a 12 mm PS foam panel, the fundamental frequency increases, say, from 60 Hz to 80 Hz, or the like when the exciter is added. I'm not sure if the same increase extends to higher modes, I just can't recall.
Eric