Hello,
I have added at the point where is the exciter a mass and a compliance in the FDM script. Here is a first result for a material similar to plywood.
The simulation was for different type of "exciters" starting from case 0 no exciter, increasing the mass and various stiffness in the range of the Dayton Audio models. Case 1 is only a mass added.
For each case, the mode frequencies and an evaluation of the productivity of the mode.
It shows the modes are changed by the exciter and also the balance of the different areas. For simplicity, edges are here simply supported. I choose here the example of plywood to show there is an impact with on of the most stiffer and heavier material.
In the case of plywood, the exciter mass is low effect. The stiffness more.
Opinion and comments welcome
@pway @Veleric : is there a possibility to check the results are plausible with FEM?
The plate is Lx = 0.4 m by Ly = 0.6 m
The mesh is Nx = 21 cells by Ny = 31 cells
with a grid dx = 20 mm by dy = 20 mm
Bending stiffness B = 20.0 Nm, Areal density µ = 1.6 kg/m²
Case 0
Lumped element at 0.55 0.55
Mass = 0 kg, Stiffness = 0 N/m
Case 1
Lumped element at 0.55 0.55
Mass = 0.002 kg, Stiffness = 0 N/m
Case 2
Lumped element at 0.55 0.55
Mass = 0.002 kg, Stiffness = 4000 N/m
Case 3
Lumped element at 0.55 0.55
Mass = 0.003 kg, Stiffness = 50000 N/m
Case 4
Lumped element at 0.55 0.55
Mass = 0.005 kg, Stiffness = 20000 N/m
I have added at the point where is the exciter a mass and a compliance in the FDM script. Here is a first result for a material similar to plywood.
The simulation was for different type of "exciters" starting from case 0 no exciter, increasing the mass and various stiffness in the range of the Dayton Audio models. Case 1 is only a mass added.
For each case, the mode frequencies and an evaluation of the productivity of the mode.
It shows the modes are changed by the exciter and also the balance of the different areas. For simplicity, edges are here simply supported. I choose here the example of plywood to show there is an impact with on of the most stiffer and heavier material.
In the case of plywood, the exciter mass is low effect. The stiffness more.
Opinion and comments welcome
@pway @Veleric : is there a possibility to check the results are plausible with FEM?
The plate is Lx = 0.4 m by Ly = 0.6 m
The mesh is Nx = 21 cells by Ny = 31 cells
with a grid dx = 20 mm by dy = 20 mm
Bending stiffness B = 20.0 Nm, Areal density µ = 1.6 kg/m²
Case 0
Lumped element at 0.55 0.55
Mass = 0 kg, Stiffness = 0 N/m
Case 1
Lumped element at 0.55 0.55
Mass = 0.002 kg, Stiffness = 0 N/m
Case 2
Lumped element at 0.55 0.55
Mass = 0.002 kg, Stiffness = 4000 N/m
Case 3
Lumped element at 0.55 0.55
Mass = 0.003 kg, Stiffness = 50000 N/m
Case 4
Lumped element at 0.55 0.55
Mass = 0.005 kg, Stiffness = 20000 N/m
Christian,
Sadly, all I can do with LISA is to add mass. There's no option to add a spring too. But I can at least do cases 0 and 1. But probably you should increase the mass (even if it's much more than is reasonable for voice coil), just to compare the models for plausibility.
Qualitatively, I can confirm that adding an actual exciter to an actual panel tends to increase the fundamental, at least for PS foam panels. I have not seen much effect on plywood panels, but I'd have to double check to see if I can see even a tiny effect.
I assume the "average" is your productivity measure. Can you describe how you calculated it? Can you add the mode indices to the table?
Eric
Sadly, all I can do with LISA is to add mass. There's no option to add a spring too. But I can at least do cases 0 and 1. But probably you should increase the mass (even if it's much more than is reasonable for voice coil), just to compare the models for plausibility.
Qualitatively, I can confirm that adding an actual exciter to an actual panel tends to increase the fundamental, at least for PS foam panels. I have not seen much effect on plywood panels, but I'd have to double check to see if I can see even a tiny effect.
I assume the "average" is your productivity measure. Can you describe how you calculated it? Can you add the mode indices to the table?
Eric
Hi Christian
Masses and springs can be added to shell solver as boundary conditions, but I've not tried it.
Let's see if I understand what you are doing:
I'm dont know whether the displacements output by an eigen analysis can be compared across eigenfrequencies. My understanding is that the outputs are the frequencies, and the mode shapes at each frequency. Without damping, displacements can go to infinity, so I presume that the mode shapes are normalised somehow and not comparable to each other. I have also not calculated average displacement from the vtu file yet (but see my description below of where I'm headed). Im willing to try something for you if you describe what you think is a reasonable setup.
Until I include harmonic analysis into a script, every frequency is a separate run of the solver.
My direction at the moment:
Ive been looking at dynamic/harmonic analysis, where a force is applied and the displacement response calculated at any particular frequency. There, I think the displacements and shapes across different eigenfrequencies can be meaningfully compared. Estimates of exciter force and panel damping are required, and I dont have either. It seems only the mass-proportional damping constant (Rayleigh damping Alpha) is implemented, not stiffness-proportional (Beta). However I could change alpha for every frequency if need be.
Currently Ive just done some runs modelling the exciter with a 30mm circular internal boundary. If I use a pressure of ~50 grams force over that area and damping alpha of 0.2, displacement looks to be in the right order of magnitude. That's as far as I have gotten. I presume I can apply extra mass and spring force to that boundary also, but unless the other bits are calibrated, I dont think it would mean much.
I think that the dynamic/harmonic analysis enables us to start creating estimates of frequency response (albeit for an infinite baffled panel). For that I do need to try to implement the Rayleigh integral. At LF this is sort of just an average over the panel, because the wavelength is long and so sound from all points on the panel reach the observation point with negligible change in phase. Im still deciding whether it's worthwhile to go to the trouble of accounting for phase at higher frequencies. On the one hand, it would give you those very deep nulls that we see in the real spectrum, but they are very sensitive to observation point anyhow, so you end up doing an average over multiple observation points anyway. I dont know if this is equivalent to just averaging over the panel in the first place. Probably is 🙂.
The other assumption I'd be making is that velocity is just scaled displacement at any frequency. I see no problem with this, because all points on the panel are moving sinusoidally (steady state), so the points of maximum displacement are also the points of maximum velocity half a cycle later.
I think this sort of 'pseudo-frequency response' could be very useful for optimising designs.
(Any comments on the usefulness/realism of this proposed model appreciated)
Paul
Masses and springs can be added to shell solver as boundary conditions, but I've not tried it.
Let's see if I understand what you are doing:
- You are modelling the effective mass of the moving bits of the exciter - voice coil and coil former, plus some effective portion of the suspension mass, correct? (The largest part of the exciter mass is fixed to the frame, and I doubt its inertia can be modelled in this way.)
- You are modelling the spring suspension of the exciter as an added spring at it's location on the panel?
I'm dont know whether the displacements output by an eigen analysis can be compared across eigenfrequencies. My understanding is that the outputs are the frequencies, and the mode shapes at each frequency. Without damping, displacements can go to infinity, so I presume that the mode shapes are normalised somehow and not comparable to each other. I have also not calculated average displacement from the vtu file yet (but see my description below of where I'm headed). Im willing to try something for you if you describe what you think is a reasonable setup.
Until I include harmonic analysis into a script, every frequency is a separate run of the solver.
My direction at the moment:
Ive been looking at dynamic/harmonic analysis, where a force is applied and the displacement response calculated at any particular frequency. There, I think the displacements and shapes across different eigenfrequencies can be meaningfully compared. Estimates of exciter force and panel damping are required, and I dont have either. It seems only the mass-proportional damping constant (Rayleigh damping Alpha) is implemented, not stiffness-proportional (Beta). However I could change alpha for every frequency if need be.
Currently Ive just done some runs modelling the exciter with a 30mm circular internal boundary. If I use a pressure of ~50 grams force over that area and damping alpha of 0.2, displacement looks to be in the right order of magnitude. That's as far as I have gotten. I presume I can apply extra mass and spring force to that boundary also, but unless the other bits are calibrated, I dont think it would mean much.
I think that the dynamic/harmonic analysis enables us to start creating estimates of frequency response (albeit for an infinite baffled panel). For that I do need to try to implement the Rayleigh integral. At LF this is sort of just an average over the panel, because the wavelength is long and so sound from all points on the panel reach the observation point with negligible change in phase. Im still deciding whether it's worthwhile to go to the trouble of accounting for phase at higher frequencies. On the one hand, it would give you those very deep nulls that we see in the real spectrum, but they are very sensitive to observation point anyhow, so you end up doing an average over multiple observation points anyway. I dont know if this is equivalent to just averaging over the panel in the first place. Probably is 🙂.
The other assumption I'd be making is that velocity is just scaled displacement at any frequency. I see no problem with this, because all points on the panel are moving sinusoidally (steady state), so the points of maximum displacement are also the points of maximum velocity half a cycle later.
I think this sort of 'pseudo-frequency response' could be very useful for optimising designs.
(Any comments on the usefulness/realism of this proposed model appreciated)
Paul
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Christian,
Setting up a similar test,
If I assume a thickness of 5mm and Poisson of 0.3 I get
But there is a problem, even with no spring or mass, the locations is constrained.
Eg eigenmode 2 with Mass=0 and Spring = 0, I still get a distorted emode shape.
I have attached the sif file in case you can see a problem.
I will try tomorrow with the plate solver.
Other Questions:
Paul
Setting up a similar test,
If I assume a thickness of 5mm and Poisson of 0.3 I get
- Volumetric density = 320
- E = 12*20*(1-0.3^2)/(0.005^3) = 1.747e9
- Assume simple support
But there is a problem, even with no spring or mass, the locations is constrained.
Eg eigenmode 2 with Mass=0 and Spring = 0, I still get a distorted emode shape.
I have attached the sif file in case you can see a problem.
I will try tomorrow with the plate solver.
Other Questions:
- Exciter location : by 0.55 I assume you mean 0.55 * width, 0.55 * height. (See image for location - seems right)
- The stiffness values look quite high. 20000 N/m = 20 N/mm ~ 2 kg force/ mm
- Why is the the average displacement of the fundamental sometimes positive and sometimes negative? Some sort of normalisation across modes?
Paul
Hello Eric
- Yes "average" is a productivity measure. A bit different from before. In the first test in previous posts it was simply the mean value of the mode shape, here based on the idea from different papers the sound production is related to the mean plate speed, I have adopted a product mode frequency by the average value of the mode displacement. More intuitive for now than scientific. The displacement being a cos(w*t), w=2pif, the speed is the derivative and gets an w factor (w.sin(wt) .
- Possibility to increase the mass : for test cases of course no problem. Here I stays in what I estimated possible in the range of the 25 to 32mm exciters.
- Adding the modes : hmm, easy in case 0 (already done previously). For the other cases, not sure of the result as the added mass, spring change the mode shape. See below
- About the dependence to the plate material, I agree that EPS is expected more sensitive. I made the choice of a quite rigid one (similar to plywood in 3mm? rq : orthotropic material is not supported by the current script)
case 3 : see raw 2, column 4 and 5 for example. Red point is the exciter position
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Nice progress Christian
- Yes, I also will estimate velocity as displacement times frequency, makes sense to me. Just average in the first instance but may try Rayleigh integral too over each point. For this test I wont even bother calculating area, I’ll assume equal weigh for each point.
- Those strange shapes will be degenerate modes with the same eigenfrequencies. There are two more in case 3. Degenerate modes can form any linear combination of the two.
- With a clearer head I think I know what my problem is so will report later today, hopefully with similar results
Yes the exciter is modeled by a moving mass attached to one point of the mesh. It is mainly the mass of the voice coil. You can find its value for some exciter in Dayton Audio specification (ie DAEX25FHE)
The spider is modeled as a spring at the same location. It is basically the simple model of any electrodynamic loudspeaker in small signal (TS parameters and so one)
Sorry probably wrote too quickly. 1st column of each case are the frequency (Hz) of each mode, the avg column is an estimation of the production (how much sound?) of the mode. See answer before to Eric for more details.
Yes, it is my understanding too. My option here is to say to be productive a mode as a non zero mean value. Modes in a case can't be compared to the other but the same mode can be compared from on case to an other one. For example some modes null without exciter are non null with exciter due to an alteration of the symmetry.
Thank you. I don't want to push you too far in a direction too different of yours. I propose to think more about that, to take a time for reflexion.
It is also the way I am on. Adding damping is a step that seems possible with FDM as the calculation at different frequencies but my current opinion would be more to make a calculation over the time and then applied a time to frequency transformation. All those simulations are CPU time consuming.
All of that lead me also to think we need an electromechanical model... Probably a bit long too explain here. It is a topic by itself. Have you already think about voltage or current driven loudspeaker? When a loudspeaker is current driven (which is unusual) the force is proportional to the audio signal. The result is peaks at the resonance frequency. When a loudspeaker is voltage driven which is the standard situation (amplifiers are almost voltage sources), the mechanical impedance interacts with the electrical side leading to a reduction of the force at resonance so reducing it. Probably to compact as explanation?
That is an important step you reach here
Just an average : yes of the speed (not the displacement). So need some derivative.
Too mathematics at the moment for me! Pragmatically even if it is CPU consuming, I would intergrate in 2 steps... or try some math?
Yes. Probably with a frequency dependent coefficient (see before the relation speed to displacement).
Yes; Clearly the goal
Christian
I think that's a case of degenerate modes, due to the shape being in the aspect ratio of integer multiples (2:3). I bet if you perturb the aspect ratio even a small bit that will go away. I couldn't get the same thing to happen in LISA, but I think that I have before in some particular cases. Probably in real life it rarely happens because the aspect ratio would not be exactly 2:3 or the material would not be truely isotropic.
Eric
Yes!
Let's check. The stiffness k is so the characteristic of the spring is F = kx (x=displacement).
For a spring mass oscillator, the resonance frequency is fs = 1/2/pi . sqrt(k/m) m being the mass (wikipedia says the same as my memory!). So lets apply : fs = 1/2/pi . sqrt(20000/0.005) = 318Hz which sounds ok according to the frequency of resonance seen in the Dayton audio specs
Good question; You are touching here the limit of my current knowledge of eigen values extraction. It means the mean value of the mode shape is sometimes positive, some times negative. Is it a kind of jump in phase?... don't know. I think if we can have results from an harmonic or time simulation, we will have the answer or at least, the question will disappear. In Elmer or Lisa, are the mode shape of the first mode always positive?
Christian
+ @pway
You are right. Change from 400x600 to 440x600, no more degenerative modes. Easy! By the way be careful from one figure to an other one, the same color doesn't show the same displacement (boundaries are 0). To put in the do list of improvements.
Yes, I didnt mention but you need to scale by omega. This is pretty clear, even if you ignore the calculus. Intuitively the velocity is distance over time - the displacement gives the distance, the frequency determines the time available.
Harking back to the equations we discussed before, the jw factor in Umn doing just that - scaling by freq and applying a 90 degrees phase shift.
The remainder of this formula is provided by the harmonic analysis - the forcing, the damping, and the modal mass.
Oh, OK. I just assumed you would know, since you wrote the code 😉
Probably its just random due to the symmetry of the plate. If you introduce some small bias to break the symmetry, it would probably always be one or the other.
Christian
I played with thickness, stiffness etc to try to adjust the response to match your eigenfrequencies more closely.
Currently thickness = 1mm, Youngs = 233e9, density 1650, Poisson zero (you use Poisson 0 is that right?)
I used 1mm in the end thinking that thinner would be closer to your setup, but other values of course become unrealistic as a result.
I scaled the case zero fundamental to get the same avg velocity value as you, then just left that constant in place for the other trials.
I am just averaging data points with the assumption that each represents equal area. I had to remesh to use the same mesh density throughout, otherwise I got skewed results. Still you will see residuals where they should add up to zero.
I checked the first 10 mode shapes and they are the same.
Results below. The small masses dont seem to do anything, but I did confirm a large effect from a 0.5 kg mass, so the effect is there at least. I'll leave it to you to see if anything is meaningful yet, I have a lot of trouble squinting at numbers, we need some graphs I think separating added mass from stiffness. I'm not really set up yet for this, but we are definitely getting closer to something worthwhile I think.
So case 0 is:
49 -30
93 2
150 2
169 43
194 1
270 1
274 -2
319 80
364 1
375 -2
410 -60
440 38
511 2
546 0
558 -2
577 3
603 -2
679 0
680 6
683 -33
776 -85
785 -3
851 -7
868 -129
878 10
Case 1 (0.002/0)
49 -30
93 2
150 2
169 43
194 1
270 1
274 -2
319 80
364 1
375 -2
410 -60
440 38
511 2
546 0
558 -2
577 3
603 -2
679 0
680 6
683 -33
776 -85
785 -3
851 -7
868 -129
878 10
Case 2 (0.002/4000)
49 -30
94 1
150 2
169 43
194 1
270 1
274 -2
319 80
364 1
375 -2
410 -61
440 38
511 2
546 0
558 -2
577 3
603 -2
679 -0
679 3
683 -34
775 -85
785 -3
851 -4
867 -129
878 9
Case 3(0.003 / 50000)
58 -36
94 -2
150 -1
171 40
194 1
270 1
275 0
320 80
364 0
375 -2
410 -60
441 39
511 2
546 0
558 -2
577 3
603 -2
679 0
680 5
683 -33
776 -85
785 -3
851 -5
868 -129
878 9
Case 4(0.005 20000)
53 -33
94 0
150 1
169 42
194 1
270 1
274 -1
319 80
364 1
375 -2
410 -61
440 38
511 2
545 1
558 -2
576 4
603 -2
679 -0
679 1
682 -34
775 -85
785 -1
850 0
867 -129
878 8
I played with thickness, stiffness etc to try to adjust the response to match your eigenfrequencies more closely.
Currently thickness = 1mm, Youngs = 233e9, density 1650, Poisson zero (you use Poisson 0 is that right?)
I used 1mm in the end thinking that thinner would be closer to your setup, but other values of course become unrealistic as a result.
I scaled the case zero fundamental to get the same avg velocity value as you, then just left that constant in place for the other trials.
I am just averaging data points with the assumption that each represents equal area. I had to remesh to use the same mesh density throughout, otherwise I got skewed results. Still you will see residuals where they should add up to zero.
I checked the first 10 mode shapes and they are the same.
Results below. The small masses dont seem to do anything, but I did confirm a large effect from a 0.5 kg mass, so the effect is there at least. I'll leave it to you to see if anything is meaningful yet, I have a lot of trouble squinting at numbers, we need some graphs I think separating added mass from stiffness. I'm not really set up yet for this, but we are definitely getting closer to something worthwhile I think.
So case 0 is:
49 -30
93 2
150 2
169 43
194 1
270 1
274 -2
319 80
364 1
375 -2
410 -60
440 38
511 2
546 0
558 -2
577 3
603 -2
679 0
680 6
683 -33
776 -85
785 -3
851 -7
868 -129
878 10
Case 1 (0.002/0)
49 -30
93 2
150 2
169 43
194 1
270 1
274 -2
319 80
364 1
375 -2
410 -60
440 38
511 2
546 0
558 -2
577 3
603 -2
679 0
680 6
683 -33
776 -85
785 -3
851 -7
868 -129
878 10
Case 2 (0.002/4000)
49 -30
94 1
150 2
169 43
194 1
270 1
274 -2
319 80
364 1
375 -2
410 -61
440 38
511 2
546 0
558 -2
577 3
603 -2
679 -0
679 3
683 -34
775 -85
785 -3
851 -4
867 -129
878 9
Case 3(0.003 / 50000)
58 -36
94 -2
150 -1
171 40
194 1
270 1
275 0
320 80
364 0
375 -2
410 -60
441 39
511 2
546 0
558 -2
577 3
603 -2
679 0
680 5
683 -33
776 -85
785 -3
851 -5
868 -129
878 9
Case 4(0.005 20000)
53 -33
94 0
150 1
169 42
194 1
270 1
274 -1
319 80
364 1
375 -2
410 -61
440 38
511 2
545 1
558 -2
576 4
603 -2
679 -0
679 1
682 -34
775 -85
785 -1
850 0
867 -129
878 8
I guess its important to distiguish exciters with spine support vs those supported by the panel. The first you would expect a small increase in fundamental, the second maybe a decrease due to large added mass, I guess...
Christian,
Just looking at the Rayleigh integral formula again. There is a k term out the front. So we need another factor of omega (radian freq) for our calculation of frequency response.
We also will need to account for equal loudness contours. Presumably we want to produce a graph of decibels (dbA) vs log frequency, so we need to apply (large!) corrections to log SPL at frequencies below and above 1000 Hz as per the dbA weighting curve.
No wonder we need subwoofers. Low frequencies cant catch a single break...
Just looking at the Rayleigh integral formula again. There is a k term out the front. So we need another factor of omega (radian freq) for our calculation of frequency response.
We also will need to account for equal loudness contours. Presumably we want to produce a graph of decibels (dbA) vs log frequency, so we need to apply (large!) corrections to log SPL at frequencies below and above 1000 Hz as per the dbA weighting curve.
No wonder we need subwoofers. Low frequencies cant catch a single break...
Magic of Python : just one simple line in the script to call a powerful function from a library... and limit of the use!
Your proposal is intersting to "randomize" a bit not to be in a more "natural" situation. To see.
I focused on exciter with spine as it is what I do and yes a small increase in fundamental is expected.
Without spine, I am not sure of what we get because the mass of the exciter body is liked to the membrane by the spider stiffness. so what is the stiffness/body mass mimimum to lower the fundamental. Is it as simple as having the stiffness/body mass resonance frequency below the fundamental without exciter? Making a quick calculation, for a DAEX23FHE, body mass is 110g and spider stiffness is 3300N/m (according to spec) so an fs = 28Hz so below most of the panel first mode without exciter. We can expect with an exciter without spine, the fundamental will be lowered.
I think I have to go deeper now in the Rayleigh integral and start real use!
Log of frequency and SPL in dB yes it is usual. dBA correction is unusual. Basically the target is a flat FR in dB (dBC in fact = no correction, as REW for example make a measurement).
Oh ok, good, music content already ‘EQed’ in that sense. But it raises a question I’ve never thought about - what is a typical base level gain in music? They must be set with some assumptions about a standard listening setup.
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I had a look to that some years ago, a time where I was not involve in this forum. If remember there are standards from the disc or movie industry. The idea is to adjust the level with a pink noise having a defined margin to the full scale (numeric signal, for example 14dB or more). The targeted level are in the range 73 to 85dB. I am pretty sure this have been discuss on this forum.
A copy of my notes below.
A site : TC electronic
The link in Audio DIY below : -18dB Pink Noise SMPTE PR-200 standard level (85dB)