## [SOLVED] Modelling thin membranes

Numerical methods and mathematical models of Elmer
CrocoDuck
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Joined: 12 May 2016, 13:15
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### [SOLVED] Modelling thin membranes

Hi guys! I am approaching the modelling of a headphone loudspeaker membrane, which is 20 um thick. I was wondering what is the best way to make the CAD model of the membrane. Is it advisable to model it just as a surface or I should make a whole solid?
Last edited by CrocoDuck on 02 Aug 2016, 19:00, edited 1 time in total.

CrocoDuck
Posts: 52
Joined: 12 May 2016, 13:15
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### Re: Modelling thin membranes

I ended up modelling the membrane as a shell and I tried to use this (although the shape is not flat). It works but it gives weird results. I expect the fundamental eigenvalue to be around few kHz, while I get one as low as 16 Hz. The linear elasticity solver gives values more in the range I expect, but that way I cannot set the thickness... Using a full 3D model of the membrane, with thickness, seems not to be wise as the meshing becomes hard to get right...

EDIT: Seems like I am not the first one thinking about these problems. I will try that. Sorry for the duplicate thread...

CrocoDuck
Posts: 52
Joined: 12 May 2016, 13:15
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### Re: Modelling thin membranes

Hey there! This seems to give a solution (I am searching for eigenvalues of the speaker membrane):

Code: Select all

``````Header
CHECK KEYWORDS Warn
Mesh DB "." "."
Include Path ""
Results Directory ""
End

Simulation
Max Output Level = 5
Coordinate System = Cartesian
Coordinate Mapping(3) = 1 2 3
Steady State Max Iterations = 1
Output Intervals = 1
Timestepping Method = BDF
BDF Order = 1
Solver Input File = case.sif
Post File = case.vtu
Coordinate Scaling = 0.001
End

Constants
Gravity(4) = 0 -1 0 9.82
Stefan Boltzmann = 5.67e-08
Permittivity of Vacuum = 8.8542e-12
Boltzmann Constant = 1.3807e-23
Unit Charge = 1.602e-19
End

Body 1
Target Bodies(1) = 2
Name = "Body 1"
Equation = 2
Material = 1
End

Solver 1
Equation = ShellSolver
Eigen System Values = 10
Eigen System Select = Smallest magnitude
Procedure = "ShellSolve" "ShellSolver"
Eigen Analysis = True
Exec Solver = Always
Stabilize = True
Bubbles = False
Lumped Mass Matrix = False
Optimize Bandwidth = True
Steady State Convergence Tolerance = 1.0e-6
Nonlinear System Newton After Tolerance = 1.0e-3
Nonlinear System Newton After Iterations = 20
Nonlinear System Max Iterations = 1
Nonlinear System Convergence Tolerance = 1.0e-5
Nonlinear System Relaxation Factor = 1.0
Linear System Solver = Direct
Linear System Iterative Method = UmfPack
End

Equation 1
Name = "Elasticity"
End

Equation 2
Name = "Plates"
Active Solvers(1) = 1
End

Material 1
Name = "Mylar"
Tension = 0.0
Poisson ratio = 0.35
Youngs modulus = 620000000
Thickness = 0.00002
Youngs modulus = 620000000
Poisson ratio = 0.35
Porosity Model = Always saturated
Density = 1360
End

Boundary Condition 1
Target Boundaries(1) = 1
Deflection 1 = Real 0
Deflection 2 = Real 0
Deflection 3 = Real 0
Deflection 4 = Real 0
Deflection 5 = Real 0
Deflection 6 = Real 0
End

``````
However, the eigenvalues are still not in the range I expect:

Code: Select all

``````EigenSolve: 1:    9.836939E+05   0.000000E+00
EigenSolve: 2:    2.162239E+06   0.000000E+00
EigenSolve: 3:    2.171148E+06   0.000000E+00
EigenSolve: 4:    8.896226E+06   0.000000E+00
EigenSolve: 5:    8.914394E+06   0.000000E+00
EigenSolve: 6:    9.561661E+06   0.000000E+00
EigenSolve: 7:    9.612378E+06   0.000000E+00
EigenSolve: 8:    1.017967E+07   0.000000E+00
EigenSolve: 9:    1.025543E+07   0.000000E+00
EigenSolve: 10:    1.338364E+07   0.000000E+00
``````
As the ShellSolver is not documented I have few questions:
• First of all, the eigenvalues are squared angular frequency (w^2) for all solvers, right?
• Second, what are the variables of the study? I can hardly make sense of Deflection 1, Deflection 2, ... Deflection 6. They appear both in the boundary condition and in the solution. I would like to deform the mesh with ParaView to have a look at the modes but there seems there is not any readily usable vector field.
• Did anything change about this? If not, I guess I will have to try something else: at some point I will need to make the membrane to vibrate against air, to simulate acoustic radiation.
Also, here a picture of the meshing I am attempting to solve on:

According to my understanding, the Elastic Plates solver should not work unless the mesh is on the XY plane. Why I was able to have the solver delivering a solution?

EDIT (Embarrassing ) So, I checked my acoustics textbook... Although we refer to the physical object as "speaker membrane" I am actually trying to model it as a plate, since it is not under tension. I checked the formula for circular plates eigenfrequencies for rigidly clamped boundary. It gives a fundamental of 18.77 Hz. The Elastic Plates solver says 17.48 Hz...

Code: Select all

``````EigenSolve: 1:    1.205945E+04   0.000000E+00
EigenSolve: 2:    5.338884E+04   0.000000E+00
EigenSolve: 3:    5.339178E+04   0.000000E+00
EigenSolve: 4:    1.492115E+05   0.000000E+00
EigenSolve: 5:    1.492209E+05   0.000000E+00
EigenSolve: 6:    1.803310E+05   0.000000E+00
EigenSolve: 7:    3.269595E+05   0.000000E+00
EigenSolve: 8:    3.269882E+05   0.000000E+00
EigenSolve: 9:    4.315564E+05   0.000000E+00
EigenSolve: 10:    4.316478E+05   0.000000E+00
``````
However I would expect the higher modes to grow as 72.82 Hz, 163.28 Hz... Still, probably the Elastic Plates is being much closer to actual physics than what I imagine. Still surprised to seeing it working tho.

raback
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### Re: Modelling thin membranes

Hi

Without the eigensolver the components of Deflection are to my knowledge
1) Normal displacement
2) Rotation with respect to the 1st tangent direction
3) Rotation with respect to the 2nd tangent direction

For eigensolver there are six components since each of these have real and imaginary components. Some solvers name the components more wisely but I suspect that this hasn't much been used like this before.

-Peter

CrocoDuck
Posts: 52
Joined: 12 May 2016, 13:15
Antispam: Yes

### Re: Modelling thin membranes

Thank you raback, that makes it clearer. I guess I can mess with ParaView and get vectors out of those. I have made further experiments and seems that making a full solid for a thin membrane is not really wise. I made my model simpler (a simply supported aluminium rectangular plate) and I can get the same eigenfrequencies I expect by using the Elastic Plates solver (SmitcSolver). I am gonna mark this thread as Solved as my main question has been answered. However, I have few more questions:
• I would like to apply a force to a point (or area) of an Elastic Plate and then couple it to air, to have it radiating sound. Is this possible? Or better, can the SmitcSolver be coupled with other solvers?
• Do you confirm that the SmitcSolver works for not flat surfaces? It seems so to me but I am not sure.
My ultimate goal is to have a full model of a miniature dynamic loudspeaker, involving magnets, coils and membrane. However, I will move baby steps first simulating a simple rectangular DML loudspeaker radiating into free field, modelled as an elestic plate driven at a point and radiating into air. I have already solved this problem through a different method and I want to compare the Elmer solution with it.