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Cylindrical Subwoofer
Introduction
This is an introduction to modeling with the Acoustics Module. The step-by-step instructions take you through the process of setting up a model of the sound field created by a cylindrical down-firing subwoofer placed on a floor. The important result of the simulation is the contribution to the system’s mechanical impedance induced by the coupling between the speaker membrane and the air inside and outside the speaker enclosure. A polar plot of the exterior radiation pattern as well as a directivity plot are also presented to demonstrate the uniform intensity distribution of the subwoofer.
Model Definition
Figure 1: The geometry of the subwoofer with a slice removed to show its interior. In the illustration the floor is cut off but in the model it is assumed to extend to infinity.
Figure 1 shows the geometry simulated in this model. A down-firing cylindrical subwoofer is placed on a floor bounding an infinite half-space. The subwoofer enclosure has a height of 0.7 m and a diameter of 0.4 m. The bottom of the subwoofer is dominated by a 12-inch (0.3 m diameter) membrane. The top has a vent with a diameter of 0.1 m.
The membrane vibrates harmonically with the acceleration a = a0eiωt where
a0 = 10 m/s2 and ω = 2πf is the angular frequency (rad/s). The frequencies considered are f = 50–70 Hz, which are in the high end of the frequency range that subwoofers are typically used for. It is assumed that the walls of the enclosure are perfectly rigid. The acoustic medium is air.
The feet that the speaker would need to stand on are judged to have a negligible effect on the sound field. With this assumption, all geometric features and physics have a rotational symmetry with respect to the axis of the speaker. This makes it natural to set up the model in a 2D axisymmetric physics interface.
Results and Discussion
By assuming the membrane is inflexible, you can write Newton’s second law for the voice coil and membrane as
where the forces acting on the system have been split into four components: the applied external force Fext, an electromagnetic force opposing the movement of the coil Femf, a mechanical part Fm independent of the acoustic environment (elastic behavior of the suspensions, spider and surround in a speaker), and a term Fa containing only the effects of the fluid loading on the membrane. The external force is typically proportional to the applied voltage, while the latter three contributions are proportional to the velocity and directed to oppose the movement. Introducing corresponding mechanical impedances (with proper normal directions assumed), and using the time-harmonic assumption, it holds that
If you know the three impedances as functions of frequency, you can predict the system’s electrical and mechanical behavior. The electrical force can be measured or simulated with a blocked voice coil, while Zm is directly related to the mass of the voice coil and membrane as well as to the stiffness of the baffle keeping the voice coil centered. These contributions are therefore independent of the environment in which the speaker operates.
The acoustic contribution to the mechanical impedance Za, on the other hand, depends on the enclosure geometry and on reflecting surfaces in the immediate surroundings. In Figure 2 you can study the real and imaginary parts of  Za as functions of frequency in the simulated range. The real part represents radiation and acts as a resistance in the electromechanical system while the imaginary part represents a reactance.
Figure 2: The resistive (real) and reactive (imaginary) parts of the contribution of the air to the mechanical impedance of the speaker.
Note that the reactance switches sign from positive to negative at approximately 60 Hz. This means that at frequencies below this visible resonance, the surrounding air acts as an added mass on the membrane, while at higher frequencies its action is spring-like.
Figure 3 displays the exterior-field sound pressure level in the rz-plane as a polar plot evaluated at 2 m from the speaker.
Figure 3: Exterior-field sound level in dB at a distance of 2 m from the floor below the subwoofer, 0o corresponds to the vertical z-direction. Note the 77 dB offset. The large wavelength compared to the size of the speaker system makes the response rather uniform.
Notes About the COMSOL Implementation
This model is set up in 2D axisymmetry using the Pressure Acoustics, Frequency Domain interface, which is described in the Acoustics Module User’s Guide. The modeled physical domain is a hemisphere with a radius of 1 m. To minimize the effect of nonphysical reflections at the exterior boundary of this domain, an absorbing perfectly matched layer (PML) is added outside of it. For more information about PMLs see the COMSOL Multiphysics Reference Manual.
The membrane is modeled with zero thickness. The Interior Normal Acceleration boundary condition allows the pressure field to be discontinuous across the membrane.
Application Library path: Acoustics_Module/Tutorials,_Pressure_Acoustics/cylindrical_subwoofer
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D Axisymmetric.
2
In the Select Physics tree, select Acoustics>Pressure Acoustics>Pressure Acoustics, Frequency Domain (acpr).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies>Frequency Domain.
6
Click  Done.
Geometry 1
Create the geometry.
To simplify this step, you can insert a prepared geometry sequence. In the Geometry toolbar, click Insert Sequence. Browse to the model’s Application Libraries folder and double-click the file cylindrical_subwoofer.mph. Click Build All in the Geometry toolbar. Then, continue with the instruction after the geometry plot below.
Otherwise, proceed with the following instructions to create the geometry from scratch:
Begin by drawing the air domain and the surrounding PML.
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 1.1.
4
In the Height text field, type 1.8.
5
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
0.5
6
Select the Layers to the right check box.
7
Clear the Layers on bottom check box.
8
Select the Layers on top check box.
9
Click  Build Selected.
Next, add the walls of the subwoofer.
Rectangle 2 (r2)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 0.02.
4
In the Height text field, type 0.66.
5
Locate the Position section. In the r text field, type 0.18.
6
In the z text field, type 0.12.
Rectangle 3 (r3)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 0.05.
4
In the Height text field, type 0.02.
5
Locate the Position section. In the r text field, type 0.15.
6
In the z text field, type 0.1.
Rectangle 4 (r4)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 0.15.
4
In the Height text field, type 0.02.
5
Locate the Position section. In the r text field, type 0.05.
6
In the z text field, type 0.78.
Difference 1 (dif1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object r1 only to add it to the Objects to add list.
3
In the Settings window for Difference, locate the Difference section.
4
Find the Objects to subtract subsection. Select the  Activate Selection toggle button.
5
Select the objects r2, r3, and r4 only.
6
Click  Build All Objects.
Finally, create the membrane.
Polygon 1 (pol1)
1
In the Geometry toolbar, click  Polygon.
2
In the Settings window for Polygon, locate the Object Type section.
3
From the Type list, choose Open curve.
4
Locate the Coordinates section. From the Data source list, choose Vectors.
5
In the r text field, type 0 0.06 0.06 0.15.
6
In the z text field, type 0.2 0.2 0.2 0.12.
Global Definitions
This model uses a parameter to represent the membrane’s peak acceleration and variables for the total force on the membrane and the mechanical impedance of the membrane-air system. An integral operator takes part in the force computation.
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
a0
10[m/s^2]
10 m/s²
Peak acceleration
Definitions
Integration 1 (intop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, type mem_int in the Operator name text field.
3
Locate the Source Selection section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 4 and 11 only.
5
Locate the Advanced section. Clear the Compute integral in revolved geometry check box.
Integration 2 (intop2)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, type pml_int in the Operator name text field.
3
Locate the Source Selection section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 6 and 19 only.
Variables 1
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
F_a
mem_int(2*pi*r*nz*(down(p)-up(p)))
N
Net force on membrane
Z_a
-F_a/(a0/acpr.iomega)
N·s/m
Mechanical impedance
P_AR
-pml_int(up(acpr.Ir)*nr+up(acpr.Iz)*nz)
W
Radiated Power
Perfectly Matched Layer 1 (pml1)
1
In the Definitions toolbar, click  Perfectly Matched Layer.
2
Select Domains 2–4 only.
3
In the Settings window for Perfectly Matched Layer, locate the Geometry section.
4
From the Type list, choose Cylindrical.
5
Locate the Scaling section. From the Coordinate stretching type list, choose Rational.
The evanescent part of the wave leaving from the vent is expected to decay with a characteristic length much shorter than the wavelength. We use the Rational stretching type which is good for open radiation problems.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Air.
4
Click Add to Component in the window toolbar.
5
In the Home toolbar, click  Add Material to close the Add Material window.
Pressure Acoustics, Frequency Domain (acpr)
Interior Normal Acceleration 1
1
In the Model Builder window, under Component 1 (comp1) right-click Pressure Acoustics, Frequency Domain (acpr) and choose Interior Conditions>Interior Normal Acceleration.
2
Select Boundaries 4 and 11 only.
3
In the Settings window for Interior Normal Acceleration, locate the Interior Normal Acceleration section.
4
Specify the a0 vector as
0
r
a0
z
The membrane boundaries are considered interior because they have a pressure distribution on both sides. The Sound Hard Boundary (Wall) condition applies per default to all the remaining exterior boundaries. This includes the surface of the enclosure.
Exterior Field Calculation 1
1
In the Physics toolbar, click  Boundaries and choose Exterior Field Calculation.
2
Select Boundaries 6 and 19 only.
3
In the Settings window for Exterior Field Calculation, locate the Exterior Field Calculation section.
4
From the Condition in the z = z^0 plane list, choose Symmetric/Infinite sound hard boundary.
The hard floor is excluded from the integral by noting that the radiation pattern is exactly the same as if replacing the floor with a mirror image of the subwoofer that is instead included in the integral. The exterior-field integral is set to use the full integral version; this means that the exterior field can be evaluated in any point outside the computational selection. This option will give the full information about both amplitude and phase, not only in the mathematical far-field.
Mesh 1
In acoustic models, the wavelength must be resolved by the mesh. In addition, small features in the geometry can induce high local pressure gradients that must be resolved properly to obtain a consistent global solution. In this particular model, the wavelength is greater than 5 m, so the focus is on resolving the geometry. Use a mapped mesh in the PML domain and a boundary layer to accurately compute the exterior field.
Free Triangular 1
In the Mesh toolbar, click  Free Triangular.
Size
1
In the Model Builder window, click Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Fine.
Free Triangular 1
1
In the Model Builder window, click Free Triangular 1.
2
In the Settings window for Free Triangular, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 1 only.
Size 1
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 4 and 11 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section. Select the Maximum element size check box.
7
In the associated text field, type 0.01[m].
Boundary Layers 1
1
In the Mesh toolbar, click  Boundary Layers.
2
In the Settings window for Boundary Layers, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 1 only.
5
Click to expand the Transition section. Clear the Smooth transition to interior mesh check box.
Boundary Layer Properties
1
In the Model Builder window, click Boundary Layer Properties.
2
Select Boundaries 6 and 19 only.
3
In the Settings window for Boundary Layer Properties, locate the Boundary Layer Properties section.
4
In the Number of boundary layers text field, type 1.
Mapped 1
1
In the Mesh toolbar, click  Mapped.
2
In the Settings window for Mapped, click  Build All.
Study 1
Step 1: Frequency Domain
1
In the Model Builder window, under Study 1 click Step 1: Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Study Settings section.
3
In the Frequencies text field, type range(30,0.25,70).
This gives you frequencies from 30 to 70 Hz with a pitch of 0.25 Hz.
4
In the Home toolbar, click  Compute.
Results
Acoustic Pressure (acpr)
The first default plot shows the pressure field for the last parameter value, 70 Hz. The postprocessing algorithms automatically try to smooth discontinuous fields. You can see this effect if you zoom in on the membrane, where the field really should be discontinuous. To turn off the smoothing, follow these steps:
Surface 1
1
In the Model Builder window, expand the Acoustic Pressure (acpr) node, then click Surface 1.
2
In the Settings window for Surface, click to expand the Quality section.
3
From the Smoothing list, choose None.
4
In the Acoustic Pressure (acpr) toolbar, click  Plot.
Note that the pressure drops rapidly toward zero in the PML domain.
For a better view of the damping, look at the second default plot, which shows the sound pressure level in dB.
Sound Pressure Level (acpr)
1
In the Model Builder window, click Sound Pressure Level (acpr).
2
In the Sound Pressure Level (acpr) toolbar, click  Plot.
With a visual inspection you can see that the pressure drop in the PML is roughly 80 dB. This means that the part of the wave that is reflected at the exterior boundary experiences a total of 160 dB damping before it returns to the physical domain. In other words, the reflected wave has virtually no effect on the solution.
Study 1/Solution 1 (sol1)
For a 3D representation of the sound pressure level, look at the fourth default plot.
First create a selection to hide the PML domain:
1
In the Model Builder window, expand the Results>Datasets node, then click Study 1/Solution 1 (sol1).
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 1 only.
Contour 1
1
In the Model Builder window, right-click Sound Pressure Level, 3D (acpr) and choose Contour.
2
In the Settings window for Contour, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Pressure Acoustics, Frequency Domain>Pressure and sound pressure level>acpr.Lp_t - Total sound pressure level - dB.
3
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
4
From the Color list, choose White.
5
Clear the Color legend check box.
6
In the Sound Pressure Level, 3D (acpr) toolbar, click  Plot.
Exterior-Field Sound Pressure Level (acpr)
The last two default plots represent the exterior-field sound pressure level and the exterior-field pressure evaluated at 1 m in the rz-plane (change the evaluation to 2 m, to be outside the computational domain). Look at the first of those to check that the exterior-field pressure distribution is indeed as uniform as you would expect from a subwoofer.
Initially the spatial response is depicted for all azimuthal angles (0 deg to 360 deg). The radiation-pattern plot has the angle defined such that 0 deg is along the z-axis; the reference direction is per default in the z-axis direction.
Radiation Pattern 1
1
In the Model Builder window, expand the Exterior-Field Sound Pressure Level (acpr) node, then click Radiation Pattern 1.
2
In the Settings window for Radiation Pattern, locate the Evaluation section.
3
Find the Angles subsection. From the Restriction list, choose Manual.
4
In the φ start text field, type -90.
5
In the φ range text field, type 180.
6
Find the Evaluation distance subsection. In the Radius text field, type 2.
7
In the Exterior-Field Sound Pressure Level (acpr) toolbar, click  Plot.
The plot indicates a maximum in the vicinity of 79 dB along the floor and a minimum of 77 dB straight above the speaker, and is depicted in Figure 3.
Proceed to remove the Exterior-Field Pressure plot as it is not relevant for this model.
Exterior-Field Pressure (acpr)
In the Model Builder window, right-click Exterior-Field Pressure (acpr) and choose Delete.
Proceed to display the previously defined impedance variable Z_a as function of frequency.
Mechanical Impedance
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Mechanical Impedance in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section. Select the y-axis label check box.
5
In the associated text field, type Impedance (N·s/m).
Global 1
1
Right-click Mechanical Impedance and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(Z_a)
N*s/m
Re(Z_a)
imag(Z_a)
N*s/m
Im(Z_a)
4
Click to expand the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Cycle.
5
In the Mechanical Impedance toolbar, click  Plot.
Your plot should be similar to Figure 2. Now proceed to generate a plot of the acoustic radiated power.
Energy Radiated
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Energy Radiated in the Label text field.
3
Locate the Axis section. Select the x-axis log scale check box.
4
Select the y-axis log scale check box.
Global 1
1
Right-click Energy Radiated  and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
P_AR
W
Radiated Power
4
Click to expand the Legends section. Clear the Show legends check box.
5
In the Energy Radiated toolbar, click  Plot.
Checking that the acoustic radiated energy is positive in all the frequency range is a good way to check that the PML is working as expected. Your plot should be similar to this.