TMM - infinite Layer

In contrast to pipe networks the chance for side paths is much lower in infinite layer systems. A typical example application is the design of sound absorption systems.

Infinite layers are a special version of one-dimensional system. Such system are one-dimensional in that sense, that the properties change in one dimension and remain constant for the other two space dimensions. The systems or layers are supposed to be infinite in those remaining two dimensions.

In contrast to acoustic networks from section Acoustic network there is an additional variable required to describe the propagation parallel to the layers: the wavenumber perpendicular to the main dimension.

_images/infinite_layer.png

Sketch of connected infinite layers.

In figure Sketch of connected infinite layers. such a set-up is shown. The use and application of the pyva.models.TMmodel is shown is this section.

In the first version of pyva TMM was restricted to fluid layer, i.e. layers that can be described by fluid state variables pressure and velocity. Note, that a this plate can also be described by fluid variable when no friction is considered. The current version of pyva has implemented SolidLayer and PoroElasticLayer. In addition, the Allard approach is implemented that allows the combination of multiple layers of different nature. In the last examples of this section the new possibilities will be shown.

Absorber design

_images/absorber.png

Absorber configurations

In our example we start with the definition of air and the usual fibre material

import pyva.TMmodel as TMM
import pyva.systems.infiniteLayers as iL

import pyva.properties.materialClasses as matC
import pyva.useful as uf

air    = matC.Fluid(eta = 0.0)
fibre1 = matC.EquivalentFluid(porosity = 0.98, \
                               flow_res = 25000.,\
                               tortuosity = 1.02, \
                               length_visc = 90.e-6, \
                               length_therm = 180.e-6,\
                               rho_bulk = 0.98*1.20 + 30.,\
                               rho0 = 1.208, \
                               dynamic_viscosity = 1.81e-5 )

We will design three absorbers:

  1. A pure fibre absorber with large thickness (and lot of required space)

  2. The pure thinner fibre absorber

  3. A thinner fibre absorber with a perforate front layer

The components or layers of our absorber are created as infinite-layer:

fibre_10cm  = iL.FluidLayer(h1,fibre1)
fibre_20cm  = iL.FluidLayer(h2,fibre1)
perforate   = iL.PerforatedLayer(0.005, 0.001, distance = 0.02)

And the set-up are created as transfer matrix

TMM_fibre_10      = mods.TMmodel((fibre_10cm,))
TMM_fibre_20      = mods.TMmodel((fibre_20cm,))
TMM_perf_fibre_20 = mods.TMmodel((perforate, fibre_10cm,))

The diffuse field absorption is determined with the appropriate method

alpha_fibre_10      = TMM_fibre_10.absorption_diffuse(omega,theta_max=np.pi/2,signal = False)
alpha_fibre_20      = TMM_fibre_20.absorption_diffuse(omega,theta_max=np.pi/2,signal = False)
alpha_perf_fibre_10 = TMM_perf_fibre_20.absorption_diffuse(omega,theta_max=np.pi/2,signal = False)

The result can be plotted and shows the typical effects. The thick fibre layer provides excellent absorption even at low frequencies. When we take the thinner version we loose low frequency performance. In order to improve this we take the perforate cover, but here we loose the high frequency performance due to impedance mismatch at higher frequencies. However, the design of absorbers at given space and weight restrictions is an art. Pyva can support here and provide the tools for absorber design.

_images/TMM_absorber_abs_diffuse.png

Diffuse absorption of different absorber configurations.

Double walls

Double walls are the master tool set for efficient acoustic isolation. Infinite layers are an excellent option to calculate the performance of double walls even though the theory fails for example for single plates at the coincidence frequency. In such cases a full SEA model in twin chamber configuration as in Two rooms separated by a wall section is recommended.

However, for fast and comparative studies on sound isolation the infinite layer in combination with the transfer matrices is very useful.

_images/DW.png

Double wall set-up.

In this example the materials from above are used with an additional aluminium plate layer

# Create props
alu = matC.IsoMat()
alu1mm = stPC.PlateProp(0.001,alu)

that is used in combination with several FluidLayer and MassLayer

# Fluid layer
air_5cm    = iL.FluidLayer(0.05,air)
fibre_5cm  = iL.FluidLayer(0.05,fibre1)
# Limp mass layer
heavy_1kg  = iL.MassLayer(0.001, 1000)
heavy_2kg7 = iL.MassLayer(0.001, 2700)
# Plate layer
iL_alu1mm  = iL.PlateLayer(alu1mm,)

All these layers are compiled in four different versions:

T_alu            = mds.TMmodel((iL_alu1mm,))
T_alu_air_mass   = mds.TMmodel((iL_alu1mm,air_5cm,heavy_1kg))
T_alu_fibre_mass10 = mds.TMmodel((iL_alu1mm,fibre_5cm,heavy_1kg))
T_alu_fibre_mass27 = mds.TMmodel((iL_alu1mm,fibre_5cm,heavy_2kg7))

Once the TMmodels are created the transmission coefficient is calculated by the transmission_diffuse() method:

tau_alu = T_alu.transmission_diffuse(omega,signal = False)
tau_alu_air_mass = T_alu_air_mass.transmission_diffuse(omega,theta_step=np.pi/1000,signal = False)
tau_alu_fibre_mass10 = T_alu_fibre_mass10.transmission_diffuse(omega,signal = False)
tau_alu_fibre_mass27 = T_alu_fibre_mass27.transmission_diffuse(omega,signal = False)

Plotting the results in the following figure shows the tremendous increase of isolation above the double wall resonance. However, in addition is becomes obvious that the inner cavity requires damping and that the lower resonance frequencies provide a much better performance in the mid-frequency.

_images/TMM_DW_transmission.png

Transmission loss of different single and double wall configurations.

Absorber Design with Poroelastic Materials

The implementation of Brouards and Allards theory that allows layups of different nature. Thus examples for such set-ups are required. We used the carpet-impervious screen-fibre layup of [All2009] in section 11.7.2. The layup is shown in the following figure.

_images/carpet_screen_fibre_layup.png

Layup from [All2009] figure 11.16

The material parameters and thicknesses are taken from Table 11.7 but with corrected thickness of the fibre layer.

Material

Unit
h

mm
\Phi

.
\sigma

Ns/m^4
\alpha_\infty

.
\Lambda

µm
\Lambda'

µm
\rho_1

kg/m³
E

kPa

\eta_s

Carpet 1

3.5

0.99

5000

1

0.5

Carpet 2

3.5

0.99

5000

1

0.5

Screen

3.0

Carpet 1

12.5

0.98

33000

1.1

0.88

Two carpet layers are mentioned but with similar properties. Thus, the original paper should be checked.

The code is given in Absorber Design with Poroelastic Materials and explained in detail here. The carpets material is created by

# Carpet as Poroelastic
carpet_solid = matC.IsoMat(E=20000.,rho0=60,nu=0.,eta=0.5)
carpet_mat   = matC.PoroElasticMat(carpet_solid, \
                            flow_res = 5000., \
                            porosity = 0.99, \
                            tortuosity = 1., \
                            length_visc = 23.E-6, length_therm = 28.E-6)

The impervious screen uses the plate property

# Impervious scree as PlateProp
screen_mat  = matC.IsoMat(E=30000.,rho0 = 2000, nu=0.49)
screen_prop = stPC.PlateProp(0.003, screen_mat)

And the fibre material is also defined as poroelastic material.

# Fibre as Poroelastic
fibre_solid = matC.IsoMat(E=100000.,rho0=60,nu=0.,eta=0.88)
fibre_mat   = matC.PoroElasticMat(fibre_solid, \
                            flow_res = 33000., \
                            porosity = 0.98, \
                            tortuosity = 1.1, \
                            length_visc = 50.E-6, length_therm = 110.E-6)

The single layers are created using the infiniteLayers module with the related classes

# Define infiniteLayers
carpet1  = iL.PoroElasticLayer(carpet_mat, 0.0035)
carpet2  = iL.PoroElasticLayer(carpet_mat, 0.0035)
screen   = iL.ImperviousScreenLayer(screen_prop)
fibre    = iL.PoroElasticLayer(fibre_mat, 0.0125)

and are connected as TMmodel object:

# Create lay-up as TMmodel
TMM_layup = mds.TMmodel((carpet1,carpet2,screen,fibre))

The impedance is calculated using the Allard version of the surface impedance calculation:

# Calculate normal surface impedance
Z          = TMM_layup.impedance_allard(omega,kx=0,signal = False)

which gives the following result in accordance with [All2009] except the frequency unit.

_images/carpet_fibre_impedance.png

Normal impedance of layup from [All2009] as in figure 11.17

With the following command the normal and diffuse absorption can be caluclated

# Calculate normal absorption
alpha0     = TMM_layup.absorption(omega,kx=0.0,signal = False,allard=True)
# Calculate diffure absorption
alpha_diff = TMM_layup.absorption_diffuse(omega,theta_max=np.pi*78/180,theta_step = np.pi/180, signal = False, allard=True)

Note the allard = True keyword argument required to use the Allard method of the absorption calculation. The figure reveals that there are better absorbers in the world.

_images/carpet_fibre_absorption.png

Normal and diffuse absorption of layup

Acoustic transmission design with poroelastic foam and rubber

A practical version double wall systems is a so called mass-spring system. Such layup consist of a soft foam covered by a heavy layer, e.g. rubber. With the Allard theory poroelaxtic foams can be investigated in detail. An example layup is shown in the following figure.

_images/alu_melamin_rubber_layup.png

Mass-spring layup

In this example the impact of different modelling approaches and assumptions is shown. The code is given in Acoustic transmission design with poroelastic foam and rubber, please refer to the code for the material details.

The rubber and the Aluminium of the base plate are given as IsoMat.

# Isotropic materials
alu     = matC.IsoMat(eta = 0.1)
rubber  = matC.IsoMat(E=2.6e6,rho0=1200,nu=0.49,eta=0.00)

The melamin foam is given by

# Melamin foam paramters
melamin_vac = matC.IsoMat(E=3.0e5,rho0=12.0,nu=0.4,eta=0.1) # Frame in vaccuum    E6
melamin = matC.PoroElasticMat(melamin_vac, \
                            flow_res = 30000., \
                            porosity = 0.99, \
                            tortuosity = 1.01, \
                            length_visc = 250.E-6, length_therm = 550.E-6)

Every layer is given as plate, so that we can try different options:

# plate properties
alu_1mm        = stPC.PlateProp(0.001, alu)
rubber_2mm     = stPC.PlateProp(0.002, rubber)
# test the foam as solid
foam_3cm_solid = stPC.PlateProp(0.03, melamin_vac)

We define the melamin foam either as PoroElasticLayer or SolidLayer

# Foam Layers
iL_foam_3cm = iL.PoroElasticLayer(melamin, 0.03)
iL_foam_3cm_solid = iL.SolidLayer(foam_3cm_solid)

whereas the aluminium plate and the rubber layer are given as SolidLayer or ImperviousScreenLayer

# rubber and alu as solid- and screen layer
iL_rubber_solid_2mm = iL.SolidLayer(rubber_2mm)
iL_rubber_imper_2mm = iL.ImperviousScreenLayer(rubber_2mm)
iL_alu_solid_1mm    = iL.SolidLayer(alu_1mm)
iL_alu_imper_1mm    = iL.ImperviousScreenLayer(alu_1mm)

For the decoupling we create a super thin layer of low mass

# Mass of Fluid as gap
iL_nothing    = iL.MassLayer(1e-6,1.)

First we are interested in the impact of the solid or screen layer modelling

# TMmpodel using the solid or impervious screen formulation
alu_melamin_rubber_solid = mds.TMmodel((iL_alu_solid_1mm,iL_foam_3cm,iL_rubber_solid_2mm))
alu_melamin_rubber_imper = mds.TMmodel((iL_alu_imper_1mm,iL_foam_3cm,iL_rubber_imper_2mm))

and two other variations that decouple the alu plate of model the melamin foam as solid - neglecting the fluid waves.

alu_melamin_rubber_decoup = mds.TMmodel((iL_alu_imper_1mm,iL_nothing,iL_foam_3cm,iL_rubber_imper_2mm))
alu_melamin_rubber_foam_as_solid = mds.TMmodel((iL_alu_imper_1mm,iL_foam_3cm_solid,iL_rubber_imper_2mm))

Calculating the transmission loss of all options is done by:

tau_imper = alu_melamin_rubber_imper.transmission_diffuse(omega,theta_max=78/180*np.pi,allard=True,signal=False)
tau_solid = alu_melamin_rubber_solid.transmission_diffuse(omega,theta_max=78/180*np.pi,allard=True,signal=False)
tau_decoup = alu_melamin_rubber_decoup.transmission_diffuse(omega,theta_max=78/180*np.pi,allard=True,signal=False)
tau_foam_as_solid = alu_melamin_rubber_foam_as_solid.transmission_diffuse(omega,theta_max=78/180*np.pi,allard=True,signal=False)

The results are shown in the following figure.

_images/allard_DW_TL.png

Transmission loss of alu + mass-spring system using different modelling options

First, the difference between the versions modelling both skins as solid or screen is low. Second, decoupling the alu from the foam leads to very different results. As a consequence this means that the connection iof the noise control treatment must be exactly known to get correct results. This is also true for experiments where the treatment must be carefully glued to the plate to get correct results. Third, when the foam is modelled as solid the result is different, but the difference is lower than compared to the decoupling effect.

Add perforation to plates and impervious screens

In order to show the influence of perforation, we slightly modify our 10mm fibre example of Absorber design. The full code can be found in Perforated Absorber created with plates with bending wave motion.

We create an additional steel plate of 5mm thickness that is perforated as given in the example. We use the plate and the impervious screen model:

steel    = matC.IsoMat(E=15.E10,rho0=7800.,nu=0.27)
steel5mm = stPC.PlateProp(0.005, steel)
il_steel_5mm_perf = iL.PlateLayer(steel5mm,perforation = il_perforate)
il_steel_5mm_s_perf = iL.ImperviousScreenLayer(steel5mm,perforation = il_perforate)

As a further test we create a one micron thin:

nothing  = stPC.PlateProp(0.000001, steel)
il_nothing_perf  = iL.PlateLayer(nothing,perforation = il_perforate)

With these additional layers we create the following layups:

TMM_steel_perf_fibre_10   = mds.TMmodel((il_steel_5mm_perf, il_fibre_10cm,))
TMM_steel_s_perf_fibre_10   = mds.TMmodel((il_steel_5mm_s_perf, il_fibre_10cm,))
TMM_nothing_perf_fibre_10   = mds.TMmodel((il_nothing_perf, il_fibre_10cm,))

Calculating the diffuse field absorption coefficient and plotting the results gives the following figure:

_images/TMM_perforate_test_abs_diffuse.png

Effect of plates that are perforated and allow bending dynamics

We see that due to the high mass of the steel plate the absorption is very similar except some small coincidence effects at \omega\approx 15000 s^{-1}. The thin micron foil has nearly no impact as it moved with the flow and therefore there is no flow through the perforation. There is only a minor impact at high frequencies.