A Comparison of the Transfer Matrix Method and the Finite Element
Method for the Calculation of the Transmission Loss in a Single
Expansion Chamber Muffler
by
Kevin J. McMahon
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2014
i
© Copyright 2014
by
Kevin J. McMahon
All Rights Reserved
ii
CONTENTS
A Comparison of the Transfer Matrix Method and the Finite Element Method for the
Calculation of the Transmission Loss in a Single Expansion Chamber Muffler ......... i
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
NOMENCLATURE ........................................................................................................ vii
GLOSSARY ................................................................................................................... viii
KEY WORDS ................................................................................................................... ix
ABSTRACT ...................................................................................................................... x
1. Introduction.................................................................................................................. 1
2. Methodology ................................................................................................................ 3
2.1
Model Definitions .............................................................................................. 3
2.2
Setup of Transfer Matrix Method for Muffler System....................................... 5
2.3
2.2.1
Equations for a Straight Pipe ................................................................. 6
2.2.2
Equations for an Expansion and Contraction ......................................... 7
2.2.3
Overall Transfer Matrix ......................................................................... 8
Setup of Finite Element Model for Muffler System .......................................... 9
3. Results and Discussion .............................................................................................. 13
3.1
Transfer Matrix Method Results ...................................................................... 13
3.2
Finite Element Model Results .......................................................................... 14
3.3
Comparison of Transfer Matrix Model to Finite Element Model .................... 16
3.3.1
Configuration 1: TMM Compared to FEM .......................................... 16
3.3.2
Configuration 2: TMM Compared to FEM .......................................... 17
3.3.3
Configuration 3: TMM Compared to FEM .......................................... 18
3.3.4
Configuration 4: TMM Compared to FEM .......................................... 19
4. Conclusion ................................................................................................................. 20
4.1
Applicability of TMM Compared to FEM ....................................................... 20
iii
5. References.................................................................................................................. 21
6. Appendix A: MATLAB Code for Configuration 1 ................................................... 22
7. Appendix B: MATLAB Code for Configuration 2 ................................................... 24
8. Appendix C: MATLAB Code for Configuration 3 ................................................... 26
9. Appendix D: MATLAB Code for Configuration 4 ................................................... 28
iv
LIST OF TABLES
Table 1: Expansion Chamber Configurations.................................................................... 4
Table 2: Fluid Properties of Air used in the Muffler System ............................................ 4
Table 3: MATLAB Code User Defined Input Parameters .............................................. 13
Table 4: FEM Input Parameter ........................................................................................ 15
v
LIST OF FIGURES
Figure 1: Schematic of Single Expansion Chamber Muffler System ................................ 3
Figure 2: Schematic of Muffler System Dimension .......................................................... 4
Figure 3: Schematic of Subsystem Components Which Define Muffler System ............. 5
Figure 4: Schematic of Muffler System Boundaries with Nodes ...................................... 5
Figure 5: Applied Boundary Conditions to Finite Element Model ................................. 10
Figure 6: Representative Mesh of Muffler System Model .............................................. 11
Figure 7: Calculated Transmission Losses for Muffler Configurations .......................... 14
Figure 8: Computed Transmission Losses for Muffler Configurations........................... 15
Figure 9: Comparison of TMM and FEM Results for 4 Inch Diameter Expansion
Chamber ........................................................................................................................... 16
Figure 10: Comparison of TMM and FEM Results for 8 Inch Diameter Expansion
Chamber ........................................................................................................................... 17
Figure 11: Comparison of TMM and FEM Results for 12 Inch Diameter Expansion
Chamber ........................................................................................................................... 18
Figure 12: Comparison of TMM and FEM Results for 16 Inch Diameter Expansion
Chamber ........................................................................................................................... 19
vi
NOMENCLATURE
Symbol
abs
acpr.c
acpr.rho
c
C1
C2
d
dB
Dc
Di
Do
f
FEM
intop1
intop2
j
k
Lc
Li
Lo
m
M
ρ
p0
pi
S
t
TMM
ui
z
Quantity
COMSOL Absolute Value Operator
COMSOL Variable for Speed of Sound
COMSOL Variable for Density
Speed of Sound
Constant
Constant
Diameter
Decibel
Expansion Chamber Diameter
Inlet Pipe Diameter
Outlet Pipe Diameter
Frequency
Finite Element Method
COMSOL Integral Operator
COMSOL Integral Operator
Imaginary Number
Wavenumber
Expansion Chamber Length
Inlet Pipe Length
Outlet Pipe Length
ratio
Average Mach Number
Density
Pressure Amplitude
Pressure
Cross-Sectional Area
Time
Transfer Matrix Method
Particle Velocity
Distance in z-Direction
vii
Units
in/s
lb/in3
in/s
--in
-in
in
in
Hz
----in-1
in
in
In
--lb/in3
lb/(in-s2)
lb/ in2
in2
s
-in/s
in
GLOSSARY
Acoustics
Study of sound and vibration in structures and fluids
COMSOL
Multi-physics computer modeling software
Cutoff Frequency
Boundary in a system’s frequency response in which wave energy
begins to be reduced
MATLAB
Mathematical computational software
Node
Point of intersection
Particle Velocity
Velocity of a particle in a given material as it transmits a wave
Speed of Sound
Speed at which sound travels through a given material
Transmission Loss
Decrease in acoustic pressure wave intensity
viii
KEY WORDS
 Acoustics
 COMSOL
 Exhaust System
 Expansion Chamber
 Finite Element Method
 MATLAB
 Muffler
 Noise Reduction
 Plane Wave Theory
 Pressure Wave
 Sound Silencing
 Transfer Matrix Method
 Transmission Loss
ix
ABSTRACT
A single expansion chamber muffler system with four different expansion chamber
diameters was analyzed using the transfer matrix method (TMM) in MATLAB and finite
element method (FEM) in COMSOL Multiphysics to determine the muffler transmission
loss. The calculated transmission loss from the TMM was compared to the computed
transmission loss from the FEM for each single expansion chamber configuration.
Differences in the results were analyzed and compared to plane wave theory in mufflers
to better validate the effectiveness of using the TMM for this study.
x
1. Introduction
A muffler is a device which is commonly used in an exhaust system to provide a certain
level of acoustic performance by reducing the amount of noise (unwanted sound)
generated by an upstream source, such as a combustion engine. Sound pressure
originating from the upstream source propagates downstream through the pipe until
reaching the muffler. The muffler is typically an enclosure which consists of a variety of
geometries, such as an expansion chamber, which can be tuned to provide noise
reduction in the means of reduced sound pressure levels. In acoustics, this reduction of
sound pressure is generally described as transmission loss, or the decrease in acoustic
pressure wave intensity as the pressure wave propagates away (downstream) from the
source.
The Transfer Matrix Method (TMM) can be used when a system is characterized by a
series of subsystems, or changes in geometries, which interact with the preceding and
proceeding systems. This process allows the subsystems to be linked together via
transfer matrices in order to represent the overall system. For a muffler, each subsystem
represents an acoustic impedance, which when combined provides the overall noise
reduction, or transmission loss, of the system.
Muffler design can be an in-depth iterative process relying heavily on overall geometry
and acoustic performance. The computational time in which it takes to iteratively
analyze changes in geometry relative to acoustic performance can become quite
cumbersome using a finite element analysis approach. In specific, a system requiring a
specific decibel (dB) reduction across particular frequency ranges might require a
significant amount of fine tuning in order to reach an optimal design. Consequently,
developing several finite element models may be required in order to achieve the desired
end result. Analytical approaches which are used to characterize the acoustic
performance relative to changing geometries could be more time effective and feasible
in producing the approximate geometries of the muffler. The effect of the analytical
approach depends heavily on the accuracy of results relative to the appropriate
application (i.e. good enough for engineering design purposes).
1
Researching this topic revealed that the TMM is a commonly studied analytical
approach used to analyzing the transmission loss of a muffler system. Specifically, this
approach is widely used in validating laboratory validation cases, such as those
presented in [1], [2], and [3]. However, many of the available studies tend to analyze a
defined system with a wide variety of internal (muffler) geometries. The study presented
in this report focuses on a system containing a single geometry (expansion chamber) and
the effect on the results as the expansion chamber diameter varies across four different
diameters.
2
2. Methodology
This project investigates the effectiveness of using the TMM to calculate the
transmission loss of a finite element model of a muffler system consisting of a single
expansion chamber. A schematic of this muffler system is shown below in Figure 1. The
analysis will study the acoustic performance of this type of muffler across a frequency
range of 0 to 2000 Hz for each analytical model, and 2 to 2000 Hz for each
computational model. Each case studied will be compared in order to evaluate the
effectiveness of the TMM.
Figure 1: Schematic of Single Expansion Chamber Muffler System
2.1 Model Definitions
The muffler system under study consists of an upstream (inlet) pipe of length, Li, 6
inches and diameter, Di, of 2 inches, and downstream (outlet) pipe of length, Lo, 6 inches
and diameter, Do, of 2 inches. The muffler, which is located between the upstream pipe
and downstream pipe, consists of a single expansion chamber of length, Lc, 24 inches
and diameter, Dc, of varying sizes. The expansion chamber is evaluated in four
configurations, which consist of the diameters presented below in Table 1. The
dimensions in this table are also figuratively shown on the muffler system below in
Figure 2.
3
Table 1: Expansion Chamber Configurations
Configuration Number
Diameter Value
Unit
1
4
in
2
8
in
3
12
in
4
16
in
Figure 2: Schematic of Muffler System Dimension
Fluid properties at 70 degrees Fahrenheit from [4] and [5] were used to replicate
conditions which could be achievable in a laboratory environment. The properties used
in the model are presented below in Table 2.
Table 2: Fluid Properties of Air used in the Muffler System
Property
Value
Density, ρ
0.00004335
Speed of Sound, c
13536
Unit
lb/in3
in/s
4
2.2 Setup of Transfer Matrix Method for Muffler System
The TMM approach takes the muffler system under study and separates it into individual
components (subsystems) consisting of straight pipes, an expansion, and a contraction,
which is shown below in Figure 3.
Figure 3: Schematic of Subsystem Components Which Define Muffler System
As expressed in [6], [7], and [8], these subsystem components can be described as 2x2
matrices in terms of the pressure and particle velocity at each boundary by taking into
consideration plane wave theory and average flow velocity. This is best presented by
applying nodes along each subsystem boundary and defining the respective pressure, pi,
and particle velocity, ui, at each node. This application of nodes on the boundaries of the
muffler system under study is shown below in Figure 4.
Figure 4: Schematic of Muffler System Boundaries with Nodes
5
2.2.1
Equations for a Straight Pipe
The muffler system in this project consists of three straight pipe sections which are
defined in sections I, III and V of Figure 4. Assuming a one-dimension propagating
wave in section of straight pipe, the acoustic pressure and particle velocities from [7] can
be presented as:
[1]
[2]
Applying the boundary conditions at each node in the straight pipe (arbitrarily at z = 0
and z = L) yields:
[3]
[4]
Combining these two equations and using Euler’s formula provides the following
relationship between the two nodes:
[5]
Consequently, defining sections I, III, and V as a-b, c-d, and e-f, respectively the
following transfer matrices are developed for:
Section a-b:
[6]
Section c-d:
6
[7]
Section e-f:
[8]
2.2.2
Equations for an Expansion and Contraction
The muffler system in this project consists of one expansion section and one contraction
section which are defined in sections II and IV, respectively, and shown in Figure 4.
Assuming a one-dimensional propagating plane wave across each discontinuity, as
discussed in [7], the acoustic pressure, pi, and particle (mass) velocity, vi, will remain
constant. Consequently, it holds true that for an arbitrary set of points at a discontinuity:
[9]
[10]
Where the mass velocity is defined as:
[11]
Therefore, relative the arbitrary set of points in matrix form results in:
[12]
Applying the definition of mass velocity to the abovementioned relation yields:
[13]
7
Consequently, defining sections II and IV as b-c and d-e, respectively the following
transfer matrices are developed for:
Section b-c:
[14]
Section d-e:
[15]
2.2.3
Overall Transfer Matrix
The overall transfer matrix is obtained by taking the equations presented above in
Section 2.2.1 and 2.2.2 and applying it to the muffler system in Figure 4. This is
achieved by multiplying each muffler system subsystem matrix in the order which they
appear in the system.
[16]
Where P is a substitute made for the term:
[17]
I is a substitute made for the transfer matrix straight pipe Section a-b:
[18]
II is a substitute made for the transfer matrix expansion Section b-c:
[19]
III is a substitute made for the transfer matrix straight pipe Section c-d:
8
[20]
IV is a substitute made for the transfer matrix of contraction Section d-e:
[21]
And V is a substitute made for the transfer matrix of straight pipe Section e-f:
[22]
Assuming there is no flow (M=0) in the muffler, the overall transfer matrix of Equation
[16] can be reduced further to:
[23]
Therefore, the overall transfer matrix is defined from Equation [23] as:
[24]
And, as presented in [4], the transmission loss of the muffler system can be expressed as:
[25]
2.3 Setup of Finite Element Model for Muffler System
The COMSOL Multiphysics Pressure Acoustics module was used to create the muffler
system under study. The muffler was created using a 2-dimensional axial symmetric
model and the dimensions derived from those provided in Figure 2. The COMSOL
Multiphysics built-in geometry feature was used to develop the axial symmetric model.
The model was the constrained using an axial symmetric boundary along the z-direction
9
(muffler system centerline), a sound hard boundary along the exterior walls of the
muffler system, a reflective pressure wave at the inlet and outlet, and an incident
pressure wave at the inlet. The incident pressure wave boundary condition acts as the
source, hence it was applied at the muffler system inlet. The boundary conditions
described above are presented below in Figure 5 and are outlined in the color blue.
Figure 5: Applied Boundary Conditions to Finite Element Model
A free triangular mesh was applied to the model using a custom mesh size. The custom
mesh constrained the maximum element size to a user defined value of
13536[in/s]/2000[Hz]/10, where 13536 in/s is the speed of sound in air, 2000 Hz is the
maximum frequency value of the study, and 10 is the number of elements per
wavelength. For frequency-dependent studies, a general rule is to have at least 5
elements per wavelength in order ensure there are enough elements to characterize shape
of the highest frequency’s wavelength. A representative mesh of the muffler system
under study is presented in Figure 6.
10
Figure 6: Representative Mesh of Muffler System Model
Once the boundary conditions, material property, and mesh were defined and applied to
the model, a frequency-dependent study was applied covering 10 to 2000 Hz.
To evaluate the transmission loss in the finite element model, two variables were defined
as suggested in [9] at the inlet and outlet of the muffler which defined the power of the
incoming and outgoing waves, respectively. These equations were defined in COMSOL
as follows:
Power of the incoming pressure wave:
[26]
Power of the outgoing pressure wave:
[27]
Transmission loss:
11
[28]
Once a frequency-dependent study is performed for a given muffler configuration, a 1-D
graph can be generated that plots transmission loss versus frequency based on Equation
[28].
12
3. Results and Discussion
3.1 Transfer Matrix Method Results
The TMM analysis was performed as described in Section 2.2.3 and using the MATLAB
code provided in Appendix A through Appendix D. The model definitions were assigned
as provided in Section 2.1, and iterated across the four different chamber diameters from
0 to 2000 Hz using 2 Hz resolution. The MATLAB code input parameters are also
provided below in Table 3, for information.
Table 3: MATLAB Code User Defined Input Parameters
MATLAB Variable
MATLAB Value
Unit
maxfreq
2000
Hz
res
2
Hz
freq
(0:res:maxfreq)
Hz
c
13536
in/s
rho
0.00004335
LC
24
in
LI
6
in
LO
LC
in
RI
1
in
SI
pi*(RI^2)
in2
m
(varies on model)
--
SC
m*SI
in2
SO
SI
in2
lb/in3
Computing the transmission loss for this muffler configuration provided the results
shown in Figure 7.
13
Single Expansion Chamber TMM Results
40
Transfer Matrix
Transfer Matrix
Transfer Matrix
Transfer Matrix
Method (TMM),
Method (TMM),
Method (TMM),
Method (TMM),
4 Inch Diameter
8 Inch Diameter
12 Inch Diameter
16 Inch Diameter
35
30
Transmission Loss, TL (dB)
25
20
15
10
5
0
0
200
400
600
800
1000
Frequency (Hz)
1200
1400
1600
1800
2000
Figure 7: Calculated Transmission Losses for Muffler Configurations
The blue, red, green and magenta data points represent the transmission loss for the 4
inch, 8 inch, 12 inch and 16 inch diameter expansion chambers, respectively, from 0 to
2000 Hz. This figure shows that the transmission loss characteristic for each muffler
configuration is repetitive across the analyzed frequency range and each repetitive
frequency span is approximately 282 Hz wide. Additionally, this figure is shown that
increasing the chamber diameter increases the maximum transmission loss for a given
muffler configuration. The 4 inch, 8 inch, 12 inch and 16 inch diameter expansion
chambers have maximum transmission losses of approximately 6.5 dB, 18.1 dB, 25.1 dB
and 30.1 dB, respectively.
3.2 Finite Element Model Results
FEM analyses were performed as described in Section 2.3 and using the input parameter
provided below in Table 4.
14
Table 4: FEM Input Parameter
Property
Value
Unit
Incident Pressure Wave
1
Pa
Start Frequency
2
Hz
Stop Frequency
2000
Hz
Step Frequency
2
Hz
A FEM was built for each muffler configuration and analyzed from 2 to 2000 Hz using 2
Hz resolution. A 1-D graph was then created using the output results of the respective
FEM and Equation [28]. The results from the 1-D graph were exported to a text file
using the built-in COMSOL export function, then read into MATLAB and saved off as a
variable containing frequencies and transmission loss values. The results of the FEM
analyses are shown below in Figure 8.
Single Expansion Chamber, FEM Results
40
Finite Element
Finite Element
Finite Element
Finite Element
Model
Model
Model
Model
(COMSOL),
(COMSOL),
(COMSOL),
(COMSOL),
4 Inch Diameter
8 Inch Diameter
12 Inch Diameter
16 Inch Diameter
35
30
Transmission Loss, TL (dB)
25
20
15
10
5
0
0
200
400
600
800
1000
Frequency (Hz)
1200
1400
1600
1800
2000
Figure 8: Computed Transmission Losses for Muffler Configurations
In this figure, the blue, red, green and magenta data points represent the transmission
loss for the 4 inch, 8 inch, 12 inch and 16 inch diameter expansion chambers,
respectively, from 0 to 2000 Hz. This figure shows that the characteristic of the
transmission loss for the 4 inch and 8 inch diameter muffler configurations is repetitive
across the frequency range of 0 to 2000 Hz and each repetitive frequency span is
15
approximately 282 Hz wide. The results for the 8 inch muffler configuration show that
as the frequency increases (above nominally 1000 Hz) the transmission loss amplitudes
increase from nominally 18.7 dB at 988 Hz to nominally 22.3 dB at 1852 Hz.
The transmission loss characteristic for the 12 inch diameter expansion is repetitive from
0 to 1190 Hz, and acts irregular and non-repetitive above 1190 Hz. The maximum
amplitude of transmission loss in the repetitive frequency range is nominally 25.1 dB.
Similarly, the transmission loss characteristic for the 16 inch diameter expansion
chamber is repetitive from 0 to 950 Hz, and acts irregular and non-repetitive above 950
Hz. The maximum amplitude of transmission loss in the repetitive frequency range is
nominally 30.2 dB.
3.3 Comparison of Transfer Matrix Model to Finite Element Model
3.3.1
Configuration 1: TMM Compared to FEM
The comparison of the TMM and FEM results of the 4 inch diameter expansion chamber
are presented below in Figure 9.
Single Expansion Chamber, Diameter = 4 Inches
40
Transfer Matrix Method (TMM)
Finite Element Model (COMSOL)
35
30
Transmission Loss, TL (dB)
25
20
15
10
5
0
0
200
400
600
800
1000
Frequency (Hz)
1200
1400
1600
1800
2000
Figure 9: Comparison of TMM and FEM Results for 4 Inch Diameter Expansion Chamber
The TMM results are displayed as the blue data points, and the FEM results are
displayed as the red data points. This figure shows good correlation between the TMM
16
and FEM across the entire frequency range analyzed, with the best correlation from 0 to
1000 Hz. Above 1000 Hz, minor differences are observed between the two data sets,
specifically TMM results begin to lag the FEM results (the FEM frequency span
increases slightly). Additionally, the TMM transmission loss amplitude remains constant
at nominally 6.5 dB throughout the entire frequency range, whereas the FEM
transmission loss amplitude increases by nominally 0.2 dB every 300 Hz above 1000 Hz.
3.3.2
Configuration 2: TMM Compared to FEM
The TMM and FEM results of the 8 inch diameter expansion chamber are presented
below in Figure 10.
Single Expansion Chamber, Diameter = 8 Inches
40
Transfer Matrix Method (TMM)
Finite Element Model (COMSOL)
35
30
Transmission Loss, TL (dB)
25
20
15
10
5
0
0
200
400
600
800
1000
Frequency (Hz)
1200
1400
1600
1800
2000
Figure 10: Comparison of TMM and FEM Results for 8 Inch Diameter Expansion Chamber
Again, the TMM results are displayed as the blue data points, and the FEM results are
displayed as the red data points. This figure shows good correlation between the TMM
and FEM across most of the frequency range analyzed, with the best correlation from 0
to nominally 1200 Hz. Above nominally 1200 Hz, the transmission loss amplitudes for
the TMM results remain constant at nominally 18.1 dB, whereas the FEM results begin
to increase and diverge away from the TMM results. This appears to be a more drastic
case as compared to the results obtained in Section 3.3.1 for the 4 inch diameter
expansion chamber.
17
3.3.3
Configuration 3: TMM Compared to FEM
The TMM and FEM results of the 12 inch diameter expansion chamber are presented
below in Figure 11.
Single Expansion Chamber, Diameter = 12 Inches
40
Transfer Matrix Method (TMM)
Finite Element Model (COMSOL)
Cutoff Frequency = 1376 Hz
35
30
Transmission Loss, TL (dB)
25
20
15
10
5
0
0
200
400
600
800
1000
Frequency (Hz)
1200
1400
1600
1800
2000
Figure 11: Comparison of TMM and FEM Results for 12 Inch Diameter Expansion Chamber
Again, the TMM results are displayed as the blue data points, and the FEM results are
displayed as the red data points. This figure shows good correlation between the TMM
and FEM across the frequency range 0 to nominally 1190 Hz (similar to the results
shown in Section 3.3.2 for the 8 inch diameter expansion chamber). Above nominally
1190 Hz, the TMM results do not correlate well as compared to the FEM results.
The cutoff frequency, or the lowest frequency at which a plane wave can be transmitted
without attenuation, for this size expansion chamber is described in [10] and is
calculated by:
[29]
For the 12 inch diameter expansion chamber, the cutoff frequency is calculated as 1376
Hz when using Equation [29]. This calculated value correlates well with the data
18
produced with the FEM model, since the transmission loss characteristic behaves
irregular and does not have good agreement with the TMM results above this frequency.
3.3.4
Configuration 4: TMM Compared to FEM
The TMM and FEM results of the 16 inch diameter expansion chamber are presented
below in Figure 12.
Single Expansion Chamber, Diameter = 16 Inches
40
Transfer Matrix Method (TMM)
Finite Element Model (COMSOL)
Cutoff Frequency = 1032 Hz
35
30
Transmission Loss, TL (dB)
25
20
15
10
5
0
0
200
400
600
800
1000
Frequency (Hz)
1200
1400
1600
1800
2000
Figure 12: Comparison of TMM and FEM Results for 16 Inch Diameter Expansion Chamber
In this figure, the TMM results are displayed as the blue data points, and the FEM results
are displayed as the red data points. The TMM results predict the transmission loss of
the FEM results up to approximately 950 Hz with good correlation. Above this
frequency, the TMM results do not predict the FEM results well; the FEM results behave
irregular and non-repetitive. The frequency at which this behavior in the data begins is
comparable with the calculated cutoff frequency of 1032 Hz.
19
4. Conclusion
4.1 Applicability of TMM Compared to FEM
The calculated results obtained using the TMM were compared to computed results
using the FEM. For the muffler configuration studied, the TMM produced relatively
accurate results as compared to the FEM across the entire frequency range analyzed for
expansion chambers of diameter 4 inches and 8 inches. For the larger expansion
chamber diameters studied, namely the 12 inch diameter and the 16 inch diameter
expansion chambers, the TMM was only able to calculate the transmission loss for
approximately half the frequency range analyzed. The difference in results of the
different sized expansion chambers is consistent with the theory of plane wave
propagation and is shown by the calculated cutoff frequency values for the larger
diameter models. Consequently, the TMM equations for transmission loss do not take
the cutoff frequency into consideration and can provide erroneous results.
Based on these results, the TMM proved to be an effective way to calculate the
transmission loss for a single expansion chamber system when compared to the FEM.
However, this study revealed that for a given muffler geometry, the TMM can be
frequency limited based on the calculated cutoff frequency for the system.
20
5. References
[1]
Andersen, K.S., Analyzing Muffler Performance Using the Transfer Matrix
Method, COMSOL Conference Hanover, 2008, page 1-7.
[2]
Gerges, S.N.Y., et. al., Muffler Modeling by Transfer Matrix Method and
Experimental Verification, Journal of the Brazilian Society of Mechanical Science
and Engineering, Volume XXVII, No. 2, April-June 2005, page 132-140.
[3]
Tao, Z., Seybert, A.F., A Review of Current Techniques for Measuring
Transmission Loss, Society of Automotive Engineers, Inc., 2003, page 1-5.
[4]
Air- Speed of Sound, The Engineering Toolbox,
http://www.engineeringtoolbox.com/air-speed-sound-d_603.html.
[5]
Air- Temperature, Pressure and Density, The Engineering Toolbox,
http://www.engineeringtoolbox.com/air-temperature-pressure-density-d_771.html.
[6]
Chen, Feng, Optimization Design of Muffler Based on Acoustic Transfer Matrix
and Genetic Algorithm, Journal of Vibroengineering, Volume 16, Issue 5, August
2014, page 2216-2223.
[7]
Yeh, L.J., Chang, Y.C., Chiu, M.C., Shape Optimal Design on Double-Chamber
Mufflers Using Simulated Annealing and a Genetic Algorithm, Turkish Journal of
Engineering and Environmental Sciences 29, 2005, page 207-224.
[8]
Kanade, S.V., Bhattu, A. P., Optimization of Sound Transmission Loss and
Prediction of Insertion Loss of Single Chamber Perforated Plug Muffler with
Straight Duct, American International Journal of Research in Science, Technology,
Engineering and Mathematics, 2014, Page 13-19.
[9]
Absorptive Muffler, Model 1367, COMSOL Model Gallery, 2011, page 10.
[10] Davis, D.D, et. al., Theoretical and Experimental Investigation of Mufflers with
Comments on Engine-Exhaust Muffler Design, National Advisory Committee for
Aeronautics Report 1192, 1954, page 1-47.
21
6. Appendix A: MATLAB Code for Configuration 1
% Validation Case for Single Expansion Chamber, 4 Inch Diameter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Closing, Clearing, etc:
close all
clear all
clc
%Define Additional Parameters
maxfreq = 2000;
res = 2;
freq = (0:res:maxfreq);
vecl = length(freq);
mat = zeros(vecl);
val = mat(1,:);
c = 13536;
rho = .00004335;
DETAIL
%Maximum Frequency, Hz
%Resolution, Hz
%Frequency Vector
%Length of <<freq>> vector
%Zero matrix sized to <<freq>> vector
%Zero vector from <<mat>> matrix
%Speed of sound (air) in/s
%Density of fluid, lbm/in^3
LC = 24;
LI = 6;
LO = LI;
%Length of Expansion Chamber, in
%Length of Upstream Pipe, in
%Length of Downstream Pipe, in
RI = 1;
SI = pi*(RI^2);
m = 4;
SC = m*SI;
SO = SI;
%Radius of Upstream Pipe, in
%Cross-Sectional Area of Upstream Pipe, in^2
%Cross-Sectional Area Ratio
%Cross-Sectional Area of Expansion Chamber, in^2
%Cross-Sectional Area of Downstream Pipe, in^2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for kk = 1:vecl
k = (2*pi*freq(kk))/c;
T1
T2
T3
T4
T5
T =
T11
T12
T21
T22
=
=
=
=
=
%Denoting Wavenumber, k
[cos(k*LI) i*sin(k*LI); i*sin(k*LI) cos(k*LI)]; %TM,Section I
[1 0; 0 SC/SI];
%TM,Section II
[cos(k*LC) i*sin(k*LC); i*sin(k*LC) cos(k*LC)];%TM,Section III
[1 0; 0 SO/SC];
%TM,Section IV
[cos(k*LO) i*sin(k*LO); i*sin(k*LO) cos(k*LO)]; %TM,Section V
T1*T2*T3*T4*T5;
= T(1,1);
= T(1,2);
= T(2,1);
= T(2,2);
%Overall
%Overall
%Overall
%Overall
TM,
TM,
TM,
TM,
%Overall TM
Element T11
Element T12
Element T21
Element T22
QQ = (T11+T12+T21+T22);
%Arbitrary Variable
RealPart = real(QQ);
%Real Part of Quantity QQ
ImagPart = imag(QQ);
%Imaginary Part of Quantity QQ
mag = sqrt((RealPart^2)+(ImagPart^2));
%Solving for the Magnitude
Term1 = (mag/2);
%Arbitrary Variable
Term2 = (SI/SO);
%Arbitrary Variable
22
val(kk) = 20*log10(Term1)+10*log10(Term2);
%Computing TL
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Plotting Results:
figure(1)
plot(freq, val,'*')
title ('Single Expansion Chamber, Diameter = 4 Inches',...
'FontWeight','bold')
axis([0 maxfreq 0 40])
xlabel('Frequency (Hz)')
ylabel('Transmission Loss, TL (dB)')
legend('Transfer Matrix Method (TMM)')
grid on
23
7. Appendix B: MATLAB Code for Configuration 2
% Validation Case for Single Expansion Chamber, 8 Inch Diameter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Closing, Clearing, etc:
close all
clear all
clc
%Define Additional Parameters
maxfreq = 2000;
res = 2;
freq = (0:res:maxfreq);
vecl = length(freq);
mat = zeros(vecl);
val = mat(1,:);
c = 13536;
rho = .00004335;
DETAIL
%Maximum Frequency, Hz
%Resolution, Hz
%Frequency Vector
%Length of <<freq>> vector
%Zero matrix sized to <<freq>> vector
%Zero vector from <<mat>> matrix
%Speed of sound (air) in/s
%Density of fluid, lbm/in^3
LC = 24;
LI = 6;
LO = LI;
%Length of Expansion Chamber, in
%Length of Upstream Pipe, in
%Length of Downstream Pipe, in
RI = 1;
SI = pi*(RI^2);
m = 16;
SC = m*SI;
SO = SI;
%Radius of Upstream Pipe, in
%Cross-Sectional Area of Upstream Pipe, in^2
%Cross-Sectional Area Ratio
%Cross-Sectional Area of Expansion Chamber, in^2
%Cross-Sectional Area of Downstream Pipe, in^2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for kk = 1:vecl
k = (2*pi*freq(kk))/c;
T1
T2
T3
T4
T5
T =
T11
T12
T21
T22
=
=
=
=
=
%Denoting Wavenumber, k
[cos(k*LI) i*sin(k*LI); i*sin(k*LI) cos(k*LI)]; %TM,Section I
[1 0; 0 SC/SI];
%TM,Section II
[cos(k*LC) i*sin(k*LC); i*sin(k*LC) cos(k*LC)];%TM,Section III
[1 0; 0 SO/SC];
%TM,Section IV
[cos(k*LO) i*sin(k*LO); i*sin(k*LO) cos(k*LO)]; %TM,Section V
T1*T2*T3*T4*T5;
= T(1,1);
= T(1,2);
= T(2,1);
= T(2,2);
%Overall
%Overall
%Overall
%Overall
TM,
TM,
TM,
TM,
%Overall TM
Element T11
Element T12
Element T21
Element T22
QQ = (T11+T12+T21+T22);
%Arbitrary Variable
RealPart = real(QQ);
%Real Part of Quantity QQ
ImagPart = imag(QQ);
%Imaginary Part of Quantity QQ
mag = sqrt((RealPart^2)+(ImagPart^2));
%Solving for the Magnitude
Term1 = (mag/2);
%Arbitrary Variable
Term2 = (SI/SO);
%Arbitrary Variable
24
val(kk) = 20*log10(Term1)+10*log10(Term2);
%Computing TL
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Plotting Results:
figure(1)
plot(freq, val,'*')
title ('Single Expansion Chamber, Diameter = 8 Inches',...
'FontWeight','bold')
axis([0 maxfreq 0 40])
xlabel('Frequency (Hz)')
ylabel('Transmission Loss, TL (dB)')
legend('Transfer Matrix Method (TMM)')
grid on
25
8. Appendix C: MATLAB Code for Configuration 3
% Validation Case for Single Expansion Chamber, 12 Inch Diameter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Closing, Clearing, etc:
close all
clear all
clc
%Define Additional Parameters
maxfreq = 2000;
res = 2;
freq = (0:res:maxfreq);
vecl = length(freq);
mat = zeros(vecl);
val = mat(1,:);
c = 13536;
rho = .00004335;
DETAIL
%Maximum Frequency, Hz
%Resolution, Hz
%Frequency Vector
%Length of <<freq>> vector
%Zero matrix sized to <<freq>> vector
%Zero vector from <<mat>> matrix
%Speed of sound (air) in/s
%Density of fluid, lbm/in^3
LC = 24;
LI = 6;
LO = LI;
%Length of Expansion Chamber, in
%Length of Upstream Pipe, in
%Length of Downstream Pipe, in
RI = 1;
SI = pi*(RI^2);
m = 36;
SC = m*SI;
SO = SI;
%Radius of Upstream Pipe, in
%Cross-Sectional Area of Upstream Pipe, in^2
%Cross-Sectional Area Ratio
%Cross-Sectional Area of Expansion Chamber, in^2
%Cross-Sectional Area of Downstream Pipe, in^2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for kk = 1:vecl
k = (2*pi*freq(kk))/c;
T1
T2
T3
T4
T5
T =
T11
T12
T21
T22
=
=
=
=
=
%Denoting Wavenumber, k
[cos(k*LI) i*sin(k*LI); i*sin(k*LI) cos(k*LI)]; %TM,Section I
[1 0; 0 SC/SI];
%TM,Section II
[cos(k*LC) i*sin(k*LC); i*sin(k*LC) cos(k*LC)];%TM,Section III
[1 0; 0 SO/SC];
%TM,Section IV
[cos(k*LO) i*sin(k*LO); i*sin(k*LO) cos(k*LO)]; %TM,Section V
T1*T2*T3*T4*T5;
= T(1,1);
= T(1,2);
= T(2,1);
= T(2,2);
%Overall
%Overall
%Overall
%Overall
TM,
TM,
TM,
TM,
%Overall TM
Element T11
Element T12
Element T21
Element T22
QQ = (T11+T12+T21+T22);
%Arbitrary Variable
RealPart = real(QQ);
%Real Part of Quantity QQ
ImagPart = imag(QQ);
%Imaginary Part of Quantity QQ
mag = sqrt((RealPart^2)+(ImagPart^2));
%Solving for the Magnitude
Term1 = (mag/2);
%Arbitrary Variable
Term2 = (SI/SO);
%Arbitrary Variable
26
val(kk) = 20*log10(Term1)+10*log10(Term2);
%Computing TL
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Plotting Results:
figure(1)
plot(freq, val,'*')
title ('Single Expansion Chamber, Diameter = 12 Inches',...
'FontWeight','bold')
axis([0 maxfreq 0 40])
xlabel('Frequency (Hz)')
ylabel('Transmission Loss, TL (dB)')
legend('Transfer Matrix Method (TMM)')
grid on
27
9. Appendix D: MATLAB Code for Configuration 4
% Validation Case for Single Expansion Chamber, 16 Inch Diameter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Closing, Clearing, etc:
close all
clear all
clc
%Define Additional Parameters
maxfreq = 2000;
res = 2;
freq = (0:res:maxfreq);
vecl = length(freq);
mat = zeros(vecl);
val = mat(1,:);
c = 13536;
rho = .00004335;
DETAIL
%Maximum Frequency, Hz
%Resolution, Hz
%Frequency Vector
%Length of <<freq>> vector
%Zero matrix sized to <<freq>> vector
%Zero vector from <<mat>> matrix
%Speed of sound (air) in/s
%Density of fluid, lbm/in^3
LC = 24;
LI = 6;
LO = LI;
%Length of Expansion Chamber, in
%Length of Upstream Pipe, in
%Length of Downstream Pipe, in
RI = 1;
SI = pi*(RI^2);
m = 64;
SC = m*SI;
SO = SI;
%Radius of Upstream Pipe, in
%Cross-Sectional Area of Upstream Pipe, in^2
%Cross-Sectional Area Ratio
%Cross-Sectional Area of Expansion Chamber, in^2
%Cross-Sectional Area of Downstream Pipe, in^2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for kk = 1:vecl
k = (2*pi*freq(kk))/c;
T1
T2
T3
T4
T5
T =
T11
T12
T21
T22
=
=
=
=
=
%Denoting Wavenumber, k
[cos(k*LI) i*sin(k*LI); i*sin(k*LI) cos(k*LI)]; %TM,Section I
[1 0; 0 SC/SI];
%TM,Section II
[cos(k*LC) i*sin(k*LC); i*sin(k*LC) cos(k*LC)];%TM,Section III
[1 0; 0 SO/SC];
%TM,Section IV
[cos(k*LO) i*sin(k*LO); i*sin(k*LO) cos(k*LO)]; %TM,Section V
T1*T2*T3*T4*T5;
= T(1,1);
= T(1,2);
= T(2,1);
= T(2,2);
%Overall
%Overall
%Overall
%Overall
TM,
TM,
TM,
TM,
%Overall TM
Element T11
Element T12
Element T21
Element T22
QQ = (T11+T12+T21+T22);
%Arbitrary Variable
RealPart = real(QQ);
%Real Part of Quantity QQ
ImagPart = imag(QQ);
%Imaginary Part of Quantity QQ
mag = sqrt((RealPart^2)+(ImagPart^2));
%Solving for the Magnitude
Term1 = (mag/2);
%Arbitrary Variable
Term2 = (SI/SO);
%Arbitrary Variable
28
val(kk) = 20*log10(Term1)+10*log10(Term2);
%Computing TL
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Plotting Results:
figure(1)
plot(freq, val,'*')
title ('Single Expansion Chamber, Diameter = 16 Inches',...
'FontWeight','bold')
axis([0 maxfreq 0 40])
xlabel('Frequency (Hz)')
ylabel('Transmission Loss, TL (dB)')
legend('Transfer Matrix Method (TMM)')
grid on
29