Muffler Modeling by Transfer Matrix Method and Experimental Verification
Muffler Modeling by Transfer Matrix Method and Experimental Verification
Muffler Modeling by Transfer Matrix Method and Experimental Verification
Gerges et al
S. N.Y. Gerges
and R. Jordan
Federal Univ. of Santa Caterina
Mechanical Engineering Dept
CP 476
Florianpolis, Brazil
samir@emc.ufsc.br
jordan@emc.ufsc.br
F. A. Thieme
Eberspcher Tuper Sistemas de Exaustao
89290-000 Sao Bento do Sul, SC. Brazil
fabeme15@zaz.com.br
J. L. Bento Coelho
CAPS, Instituto Superior Tcnico
1049-001, Lisbon, Portugal
bcoelho@ist.utl.pt
J. P. Arenas
Institute of Acoustics, Univ. Austral de Chile
PO Box 567, Valdivia, Chile
jparenas@uach.cl
Introduction
Mufflers are commonly used in a wide variety of applications.
Industrial flow ducts as well as internal combustion engines
frequently make use of silencing elements to attenuate the noise
levels carried by the fluids and radiated to the outside atmosphere by
the exhausts. Restrictive environmental legislation require that
silencer designers use high performance and reliable techniques.
Various techniques are currently available for the modeling and
testing of duct mufflers. Empirical, analytical and numerical
techniques have been used and proved reliable under controlled
conditions.1
Design of a complete muffler system is, usually, a very complex
task. Each element is selected by considering its particular
performance, cost and its interaction effects on the overall system
performance and reliability.
Numerical techniques, such as the Finite Element Method
(FEM) and the Boundary Element Method (BEM) have proven to be
convenient for complex muffler geometries. Although these
methods are applicable to any muffler configuration, when the
silencer shape becomes complex, the three-dimensional FEM
requires a very large number of elements and nodes. This results in
lengthy and time-consuming data preparation and computation.
Although high speed computational and storage machines exist, the
use of FEM or BEM for muffler design is restricted to trained
personnel and is commercially expensive, in particular for
preliminary design evaluation. Most muffler manufacturers are
small and medium companies with a limited number of resources.
Paper accepted April, 2005. Technical Editor: Jos Roberto de Frana Arruda.
They thereby require fast and low cost methods for preliminary
muffler design.
A large amount of work has been published in the last five
decades on the prediction of muffler performance. The NACA
report by Davis et al. (1954) was one of the first comprehensive
attempts to model mufflers. They used the transmission line theory
by assuming both continuity of pressure and continuity of volume
velocity at discontinuities. In the late fifties and sixties, equivalent
electrical circuits based on the four-pole transmission matrices were
widely used to predict the muffler performance through the use of
analogue computers (Igarashi and Toyama, 1958; Davies, 1964). In
the seventies and eighties Alfredson (1970), Munjal (1970 and
1987), Thawani and Noreen (1988), Sullivan and Crocker (1978)
and Jayaraman and Yam (1981) presented approaches to the partial
and full modeling of mufflers. In addition, Craggs (1989) reported a
technique that combines the use of transfer matrix approach and
finite elements in the study of duct acoustics. Later, Davies (1993)
has cautioned that approaches using electrical analogies need to be
used with great care where the influence of mean flow on wave
propagation is important.
During the last fifteen years a great body of research has been
published on the sound propagation in curved ducts and bends used
as muffler elements. Kim and Ih (1999) studied the problem and
presented expressions for the transfer matrix of such systems. They
found that the classical plane wave theory for the straight duct of
equivalent length is not applicable when dealing with high sound
pressures. Flix and Pagneux (2002) presented an exact multimodal
formalism for acoustic propagation in three-dimensional rigid bends
of circular cross-section. They defined an impedance matrix as a
Riccati equation which can be solved numerically. The approach
was applied to calculate the reflection and transmission of a typical
bend, and also to obtain the resonance frequencies of closed tube
systems.
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(4)
(5)
(6)
and
Theory
(7)
A B
T1 =
,
C D
(8)
p1 = Ap 2 + B 2 ,
(1)
1 = Cp 2 + D 2 ,
(2)
and
(3)
S. N. Y. Gerges et al
(9)
(10)
q 0 = T0 T1T2 T3 T4 T5 T6 q 7 =
T q
i 7
(11)
i =0
which relates the convective state variables at two points (inlet and
exhaust) in a muffler.
Now, at a point n, the vector of convective state variables is
related to the vector of classical state variables (for stationary
medium) by means of the linear transformation
qn = Cn u n ,
M n n cn / S n
.
1
(13)
(14)
~
where T1 = C11T1C 2 is the transfer matrix for stationary medium.
Therefore, it is possible to use convective or classical state
variables to formulate a particular problem. Later, the proper
transfer matrix can be obtained by using two transformation
matrices, one evaluated at the upstream end and the other evaluated
at the downstream end of a muffler element.
It can be seen that this matrix formulation is very convenient
particularly for use with a digital computer. The only information
needed to model any complex muffler are the transfer matrices
entries. The four-pole constants (entries for the transfer matrix) can
be found easily for simple muffler elements such as straight pipes
and expansion chambers. But for a complex element its four-pole
constants can take very complicated forms which are not easily
determined mathematically. An alternative is to use the finite
element method to obtain numerically each constant (Craggs, 1989).
Fortunately, a large number of transfer matrices have been
theoretically developed and reported in the literature. Some of them
include the convective effects of a mean flow. Several formulas of
transfer matrices of different elements constituting commercial
mufflers have been compiled in the book by Munjal (1987) and,
more recently, in the book by Mechel (2002). They also included an
extensive list of relevant references.
It has to be noticed that, in addition to the geometric
characteristics and flow conditions, some transfer matrices depend
on additional parameters such as the drop in stagnation pressure and
(12)
where u n = [ ~
p n ~n ]T , ~
p and ~ are, respectively, the sound
pressure and volume velocity for stationary medium, and Cn is a non
singular 22 transformation matrix given by
Cn =
M n S n / n c n
Transmission loss does not involve neither the source nor the
radiation impedance. It is thus an invariant property of the element.
Being made independent of the terminations, TL finds favor with
researchers who are sometimes interested in finding the acoustic
transmission behavior of an element or set of elements in isolation
of the terminations (Munjal, 1987). In the past, measurement of the
incident wave in a standing wave field required use of the
impedance tube technology, leading to quite laborious experiments.
However, use of the two-microphone method with modern
instrumentation allows faster and accurate results.
Considering the muffler system sketched in Fig. 3 and taking
into account the effect of a mean flow, Transmission Loss can be
calculated in terms of the four-pole constants as (Mechel, 2002)
1+ M
1
TL = 20 log
+
1
M
2
2 c 2 S1
c S
1 1 2
1/ 2
(15)
BS 2 C 1c1 D1c1 S 2
1
+
+
.
A+
2
2 c2
S1
2 c 2 S1
(16)
Experimental Method
In general, experimental results are required for supplementing
the analysis by providing certain basic data or parameters that
cannot be predicted precisely, for verifying the analytical/numerical
predictions, and also for evaluating the overall performance of a
system configuration so as to check if it satisfies the design
requirements.
ABCM
Arenas, 2004; Chung and Blaser, 1980, and Abom and Bodn,
1988)
0.1c / 2 s < f < 0.8c / 2 s .
(19)
(20)
(21)
e = 1 b2 / a 2
exp( jks) H 12
G
S
+ 10 log 11 + 10 log 1 ,
exp( jks) H 34
G33
S2
H 12 = H 1F H F 2 ,
(18)
where H1F is the complex transfer function between the signal from
the microphone placed at position 1 and the source signal, and HF2 is
the complex transfer function between the source signal and the
microphone placed at position 2. Of course, H34 can be estimated in
a similar way.
1/ 2
(22)
where a is half the length of the largest axis (m) and b is half the
length of the smallest axis (m). Equation (21) gives results being
quite similar to those obtained by means of more complex
approaches involving Mathieu functions (Denia et al., 2001).
Table 1. Relationship between the eccentricity and in Eq. (21).
Eccentricity e
0.10320
0.20490
0.30050
0.39932
0.50032
0.60057
0.69953
0.79911
0.89999
0.90157
(17)
Factor
3.0624
3.0858
3.1208
3.1677
3.2228
3.2798
3.3342
3.3857
3.4358
3.4366
For practical reasons the distance between the sound source and
the muffler and the distance between the muffler and the termination
should be small. A distance of five to 10 times the diameter of the
tube is recommended (Abom and Bodn, 1988). In addition, the
measurement point 1 (see Fig. 4), should be positioned as close as
possible to the muffler, although a minimum distance of about 10
mm should be considered in order to avoid the influence of nearby
fields.
A distance s=50 mm was used for the experiments. Therefore,
all the results of measurements presented in the following sections
can be considered as valid for frequencies above 343 Hz.
Results
Eight muffler configurations were carefully manufactured and
tested using the experimental set up described above. The
configurations include a simple elliptic expansion chamber, an
expansion chamber with extended tube at the inlet/outlet, a
reflective muffler with extended tube at the inlet/outlet, a reflective
muffler with extended tube at both the inlet and outlet, a concentric
tube resonator, an expansion chamber with perforated tube at the
inlet, an expansion chamber with perforated tube at the inlet and
S. N. Y. Gerges et al
outlet, and a complex muffler with several elements. For each one,
the theoretical results using the Transfer Matrix Method were
numerically evaluated by means of a digital computer. Since the
experiments were conducted in a stationary medium and the fact
that, most of the time, the transfer matrices are defined in the
literature with respect to convective variables, the corresponding
transfer matrices for stationary medium were obtained by means of
the transformation matrix procedure described by Eqs. (13) and
(14).
The results obtained experimentally and predicted by the TMM
modeling of each muffler configuration are presented in this section.
(23)
f min = 2nc / 4 L ,
(24)
and
Expansion Chambers
Figure 5 shows the geometry of a simple elliptic expansion
chamber and the comparison between the experimental values of TL
and the numerical results of TL obtained from TMM modeling. The
units for the dimensions of the chamber shown in Fig. 5 and for all
subsequent figures are given in mm, unless otherwise is stated. A
good agreement, in general, is shown. Differences of up to 3 dB
were obtained at the first and last TL loops. These are located close
to the limits of the valid frequency range, determined by the
microphone distance limitation given by Eq. (19) and the cut-off
frequency of the chamber given by Eq. (21), where some errors are
expected. Therefore, for this particular chamber the measurements
are valid for frequencies above 343 Hz and below 2100 Hz.
(25)
where L is the total length of the extended tube (m) and d is its
internal diameter (m). The factor 0.315 corresponds to an end
correction (Pierce, 1981). Therefore, for this particular muffler
geometry, the first TL peak occurs at fr=388 Hz. Additional peaks
appear at odd multiples of this frequency, as confirmed during the
experiments (see Fig. 7).
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Reflective Systems
Reflective systems are non straight-through mufflers. They can
have extended tubes at the inlet, extended tubes at the outlet or both.
Figures 10, 11 and 12 show the geometry and the results of TL
obtained from TMM and measurements for three reflective systems.
with extended tube at the inlet show a similar behavior than the
reflective system with extended tube at the outlet. According to
Kimura (1995), the first TL peak can be calculated as
f r = c /[4(l 0.315d )] ,
(26)
where l is the distance between the ends of the tube and the
reflection wall (m) and d is the internal diameter of the extended
tube (m). Therefore, for this case, the first TL peak is located at
fr=820 Hz, which agrees quite well with the experimental result.
S. N. Y. Gerges et al
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Real Muffler
A more complex geometry, as found in real mufflers, was
finally considered for modeling using TMM and experimental
testing. The results presented in Fig. 17 show a good agreement
between theoretical predictions and the experimental measurements.
It is observed that this complex muffler presents three
significant TL peaks below 2 kHz. The TMM modeling results
predicted quite reasonably the first two of them. In addition, a good
prediction of the drop in TL around 1100 Hz is also observed.
Conclusions
S. N. Y. Gerges et al
Acknowledgments
The authors acknowledge the support given by the Brazilian
National Council for Scientific Research, Development, and
Technology (CNPq), the Portuguese (ICCTI), and the Chilean
CONICYT-FONDECYT under grant No 7020196.
References
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Silencers, Ph.D. thesis, University of Southampton, UK.
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