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1
TENTH INTERNATIONAL CONGRESS ON SOUND AND VIBRATION
Universidad Politécnica de ValenciaDepartamento de Ingeniería Mecánica y de Materiales
F. D. Denia, L. Baeza, J. Albelda, F. J. FuenmayorAuthors
Acoustics of Ducts and Mufflers II
ACOUSTIC BEHAVIOUR OF ELLIPTICAL MUFFLERS WITH SINGLE-INLET AND DOUBLE-OUTLET
2
CONTENTS
1. Introduction.
2. Mathematical approach:Mode-matching method.
3. Experimental measurement.
4. Results and discussion.
5. Conclusions.
3
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
1. Introduction
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Elliptical muffler with single-inlet and double-outlet
� Reduces flow noise and back pressure
� Similarities and differences in comparison with circular geometries. Dependence on the eccentricity.
� Three-dimensional acoustic modelling: Mode-matching methodFEM
4
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
1. Introduction
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Acoustic behaviour
Analytical models
Numerical models
Measurements
Plane wave:- Important limitations
Three-dimensional:- Simple geometries
FEM:- Computational effort- Validation
Experimental set-up:- Expensive- Validation
r
z
r
z
5
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
2. Mathematical approach: MMM
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Solutions of the wave equationCircular ducts
Elliptical chamber
2R1
L
z1
zA B
2R3
2R2z2
z3
C
D
2R1
L
z1
zA B
2R3
2R2z2
z3
C
D
δ2 θ2
δ3 θ3
δ1 θ1
r1 ϕ1
r2 ϕ2
r3 ϕ3
v ur
θ
2b
2a
u = uwδ2 θ2
δ3 θ3
δ1 θ1
r1 ϕ1
r2 ϕ2
r3 ϕ3
v ur
θ
2b
2a
u = uw
θi = 90, 180 degrees
( ) ( ) ( )��∞
=
∞
=
−−+���
����
�+=
0 01
1
1,
j,
j,111 cosJee,, 1,,1,,
m nnmm
zknm
zknmA m
RrAAzrP nmAnmA ϕαϕ
� Mode-matching method
( ) ( ) ( ) ( )nmmm n
nmmzk
nmzk
nmB quqvBBzuvP nmBnmB,
0 0,
j,
j, ,Ce,ceee,, ,,,,��
∞
=
∞
=
−−+ +=
6
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
2. Mathematical approach: MMM
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Second step. Multiplication by weighting functions andintegration over cross-section
Pressure conditions: modes of circular ducts
Velocity conditions: modes of elliptical chamber
ABzB
AzBzAzBzA
SSU
SUUPP
−=
==
=
====
on0
on,
0
0000 11
� First step. Pressure and velocity conditions at discontinuitiesExpansion
Contraction
DCBLzB
DzDLzBzDLzB
CzCLzBzCLzB
SSSU
SUUPP
SUUPP
−−=
==
==
=
====
====
on0
on,
on,
00
00
33
22
( ) ( )iiistt tRr ϕα cosJ ,
( ) ( )sttstt quqv ,, ,Ce,ce
7
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
2. Mathematical approach: MMM
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Second step. Details
VelocityContraction
VelocityExpansion
PressureExpansion A. i = 1Contraction C. i = 2Contraction D. i = 3
Integrate overMultiply byCondition
( ) ( )∞=∞= ...,,1,0;...,,1,0
cosJ ,
sttRr iiistt ϕα
( ) ( )∞=∞= ...,,1,0;...,,1,0
,Ce,ce ,,
stquqv sttstt
Duct Chamber
ASCSDS
ASCSDS
( ) ( )∞=∞= ...,,1,0;...,,1,0
,Ce,ce ,,
stquqv sttstt
AS BS
CSBS
DS
Pressure conditions: 3 (t + 1) (s + 1) equationsVelocity conditions: 2 (t + 1) (s + 1) equations
8
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
2. Mathematical approach: MMM
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Third step. Evaluation of integrals
Pressure. Circular ducts
- Orthogonality of Fourier-Bessel functions
- Analytical integrals( ) ( )
( ) ( ) ( ) ( )( )
( ) ( )��
�
��
�
�
=���
���
���
���
−+′
≠′−′−
=�µλλ
λλ
µλλµλµλµµλ
µλ2
22
22
2
22
0J1J
2
JJJJdJJ
rRsrR
rRrRR
rrrrss
ssssR
ss
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
( ) π
ππ
π
π
π
π
=
+=
��
���
� +=
�
��
��∞
=
++
++
++
∞
=+
2
0
2
0
1232
1212
2121
2
0
212
0
222
22
22
20
2
0
22
d,ce
2d2cos,ce
d2cos,ce
vqv
AAAvvqv
AAAAvvqv
s
r
sr
sr
ss
r
sr
sr
sss
Velocity. Elliptical chamber- Orthogonality of Mathieu functions- Analytical integrals
- Numerical integrals ( ) ( ) ( )��ww u
s
u
s uquuuqu0
2
0
2 d,Ce;d2cosh,Ce
9
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
2. Mathematical approach: MMM
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Third step. Evaluation of integrals
Pressure. Elliptical chamber
- Mathieu functions in terms of Fourier-Bessel functions
- Graf’s addition theoremVelocity. Circular ducts
- Applicate the same procedure� Fourth step. Truncation of the algebraic system of equations
Incident plane wave Anechoic terminations5 (t + 1) (s + 1) equations 5 (m + 1) (n + 1) unknownst = m = p s = n = q
10,0 =+A 0,, == −−nmnm DC
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )θ
ρπ
θρ
π
12cos2
J1,2ec,0ce,Ce,ce
2cos2J1,2ce,0ce,Ce,ce
012
121212
1
12121212
02
222
0
2222
+���
����
�−
′=
���
����
�−=
�
�
∞
=+
+++
++++
∞
=
trqAAq
qqquqv
trqAA
qqquqv
tt
st
tsss
ss
tt
st
tsss
ss
( ) ( )
( ) ( ) ( )�∞
−∞=− +
=
jiiijijm
m
mjr
mr
θϕµδµ
θµ
cosJJ
cosJ
10
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
3. Experimental measurement
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Experimental set-up
Transfer function method
� Acoustic attenuation performance
Transmission loss
��
�
�
��
�
�
−−+−
−=Ar
DrCr
SHHSSHHSSHHS
TL 21211
2'4'3'3'3
23433log10
1
PowerAmplifier
Loudspeaker ConicalAdapter
2 3 4
3’ 4’
AnechoicTerminationsMicrophones
PC with NI4451/4454 boards
Conditioning Amplifier
Muffler
1
PowerAmplifier
Loudspeaker ConicalAdapter
2 3 4
3’ 4’
AnechoicTerminationsMicrophones
PC with NI4451/4454 boards
Conditioning Amplifier
Muffler
skjrH 0e=
11
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
4. Results and discussion
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Validation: Transmission Loss vs frequencyL = 0.3 m, a = 0.23/2 m, b = 0.13/2 m, R1 = R2 = R3 = 0.051/2 m, δδδδ1= 0 m, δδδδ2 = δδδδ3= 0.073 m, θθθθ1 = θθθθ2 = 0º, θθθθ3 = 180º
0 800 1600 2400 32000
10
20
30
L = 0.3 m, a = 0.23/2 m, b = 0.13/2 m, R1 = R2 = R3 = 0.051/2 m, δδδδ1= δδδδ2 = δδδδ3= 0.073 m, θθθθ1 = θθθθ2 = 0º, θθθθ3 = 180º
─── MMM+++ FEMooo Experimental
0 800 1600 2400 32000
5
10
15
20
25 ─── MMM+++ FEMooo Experimental
12
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
4. Results and discussion
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Effect of lengthL = 0.05 m, a = 0.23/2 m, b = 0.13/2 m, R1 = R2 = R3 = 0.051/2 m
L = 0.3 m and L = 0.15 m, a = 0.23/2 m, b = 0.13/2 m, R1 = R2 = R3 = 0.051/2 m
0 800 1600 2400 32000
10
20
30─── MMM+++ FEM
─── MMM+++ FEM
0 800 1600 2400 32000
10
20
30
40
50─── MMM+++ FEM
13
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
4. Results and discussion
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Effect of eccentricityL = 0.25 m, R1 = R2 = R3 = 0.051/2 m, δδδδ1= 0 m, δδδδ2 = δδδδ3= 0.06 m, θθθθ1 = θθθθ2 = 0º, θθθθ3 = 180ºFirst case: a = 0.23/2 m, b = 0.13/2 m, εεεε1 = 0.8249Second case: a = 0.1858/2 m, b = 0.1609/2 m, εεεε2 = 0.5Third case: circular, R = 0.1729/2 m
� Cut-off frequenciesElliptical
20,20
0,2,4
2 ρπqcfc =
aερ =
Circular
Rcfc 2
05.3 00,2, π=
1599 Hz 1877 Hz 1909 Hz
0 800 1600 2400 32000
5
10
15
20
25
30
35─── MMM+++ FEM
14
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
4. Results and discussion
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� Comparison double-outlet/single-outlet
� Location of ductsL = 0.3 m, a = 0.23/2 m, b = 0.13/2 m, R1 = R2 = R3 = 0.051/2 mFirst case: δδδδ1= 0.05 m, δδδδ2 = 0.06 m, δδδδ3= 0.03 m,θθθθ1 = θθθθ2 = 0º, θθθθ3 = 180ºSecond case: δδδδ1= 0 m, δδδδ2 = 0.06 m, δδδδ3= 0.03 m,θθθθ1 = θθθθ2 = 0º, θθθθ3 = 180ºThird case: δδδδ1= 0 m, δδδδ2 = 0.0479 m, δδδδ3= 0.0479 m,θθθθ1 = θθθθ2 = 0º, θθθθ3 = 180º
0 800 1600 2400 32000
5
10
15
20
25
30─── MMM+++ FEM
0 800 1600 2400 32000
5
10
15
20
25
30
─── MMM+++ FEM
15
1. Introduction
2. Mathematicalapproach: M M M
3. Experimental measurement
4. Results anddiscussion
5. Conclusions
ICSV10 —
Stockholm
2003
5. Conclusions
Acoustic Behaviour of Elliptical Mufflers with Single-Inlet and Double-Outlet
� The mode-matching method has been applied to the acousticanalysis of elliptical mufflers with single-inlet and double-outlet.
� Comparisons with FEM and experimental measurements show good agreement.
� The effect of the chamber length, the eccentricity and thelocation of the ducts has been illustrated.
� The use of a second outlet pipe modifies the behaviour of short chambers, from transversal resonance to dome-like behaviour.
� The location of the outlet ducts on the nodal line of the modem = 2 and n = 0 improves the acoustic attenuation.
� For similar configurations, the attenuation is slightly reduced in comparison with the single-outlet chamber.
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