TENTH INTERNATIONAL CONGRESS ON SOUND AND VIBRATION · TENTH INTERNATIONAL CONGRESS ON SOUND AND...

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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




5. Conclusions

ICSV10 —

Stockholm

2003

1. Introduction


� 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




5. Conclusions

ICSV10 —

Stockholm

2003

2. Mathematical approach: MMM


� 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,, ,,,,��

∞

=

∞

=

−−+ +=

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1. Introduction




5. Conclusions

ICSV10 —

Stockholm

2003



� 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

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1. Introduction




5. Conclusions

ICSV10 —

Stockholm

2003



� 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

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1. Introduction




5. Conclusions

ICSV10 —

Stockholm

2003



� 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

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1. Introduction




5. Conclusions

ICSV10 —

Stockholm

2003



� 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

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1. Introduction




5. Conclusions

ICSV10 —

Stockholm

2003



� 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=

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1. Introduction




5. Conclusions

ICSV10 —

Stockholm

2003

4. Results and discussion


� 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

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1. Introduction




5. Conclusions

ICSV10 —

Stockholm

2003



� 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

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1. Introduction




5. Conclusions

ICSV10 —

Stockholm

2003



� 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

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1. Introduction




5. Conclusions

ICSV10 —

Stockholm

2003



� 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




5. Conclusions

ICSV10 —

Stockholm

2003

5. Conclusions


� 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.

TENTH INTERNATIONAL CONGRESS ON SOUND AND VIBRATION · TENTH INTERNATIONAL CONGRESS ON SOUND AND...

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