17_design of mufflers and silencers2
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Design of Mufflers and Silencers
Slides to accompany lectures in ME 599/699:
Vibro-Acoustic Design in Mechanical Systems 2003 by A. F. Seybert
Department of Mechanical Engineering
University of Kentucky
Lexington, KY 40506-0108
Tel: 859-257-6336 x 80645Fax: 859-257-3304
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Goals of the Lecture
To understand how the components and geometry of amuffler (silencer) determine its performance
To design a muffler for a specific attenuation andfrequency range
To learn and use muffler terminology
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Types of Mufflers
1. Dissipative (absorptive) silencer:
Sound is attenuated dueto absorption (conversion
to heat)
Sound absorbing material(e.g., duct liner)
Duct or pipe
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Types of Mufflers (cont.)
2. Reactive muffler:
Sound is attenuated by reflection andcancellation of sound waves
Compressor discharge details
40 mm
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Types of Mufflers (cont.)
3. Combination reactive and dissipative muffler:
Sound is attenuated by
reflection and cancellation ofsound waves + absorption ofsound
Sound absorbing material
Perforated tubes
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Performance Measures Transmission Loss
Transmission loss (TL) of the muffler:
Wi
Wr
Wt AnechoicTerminationMuffler
TL dB Log WW
i
t
( )=10 10
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Performance Measures Insertion Loss
IL (dB) = SPL1 SPL2
Insertion loss depends on : TL of muffler Lengths of pipes Termination (baffled vs. unbaffled) Source impedance
Muffler
SPL1
SPL2
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Example TL and IL
24
12
122 6Source
-50
-40
-30
-20
-10
0
10
20
0 200 400 600 800 1000
Frequency (Hz)
TL
and
IL
(dB)
Insertion Loss
Transmission Loss
Pipe resonances
Inlet Pipe Outlet Pipe
Expansion Chamber Muffler
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Zeffort variable
flow variable
W = effort variable x flow variable *
SYSTEM
Flow variable
Effort variable
+
_
The Concept of Impedance
Generalized impedance of a systemZ and power supplied W :
PAV AS = P2 AS /ZP/VVelocity (V)Pressure (P)Acoustic
T/Q
P/Q
F/V
E/I
Impedance Z
PAQ = P2/ZVolume Flow Rate (Q)Pressure (P)Fluid
TAQ = T2/ZHeat Flow Rate/Deg(Q)
Temperature(T)
Thermal
FAV = F2/ZVelocity (V)Force (F)Mechanical
EAI = E2/ZCurrent (I)Voltage (E)Electrical
Power supplied WFlow VariableEffort VariableSystem Type
* For acoustic systems we must multiply by the area S
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Network Interpretation: Electrical to Acoustical
ZV
P
source loadZI
E
source load
Electrical System Acoustical System
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Source VAny acoustic
system
V
P(sound
pressurereaction)
Zt
z
P
V r jx= = + zP
V r xtt
t
t t= = +Input or loadimpedance
Terminationimpedance
Acoustic System Components
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Summary 1
Dissipative mufflers attenuate sound by convertingsound energy to heat via viscosity and flow resistance this process is called sound absorption.
Common sound absorbing mechanisms used indissipative mufflers are porous or fibrous materials orperforated tubes.
Reactive mufflers attenuate sound by reflecting aportion of the incident sound waves back toward thesource. This process is frequency selective and mayresult in unwanted resonances.
Impedance concepts may be used to interpret reactivemuffler behavior.
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The Helmholtz Resonator
Named for:
Hermann von Helmholtz, 1821-1894, Germanphysicist, physician, anatomist, and physiologist.
Major work: Book, On the Sensations of Tone asa Physiological Basis for the Theory of Music,1862.
von Helmholtz, 1848
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Helmholtz developed a set ofresonators for studying the auditory
response of humans to tones.
Modern applications:
fundamental resonance of stringed instruments base-reflex (ported) loudspeakers
muffler components
History
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Lumped Parameter Model
Mx Kx PS x j v v
j
j M
K
v PS
z P
v j
S M
K K
M c
S
L V
B B
B
B B
B
B B
B
&& &&
'
+ = = =
=
= =
= =
x
when
1
0
Kc S
V
M S L
o B
o B
=
=
2 2
'
F = PSB
x
V
SB
L
L is the equivalentlength. Losses due toviscosity in the neck and
radiation are neglected.
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Example HR Used as a Side Branch
V = 0.001 m3
L = 25 mmSB= 2 x 10
-4 m2
S= 8 x 10-4 m2
fn = 154 Hz
Anechoic termination
TL dBc S
L S c VB( ) log
/
'/ /= +
10 12
10 2
2
0
5
10
15
20
0 50 100 150 200 250 300
Frequency (Hz)
TL
(dB)
35 Hz
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A Tuned Dynamic Absorber
K1
M1xF
K1
M1xF
K2
M2
" "Tune K
M
K
M2
2
1
1
=
Original System
Tuned Dynamic Absorber
T/T1
|x/F|
Original system
Tuned dynamic absorberM2/M1=0.5
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Resonances in an Open Pipe
P = 1 Pa
Lp= 1 msource
First mode
Second Mode
11
12343
2 1 1715= = = =L
c
f fp ( )
. Hz
22
2
343
1
343= = = =Lc
f
fp Hz
etc.
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SPL at Pipe Opening No Resonator
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Example HR Used as a Side Branch
V = 750 cm3
L = 2.5 cm (L= 6.75 cm)DB= 5 cm (SB= 19.6 cm
2)D= 10 cm (S = 78.5 cm2)
fn = 340 Hz
Anechoic termination
TL dBc S
L S c VB( ) log
/
'/ /= +
10 12
10 2
2
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SPL at Pipe Opening with Resonator
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Network Interpretation
z z z
z zB A
B A
=+
Can we make ZB zero?
zAV
P
zB
z
z zA
zB(any system)
z Pv
jS
M K KM
c SL VB B B
B= =
= =
1 0
when'
(Produces a short circuit and P is theoretically zero.)
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The Quarter-Wave (QW) Resonator
z j c L c L c n
n cL
f nc
L
nc
f n
B o
n
n
= = = =
=
= = =
cot( / ) / / , , ...0 2 13 5
2
4 4 4
when n
or L
The Quarter-Wave Resonator has an effect similar to the HelmholtzResonator:
( )( )
+=2
22
104
4tan10
b
b
SS
SSkLLogTL
zB
L
S
Sb
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Summary 2
The side-branch resonator is analogous to thetuned dynamic absorber.
Resonators used as side branches attenuate
sound in the main duct or pipe.
The transmission loss is confined over arelatively narrow band of frequencies centeredat the natural frequency of the resonator.
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The Simple Expansion Chamber
TL Log kL mm
kL= + +
10
1
4 4
110
2 2 2cos ( ) sin
18
2 26
where m is the expansion ratio (chamber area/pipearea) = 9 in this example and L is the length of the
chamber.
0
5
10
15
20
25
30
0 100 200 300 400 500 600 700 800
Frequency (Hz)
TL
(dB)
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QW Tube + Simple Expansion Chamber
0
5
10
15
20
25
30
0 100 200 300 400 500 600 700 800
Frequency (Hz)
TL
(dB)
2
918
2 26
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Extended Inlet Muffler
18
2 269
0
5
10
15
20
25
30
0 100 200 300 400 500 600 700 800
Frequency (Hz)
TL
(dB) ( same for
extendedoutlet )
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Two-Chamber Muffler
0
10
20
30
40
50
0 100 200 300 400 500 600 700 800
Frequency (Hz)
TL
(dB)
9 9
4 6
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Source
Engine
PumpCompressor(intake or exhaust)
Area change
Expansion chamber
Helmholtz Resonator
Quarter-wave resonator
termination
Complex System Modeling
We would like to predict the sound pressure level at the termination.
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The sound pressure p and the particle velocity v are the acoustic state variables
any acousticcomponent
2
1
p2, v2
p1, v1
For any passive, linear component:
p Ap Bv
v Cp Dv
or
p
v
A B
C D
p
v
2 1 1
2 1 1
2
2
1
1
= += +
=
Basic Idea
Transfer, transmission, or four-pole matrix
(A, B, C, and D depend on the component)
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Combining Component Transfer Matrices
[ ] [ ] [ ][ ] [ ]
[ ]
[ ]
TA B
C D
pv
T T T T pv
T pv
TA B
C D
i
i i
i i
n
nn i system
system
system system
system systemi x
=
=
=
=
2x2
th = transfer matrix of i component
3 2
1
1
1
1
2 2
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Performance Measures Transmission Loss
Transmission loss (TL) of the muffler:
TL dB Log W
W
i
t
( )=10 10
TL Log S
S A cC
B
c
S
S D
oo
=
+ + +
10 1
4102
1
1
2
2
Wi
Wr
Wt AnechoicTermination
2 1
A B
C D
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The Straight Tube
p2, v2 p1, v1
S
LA
B
(x = 0) (x = L)
p x Ae Be v xjk c
dp
dx
p p A B
v vA B
c
p L p Ae Be
v L vAe Be
c
p p kL v j c kL
v p j c kL v kL
p
v
kL j c kL
j c kL kL
jkx jkx
o
o
jkL jkL
jkL jkL
o
o
o
o
o
( ) ( )
( )
( )
( )
( )
cos ( )sin
( / )sin cos
cos sin
( / )sin cos
= + =
= = +
= =
= = +
= =
= +
= +
=
1
0
0
2
2
1
1
2 1 1
2 1 1
2
2
p
v
1
1
Solve for A, Bin terms of p1, v1then put into
equations for p2,v2.
(note that the determinant A1D1-B1C1 = 1)
must have plane waves
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Straight Tube with Absorptive Material
p
v
k L jz k L
j z k L k L
p
v
c
c
2
2
1
1
=
cos ' sin '
( / )sin ' cos '
L
k,zc
(complex wave number andcomplex characteristic
impedance)
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Area Change
S2
S1
2 1
p p
S v S v
2 1
2 2 1 1
==
p
v S S
p
v
2
2 1 2
1
1
1 0
0
=
/
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Expansion Chamber Muffler
L
S SSstraight
tube
area changes
[ ]'/
cos sin
( / )sin cos / '
[ ]cos ( / ) sin
( / )sin cos
TS S
kL j c kL
j c kL kL S S
TkL j c m kL
m j c kL kL
o
o
o
o
=
=
1 0
0
1 0
0
(m = S/S is called the expansion ratio of the muffler)
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Expansion Chamber Muffler - Example
TL Log S
S A cC
B
c
S
S D
TL Log kLm
kL
oo
=
+ + +
= + +
10 1
4
10 1
4 4
1
10
2
1
1
2
2
10
2 2 2
cos (m ) sin
Recall:
18
2 26
(m = 9)
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Transfer Matrix for a Side-Branch
SB
S
2 1
p p p
Sv S v Sv
z p v p vSv S z p Sv
B
B B
B B B B
B B
2 1
2 1
1
2 1 1
= =
= +
= == +/ /
( / )p
v S Sz
p
vB B
2
2
1
1
1 0
1
=
/ ( )
TL LogS
S A cC
B
c
S
S D Log
cS
Szo o
o B
B
=
+ + +
= +10
1
4 10 1
2102
1
1
2
2
10
2
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Helmholtz Resonator
z jS
M K
B
B
=
1
K c S
V
M S L
o B
o B
=
=
2 2
'
V
SB
L
L is the equivalentlength. Losses due toviscosity in the neck andradiation are neglected.
TL Log cSSz
o B
B
= + =10 1210
2
10 1 210 22
log/
'/ /+
c S
L S c VB
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Summary 3
The transfer matrix method is based on plane wave (1-D)acoustic behavior (at component junctions).
The transfer matrix method can be used to determine thesystem behavior from component transfer matrices.
Applicability is limited to cascaded (series) componentsand simple branch components (not applicable to successivebranching and parallel systems).