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D. W. Herrin, Ph.D., P.E. University of Kentucky
Department of Mechanical Engineering
Design of Mufflers and Silencers
Design of Mufflers and Silencers
Noise and Vibration Short Course
2 Dept. of Mech. Engineering University of Kentucky
1. Dissipative (absorptive) silencer:
Sound is attenuated due to absorption (conversion to
heat)
Sound absorbing material (e.g., duct liner)
Duct or pipe
Types of Mufflers
Design of Mufflers and Silencers
Noise and Vibration Short Course
3 Dept. of Mech. Engineering University of Kentucky
2. Reactive muffler:
Sound is attenuated by reflection and “cancellation” of sound waves
Compressor discharge details
40 mm
Types of Mufflers
Design of Mufflers and Silencers
Noise and Vibration Short Course
4 Dept. of Mech. Engineering University of Kentucky
3. Combination reactive and dissipative muffler:
Sound is attenuated by reflection and “cancellation” of sound waves + absorption of sound
Sound absorbing material
Perforated tubes
Types of Mufflers
Design of Mufflers and Silencers
Noise and Vibration Short Course
5 Dept. of Mech. Engineering University of Kentucky
Transmission loss (TL) of the muffler:
Wi
Wr
Wt Anechoic Termination Muffler
Performance Measures Transmission Loss
( )t
i
WWTL 10log10dB =
Design of Mufflers and Silencers
Noise and Vibration Short Course
6 Dept. of Mech. Engineering University of Kentucky
IL (dB) = SPL1 – SPL2
Insertion loss depends on : TL of muffler Lengths of pipes Termination (baffled vs. unbaffled) Source impedance
Muffler
SPL1
SPL2
Note: TL is a property of the muffler; IL is a “system” performance measure.
Performance Measures Insertion Loss
Design of Mufflers and Silencers
Noise and Vibration Short Course
7 Dept. of Mech. Engineering University of Kentucky
24” 12”
12” 2” 6” Source
-50
-40
-30
-20
-10
0
10
20
0 200 400 600 800 1000
Frequency (Hz)
TL a
nd IL
(dB
)
Insertion LossTransmission Loss
Pipe resonances
Inlet Pipe Outlet Pipe
Expansion Chamber Muffler
Example TL and IL
Design of Mufflers and Silencers
Noise and Vibration Short Course
8 Dept. of Mech. Engineering University of Kentucky
Source Su Any acoustic
system Su
P (sound pressure
reaction)
Zt
Input or load impedance
Termination impedance z = P
Su= r + jx zt =
PtSut
= rt + jxt
Acoustic System Components
Design of Mufflers and Silencers
Noise and Vibration Short Course
9 Dept. of Mech. Engineering University of Kentucky
Dissipative mufflers attenuate sound by converting sound energy to heat via viscosity and flow resistance – this process is called sound absorption.
Common sound absorbing mechanisms used in
dissipative mufflers are porous or fibrous materials or perforated tubes.
Reactive mufflers attenuate sound by reflecting a portion
of the incident sound waves back toward the source. This process is frequency selective and may result in unwanted resonances.
Impedance concepts may be used to interpret reactive muffler behavior.
Summary 1
Design of Mufflers and Silencers
Noise and Vibration Short Course
10 Dept. of Mech. Engineering University of Kentucky
Named for: Hermann von Helmholtz, 1821-1894, German physicist, physician, anatomist, and physiologist. Major work: Book, On the Sensations of Tone as a Physiological Basis for the Theory of Music, 1862.
von Helmholtz, 1848
The Helmholtz Resonator
Design of Mufflers and Silencers
Noise and Vibration Short Course
11 Dept. of Mech. Engineering University of Kentucky
F = PSB
x
V
SB
L L’ is the equivalent length of the neck (some air on either end also moves).
Damping due to viscosity in the neck are neglected
(resonance frequency of the Helmholtz resonator)
Helmholtz Resonator Model
Mx +Kx = PSB x = jωuB x = uBjω
j ωM −Kω
"
#$
%
&'uB = PSB
zB =PSBuB
= j 1SB2
"
#$
%
&' ωM −
Kω
"
#$
%
&'
VScK Bo22ρ
=
LSM Bo ʹ′= ρ
VLSc
MKz B
B ʹ′==→ ωwhen0
Design of Mufflers and Silencers
Noise and Vibration Short Course
12 Dept. of Mech. Engineering University of Kentucky
A 12-oz (355 ml) bottle has a 2 cm diameter neck that is 8 cm long. What is the resonance frequency?
Helmholtz Resonator Example
( )( )( )
Hz1821035508.0402.0
2343
2 6
2
=
×=
ʹ′=
−
n
Bn
fVLScf π
ππ
Design of Mufflers and Silencers
Noise and Vibration Short Course
13 Dept. of Mech. Engineering University of Kentucky
V = 0.001 m3
L = 25 mm SB = 2 x 10-4 m2
S = 8 x 10-4 m2
fn = 154 Hz
Anechoic termination
0
5
10
15
20
0 50 100 150 200 250 300
Frequency (Hz)
TL (d
B)
35 Hz
Helmholtz Resonator as a Side Branch
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
−ʹ′+=
2
21021log10dB
VcSLScTL
B ωω
Design of Mufflers and Silencers
Noise and Vibration Short Course
14 Dept. of Mech. Engineering University of Kentucky
Can we make ZB zero?
zA V
P
zB
z
z zA
zB
(any system)
(Produces a short circuit and P is theoretically zero.)
Network Interpretation
AB
AB
zzzzz+
=
zB =PSBuB
= j 1SB2
!
"#
$
%& ωM −
Kω
!
"#
$
%&
VLSc
MKz B
B ʹ′==→ ωwhen0
Design of Mufflers and Silencers
Noise and Vibration Short Course
A Tuned Dynamic Absorber
K1
M1 x F
K1
M1 x F
K2
M2
Original System
ω/ω1
|x/F|
Original system
Tuned dynamic absorber M2/M1=0.5
K2
M2
=K1M1
tune
Tuned Dynamic Absorber
15 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
Resonances in an Open Pipe
P = 1 Pa
Lp= 1 m source
First mode
Second Mode
etc.
λ1 = 2Lp =cf1→ f1 =
3432 1( )
=171.5 Hz
λ2 = Lp =cf2→ f2 =
3431 1( )
= 343 Hz
16 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
SPL at Pipe Opening – No Resonator
17 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
Example – HR Used as a Side Branch*
V = 750 cm3
L = 2.5 cm (L’= 6.75 cm) DB = 5 cm (SB= 19.6 cm2) D = 10 cm (S = 78.5 cm2)
fn = 340 Hz
Anechoic termination
_____ * e.g., engine intake systems
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
−ʹ′+=
2
21021log10dB
VcSLScTL
B ωω
18 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
SPL at Pipe Opening – with Resonator
19 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
20 Dept. of Mech. Engineering University of Kentucky
The Quarter-Wave Resonator has an effect similar to the Helmholtz Resonator:
zB
L
S
SB
The Quarter Wave Resonator
( ) ( )( ) ⎟
⎟⎠
⎞⎜⎜⎝
⎛ += 2
22
10 44tanlog10B
B
SSSSklTL
zB = −jρocSB
cot ωL c( ) = 0 when ωL c = nπ 2 n =1,3, 5...
ωn =nπc2L
fn =nc4L
or L = nc4 f
= n λ4"
#$
%
&'
Design of Mufflers and Silencers
Noise and Vibration Short Course
21 Dept. of Mech. Engineering University of Kentucky
The side-branch resonator is analogous to the tuned dynamic absorber.
Resonators used as side branches attenuate sound
in the main duct or pipe.
The transmission loss is confined over a relatively narrow band of frequencies centered at the natural frequency of the resonator.
Summary 2
Design of Mufflers and Silencers
Noise and Vibration Short Course
22 Dept. of Mech. Engineering University of Kentucky
18”
2” 2” 6”
where m is the expansion ratio (chamber area/pipe area) = 9 in this example and L is the length of the chamber.
The Simple Expansion Chamber
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛ ++= klm
mklTL 22
210 sin1cos441log10
0
5
10
15
20
25
30
0 100 200 300 400 500 600 700 800
Frequency (Hz)
TL (d
B)
Design of Mufflers and Silencers
Noise and Vibration Short Course
23 Dept. of Mech. Engineering University of Kentucky
2”
9” 18”
2” 2” 6”
Quarter Wave Tube + Expansion Chamber
0
5
10
15
20
25
30
0 100 200 300 400 500 600 700 800
Frequency (Hz)
TL (d
B)
Design of Mufflers and Silencers
Noise and Vibration Short Course
24 Dept. of Mech. Engineering University of Kentucky
18”
2” 2” 6” 9”
(same for extended outlet)
Extended Inlet Muffler
0
5
10
15
20
25
30
0 100 200 300 400 500 600 700 800
Frequency (Hz)
TL (d
B)
Design of Mufflers and Silencers
Noise and Vibration Short Course
25 Dept. of Mech. Engineering University of Kentucky
9” 9”
4” 6”
Two-Chamber Muffler
0
10
20
30
40
50
0 100 200 300 400 500 600 700 800
Frequency (Hz)
TL (d
B)
Design of Mufflers and Silencers
Noise and Vibration Short Course
26 Dept. of Mech. Engineering University of Kentucky
Source
Engine Pump Compressor (intake or exhaust)
Area change
Expansion chamber
Helmholtz Resonator
Quarter-wave resonator
termination
We would like to predict the sound pressure level at the termination.
Complex System Modeling
Design of Mufflers and Silencers
Noise and Vibration Short Course
The sound pressure p and the particle velocity v are the acoustic state variables
any acoustic component
1
2
p1, u1
p2, u2
For any passive, linear component:
Transfer, transmission, or four-pole matrix (A, B, C, and D depend on the component)
The Basic Idea
p1 = Ap2 +BS2u2S1u1 =Cp2 +DS2u2
p1S1u1
!"#
$#
%&#
'#= A B
C D
(
)*
+
,-
p2S2u2
!"#
$#
%&#
'#
or
27 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
p1, u1 p2 ,u2
S
L
A B
(x = 0) (x = L)
Solve for A, B in terms of p1, u1 then put into equations for p2, u2.
(note that the determinant A1D1-B1C1 = 1)
must have plane waves
The Straight Tube
p x( ) = Ae− jkx +Be+ jkx u x( ) = −1jkρoc
dpdx
p 0( ) = p1 = A+B
u 0( ) = u1 =A−Bρoc
p L( ) = p2 = Ae− jkL +Be+ jkL
u L( ) = u2 =Ae− jkL −Be+ jkL
ρocp1 = p2 cos kL( )+u2 jρoc( )sin kL( )u1 = p2 j ρoc( )sin kL( )+u2 cos kL( )
p1S1u1
"#$
%$
&'$
($=
cos kL( ) jρocS2
sin kL( )
jS1ρoc
sin kL( ) S1S2cos kL( )
)
*
+++++
,
-
.
.
.
.
.
p2S2u2
"#$
%$
&'$
($
28 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
29 Dept. of Mech. Engineering University of Kentucky
Combining Component Transfer Matrices
[ ]22×
⎥⎦
⎤⎢⎣
⎡=
ii
iii DC
BAT Transfer matrix of ith component
p1S1u1
!"#
$#
%&#
'#= T1[ ] T2[ ] T3[ ] Tn[ ]
p2S2u2
!"#
$#
%&#
'#= Tsystem() *+
p2S2u2
!"#
$#
%&#
'#
[ ]22systemsystem
systemsystemsystem
×
⎥⎦
⎤⎢⎣
⎡=
DCBA
T
Design of Mufflers and Silencers
Noise and Vibration Short Course
L
k’,zc
(complex wave number and complex characteristic impedance)
Straight Tube with Absorptive Material
p1S1u1
!"#
$#
%&#
'#=
cos k 'L( ) jzcS2sin k 'L( )
jS1zcsin k 'L( ) S1
S2cos k 'L( )
(
)
*****
+
,
-----
p2S2u2
!"#
$#
%&#
'#
30 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
31 Dept. of Mech. Engineering University of Kentucky
S1 S2
1 2
Area Change
p1 = p2S1u1 = S2u2
p1S1u1
!"#
$#
%&#
'#= 1 0
0 1
(
)*
+
,-
p2S2u2
!"#
$#
%&#
'#
Design of Mufflers and Silencers
Noise and Vibration Short Course
32 Dept. of Mech. Engineering University of Kentucky
L
S S S’ straight
tube
area changes
Expansion Chamber Muffler
T[ ] = 1 00 1
!
"#
$
%&
cos kL( ) jρocS '
sin kL( )
jS 'ρoc
sin kL( ) cos kL( )
!
"
#####
$
%
&&&&&
1 00 1
!
"#
$
%&
T[ ] =cos kL( ) jρoc
S 'sin kL( )
jS 'ρoc
sin kL( ) cos kL( )
!
"
#####
$
%
&&&&&
Design of Mufflers and Silencers
Noise and Vibration Short Course
33 Dept. of Mech. Engineering University of Kentucky
18”
2” 2” 6”
Expansion Chamber Muffler
S 'S= 9
Design of Mufflers and Silencers
Noise and Vibration Short Course
34 Dept. of Mech. Engineering University of Kentucky
SB
S
1 2
Transfer Matrix of a Side Branch
p1Su1
!"#
$#
%&#
'#=
1 01 zB 1
(
)**
+
,--
p2Su2
!"#
$#
%&#
'#
p1 = p2 = pBSu1 = SBuB + Su2zB = pB SBuB = p2 SBuBSu1 = p2 zB( )+ Su2
Design of Mufflers and Silencers
Noise and Vibration Short Course
35 Dept. of Mech. Engineering University of Kentucky
F = PSB
x
V
SB
L L’ is the equivalent length of the neck (some air on either end also moves).
Damping due to viscosity in the neck are neglected
(resonance frequency of the Helmholtz resonator)
Helmholtz Resonator Model
VScK Bo22ρ
=
LSM Bo ʹ′= ρ
VLSc
MKz B
B ʹ′==→ ωwhen0
Mx +Kx = PSB x = jωuB x = uBjω
j ωM −Kω
"
#$
%
&'uB = PSB
zB =PSBuB
= j 1SB2
"
#$
%
&' ωM −
Kω
"
#$
%
&'
Design of Mufflers and Silencers
Noise and Vibration Short Course
36 Dept. of Mech. Engineering University of Kentucky
Transmission loss (TL) of the muffler:
Wi
Wr
Wt Anechoic Termination
1 2
⎥⎦
⎤⎢⎣
⎡
DCBA
( )t
i
WWTL 10log10dB =
TL =10 log10Sin4Sout
A+ SoutBρc
+ρcCSin
+SoutSin
D2!
"#
$#
%&#
'#
Performance Measures Transmission Loss
Design of Mufflers and Silencers
Noise and Vibration Short Course
37 Dept. of Mech. Engineering University of Kentucky
IL = 20 log10A ZS +B ZTZS +C +D ZT
A0 ZS +B0 ZTZS +C0 +D0 ZT
!"#
$#
%&#
'#
Performance Measures Insertion Loss
Muffler
SPL1
SPL2
ZS ZT
T0[ ] =A0 B0C0 D0
!
"##
$
%&&
ZS ZT
T[ ] = A BC D
!
"#
$
%&
Design of Mufflers and Silencers
Noise and Vibration Short Course
38 Dept. of Mech. Engineering University of Kentucky
Sound Wave Reflections in Engines
Muffler
Engine
Waves leaving engine
Reflected from muffler
Reflected from engine
Waves leaving muffler
Reflected from open end
Reflected from muffler
Resonances can form in the exhaust and tail pipes as well as within the muffler.
Design of Mufflers and Silencers
Noise and Vibration Short Course
Source Impedance
Source
ps pL
Load
zs
zL
pszs + zL
=pLzL
Acoustic Source
Waves Leaving Source
Reflected from Attenuating Element
Attenuating Element
(i.e. Load)
Reflected from Source
uL
39 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
Transfer Impedance
1p 2p
21 uu =
ztr
ztr =p1 − p2Su
Incident Wave
Reflected Wave
Transmitted Wave
1p 2pu
40 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
Source/Load Concept
L1
Source zs , ps
ps
IL = f TL, zs, zt( )pt = f TL, zs, zt, ps( )pL
L2
zt , pt
Load zL , pL
Muffler
zs
zL
41 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
Insertion Loss Prediction
-30
-20
-10
0
10
20
30
40
50
60
0 200 400 600 800 1000Frequency (Hz)
IL (d
B)
Actual source impedancePressure source (Zs=0)Velocity source (Zs=infinite)Anechoic source (Zs=rho*c)
42 Dept. of Mech. Engineering University of Kentucky
Design of Mufflers and Silencers
Noise and Vibration Short Course
43 Dept. of Mech. Engineering University of Kentucky
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 the
system behavior from component “transfer matrices.” Applicability is limited to cascaded (series) components and
simple branch components (not applicable to successive branching and parallel systems).
Summary 3