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TRANSCRIPT
1
Slides to accompany lectures in
Vibro-Acoustic Design in Mechanical Systems © 2012 by D. W. Herrin
Department of Mechanical Engineering University of Kentucky
Lexington, KY 40506-0503 Tel: 859-218-0609
Chapter 10 – Sound in Ducts
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design 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 =
ME 510 Vibro-Acoustic Design 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
2
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design 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
3
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design
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 ME 510 Vibro-Acoustic Design
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
ME 510 Vibro-Acoustic Design
SPL at Pipe Opening – No Resonator
17 Dept. of Mech. Engineering University of Kentucky ME 510 Vibro-Acoustic Design
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
4
ME 510 Vibro-Acoustic Design
SPL at Pipe Opening – with Resonator
19 Dept. of Mech. Engineering University of Kentucky ME 510 Vibro-Acoustic Design 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"
#$
%
&'
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design 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)
ME 510 Vibro-Acoustic Design 23 Dept. of Mech. Engineering University of Kentucky
2”
9” 18”
2” 2” 6”
Quarter Wave Tube + Helmholtz Resonator
0
5
10
15
20
25
30
0 100 200 300 400 500 600 700 800
Frequency (Hz)
TL (d
B)
ME 510 Vibro-Acoustic Design 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)
5
ME 510 Vibro-Acoustic Design 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)
ME 510 Vibro-Acoustic Design 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
ME 510 Vibro-Acoustic Design
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
!"#
$#
%&#
'#
or
27 Dept. of Mech. Engineering University of Kentucky ME 510 Vibro-Acoustic Design
p1, u1 p2 ,u2
S
L
A B
(x = 0) (x = L)
Solve for A, B in terms of p1, u1 then put into equations for p2, u2.
(note that the determinant A1D1-B1C1 = 1)
must have plane waves
The Straight Tube
p x( ) = Ae− jkx +Be+ jkx u x( ) = −1jkρoc
dpdx
p 0( ) = p1 = A+B
u 0( ) = u1 =A−Bρoc
p L( ) = p2 = Ae− jkL +Be+ jkL
u L( ) = u2 =Ae− jkL −Be+ jkL
ρocp1 = p2 cos kL( )+u2 jρoc( )sin kL( )u1 = p2 j ρoc( )sin kL( )+u2 cos kL( )
p1S1u1
"#$
%$
&'$
($=
cos kL( ) jρocS2
sin kL( )
jS1ρoc
sin kL( ) S1S2cos kL( )
)
*
+++++
,
-
.
.
.
.
.
p2S2u2
"#$
%$
&'$
($
28 Dept. of Mech. Engineering University of Kentucky
ME 510 Vibro-Acoustic Design 29 Dept. of Mech. Engineering University of Kentucky
Combining Component Transfer Matrices
[ ]22×
⎥⎦
⎤⎢⎣
⎡=
ii
iii DC
BAT Transfer matrix of ith component
[ ] [ ] [ ][ ][ ] [ ]⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧
1
1system
1
1123 v
pT
vp
TTTTTvp
inn
n
[ ]22systemsystem
systemsystemsystem
×
⎥⎦
⎤⎢⎣
⎡=
DCBA
T
ME 510 Vibro-Acoustic Design
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
6
ME 510 Vibro-Acoustic Design 31 Dept. of Mech. Engineering University of Kentucky
S1 S2
1 2
Area Change
p1 = p2S1u1 = S2u2
p1S1u1
!"#
$#
%&#
'#= 1 0
0 1
(
)*
+
,-
p2S2u2
!"#
$#
%&#
'#
ME 510 Vibro-Acoustic Design 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( )
!
"
#####
$
%
&&&&&
ME 510 Vibro-Acoustic Design 33 Dept. of Mech. Engineering University of Kentucky
18”
2” 2” 6”
Expansion Chamber Muffler
S 'S= 9
ME 510 Vibro-Acoustic Design 34 Dept. of Mech. Engineering University of Kentucky
SB
S
1 2
Transfer Matrix of a Side Branch
p1Su1
!"#
$#
%&#
'#=
1 01 zB 1
(
)**
+
,--
p2Su2
!"#
$#
%&#
'#
p1 = p2 = pBSu1 = SBuB + Su2zB = pB SBuB = p2 SBuBSu1 = p2 zB( )+ Su2
ME 510 Vibro-Acoustic Design 35 Dept. of Mech. Engineering University of Kentucky
F = PSB
x
V
SB
L L’ is the equivalent length of the neck (some air on either end also moves).
Damping due to viscosity in the neck are neglected
(resonance frequency of the Helmholtz resonator)
Helmholtz Resonator Model
VScK Bo22ρ
=
LSM Bo ʹ′= ρ
VLSc
MKz B
B ʹ′==→ ωwhen0
Mx +Kx = PSB x = jωuB x = uBjω
j ωM −Kω
"
#$
%
&'uB = PSB
zB =PSBuB
= j 1SB2
"
#$
%
&' ωM −
Kω
"
#$
%
&'
ME 510 Vibro-Acoustic Design 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
7
ME 510 Vibro-Acoustic Design 37 Dept. of Mech. Engineering University of Kentucky
Wi
Wr
Wt Anechoic Termination
1 2
⎥⎦
⎤⎢⎣
⎡
DCBA
Derivation Transmission Loss
p1S1u1
!"#
$#
%&#
'#= A B
C D
(
)*
+
,-
p2S2u2
!"#
$#
%&#
'#
Wi =p+a2
ρcS1
Wt =p+b2
ρcS2
TL =10 logWi
Wt
p1 = p+a + p−a
u1 =p+a − p−aρc
p2 = p+b
u2 =p+bρc
Express p1, p2, u1 and u2 in terms of incident reflected waves
ME 510 Vibro-Acoustic Design 38 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
!
"#
$
%&
ME 510 Vibro-Acoustic Design 39 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.
ME 510 Vibro-Acoustic Design
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
40 Dept. of Mech. Engineering University of Kentucky
ME 510 Vibro-Acoustic Design
Transfer Impedance
1p 2p
21 uu =
ztr
ztr =p1 − p2Su
Incident Wave
Reflected Wave
Transmitted Wave
1p 2pu
41 Dept. of Mech. Engineering University of Kentucky ME 510 Vibro-Acoustic Design
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
42 Dept. of Mech. Engineering University of Kentucky
8
ME 510 Vibro-Acoustic Design
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)
43 Dept. of Mech. Engineering University of Kentucky ME 510 Vibro-Acoustic Design
Source
ps pL
Load
zs
zL
uL
pszs + zL
=pLzL= SuL
ps = SuLzs + pL
zs =ps − pLSuL
Source Impedance Series Impedance
44 Dept. of Mech. Engineering University of Kentucky
ME 510 Vibro-Acoustic Design
Source
us pL
Load
zs zL
uL
SuLzL = SuszszLzs + zL
= pL
zs =pL
S us −uL( )
Source Impedance Parallel Impedance
45 Dept. of Mech. Engineering University of Kentucky ME 510 Vibro-Acoustic Design 46 Dept. of Mech. Engineering
University of Kentucky
Derivation Insertion Loss
Muffler SPL2
ZS ZT
p1S1u1
!"#
$#
%&#
'#= A B
C D
(
)*
+
,-
p2S2u2
!"#
$#
%&#
'#
zS =ps − p1S1u1
⇒ p1 = ps − S1u1zs
zT =p2S2u2
⇒ S2u2 =p2zT
1 2
p1 = Ap2 +BzTp2
p1 = ps − S1u1zS
= ps − zS Cp2 +Dp2zT
"
#$
%
&'
Ap2 +BzTp2 = ps − zS Cp2 +D
p2zT
"
#$
%
&'
p2ps=
1
A+ 1zTB+ zSC +
zSzTD
ME 510 Vibro-Acoustic Design 47 Dept. of Mech. Engineering University of Kentucky
Derivation Insertion Loss
p2ps=
1
A0 +1zTB0 + zSC0 +
zSzTD0
SPL1
ZS ZT
T0[ ] =A0 B0C0 D0
!
"##
$
%&&
Determined in same manner as prior slide
ME 510 Vibro-Acoustic Design 48 Dept. of Mech. Engineering University of Kentucky
Derivation Insertion Loss
IL = 20 logp2,nomufflerp2,muffler
p2,nomufflerps
=1
A0 +1zTB0 + zSC0 +
zSzTD0
p2,mufflerps
=1
A+ 1zTB+ zSC +
zSzTD
IL = 20 log10A ZS +B ZTZS +C +D ZT
A0 ZS +B0 ZTZS +C0 +D0 ZT
!"#
$#
%&#
'#
9
ME 510 Vibro-Acoustic Design 49 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