07 Sound Absorbing Materials · 2021. 2. 4. · DUnder hood applications for engine noise DVehicle...

D. W. Herrin, Ph.D., P.E. University of Kentucky

Department of Mechanical Engineering

Sound Absorbing Materials

Sound Absorption by Porous Materials

Noise and Vibration Short Course

2 Dept. of Mech. Engineering University of Kentucky

Overview

  The Basics   Impedance and Absorption

  Transfer Matrix Approach   Flow Resistivity




Relatively massive, stiff barrier having high damping

Most sound is reflected

Some sound is transmitted

Less sound is reflected

Less sound is transmitted

( These two objectives can be combined)

Sound Blocking Versus Sound Absorption




Bedroom Noisy family or playroom

Ways to reduce noise level in the bedroom:

Which use sound blocking and which use sound absorption?

Example




Sound Absorbing Materials




NVH Applications Driven by Auto Industry




Sound Absorbing Materials in Car




a: fully reticulated plastic foam (x14)

b: partially reticulated

plastic foam (x14) c: glass fiber (x14) d: mineral (rock) wool (x14)

Examples of Sound Absorbing Materials




  Foams are made of various materials including polyurethane, polyethylene, and polypropylene.   Foams are created by pouring premixed liquid products onto a conveyor and allowing the chemical process to create cells (voids) of various size as the foam cures and hardens. Similar to bread rising.   If the cell walls are fractured, the foam is called “open cell”   If the cell walls remain intact, the foam is called “closed cell”   Coverings such vinyl, aluminum, urethane, or aluminized mylar protect the surface, improve appearance, and reduce absorption of liquids, dirt, etc.   Coverings may also act as a barrier, e.g., loaded vinyl

Foam Manufacture and Foam Types




 Suspended baffles in gymnasiums or factories  Under hood applications for engine noise  Vehicle interiors   Inside building walls – improves transmission loss   Inside office and computer equipment – reduces reverberant buildup of sound  Ceiling tiles and carpeting  HVAC applications – duct liner

Applications of Sound Absorbing Materials




Sound is “absorbed” by converting sound energy to heat within the material, resulting in a reduction of the sound pressure. Two primary mechanisms:

  vibration of the material skeleton - damping   friction of the fluid on the skeleton - viscosity

Mechanisms of Sound Absorption




Vibration of the material matrix is caused by sound pressure and velocity fluctuations within the material. Damping of the material converts sound to heat. Important for light materials at low frequencies. Difficult to model and measure (ignored here)

Vibration of Material Skeleton




The oscillating fluid particles within the material rub against the matrix and create heat by friction (viscosity). Primary material parameters affecting absorption:   porosity (fraction of air volume in the material)   structure factor (orientation of fibers, tortuosity)   flow resistance

Friction of the Fluid on the Skeleton




Overview






  The wave component within the material parallel to the surface is attenuated rapidly

  The material is similar to a set of rigid-wall, parallel capillaries

  The particle velocity un in the material is normal to the surface and only a function of the local sound pressure ps at the surface (un is independent of the form of the incident wave)

The Local Reaction Model

un

φ

φ ps

pi

pr

n

s

upz = (independent of pi)




Specific Boundary Impedance

resistance reactance

(named in honor of Lord Rayleigh)

Absorption coefficient:

jxrupzn

+==surface

raylsmkgmsPa 12

1 ≡=−−

−Units:

( )IncidentEnergySoundAbsorbedEnergySound

=φα




un jkxAe−

jkxBe+

R is called the pressure reflection coefficient

Specific Boundary Impedance

p, un

x = 0 ( )( )ABABc

cBABA

upz o

o

xnn −

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

+==

=11

0

ρ

ρ

RR

cz

o

n

−

+=11

ρ

ABR =




While the specific boundary impedance is independent of angle of incidence, the absorption coefficient is not:

An angle of maximum absorption exists:

The maximum absorption here is:

Absorption Coefficient and Impedance

( )( ) ( ) c

xxcrr

xrr

o

nn

o

nn

nn

n

ρρφφφ

φα =ʹ′=ʹ′ʹ′+ʹ′+

ʹ′= where

coscos1cos4

22

( ) ( ) ⎟⎠⎞⎜

⎝⎛ ʹ′+ʹ′= − 221max cos nn xrφ

( )nnn

rzrʹ′+ʹ′

ʹ′=

2maxα

( ) 10 →°α 1→ʹ′nr 0→ʹ′nxif and




Example Impedance of Foam

-6

-4

-2

0

2

4

6

8

0 1000 2000 3000 4000

Frequency (Hz)

Dim

ensi

onle

ss B

ound

ary

Impe

danc

e

Resistance r'Reactance x'




Example Sound Absorption Coefficient

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000Frequency (Hz)

Abso

rptio

n C

oeffi

cien

t

phi = 0 degreesphi = 30 degreesphi = 60 degrees




0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000Frequency (Hz)

Abs

orpt

ion

Coe

ffici

ent

Glass Fiber

Common Glass Fiber




Closed Cell vs. Open Cell Foam

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000Frequency (Hz)

Abs

orpt

ion

Coe

ffici

ent

Closed Cell FoamOpen Cell Foam




Closed-cell Open-cell

Foam Absorbers must be Open Cell




0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000

Frequency (Hz)

Abs

orpt

ion

Coe

ffici

ent

2 in. thick

1 in. thick

glass fiber

Effect of Thickness




cover

Effect of Covering an Absorber

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

Frequency (Hz)

Abs

orpt

ion

Coe

ffici

ent

Foam without Cover

Foam with Cover




0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

Frequency (Hz)

Abso

rptio

n C

oeffi

cien

t

2” 1.4”

2”

air gap

Layering of Materials




vibrating surface

Lining of Partial Enclosures

40

50

60

70

80

90

100

0 1000 2000 3000 4000 5000

Frequency (Hz)

Soun

d Po

wer

Lev

el (d

B)

No Absorption (108.2 dBA)

With Absorption (97.6 dBA)




Placement of material is often flexible

Plexiglass enclosure

loudspeaker

Full Enclosure

50

55

60

65

70

75

80

85

90

95

100

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Frequency (Hz)

SP

L (d

B)

Top and BottomFront and RearSidesEmpty




Overview





Plane waves in a porous material:   amplitude decreases with distance   p and u are not in phase   characteristic impedance z’c is a complex number   wave number k’ is a complex number

x

xkjoePxp

ʹ′−=)(

czxuxp

ʹ′=)()(

complex

γβ jk −='

Attenuation constant (responsible for wave attenuation)

Sound Propagation – Porous Material




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




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

"#$

%$

&'$

($




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

!"#

$#

%&#

'#




Transfer Matrix Approach

piSui

!

"##

$

%&&=

Ai BiCi Di

!

"##

$

%&&

pi+1Sui+1

!

"##

$

%&&

[ ] [ ][ ][ ] [ ] ⎥⎦

⎤⎢⎣

⎡==

TT

TTntotal DC

BAT...TTTT 321

Layer 1 Layer 2 Perforate Layer n Z = p1u1=SATCT

For each layer:

air air

Plane Wave Assumption

Overall:





0

0.2

0.4

0.6

0.8

1

0 500 1000 1500

Frequency (Hz)

Abs

orpt

ion

Coe

ffici

ent

Predicted

Measured

Example - Layered Materials

51 36 51

Units: mm




Overview






Bulk k’ and z’c

Surface z and α

Layered z and α

Designing the Absorber from Scratch

????




Absorption Coefficient vs. Frequency

(Mechel, 1988)

Different densities and thicknesses

Frequency (Hz)

αo




Flow Resistivity and Absorption

αo

ρof/σ




Flow Resistance and Flow Resistivity σ

Vacuum source

u (velocity)

ΔP Sample (thickness t )

Flow resistance:

uPrs

Δ=

Flow resistivity:

trs=σ




Vacuum pump Flow Meter

Valve Specimen

Pipe Manometer

Specimen Holder

Specimen

Valve Flow Meters

Manometers Pump

ASTM C522-03

Flow Resistance Measurement




φ

φ

pi

pr

Absorption Coefficient

α φ( ) =sound energy absorbedsound energy incident




Flow Resistivity

Abs

orpt

ion

Coe

ffici

ent

solid fluid

Absorption Coefficient and Flow Resistivity

σ tρc

≈ 2




Three Empirical Models

732.0754.0

595.0700.0

087.0)0571.01(189.0)0978.01(

−−

−−

−+=ʹ′

−+=ʹ′

XjXzzXjXkk

oc

Delaney and Bazly, 1970 – fibrous materials; f = 250-4000 Hz; σ = ?; 0.012 < X < 1.21

607.0548.0

592.0554.0

105.0)209.01(163.0)188.01(

−−

−−

−+=ʹ′

−+=ʹ′

XjXzzXjXkk

oc

Wu, 1988 – 17 plastic foam materials;

f = 200-2000 Hz; 2900 ≤ σ ≤ 24300 rayls; 0.01 < X < 0.83

655.0725.0

663.0716.0

556.0699.0

502.0641.0

127.0)0563.01(179.0)103.01(

:025.0for 191.0)081.01(

322.0)136.01(:025.0for

−−

−−

−−

−−

−+=ʹ′

−+=ʹ′

>

−+=ʹ′

−+=ʹ′

<

XjXzzXjXkk

XXjXzzXjXkk

X

oc

oc Mechel (after Fahy) – fibrous materials; f = ?; σ = ?; 0.002 < X < 0.5

σρ fX o=

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07 Sound Absorbing Materials · 2021. 2. 4. · DUnder hood applications for engine noise DVehicle...

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