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Title NOVI - Advanced functional solutionsfor Noise and Vibration reduction ofmachinery, Deliverable D1.1 Physicalmechanisms of absorption and soundinsulation

Author(s) Uosukainen, SeppoCitation VTT (2014), 45 p.Date 2014Rights This report may be downloaded for

personal use only.

NOVI - Advanced functional solutions forNoise and Vibration reduction of machinery

D1.1 Physical mechanisms of absorption and soundinsulation, VTT

Author: Seppo UosukainenConfidentiality: Public

216.01.2012

SummaryProject name Project number/Short nameNOVI SP1 71902 – 1.1.2Author(s) PagesSeppo Uosukainen 45Keywords

Summary

• Absorption of acoustic wave is due to viscous and thermal losses andrelaxation processes in fluid

• Absorption is emphasized near boundaries, especially in small holes,slits and narrow tubes

• Viscous losses dominate with small free space dimensions• Perforated / microperforated plates• Absorbing materials

• “Holes” and “slits” described by fluid bulk properties (flowresistivity, porosity, tortuosity, viscous and thermalcharacteristic lengths)

• Sound transmission of a wall is due to two parallel transmission paths• Non-resonant and resonant transmission• Below coincidence frequency non-resonant transmission dominates

except in the resonance region• With double walls above double-wall resonance, sound insulations in dBs

are summed with reasonable amount of absorption in cavity if mechanicalcoupling is insignificant

Confidentiality Public

Espoo 16.1.2012

Written bySeppo UosukainenSenior Scientist

Reviewed byHannu NykänenPrincipal Scientist

Accepted byJohannes HyrynenDeputy Technology Manager

316.01.2012

Table of content

Physical mechanisms of sound absorptionBasic conceptsLosses in mediumLosses at boundariesLosses in narrow tubes and in small holes and slits

Physical mechanisms of sound insulationBasic conceptsSound insulation of a single wall in diffuse fieldSound insulation of a double wall in diffuse fieldSound insulation of sandwich structures

416.01.2012

Table of content

Physical mechanisms of sound absorptionBasic concepts

516.01.2012

Basic concepts

Physical properties of fluid are responsiblefor acoustic absorption

ViscosityThermal conductivity (Prandtl number)

In sound absorbing structuresFluid bulk properties affect theefficiency of absorptionElastic bulk properties increaseabsorption

Energy is transferred to vibrationenergy which is decreasedthrough internal lossesNot handled here

Va One 2010.5 Foam Module – User’s Guide, Theory & QA

616.01.2012

Fluid bulk properties

Flow resistivity r (specific flow resistance)Based on ratio of pressure difference p betweensurfaces of material sample, due to volume velocity Qthrough the sample

PorosityVolume of fluid phase Vfluid compared to total volume Vtotof material sample

TortuosityUnitless quantity relating average fluid path length throughmaterial sample normalized by sample thickness

Viscous and thermal characteristic lengths and ’Macroscopic shell dimensions related to viscous and thermallosses (average radius of smaller and larger pores)

tot

fluid

VVQhpSr

S : sample areah : sample thickness

716.01.2012

Absorption coefficient

Absorptive properties of a structure isdefined by absorption coefficient

Includes transmitted sound andvibration power in addition toabsorbed soundFunction of mounting conditionsFunction of angle of incidenceAbsorption coefficient is nothandled here, only physicalmechanisms involved in soundabsorption phenomenon

0

0

21

ZZZZR

RPP

s

s

inc

abs

Pabs = absorbed sound powerPinc = incoming sound powerR = reflection coefficient of structure front endZs = impedance of structure front end

(ratio of sound pressure and particle velocity)Z0 = impedance of air (= 0c0, density times speed of sound)

816.01.2012

Table of content

Physical mechanisms of sound absorption

Losses in medium

916.01.2012

Losses in medium (1)

PhenomenaViscous losses

Involved in strain of fluid elements caused by particlevelocity of sound wave

Thermal lossesInvolved in thermal conduction from sound pressuremaxima to sound pressure minima in sound wave

RelaxationVibration state of molecules is excited due to sound waveExcited state is relaxed after a while

1016.01.2012


Sound wave propagates in successivecondensations and dilatations (rarefactions)

Fluid is compressed in condensations andrarefied in dilatations

Proportional motion between condensations anddilatations causes viscous losses

Frictional lossesHeat transfer from higher temperatures incondensations to lower temperatures in dilatationscauses thermal lossesIn relaxation process part of molecular kinetic energyis changed to other molecular energy types causinglosses

Rossing, T. D. R. The Science of Sound

1116.01.2012


Loss power / unit volumeProportional to frequencysquared

Viscous loss power Pv

Maximum at particle velocitymaximum

Thermal loss power Pl

Maximum at sound pressuremaximum

Pv and Pl of the same orderImportant only at highfrequencies and in large spaces

234

2

0

uc

Pv

220

20

0

2

0

1 pcc

kc

PP

l

: angular frequency, c0 : speed of sound, : viscosity factor,u : particle velocity, k0 : thermal conductivity,

: adiabatic constant, 0 : density,cP : specific heat in constant pressure, p : sound pressure

Uosukainen, S. Acoustic Field Theory

1216.01.2012

Table of content


Losses at boundaries

1316.01.2012

Losses at boundaries (1)

Based on same mechanisms as aboveBoundaries emphasizes the phenomenaViscosity wave u2 (particle velocity)near boundaryThermal wave T’ near boundaryReal part of exponent associated to lossesWaves couple to acoustic waves u and T

Total waves disappear at boundary

2tot

j102 ee

uuuuu x

y v

0

2v

TTTTT ly

tot

j11 e

lP

kc

2 0

0

u0 : amplitude of viscous velocity, u : acoustic particle velocity,T1 : amplitude for thermal wave, T : acoustic temperature(temperature fluctuations associated to acoustic fields),

utot, Ttot : total fieldsy : distance from boundary, ex : unit tangential vector on the boundary,

v : boundary layer thickness for viscous wave,l : boundary layer thickness for thermal wave


y

x

ux

1416.01.2012

Losses at boundaries (2)

Boundary layer thicknesses v and l very smallPhenomena important only in close vicinity of boundaries

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y /

abs(

u/u

0),ab

s(T

/T1)

u2, T'

utot, Ttot

1516.01.2012

Table of content


Losses in narrow tubes and in small holes and slits

1616.01.2012

Losses in narrow tubes and in small holes and slits (1)

The viscosity and thermal wave effects are emphasized whenReflecting surfaces form a closed surface

TubesViscosity and thermal waves are attached to allsurfaces decaying exponentially towards the middle ofthe tube cross section and growing exponentiallytowards the opposing surface

The size of the space between surfaces is of the same orderas the boundary layer thicknesses

Narrow tubes, small holes and slitsPhenomenon affects everywhere betweensurfaces

Allard, J. F. Propagation of Sound in Porous Media

1716.01.2012


Viscous and thermal boundary layer thicknesses as functions offrequency


0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f [Hz]

[mm

]ViscousThermal

1816.01.2012


Equivalent complex density in circulartube or hole for viscous wave

Equivalent complex compressibility incircular tube or hole for thermal wave

Imaginary parts associated to losses

Also end corrections to real andimaginary parts to be taken into account

1

0

100

21RKJRKJ

RK v

v

v

202 2jj

vvK

RKJRKJ

RKQQ

l

l

l 0

100

121

20

02 2jjl

Pl k

cKJ0, J1 : Bessel functions of order 0 and 1

R : radius of cross sectionQ0 : compressibility of air

References:Maa, D.-Y. Noise Control Eng. J. 29(1987)3Maa, D.-Y. J. Acoust. Soc. Am. 104(1998)5Allard, J. F. Propagation of Sound in Porous MediaUosukainen, S. Acoustic Field Theory

1916.01.2012


At low frequencies (cross section small compared to wavelength):

Viscous losses grow without limit when R 0Thermal losses approach zero when R 0Equivalent density approaches Q0 when R 0

Viscous losses dominate and isothermal conditionsapproached instead of adiabatic

2

34

002jR

v2

00 21j

l

RQQ


2016.01.2012


R = 0.5 cmreal partimaginary part


10-1

100

101

102

103

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Q/Q

0

Proportional complex compressibility

f [Hz]10

-110

010

110

210

3-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

f [Hz]

/0

Proportional complex density

2116.01.2012


Applications of above:Perforated / microperforated platesAbsorbing materials

Most important loss mechanism”Holes” and ”slits” described by fluid bulk properties

Flow resistivity most importantPorosity, tortuosity, viscous and thermalcharacteristic lengths

Irreversible effects in the skeleton affect also

Viscosity of air is the most important phenomenon causingacoustic losses in (micro)perforated plates and absorbing materials

2216.01.2012


Slits:Same conclusions thanwith tubes and holes canbe deduced

1

00 2/2/tanh1

dKdK

v

v

2/2/tanh1100 dK

dKQQl

l

202 2j

vvK

20

02 2jjl

Pl k

cKAllard, J. F. Propagation of Sound in Porous Media

d : slit width

2316.01.2012

Table of content

Physical mechanisms of sound insulationBasic concepts

2416.01.2012

Basic concepts (1)

Coincidence frequency fc(critical frequency) of a plate

The frequency at which thespeed of bending wave(~ f) in plate is equal tosound speed in airPlate vibration and soundfield are well coupled atfrequencies abovecoincidence frequency andespecially at fc

Also modal coincidencefrequencies for each mode

Ehc

Dmcf s

c)1(12

22

220

20

Plate parameters:m = mass per unit area

s = densityD = bending stiffnessE = Young’s modulush = thickness = Poisson’s ratio

100 101 1020

5

10

15

20

25

30

35

Coin

cide

nce

frequ

ency

[kH

z]Thickness [mm]

Gypsum, brickstone, chipboardConcreteSteel, aluminium, glass

2516.01.2012

Basic concepts (2)

Radiation ratio of vibrationMeasure of proportional radiation efficiencyDepends on vibration types

Proportions of resonant and non-resonant vibration

Depends on excitation(mechanical / acoustical)

200 vScP

Plate parameters:P = radiated sound powerS = areav = vibration velocity< > = mean value

2616.01.2012

Basic concepts (3)

Radiation ratio of non-resonant vibration of a single modeTypically constant and > 1 at frequencies above modalcoincidence frequency

Modal averaged radiation ratio of resonant vibration:With edge and corner modes, radiation from inefficientsurface parts cancel each other

10 lg

fc

0

f

Surface modes 1

Edge modes 2

Corner modes 3

Efficient radiation from:1 whole surface2 plate egdes3 plate corners

2716.01.2012

Basic concepts (4)

With resonant vibration below fc, acoustic wavelength > structuralwavelength s

Near-by vibration zones with opposite signs cancel soundradiation of each other acoustical short circuitOnly edges or corners remain unaffected

Regions of uncancelled sound radiation below fcFahy, F. Sound and Structural Vibration

2816.01.2012

Basic concepts (5)

Below coincidence frequency, airborne noise is transmitted via aplate mainly by non-resonant vibration except in the resonanceregion (presented later)

Low vibration modes, especially first (volume velocity mode),most important especially well below coincidence frequency

Above coincidence frequency, both vibration types affect

Definition of sound insulation (sound transmission loss) R [dB]

t

i

PPR 10log10

Pi = sound power incident to insulating structurePt = sound power transmitted through structure

2916.01.2012

Table of content

Physical mechanisms of sound insulation

Sound insulation of a single wall in diffuse field

3016.01.2012

Sound insulation of a single wall in diffuse field (1)

Two basic phenomena in sound transmission working as paralleltransmission paths:

Non-resonant transmissionTransmission coefficient n

Resonant transmissionTransmission coefficient r

Total transmission coefficientSound insulations Rn, Rr, R

In practice, lowest of Rn and Rrdetermines total sound insulation R

Wave impedance law for continuums rejectssound insulation at high frequencies

/1log10/1log10/1log10

10

10

10

RRR

rr

nn

rn

3116.01.2012

Sound insulation of a single wall in diffuse field (2)

Non-resonant transmissionLowest modes above their eigenfrequenciesExtra stiffness effects below lowesteigenfrequency and abovecoincidence frequency

Resonant transmission

Auxiliary variable 0= non-resonant transmission coefficient

with normal incidence (”mass law”)

2

00

0

21

1

cm

0n

0

2

0 2 ffc

r

Wall parameters:m = mass per unit areafc = coincidence frequency

= radiation ratio of resonant vibration= loss factor

References:Beranek, L. L. Noise and Vibration ControlUosukainen, S. & Pesonen, K. Ääntäeristävienkoteloiden äänitekninen mitoitus ja valinta

3216.01.2012

Non-resonant transmission

Rn

f

- 6 dB/oct.

+ 6 dB/oct.

+ 18 dB/oct. + 6 dB/oct.

stiffnessof lowest

mode

mass

stiffness

f1 = lowest eigenfrequencyfc = coincidence frequency

f1 fccff1

coincidence frequencyof lowest mode

3316.01.2012

Resonant transmission

Rr

f

+ 9 dB/oct.

+ 9 dB/oct.

f1 fc

coincidence

cff1

3416.01.2012

Non-resonant and resonant transmission

Rn,r

ff1 fccff1 fc/2

wave impedance law


= loss factor

Rn

Rr

3516.01.2012

Total transmission

R

ff1 fccff1 fc/2

mass region

coincidenceregion

wave impedanceregion

resonanceregion

stiffnessregion


= loss factor

- 6 dB/oct.

+ 6 dB/oct.

+ 6…9 dB/oct

3616.01.2012

Effects of material parameters on sound insulation

R

ff1 fccff1 fc/2

Zonesimproved byincrease in:

massstiffnesslosses

Increase in:massstiffnesslosses


= loss factor

Loss increase:Lossy edge mountingViscoelastic layers

Tanttari, J. & Saarinen, K. Työkoneiden melun vähentäminen - perusteet

Sound insulation poorwith light and stiff plates

Stiffness increase:Pretensioned or curved platesStiffeners

Mass increase:Barrier mats

3716.01.2012

Table of content


Sound insulation of a double wall in diffuse field

3816.01.2012

Sound insulation of a double wall in diffuse field (1)

Double-wall resonance frequency f0

Obtained from Eq. (10) in D2.2, mdisplaced by equivalent mass me

Mass-air-mass resonanceThree frequency regions with differentbehavior:

f > f0

f f0

f < f0

21

000

/1/11

2

mmm

hmcf

e

e

h = depth of air cavitysubscripts 1 and 2 = indices of walls 1 and 2

10 12 14 16 18 20 22 24 26 28 30100

150

200

250

300

350

Air cavity depth [mm]

f 0[H

z]

Two similar steel plates with thicknesses 1 - 3 mm

1.0 mm1.5 mm2.0 mm2.5 mm3.0 mm

3916.01.2012


At frequencies above double-wall resonanceWith reasonable amount of absorption in cavity, R1 and R2 indBs are summed in total sound insulation R12(max. R1 + R2 + 6)

)1(

/1

41log10

21102112

AR

SRRRR

v

v

Rv = room constant of air cavityS = wall area (one side)

A = absorption area of air cavity= mean absorption coefficient inside air cavity

subscripts 1 and 2 = indices of walls 1 and 2

References (adapted):Beranek, L. L. AcousticsBeranek, L. L. Noise and Vibration ControlUosukainen, S. & Pesonen, K. Ääntäeristävienkoteloiden äänitekninen mitoitus ja valinta

effect of direct field in cavity

Rv = room constant of air caS = wall area (one side)

A = absorption area of air ca= mean absorption coefficient inside air ca

subscripts 1 and 2 = indices of walls 1 and 2

effect of reverberant field in cavity

4016.01.2012


At double-wall resonance frequencySound insulation is close to 0 dB without absorbents in cavityand with a small amount of mechanical damping

Increasing internal losses to double increases soundinsulation 6 dBUsing walls of different mass per unit area increasessound insulation

This degrades sound insulation at higher frequenciesAt frequencies below double-wall resonance

Sound insulation is as with a single wall, m replaced by totalmass m1 + m2

Fahy, F. & Gardonio, P. Sound and Structural Vibration

4116.01.2012


Double wall with identical walls

Lower dashed curve:single wall R = R1

Upper dashed curve: R = 2R1

(a) No absorbent in cavity(b) Some absorbent in cavity(c) Cavity uniformly filled with

absorbent

Beranek, L. L. Noise and Vibration Control

18 dB/octave

R

4216.01.2012


Mechanical coupling via supporting structures or absorbent incavity degrades sound insulation at whole frequency range,especially with resonant vibration

Double-wall resonance occurs at higher frequenciesCoupling decreases with resilient mounting

Effects of coincidence and lowest modes can be splitted in twofrequencies with lower dips using different walls

Sound insulation at high frequencies decreasesAir cavity resonances degrade sound insulation at high frequencies

Effects decrease with using absorbent in air cavityEffects of absorbent (increase of losses) are higher with deepercavities


4316.01.2012

Table of content


Sound insulation of sandwich structures

4416.01.2012

Sound insulation of sandwich structures

Light and stiff (in normal direction) structuresSound insulation properties between single and double wallCoincidence frequency low

Coincidence can be avoided by tuning shear wavespeed cs in core material below speed of sound c0

Rejects bending wave speed below c0

Double-wall resonance frequency typically higher than with adouble wall with air cavity

Stiff core material and light masses at surfacesCan be tuned above relevant frequencies by more stiffcore structure in normal direction (e.g., honeycomb)

mGhcs

G = shear modulus of core materialh = thickness of corem = total surface mass of surface plates


4516.01.2012

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