Acoustic radiation of a vibrating wall covered by a porous layer: Transfer impedance concept and effect of compression

(1)

Nicolas DAUCHEZ

Supméca – Institut Supérieur de Mécanique de Paris, Saint Ouen, France

Olivier DOUTRES, Jean-Michel GENEVAUX Laboratoire d’Acoustique UMR CNRS 6613

Université du Maine, Le Mans, France

Acoustic radiation of a vibrating wall covered by a porous layer

Transfer impedance concept and effect of compression

29 october 2009

158th Meeting of the Acoustical Society of America San Antonio, Texas

(2)

2 Introduction

Part I Part II Part III

Conclusion

Context

• Porous materials used in industrial applications

(automotive, aeronautics,…)

for noise reduction

sound absorption

(trim panel, floor,^…) vibration damping

(fuselage)

sound insulation

(fuselage)

(3)

3 Introduction

Conclusion

Context

• Porous material attached to a vibrating structure Influence of a porous layer on the acoustic

radiation of a plate ?

• Analytical model using transfert impedance concept

• Experimental validation

Method

Porous layer impedance applied to a moving wall:

Application to the radiation of a covered piston,

Doutres, Dauchez, Genevaux, J. Acoust. Soc. Am. 121(1), 2007

(4)

4 Introduction

Conclusion

Introduction

1. Transfert impedance concept 2. Acoustic radiation efficiency

2.1. Infinite plate 2.2. Flat piston 2.3. Circular plate

3. Application to multilayer

Conclusion

(5)

5 Introduction Part I

Part II Part III

Conclusion

Problem to be solved

0 ²

² ,

in Ω ∇ p + k p =

) ( ,

at h r

n B p

p

A =

∂ + ∂

Γ

Dirichlet

(imposed pressure)

Neumann

(imposed velocity)

Source at boundary

(6)

Conclusion

Problem divided into 2 cases :

a) Acoustic excitation : amplitude of reflected wave ?

(7)

Conclusion

a) Acoustic excitation : amplitude of reflected wave ? b) Vibratory excitation : amplitude of transmitted wave ?

Problem divided into 2 cases :

(8)

Conclusion

Surface Impedance Z

_S

The reflected wave is function of surface impedance Z_S:

) 0 (

) 0 ( v Z

_S

= p

0 0

Z Z

Z R Z

S S

+

= −

Z_Sis measured in a Kundt tube

Porous layer characterized by 1

R jkz jkz

e R e

z

p ( ) = 1 +

⁻

( ^e

^jkz

^R ^e

^jkz

)

z Z

v = −

⁻

0

) 1 (

0 0

with Z

0

= ρ c

(9)

Conclusion

Transfert impedance Z

_T

The transmitted wave is function of transfert impedance Z_T:

) 0 ( ) 0 (

v v

Z p

p

T

= −

p T

T

Z v

Z Z

T Z

₀

+

0

=

gives

Can Z_T be measured in a Kundt tube ? Is Z_T equivalent to Z_S ?

Porous layer characterized by T 1

e

jkz

T z

p ( ) =

⁻

Non-porous massless coating :

Thickness modulus Bulk

j ω

Z

_T

=

(10)

Conclusion

A simple coating : Spring

Case b:

0 )

( )

0 ( − v − v

_p

= j

p K

ω

ω j v K

v Z p

p

T

( 0 ) /

− =

=

Static law:

Case a:

ω j v K

Z

_S

p ( 0 ) /

=

K

= 0 v

p

L j

j Z K

Z

_T _S

ω ω

modulus Bulk

=

Elastic layer

(11)

Conclusion

Adding a layer defined by its mass

Case b:

) (

) 0

(

_p

S

v v

j p K

v M

j = − −

ω ω

s

T

Z j M

Z j

Z K

ω ω −

=

⇒

0

Z_T ≠ Z_S excepted when Dynamic law:

Case a:

ω ω ^M ^K ^j v j

Z

_S

p ( 0 )

_s

/ +

=

K M_s

= 0 v

p

0

) 0

( Z

v

p =

:

→ 0 ω

L j

j Z K

Z

_T _S

ω ω

modulus Bulk

=

Elastic layer at low frequency

(12)

Conclusion

Monophasic continuous layer

Case b:

0 0

1 cZ a

Z

_T

Z

−

= −

⇒

Z_T ≠ Z_S excepted when Transfert matrix:

0

) 0

( Z

v

p =

Z

e

j kd c

a → →

→ 0 : 1 and ω

L j

kd j Z Z

Z

_T _S ^e

ω

modulus Bulk

=

^Monophasiclayer at low

frequency

k Z

_e

,

 

 



= 

 

 





kd kd

Z j

kd jZ

kd d

c

b a

e

cos sin

/

sin cos

 

 





−

 −



 



= 

 

 





) (

) 0 (

L v

L p

d c

b a

v p

Case a:

c Z

_S

= a 0 ⇒

p

=

v

(13)

Conclusion

Real part Imaginary part

Z_T transfert impedance Impedance curves

≠

_Z

S surface impedance 30 kg.m^-3

Density

140(1+j0.1) kPa Bulk modulus

20 mm Thickness

10² 10³ 10⁴

-4000 -2000 0 2000 4000 6000 8000 10000 12000 14000

Frequency (Hz)

real part of impedance

10² 10³ 10⁴

-12000 -10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000

Frequency (Hz)

imaginary part of impedance

(14)

Conclusion

Poroelastic layer: Surface impedance

Z_S

d x 0

1 2

A B E

Z₂,k₂

C D

v(d)

F Excitation

p(d)

) (

d v

d Z

_S

= p

Boundary conditions : Continuity of Stress and displacement at x=0

at x=d

Biot-Allard theory in 1D in the porous layer : 2 waves travelling in each direction Two waves in fluid medium

0 )

0 ( )

0 ( =

^f

=

s

v

) ( ) 1

( )

( and

) ( )

( d p d

^s

d p d

f

φ σ φ

σ v ( d ) = = ( 1 − − φ ) v

^s

( 0 ) + φ v

^f

( 0 ) = − −

(15)

Conclusion

d x 0

v_P

1 2

A B E

Z₂,k₂

C D

Excitation v(d)

) (

d v v

d Z p

p

T

= −

Poroelastic layer: Transfert impedance

Z_T

Boundary conditions : Continuity of Stress and displacement at x=0

at x=d

Biot-Allard theory in 1D in the porous layer : 2 waves travelling in each direction

p f

s

v v

v ( 0 ) = ( 0 ) =

) ( ) 1

( )

( and

) ( )

( d p d

^s

d p d

f

φ σ φ

σ v ( d ) = = ( 1 − − φ ) v

^s

( 0 ) + φ v

^f

( 0 ) = − −

)

0

( )

( Z

d v

d

p =

One waves in fluid medium

⇒

(16)

Conclusion

Real part Imaginary part

Z_T transfert impedance Impedance curves

≠

_Z

S surface impedance

(17)

Conclusion

Boundary conditions at excitation interface

Fluid-porous interface :

f

s

u

u j

v ( 0 ) / ω = ( 1 − φ ) + φ

Wall-porous interface :

f s

p

j u u

v ^/ ω = =

The skeleton is much more excited in case b)

⇒

(18)

Conclusion

Relative skeleton velocity for both excitation at fluid-porous interface

No strain in skeleton

Strong influence of frame borne

wave

Skeleton almost at rest vibratory acoustical

(19)

Conclusion

Excepted at low frequency,

S

T Z

Z ≠

Z_T can not be measured in a Kundt tube

Z_S Z_T

Conclusion of Part I

(20)

Conclusion

1 impedance for each problem

Z 0

) a (

S

0

=

∂ + ∂

z p p

jk Z

p T

v z j

p p Z

jk Z

⁰ ₀

)

b

( = ωρ

∂

− ∂

(21)

21

Introduction Part I

Part II

Part III

Conclusion

Introduction

Conclusion

(22)

22

Part II

Part III

Conclusion

2. Acoustic radiation efficiency

a R

v

σ = ^Π Π

Radiated acoustic power

2

sin

a

S

I r ϑ ϑ ϕ d d Π = < > ∫∫

2

0 0

I p

ρ c

< >=

with:

Injected vibratory power

2 0 0

( )

v

2

S

c w r dS ρ

Π = ∫∫

r

(23)

23

Part II

Part III

Conclusion

(0 dB)

with Z_T with Z_S

no effect increase decrease

2.1. Acoustic radiation efficiency

Infinite plate at normal incidence (1D) :

2

0 2

Z Z

Z v

Z T

T T p

R

= = +

σ

(24)

24

Part II

Part III

Conclusion

2.2. Acoustic radiation of a flat piston in semi-infinite field

Calculation of the far field pressure: Rayleigh Integral

Z

_T

with

Z₀

For a flat piston of radius a in far field :

Reflective baffle

Z₀

(25)

25

Part II

Part III

Conclusion

2.2. Acoustic radiation of a flat piston in semi-infinite field

Measurement :

2 materials :

A : polymer foam B : soft fibrous

(26)

26

Part II

Part III

Conclusion

2.2. Acoustic radiation of a flat piston in semi-infinite field

A : polymer foam

with porous

(27)

27

Part II

Part III

Conclusion

2.2. Acoustic radiation of a flat piston in semi-infinite field

B : soft fibrous

with porous

(28)

28

Part II

Part III

Conclusion

Introduction

Conclusion

(29)

29

Part II

Part III

Conclusion

2.3. Radiation efficiency of a circular plate

Rayleigh integration

Acoustic power

Vibratory power Plate equation

Axisymetrical modes 0 0 ⁰ ⁰ 0 0

0 0

( )

( ) ( ) ( )

( )

n

n n n

n

J a

w r J r I r

I a

β β β

= − β

4 2

0n n

h D β = ρ ω

with:

2

4 h 0

w w

D

∇ − ρ ω =

Modal synthesis

4 4 2

( ) ( )

n s n

n n n

w r w r w r F

D β β π a

= ∑ − Λ

(30)

30

Part II

Part III

Conclusion

Modal contribution to the acoustic radiation

Frequency (Hz)

Radiation efficiency (dB)

Mode 1 Mode 2 Mode 3 Synthesis

(31)

31

Part II

Part III

Conclusion

Influence of the porous material on the radiation of the plate

Frequency (Hz)

Radiation efficiency (dB)

Plate + porous layer

Bare plate

(32)

32

Part II

Part III

Conclusion

Relative radiation efficiency

plate R

porous plate

R

Rn

σ

σ = σ

⁺

Frequency (Hz)

Relative radiation efficiency (dB)

(33)

33

Part II

Part III

Conclusion

Experimental validation

(34)

34

Part II

Part III

Conclusion

Results: Radiation efficiency of the covered plate

Frequency (Hz)

Relative radiation efficiency (dB)

∆pi > 10

Good prediction of the acoustic radiation

∆pi < 10

model

measurement

(35)

Conclusion

Introduction

Conclusion

(36)

Conclusion

Calculation of Z

_T

for a multilayer

• Use of transfert matrix method

• Maine3A software

R₁

T_n

Impervious film

⇒

⇒⇒

⇒

u

^s

=u

^f

n n

T

R T

T Z Z

−

= −

1 0

1

(37)

Conclusion

Effect of fibrous material compression on the radiation

• Fibrous material compressed to 20% and 50%

10² 10³

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Frequency (Hz) Sigma R

no compression 20% compressed 50% compressed

• Increase of the radiation with compression

(38)

Conclusion

Effect of a light film on the radiation

• Increase of the radiation for the foam

• Almost no effect for fibrous

10² 10³

10^-1 10⁰ 10¹ 10²

fiber fiber + film foam foam + film

(39)

Conclusion

Effect of a septum on the radiation

10² 10³

10^-2 10^-1 10⁰ 10¹ 10²

fiber + septum foam + air + septum foam + septum foam

• Decrease of the frequency

• Effect of air layer for the foam skeleton bypass

(40)

Conclusion

Conclusion

• Acoustic radiation of a covered piston and plate good agreements with measurements

• using transfert impedance concept for porous materials

Prospects

• Effect of mounting conditions ?

• Absorption versus transmission coefficient ?

(NOT SURFACE IMPEDANCE !)

(41)

Nicolas DAUCHEZ

Supméca – Institut Supérieur de Mécanique de Paris, Saint Ouen, France

Olivier DOUTRES, Jean-Michel GENEVAUX Laboratoire d’Acoustique UMR CNRS 6613

Université du Maine, Le Mans, France

Acoustic radiation of a vibrating wall covered by a porous layer

Transfer impedance concept and effect of compression

29 october 2009

158th Meeting of the Acoustical Society of America San Antonio, Texas

(42)

42

Acoustic radiation of a vibrating wall covered by a porous layer: Transfer impedance concept and effect of compression