Acoustic radiation of a vibrating wall covered by a porous layer - - PowerPoint PPT Presentation

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

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

Introduction Part I Part II Part III 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

Introduction Part I Part II Part III 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

Introduction Part I Part II Part III Conclusion

Problem to be solved

² ² , in = + ∇ Ω p k p

) ( , at r h n p B p A = ∂ ∂ + Γ

Dirichlet (imposed pressure) Neumann (imposed velocity) Source at boundary

6

Introduction Part I Part II Part III Conclusion

Problem divided into 2 cases :

a) Acoustic excitation : amplitude of reflected wave ?

7

Introduction Part I Part II Part III Conclusion

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

Problem divided into 2 cases :

8

Introduction Part I Part II Part III Conclusion

Surface Impedance ZS

The reflected wave is function of surface impedance ZS :

) ( ) ( v p ZS = Z Z Z Z R

S S

+ − =

ZS is measured in a Kundt tube Porous layer characterized by 1 R

jkz jkz

e R e z p

−

+ =1 ) (

( )

jkz jkz

e R e Z z v

−

− = 1 ) ( with c Z ρ =

9

Introduction Part I Part II Part III Conclusion

Transfert impedance ZT

The transmitted wave is function of transfert impedance ZT :

) ( ) ( v v p Z

p T

− =

p T T

v Z Z Z Z T + =

gives Can ZT be measured in a Kundt tube ? Is ZT equivalent to ZS ? Porous layer characterized by 1 T

jkz

e T z p

−

= ) (

Non-porous massless coating :

Thickness modulus Bulk jω ZT =

10

Introduction Part I Part II Part III Conclusion

A simple coating : Spring

Case b:

) ( ) ( = − −

p

v v j K p ω ω j K v v p Z

p T

/ ) ( = − =

Static law: Case a:

ω j K v p ZS / ) ( = =

K

=

p

v L j j K Z Z

S T

ω ω modulus Bulk = = =

Elastic layer

11

Introduction Part I Part II Part III Conclusion

Adding a layer defined by its mass

Case b:

) ( ) (

p S

v v j K p v M j − − = ω ω

s T

M j Z Z j K Z ω ω − = ⇒

ZT ≠ ZS excepted when Dynamic law: Case a:

ω ω j K M j v p Z

s S

/ ) ( + = =

K Ms

=

p

v ) ( Z v p = : → ω L j j K Z Z

S T

ω ω modulus Bulk = = =

Elastic layer at low frequency

12

Introduction Part I Part II Part III Conclusion

Monophasic continuous layer

Case b:

1 cZ a Z ZT − − = ⇒

ZT ≠ ZS excepted when Transfert matrix:

) ( Z v p =

e

Z kd j c a → → → and 1 : ω L j kd Z j Z Z

e S T

ω modulus Bulk = = =

Monophasic layer at low frequency

k Ze ,       =       kd kd Z j kd jZ kd d c b a

e e

cos sin / sin cos         − −       =         ) ( ) ( ) ( ) ( L v L p d c b a v p

Case a:

c a ZS = ⇒ =

p

v

13

Introduction Part I Part II Part III Conclusion

Real part Imaginary part ZT transfert impedance Impedance curves

≠

ZS surface impedance 30 kg.m-3 Density 140(1+j0.1) kPa Bulk modulus 20 mm Thickness

10

2

10

3

10

4

• 4000
• 2000

2000 4000 6000 8000 10000 12000 14000 Frequency (Hz) real part of impedance 10

2

10

3

10

4

• 12000
• 10000
• 8000
• 6000
• 4000
• 2000

2000 4000 6000 8000 Frequency (Hz) imaginary part of impedance

14

Introduction Part I Part II Part III Conclusion

Poroelastic layer: Surface impedance ZS

x d 1 2 A E B Z2,k2 C D v(d) F

Excitation

p(d)

) ( ) ( d v d p ZS =

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

) ( ) ( = =

f s

v v ) ( ) 1 ( ) ( and ) ( ) ( d p d d p d

s f

φ σ φ σ − − = − = ) ( ) ( ) 1 ( ) (

f s

v v d v φ φ + − =

15

Introduction Part I Part II Part III Conclusion

x d vP 1 2 A E B Z2,k2 C D v(d)

Excitation

) ( ) ( d v v d p Z

p T

− =

Poroelastic layer: Transfert impedance ZT

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 = = ) ( ) ( ) ( ) 1 ( ) ( and ) ( ) ( d p d d p d

s f

φ σ φ σ − − = − = ) ( ) ( ) 1 ( ) (

f s

v v d v φ φ + − =

) ( ) ( Z d v d p = ⇒

One waves in fluid medium

16

Introduction Part I Part II Part III Conclusion

Real part Imaginary part ZT transfert impedance Impedance curves

≠

ZS surface impedance

17

Introduction Part I Part II Part III Conclusion

Boundary conditions at excitation interface

Fluid-porous interface :

f s

u u j v φ φ ω + − = ) 1 ( / ) (

Wall-porous interface :

f s p

u u j v = = ω /

The skeleton is much more excited in case b)

⇒

18

Introduction Part I Part II Part III 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

Introduction Part I Part II Part III Conclusion

Excepted at low frequency,

S T

Z Z ≠

ZT can not be measured in a Kundt tube ZS ZT

Conclusion of Part I

20

Introduction Part I Part II Part III Conclusion

1 impedance for each problem

Z ) a (

S

= ∂ ∂ + z p p Z jk

p T

v j z p p Z Z jk ) b ( ωρ = ∂ ∂ −

21

Introduction Part I Part II Part III 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

22

Introduction Part I Part II Part III Conclusion

• 2. Acoustic radiation efficiency

a R v

σ Π = Π

Radiated acoustic power

2 sin a S

I r d d ϑ ϑ ϕ Π = < >

∫∫

2 0 0

2 p I c ρ < >=

with: Injected vibratory power

2 0 0

( ) 2

v S

w r c dS ρ Π =

∫∫

23

Introduction Part I Part II Part III Conclusion

(0 dB)

with ZT with ZS

no effect increase decrease

2.1. Acoustic radiation efficiency

Infinite plate at normal incidence (1D) :

2 2 2

Z Z Z v Z T

T T p R

+ = = σ

24

Introduction Part I 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

ZT

with

Z0

For a flat piston of radius a in far field : Reflective baffle

Z0

25

Introduction Part I 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

Introduction Part I Part II Part III Conclusion

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

A : polymer foam with porous

27

Introduction Part I Part II Part III Conclusion

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

B : soft fibrous with porous

28

Introduction Part I Part II Part III 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

29

Introduction Part I Part II Part III Conclusion

2.3. Radiation efficiency of a circular plate

Rayleigh integration Acoustic power Vibratory power Plate equation Axisymetrical modes

( ) ( ) ( ) ( ) ( )

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 ρ β ω =

with:

2 4

h w w D ρ ω ∇ − =

Modal synthesis

4 4 2

( ) ( ) ( ) ( )

n s n n n n

w r w r F w r D a β β π = − Λ

∑

30

Introduction Part I Part II Part III Conclusion

Modal contribution to the acoustic radiation

Frequency (Hz) Radiation efficiency (dB) Mode 1 Mode 2 Mode 3 Synthesis

31

Introduction Part I 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

Introduction Part I Part II Part III Conclusion

Relative radiation efficiency

plate R porous plate R Rn

σ σ σ

+

=

Frequency (Hz) Relative radiation efficiency (dB)

33

Introduction Part I Part II Part III Conclusion

Experimental validation

34

Introduction Part I 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 pi < 10

model measurement

35

Introduction Part I Part II Part III 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

36

Introduction Part I Part II Part III Conclusion

Calculation of ZT for a multilayer

• Use of transfert matrix method
• Maine3A software

R1 Tn Impervious film ⇒ ⇒ ⇒ ⇒ us=uf

n n T

T R T Z Z − − =

1

1

37

Introduction Part I Part II Part III Conclusion

Effect of fibrous material compression on the radiation

• Fibrous material compressed to 20% and 50%

10

2

10

3

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) SigmaR no compression 20% compressed 50% compressed

• Increase of the radiation with compression

38

Introduction Part I Part II Part III Conclusion

Effect of a light film on the radiation

• Increase of the radiation for the foam
• Almost no effect for fibrous

10

2

10

3

10

• 1

10 10

1

10

2

Frequency (Hz) SigmaR fiber fiber + film foam foam + film

39

Introduction Part I Part II Part III Conclusion

Effect of a septum on the radiation

10

2

10

3

10

• 2

10

• 1

10 10

1

10

2

Frequency (Hz) SigmaR fiber + septum foam + air + septum foam + septum foam

• Decrease of the frequency
• Effect of air layer for the foam

skeleton bypass

40

Introduction Part I Part II Part III 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 !)

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