Introduction to the numerical modeling and experimental

Public Technical Course 1, July 1-2, 2014, Graz, Austria
Introduction to the numerical
modeling and experimental
characterization of porous materials
Pr Noureddine Atalla
Université de Sherbrooke
www.gaus.ca
[email protected]
References
Extended version of this talk and more info about the topic can
be found in :
• ASME/NCAD workshop :
http://files.asme.org/Divisions/NCAD/33112.pdf
•
Winter school on the Acoustics of Poro-Visco-Elastic Media.
Lyon, 12-14 Feb. 2014 ([email protected])
• Allard & Atalla: Propagation of Sound in Porous Media. 2nd
Edition, Wiley, 2009
2
Objectives
1. Define sound absorbing materials an their classes
2. Review models of porous materials and methods to
measure their input parameters
3. Introduce Transfer Matrix Method based models to
simulate the vibration and acoustic response of sound
packages
3
4
Outline
 Sound packages and porous materials
 Modeling porous materials
 Experimental characterization of porous materials
 Transfer Matrix Method based modeling methodology
 Conclusion
6
Sound packages ?
 Sound packages are made
up from a combination of
materials, notably:



Porous materials
 Foams
 Fibers
 Felts
 Carpets
 Porous films
 Fabrics
Damping materials
Limp and elastic (solid)
layers
7
Porous materials and their characteristics
 Porous materials
 Two phases : solid and fluid
 Elastic coupling
 Visco-inertial coupling
Fiberglass
Urethane
 What do they do ?
The behavior of porous materials depends on their nature, loads and the coupled
 Transform
acoustic
energy into heat
structures.
The selection
of a model/methodology
should account for this fact
 How do they dissipate energy ?
 viscous effect
 thermal effect
 structural damping
Melamine
Double porosity
8
What about their performance?
9
What about their performance?
 Typical diffuse field results
10
What about their performance?
 Typical normal incidence results
11
What about their frame ?
Elastic frame
Rigid frame
Limp frame
12
Types of porous materials
Glass wool / glass fiber
Granular
Woven fabrics
Felt
Rigid
Elastic polymeric foams
Recycled
Perforated
Perforated plates
Non woven
Scims & screens
Multilayer / Laminates
13
Outline
 Objectives
 Sound packages and porous materials
 Modeling porous materials
 Experimental characterization of porous
materials
 Transfer Matrix Method based modeling
methodology
 Conclusion
14
Modeling Porous materials
 Mechanical model
(Equivalent mechanical systems)
The membrane mass
K=Stiffness
of the
foam/Fiber
c = Kinetic
viscous
coefficient
between
moving frame
and air
 Admittance model
Example Delaney and Basley model for
fibrous materials, …)
0.754
0.732


ρ f 
ρ f 
ZC  ρ0C0 1  0.0571  0 
 j 0.087  0 
 ,

σ
σ







ρ f 
k  1  0.0978  0 
C0 
 σ 
0.700
ρ f 
 j 0.189  0 
 σ 
0.595

 ,

Microstructural models (Biot)
Rigid/limp frame
(Classical codes; …)
P
 
 j ~o P  0
n
Zn
Empirical models
(limited to certain type of materials
and can be inaccurate; D&B, Miki, …
(Equivalent fluid models)
= Rigide
 Flexible
Elastic frame.
Coupled poroelastic models
15
Modeling of porous materials
They are modeled at a homogeneous macroscopic scale
Elastic or acoustic
wave
Macroscopic scale H1
h <<
<<
1 HH1
H 1<<
2H 
<<

ux
h
uz
H1
Poroelastic
material
Ux
Macroscopic
element
uy
Homogeneous
solid phase
2
Uz
Uy
Homogeneous
fluid phase
16
16
Biot based models
Example: BIOT (u,p) formulation governs the propagation of
the coupled elastic waves (compression and shear) and
acoustic wave (compression).
Biot's macroscopic
parameters
2ρ
~ u  γ~ p,
~u   λ~  μ
~ u

μ
ω


s i
i,jj 
i
 j,ij
1
1
p,ii  ~ p  γ~ u i,i
~
Kf
ω2ρ
f
Elasto-dynamic equation
Helmholtz equation
u : solid phase macroscopic displacement vectors
p : fluid phase macroscopic pressure
~:
denotes a complex and frequency dependent quantity (Dynamic properties)
, : Effective solid phase Lamé coefficients
Kf : Effective fluid phase bulk modulus
s : Effective solid phase density
f : Effective fluid phase density
 : Fluid-solid coupling coefficient
Ref.: Allard & Atalla: Propagation of Sound in Porous Media. 2nd Edition, Wiley, 2009
17
Equivalent fluid model
Rigid frame (Limit of Biot’s model when frame is motionless)
Helmholtz equation with equivalent density and bulk modulus
p  
2
eq
p0
K eq
Limp frame (Limit of Biot’s model when frame is limp)
Helmholtz equation with equivalent density and bulk modulus
 e,l
p  
p0
K
2
e
 e M 02
 e,l 
M  e  20
M  1  0
Total apparent mass
of the bulk volume
Ref.: http://dx.doi.org/10.1121/1.2800895
Panneton: Comments on the limp frame equivalent fluid model for porous media, J. Acoust. Soc. Am. 122 (2007)
18
Equivalent fluid model
Equivalent dynamic density
• Takes into account viscous and inertial effects
• Some recent models:
• Johnson et al. (1987) - , , , 
• Pride et al. (1993) - , , , , 0
e


 EQ 

K e

K EQ 

Equivalent dynamic bulk modulus
• Takes into account thermal effects
• Some recent models:
• Champoux and Allard (1991) - , 
• Lafarge I (1993,1997) - , , k0
• Lafarge II (1993) - , , k0, 0
JCA model
JL model
Macroscopic parameters

- Open porosity
 - Tortuosity
 - Static airflow resistivity
 - Viscous characteristic length
0 - Static viscous tortuosity
k0 - Static thermal permeability
 - Thermal characteristic length
0 - Static thermal tortuosity
19
Input Parameters for the Biot model
The Biot-Allard model: 9 parameters
Source: GUI from Nova software
20
20
Bulk density [1]
Following the Biot theory, the bulk density is the ratio
between the in-vacuum mass of the porous aggregate
and its bulk volume.
M
1 
Vt
M
1 
2
r h
r
h
21
Open Porosity []
Open porosity is defined as the fraction of the
interconnected pore fluid volume to the total bulk
volume of the porous aggregate.

Vf
Vt
Vs
1
 1
1
s
Vt
Typical values
For perforates : < 50%
For light fibrous: ~ 99%
For foams: > 90%
Vs : Solid phase volume
s : Density of solid phase
material
22
Static airflow resistivity []
Static airflow resistivity governs the low-frequency viscous
effects in open-cell porous media, where the viscous skin
depth is of the order of magnitude of the characteristic size of
the cells.
It is defined as the limit, when flow tends to zero, of the
quotient of the air pressure difference across a specimen
divided by its thickness and the velocity of airflow through it.
P A

Q h
[Ns/m4 or MKS Rayls/m]
P
23
Tortuosity []
Tortuosity accounts for the apparent
increase in the fluid density when the
fluid saturates a porous structure.
It can be seen as the effective length
of the path follows by acoustical wave
through the material.

 L 
 

 L 
  1
2
Typical values
- Low density fibrous:
 = 1.00
- Mid/High density fibrous: 1.00    1.45
- Reticulated foams:
1.00    2.0
- Partially reticulated foams : 2.0    3.0
24
Thermal characteristic length []
The thermal characteristic length describes the thermal
dissipation effects at medium and high frequencies.
It is of the order of magnitude of the average radius of the
larger cells where thermal losses dominate viscous losses.

  2

V
dV
f

dS
Vf : Pore volume
: Wet surface boundary
Typical values
From 10 µm to 500 µm
25
Viscous characteristic length []
The viscous characteristic length describes the viscous
dissipation effects at medium and high frequencies.
It is of the order of magnitude of the average radius of the
smaller cells and necks where viscous losses dominate
thermal losses.


 2

v dV
2
Vf
v dS
2

 
Vf : Pore volume
: Wet surface boundary
Typical values
From 10 µm to 500 µm
26
Microstructure based models
Alternative models  microstructure based: needed to optimize
foam’s fabrication process to target specific vibroacoustic
applications (dash insulator, floor insulator,…)
link micro/macro for
polyurethane foams
Chemists
Fabrication
Microstructure
cell size,…
Acousticians
Non-Acoustic
properties
porosity, tortuosity,…
Acoustic behavior
absorption, transmission,…
27
Microstructure based models
An example: semi-empirical model*
Fabricatio
n
Microstructure
cell size,reticulation rate
Non-Acoustic
properties
Acoustic behavior
porosity,
tortuosity,…
absorption,
transmission,…
2
  t
  C C 2 
 l 

Polyurethane foam
•
Tetrakaidecahedra Unit cell
Cell size (Cs)
Ligament length (l )
Ligament thickness
(t)
•
Reticulation rate (Rw)
2V
Sound
JCA
model Absorption
f
A  (1  R ) A
S
w w
0.676
close pore
ligament
0.380
 1

 Rw 
  1.05
' 
Cell
size
1.116
 1
 
 Rw 
open pore
 1 
  ' / 1.55 
 Rw 
2
t
  1C  
l 
*O. Doutres, N. Atalla, “A semi-empirical model to predict the acoustic behavior of fully and partially reticulated polyurethane foams based
on microstructure properties” Acoustics 2012
28
Microstructure based models
Validation
Foam P2:
CS= 616 ± 36 m
l = 209 ± 14 m
t =50 ± 4 m
Rw= 32 ± 11 %
Foam P3:
CS=1710 ± 161
m
l = 554 ± 39 m
t =151 ± 8 m
Rw=5 ± 2 %
simulation
expanded uncertainty
measurements
29
Predictive methods for porous media
Accuracy
Mechanical
Admittance
Empirical
Limp
Rigid
Poroelastic
Assumptions
Set up, Needed
parameters and
computation time
30
Experimental characterization of porous materials
31
Characterization Methods
Direct
Viscous length
Thermal length
Tortuosity
Resistivity
Porosity
Bulk density
Young’s modulus
Poisson’s ratio
Loss factor
Inversion
Time
Viscous length
Thermal length
Tortuosity
Resistivity
Porosity
Frequency
Ultrasound
Viscous length
Thermal length
Tortuosity
Good compromise easy/robust
Number of searched parameters
Accuracy
Audio
Iterative
Direct
impedance tube
transmission tube
Viscous length
Thermal length
Tortuosity
Resistivity
Porosity
Viscous length
Thermal length
Tortuosity
Resistivity
Number of searched parameters
Accuracy
32
Measurement of Open porosity and bulk density
 Method:






Pressure/Mass method
Perfect gas law
Assume isothermal process
Heavier gas reduces uncertainty:
Air, Argon, Krypton, Xenon
Uncertainty mainly control by bulk
volume of porous aggregate
Bulk volume larger than 350 cm for
absolute error less than 1%
RT  m2  m1 m4  m3 
  1



Vt  P2  P1
P4  P3 
m3  m1
1 
Vt
Air
Vacuum
PVacuum
1 , m1 ,V
P1, m1, V
Argon
GAS
High
pressure
P2 , m
2 ,V
P2, m2, V
M1
M2
Air
Vacuum
P3 , m3 ,V  Vs
Argon
GAS
P4 , m4 ,V  Vs
Vs
Vs
M3
M4
Ref.: http://dx.doi.org/10.1063/1.2749486
Salissou and Panneton: Pressure/mass method to measure open porosity of porous solids, J. Appl. Phys. 101 (2007)
33
Measurement of Static airflow resistivity






Direct method based on ASTM C522
or ISO 9053
Ideally on 100-mm diameter specimen
Minimum of 3 specimens
If pressure drop too small, stack specimens up to maxium
of 5
Measurement at 0.5 mm/s
(~ sound pressure of 80 dB-ref20µPa) or stepwise down to
lower limit of system and extrapolated to 0.5 mm/s.
0.5 mm/s correspond to a flow of 240 CCM for 100-mm
diameter
 Special cares to prevent leaks
P
Q

P A
Q h
Wall of mounting rings should be thin; however diameter correction may be
applied in the calculations
34
Tortuosity measurements
Principle of the transmission method
Transducer
Source
Transducer
Receiver
Without sample
With sample
t
(1)
Awo
Aw
(2)
0
d
L
L
time (s)
Refraction index
nr ( ) 
c0
c  
 1  c0
t ( )
d

 1   1 
nr2 ( )    1    

  B  

M
 
=   n 
f 
f
 ( ) 
2
0
=
 1
 Viscous layer thickness
 f
35
Tortuosity measurements
Principle of the reflection method: Measure the reflection
coefficient at various angles
r t,  
Am (t , )
Ar (t , )
Transducer
Receiver
With rigid plate
With sample
Transducer
Source
d
Arigid
Amaterial
Amplitude


L
0

z2 
 
1
2

L
time (s)
2 
 2
1 r 
 
1   sin    with z 
1  r cos 
 
 z

Necessitates the value of the porosity
36
Characteristic lengths estimation
Based in the transmission method in two gases
Transducer
Source
Transducer
Receiver
Without sample
With sample
t
(1)
Aw o
Aw
(2)
0
time (s)
L
L
d
Measure equivalent lengths in air (gas 1)
and argon (gas 2) and solve:
1

1
 1  1
  2  1
Pr1   1   1 Leq ,1 




Pr 2  1   1 Leq ,2 
However, the only robust method for the characterization of
tortuosity and characteristic lengths is iterative inversion
37
Iterative method
 Optimization process
Iteratively adjust model parameters so that the model
predicts impedance tube measurements
Poro‐acoustical
model
direct
  
 ’
inverse
Direct
characterization
optional
Inverse
characterization
Acoustical
model
Acoustical
indicator
Impedance
tube
Ref.: http://www.mecanum.com/files/InversePaper.pdf
Y. Atalla, R. Panneton: Inverse Acoustical Chararacterization…, Canadian Acoustics 33 (2001)
38
But rules must be observed
For inverse (iterative or direct) characterizations,
impedance tube measurements must verify following tests :
1. Sample is saturated by air at rest.
2. Linear acoustics
3. The resistivity and open porosity of the sample are known.
4. Acoustical response mostly follows that of an equivalent fluid.
5. All the physics is captured by the absorption curves.
6. Sample is homogeneous (~symmetric).
7. Boundary conditions do not influence tube measurements.
39
Quality of test sample
40
Is all the physics captured in the absorption curves?
Large number of
open-cell
porous media follow
this typical behavior
Exception: high
resistivity materials,
where viscous length
dominate viscous
forces over resistivity
Zones I + II are necessary
Zone III is highly recommended
for accurate estimations of the parameters
41
Is the material homogeneous along thickness (through-thickness
symmetry) ?
x  RD   
Z AB    Z BA  

max Z AB   , Z BA  

Z AB
Non symmetric
Z BA
Ref.: http://dx.doi.org/10.1121/1.2947625
Salissou, Panneton: Quantifying through-thickness asymmetry of sound absorbing materials, JASA 124 (2008)
42
Is the material homogeneous along thickness (through-thickness
symmetry) ?
 Example :
ok
Cast foam
Face A versus Face B
< 10 %
43
Are measurements too sensitive to boundary conditions in tube ?
What is done
What we
want to
measure
Acoustical measurements in
the Standing Wave Tube
(SWT)


What we
actually
measure
or

2)
or
1)
or

1) Check for sensitivity to edge constraints
2) Check for sensitivity to edge acoustical leaks
Ref.: http://dx.doi.org/10.1121/1.2947625
Pilon, Panneton, Sgard: Behavioral criterion quantifying …, JASA 124 (2008)
44
44
Effects of mounting conditions
Experimental results
Transfer Matrix
Method



Axisymmetric Biot (u,P)
model: Bonded all
around
difficulty of Tube based methods to determine
mechanical properties
45
Effects of lateral gaps (leaks)
The effect of lateral air gaps (leaks) are important for thick highly resistive materials
An axisymetric finite element solver helps to verify this effect or used for design.
No leak
With 1% leak
Air
gap
Rigid
backing
F. Castel, Ph.D. thesis, Sherbrooke (2005)
46
Elastic properties [E,,]
Young’s modulus [Pa], Poisson’s ratio and damping loss
factor follow the same definitions as for elastic materials.
Porous materials are generally not isotropic.
Most of the time, only the normal properties (under
tension or compression) are used.
F
Typical values
- Low density fibrous:
- Mid/High density fibrous:
- Elastic foams :
- Rigid foams :
- Metal foams :
E  10 kPa; =0
10 kPa  E  150 kPa
50 kPa  E  500 kPa
500 kPa  E  2 Mpa
E  30 MPa
F
47
Measurement of Elastic properties
 Method:


Based on compression tests using disk shaped samples
Properties:
Gives true elastic properties (E,,)
 Account for boundary conditions


Excitation frequencies below 1st resonance of the system (5Hz - 60Hz)

A minimum of two (2) samples of different shape factors are required
R
L
s
R
2L
Ref.: http://dx.doi.org/10.1121/1.1419091
Langlois, Panneton, Atalla: Mechanical characterization of poroelastic materials, J. Acoust. Soc. Am. 110 (2001)
48
Direct measurement of Elastic properties
 Method:


The compression test yields the mechanical impedance Z of the sample
From Z, loss factor and apparent Young’s modulus are found
F  
Z    K    jX   
u  
Im( Z )
   
Re( Z )

KL
 F  E A

E 
K  Re   
L
A
u

Apparent Young’s modulus
49
A large correction is needed for large shape
factors or large Poisson’s ratio.
Measured (Apparent)
True
P(s,)
E
E
P( s, )

Bulge out effect
Shape factor - s
The correction factor P depends on:
-Shape factor “s=R/2L”
-Boundary conditions
-Poisson’s ratio “”
An Axisymmetrical high-order solid FEM
model of the experimental set-up is used to
solve and tabulate the correction factors for
various (s,).
50
Measuring mechanical impedance on two samples with different
shape factors yields…
E1
E2

0
P( s1 , ) P( s2 , )
1 equation
1 unknown ()


From the Tabulated Correction Values, and
polynomial curvefits, the correction factors
Ps1 and Ps2 are found.
E1
E2
E
or
P ( s1 , )
P( s2 , )

51
Transfer Matrix Method based modeling methodology
52
52
Modeling sound packages
Noise control materials models are usually implemented using the Transfer
Matrix Method (TMM) for planar multilayer systems and FE/BEM approach for
general configurations
FE module
FEM/BEM
TMM module
Finite size correction
for the transfer
matrix method
Generalized
Transfer matrix
method (TMM) with
size effects


Hierarchical finite
element for speed
and accuracy
LF
MF

HF
53
Principle of Transfer Matrix Method
- Assumes planar infinite systems (1D problems)
- The global matrix is constructed from constituent transfer matrices, the coupling
conditions and the termination conditions :
Fluid 1
[ D ]V  0
Structure
Thin panel
Solid
Septum
Sandwich
General laminates
(1)
(2)
(n)
...

A M1
M2 M3
M4
M2n-1
M2n
Porous layer
Rigid /limp frame
Porous-elastic
Perforated plates & screens
Double porosity
…
…
54
Example : TMM modeling of sound absorbers
(equivalent fluid models)
Single layer – Transfer Matrix Method (TMM)
 p
a b p

v
 
  front  c a   v back
Multilayer – TMM in series
 p
 A B  p

v
C D v 
  front 
  back
 A B  a b a b a b a b a b
C D   c a   c a   c a   c a   c a 

 
1 
2 
3 
4 
5
55
Transmission of a layer (equivalent fluid)
1
 p1 
 
 v1 
p
p

2
3
4
 p2 
 
 v2 
 p3 
 
 v3 
 p4 
 
 v4 
p  x   Ae  jk1x  Be jk1x

v  x 
x
x1
0

A  jk1x B jk1x
e
 e

Z1
Z1
p1  p  0   A  B
v1  v  0  
1
 A  B
Z
 cos k1h1
 p1  
  1

 v1   j  sin k1h1
 Z1
…
k1 : Complex wave number of medium 1
Z1 : Characteriztic impedance of medium 1
p2  p  x1   Ae  jk1x1  Be jk1x1

v2  v  x1  

A  jk1x1 B jk1x1
e
 e

Z1
Z1
jZ1 sin k1h1 
  p2 
 
cos k1h1   v2 

 p1 
 p2 

T
 

1
v
v
 1
 2
56
Transmission through a series of layers (multilayer)
1
 p1 
 
 v1 
p
p
0
 p1 
 p2 

T
 

1
 v1 
 v2 

x1
2
3
4
 p2 
 
 v2 
 p3 
 
 v3 
 p4 
 
 v4 
x2
 p3 
 p2 

T

 
2
 v2 
 v3 
 p2 
 p4 
   T1T2 T3  
 v2 
 v4 
…
x3 x
 p3 
 p4 

T
 

3
 v4 
 v3 
Transfer matrix combined
in series
 p2 
 p4 

T
 

G
 v2 
 v4 
57
Use of the Transfer Matrix Method
Calculation of vibroacoustic indicators under various excitations:
• Impedance / Absorption
• Transmission Loss
• Added damping
• Radiation efficiency
• Acoustic/Vibration transmissibility
•…
Can also be used within SEA framework to handle complicated geometries and
systems
Principle: assumes the sound package planar
and uses TMM to correct (i) non-resonant
path; (ii) radiation efficiency; (iii) absorption
and (iv) added damping
58
58
Typical application
TMM can account
Absorption
Problem:for Size effects (FTMM) for TL and Absorption predictions


4  Z A 
lim
Z A  Z R ,avg
0
 f , st ,avg 

lim
0

1
Z R ,avg   
2

2
sin  d
cos  sin  d

2
0
Z R  ,   d
Transmission Problem:
2
 diffuse 
max
 
0
min
 ( , )  Re ZR  c cos  sin  cos d d

2
 
0
ZR 
ik
S

S
S
e
 jk sin   cos  x0  sin  y0 
1
Z R ,avg   
2
0
max

0
sin  cos d d
min
G ( M , M 0 )e
2

jk sin   cos  x  sin  y 
Z R  ,   d
dS  M 0  dS  M 
Z R ,avg   
 
 0 c0
cos 
Double surface Integral to evaluate ZR is shown to reduce to 1D  quick estimation
59
Illustration of size correction
60
Absorption of Multilayer systems
Experimental validation for multilayer system with impervious film
(septum)
.
Plane wave
Fibrous
Septum
Foam 2
Rigid wall
61
Absorption of foam-screen systems
Perforated screen (h=0.47 mm
=137700 Nsm-4 ; =0.08)
M3
M1
Screen
(=17000 Nsm-4 ;=0.059)
1
Coefficient d'absorption
0,9
0,8
0,7
0,6
0,5
M1 (25.06 mm)
écran poreux (0.79 mm)
M3 (12.57 mm)
0,4
0,3
Fond rigide
0,2
Prédiction
0,1
0
100
Mesure
1100
2100
3100
4100
5100
6100
Frequences (Hz)
Use of an equivalent fluid model for the perfortaed facing  Good agreement using tortuosity
correction formulation (Atalla & Sgard. JSV 2005)
62
Modeling DWL systems
Transmission Loss estimation of a DWL system
2 mm
AL
25 mm
Foam
1mm
AL
38 mm
Air
Excellent agreement (the challenge is rather the test reduce flanking paths)
63
Radiation of a piston in a Tube
Vibrating Piston
283 mm
400 mm
Microphone
Anechoic termination
20
10
0
Bare Piston
Piston with Foam (test)
Analysis (IL)
-10
-20
-30
-40
-50
-60
20
120 220 320 420 520 620 720
64
Double porosity materials
 Allows for materials with two scales of porosity (e.g. perforated
foams).
65
Patch-work type materials
 Allows for non-homogenous materials (patch-works) and
parallel assemblies using an advanced Parallel/Series
implementation of the TMM.
Verdiere et al. (2013) Transfer matrix method applied to the parallel assembly of sound
absorbing materials. JASA 134(6).
66
Embedded resonators
67
Transmission through mounts
Validation vs. FE & Tests for a point load (ROF) excitation
Excitation: ROF
Metric: Acoustical to
Meahanical Conversion
Efficiency (AMCE)
2in FG
Sandwich
trim
Stiffened
panel
Rigid or flexible
(mount)
connection
 
AMCE  10log10  in 
 rad 
68
Conclusion
 Use of Biot based models for porous material is nowadays
widely in industry
 Many Direct and iterative (Indirect) methods exist for the
characterization of the 9 material properties
 Fine characterization depends on many factors:
 Quality of specimens (geometry, homogeneity, …)
 Control of edge effects
 Expertise of experimenter
Perspectives: Need for standards for full poroelastic
characterization:
 Impedance, absorption measurement (Ok but not complete)
 Resistivity (Ok)
 The rest of the properties (standards & QA needed)
69
Conclusion
 TMM based models are Quick & accurate enough to be used for
design and what if questions…
 Comparison with Tests & FE simulations show that they are
even acceptable for large curved panel (aerospace applications;
low ring frequency) and various types of excitations
Perspectives:
• More studies with various panel constructions, radius of
curvatures & excitations (DAF & TBL)
• Efficient combination with FE models  hybrid models (see
my keynote presentations at ISNVH14)
• Accounting for input parameters variance, anisotropy and
mounting conditions
70
References
Extended version of this talk and more info about the topic can
be found in :
• ASME/NCAD workshop :
http://files.asme.org/Divisions/NCAD/33112.pdf
•
Winter school on the Acoustics of Poro-Visco-Elastic Media.
Lyon, 12-14 Feb. 2014 ([email protected])
• Allard & Atalla: Propagation of Sound in Porous Media. 2nd
Edition, Wiley, 2009
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