Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 4, pp. 895-904, Warsaw 2014

VIBRO-ACOUSTIC ANALYSIS OF LAMINATED DOUBLE GLAZING USING

THE FORCE IDENTIFICATION METHOD

Mounir Ben Jdidia, Ali Akrout, Dhouha Tounsi, Tahar Fakhfakh,

Mohamed Haddar

Mechanics, Modelling and Production Research Laboratory, Mechanical Engineering Department, National School

of Engineers of Sfax, Sfax, Tunisia

e-mail: mounir.benjdidia@isetsf.rnu.tn; ali.akrout@enit.rnu.tn and ali akrout2005@yahoo.fr; dhouha.ing@gmail.com;

tahar.fakhfakh@enis.rnu.tn; mohamed.haddar@enis.rnu.tn

This paper presents a procedure for identifying wave forms and excitation frequencies of
some forces applied on a given complex fluid-structure coupled system by using only its
vibro-acoustic response. The considered concept is called the Independent ComponentAna-
lysis (ICA) which is based on the Blind Source Separation (BSS). In this work, the ICA
method is exploited in order to determine the excitation force applied to a thin-film lamina-
ted double glazing system enclosing a thin fluid cavity and limited by an elastic joint. The
dynamic response of the studied fluid-structure coupled system is determined by finite ele-
ment discretization and minimization of the homogenized energy functional of the coupled
problem. This responsewill serve as the input for the ICA algorithm in order to extract the
applied excitation.

Keywords: fluid-structure interaction, thin-film laminated plate, ICA, excitation force,
Kurtosis

1. Introduction

Laminated and sandwich plates present their advantage when they are used to reduce noise
and vibration due to their high structural damping. In fact, the presence of viscoelastic core or
ultra-thin film gives more rigidity to laminates with amuch reduced vibration deflection.

That is why the resolution of direct fluid-structure coupled problems, such as the study of
acoustic transparency of double panel systems, has been carried out in several research works
such as those developed by Cheng et al. (2005), Abdennadher et al. (2005) and Akrout et al.
(2010).

In thiswork,we deal with a system composed of a thin-film laminated double glazing system
enclosing a thin fluid cavity and limited by an elastic joint, to which we use the ICA concept in
order to extract the applied force.

In fact, theBlind Source Separation (BSS) is an important research area in signal processing
and data analysis.

The first formulation of the problem was made in 1985 by researchers in neuroscience and
signal processing to model biologically coding of motion. In fact, the source separation problem
was developed byHérault andAns (1984) andHérault et al. (1985). Then, Comon (1994) made
the link between the Independent Component Analysis (ICA) and the Blind Source Separation
(BSS).

The independent component analysis (ICA) is one of themajor pathways of sources separa-
tion concept (Hyvärinen and Oja, 2000; Antoni, 2005; Zarzoso and Comon, 2008; Abbes et al.,
2011; Akrout et al., 2012b). It extracts from the observed signal components as independent as
possible. In the recent years, this method (ICA) was investigated for extracting signals such as



896 M. Ben Jdidia et al.

excitation forces and internal defaults in mechanical systems (Akrout et al., 2012b; Taktak et
al., 2012).
In this work, dynamical study is carried out in order to model the vibratory excitation of

a vibro-acoustic problem defined by a thin fluid cavity coupled to the thin-film laminated glass
plate structural model developed in our previous work (Akrout et al., 2012a). Then, the ICA
concept is applied to the finite element signals defined by the displacement vector of the studied
system in order to extract the wave form and the excitation frequencies of the external applied
forces. So, the main original contribution of this work is based on the developed fluid-structure
laminated double glazingmodel which can be exploited for identifying excitation sources by the
inverse method (ICA).

2. Description of the studied fluid-structure coupled system

The studied system is composed of two ultra-thin film laminated glass plates coupled to a thin
fluid cavity and related with an elastic joint, as presented in Fig. 1. k0, k1 and k2 represent the
laminate edge stiffness and F0 is the harmonic uniform distributed force applied to thin-film
laminated plate 1.

Fig. 1. Ultra-thin film laminated double glazing system (the xz-plane)

3. Resolution of the direct vibro-acoustic problem

3.1. Dynamic equation of the coupled problem

Discretization by the finite element method andminimization of the coupled system energy
functional give the following coupled matrix system (Akrout et al., 2010, 2012a)







K1+J1+J0−ω
2M1 −J012 −C1

−JT012 K2+J2+J0−ω
2M2 C2

−CT1 C
T
2

hf
ρfω

2(H−k
2
fQ)

















U1
U2
P











=











Fext

0

0











(3.1)

where K1+J1+J0, K2+J2+J0 and J012 are the stiffnessmatrices of the structural part of the
coupled system. M1 and M2 are themass matrices of the laminates. C1 and C2 are the fluid-



Vibro-acoustic analysis of laminated double glazing... 897

-structure coupling matrices. H and Q represent the acoustic matrices. hf, ρf and kf =ω/cf
represent the cavity thickness, the fluid density and the acoustic wave number, respectively.
cf and ω are the speed of sound and the angular frequency, respectively. U1,U2 and P are the
nodal response vectors. Fext =F0exp(−iωt) represent the nodal force vector to be reconstituted
by the developed inverse method (ICA).
The film-laminate finite element presents seven degrees of freedom at each node

(um,vm,uτ ,vτ,w,βx,βy), whereas the fluid cavity finite element presents one degree of freedom
at each node (pressure p).

3.2. Resolution of the dynamic equation

The resolution of the direct problemdefined by equation (3.1) is based on amodal approach
(Akrout et al., 2010, 2012a). Then, the following eigenvalue problems to be solved are considered

(K1+J1+J0−ω
2
M1)U1 =0 (K2+J2+J0−ω

2
M2)U2 =0

hf
ρfω

2
(H−k2fQ)P=0

(3.2)

By resolving these equations, three modal bases could be constructed: Φs1 for the structure
defined by (laminate1+joint1+joint0), Φs2 for the structure (laminate2+joint2+joint0) and
Φf for the fluid cavity. The second step consists on reducing the size of system (3.1) by modal
projection on these modal bases. So, matrix system (3.1) becomes (Akrout et al., 2010, 2012a)








(K1+J1+J0)−ω
2M1 −J012 −C1

−J
T

012 (K2+J2+J0)−ω
2M2 C2

−C
T

1 C
T

2
hf
ρfω

2(H−k
2
fQ)



















U1
U2
P











=











F
ext

0

0











(3.3)

where (Ki+Ji+J0) =Φ
T
si(Ki+Ji+J0)Φsi, Mi =Φ

T
siMiΦsi (i=1,2) are respectively the

reduced stiffness and mass matrices of the coupled system. J012 = Φ
T
s1J012Φs2 is the reduced

stiffness matrix due to the structural coupling between laminate 1 and 2. H = ΦTfHΦf and

Q = ΦTfQΦf represent the reduced acoustic matrices. Ci = Φ
T
siCiΦf is the reduced fluid-

-laminate (i=1,2) coupling matrix. U1 =Φ
T
s1U1, U2 =Φ

T
s2U2 and P=Φ

T
fP represent the

modal response vectors and F
ext
=ΦTs1F

ext is the modal force vector.
The coupledmodal basis Φc = [φ1, . . . ,φN] (N is the number of retained eigenmodes)which

contains the coupled eigenmodes, is obtained from the resolution of the eignemode reduced
coupled problem as presented in our previous works (Akrout et al., 2010, 2012a).
Themodal variables αr(ω) of the r-th eigenmode,which are obtainedbyprojecting equation

(3.5) on the coupled eigenmode basis Φc, could have the following expression (Akrout et al.,
2010)

αr(ω)=
fr
kr

[

1−
(ω

ωr

)2]−1

r=1,2, . . . ,N (3.4)

where ωr =
√

kr/mr is the r-th eigenfrequency of the coupled system. kr and mr are the r-th
generalizedmass and the r-th generalized stiffness of the coupled system, respectively. fr is the
generalized force of the r-th eignmode.
So, the dynamic response is determined by modal recombination (Hammami et al., 2005;

Akrout et al., 2010)

U(ω) = [φ1, . . . ,φN]











α1(ω)
...

αN(ω)











(3.5)



898 M. Ben Jdidia et al.

Now, after determining thedynamic responsewhich is definedby the displacement of the studied
structure, wewill study the inverse problem. In this case, the dynamic responsewill serve as the
input for the ICA algorithm in order to determine the wave form of the applied force signal.

4. The concept of the Independent Component Analysis: ICA

4.1. Definition and principle hypothesis

ICA is a statistical technique that aims to break a random signal multivariate X (measu-
red signal) in a multivariable linear combination of independent signals (the source signals) to
highlight the signals as independent as possible from the measured signals.
It was developed by Hérault et al. (1985) and it is defined by the following equation

X(t)=AS(t) (4.1)

where X(t) and S(t) are respectively theobserved signals through sensors and the source signals,
A is the mixture matrix.
The source separation principle consists of determining a matrix B in order to estimate N

source signals defined by the vector Y(t)= [Y1(t), . . . ,YN(t)]
T as follows (Antoni, 2005; Abbès

et al., 2011; Akrout et al., 2012b)

Y(t)=BX(t) (4.2)

In order to achieve this goal, general assumptionsmustbe considered (HyvärinenandOja, 2000).
The principal assumption is the statistical independency of the source signals. The second one
imposes non-Gaussian distributions on the source signals (Comon, 1991; Moreau and Macchi,
1993). The last one is defined by the principle of uncorrelated sources. In fact two variables
Y1 and Y2 are uncorrelated if their covariance is equal to zero. This can be expressed by the
following relation

E{Y1Y2}−E{Y1}E{Y2}=0 (4.3)

4.2. Separation concept

The principle object of the presentedmethod is to extract the source signals from amixture
of the observed signals. In order to achieve this goal, the observed signals of the system should
be centred (substrate its mean vector), then whitened (which consist on eliminating the noise,
so we obtain a new signal with an uncorrelated component and variance equal to unity). So, the
estimated source is defined by Antoni (2005)

Y=WHX (4.4)

where WH is the separating matrix, (·)H denotes the conjugate-transpose operator.
Inorder to guarantee thenon-gaussianity of the signals, the estimated sourcesmustmaximize

thecontrast function.Thecontrast functionutilized in the ICAalgorithm is theKurtosis function
defined by Zarzoso and Comon (2010). This forth order cumulant can be normalized as follows

K(ω)=
E{|y|4}−2E2{|y|2}−|E{y2}|2

E2{|y|2}
(4.5)

Finally, after determining the first column of the separating matrix, the deflation approach is
applied in order to extract the corresponding source vector from the original mixture related to
the determined column of the separating matrix.
So, each source will be chosen once with the multiplying factor (Hyvärinen andOja, 2000).



Vibro-acoustic analysis of laminated double glazing... 899

5. Numerical results

The geometrical and physical features of the ultra-thin film laminated glass panel are given as
follows (Akrout et al., 2010):

• two identical skins (glass) for each laminate:

– Young’s modulus E
p1
1 =E

p1
2 =E

p2
1 =E

p2
2 =7.2 ·10

4MPa

– density ρ
p1
1 = ρ

p1
2 = ρ

p2
1 = ρ

p2
2 =2500Kg/m

3

– Poisson’s ratios ν
p1
1 = ν

p1
2 = ν

p2
1 = ν

p2
2 =0.22

– skin thicknesses for the symmetrical system h
p1
1 =h

p1
2 =h

p2
1 =h

p2
2 =3mm

– skin thicknesses for the asymmetrical system h
p1
1 =h

p1
2 =3mm, h

p2
1 =h

p2
2 =4mm

• film stiffness kfilm =1.362 ·10
7N/mm3 (Araldite), kfilm =1.1 ·10

4N/mm3 (Epoxy)

• in-plane (x,y) laminate half dimensions ℓx =0.6m, ℓy =0.4m’

The geometrical and physical parameters of the air cavity are given as follows (Akrout et al.,
2010): cf =340m/s, hf =1mm, ρf =1.2Kg/m

3.
Only the translational joint is considered (Akrout et al., 2012a): kjoint =0.264·10

10N/mm2.

5.1. Parametric study

Three configurations for the coupled studied system are considered:

• model 1 defined by two identical laminates (h
p1
1 = h

p1
2 = h

p2
1 = h

p2
2 = 3mm) with an

ultra-thin film of Araldite (kfilm =1.362 ·10
7N/mm3)

• model 2 characterized by the same laminates as model 1, but another material is chosen
for the adhesive film (kfilm =1.1 ·10

4N/mm3, Epoxy)

• model 3 distinguished by an asymmetrical double glazing system (laminate 1: h
p1
1 =h

p1
2 =

3mm, laminate 2: h
p2
1 =h

p2
2 =4mm, kfilm =1.362 ·10

7N/mm3).

For these three different models, two types of loads are applied on laminate 1: the first one is
defined by a uniformly distributed force and the second one is a punctual force.

5.2. Observed signals

For each model defined above, the FE coupled system vibratory responses (transversal di-
splacement on themiddle of thefirst and second laminate: w1,w2) are determinedandpresented
in Figs. 2-7.

Fig. 2. Model 1: observed signals for a punctual force



900 M. Ben Jdidia et al.

Fig. 3. Model 1: observed signals for a distributed force

Fig. 4. Model 2: observed signals for a punctual force

Fig. 5. Model 2: observed signals for a distributed force

Fig. 6. Model 3: observed signals for a punctual force



Vibro-acoustic analysis of laminated double glazing... 901

Fig. 7. Model 3: observed signals for a distributed force

Then, after determining the observed signals which represent the displacements in the centre
of the first and the second laminate, the obtained results will serve as the input in the ICA
algorithm.

5.3. Estimated sources

The wave form and excitation frequency of the punctual and distributed forces applied to
the systemwill be constructed using the ICA algorithm.
The time-evolution of the applied force and the corresponding spectrum(FFT) are presented

in Figs. 8a and 8b, respectively. In this case, we have mentioned that this applied excitation
can be distributed on the whole nodes of laminate 1. So, it can be also reconstituted by the
developed inverse method.
Then, from Fig. 9 to Fig. 14, we present the corresponding ICA results.

Fig. 8. (a) Time-evolution of the applied excitation source, (b) FFT of the applied excitation source

From Fig. 9 to Fig. 14, we can deduce that the adopted inverse method can be applied to
identify the dynamic excitation (the punctual and distributed forces and their spectrum) for
analyzing the vibro-acoustic behaviour of a fluid-structure coupled system. In this case, the
excitation frequencies of each force are localized and determined.

6. Conclusions

In this paper, one of the major techniques of the Blind Source Separation (BSS) called the
Independents Components Analysis (ICA) is presented and exploited in order to extract the



902 M. Ben Jdidia et al.

Fig. 9. Model 1: (a) estimated punctual source, (b) FFT of the estimated punctual source

Fig. 10. Model 1: (a) estimated distributed source, (b) FFT of the estimated distributed source

Fig. 11. Model 2: (a) estimated punctual source, (b) FFT of the estimated punctual source

Fig. 12. Model 2: (a) estimated distributed source, (b) FFT of the estimated distributed source



Vibro-acoustic analysis of laminated double glazing... 903

Fig. 13. Model 3: (a) estimated punctual source, (b) FFT of the estimated punctual source

Fig. 14. Model 3: (a) estimated distributed source, (b) FFT of the estimated distributed source

force applied to a fluid-structure coupled system composed of a thin-film laminated double
glazing system enclosing a thin fluid cavity and limited by an elastic joint. The vibro-acoustic
responses are determined by the modal recombination method which is applied to the Finite
Element (FE) coupled matrix system.

As a continuation of our previous published works, this method allows us to determine the
wave form of any external or internal forces applied to the structure, so it is a useful method to
study complex fluid-structure coupled systems.

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Manuscript received February 21, 2014; accepted for print April 9, 2014