numerical vibroacoustic analysis of plates with constrained-layer damping patches

Numerical vibroacoustic analysis of plates withconstrained-layer damping patches

Alexandre Loredo,a) Arnaud Plessy, and Ali El HafidiUniversite de Bourgogne, Laboratoire de Recherche en Mecanique et Acoustique, 49 rue Mlle Bourgeois,B.P. 31, 58027 Nevers Cedex, France

Nacer HamzaouiINSA Lyon, Laboratoire Vibrations Acoustique, 25 bis avenue Jean Capelle, F-69621 VilleurbanneCedex, France

(Received 20 November 2009; revised 2 December 2010; accepted 29 December 2010)

A numerical vibroacoustic model that can manage multilayered plates locally covered with damping

patches is presented. All the layers can have an on-axis orthotropic viscoelastic behavior. Continuity

of displacements and transverse shear stresses at each interface is enforced, which permits to write

the entire displacement field in function of the displacements of the—common—first layer, leading

to a two-dimensional plate model. The problem is then discretized by Rayleigh–Ritz’s method

using a trigonometric basis that includes both sine and cosine functions in order to treat various

boundary conditions. The excitation can be of mechanical kind (concentrated or distributed forces)

or of acoustic kind (plane wave of any incidence, diffuse field, etc.). The model permits to compute

different vibroacoustic indicators: the mean square velocity of the plate, the radiation efficiency,

and the transmission loss. Comparisons between the present model and numerical results from

literature or finite element computations show that the model gives good results in both mechanical

and acoustical aspects. Then, a comparison of the effects of different distributions of patches is

presented. The role of the surface covering rate is first discussed, followed by a study involving

different geometries for the same surface covering rate. VC 2011 Acoustical Society of America.

[DOI: 10.1121=1.3546096]

PACS number(s): 43.40.Dx, 43.20.Tb [JHG] Pages: 1905–1918

I. INTRODUCTION

Noise reduction is of great importance in many indus-

tries. A common way to reduce the noise radiated by the

vibrations of a structure is to add viscoelastic materials. For

plate or shell structures they are generally used in the form of

adhesive mono or multilayer products. An upper layer made

of high module elastic material can be superimposed to the

low module viscoelastic product in order to force shear defor-

mation in the viscoelastic layer, what increases the energy dis-

sipation. These complexes, called passive constrained-layer

damping (PCLD) patches, are being increasingly used in auto-

motive and aircraft applications.1 Among the first studies

dealing with viscoelastic damping, Kerwin2 developed a sim-

plified theory to calculate the loss factor of a three-layer beam

with a damping core layer. The theory takes into account the

shear strain of the damping layer that occurs during the trans-

verse motion of the structure. His work was followed by a

number of investigations. Di Taranto,3 and Mead and Markus4

extended Kerwin’s work and developed a sixth order equation

of motion governing the transverse displacement of a sand-

wich beam with arbitrary boundary conditions.

For the study of multilayered plates damped with local

PCLD patches, a three-dimensional (3D) numerical simula-

tion can be done by the finite element method. However, to be

accurate with aspect ratio a=h up to 500 and for—relatively—

short flexural wavelengths, the required mesh leads to a very

large number of degrees of freedom. This is acceptable for

the simulation of the response for a few frequencies but leads

to huge computational time if frequency response studies are

envisaged and optimization studies are definitely out of

reach. Fortunately, plates can be studied by two-dimensional

(2D) models, plate theories, whose aim is to make the z—across the thickness—coordinate vanish. Even if other types

of displacements appear like rotations, the size of discrete

structures is strongly reduced.

Several theories of multilayered plates exist. These the-

ories are based on variational approaches and are solved ei-

ther by the Rayleigh–Ritz method or by the finite element

method. The hypothesis of 2D elasticity theory with plane

stress, which assumes no stress in the transverse direction of

the plate, is used in numerous works to modelize multilay-

ered plate. First, classical lamination theory (CLT), based on

Love–Kirchoff’s assumptions, is the simplest multilayered

plate model. It takes into account in-plane (membrane) de-

formation and flexural motion of the structure but neglects

the transverse shear. Classical bending theory produces

errors when shear deformation becomes important compared

to flexural deformation. It is the case for thick plates, for

composites plates, and especially for sandwich plates. To

overcome this problem, (first order) shear deformation theo-

ries (SDTs) for multilayered plates have been built using

the Reissner–Mindlin’s assumptions.7,8 Therefore, in-plane

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]

J. Acoust. Soc. Am. 129 (4), April 2011 VC 2011 Acoustical Society of America 19050001-4966/2011/129(4)/1905/14/$30.00

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.63.180.147 On: Tue, 12 Aug 2014 14:28:53

displacements (ux and uy) depends both on bending (first de-

rivative of the transverse displacement) and on shear deforma-

tion. These theories imply constant values of transverse

shear strains through the thickness of the plate and thus

require shear correction factors to work properly. Finally,

five unknowns are involved: in-plane displacements in x and

y directions, the rotations around x and y axis, and the trans-

verse displacement. Several finite element models are based

on this theory. Other models,9,10 based in higher-order SDTs,

are defined with polynomials (e.g., of order 2 or 3 in z) for the

in-plane displacements. In these cases the number of

unknowns is greater than five.

We turn our attention to a particular approach, first initi-

ated by Sun and Whitney11 and by Srinivas,12 which uses a

piecewise linear displacement field across the thickness. This

method, classed as a discrete layer theory, satisfies the continu-

ity of the displacement at each layer interfaces and leads to

2N þ 3 unknowns, where N is the number of layers. In the first

of these two studies,11 another model derived from the discrete

model was presented, in which the shear stresses continuity

was enforced at each interface, leading to a five unknown

model. The two models were compared to the classical SDT

with and without shear correction coefficients. The authors

made the conclusion that when laminates with large ratio of

shear moduli are studied, theories assuming that plane sections

remain plane after deformation and excluding local rotations

are unable to predict correct fundamental modes of vibration.

When constrained-layer damping is considered, the ratio of

transverse shear moduli can reach 105. The five unknown

model, in which the global motion of the structure is written in

terms of displacements of the first layer, appears to be a very

good compromise between precision and number of unknowns.

This five unknown mechanical model has been used by

Guyader and Lesueur13–15 and by Woodcock and Nicolas16

to study acoustic transmission through multilayered plates.

Recent works, only dealing with vibrations, also use this

model.17,18 This model is adapted in this study to simulate

the mechanical behavior of the damped plate. In Ref. 18,

Woodcock proposed a generalization of this mechanical

model for laminates made up with off-axis orthotropic layers.

The sound radiation of plates is sensible to the nature of

the boundary conditions, as shown by Gomperts5 and Berry.6

Many authors have proposed different approaches to take

into account arbitrary boundary conditions.

In this study, the following choices are done:

(1) the mechanical plate model of Refs. 11, 13, and 18 is

adopted, hence, the global motion of the structure is

expressed in terms of the motion of its first layer;

(2) a 2D Rayleigh–Ritz discretization of the structure is

done, using a special trigonometric basis which permits

to treat various boundary conditions;

(3) the local patches are taken into account adding their

energy contributions to the structure with the help of a

special substructuration technique based on the fact—as

shown later—that the first layer is common to the cov-

ered and uncovered zones;

(4) boundary conditions are taken into account by means of

complex-stiffness translational and rotational springs,

which permits to cover all the cases (among which the

simply supported, the clamped, and the free boundary),

with a dissipative behavior if needed;

(5) both mechanical and acoustical excitation can be made;

the fluid inertia is neglected (light fluid hypothesis);

(6) the far-field acoustic radiation is computed from the

deflection field, leading to classical vibroacoustic indica-

tors: means square velocity, TL, and radiation efficiency.

II. THEORETICAL ANALYSIS

A. Statement of the problem

The studied plate is assumed to be rectangular and

baffled. The baffled plate separates two semi-infinite mediums.

Fluid loading is considered light and is not accounted in the

equation of motion. In-plane displacements, transverse dis-

placement, and transverse shear strains are taken into account.

The proposed model describes the vibroacoustic behavior of a

multilayered plate with multilayered rectangular patches.

These patches can have any number of layers, which permits

to treat both free-layer (elastic–viscoelastic) and constrained-

layer (elastic–viscoelastic–elastic) cases.

For practical reasons, as explained in Sec. II D, the plate

is divided into two parts: the uncovered part and the coveredpart. Geometrical characteristics of the plate are shown in

Figs. 1 and 3. Several rectangular patches can be put on the

same plate. The covered part then corresponds to all the

patches and the uncovered part corresponds to the comple-

mentary surface. Integration over rectangular parts leads to

analytical formulas which are fast to compute. This was par-

ticularly useful to simulate the strips by means of rectangular

patches (see Sec. III B). In this case, to ensure continuity,

the patches are supposed to be made with the same layers

and materials.

The plate is excited by sound, an incident plane wave

for example, as shown in Fig. 2.

B. Integration of the multilayer behavior

Each part of the structure (uncovered and covered) is

composed of multilayered materials with an arbitrary

FIG. 1. Geometrical elements of the damped plate—single rectangular

patch case.

1906 J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011 Loredo et al.: Numerical analysis of patch-damped plates


number of layers. In the simplest case, we consider two mul-

tilayered materials: the first one is named support and con-

sists of layers of the uncovered part; the other is named

damped and includes all the layers. Several damped multi-

layered materials can be considered if several patches of dif-

ferent nature have to be simulated. The layers of the supportmultilayered material are common with the base layers of

the damped multilayered materials.

The considered formulation is based upon previous

works.11,13 It is a 2D plate theory with the classical five dis-

placement unknowns, but it differs from classical laminate

theories CLT and SDT because the assumed displacement

variation, with respect to z, is piecewise linear. This is the

result of writing continuities of both displacements and shear

stresses at each interface, as shown below. For practical rea-

sons, the displacement field of each layer ‘ 2 ½2 … N� is

linked to the displacement field of the first layer. Thick-

nesses and elevations for an N layer material are presented in

Fig. 3. The displacement field of each layer is written as

follows:

u‘ðx; y; zÞ ¼ w‘xðx; yÞ þ ðR‘ � zÞl‘x;

v‘ðx; y; zÞ ¼ w‘yðx; yÞ þ ðR‘ � zÞl‘y;

w‘ðx; y; zÞ ¼ wðx; yÞ

8>><>>: (1)

with

l‘x ¼@w x; yð Þ@x þ u‘

x x; yð Þ;l‘y ¼

@w x; yð Þ@y þ u‘

y x; yð Þ

(

where w is the transverse displacement, w‘x and w‘

y are the mem-

brane displacements along x and y axis, u‘x and u‘

y are the rota-

tions caused by shear effects around x and y axis for the layer l.The R‘ coordinate connects all the upper layers to the first layer.

The strain field is given by

e‘ij ¼1

2

@u‘i@xjþ@u‘j@xi

!; (2)

where u‘i and xi, respectively, denote u‘, v‘, w‘ and x, y, zwhen subscript i takes values 1,2,3.

In most of the plate theories including this one, the gener-

alized plane stress r‘33 ¼ 0 hypothesis is considered, leading

to plane stress elastic constants lower than the corresponding

3D ones. On the other hand, the assumed displacement field

leads to e‘33 ¼ 0. Even if most of the theories work with these

two hypotheses, they are not compatible. However, in many

cases, the small values of e‘33 compared to other strains permit

to keep this kind of displacement field. In our study, a special

attention must be paid to these hypotheses because the visco-

elastic layer has a modulus much lower than that of the other

layers. Two-dimensional finite element computations over the

thickness (an equivalent beam) have been done. They show

that this hypothesis is valid for our studies for frequencies up

to 1.5 kHz and is still good for frequencies in the (1.5–2 kHz)

range except for a few frequencies. However, for frequencies

higher than 2 kHz, this hypothesis need to be rejected, and a

new model must be proposed.

With the generalized plane stress hypothesis and consid-

ering that orthotropic axes coincide with the natural axes of

the plate, Hooke’s law is as follows:

r‘11

r‘22

r‘23

r‘13

r‘12

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;¼

C‘1111 C‘

1122 0 0 0

C‘1122 C‘

2222 0 0 0

0 0 C‘2323 0 0

0 0 0 C‘1313 0

0 0 0 0 C‘1212

2666666664

3777777775

e‘11

e‘22

2e‘23

2e‘13

2e‘12

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;(3)

with C‘1111 ¼ E‘1=ð1� m‘12m

‘21Þ, C‘

1122 ¼ E‘2m‘12=ð1� m‘12m

‘21Þ,

C‘2222 ¼ E‘2=ð1� m‘12m

‘21Þ, C‘

2323 ¼ G‘23, C‘

1313 ¼ G‘13, and

C‘1212 ¼ G‘

12, and where E‘1 and E‘2 are Young’s modulus in xand y directions, G‘

23 and G‘13 are the transverse shear modu-

lus, G‘12 is the shear modulus in the Oxy plane, and m‘12 and

m‘21 are Poisson’s ratios of the layer l.

FIG. 3. Geometrical parameters of the multilayer structure (represented on

the left side in an undeformed state) and displacements of the first layer (rep-

resented on the right side after deformation).

FIG. 2. (Color online) Geometrical representation of a baffled plate with a

patch submitted to a plane wave.

J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011 Loredo et al.: Numerical analysis of patch-damped plates 1907


In the case of an isotropic layer, the elasticity matrix

reduces according to E‘ ¼ E‘x ¼ E‘y, m‘ ¼ m‘12 ¼ m‘21, and

G‘ ¼ G‘12 ¼ E‘=2ð1þ m‘Þ). Note that for a damped material

and for periodic solicitations, which will be envisaged fur-

ther in this study [from Eq. (25)], it is convenient to define a

complex Young’s modulus. In this study, viscoelastic mate-

rial behavior is considered linear with respect to strains but

depends on frequency. Then Young’s modulus and the shear

modulus will be taken as follows:

~E‘ðxÞ ¼ E‘ðxÞ 1þ jg‘ðxÞ� �

;~G‘ðxÞ ¼ G‘ðxÞ 1þ jg‘ðxÞ

� ��(4)

where x is the angular frequency in rad/s and g‘ðxÞ is the

loss factor.

As seen before, this model differs from CLT or SDT

because assumptions on the displacement field with respect

of the z coordinate are made with the help of interface condi-

tions. These conditions link the displacement fields of each

layer to the base layer, regardless of the number of layers.

Conditions that must be enforced are

• the continuity of displacements

u‘ðx; y;R‘ þ h‘=2Þ ¼ u‘þ1ðx; y;R‘þ1 � h‘þ1=2Þ;v‘ðx; y;R‘ þ h‘=2Þ ¼ v‘þ1ðx; y;R‘þ1 � h‘þ1=2Þ;w‘ðx; y; zÞ ¼ wðx; yÞ

8><>: (5)

• the continuity of transverse shear stresses

r‘23 ¼ r‘þ123 ;

r‘13 ¼ r‘þ113 :

((6)

Equations (5) and (6) allow to link the displacements

field in the (l þ 1)th layer with the one of the lth layer, which

can be written by means of a transfer matrix following,13,18

fV‘þ1x g ¼ ½T‘x�fV‘

xg with fV‘xg ¼

@w‘

@x

u‘x

w‘x

8<:9=; (7)

and a similar ½T‘y� matrix for the y direction. Recursively, the

(l þ 1)th layer displacement field can be linked to the first

one,

fV‘þ1i g ¼ ½K‘

i �fV1i g with ½K‘

i � ¼ ½T‘i �½T‘�1i �…½T1

i � (8)

with i¼ x or y. Terms of matrices ½T‘i � and ½K‘i � are explicited

in Refs. 13 and 18. Note that the most recent one18 proposes

a generalization which permits to handle off-axis orthotropic

layers.

The 2� (N� 1) matrices ½K‘i � allow to write the global

motion of a multilayered material in terms of the displace-

ments of its first layer. As the support and damped multilay-

ered materials have the same first layer, only the

displacement field of this single equivalent layer, including

final five degrees of freedom, is used to write the Lagrangian

function.

C. Contribution of the boundary conditions

Boundary conditions are prescribed along the sides Cq

(q¼ 1, 2, 3, or 4) by means of springs, as shown in Fig. 4.

Translational stiffnesses per unit length Kqi ðPÞ and rotational

stiffnesses per unit length Cqi ðPÞ are considered at point P on

the side q of the plate, where subscript i denotes the consid-

ered axis. The force and the bending moment induced by the

edge springs on the plate are given by

FðP; tÞ ¼ � Kqx ðPÞwxðP; tÞex � Kq

y ðPÞwy

ðP; tÞey � Kqz ðPÞwðP; tÞez; (9)

MðP; tÞ ¼ CqxðPÞ

@wðP; tÞ@y

þ uyðP; tÞ� �

ðn � exÞ ex

� CqyðPÞ

@wðP; tÞ@x

þ uxðP; tÞ� �

ðn � eyÞ ey; (10)

where ex, ey, and ez are orthonormal basis vectors of the

physical space.

This formulation allows to model the classical cases of

simply supported, clamped, and free boundary conditions

and permits to simulate intermediate (e.g., an arbitrary

choice of Kqi and Cq

i ) and mixed (e.g., C1¼ clamped,

C2¼ simply supported, C3¼ free, and C4¼ intermediate)

boundary conditions. Complex values of stiffnesses also per-

mit to simulate damping at boundaries.

Ideal values of stiffnesses depend on the type of bound-

ary conditions, on the structure geometry, on the material

properties, and on frequency. The choice is not so easy, but

it is always possible to confirm a posteriori if a good choice

has been done, looking at displacements and/or forces in the

springs. The sensitivity on the stiffnesses at the boundaries

has been discussed in Woodcock’s work.18 The author, who

was searching for natural frequencies, found some problems

due to ill-conditioned matrices. It confirms that special atten-

tion must be paid to this point.

D. Contribution of the patch and energy formulation

The equations of motion of the structure are derived

from the Lagrange–d’Alembert variational principle. It is a

generalization of Hamilton’s principle for systems with

external forces that do not derive from a potential. For the

present (undamped) problem, it consists on solving,

FIG. 4. (Color online) Description of the boundary conditions.



dðt2

t1

Lðqi; _qiÞdtþðt2

t1

Qexti dqidt ¼ 0; (11)

where Lðqi; _qiÞ ¼ T � V is the Lagrangian, T and V are,

respectively, the kinetic and deformation energy of the plate,

qi and _qi are the generalized displacements and their time

derivatives, and Qexti are the generalized external forces.

This equation must be solved for any virtual displacement

dqi.

Bringing all the deformation effects back to the base

layer permits to define a surface energy density. The Lagran-

gian term can then be split into three parts corresponding to

the covered and the uncovered parts, and to the contribution

of the springs. Then Eq. (11) becomes

dðt2

t1

ð ðSu

est � es

v

� �dS|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

Lagrangian of the uncovred part

þð ð

Sc

edt � ed

v

� �dS|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

Lagrangian of the covered part

�ð

CebdC|fflfflfflffl{zfflfflfflffl}

energy of springs at boundaries

!dtþ

ðt2

t1

Q exti dqi|fflfflffl{zfflfflffl}

virtual work done by external forces

dt¼ 0; (12)

where Si is the surface with i¼ u for the uncovered part and

i = c for the covered part, C is the contour of the plate, ejt and

ejv are, respectively, the surface density of kinetic energy

and the surface density of deformation energy for the multi-

layered material j, with j¼ s for the support (base layers)

material and j¼ d for the damped (all the layers) material,

and eb is the linear density of potential energy of the springs.

Since the energy deformation of the covered part

includes the contribution of the first layers, and integrals

over rectangular zones are easy to compute by means of ana-

lytical formulas, it is convenient to apply the principle used

by Letourneaux.19 The integral on the uncovered part is

computed by subtracting energy contribution of the common

layers on the covered part to the energy contribution of the

common layers on the entire plate (Su¼ S�Sc). Then the

total energy contribution of the covered part is added, as

shown in Fig. 5. The kinetic energy T of each part of the

structure (uncovered and covered) can be calculated from

the surface density of kinetic energy of corresponding multi-

layered materials,

ejt ¼

1

2

XNj

‘¼1

ðh‘;j=2

�h‘;j=2

q‘;j _u‘;j�� 2dðR‘;j � zÞ (13)

with j _u‘;jj2 ¼ j _u‘;jj2 þ j _v‘;jj2 þ j _w‘;jj2 and q‘;j is the density of

the layer ‘ of the multilayered material j.In the same way, surface density of deformation energy

for material j is required for the calculation of the deforma-

tion energy V. It is given by

ejv ¼

1

2

XNj

‘¼1

ðh‘;j=2

�h‘;j=2

r‘;jik e‘;jik d R‘;j � z� �

: (14)

After substitution of the in-layer displacements according to

formulas (7) and (8), expressions of the surface densities of

kinetic energy and deformation energy of each part only

depends on displacements of the first layer and their first and

second order spatial derivatives. These expressions are pre-

sented in the Appendix.

The contribution to the deformation energy V of the linear

density of potential energy due to elastic springs is given by

eb ¼1

2

Xq

ðCq

�Kq

z ðwÞ2 þ Kq

x ðwxÞ2 þ Kq

y ðwyÞ2

þ Cqx

�@w

@xþ ux

�2

þ Cqy

�@w

@yþ uy

�2dCq (15)

with q from 1 to 4, in order to account the individual contri-

butions of the four edges of the plate.

In this work, only normal applied forces are considered.

The corresponding displacement is the normal displacement

w. Thus the virtual work done by these applied forces can be

formally written as

dWext ¼ðð

S

Fwðx; y; 0; tÞdwðx; y; tÞdS: (16)

Once the approximation basis functions are chosen (see Sec.

II E) these energies [formulas (A1), (A2), (15), and (16)]

will be expressed in function of the displacement field com-

ponents with respect to the basis.

E. Choice of the basis functions and equation ofmotion

The differential equations of motion are obtained solv-

ing Eq. (12) for each generalized displacements variation

dqi, where formally

fdqig ¼ fdw; dwx; dwy; dux; duyg: (17)

This gives the classical Lagrange’s equations of motion with

external forces.

These equations cannot be solved in general. In this study,

a Rayleigh–Ritz method is used to find an approximateFIG. 5. (Color online) Principle of superposition applied to the patched

plate.



solution. The method consists in selecting a basis of trial

functions satisfying the geometric boundary conditions. Each

function (displacement, load, etc.) is approximated by its pro-

jection on the basis. In this study, a trigonometric basis of

functions is adopted. Both sine and cosine functions are con-

sidered, with a maximum order m for the x direction and n for

the y direction. There is no constant function in the basis. This

leads to 2 � m basis functions for the x direction and 2� n ba-

sis functions for the y direction.

The classical Fourier’s basis with sine and cosine func-

tions of periods equal to a, a=2, a=3,… is enriched with sine

and cosine functions of period equal to 2a=1, 2a=3, 2a=5,…

In the case of a plate of length a, this enriched basis, which

does not remain orthonormal, permits to consider functions

of period equal to 2a, 2a=3, 2a=5,… and can give very good

approximations of functions of period 4a, 4a=3, 4a=5, which

is of practical interest for our concern because it permits to

deal with mixed boundary conditions like simply supported

at one end and free at the other end, for example.

So, the displacement w is written as

w x; y; tð Þ ¼X

1�p�m1�q�n

½Apq1 tð Þ sin ppx=að Þ sin qpy=bð Þ

þ Apq2 tð Þ sin ppx=að Þ cos qpy=bð Þ

þ Apq3 tð Þ cos ppx=að Þ sin qpy=bð Þ

þ Apq4 tð Þ cos ppx=að Þ cos qpy=bð Þ�: (18)

Note that this basis is not orthogonal, which may lead to

errors if one applies the classical rules for projection and

handling components. We have to distinguish contravariant

and covariant vector components. The above Apqr ðtÞ com-

ponents are contravariant ones. The metric matrix built

with scalar products like $$sin(mpx=a)cos(npx=a)sin(ppy=b)

�sin(qpy=b)dxdy and its inverse can be used to change con-

travariant components into covariant ones and vice versa.

The present model is built in a manner that it handles these

contravariant components which are the unknowns. It per-

mits to regenerate directly any output function by means of

formulas like (18), but carefully attention must be payed

when extra computations are made.

The Apqr ðtÞ can be stacked up with appropriate order into

a 4�m� n vector called {A(t)} and the corresponding base

functions stacked up into a 4�m� n vector called {/(x,y)}.

Hence Eq. (18) is now written as

wðx; y; tÞ ¼ fAðtÞgTf/ðx; yÞg: (19)

In the same way, using the same basis, we can define similar

expressions for the other displacement variables,

wxðx; y; tÞ ¼ fBðtÞgTf/ðx; yÞg;

wyðx; y; tÞ ¼ fCðtÞgTf/ðx; yÞg;

uxðx; y; tÞ ¼ fDðtÞgTf/ðx; yÞg;uyðx; y; tÞ ¼ fEðtÞgTf/ðx; yÞg:

8>>>><>>>>: (20)

These vectors can be stacked up to build a global 5 � 4 � m� n vector {X(t)} of unknowns,

fXðtÞg ¼

AðtÞBðtÞCðtÞDðtÞEðtÞ

8>>>><>>>>:

9>>>>=>>>>; ¼A11

1 ðtÞ...

..

.

Emn4 ðtÞ

8>>>><>>>>:

9>>>>=>>>>;: (21)

The use of polynomial functions is classically encountered in

the literature, but trigonometric functions present some advan-

tages. This trigonometric set can be easily implemented on a

computer and it has been shown20 that the trigonometric set

presents a better convergence rate than polynomials when

predicting high order natural flexural modes of rectangular

plates with any boundary conditions.

Introducing these approximated fields in the different

energy terms [formulas (A1) and (A2)] for each part, in the

boundary potential energy term [Eq. (15)], and in the exter-

nal forces work [Eq. (16)] permits to express Eq. (12) in

terms of the components and their time derivatives. Omitting

the time variable in {X} for clarity, this leads to

dðt2

t1

LðfXg; f _XgÞdtþðt2

t1

fFgTfdXgdt ¼ 0; (22)

where {F} is the generalized discrete external forces vector

(see Sec. II F). In this equation, damping is not taken into

account because of the lack of time explicit dependence of ma-

terial characteristics. It is however possible to do it if dissipa-

tive forces derivate from a Rayleigh’s potential as it is

explained in Ref. 21. Indeed, the damping characteristics are

known as function of the frequency for periodic solicitations.

Hence, a straightforward way to take them into account is to,

first, build the time equation without damping, second, derive

the corresponding harmonic equation, and, finally, consider

complex material properties [formulas (4)] instead of real ones.

Solving Eq. (22) leads to Lagrange’s equations for the

components of the generalized displacements Xj:

d

dt

@T

@ _Xj

� @V

@Xj¼ Fj: (23)

This leads to the d � d linear equation system of forced

motion in a matrix form:

½M�fX::

g þ ½K�fXg ¼ fFg (24)

with d¼ 20 � m � n and where [M] is the mass matrix, [K]

the real stiffness matrix, and {F} the vector of external forces.

For periodic solicitations of given angular frequency x,

Eq. (24) becomes the well known harmonic system. As men-

tioned above, introduction of damping properties by means

of complex material properties leads finally to

ð�x2½M� þ ½~K�Þf~Xg ¼ f~Fg: (25)

We use tilde to denote complex variables throughout this

document. This system can be solved by a standard linear sys-

tem solving algorithm.

F. External forces vector

The generalized force f~Fg vector can be correlated by

the derivation scheme to the displacement variables writing



f~Fg ¼

~Fw

~Nx

~Ny

~Mx

~My

8>>>><>>>>:

9>>>>=>>>>;; (26)

where f~Fwg is the normal loading, f~Nxg and f~Nyg are gener-

alized in-plane tensions and f ~Mxg and f ~Mxg are generalized

bending moments. As we only consider transverse loading of

the plate [see Eq. (16) in Sec. II E], these last four blocks are

null. Then the generalized force vector reduces to

f~Fg ¼

~Fw

0

0

0

0

8>>>><>>>>:

9>>>>=>>>>;: (27)

1. Punctual excitation

Suppose the plate is excited by a point force ~f at point

(x0, y0). A complex value of ~f may be useful to set a phase.

We recall here that time has been changed into frequency by

the time Fourier transform. Using Eqs. (23) and (16), the

generalized force f~Fwg is found to be

f~Fwg¼ ~f

ða

0

ðb

0

dðx�x0Þdðy� y0Þf/ðx;yÞgdS (28)

which gives

f~Fwg ¼ ~ff/ðx0; y0Þg (29)

2. Plane wave excitation

The plate separates two semi-infinite fluid media as

shown in Fig. 2. In the case of an incident plane wave, the

forces acting on the plate are due to a difference of pressure.

With the assumption of a light fluid for the two sides of the

plate, we can neglect the pressure caused by the radiation

impedances of the fluid mediums. Also, the radiated pressure

in the exciting fluid medium is in general ignored because it

is small compared to the blocked pressure (sum of the inci-

dent pressure and reflected one when the plate is blocked).

With these assumptions, the incident plane wave gener-

ates the incident pressure field ~Pinc,

~Pincðx; yÞ ¼ 2~pince�jðk1xðx�a=2ÞÞ � e�jðk1yðy�b=2ÞÞ; (30)

where ~pinc is the amplitude of the incident plane wave that is

considered as a complex number to set a phase if necessary,

k1x¼ k1sin(h)cos(u), k1y¼ k1sin(h)sin(u) are the projections

of the wave vector k1, k1¼x=c1 is the wave number, and c1

is the sound celerity into the exciting (1) fluid.

Using Eqs. (23) and (16), the generalized force f~Fwg is

found to be the projection of ~Pincðx; yÞ on the basis {/(x, y)}

f~Fwg ¼ða

0

ðb

0

~Pincðx; yÞf/ðx; yÞgdS (31)

which may also be written as

f~Fwg ¼ 2~pincf~wðk1Þg: (32)

Building the vector f~wðk1Þg requires to compute the follow-

ing integrals:

Ispxðk1xÞ ¼

ða

0

e�jðk1xðx�a=2ÞÞ sinðppx=aÞdx;

Icpxðk1xÞ ¼

ða

0

e�jðk1xðx�a=2ÞÞ cosðppx=aÞdx (33)

and similar terms Isqyðk1yÞ and Icq

yðk1yÞ for the y direction.

This is done using following formulas:

for p > 0;

Ispxðk1xÞ ¼

ppa½eðjk1xa=2Þ � ð�1Þpeð�jk1xa=2Þ�ðppÞ2 � ðk1xaÞ2

;

Icpxðk1xÞ ¼

jk1xa2½eðjk1xa=2Þ � ð�1Þpeð�jk1xa=2Þ�ðppÞ2 � ðk1xaÞ2

;

for p ¼ 0;

Ispxðk1xÞ ¼ 0; Icp

xðk1xÞ ¼2 sinðk1xa=2Þ

k1x:

(34)

Replacing p by q, a by b and k1x by k1y in these formulas

leads to the required terms for the y direction.

3. Diffuse field excitation

A sound field in which the time average of the meansquare sound pressure is everywhere the same and the flowof acoustic energy in all directions is equally probabledefines an acoustic diffuse field according to the Institute of

Noise Control Engineering (INCE-USA). Mathematically,

an acoustic diffuse field of frequency x is described by a

superposition of (many) plane waves of random (equiprob-

able) propagation direction. For numerical considerations, it

can be taken into account in two ways.

As the real resultant pressure field is chaotic, it is possi-

ble to build a spatially random time-periodic pressure field

which represents the physical reality, sometimes called theincident diffuse field. This method can be regarded as ran-

dom excitation models.

A more classical way consists in doing the mechanical

computation for a set of plane waves of different incidences

and summing energy-types outputs (typically, the mean square

velocity). As this computation is non linear, it needs as many

computations than the number of considered plane waves. The

directional discretization of the half space is then of crucial im-

portance to give a good (precision/computational time) ratio.

Random models lead to an explicit force vector f~Fwgwhose construction is not detailed here. For the discretized

method, the system is solved for each k1 plane wave whose

vector f~Fwg is given by formula (32).

G. Sound power radiation and vibroacoustic indicators

In the case of a light fluid, the sound power can be cal-

culated from the far-field sound pressure distribution, since



the computation of the pressure distribution at the surface of

the plate is not needed. This approach is also easier to imple-

ment. With these assumptions, Rayleigh’s integral simpli-

fies, giving at point (r, h, u) the acoustic pressure,

~P2ðr; h;uÞ ¼�x2q2ejk2r

2prbwðk2x; k2yÞ; (35)

where k2x¼ k2sin(h)cos(u), k2y¼ k2sin(h)sin(u) are the pro-

jections of the wave vector k2, k2¼x=c2 is the wave num-

ber, c2 is the sound celerity into the receiving (2) fluid and

q2 is the density of this fluid and bwðk2x; k2yÞ is the double

Fourier transform of the plate’s displacement,

wðk2x; k2yÞ ¼ða

0

ðb

0

~wðx; yÞe�jðk2xðx�a=2ÞÞe�jðk2yðy�b=2ÞÞdydx: (36)

As the displacement field ~wðx; yÞ is given by formula (18),

terms similar than those of Eq. (33) appear in the above inte-

gral, so their analytical expressions may be reused. Thus, thebwðk2x; k2yÞ term can be directly computed with the formula,

wðk2x; k2yÞ ¼ f~AgTf~wðk2Þg: (37)

According to the far-field hypothesis, the radiated

acoustic power is obtained from the integration of the radial

intensity over a hemisphere of infinite radius and leads to the

equation,

Wt ¼q2x

4

8c2p2

ð2p

0

ðp=2

0

jwðk2x; k2yÞj2 sinðhÞdhdu: (38)

In the case of an excitation by a plane wave ph,u, the

resulting normal displacement components f~Ag are trans-

formed by means of Eqs. (37) and (38) to give the radiated

acoustic power Wt(ph,u) for this (h, u) incident plane wave.

This permits to define an oblique incidence transmission

coefficient s(h, u) as

sðh;uÞ ¼ WtðPh;uÞWincðh;uÞ

; (39)

where Winc(h, u) is the incident power of this plane wave,

given by

Wincðh;uÞ ¼~pincj j2cosðhÞS

2q1c1

: (40)

In order to characterize the acoustic transparency of the

plate, the classical transmission loss (TL ) is calculated by

means of the following formula:22

TL ¼ 10 log1

sðh;uÞ

� �¼ 10 log

Wincðh;uÞWtðPh;uÞ

� �: (41)

For a good understanding of the physical phenomena,

the other indicators used in the present work are the mean

square velocity hV2i and the radiation efficiency r. The

mean square velocity is defined as a space and time average

of the structure velocity,

V2 �

¼ 1

S

ða

0

ðb

0

1

2

d ~wðx; yÞdt

�� 2dydx (42)

thus can be developed, using Eq. (18), as

V2 �

¼x2

2Sf~AgT

ða

0

ðb

0

f/ðx; yÞgf/ðx; yÞg�Tdydx

� f~Ag�; (43)

where * denotes the complex conjugate operation.

The radiation efficiency is defined as a non-dimensional

ratio of the radiated power to the mean square velocity of the

structure,

r ¼ Wt

qcS V2h i : (44)

For a diffuse field, as seen in Sec. II F 3, the computa-

tion is made with waves of same amplitudes and incidence

angle (hi, uj) [ [0, p=2] � [0, 2p]. Suppose the half space is

divided into nh parts for the h angle and nu parts for the uangle. The indicators K(Phi,ui) (where K stands for: the

mean square velocity hV2i, the incident power Winc and the

radiated power Wt) corresponding to unit plane wave excita-

tions of incidence (hi, uj) can be summed up to obtain the

equivalent indicator Kd for the diffuse field excitation,

Kd ¼Xnh

i¼1

Xnu

j¼1

KðPhi;uiÞDXðhi;uiÞ; (45)

where DX(hi, ui)¼ cos(hi)DhDu is the solid angle corre-

sponding to the plane wave excitation of incidence angle

(hi, ui) (Dh¼ p=2r and Du¼ 2p=s). We recall that in this

case, the linear system [Eq. (25)] must be solved for each

orientation (hi, ui).

A limit angle of incidence hlim instead of p=2 is often

introduced. It permits not to take grazing-waves into account

which is of practical interest when comparisons with experi-

ments are done. Indeed, this limit angle results of geometri-

cal consideration of experimental devices in particular the

window in reverberation chambers.

A total of 400 plane waves have been used to simulate

the diffuse field. It has been verified that taking more plane

waves does not affect significantly the diffuse field response

computed from Eq. (45) in 10 Hz to 2000 Hz range.

III. NUMERICAL VALIDATION AND RESULTS

This section presents several studies for which the pro-

posed approach has been implemented. First, the validity of

the proposed model is tested by comparison with different

existing models in the literature. As an assumption of negli-

gible fluid load has been used in this work, its validity has

been verified by comparison with the model considered in

the work of Foin et al.,23 which takes into account the



radiation impedances of the fluids (this model was initially

proposed by Sandmann24 and Nelisse25). The ability of the

model to manage multilayer behavior is tested by compari-

son with experimental results in Ref. 23 for a mechanically

excited fully patched plate. The viscoelastic material used by

these authors is considered again in our last study. To vali-

date the vibroacoustic behavior of the plate, a comparison

with Woodcock’s model16 is made in terms of the vibroa-

coustic indicators presented in Sec. II G. Finally, a paramet-

ric study is done, which consists in varying first the cover

ratio, and then the patch surface repartition.

In these studies, orders m and n of the basis were

adapted to the minimal possible wavelength, which is differ-

ent for each study. A maximal 13 � 13 order had been used.

Convergence can be tested, on a small band (100 kHz) of the

higher considered frequencies, for growing values of the

orders m and n.

A. Validation of the model

1. Fluid load relative importance

In this study, the fluid load was considered negligible.

To evaluate the importance of the fluid load, a comparison

with Foin et al.23 was made. The author has computed the

radiation impedances of the exciting and receiving fluid

mediums, without approximation. A simple supported plate

with dimensions a¼ 0.48 m, b¼ 0.42 m and thickness

h = 3.22 mm is studied. The plate has the following proper-

ties: Young’s modulus E¼ 6.6 � 1010 Pa, density qs¼ 2680

kg/m3, Poisson’s ratio m¼ 0.33, and loss factor gs¼ 0.005.

The excitation is a point force applied at x¼ 0.08 m,

y¼ 0.07 m from a corner of the plate. For the present study,

a total of 13 functions in each direction (u and v) have been

used to expand the plate deflection. Figure 6 shows the radia-

tion efficiency for a plate immersed in air computed with our

model. This curve is very similar to the curve of the article23

in which the mass of the fluid is taken into account. This

computation shows that, for our study, the fluid-structure

coupling is weak. It is also neglected in the following studies

even if the thickness and the density of the considered plates

are different, which may have an influence.

2. Multilayer behavior

In order to validate the ability of the model to manage

multilayer behavior, a comparison with experimental results

obtained by Foin et al.23 on a plate with and without a patch

is done. The plate, boundary conditions, and excitation are

the same as those of Sec. III A 1. For the damped configura-

tion, the plate is completely covered by a viscoelastic mate-

rial ISD 112 manufactured by 3M and a constraining thick

layer of 0.5 mm that has the same properties as the base

plate. The viscoelastic material ISD 112 is 0.25 mm thick

with a density of 1015 kg/m3 and Poisson’s ratio of 0.3.

Table I presents the frequency dependence of the visco-

elastic material properties for a given temperature.

In the experimental study detailed in Ref. 23, the mean

square velocity of a plate covered or not by a patch (ISD 112

þ constraining layer) and immersed in air is measured. The

comparison of the proposed model (see Fig. 7) with experi-

ments of Ref. 23 present excellent agreement. The damped

case is slightly different but the agreement is still very good.

3. Vibroacoustic indicators

In order to validate the results in terms of vibroacoustic

indicators, a comparison with Woodcock’s model16 is made.

Woodcock’s study uses a Love–Kirchoff model with polyno-

mial basis functions, instead of trigonometric basis functions

for the present model. The Love–Kirchoff plate model is

known to be accurate for thin homogeneous plates, which

FIG. 6. (Color online) Radiation efficiency of the proposed model com-

pared with the results of article (Ref. 23) in which the radiation impedances

of the fluids are taken into account.

TABLE I. Frequency dependence of the mechanical properties of the visco-

elastic material ISD 112 ( T¼2rC).

Frequency (Hz) Young’s modulus (Pa) Loss factor

10 7.28 � 105 0.90

100 2.34 � 106 1.00

500 5.20 � 106 1.00

1000 7.28 � 106 0.90

2000 9.88 � 106 0.80

3000 1.17 � 107 0.75

4000 1.38 � 107 0.70

FIG. 7. (Color online) Mean square velocity of an aluminum and a three-

layered (aluminum–ISD 112–aluminum) simply supported plates. The exci-

tation is a point force applied at x = 0.08 m, y = 0.07 m. These curves show

good agreement with both model and experimental results of the work of

Foin et al. (Ref. 23).



correspond to the case studied here. Hence, the presented

model is compared to Woodcock’s model for a plate excited

by acoustic plane wave of any incidence. All the results pre-

sented hereafter concern a rectangular steel plate having

the following physical constants: a¼ 0.455 m, b¼ 0.376 m,

h = 1 mm, E¼ 2.1 � 1011 Pa, qs¼ 7800 kg/m3, m¼ 0.3, and

gs¼ 0.01.

Figure 8(a) presents the mean square velocity hV2i and

Fig. 8(b) presents the TL results for two angles of incidence

h¼ 85� and h¼ 0�. They show that a very good agreement is

obtained with Woodcock’s work.16

B. Parametric study

In this section, the results of a parametric study per-

formed on a plate with different distribution of patches are

presented. All the following computations are done for a

simply supported plate, immersed in air and receiving an

acoustic excitation by means of a h¼ 85� incidence plane

wave. The characteristics of the base steel plate are taken

identical than the Woodcock’s plate considered previously.

The constraining layer is also considered to be made with

the same steel, and its thickness is set to 0.5 mm. The visco-

elastic layer material is the ISD 112 whose properties are

presented in Table I. The thickness is set to 0.25 mm.

The first damped configuration considered is a full cov-

ered plate. In this case, the patch increases the mass of the

plate by 53%. All the other configurations cover 40% of the

plate surface and add 21% of mass.

An endless number of surfacic distributions of patches

which cover 40% of the plate surface could be defined. One

way is to distribute patches by means of strips along the

sides of the plate, taking into account the ratio a=b to define

the dimensions of the patch(es). In this study, four distribu-

tions of patches have been choosen to cover 40% of the plate

surface. One is a patch of dimensions 288 mm � 238 mm

centered on the plate surface and the others are strip-type

distributions, respectively: 1 � 1 strips, 2 � 2 strips, and

3 � 3 strips. These distributions are presented in Fig. 9 and

the geometric parameters, which localize the patches, are

defined in Table II.

Analysis is made in two times. The comparison of the

first three configurations of Table III points out the effect of

the patch covering ratio, while the comparison of the last

four configurations of Table III helps us to understand the

patch shape influence.

Comparison of the first three cases is made in terms of

global vibration and acoustic levels (Table III), mean square

velocity [Fig. 10(a)], TL [Fig. 10(b)], and radiation effi-

ciency [Fig. 10(c)]. In Fig. 10(a), it can be seen that the plate

is much damped than the 40% covered one. The vibration

and acoustic gains due to the 100% covering are, respec-

tively, of 15.3 dB and 9.7 dB compared to the plate alone

(see Table III). Considering the 40% covering, the vibration

and acoustic gains are, respectively, of 11.2 dB and 7.6 dB

FIG. 8. (Color online) Vibroacoustic indicators for a steel plate for plane

wave excitations of incidence h = 85� and h = 0� compared with Wood-

cock’s model (Ref. 16): (a) Mean square velocity and (b) TL.

FIG. 9. (Color online) Considered distributions of patches. They all cover

40% of the plate surface. The first one is centered and the others are strip-

type distributions: 1 � 1 strips, 2 � 2 strips, and 3 � 3 strips.

TABLE II. Geometric parameters of the patches’ distributions.

ap (mm) bp (mm) Dx (mm) Dy (mm)

1 � 1 strips 102.6 84.8 176.2 145.6

2 � 2 strips 51.3 42.4 117.5 97.1

3 � 3 strips 34.2 28.3 88.1 72.8

TABLE III. Global vibration and acoustic levels of the different

configurations.

LV (dB)a Lwt (dB)b

Base plate alone �27.71 89.63

Patch 100% �43.01 79.89

Patch 40% �38.87 82.07

Patches 1 � 1 strips 40% �39.15 82.40

Patches 2 � 2 strips 40% �38.18 82.52

Patches 3 � 3 strips 40% �38.01 82.56

aLV : Global vibration level (dB, ref. 1 m2s�2).bLWt: Global acoustic level (dB, ref. 10�12 W).



compared to the plate alone with an added mass of 21%

instead of 53% for the full covered plate.

Figure 10(c) presents the radiation efficiency, which

expresses the portion of the vibration energy transformed into

sound, for the three above configurations. One can keep in

mind that the radiation efficiency is defined relatively to a unit

of mechanical power, then, it is not illogical to find the more

damped case having the best radiation efficiency. Indeed,

damping strongly reduces energy levels of modes, which are

precisely characterized by a high level of mechanical power

and a more or less poor sound radiation level caused by inter-

ferences between the different parts of the modal shape.

Hence, for a unit of mechanical power, radiated acoustic

power is generally higher in the damped case. Results of the

same kind are usually encountered in the literature.

For all distributions of patches covering 40% of the plate

surface, the vibration and acoustic gains compared to the plate

alone are roughly the same. However, Figs. 11(a) and 11(b)

show that for some frequency ranges, differences between the

considered patch(es) distributions can be noticed. For the TL

indicator, the distribution 1 � 1 strips is less efficient than the

other distributions in the frequency range 170–420 Hz [see

Fig. 11(b)]. On the contrary, the distribution 3 � 3 strips is

more efficient than the other distributions in the frequency

range 570–830 Hz. This phenomena occurs because the surfa-

cic patches’ distributions allow different modes to appear,

with bending wavelength less or more compatible with the

dimensions Dx and Dy of the naked areas.

Indeed, if we observe the spatial distribution of the

square velocity at a given frequency for each configuration,

low values of TL—high values of sound power radiation—

are associated with two kinds of deformations: either defor-

mation of the whole damped structure or deformation in the

localized naked areas (Dx � Dy).

To illustrate these phenomena, Fig. 12(a) presents the

spatial distribution of the square velocity at a frequency of

170 Hz for the 1 � 1 strips distribution. At this frequency,

FIG. 10. (Color online) Vibroacoustic indicators for three configurations:

the base plate alone, the plate with a 100% covering patch (ISD 112 þsteel), and the plate with a centered 40% covering patch (ISD 112 þ steel):

(a) Mean square velocity, (b) TL, and (c) radiation efficiency.

FIG. 11. (Color online) Vibroacoustic indicators of plates with different

patches’ (ISD 112 þ steel) distributions covering 40% of the plate surface: a

centered patch, 1 � 1 strips, 2 � 2 strips, and 3 � 3 strips: (a) Mean square

velocity and (b) TL.



the structure radiates much more than other strip-type distri-

butions, leading to a trough on the TL curve [see Fig. 11(b)].

Clearly, the structure is bending over its entire surface with

high values of velocity. This shape corresponds to the (1, 3)

mode of the base plate.

Figure 12(d) shows that a very similar behavior is

observed for the 2 � 2 strip configuration around the 175 Hz

frequency. It seems that neither of these two configurations

are able to trap the (1, 3) mode of the plate.

The 1 � 1 strips configuration is still the most radiant at

258 Hz, but this results, this time, from strong transverse dis-

placements of the naked areas [see Fig. 12(b)]. These areas

seem to deform like a plate in its first bending mode.

An idea would be to correlate the frequency of 258 Hz

with the frequency of the first mode of a tiny plate of dimen-

sions Dx and Dy with boundary conditions that must be adjusted.

As the real boundary conditions of each naked area (that are dif-

ferent for each considered case!) are difficult to formalize, let us

consider the simply supported case and the clamped case. Cal-

culations for both cases are made to find the first bending mode

for a plate of dimensions Dx � Dy. Different values of Dx and

Dy are considered corresponding to each strip-type distributions:

1� 1, 2� 2, and 3� 3. They are summarized in Table IV.

The frequency of 258 Hz, corresponding to a trough on

the TL curve, fits in the frequency range 195.7–360.1 Hz of

the first bending mode of a simple supported and clamped

plate (see Table IV). Figures 12(e) and 12(f) show the same

phenomenon for the 2 � 2 and 3 � 3 strips configurations,

respectively. The corresponding frequencies of 600 and

1004 Hz also fits in correspondant frequency ranges of Table

IV. It shows that the dimensions of the naked areas condition

the establishment of bending waves which may result in

high vibration level and high acoustic radiation.

On the contrary, the 2 � 2 strips configuration shows a

high damping effect at the frequency of 258 Hz with maximum

velocities approximately seven times lower than for the 1 � 1

case as can be seen in Fig. 12(c). In this case and for this fre-

quency, flexural deformations happen in the patch’ zone which

seems to mean that the patch is doing a high dissipative work.

FIG. 12. (Color online) Mean square

velocity of plates with three different

patches’ distributions and for some

particular frequencies: (a) 1 � 1

strips at 170 Hz, (b) 1 � 1 strips at

258 Hz, (c) 2 � 2 strips at 258 Hz,

(d) 2 � 2 strips at 177 Hz, (e) 2 � 2

strips at 600 Hz, and (f) 3 � 3 strips

at 1004 Hz. The same arbitrary unit

is used in all figures.

TABLE IV. Natural frequency (Hz) of the mode shape 1 of a homogeneous

plate of dimensions D x � D y with two kinds of boundary conditions: simply

supported or clamped.

Fundamental freq. (Hz)

Dx (mm) Dy (mm) Simply supp. Clamped

1 � 1 strips 176.2 145.6 195.7 360.1

2 � 2 strips 117.5 97.1 438.3 809.5

3 � 3 strips 88.1 72.8 779.2 1435



This parametric study shows that a detailed analysis is

necessary to understand the noise reduction impact of

patches. Naked areas permit strong vibrations to appear at

frequencies corresponding to their own fundamental mode

(or harmonics), but this does not necessarily imply low val-

ues of TL because spatial and time coincidence are, in this

case, also dependent of the dimensions and of the positions

of these naked areas.

IV. CONCLUSION

The main objective of this paper was to propose a model

that describes the vibroacoustic behavior of a baffled plate

with PCLD patches. A theoretical model able to manage a

damped multilayered plate with patches, taking into account

bending, membrane, and shear motions has been developed.

The boundary conditions are modeled as a continuous distri-

bution of springs acting against both deflection and rotation of

the plate’s contour. The fluid loading of the plate is neglected

in this study. The formulation is based on Rayleigh–Ritz’s

method and the Lagrange–d’Alembert principle is used to

derive the equations of motion. The far-field approach has

been used to calculate the radiated sound power. The pro-

posed model can be applied to either fully or partially cov-

ered multilayered plate. This model can manage the

integration of multiple PCLD patches, different in size and

constitution and localized anywhere on the plate.

Simulations of basic problems found in previous works

(naked plate, fully patched plate) were tested with our model,

and they were found to be in very good agreement. Other con-

figurations of practical interest (partially covered plate) have

been studied and compared to the fully covered and the naked

plate. In a second study, four different patches’ distributions

with the same covering rate of 40% were compared. It has

been shown that the TL of the plate could be increased in spe-

cific frequency bands by the choice of specific distributions.

All the results show that a structure can be efficiently

damped by adding appropriate patch(es). However, the diffi-

culty to have the better damping consists first to choose the

appropriate viscoelastic treatment and second to determine the

better distribution while keeping the best ratio performance/

added mass. It has been shown in particular that noise reduc-

tion was sensible to the patch distribution and that this sensitiv-

ity was difficult to foresee without doing a detailed analysis.

Hence, further studies to develop methodologies to opti-

mize the TL in a given frequency range are necessary. Given

all fixed parameters (such as geometric and material parame-

ters of the structure, some of the material parameters of the

patches, the maximum tolerated added mass and an objective

in terms of frequency range) these methodologies will have

to give an optimal solution in terms of patch distribution.

Particular structures like stiffened plates or stiffened shells

must also be considered for this optimization process.

ACKNOWLEDGMENT

This work was supported by AIRBUS Operations SAS,

the Burgundy Region, and the European Social Fund.

APPENDIX: KINETIC AND DEFORMATIONENERGIES

The surface density of kinetic energy ejt of the multilay-

ered material j is given by

ejt ¼�

x2

2dj

1

@w

@x

� �2

þdj2 u1

x

� �2þdj3 w1

x

� �2þ2dj4

@w

@xu1

x þ 2dj5

@w

@xw1

x þ 2dj6u

1xw

1x

"

þ dj7

@w

@y

� �2

þdj8 u1

y

� 2

þdj9 w1

y

� 2

þ2dj10

@w

@yu1

y þ 2dj11

@w

@yw1

y þ 2dj12u

1yw

1y þ dj

13 wð Þ2# (A1)

and the surface density of deformation energy ejv of the multilayered material j is given by

ejv ¼

1

2kj

1

@2w

@x2

� �2

þkj2

@u1x

@x

� �2

þkj3

@w1x

@x

� �2

þ2kj4

@2w

@x2

@u1x

@x

"þ 2kj

5

@2w

@x2

@w1x

@xþ 2kj

6

@u1x

@x

@w1x

@xþ kj

7

@2w

@y2

� �2

þkj8

@u1y

@y

!2

þ kj9

@w1y

@y

!2

þ2kj10

@2w

@y2

@u1y

@yþ 2kj

11

@2w

@y2

@w1y

@yþ 2kj

12

@u1y

@y

@w1y

@yþ 2kj

13

@2w

@x2

@2w

@y2þ 2kj

14

@2w

@x2

@u1y

@yþ 2kj

15

@2w

@x2

@w1y

@y

þ 2kj16

@2w

@y2

@u1x

@xþ 2kj

17

@u1x

@x

@u1y

@yþ 2kj

18

@u1x

@x

@w1y

@yþ 2kj

19

@2w

@y2

@w1x

@xþ 2kj

20

@w1x

@x

@u1y

@yþ 2kj

21

@w1x

@x

@w1y

@yþ kj

22

@2w

@x@y

� �2

þ kj23

@u1x

@y

� �2

þkj24

@u1y

@x

!2

þkj25

@w1x

@y

� �2

þkj26

@w1y

@x

!2

þ2kj27

@2w

@x@y

@u1x

@yþ 2kj

28

@2w

@x@y

@u1y

@xþ 2kj

29

@2w

@x@y

@w1x

@y

þ 2kj30

@2w

@x@y

@w1y

@xþ 2kj

31

@u1x

@y

@u1y

@xþ 2kj

32

@u1x

@y

@w1x

@yþ 2kj

33

@u1x

@y

@w1y

@xþ 2kj

34

@w1x

@y

@u1y

@xþ 2kj

35

@w1y

@x

@u1y

@xþ 2kj

36

@w1x

@y

@w1y

@x

þ kj37 u1

x

� �2þkj38 u1

y

� 2: (A2)



The terms dji (from i¼ 1–13 ) and kj

i (from i¼ 1–38) are

related to the 2N� 2 transfer matrices K‘x and K‘

y which link

the (l þ 1)th layer at the first layer of each multilayer mate-

rial j. They include information (material properties and

thicknesses) of each layer. The expressions of these coeffi-

cients are developed in Ref. 18.

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numerical vibroacoustic analysis of plates with constrained-layer damping patches

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