damping analysis of laminated plates and beams using ritz method
TRANSCRIPT
Composite Structures 74 (2006) 186–201
www.elsevier.com/locate/compstruct
Damping analysis of laminated beams and platesusing the Ritz method
Jean-Marie Berthelot *
ISMANS, Institute for Advanced Materials and Mechanics, 44 Avenue Batholdi, 72000 Le Mans, France
Available online 14 June 2005
Abstract
The paper presents an analysis of the damping of unidirectional fibre composites and different laminates. Damping characteristics
of laminates are analysed experimentally using cantilever beam test specimens and an impulse technique. Damping modelling of
unidirectional composites and laminates is developed using the Ritz method for describing the flexural vibrations of beams or plates.
The influence of the beam width as well as the influence of the vibration frequency are considered. Next, the paper presents an eval-
uation of the damping of laminate plates with two different boundary conditions.
� 2005 Elsevier Ltd. All rights reserved.
Keywords: Laminate composites; Damping; Beam vibrations; Plate vibrations
1. Introduction
Damping in composite materials is an important fea-
ture of the dynamic behaviour of structures, controlling
the resonant and near-resonant vibrations and thus pro-
longing the service life of structures under fatigue load-
ing and impact. Composite materials generally have ahigher damping capacity than metals. At the constituent
level, the energy dissipation in fibre-reinforced com-
posites is induced by different processes such as the
viscoelastic behaviour of matrix, the damping at the
fibre–matrix interface, the damping due to damage,
etc. At the laminate level, damping is depending on
the constituent layer properties as well as layer orienta-
tions, interlaminar effects, stacking sequence, etc.The initial works on the damping analysis of fibre
composite materials were reviewed extensively in review
papers by Gibson and Plunket [1] and by Gibson and
Wilson [2]. Viscoelastic materials combine the capacity
0263-8223/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruct.2005.04.031
* Tel.: +33 243 2140 00; fax: +33 243 2140 39.
E-mail address: [email protected]
of an elastic type material to store energy with the
capacity to dissipate energy. The most general treatment
has been given by Gross [3] considering the various
forms that viscoelastic stress–strain relations can take.
A form of the viscoelastic stress–strain relations is that
involving the complex moduli, where the stress field is
related to the strain field introducing the complex stiff-ness matrix. Thus, the static elastic solutions can be con-
verted to steady state harmonic viscoelastic solutions
simply by replacing elastic moduli by the corresponding
complex viscoelastic moduli. The elastic–viscoelastic
correspondence principle was developed by Hashin
[4,5] in the case of composite materials. Furthermore,
Sun et al. [6] and Crane and Gillespie [7] applied the cor-
respondence principle to the laminate relations derivedfrom the classical laminate theory. Following this pro-
cess, the effective bending modulus of a laminate beam
can be derived [8], leading to the experimental evalua-
tion of laminate damping. This complex modulus was
also considered by Yim [9]. Indeed, the experimental
analysis implemented in the case of unidirectional glass
and Kevlar fibre composites [10] shows that the complex
stiffness model leads to a rather worse description of the
J.-M. Berthelot / Composite Structures 74 (2006) 186–201 187
experimental results derived for the damping as a func-
tion of fibre orientation.
A damping process has been developed initially by
Adams and Bacon [8] in which the energy dissipation
can be described as separable energy dissipations associ-
ated to the individual stress components. This analysiswas refined in later paper of Ni and Adams [11]. The
damping of orthotropic beams is considered as function
of material orientation and the papers also consider
cross-ply laminates and angle-ply laminates, as well as
more general types of symmetric laminates. The damping
concept of Adams and Bacon was also applied by Adams
and Maheri [12] to the investigation of angle-ply lami-
nates made of unidirectional glass fibre or carbon layers.The finite element analysis has been used by Lin et al. [13]
and by Maheri and Adams [14] to evaluate the damping
properties of free–free fibre-reinforced plates. These
analyses were extended to a total of five damping param-
eters, including the two transverse shear damping param-
eters. More recently the analysis of Adams and Bacon
was applied by Yim [9] and Yim and Jang [15] to different
types of laminates, then extended by Yim and Gillespie[16] including the transverse shear effect in the case of
0� and 90� unidirectional laminates.
For thin laminate structures the transverse shear ef-
fects can be neglected and the structure behaviour can
be analysed using the classical laminate theory. The nat-
ural frequencies and mode shapes of rectangular plates
are well described using the Ritz method introduced
by Young [17] in the case of homogeneous plates. TheRitz method was applied by Berthelot and Sefrani [10]
to describe the damping properties of unidirectional
plates. The results obtained show that this analysis eval-
uates fairly well the damping properties of unidirec-
tional materials. The analysis of damping properties of
rectangular unidirectional plates is developed in the
present paper and next extended to laminated plates.
The results derived from these analyses will be first ap-plied to the evaluation of damping parameters of mate-
rials from the flexural vibrations of beam specimens and
compared to the experimental results. Next damping of
different laminates will be considered.
2. Theory
2.1. Evaluation of the damping of a rectangular
orthotropic plate
In the models of Adams–Bacon [8] and Ni–Adams
[11] for the evaluation of the beam damping, the influ-
ence of the width of beams is not considered. This influ-
ence can be analyzed by the finite element analysis, or by
the Ritz method which is considered hereafter.The analysis of the transverse vibrations of a rectan-
gular plate using the Ritz method consists in searching
the transverse displacement in the form of a double ser-
ies of the in-plane co-ordinates x and y as
w0ðx; yÞ ¼XM
m¼1
XN
n¼1
AmnX mðxÞY nðyÞ. ð1Þ
The functions Xm(x) and Yn(y) have to form func-tional bases and are chosen to satisfy the essential
boundary conditions. The coefficients Amn are deter-
mined by considering the stationarity conditions of the
total potential energy.
The strain energy U can be expressed as a function of
the strain energies related to the material directions as
U ¼ U 1 þ U 2 þ U 6; ð2Þwith
U 1 ¼1
2
Z Z Zr1e1 dxdy dz;
U 2 ¼1
2
Z Z Zr2e2 dxdy dz;
U 6 ¼1
2
Z Z Zr6e6 dxdy dz;
ð3Þ
where the triple integrations are extended over the vol-
ume of the plate.
Considering the case of a plate constituted of a single
layer of unidirectional or orthotropic material, the
strains e1, e2 and e6 are related to the strains exx, exx
and cxy in the beam directions according to the straintransformations. Next the stresses r1, r2 and r6 can be
evaluated considering the elasticity relations of plates
r1 ¼ Q11e1 þ Q12e2;
r2 ¼ Q12e1 þ Q22e2;
r6 ¼ Q66e6.
ð4Þ
It results that the strain energy U1, stored in tension–
compression in the fibre direction, can be written as
U 1 ¼ U 11 þ U 12; ð5Þwith
U 11 ¼1
2
Z Z ZQ11e
21 dxdy dz;
U 12 ¼1
2
Z Z ZQ12e1e2 dxdy dz:
ð6Þ
Expression (5) separates the energy U11 stored in the
fibre direction and the coupling energy U12 induced by
the Poisson�s effect. Introducing the strain transforma-
tions in relations (6), the energies are expressed as
U 11 ¼1
2
Z Z ZQ11 e2
xxcos4hþ e2yysin4hþ c2
xysin2hcos2h�
þ 2exxeyysin2hcos2hþ 2exxcxy sin hcos3h
þ 2eyycxysin3h cos h�
dxdy dz; ð7Þ
12
188 J.-M. Berthelot / Composite Structures 74 (2006) 186–201
U 12 ¼1
2
Z Z ZQ12 e2
xxsin2hcos2hþ e2yysin2hcos2h
h
� c2xysin2hcos2hþ exxeyy sin4hþ cos4h
� �þ exxcxy sin2h� cos2h
� �sin h cos h
þ eyycxyðcos2h� sin2hÞ sin h cos hi
dxdy dz. ð8Þ
The strain energies U11 and U12 can then be expressed
by introducing the transverse displacement (1) in theprevious relations, and next integrating over the plate
volume. Calculation of the integrals in expressions (7)
and (8) leads to the introduction [18] of coefficients
Cpqrsminj expressed as
Cpqrsminj ¼ Ipq
miJrsnj; ð9Þ
introducing the dimensionless integrals
Ipqmi ¼
Z 1
0
dpX m
dup
dqX i
duqdu; m; i¼ 1;2; . . . ;M ; p;q¼ 0;1;2;
ð10Þ
J rsnj ¼
Z 1
0
drY n
dvr
dsY j
dvsdv; n; j¼ 1;2; . . . ;N ; r; s¼ 0;1;2.
ð11ÞThe integrals Ipq
mi and J rsnj are calculated using the re-
duced co-ordinates
u ¼ xa; and v ¼ y
b; ð12Þ
where a and b are the length and the width of the plate,respectively.
Finally, the strain energies U11 and U12 can be written
in the forms
U 11 ¼1
2Ra2
XM
m¼1
XN
n¼1
XM
i¼1
XN
j¼1
AmnAijD11f11ðhÞ; ð13Þ
with
f11ðhÞ ¼ C2200minj cos4hþ C0022
minj R4sin4h
þ 4C1111minj þ C2002
minj þ C0220minj
� �R2sin2hcos2h
þ 2 C1210minj þ C2101
minj
� �R sin hcos3h
þ 2 C1012minj þ C0121
minj
� �R3sin3h cos h; ð14Þ
D11 ¼ Q11
h3
12; ð15Þ
and
U 12 ¼1
2Ra2
XM
m¼1
XN
n¼1
XM
i¼1
XN
j¼1
AmnAijD12f12ðhÞ; ð16Þ
with
f12ðhÞ ¼ C2200minj þ C0022
minj R4 � 4C1111minj R2
� �sin2hcos2h
þ 1
2C2002
minj þ C0220minj
� �R2 cos4hþ sin4h� �
þ C1210minj þ C2101
minj
� �R� C1012
minj þ C0121minj
� �R3
h i
� sin2h� cos2h� �
sin h cos h; ð17Þ
D12 ¼ Q12
h3
12. ð18Þ
These expressions introduced the length-to-width
ratio of the plate (R = a/b).
In the same way, the energy U2 stored in tension–
compression in the direction transverse to the fibre
direction is obtained as
U 2 ¼ U 21 þ U 22; ð19Þwith
U 21 ¼ U 12 ð20Þand
U 22 ¼1
2Ra2
XM
m¼1
XN
n¼1
XM
i¼1
XN
j¼1
AmnAijD22f22ðhÞ; ð21Þ
with
f22ðhÞ ¼ C2200minj sin4hþ C0022
minj R4cos4h
þ 4C1111minj þ C2002
minj þ C0220minj
� �R2sin2hcos2h
� 2 C1210minj þ C2101
minj
� �Rsin3h cos h
� 2 C1012minj þ C0121
minj
� �R3 sin hcos3h; ð22Þ
D22 ¼ Q22
h3
12. ð23Þ
Lastly, the strain energy U66 stored in in-plane shear
can be written as
U 66 ¼1
2Ra2
XM
m¼1
XN
n¼1
XM
i¼1
XN
j¼1
AmnAijD66f66ðhÞ; ð24Þ
with
f66ðhÞ¼ 4 C2200minj þC0022
minj R4� C2002minj þC0220
minj
� �R2
h isin2hcos2h
þ4C1111minj R2 cos2h� sin2h
� �2þ4 C1012minj þC0121
minj
� �R3
h
� C1210minj þC2101
minj
� �Ri
cos2h� sin2h� �
sinhcosh;
ð25Þ
D66 ¼ Q66
h3
. ð26Þ
J.-M. Berthelot / Composite Structures 74 (2006) 186–201 189
Then, the energy dissipated by damping in the mate-
rial is written in the form
DU ¼ w11U 11 þ 2w12U 12 þ w22U 22 þ w66U 66; ð27Þintroducing the damping coefficients w11, w12, w22 andw66 associated to the strain energies, respectively. The
strain energy U12 is generally negative, due to the cou-
pling between e1 and e2 and the corresponding dissipated
energy must be taken positive. In fact, this energy can be
neglected with regard to the other energies. Next, the
damping wx in the x direction of the plate along its
length is evaluated by the relation
wx ¼DUU
. ð28Þ
2.2. Evaluation of the damping of a rectangular
laminated plate
In the present subsection we consider the case of a
laminated plate constituted of n orthotropic layers
(Fig. 1). The middle surface is chosen as the reference
plane (Oxy). Each layer k is referred to by the z co-ordi-
nates of its lower face hk�1 and upper face hk. Layer can
also be characterised by introducing the thickness ek and
the z co-ordinate zk of the centre of the layer. Layer ori-entation is defined by the angle hk of layer axes with the
Ox�!
axis of the plate. For a laminate, relation (2) consid-
ered for a single orthotropic layer can be rewritten in the
axes of each layer. Thus, the strain energy of the layer k
can be expressed as function of the strain energies
related to layer directions as
U k ¼ Uk1 þ U k
2 þ U k6 ð29Þ
and the total energy of laminate is given by
U ¼Xn
k¼1
Uk1 þ U k
2 þ U k6
� �. ð30Þ
In the case of the vibrations of a rectangular plate of
length a and width b, the strain energies are expressed by
U k1 ¼
Z a
x¼0
Z b
y¼0
Z hk
z¼hk�1
r1e1 dxdy dz; ð31Þ
U k2 ¼
Z a
x¼0
Z b
y¼0
Z hk
z¼hk�1
r2e2 dxdy dz; ð32Þ
hk-1hk
h1 h0
h2
z
n
k
1 2
middle plane
layer number
Fig. 1. Laminate element.
U k6 ¼
Z a
x¼0
Z b
y¼0
Z hk
z¼hk�1
r6c6 dxdy dz. ð33Þ
As in the previous subsection, the strain energy can
be written in the form
U ¼Xn
k¼1
Xpq
U kpq; ð34Þ
with
Ukpq ¼
1
2
Z a
x¼0
Z b
y¼0
Z hk
z¼hk�1
Qkpqe
kpe
kq dxdy dz;
pq ¼ 11; 12; 22; 66.
ð35Þ
By considering the Ritz method, the transposition of
the results obtained previously in the case of a singlelayer leads to
U kpq ¼
1
2Ra2
XM
m¼1
XN
n¼1
XM
i¼1
XN
j¼1
AmnAijf kpqðhÞ
Z hk
hk�1
Qkpqz2 dz.
ð36ÞHence
U kpq ¼
1
2Ra2
XM
m¼1
XN
n¼1
XM
i¼1
XN
j¼1
AmnAijDkpqf k
pqðhÞ; ð37Þ
with
Dkpq ¼
1
3h3
k � h3k�1
� �Qk
pq ¼ ekz2k þ
e3k
12
� �Qk
pq. ð38Þ
Then, the total energy dissipated by damping in the
laminated plate is expressed as
DU ¼Xn
k¼1
wk11Uk
11 þ 2wk12U k
12 þ wk22U k
22 þ wk66Uk
66
� �;
ð39Þintroducing the specific damping coefficient wk
pq of each
layer. Next, the damping wx in the x direction of the
plate along its length is evaluated by relation (28), the
dissipated energy being estimated by relation (39) and
the total strain energy by relation (34).
The functions f kpqðhÞ of each layer are simply derived
from the functions fpq(h) expressed previously in the case
of a single layer of orthotropic material as
f kpqðhÞ ¼ fpqðhþ hkÞ; ð40Þ
where functions fpq(h) are given by (14), (17), (22) and
(25).
2.3. Plates with free or clamped edges
In the Ritz method, the functions Xm(x) and Yn(y)
introduced in expression (1) can be chosen [18] as poly-
nomials or as beam functions which have been consid-
ered by Young [17,19]. The beam functions satisfy
190 J.-M. Berthelot / Composite Structures 74 (2006) 186–201
orthogonality relations which make zero many of the
integrals (10) and (11).
In the case where one edge is clamped and the other
opposite edge is free, the beam functions are written as
• for the edge x = 0 clamped and the edge x = a free:
X iðxÞ ¼ cos jixa� cosh ji
xa� ci sin ji
xa� sinh ji
xa
� �;
ð41Þ• for the edge y = 0 clamped and the edge y = b free:
Y iðyÞ ¼ cos jiyb� cosh ji
yb� ci sin ji
yb� sinh ji
yb
� �;
ð42Þwhere the coefficients ji and ci are reported in Table 1.
In the case of free opposite edges, the transverse dis-placement can be expanded according to the beam
functions:
• for the free edges x = 0 and x = a:
X 1ðxÞ ¼ 1;
X 2ðxÞ ¼ffiffiffi3p
1� 2xa
� �;
X iðxÞ ¼ cos jixaþ cosh ji
xa� ci sin ji
xaþ sinh ji
xa
� �;
i P 3;
ð43Þ• for the free edges y = 0 and y = b:
Y 1ðyÞ¼1;
Y 2ðyÞ¼ffiffiffi3p
1�2yb
� �;
Y iðyÞ¼ cosjiybþcoshji
yb�ci sinji
ybþsinhji
yb
� �; iP3;
ð44Þwhere the coefficients ji and ci are reported in Table 2.
Considering the functions (41)–(44), integrals (10)
and (11) can be next calculated by an analytical develop-
Table 2
Values of the coefficients of the free–free beam function
i 1 2 3 4 5 6 7 8
ji – – 4.7300 7.8532 10.996 14.137 17.279 20.420
ci – – 0.9825 1.0008 1.0000 1.0000 1.0000 1.0000
Table 1
Values of the coefficients of the clamped-free beam function
i 1 2 3 4 5 6 7 8
ji 1.8751 4.6941 7.8548 10.996 14.137 17.279 20.420 23.562
ci 0.7341 1.0185 0.9992 1.0000 1.0000 1.0000 1.0000 1.0000
ment or by a numerical process and stored for evaluat-
ing the strain energies.
3. Experimental analyses
3.1. Materials
Laminates with E-glass fibres or Kevlar fibres in an
epoxy resin, considered in [10], have been studied in
the present experimental analysis. The nominal volume
fraction of fibres was equal to 0.40 and the engineering
constants of the unidirectional layers referred to the
fibre direction are reported in Table 3.
3.2. Determination of the constitutive damping
parameters
The damping characteristics of the unidirectional lay-
ers or laminates were obtained by subjecting beams to
flexural vibrations. The equipment used is shown in
Fig. 2. The test specimen is supported horizontally asa cantilever beam in a clamping block. An impulse ham-
mer is used to induce the excitation of the flexural vibra-
tions of the beam and the beam response is detected by
using a laser vibrometer. Next, the excitation and the re-
sponse signals are digitalized and processed by a dy-
namic analyzer of signals. This analyzer associated
with a PC computer performs the acquisition of signals,
controls the acquisition conditions and next performsthe analysis of the signals acquired (Fourier transform,
frequency response, mode shapes, etc.).
The damping characteristics of the beams are de-
duced from the Fourier transform of the beam response
to an impulse input by fitting this experimental response
with the analytical response of the beam which was de-
rived in [10]. This fitting is obtained by a least square
method which allows to obtain the values of the naturalfrequencies fi and the modal loss factors gi, related to the
specific damping coefficient by the relation wi = 2pgi.
3.3. Plate damping measurement
Rectangular plates with two adjacent edges clamped
with the other two free and plates with one edge
clamped and the others free were tested to determinethe damping characteristics for the first modes of flex-
ural vibrations. As in the case of beams, the excitation
of vibrations was induced by the impulse hammer and
Table 3
Engineering constants of the unidirectional materials
Material EL (GPa) ET (GPa) GLT (GPa) mLT
Glass fibre composites 29.9 5.85 2.45 0.24
Kevlar fibre composites 50.7 4.50 2.10 0.29
Fig. 2. Experimental equipment.
rη i
(%)
1.1
1.3
1.5
1.7
θ = 30˚
θ = 45˚θ = 75˚θ = 90˚
θ = 60˚
J.-M. Berthelot / Composite Structures 74 (2006) 186–201 191
the plate response was detected by using the laser vib-
rometer. The damping parameters were derived from
the Fourier transform of the plate response. Vibration
excitation and response detection were carried out at
different points of the plates so as to generate and detect
all the modes.
Frequency (Hz)
0 200 400 600 800
Los
s fa
cto
0.3
0.5
0.7
0.9
θ = 0˚
θ = 15˚
Fig. 3. Experimental results obtained for the damping as function of
the frequency for different fibre orientations, in the case of glass fibre
composites.
Frequency (Hz)0 200 400 600 800 1000 1200
Los
s fa
ctor
i
(%)
1.0
1.5
2.0
2.5
3.0
3.5
4.0
η
θ = 30˚θ = 45˚θ = 60˚θ = 75˚
θ = 15˚
θ = 90˚
θ = 0˚
Fig. 4. Experimental results obtained for the damping as function of
the frequency for different fibre orientations, in the case of Kevlar fibre
composites.
4. Results and discussion
4.1. Damping of the unidirectional materials
4.1.1. Damping parameters
The experimental evaluation of damping was per-formed on beams of different lengths: 160, 180 and
200 mm so as to have a variation of the values of the
peak frequencies. Beams had a nominal width of
20 mm and a nominal thickness of 2.4 mm.
Figs. 3 and 4 show the experimental results obtained
for damping in the case of glass fibre composites and
Kevlar fibre composites, respectively. The results are re-
ported for the first three bending modes and for the dif-ferent lengths of the beams. For a given fibre
orientation, it is observed that damping increases when
the frequency is increased. The values of the damping in-
crease when the frequency is increased from 50 to
600 Hz are reported in Table 4 for the glass fibre com-
posites and the Kevlar fibre composites. The table shows
that the damping increase is fairly the same (from 21%
to 27%) for the different fibre orientations in the caseof the glass fibre composites, when the increase depends
on the fibre orientation in the case of the Kevlar fibre
composites with values varying from about 5% to 18%.
The variations of the loss factor with fibre orientation
are given in Figs. 5 and 6, respectively, for the three fre-
quencies 50, 300 and 600 Hz. The experimental results
show that damping is maximum at a fibre orientation
Table 5
Loss factors derived from the Ritz method in the case of unidirectional
glass fibre laminates
f (Hz) g11 (%) g12 g22 (%) g66 (%)
50 0.35 0 1.30 1.80
300 0.40 0 1.50 2.00
600 0.45 0 1.65 2.22
Table 6
Loss factors derived from the Ritz method in the case of unidirectional
Kevlar fibre laminates
f (Hz) g11 (%) g12 g22 (%) g66 (%)
50 1.50 0 2.50 3.80
300 1.63 0 2.60 4.10
600 1.69 0 2.78 4.50
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
ηi (
%)
1.0
1.5
2.0
2.5
3.0
3.5
4.0
f = 50 Hzf = 300 Hzf = 600 Hzf = 50 Hzf = 300 Hzf = 600 Hz
Experimental results
Ritz analysis
Fig. 6. Variation of damping as function of fibre orientation, in the
case of Kevlar fibre composites.
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
ηi (
%)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
f = 50 Hzf = 300 Hzf = 600 Hzf = 50 Hzf = 300 Hzf = 600 Hz
Experimental results
Ritz analysis
Fig. 5. Variation of damping as function of fibre orientation, in the
case of glass fibre composites.
Table 4
Damping increase (%) in the frequency range (50, 600 Hz)
Fibre orientation (degrees) 0 15 30 45 60 75 90
Glass fibre composites 21 24 26 23 26 23 27
Kevlar fibre composites 5.4 10.2 16.5 17 18.3 18 11.6
192 J.-M. Berthelot / Composite Structures 74 (2006) 186–201
of about 60� for the glass fibre composites, when a max-
imum for about 30� fibre orientation is observed in thecase of the Kevlar composites.
The analysis using the Ritz method was applied to the
experimental results obtained for the bending of beams.
The beams were considered in the form of plates with
one edge clamped and with the others free. Damping
was evaluated by the Ritz method (28) considering the
beam functions introduced previously (41)–(44). Thus,
the present evaluation of the beam damping takesaccount of the effect of the beam width. The results
deduced from the Ritz method are reported in Figs. 5
and 6 in the case of glass fibre composites and Kevlar
fibre composites, respectively. A good agreement is ob-
tained with the experimental results. The values of the
loss factors considered for modelling are reported in
Tables 5 and 6 for the frequencies 50, 300 and 600 Hz.
It has been shown [10] that the shear damping evaluatedby using the Ritz method is fairly higher that the values
of the shear loss factor deduced from the Adams–Bacon
analysis or from the Ni–Adams analysis which do not
consider the width of the beam.
4.1.2. Influence of the width of the beams
The influence of the beam width can be analysed by
the Ritz method. Fig. 7 shows the results obtained forthe loss factor of the first mode of beams with a nominal
length of 200 mm and for different length-to-width ratio
of the beam: 100, 20, 10, 7 and 5, in the case of glass fibre
composites (Fig. 7(a)) and Kevlar fibre composites (Fig.
7(b)). These figures show that the results reach a limit for
high values of the length-to-width ratio of the beams.
Furthermore, the results deduced from the Ni–Adams
analysis, considering the values of damping derived fromthe experimental results using the Ritz method in the case
of beams with a length to width ratio equal to 10 (Tables
5 and 6), are compared in Fig. 8(a) (glass fibre compos-
ites) and (b) (Kevlar fibre composites) with the results
derived from the Ritz method in the case of a length-
to-width ratio of the beams equal to 100. The results
are rather similar. This shows that the Ni–Adams analy-
sis can be applied to the evaluation of damping proper-ties of beams with high values of the length-to-width
ratio. In fact, in order to minimize the edge effects espe-
cially for off-axis materials it is difficult to implement an
experimental analysis with a high value of the length-
to-width ratio of the beams. A ratio about 10 which leads
to a beam width of 20 mm for a length of 200 mm
appears to be a good compromise. In this case it is
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i(%
)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Experimental resultsR = 100R = 20 R = 10 R = 7 R = 5
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
ηi(%
)
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
Experimental results R = 100R = 20 R = 10 R = 7R = 5
η
a
b
Fig. 7. Unidirectional beam damping obtained with different values of
the length-to-width ratio R: (a) in the case of glass fibre composites and
(b) in the case of Kevlar fibre composites.
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
ηi (
%)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Experimental resultsNi-Adams analysisRitz analysis
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
η i(%
)
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
Experimental resultsNi-Adams analysisRitz analysis
a
b
Fig. 8. Comparison of the results deduced from the Ni–Adams
analysis and the Ritz method in the case of a length-to-width ratio of
100: (a) glass fibre beams and (b) Kevlar fibre beams.
J.-M. Berthelot / Composite Structures 74 (2006) 186–201 193
necessary to analyse the experimental results with a mod-elling which takes the width of the beams into account.
4.1.3. Damping according to modes of beam vibrations
The Ni–Adams analysis is established considering the
beam theory which considers the case of bending along
the x-axis of beams and assumes that the transverse dis-
placement of beams is a function of the x co-ordinate
only
w0 ¼ w0ðxÞ. ð45ÞAccording to this theory, only the bending modes of
beams are described and the damping of unidirectional
beams will all the more high as the beam deformation
will induce bending in the direction transverse to fibres
and in-plane shearing for intermediate orientations of fi-bres. The Ni–Adams analysis does not take account of
the effects of beam twisting which can induce notable
twisting deformation of beams for which the transverse
displacement is not anymore independent of the y
co-ordinate.
Fig. 9 shows the variations of beam damping deduced
from the Ritz method for the first four modes of unidi-
rectional beams in the case of beam length equal to
180 mm and a length-to-width ratio equal to 10: glass
fibre beams (Fig. 9(a)) and Kevlar fibre beams (Fig.
9(b)). For the damping evaluation of beams we haveconsidered that the loss factors of the materials depend
on the frequency according to the results obtained in
Section 4.1.1. The natural frequencies and modes of
the beams were first derived using the Ritz method.
Next, the damping evaluation of laminated beams was
derived according to the modelling developed in Section
2 and considering that the damping loss factors g11, g22
and g66 increased linearly in the frequency range (50,600 Hz) according to the values reported in Tables 5
and 6. The results for the first two modes are similar, dif-
fering by the increase of the damping with the
frequency.
In the case of the third mode (Fig. 9), it is observed a
high beam damping for fibre orientations of 0� and 10�with a value which is fairly near of the shear damping.
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
ηi
(%
)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
mode 1mode 2mode 3mode 4mode 1mode 2mode 3
Ritz analysis
Experimentalbending modes
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
ηi
(%
)
1.4
1.8
2.2
2.6
3.0
3.4
3.8
4.2
mode 1mode 2mode 3mode 4mode 1mode 2mode 3
Ritz analysis
Experimentalbending modes
a
b
Fig. 9. Variation of the damping of unidirectional beams of length
equal to 180 mm, derived from the Ritz method for the first four
modes: (a) glass fibre beams and (b) Kevlar fibre beams.
194 J.-M. Berthelot / Composite Structures 74 (2006) 186–201
The shapes of the modes 1 to 4 for a fibre orientation of
0� are given in Fig. 10. The results show that the shapes
of modes 1, 2 and 4 satisfy the assumption (45), whereas
an important twisting of the beam is observed for themode 3 inducing a notable in-plane shear deformation.
mode 1
mode 3
Fig. 10. Free flexural modes of unidirectional
Finally, the beam damping results from the respective
contributions of the energies induced in bending along
the x direction of the beam, bending along the trans-
verse y direction and beam twisting. These energies are
taken into account by the damping analysis based on
the Ritz method. Fig. 11 reports the mode shapes de-duced in the case of 30� fibre orientation showing the
participation of the different deformation modes. In this
case, it is observed that beam twisting of the mode 4 is
associated to a lower damping of the beam. These re-
sults show that beam twisting induces an increase of
damping for fibre orientations near the material direc-
tions: 0� direction for mode 3 and 90� direction for mode
4 (Figs. 9 and 10), resulting from the increase of in-planeshear deformation of materials. In contrast, the beam
twisting results in a decrease of damping for intermedi-
ate orientations (mode 4, Figs. 9 and 11) associated to
the decrease of in-plane shear deformation. Similar re-
sults are observed in the case of unidirectional Kevlar
composites.
The variations of beam damping deduced from the
Ritz method are compared in Fig. 9 with the experimen-tal results obtained for the first three bending modes of
beams. These bending modes were obtained by exciting
the beams by an impulse applied on the beam axis so as
to induce vibration modes without beam twisting. The
damping evaluation by the Ritz method agrees fairly
well with the experimental results when only the bending
modes of the beams are considered.
4.2. Damping of different laminates
Laminates with three different stacking sequences
were analysed: [0/90/0/90]s cross-ply laminates, [0/90/
45/�45]s laminates and [h/�h/h/�h]s angle-ply laminates
with h varying from 0� to 90�. The laminates were
prepared from 8 plies of the unidirectional materials
studied in the previous subsection. The nominal thick-ness of the laminates was 2.4 mm and the analysis was
mode 2
mode 4
glass fibre beam for 0� fibre orientation.
mode 1 mode 2
mode 3 mode 4
Fig. 11. Free flexural modes of unidirectional glass fibre beam for 30� fibre orientation.
J.-M. Berthelot / Composite Structures 74 (2006) 186–201 195
implemented in the case of beams 200 mm long and
20 mm width.
Figs. 12 and 13 show the results obtained for the
damping in the case of glass fibre laminates and Kevlar
fibre laminates, respectively. Figures report the results
deduced for the damping by the Ritz method for the first
four modes and the experimental damping measured for
the first mode. The evaluation of laminate damping bythe Ritz method takes account of the variation of the
loss factors g11, g22 and g66 with frequency (Tables 5
and 6). For the cross-ply laminates (Figs. 12(a) and
13(a)) and [0/90/45/�45]s laminates (Figs. 12(b) and
13(b)), the material damping is derived as function of
laminate orientation. For the [h/�h/h/�h]s angle-ply
laminates (Figs. 12(c) and 13(c)), damping is reported
as function of the ply orientation h. The damping de-duced from the Ritz method was evaluated by applying
the results of Section 2.2 to the different laminates.
The in-plane behaviour of the [0/90/0/90]s cross-ply
laminates is the same in the 0� and 90� directions, when
the external 0� layers of the stacking sequence leads to a
slight increase of the bending properties in the 0� direc-
tion. Thus, compared to the damping of unidirectional
composites (Figs. 5 and 6) the stacking sequence [0/90/0/90]s leads to a more symmetric variation of damping
as function of the orientation with damping characteris-
tics which are slightly higher in the 90� direction. Near
45� orientations damping of the [0/90/0/90]s laminates
is clearly reduced (about 1.2% for glass fibre laminates
and 2.6% for Kevlar fibre laminates, for the first two
modes) compared to the damping of the unidirectional
laminates (about 1.4% and 3.0%, respectively). Thisreduction results from the in-plane shear deformation
which is constrained by the [0/90] stacking sequence.
For the third mode it is observed a high damping for
directions near 0� and 90� associated to the effects of
beam twisting as in the case of the unidirectional lami-
nates. For the fourth mode the beam twisting leads to
a decrease of the beam damping. The use of the [90/0/
90/0]s stacking sequence would lead to a damping behav-
iour where the 0� and 90� directions would be inverted.
For [0/90/45/�45]s laminates (Figs. 12(b) and 13(b)),
the damping behaviour is practically symmetric as func-
tion of the fibre orientation with an in-plane shear con-
strain effect which is more important than in the case of
cross-ply laminates, leading to a reduction of the damp-
ing near 45� orientation, for modes 1 and 2: loss factorof about 0.98% for glass fibre laminates and 2.2% for
Kevlar fibre laminates in the case of mode 1.
In the case of the [h/�h/h/�h]s angle-ply laminates
and for the first three modes (Figs. 12(c) and 13(c)),
the damping for ply angles higher than 60� is practically
the same as damping observed for the unidirectional
beams with fibre orientation equal to h. For lower values
of ply angle, it is observed a reduction of laminatedamping comparatively to the unidirectional compos-
ites, associated to the in-plane constrain effect induced
by the [h/�h/h/�h]s sequence. For mode 4 the damping
reduction of angle-ply laminates is observed for all the
ply orientations, except for orientations near 0� and
90� where angle-ply laminates are similar to unidirec-
tional laminates.
4.3. Damping of laminated plates
4.3.1. Damping investigation
The damping of rectangular laminated plates with
different edge conditions can be evaluated using the Ritz
method. The results obtained in the case of glass fibre
plates and in the case of Kevlar fibre plates are very sim-
ilar and differ by the levels of the damping of plate vibra-tions. Figs. 14 and 15 show the results derived for the
damping by the Ritz analysis for the first four modes
in the case of rectangular plates of glass fibre laminates
with two edge conditions: one clamped edge and the
other edges free (Fig. 14) and two adjacent edges
clamped and the other two free (Fig. 15). The plates
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
ηi
(%)
1.5
2.0
2.5
3.0
3.5
4.0
4.5
mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
Fibre orientation (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i
(%
)
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
Fibre orientation (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i
(%)
1.5
2.0
2.5
3.0
3.5
4.0
4.5mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
η
θ
η
θ
a
b
c
Fig. 13. Damping variation as function of laminate orientation for
beams of different Kevlar fibre laminates: (a) [0/90/0/90]s cross-ply
laminates, (b) [0/90/45/�45]s laminates and (c) [h/�h/h/�h]s angle-ply
laminates.
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
ηi
(%)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
Fibre orientation (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i
(%
)
0.4
0.6
0.8
1.0
1.2
1.4
1.6mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
Fibre orientation (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i
(%
)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
θ
ηη
θ
a
b
c
Fig. 12. Damping variation as function of laminate orientation for
beams of different glass fibre laminates: (a) [0/90/0/90]s cross-ply
laminates, (b) [0/90/45/�45]s laminates and (c) [h/�h/h/�h]s angle-ply
laminates.
196 J.-M. Berthelot / Composite Structures 74 (2006) 186–201
were clamped along the width and the investigation was
performed on plates 200 mm wide and 300 mm longwith a nominal thickness of 2.4 mm. The experimental
results obtained are also reported for the first mode
and show a good agreement with the results derived
from the Ritz analysis.
4.3.2. Plates with one edge clamped and the other
edges free
The mode shapes of the unidirectional plates with one
clamped edge and the other edges free are rather similar
for the different orientations of fibres. Fig. 16 shows an
example obtained for the shapes of the first four modes
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
ηi
(%
)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
Fibre orientation (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i
(%
)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
Fibre orientation (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i
(%)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
Fibre orientation (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i
(%
)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
η
θ
η
θ
η
θ
a b
c d
Fig. 14. Damping variation as function of laminate orientation for rectangular plates with one edge clamped and the other edges free, in the case of
different glass fibre laminates: (a) unidirectional laminates, (b) [0/90/0/90]s cross-ply laminates, (c) [0/90/45/�45]s laminates and (d) [h/�h/h/�h]s angle-
ply laminates.
J.-M. Berthelot / Composite Structures 74 (2006) 186–201 197
of unidirectional plates in the case of 15� fibre
orientation.
Modes 1 and 3 correspond to plate bending in the x
and y directions of plates. So the damping variation of
plates corresponding to these modes (Fig. 14(a)) are
comparable to the damping variation observed in the
case of the first and second modes of the unidirectionalbeams considered in Section 4.1.3 (Fig. 9(a)). The differ-
ence between the results is induced by the effects of the
length-to-width ratio: 10 in the case of the beams and
1.5 in the case of the plates. The influence of in-plane
shear is lower for fibre orientations near 45� in the case
of the plates.
For modes 2 and 4, it is observed (Fig. 16) an impor-
tant twisting of plates which leads to similar effectsas the ones observed in the case of unidirectional beams:
plate twisting induces an increase of damping for
fibre orientations near the material directions and a
decrease of damping for intermediate orientations
(Fig. 14(a)).
For [0/90/0/90]s and [0/90/45/�45]s plates the shape
modes are similar to the ones observed in the case of
unidirectional plates (Fig. 16) and it is observed (Fig.
14(b) and (c)) a more symmetric variation of damping
as function of the fibre orientation. Lastly, damping var-
iation of [h/�h/h/�h]s angle-ply plates (Fig. 14(d)) is
practically the same as the variation obtained in the caseof unidirectional plates (Fig. 14(a)).
4.3.3. Plates with two edges clamped and
the other edges free
Fig. 17 gives examples of the mode shapes of unidi-
rectional plates with two adjacent edges clamped for
three fibre orientations: 0�, 30� and 60�. The mode
shapes combine bending vibrations along the two freeedges and the figures show an evolution of the mode
shapes with fibre orientation.
For mode 1 it is observed (Fig. 15(a)) a rather high
damping induced by the plate twisting for fibre orienta-
tion near 0� (Fig. 17(a)) and next the damping decreases
Fibre orientation θ (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
ηi
(%
)
0.4
0.6
0.8
1.0
1.2
1.4
mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
Fibre orientation (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i
(%
)
0.4
0.6
0.8
1.0
1.2
1.4
mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
Fibre orientation (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i
(%)
0.4
0.6
0.8
1.0
1.2
1.4
mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
Fibre orientation (˚)
0 10 20 30 40 50 60 70 80 90
Los
s fa
ctor
i
(%)
0.4
0.6
0.8
1.0
1.2
1.4mode 1mode 2mode 3mode 4mode 1
Ritz analysis
Experimental results
η
θ
η η
θ θ
a b
c d
Fig. 15. Damping variation as function of laminate orientation for rectangular plates with two adjacent edges clamped and the other two free, in the
case of different glass fibre laminates: (a) unidirectional laminates, (b) [0/90/0/90]s cross-ply laminates, (c) [0/90/45/�45]s laminates and (d) [h/�h/h/
�h]s angle-ply laminates.
mode 1 mode 2
mode 3 mode 4
Fig. 16. Flexural mode shapes of unidirectional plate with one edge clamped and the others free for 15� fibre orientation.
198 J.-M. Berthelot / Composite Structures 74 (2006) 186–201
regularly when the fibre orientation increases. The mode
shape of the mode 2 is rather similar for the different
fibre orientations resulting in a low variation of damp-
ing. For mode 3 plate damping is nearly constant for
fibre orientations from 0� to 40� and next plate damp-
ing is clearly increased up to 90� fibre orientation.
mode 1 mode 2
mode 3 mode 4 a
mode 1 mode 2
mode 3 mode 4 b
mode 1 mode 2
mode 3 mode 4 c
Fig. 17. Flexural mode shapes of unidirectional plate with two adjacent edges clamped and the others free for three fibre orientation: (a) 0�orientation, (b) 30� orientation and (c) 60� orientation.
J.-M. Berthelot / Composite Structures 74 (2006) 186–201 199
Lastly, the shapes of mode 4 (Fig. 17) show an impor-
tant twisting of the plates for 0� and 30� (Fig. 17(a)
and (b)), inducing a high damping of the plates for fibre
orientations from 0� to 40�. Then, the damping de-
creases according to a mode shape with low plate twist-
ing (Fig. 17(b)).
The shape modes of [0/90/0/90]s and [0/90/45/�45]splates with two adjacent edges clamped are very similar
200 J.-M. Berthelot / Composite Structures 74 (2006) 186–201
to the ones obtained in the case of the unidirectional
plates. As it was observed previously for beams and
plates, Fig. 15(b) and (c) show a more symmetric varia-
tion of damping with fibre orientation. Also, the damp-
ing variation of angle-ply laminates (Fig. 15(d)) is rather
similar to the variation observed in the case of unidirec-tional plates.
5. Conclusions
An evaluation of the damping of rectangular lami-
nated plates was presented based on the Ritz method
for which the analysis of the transverse vibrations con-sists in expressing the transverse displacement of the
plates in the form of a double series of the in-plane co-
ordinates of the plates. Thus, the strain energies stored
in each layer of the laminates can be evaluated and the
energy dissipated by damping in the laminates can be de-
rived as a function of the strain energies and the damping
coefficients associated to the different energies.
Damping characteristics of laminates were evaluatedexperimentally using cantilever beam specimens sub-
jected to an impulse input. Loss factors were then de-
rived by fitting the experimental Fourier responses
with the analytical motion responses expressed in modal
co-ordinates.
Damping parameters were first measured in the case
of unidirectional beams of glass fibre composites and
Kevlar fibre composites as function of fibre orientation.The experimental results obtained for the loss factors of
the unidirectional materials show a significant increase
of material damping with frequency. The influence of
the beam width was studied using the Ritz analysis
and the results obtained show that for a given beam
length the beam damping depends on the length-to-
width ratio of the beam. Thus, for the damping evalua-
tion of materials with usual length-to-width ratios equalto 10 it is necessary to take account of the beam width.
Moreover, it was shown that the Ni–Adams analysis for
the damping evaluation in bending vibrations of beams
can be applied only for high length-to-width ratios of
the beams, condition which is not really verified in the
usual tests.
The damping evaluation based on the Ritz method
can be applied for the flexural modes for which beamtwisting is induced. For the bending modes the damping
of unidirectional beams is all the more high as the beam
deformation induces bending in the direction transverse
to fibres and in-plane shearing for intermediate orienta-
tions of fibres. For the other modes it was observed that
the beam twisting induces an increase of the beam
damping for fibre orientations near the material direc-
tions (0� direction and 90� direction), damping increasewhich results from the increase of in-plane shear defor-
mation of the materials. In contrast when beam twisting
is induced for intermediate orientations it is observed a
decrease of mode damping which can be associated to
the decrease of in-plane shear deformation.
The Ritz method was then applied to the analysis of
damping of [0/90/0/90]s, [0/90/45/�45]s and [h/�h/h/�h]sbeams and to the analysis of damping of rectangularplates with two different edge conditions. The evaluation
of damping takes account of the loss factors with fre-
quency. It was observed that the stacking sequences [0/
90/0/90]s and [0/90/45/�45]s lead to practically similar
damping properties in 0� and 90� fibre directions, when
[h/�h/hh/� h]s angle-ply laminates show rather similar
properties to unidirectional laminates with fibre orienta-
tion equal to h. Beam and plate damping depends on thevibration modes and the damping evaluation was re-
lated with the mode shapes of the beam and plate
vibrations.
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