Damping Analysis of Laminated Plates and Beams Using RItz Method

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

mailto:[email protected]

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


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


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


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


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


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˚


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.


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


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


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


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


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.


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


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.


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.


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


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.


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.


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


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


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



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


Ritz analysis


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.


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


Ritz analysis



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


Ritz analysis



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


Ritz analysis


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.


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


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


Ritz analysis



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


Ritz analysis



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


Ritz analysis



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


Ritz analysis


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.


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


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


Ritz analysis



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


Ritz analysis



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


Ritz analysis



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


Ritz analysis


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.


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.


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


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.

References

[1] Gibson RF, Plunkett RA. Dynamic stiffness and damping of fiber-

reinforced composite materials. Shock Vib Digest 1977;9(2):9–17.

[2] Gibson RF, Wilson DG. Dynamic mechanical properties of fiber-

reinforced composite materials. Shock Vib Digest 1979;11(10):

3–11.

[3] Gross B. Mathematical structure of the theories of viscoelastic-

ity. Paris: Hermann; 1953.

[4] Hashin Z. Complex moduli of viscoelastic composites—I. General

theory and application to particulate composites. Int J Solids

Struct 1970;6:539–52.

[5] Hashin Z. Complex moduli of viscoelastic composites—II. Fiber

reinforced materials. Int J Solids Struct 1970;6:797–807.

[6] Sun CT, Wu JK, Gibson RF. Prediction of material damping of

laminated polymer matrix composites. J Mater Sci 1987;22:

1006–12.

[7] Crane RM, Gillespie JW. Analytical model for prediction of the

damping loss factor of composite materials. Polym Compos 1992;

13(3):448–53.

[8] Adams RD, Bacon DGC. Effect of fiber orientation and laminate

geometry on the dynamic properties of CFRP. J Compos Mater

1973;7:402–8.

[9] Yim JH. A damping analysis of composite laminates using the

closed form expression for the basic damping of Poisson�s ratio.

Compos Struct 1999;46:405–11.

[10] Berthelot J-M, Sefrani Y. Damping analysis of unidirectional

glass and Kevlar fibre composites. Compos Sci Technol 2004;64:

1261–78.

[11] Ni RG, Adams RD. The damping and dynamic moduli of

symmetric laminated composite beams. Theoretical and experi-

mental results. Compos Sci Technol 1984;18:104–21.

[12] Adams RD, Maheri MR. Dynamic flexural properties of aniso-

tropic fibrous composite beams. Compos Sci Technol 1994;50:

497–514.

[13] Lin DX, Ni R, Adams RD. Prediction and measurement of the

vibrational parameters of carbon and glass–fibre reinforced plastic

plates. J Compos Mater 1984;18:132–52.

[14] Maheri MR, Adams RD. Finite element prediction of modal

response of damped layered composite panels. Compos Sci

Technol 1995;55:13–23.

[15] Yim JH, Jang BZ. An analytical method for prediction of the

damping in symmetric balanced laminates composites. Polym

Compos 1999;20(2):192–9.


[16] Yim JH, Gillespie Jr JW. Damping characteristics of 0� and 90�AS4/3501-6 unidirectional laminates including the transverse

shear effect. Compos Struct 2000;50:217–25.

[17] Young D. Vibration of rectangular plates by the Ritz method.

J Appl Mech 1950;17:448–53.

[18] Berthelot J-M. Composite materials. Mechanical behavior and

structural analysis. New York: Springer; 1999.

[19] Timoshenko S, Young DH, Weaver W. Vibration problems in

engineering. New York: John Wiley & sons; 1974.

Documents

Damping Analysis of Laminated Plates and Beams Using RItz Method