Active control of large amplitude vibrations of smart MEE doubly curved shells

Active control of large amplitude vibrations of smartmagneto–electro–elastic doubly curved shells

S. C. Kattimani • M. C. Ray

Received: 19 February 2014 / Accepted: 4 March 2014

� Springer Science+Business Media Dordrecht 2014

Abstract This paper deals with the analysis of active

constrained layer damping (ACLD) of large amplitude

vibrations of smart magneto–electro–elastic (MEE)

doubly curved shells. The constraining layer of the

ACLD treatment is composed of the vertically/

obliquely reinforced 1–3 piezoelectric composite

(PZC). The constrained viscoelastic layer of the

ACLD treatment is modeled by using the Golla–

Hughes–McTavish method in the time domain. A

three-dimensional finite element model of the overall

smart MEE doubly curved shells has been developed

taking into account the effects of electro–elastic and

magneto–elastic couplings, while the von Karman

type nonlinear strain displacement relations are used

for incorporating the geometric nonlinearity. Influence

of the curvature ratio, the curvature aspect ratio, the

thickness aspect ratio on the nonlinear frequency

ratios of the MEE doubly curved shells has been

investigated. Effects of the location of the ACLD

patches and the edge boundary conditions on the

control of geometrically nonlinear vibrations of

paraboloid and hyperboloid MEE shells have been

studied. Particular attention has been paid to investi-

gate the performance of the ACLD treatment due to

the variation of the piezoelectric fiber orientation

angle in the 1–3 PZC constraining layer of the ACLD

treatment.

Keywords Magneto–electro–elastic � Active

constrained layer damping (ACLD) � Doubly

curved shells � Piezoelectric composites �Geometrically nonlinear vibrations

1 Introduction

Comprehensive research carried out over the recent

decades reveals that the piezoelectric materials have

been widely used as smart sensors and actuators. The

structures integrated with such piezoelectric sensors

and actuators possess self sensing, self monitoring and

diagnosing capabilities and are commonly known as

smart structures. Piezoelectric actuators can produce

the strain in the order of 1,000 micro strain and within

the linear range actuation strain is proportional to the

applied electric field/voltage. These features make

them attractive for exploiting them in case of active

control of the stability, shape, noise and vibrations of

high-functioning lightweight flexible structures

(Newnham et al. 1980; Bailey and Hubbard 1985;

Crawley and Luis 1987; Azvine et al. 1995; Moita

et al. 2002; Gao and Shen 2003; Kulkarni and Bajoria

2006; Damanpack et al. 2013). However, monolithic

piezoelectric materials possess low coupling coeffi-

cients and require high control voltage for satisfactory

S. C. Kattimani � M. C. Ray (&)

Department of Mechanical Engineering, Indian Institute

of Technology, Kharagpur 721302, India

e-mail: [email protected]

123

Int J Mech Mater Des

DOI 10.1007/s10999-014-9252-3

control of vibrations of host structures. To mitigate the

drawbacks of these low-control-authority monolithic

piezoelectric materials, the active constrained layer

damping (ACLD) treatment has been developed (Baz

and Ro 1996; Baz 1998). In a typical ACLD treatment,

the performance of the passive constrained layer

damping (PCLD) treatment is appreciably improved

by replacing the passive constraining layer with an

active constraining layer made of piezoelectric mate-

rials. The constrained layer is made of the viscoelastic

material and sandwiched between the host structure

(substrate) and the active piezoelectric constraining

layer. The performance of the low control authority

piezoelectric materials for attenuating the vibrations is

improved significantly when they are used as the

constraining layer of the ACLD treatment rather than

when they are directly attached to the base structure.

Also, if the constraining layer of the ACLD treatment

is not activated with applied voltages, the ACLD

treatment turns into the PCLD treatment. Thus, the

simultaneous passive and active damping can be

achieved by the ACLD treatment when under opera-

tion (Baz and Ro 1996). The successful experimental

work on the ACLD treatment (Baz and Ro 1996) has

motivated the researchers to carry out further inves-

tigation on the use of the ACLD treatment for smart

structures. Ray et al. (2001) experimentally and

theoretically analyzed the ACLD of cylindrical shell.

Chantalakhana and Stanway (2001) investigated the

performance of the ACLD treatment for clamped–

clamped plate. Lim et al. (2002) developed the closed

loop finite element modeling of ACLD for time

domain analysis. Illaire and Kropp (2005) studied the

quantification of damping mechanisms of active

constrained layer treatments. In recent decades, Ray

and his co-researchers (Ray and Mallik 2005; Ray and

Pradhan 2006; Ray and Faye 2009; Sarangi and Ray

2010; Kanasogi and Ray 2013; Kumar and Ray 2013)

have been performing the extensive research on the

performance of the ACLD treatment for active damp-

ing of linear and nonlinear vibrations of smart

structures and they established that the damping

characteristics of these structures can be improved

significantly by using 1–3 piezoelectric composites

(PZC) as the materials of the constraining layer of the

ACLD treatment.

An exceptional class of smart laminated composites

made of piezoelectric and magnetostrictive materials is

known as the magneto–electro–elastic (MEE) or

multiferroic composites. The MEE composites have

the capability to transform energy among magnetic,

electric and mechanical energies. These composites

earned significant interest of the researchers in recent

years because of the promising properties for the

applications in sensors, actuators, transducers, space

structures, sonar applications, ultrasonic imaging

devices etc. These composites possesses very high

electromechanical coupling effect when used in layered

forms than the bulk/fiber form due to the absence of

leakage current and the ease of poling to align the

electric dipoles (Bichurin et al. 2003). Research

concerned with the coupled electro-magnetic effect in

the elastic media has received the attention of the

researchers (Van Den Boomgaard et al. 1976; Nan

1994; Pan 2001). Pan (2001) obtained the exact solution

of simply supported multilayered MEE plates under

surface and internal loads by using modified Stroh

formalism and propagator matrix method. The same

approach was extended to the free vibration analysis by

Pan and Heyliger (2002). Buchanan (2004) computed

the natural frequencies of vibration for the MEE layered

infinite plates. Hou and Leung (2004) studied the

transient response of the MEE hollow cylinders

subjected to dynamic loads. Wang and Ding (2007)

investigated the radial vibration of piezoelectric/

magnetostrictive composite hollow sphere. Daga et al.

(2008) studied the transient response of MEE cylinder

by semi-analytical finite element method. Bhangale and

Ganesan (2005) carried out the free vibration analysis

of functionally graded (FG) MEE cylindrical shells by

the finite element method while Ramirez et al. (2006)

used the discrete layer method to study the free

vibrations of FG MEE plate. Tsai et al. (2008) presented

the static analysis of three-dimensional doubly curved

FG MEE shells by an asymptotic approach. The finite

element model based on a higher order shear deforma-

tion theory for static and free vibration analysis of MEE

plates has been developed by Moita et al. (2009).

Pradhan (2009) presented the analytical solutions for

FG shells with embedded magnetostrictive layers and

studied the vibration characteristics using higher order

shear deformation theory. Liu and Chang (2010)

derived the closed form expressions for the transverse

vibrations of MEE thin plates and presented the natural

frequencies for two-layered MEE plate. Qin (2010)

investigated a circular cylindrical shell made of trans-

versely isotropic, non-ferromagnetic, perfectly electro-

conductive material immersed in an axially aligned

S. C. Kattimani, M. C. Ray

123

magnetic fields and a thermal field. Wu and Tsai (2010)

investigated the dynamic responses of FG MEE shells

using the method of multiple scales. Lang and Xuewu

(2012) carried out the buckling and vibration analysis of

FG magneto–electro–thermo-elastic circular cylindri-

cal shell by considering the mechanical, electric,

magnetic and thermal coupling effects. Badri and Al-

Kayiem (2012) developed the model for dynamic

analysis of piezolaminated shell structure with embed-

ded smart material lamina and influenced by magneto–

thermo–electro–elastic load. Loja et al. (2014) imple-

mented the differential evolution technique for optimi-

zation of MEE composite structures. Milazzo (2013)

analyzed the large deflection of MEE laminated plates

using first order shear deformation theory and von

Karman stress function approach. Although, MEE

structures have gained remarkable attention of the

researchers in recent years, very few research on the

large deflection analysis of MEE plates have been

reported in the literature. Xue et al. (2011) presented an

analytical solution for the large-deflection of rectangu-

lar MEE thin plate under the action of a transverse static

mechanical load. Sladek et al. (2013) presented the

mesh-less local Petrov–Galerkin method for the ana-

lysis of large deformation of MEE thick plates under

static and time-harmonic mechanical load and station-

ary electromagnetic load. Most recently, carrying out a

finite element analysis, Alaimo et al. (2013) proposed

an equivalent single-layer model for the large deflection

analysis of multilayered MEE laminates. From the

literature discussed above and to the author’s best

knowledge, it appears that no work has been reported on

the analysis of geometrically non linear vibrations of

MEE doubly curved shells. Although, the 1–3 PZCs are

commercially available and implemented for active

control of structures made of conventional materials

and composite materials, the active control of large

amplitude vibrations of the MEE shells using such

PZCs has not yet been reported in the open literature.

Consequently, in the present study, the performance of

the vertically/obliquely reinforced 1–3 PZCs as the

materials of the constraining layer of the ACLD

treatment to accomplish the task of active damping of

geometrically nonlinear vibrations of the smart MEE

doubly curved shells has been investigated. For such

investigation, three dimensional analysis of the ACLD

of MEE shells integrated with the patches of the ACLD

treatment has been carried out by the finite element

method. The effects of various parameters such as the

curvature ratio, the curvature aspect ratio, the thickness

aspect ratio, the coupling coefficients, the boundary

conditions and the variation of the piezoelectric fiber

orientation angle in the 1–3 PZC constraining layer on

the attenuation of geometrically nonlinear vibrations of

the MEE doubly curved shells have been thoroughly

investigated.

2 Problem description and governing equations

A MEE doubly curved shell integrated with a patch of

the ACLD treatment on its top surface is schematically

illustrated in Fig. 1(a). The curvilinear length, the

curvilinear width and the total thickness of the MEE

shell are a, b and H, respectively. Radii of curvature of

the middle surface are assumed to be R1 and R2. The

thickness of the constraining piezoelectric layer and the

constrained viscoelastic layer of the ACLD treatment

are hp and hv, respectively. The substrate of this MEE

shell consists of three layers. The commonly preferred

layer sequences for the MEE structures used in the

literature are P/M/P and M/P/M in which P stands for

the piezoelectric material like BaTiO3 (Barium Tita-

nate) and M stands for the magnetostrictive material

such as CoFe2O4 (Cobalt Ferrite) indicating the top/

middle/bottom layers. The origin of the curvilinear

coordinate system (xyz) is fixed at one corner of the

mid-plane of the middle layer, such that the curvilinear

lines x = 0 and a and y = 0 and b represent the

boundaries of the mid-plane of the substrate. The

constrained viscoelastic layer is sandwiched between

the host MEE substrate and the constraining layer of the

ACLD treatment. The constraining layer of the ACLD

treatment is made of the vertically/obliquely reinforced

1–3 PZC material. A layer of the vertical/obliquely

reinforced 1–3 PZC material is also illustrated in

Fig. 1(c) wherein the piezoelectric fibers are coplanar

with the xz plane while their orientation angle with the

z-axis is k. Although not shown here, the piezoelectric

fibers can be coplanar with the yz-plane and their

orientation angle with the z-axis is k. In case of the

obliquely reinforced 1–3 PZC, the orientation angle (k)

is nonzero while it is zero for the vertically reinforced

1–3 PZC as shown in Fig. 1(d). Since, the overall

structure is made of layers of dissimilar materials, the

kinematics of deformations of the overall structure

cannot be described by an equivalent single layer

displacement theory because of the fact that the elastic

Active control of large amplitude vibrations

123

properties of the adjacent continua of differ in orders.

Figures 2 (a) and (b) illustrate the schematic represen-

tation of the kinematics of deformations of the unde-

formed transverse normal in the xz- and the yz-planes,

respectively. The rotations of the portions of the normal

lying in the substrate shell, the viscoelastic layer and the

piezoelectric layer are represented by hx, /x and cx,

respectively in the xz-plane, while hy, /y and cy

represent the same in the yz-plane. Accordingly, the

axial displacements u and v of any point in the overall

shell along the x- and y-directions, respectively, can be

written as:

uðx; y; z; tÞ ¼ u0ðx; y; tÞ þ z� z� h

2

� ��

hxðx; y; tÞ þ z� h

2

� �� z� hNþ2h i

� �/xðx; y; tÞ

þ hz� hNþ2icxðx; y; tÞð1Þ

vðx; y; z; tÞ ¼ v0ðx; y; tÞ þ z� z� h

2

� �� hyðx; y; tÞ

þ z� h

2

� �� hz� hNþ2i

� �/yðx; y; tÞ

þ hz� hNþ2icyðx; y; tÞ ð2Þ

in which the brackets h i are used to define the

appropriate singularity functions such that the conti-

nuity of the displacement is satisfied at the interface

between the viscoelastic layer and the PZC layer or

between the substrate shell and the viscoelastic layer.

In order to utilize the vertical actuation by the active

constraining layer of the ACLD treatment and to

achieve the accurate results a high order transverse

deformation theory has been assumed for the overall

shell as follows:

wðx; y; z; tÞ ¼ w0ðx; y; tÞ þ zhzðx; y; tÞ þ z2/zðx; y; tÞð3Þ

Fig. 1 Schematic representation of a MEE doubly curved shell: a integrated with a single patch, b integrated with two patches at edges,

c obliquely reinforced 1–3 PZC composites, d vertically reinforced 1–3 PZC composites


123

In Eqs. (1)–(3), u0, v0 and w0 are the translational

displacements at any point on the mid-plane of the

substrate along x-, y- and z-directions, respectively,

with hz and /z being the generalized rotational

displacements. To facilitate the computation, the

rotational variables {dr} are separated from the

translational displacement variables {dt} as follows:

dtf g ¼ u0 v0 w0½ �T and

drf g ¼ hx hy hz /x /y /z cx cy

h iT

ð4Þ

For alleviating the shear locking in thin structures

and computing the element stiffness matrices

concerned with the transverse shear deformations

in an explicit manner, the selective integration rule

should be utilized. In order to accomplish this

task, the state of strain at any point in the overall

shell is expressed by the strain vector {eb}

containing in-plane strains and transverse normal

strain and the vector {es} of transverse shear strains

as follows:

ebf g ¼ ex ey ez exyf g and esf g ¼ exz eyzf gð5Þ

in which ex, ey and ez are the normal strains along x-,

y- and z-directions, respectively; exy is the in-plane

shear strain, exz and eyz are the transverse shear strains.

Making use of the von Karman type nonlinear strain–

displacement relations, the strain vectors defining the

state of in-plane and transverse normal strains at any

point in the MEE shell and the piezoelectric actuator

layer, respectively, can be expressed as:

ekb

� �¼ ebtf g þ z1½ � erbf g þ etbnf g; k ¼ 1; 2; 3

ekb

� �¼ ebtf g þ z2½ � erbf g þ etbnf g; k ¼ 5 ð6Þ

Here, k designates the layer number starting from the

bottom layer of the overall shell. The strain vectors

representing the state of transverse shear strains at any

point in the substrate, the viscoelastic layer and the

piezoelectric actuator layer, respectively, can be stated as

follows:

eks

� �¼ etsf g þ z3½ � ersf g; k ¼ 1; 2; 3

eks

� �¼ etsf g þ z4½ � ersf g; k ¼ 4 and

eks

� �¼ etsf g þ z5½ � ersf g; k ¼ 5

ð7Þ

The transformation matrices [Z1]–[Z5] appearing in

Eqs. (6) and (7) are presented in the Appendix 1,

whereas the generalized strain vectors in Eqs. (6) and

(7) are given by

ebtf g ¼ ou0

oxþ w

R1

ov0

oyþ w

R2

ou0

oyþ ov0

ox0

h iT

;

etsf g ¼ ow0

ox� u0

R1

ow0

oy �v0

R2

h iT

;

erbf g ¼ohx

ox

ohy

oyohx

oy þohy

oxhz /z

o/x

ox

o/y

oy

o/x

oy þo/y

oxocx

ox

ocy

oyocx

oy þocy

ox

24

35;

etbnf g ¼ 1

2dw0

dx

� 2 dw0

dy

�2

0 2 dw0

dx

� dw0

dy

�� T

and

ersf g ¼ hx hy /x /y cx cyohz

oxohz

oyo/z

oxo/z

oy

h i

ð8Þ

Fig. 2 Kinematics of deformations of the MEE doubly curved shell integrated with the ACLD patch: a transverse cross section parallel

to xz-plane, b transverse cross section parallel to yz-plane


123

Analogous to the representation of the state of strains

given by Eq. (5), the state of stress at any point in the

overall MEE shell can be written as follows:

rbf g ¼ rx ry rxy rz½ �T and

rsf g ¼ rxz ryz½ �Tð9Þ

in which rx; ry and rz are the normal stresses along x-,

y- and z-directions, respectively; rxy is thein-plane

shear stress; rxz and ryz are transverse shear stresses.

The coupled constitutive relations for the MEE

substrate are given by

rkb

� �¼ �C

kb

h iek

b

� �� ek

b

� �Ez � qk

b

� �Hz;

rks

� �¼ �C

ks

h iek

s

� �;

Dz ¼ ekb

� �Tek

b

� �þ 2k

33Ez þ d33Hz;

and Bz ¼ qkb

� �Tek

b

� �þ d33Ez þ l33Hz ; k ¼ 1; 2; 3;

ð10Þ

where, Dz, Bz, Ez and Hz are the electric displacement,

the magnetic induction, the electrical field and the

magnetic field, respectively, along the z-direction;

[ �Ckb] and [ �C

ks ] are the transformed elastic coefficient

matrix; 2k33 and l33 are the dielectric constant and the

magnetic permeability constant, respectively; {ekb},

{qkb} and d33 are the piezoelectric coefficient matrix,

the magnetostrictive coefficient matrix and the elec-

tromagnetic coefficient, respectively. The various

matrices appearing in Eq. (10) are given by

�Ckb

h i¼

�Ck11

�Ck12

�Ck13

�Ck16

�Ck12

�Ck22

�Ck23

�Ck26

�Ck13

�Ck32

�Ck33

�Ck36

�Ck16

�Ck26

�Ck36

�Ck66

2666664

3777775;

�Cks

h i¼

�Ck55

�Ck45

�Ck45

�Ck66

" #; ek

b

� �¼

e31

e32

e33

e36

8>>><>>>:

9>>>=>>>;;

qkb

� �¼

q31

q32

q33

q36

8>>><>>>:

9>>>=>>>;;

ð11Þ

The constitutive relations for the 1–3 PZC constrain-

ing layer of the ACLD treatment adaptable with the

present method of finite element formulation are given

by

rkb

� �¼ �C

kb

h iek

b

� �� Cbs

� �ek

s

� �� ebf gEz;

rks

� �¼ �Cbs

� �Tek

b

� �þ �C

ks

h iek

s

� �� ebf gEz and

Dz ¼ �ebf gT ekb

� �þ �esf gT ek

s

� �þ �233Ez k ¼ 5ð Þ

ð12Þ

In Eq. (12), the forms of the transformed elastic

coefficient matrices [ �Ckb] and [ �C

ks ] are similar to those

given by Eq. (11). The constitutive relations for the

1–3 PZC reveal that the inclination of the piezoelectric

fibers in the vertical xz- or yz-plane is responsible for

the coupling between the transverse shear strains and

the inplane stresses. The corresponding coupling

elastic constant matrix [ �CNþ2

bs ] is given by

�CNþ2

bs

h i¼

�C15�C25

�C35 0

0 0 0 �C46

� T

or

�CNþ2

bs

h i¼ 0 0 �C56 0

�C14�C24 0 �C34

� Tð13Þ

according as the fibers are coplanar with the xz- or the

yz-plane, respectively. If the piezoelectric fibers are

coplanar with both the xz- and the yz-planes (k = 0),

the coupling matrix of Eq. (13) turns into a null matrix.

Also, the elements of the piezoelectric constant

vectors {ekb} and {ek

s } appearing in Eq. (12) are given

by

�ebf g ¼ �e31 �e32 �e33 �e36f gTand

�esf g ¼ �e35 �e34f gT ð14Þ

The present analysis of the overall MEE shells will be

carried out in the time domain. Consequently, the

Golla–Hughes–McTavish (GHM) method has been

implemented to model the viscoelastic material. For

time domain analysis, the constitutive equation for the

linear, isotropic and homogeneous viscoelastic mate-

rial is expressed in stieltjes integral form and is given

by (Mc Tavish and Hughes 1993).

rks

� �¼Z t

0

G t� sð Þo ek

s

� os

os; k ¼ 4; ð15Þ

where G(t) is the relaxation function of the viscoelas-

tic material. The principle of virtual work employed to


123

derive the governing equations of the overall system

can be written as

X3

k¼1

Z

Xk

d ekb

� �Trk

b

� �dXk þ

X4

k¼1

Z

Xk

d eks

� �Trk

s

� �dXk

þZ

X5

d e5b

� �Tr5

b

� �þ d e5

s

� �Tr5

s

� � �dX5

�ZXt

dEtzDt

zdXt �Z

Xb

dEbz Db

z dXb

�Z

Xm

dHzBzdXm �ZA

d dtf gT ff gdA

þX5

k¼1

Z

Xk

d dtf gTqk d::

t

� �dXt

¼ 0

ð16Þ

where, {f} = [0 0 p]T is the externally applied surface

traction vector acting over a surface area A with

p being the transverse pulse load, Xk (k = 1, 2,

3,…,5) indicates the volume of the relevant layer, qk is

the mass density of the kth layer and d is the symbol of

the first variation. According to the Maxwell’s equa-

tions, the transverse electric fields Etz and Eb

z and the

magnetic field Hz are related to the electric potentials

/t and /band the magnetic potential w in the

following forms:

Etz ¼ �

o/t

oz; Eb

z ¼ �o/b

ozand Hz ¼ �

owoz

ð17Þ

in which /t and /b are the electric potential functions

in the top and the bottom piezoelectric layer, respec-

tively. It is assumed that the interfaces between the

piezoelectric layer and the magnetostrictive layer are

suitably grounded. Also, since the thicknesses of the

layers of the substrate are very small, linear variations

of the electric potential and the magnetic potential

functions can be assumed across the thickness of the

layers. Thus the electric potential functions /t and /b

can be expressed as

/t ¼ z� zb

h/1 and /b ¼ � z� h2

h/2 ð18Þ

while the magnetic potential distribution field in the

magnetostrictive layer can be expressed as

w ¼ z� h2

h�w ð19Þ

where zb is the z-coordinate of the bottom surface of

the top piezoelectric layer of the substrate, h2 is the z-

coordinate of the top surface of the bottom piezoelec-

tric layer, /1 and /2 are electric potentials on the top

surface and the bottom surface of the top and the

bottom piezoelectric layers, respectively, �w is the

magnetic potential on the top surface of the magne-

tostrictive layer and h is the thickness of each layer of

the substrate. It may be noted that Eqs. (16)–(19)

should be augmented for the M/P/M shell by replacing

the top and the bottom piezoelectric layers with the

magnetostrictive layers, while the middle layer is

piezoelectric.

3 Finite element model of the magneto–electro–

elastic doubly curved shell integrated

with ACLD patches

The overall MEE shell integrated with the ACLD

patches is discretized by eight noded iso-parametric

quadrilateral elements. In accordance with Eq. (5), the

generalized displacement vectors {dti} and {dri}

associated with the ith (i = 1, 2, 3,…,8) node of the

element can be written as

dtif g ¼ u0i v0i w0i½ �T and

drif g ¼ hxi hyi hzi /xi /yi /zi cxi cyi

h iT

ð20Þ

The generalized displacement vectors, the electric

potential vector {/} and the magnetic potential �w at

any point within the element can be expressed in terms

of the nodal generalized displacement vectors ({det }

and {der}), the nodal electric potential vector {/e} and

the nodal magnetic potential vector { �we}, respec-

tively, as follows:

dtf g ¼ Nt½ � det

� �; drf g ¼ Nr½ � de

r

� �;

/f g ¼ /1 /2½ �T¼ N/� �

/ef g and �w ¼ Nw� �

�we� �ð21Þ

in which,


123

det

� �¼ de

t1

� �Tde

t2

� �T. . . de

t8

� �Th iT

;

der

� �¼ de

r1

� �Tde

r2

� �T. . . de

r8

� �Th iT

;

/ef g ¼ /11 /21 /12 /22 : : : /18 /28½ �T;�w

e� �¼ �w1

�w2 . . . �w8

� �T;

Nt½ � ¼ Nt1 Nt2 . . . Nt8½ �T;Nr½ � ¼ Nr1 Nr2 . . . Nr8½ �T;

N/� �

¼N/11 0 N/12 0 . . . N/18 0

0 N/21 0 N/22 . . . 0 N/28

� T

;

Nw� �

¼ Nw1 Nw2 . . . Nw8½ �T;Nti ¼ niIt; Nri ¼ niIr

ð22Þ

where [Nt], [Nr], [N/] and [Nw] are the (3 9 24),

(8 9 64), (2 9 16) and (1 9 8) shape function matri-

ces, respectively, It and Ir are the (3 9 3) and the

(8 9 8) identity matrices, respectively and ni is the

shape function of natural coordinates associated with

the ith node. Also, /1i, /2i(i = 1, 2, 3,…,8) are nodal

electric potential degrees of freedom and �wi (i = 1, 2,

3,…,8) are the magnetic potential degrees of freedom.

Using Eqs. (18)–(22), the transverse electric fields Etz,

Ebz and the transverse magnetic field Hz are given by

Etz ¼�

1

h1 0½ � N/

� �/ef g; Eb

z ¼�1

h0 1½ � N/

� �/ef g

and Hz ¼�1

hNw� �

�we� �

ð23Þ

Now, using Eqs. (6), (7) and (22), the generalized

strain vectors at any point within the element can be

expressed in terms of the nodal generalized displace-

ment vectors as follows:

ebtf g ¼ Btb½ � det

� �; ebrf g ¼ Brb½ � de

r

� �etbnf g

¼ 1

2B1½ � B2½ � de

t

� �;

estf g ¼ Bts½ � det

� �and ersf g ¼ Brs½ � de

r

� �ð24Þ

in which the nodal strain–displacement matrices [Btb],

[Brb], [Bts], [Brs], [B1] and [B2] are given by

Btb½ � ¼ Btb1 Btb2 . . . Btb8½ �;Brb½ � ¼ Brb1 Brb2 . . . Brb8½ �;Bts½ � ¼ Bts1 Bts2 . . . Btb8½ �;Brs½ � ¼ Brs1 Brs2 . . . Brs8½ �;

B1½ � ¼dw0

dx 0 dw0

dy 0

0 dw0

dydw0

dx 0

" #T

;

B2½ � ¼ B21 B22 . . . B28½ �

ð25Þ

The various submatrices [Btbi], [Brbi], [Btsi] and [Brsi]

(i = 1, 2, 3,…,8) appearing in Eq. (25) have been

explicitly presented in the Appendix 1. Now, on

substitution of Eqs. (6), (7), (12), (23) and (24) into

Eq. (16), the following open loop elemental equations

of motion for the MEE shell integrated with the ACLD

treatment are obtained as follows:

Me½ � €de

t

n oþ Ke

tt

� �de

t

� �þ Ke

tr

� �de

r

� �

þ Ketsv

� � Z t

0

Gðt� sÞ o

osde

t

� �ds

þ Ketrsv

� � Z t

0

Gðt� sÞ o

osde

r

� �dsþ Ke

t/

h i/ef g

þ Ketw

h iwef g ¼ Fe

t

� �� Fe

tp

n oV� Fe

tpn

n oV

ð26Þ

Ketr

� �Tde

t

� �þ Ke

rr

� �de

r

� �

þ Ketrsv

� � Z t

0

Gðt�sÞ o

osde

t

� �ds

þ Kerrsv

� � Z t

0

Gðt�sÞ o

osde

r

� �dsþ Ke

r/

h i/ef g

þ Kerw

h iwef g

¼ � Ferp

n oV

ð27Þ

Ke/t

h ide

t

� �þ Ke

r/

h iT

der

� �� Ke

//

h i/ef g ¼ 0 ð28Þ

Kewt

h ide

t

� �þ Ke

rw

h iT

der

� �� Ke

ww

h iwef g ¼ 0

ð29Þ

The various matrices and vectors appearing in Eqs.

(26)–(29) are the elemental mass matrix [Me], the

elemental elastic stiffness matrices Kett

� �; Ke

tr

� �;

�Ke

rr

� �; Ke

tsv

� �; Ke

trsv

� �and Ke

rrsv

� �Þ, the elemental

electro-elastic coupling stiffness matrices ([Ket/] and

[Ker/]), the elemental magneto–elastic coupling stiff-

ness matrices ([Ketw] and [Ke

rw]), the elemental elec-

trical stiffness matrix [Ke//], the elemental magnetic

stiffness matrix [Keww], the elemental electro-elastic

coupling load vectors ({Fetp} and {Fe

rp}) and the

elemental mechanical load vector {Fet }. Also,


123

V represents the applied voltage across the thickness

of the 1–3 PZC layer. The elemental matrices and

vectors are given by

Kett

� �¼ Ke

tb

� �þ Ke

ts

� �þ Ke

tbp

h iþ Ke

tsp

h i

þ Ketbn

� �þ Ke

tbpn

h i;

Ketr

� �¼ Ke

trb

� �þ Ke

trbn

� �þ Ke

trs

� �þ Ke

trbp

h i

þ Ketrbpn

h iþ Ke

trsp

h i;

Kert

� �¼ Ke

trb

� �Tþ 1

2Ke

trbn

� �Tþ Ketrs

� �Tþ Kertbp

h i

þ Kertbpn

h iþ Ke

trsp

h iT

;

Kerr

� �¼ Ke

rrb

� �þ Ke

rrs

� �þ Ke

rrbp

h iþ Ke

rrsp

h i;

Ket/

h i¼ Ke

t/l

h iþ Ke

t/n

h i

Ketw

h i¼ Ke

twl

h iþ Ke

twn

h i;

Ke/t

h i¼ Ke

t/l

h iT

þ 1

2Ke

t/n

h iT

;

Kewt

h i¼ Ke

twl

h iT

þ 1

2Ke

twn

h iT

Fetp

n o¼ Fe

tpb

n oþ Fe

tps

n o;

Ferp

n o¼ Fe

rpb

n oþ Fe

rps

n o;

Fetpn

n o¼Zbe

0

Zae

0

B2½ �T B1½ �T Ftbp

� �dxdy;

Me½ � ¼Zbe

0

Zae

0

�m Nt½ �T Nt½ � dxdy and

�m ¼XNþ5

k¼1

Zhkþ1

hk

qkdz

Fef g ¼Zbe

0

Zae

0

Nt½ �T ff g dxdy;

ð30Þ

The elemental stiffness matrices appearing in Eq. (30)

corresponding to the bending stretching deformations

are

Ket/l

h i¼Zae

0

Zbe

0

Btb½ �T Dt/� �

N/� �

dxdy;

Ker/

h i¼Zae

0

Zbe

0

Brb½ �T Dr/� �

N/� �

dxdy;

Ket/n

h i¼Zae

0

Zbe

0

B2½ �T B1½ �T Dt/� �

N/� �

dxdy;

Ketwl

h i¼Zae

0

Zbe

0

Btb½ �T Dtw� �

Nw� �

dxdy

Kerw

h i¼Zae

0

Zbe

0

Brb½ �T Dtw� �

Nw� �

dxdy;

Ketwn

h i¼Zae

0

Zbe

0

B2½ �T B1½ �T Dtw� �

Nw� �

dxdy;

Ke//

h i¼Zae

0

Zbe

0

N/� �T

D//� �

N/� �

dxdy;

Keww

h i¼Zae

0

Zbe

0

Nw� �T

Dww� �

Nw� �

dxdy;

Ketb

� �¼Zae

0

Zbe

0

Btb½ �T Dtb½ � Btb½ � dxdy;

Ketrb

� �¼Zae

0

Zbe

0

Btb½ �T Dtrb½ � Brb½ � dxdy;

Ketbn

� �¼Zbe

0

Zae

0

1

2Btb½ �T Dtb½ � B1½ � B2½ �

�

þ B2½ �T B1½ �T Dtb½ � Btb½ �

þ 1

2B2½ �T B1½ �T Dtb½ � B1½ � B2½ �

�dxdy;

Ketbp

h i¼Zae

0

Zbe

0

Btb½ �T Dptb

� �Btb½ �

þ2 Btb½ �T Dptbs

� �Bts½ ��

dxdy; ð31Þ


123

Ketrbp

h i¼Zae

0

Zbe

0

Btb½ �T Dptrb

� �Brb½ �

þ Btb½ �T Dptrbs

� �Brs½ �

þ Bts½ �T Dprtbs

� �TBrb½ �

�dxdy;

Kerrbp

h i¼Zae

0

Zbe

0

Brb½ �T Dprrb

� �Brb½ �

þ2 Brb½ �T Dprrbs

� �Brs½ �

�dxdy;

Fetpb

n o¼Zae

0

Zbe

0

Btb½ �T Dbtp

n odxdy;

Ferpb

n o¼Zae

0

Zbe

0

Brb½ �T Dbrp

n odxdy:

ð31Þ

and those associated with the transverse shear defor-

mations are

Kets

� �¼Zae

0

Zbe

0

Bts½ �T Dts½ � Bts½ � dxdy;

Ketrs

� �¼Zbe

0

Zae

0

Bts½ �T Dtrs½ � Brs½ � dxdy;

Kerrs

� �¼Zbe

0

Zae

0

Brs½ �T Drrs½ � Brs½ � dxdy;

Ketsp

h i¼Zae

0

Zbe

0

Bts½ �T Dpts½ � Bts½ � dxdy;

Ketrsp

h i¼Zae

0

Zbe

0

Bts½ �T Dptrs½ � Brs½ � dady;

Kerrsp

h i¼Zae

0

Zbe

0

Brs½ �T Dprrs

� �Brs½ � dxdy;

Ketsv

� �¼ hv

Zbe

0

Zae

0

Bts½ �T Bts½ � dxdy;

Ketrsv

� �¼Zbe

0

Zae

0

Bts½ �T Dtrsv½ � Brs½ � dxdy;

Kerrsv

� �¼Zbe

0

Zae

0

Brs½ �T Drrsv½ � Brs½ � dxdy

ð32Þ

The various rigidity matrices and rigidity vectors

involved in the elemental matrices of Eqs. (30)–(32)

are explicitly presented in Appendix 2. It may be noted

that in case of an element without integrated with the

ACLD treatment, the vectors {Fetp} and {Fe

rp} of Eqs.

(26) and (27) turns out to be the null vectors. The

elemental equations of motion are assembled in the

straight forward manner into the global space to obtain

the open-loop coupled global equations of motion of

the overall smart MEE shell integrated with the

patches of the ACLD treatment as follows:

M½ � X::n oþ Ktt½ � Xf g þ Ktr½ � Xrf g þ Kt/

� �Uf g

þ Ktw� �

wf g þ Ktsv½ �Z t

0

Gðt� sÞ o

osXf gds

þ Ktrsv½ �Z t

0

Gðt� sÞ o

osXrf gds

¼ Ff g �Xq

j¼1

Fjtp

n oþ Fj

tpn

n o �Vj

ð33Þ

Ktr½ �T Xf g þ Krr½ � Xrf g

þ Ktrsv½ �TZ t

0

Gðt� sÞ o

osXf gds

þ Krrsv½ �Z t

0

Gðt� sÞ o

osXrf gdsþ Kr/

� �Uf g

þ Krw� �

wf g

¼ �Xq

j¼1

Fjrp

n oVj

ð34Þ

Kt/� �T

Xf g þ Kr/� �T

Xrf g � K//� �

Uf g ¼ 0 ð35Þ

Ktw� �T

Xf g þ Krw� �T

Xrf g � Kww� �

wf g ¼ 0 ð36Þ

where [M] is the global mass matrix; [Ktt], [Ktr],

[Krr], [Ktsv], [Ktrsv] and [Krrsv] are the global elastic

stiffness matrices, [Kt/] and [Kr/] are the global

electro-elastic coupling stiffness matrices, [Ktw] and

[Krw] are the global magneto–elastic coupling stiff-

ness matrices, [K//] is the global electrical stiffness

matrix, [Kww] is the global magnetic stiffness matrix,

{Ftp} and {Frp} are the global electro-elastic coupling

vectors; {F} is the global nodal mechanical load


123

vector, respectively, {X} and {Xr} are the global

generalized nodal displacement vectors, {A} and {w}

are the global nodal generalized electrical potential

and magnetic potential vectors, Vj is the applied

control voltage to the jth ACLD patch and q is the

number of patches. In the absence of the applied

control voltage, the coupled global equations derived

above also govern the passive (uncontrolled) con-

strained layer damping of the MEE shell. After

invoking the boundary conditions, the global electric

and magnetic potential vectors {A} and {w} are

condensed and subsequently taking the Laplace

transform of resulting equations, the following global

equations in the Laplace domain are obtained:

s2 M½ � ~Xt

� �þ L K1½ � Xtf g þ K2½ � Xrf gð Þ

þ Ktsv½ �s ~GðsÞ ~Xt

� �þ Ktrsv½ �s ~GðsÞ ~Xr

� �

¼ ~F� ��Xq

j¼1

Fjtp

n o~V

j � LXq

j¼1

Fjtpn

n oVj

!

ð37Þ

L K3½ � Xtf gð Þ þ K4½ � ~Xr

� �þ Ktrsv½ �Ts ~GðsÞ ~Xt

� �þ Krrsv½ �s ~GðsÞ ~Xr

� �¼ �

Xq

j¼1

Fjrp

n o~V

j

ð38Þ

in which, the augmented matrices are given by

K1½ � ¼ Ktt½ � þ Kt/� �

K//� ��1

Kt/� �T

þ Ktw� �

Kww� ��1

Ktw� �T

;

K2½ � ¼ Ktr½ � þ Kt/� �

K//� ��1

Kr/� �T

þ Krw� �

Kww� ��1

Krw� �T

;

K3½ � ¼ Krt½ � þ Kr/� �

K//� ��1

K/t

� �Tþ Krw� �

Kww� ��1

Kwt

� �Tand

K4½ � ¼ Krr½ � þ Kr/� �

K//� ��1

Kr/� �T

þ Krw� �

Kww� ��1

Krw� �T

ð39Þ

where, L is the Laplace operator while { ~Xt}, { ~Xr},

{ ~F} and ~Vj

are Laplace transforms of {Xt}, {Xr}, {F}

and Vj, respectively. The term s ~GðsÞ is referred to as

the material modulus function of the viscoelastic

material in the Laplace domain. Making use of the

GHM model for the viscoelastic material in the time

domain, the material modulus function can be

expressed by a series of mini-oscillator terms (Mc

Tavish and Hughes 1993) as follows:

s ~GðsÞ ¼ G1 1þXN

k¼1

aks2 þ 2nkxks

s2 þ 2nkxksþ x2k

" #ð40Þ

in which, G1 is the equilibrium value of the modulus

i.e. the final value of the relaxation G(t). Every mini-

oscillator term is a second-order rational function

consisting of three positive constants ak, xk and nk.

These constants influence the shape of the modulus

function in the complex s-domain. Taking into con-

sideration of a GHM material modulus function with a

single mini-oscillator term (Mc Tavish and Hughes

1993) i.e.,

s ~GðsÞ ¼ G1 1þ as2 þ 2nxs

s2 þ 2nxsþ x2

" #ð41Þ

the auxiliary dissipation coordinates Z and Zr

are introduced as follows (Mc Tavish and Hughes

1993):

s ~GðsÞ ~Xt

� �¼ G1 1þ að Þ ~Xt

� �� a ~Z sð Þ

� �and

s ~GðsÞ ~Xr

� �¼ G1 1þ að Þ ~Xr

� �� a ~Zr sð Þ

� �ð42Þ

~ZðsÞ ¼ x2

s2 þ 2nxsþ x2Xtf g and

~ZrðsÞ ¼x2

s2 þ 2nxsþ x2Xrf g;

ð43Þ

where, ~ZðsÞ and ~ZrðsÞ are the Laplace transforms of

Z and Zr, respectively. Using Eqs. (41) and (42) in

Eqs. (37) and (38), it can be written that

s2 M½ � ~Xt

� �þ L K1½ � Xtf g þ K2½ � Xrf gð Þ

þG1 Ktsv½ � ð1þ aÞ ~Xt

� �� a~zðsÞ

� �þG1 Ktrsv½ � ð1þ aÞ ~Xr

� �� a~zrðsÞ

� �

¼ ~F� ��Xq

j¼1

Fjtp

n o~V

j � LXq

j¼1

Fjtpn

n oVj

!

ð44Þ

L K3½ � Xtf gð Þ þ K4½ � ~Xr

� �þ Ktrsv½ �Ts ~GðsÞ ~Xt

� �

þ Krrsv½ �s ~GðsÞ ~Xr

� �¼ �

Xq

j¼1

Fjrp

n o~V

j

ð45Þ


123

Taking inverse Laplace transforms of Eqs. (43)–(45)

and condensing the global degrees of freedom {Xr}

from the resulting equations in the time domain, the

following equations are obtained:

M½ � Xt

::n oþ Kx½ � Xtf g þ Kz½ � ZðtÞf g þ Kzr½ � Zrf g

¼ Ff g �Xq

j¼1

Fjp

n oVj

ð46Þ

Z::n oþ 2nx _Z

� �þ x2 Zf g � x2 Xtf g ¼ 0 ð47Þ

Zr

::n oþ 2nx _Zr

� �þ x2 K5½ � Xtf g � x2 K6½ � Zf g

� x2 K7½ � Zrf g ¼ �Xq

j¼1

Fjpz

n oVj

ð48Þ

in which,

Kx½ � ¼ �Ktt½ � � �Ktr½ � �Krr½ ��1Ktr½ �T;

Kz½ � ¼ aG1 �Ktr½ � �Krr½ ��1Ktrsv½ �T�aG1 Ktsv½ �;

Kzr½ � ¼ aG1 �Ktr½ � �Krr½ ��1Krrsv½ � � aG1 Ktrsv½ �;

Fjp

n o¼ � �Ktr½ � �Krr½ ��1

Fjrp

n oþ Fj

tp

n oþ Fj

tpn

n o;

�Ktt½ � ¼ K1½ � þG1 1þ að Þ Ktsv½ �;�Ktr½ � ¼ K2½ � þG1 1þ að Þ Ktrsv½ �T;�Krt½ � ¼ K3½ � þG1 1þ að Þ Krrsv½ �T;�Krr½ � ¼ K4½ � þG1 1þ að Þ Krrsv½ �;K5½ � ¼ �Krr½ ��1 �Ktr½ �T

K6½ � ¼ aG1 �Krr½ ��1Ktrsv½ �T;

K7½ � ¼ Izr � aG1 �Krr½ ��1Krrsv½ �;

Fjpz

n o¼ x2 �Krr½ ��1

Fjrp

n o

ð49Þ

Now, Eqs. (46)–(48) are combined ***to obtain the

global open-loop equations of motion in the time

domain as follows:

M�½ � X::n oþ C�½ � _X

� �þ K�½ � Xf g

¼ F�f g þXq

j¼1

�Fjp

n oVj; ð50Þ

where,

M�½ � ¼M½ � 0 0

0 IZ 0

0 0 IZr

264

375; C�½ � ¼

0 0 0

0 2nx 0

0 0 2nx

264

375;

K�½ � ¼Kx½ � Kz½ � Kzr½ ��x2 x2 0

x2 K5½ � �x2 K6½ � x2 K7½ �

264

375

F�f g ¼Ff g0

0

8><>:

9>=>;; �F

jp

n o¼

Fjp

n o0

Fjpz

n o

8>><>>:

9>>=>>;;

Xf g ¼Xtf gZ

Zr

8><>:

9>=>;

ð51Þ

4 Closed loop model

The control voltage Vj for activating the constraining

layer of each ACLD patch is supplied according to a

simple velocity feedback control law. Thus, the

control voltage for each patch can be represented in

terms of the derivatives of the global nodal degrees of

freedom as follows:

Vj ¼ �Kd _w0ðxj; yjÞ ¼ �Kd Ujt

h i_X

� �ð52Þ

where Kd is the control gain for the jth ACLD patch

and [Ujt] is the unit vector for expressing the transverse

velocity of a point (xj, yj) in terms of the time

derivative of the global nodal generalized translational

displacements. Substituting Eq. (52) into Eq. (50), the

final global equations of motion governing the closed

loop dynamics of the MEE shell integrated with the

ACLD system can be obtained as follows:

M�½ � X::n oþ C�d� �

_X� �

þ K�½ � Xf g ¼ F�f g ð53Þ

in which, [C�d] is the active damping matrix and is

given by

C�d� �

¼ C�½ � þXq

j¼1

Kd�F

jp

n oUj

t

h i

5 Results and discussions

In order to investigate the performance of the ACLD

treatment for controlling the geometrically nonlinear

vibrations of the MEE doubly curved shells, numerical


123

results are evaluated using the three dimensional finite

element model derived in the preceding section. The

MEE shell considered for computing the numerical

results is integrated either with a single ACLD patch as

illustrated in Fig. 1(a) or two identical patches placed

on the opposite edges of the shell as shown in

Fig. 1(b). The length and the width of the single

ACLD patch is a/2 and b/2 while that of the patches

placed on the edges are a/2 and b/4, respectively. The

total volume of the two patches is equal to that of the

single patch placed at the center. The material of the

constraining layer of the ACLD treatment is PZT-5H/

spur epoxy composite with 60 % piezoelectric fiber

volume fraction. The elastic and the piezoelectric

properties of this constraining 1–3 PZC layer are as

follows (Ray and Pradhan 2006):

CP11 ¼ 9:29GPa; CP

12 ¼ 6:18GPa;

CP13 ¼ 6:05GPa; CP

33 ¼ 35:44GPa; CP23 ¼CP

13;

CP44 ¼ 1:58GPa; CP

55 ¼ 1:54GPa;

CP55 ¼CP

44; e31 ¼�0:1902C=m2;e33 ¼ 18:4107C=m2

The thicknesses of the constraining 1–3 PZC layer and

the viscoelastic layer are considered to be 250 and

200 lm, respectively. Unless otherwise stated, the

aspect ratio (a/H) of the MEE shell is considered to be

200 and the controlled responses are computed using

the vertically reinforced 1–3 PZC (i.e. k = 0�). The

substrate shell is modeled by three layers of equal

thickness (h = 0.001 m) with commonly used stack-

ing sequences (i.e., P/M/P and M/P/M). The material

properties of the layers of the MEE shells are listed in

Table 1. The density of both the constituent materials

(BaTiO3 and CoFe2O4) of the MEE substrate is

1600 kg/m3. In order to consider single term GHM

expression, the values of a, n and x are used as 11.42,

20 and 1.0261 9 105, respectively (Young-Hun et al.

2002). The shear modulus ðG1Þ and the density of the

viscoelastic material are used as 1.822 9 106 Pa and

1140 kg/m3, respectively (Young-Hun et al. 2002).

The simply supported boundary conditions employed

in the present analysis are given by

v0 ¼ w0 ¼ hy ¼ /y ¼ cy ¼ hz ¼ /z ¼ 0;

at x ¼ 0; a and

u0 ¼ w0 ¼ hx ¼ /x ¼ cx ¼ hz ¼ /z ¼ 0; ;

at y ¼ 0; b

ð54Þ

Ta

ble

1M

ater

ial

pro

per

ties

of

Ba

TiO

3an

dC

oF

e 2O

4(P

an2

00

1)

C11

=C

22

(10

9N

/m2)

C12

(10

9N

/m2)

C13

=C

23

(10

9N

/m2)

C33

(10

9N

/m2)

C44

=C

55

(10

9N

/m2)

C66

(10

9N

/m2)

q(k

g/m

3)

BaT

iO3

16

67

77

81

62

43

44

.55

,800

Co

Fe 2

O4

28

61

73

17

0.5

26

9.5

45

.35

6.5

5,3

00

BaT

iO3

e 31

=e 3

2(C

/m2)

-4

.4

e 33

(C/m

2)

18

.6

e 24

=e 1

5(C

/m2)

11

.6

[ 11

=[ 2

2(1

0-

9C

2/N

m2)

11

.2

[ 33

(10

-9

C2/N

m2)

12

.6

l 11

=l 2

2(1

0-

6N

s2/C

2)

5

l 33

(10

-6

Ns2

/C2)

10

Co

Fe 2

O4

q31

=q

32

(N/A

m)

18

0.3

q33

(N/A

m)

69

9.7

q24

=q

15

(N/A

m)

55

0

[ 11

=[ 2

2(1

0-

9C

2/N

m2)

0.0

8

[ 33

(10

-9

C2/N

m2)

0.0

93

l 11

=l 2

2(1

0-

6N

s2/C

2)

-5

90

l 33

(10

-6

Ns2

/C2)

15

7


123

In all cases the electric and the magnetic potentials at

the boundaries are assumed to be zero. The imple-

mentation of the GHM method for modeling the

ACLD treatment in case of MEE shell is validated

following the methodology prescribed by Lim et al.

(2002). Accordingly, the GHM method and the

conventional complex modulus approach have been

implemented independently for modeling the visco-

elastic material in computing the linear frequency

responses for the transverse displacement of the

doubly curved MEE shells. First few modes of the

shells are excited by applying a time harmonic load of

amplitude 1 N. The frequency response functions

obtained by the GHM method as well as by the

standard complex modulus approach for the simply

supported MEE shell with P/M/P and M/P/M stack-

ing sequences have been demonstrated in Figs. 3 and

4, respectively. It may be observed from these figures

that the frequency response curves obtained by both

the approaches are almost overlapped. Therefore, it

may be considered that the present approach of

modeling the viscoelastic material by the GHM

method accurately estimates the damping characteris-

tics of the overall MEE doubly curved shells. The

finite element model derived here can also be used for

studying the laminated composite plates by replacing

the layers of the MEE shell with the conventional

composite layers. Hence, in order to verify the validity

of the present finite element model, the load deflection

curve for laminated composite shell with infinite radii

of curvature (R1 = 1, R2 = R1) integrated with the

inactivated ACLD patches of negligible thickness are

computed by the present model and subsequently,

compared with the identical laminated composite

plates studied by Sarangi and Ray (Sarangi and Ray

2011). Figure 5 illustrates this comparison for the

simply supported laminated composite plates. It may

be observed that the results are in excellent agreement.

0 100 200 300 400 500 600 700 800 900 100010

-6

10-4

10-2

100

102

Frequency (Hz)

w/H

Complex modulus approachGHM Approach

Fig. 3 Linear frequency

responses for the center

deflection of a simply

supported (SS) P/M/P-MEE

doubly curved shell

obtained by the GHM

method and the complex

modulus approach

(a = 200H, R1/a = 10, R2/

R1 = 10)

0 100 200 300 400 500 600 700 800 900 1000

10-4

10-2

100

102

Frequency (Hz)

w/H

Complex modulus approachGHM Approach

Fig. 4 Linear frequency

responses for the center

deflection of a simply

supported (SS) M/P/M-MEE doubly curved shell

obtained by the GHM

method and the complex

modulus approach

(a = 200H, R1/a = 10, R2/

R1 = 10)


123

It is worth mentioning that to the authors’ best

knowledge no solution has been reported in the

literature concerned with the active control of geo-

metrically nonlinear vibrations of the MEE doubly

curved shells. Thus, in order to validate further, the

results of the doubly curved shell with infinite radii of

curvature (R1 = 1, R2 = R1) are considered for the

comparison with the results of the MEE square plate

available in the literature (Sladek et al. 2013). The

variation of non dimensional vertical displacement w/

H at the center of the shell with respect to non-

dimensional load parameter q0 = qa4/C11H4 for the

MEE shell integrated with the inactivated ACLD

patches of negligible thickness are first computed and

subsequently compared with the results of identical

square plate reported by Sladek et al. (2013).

Figures 6(a) and (b) illustrate this comparison for the

simply supported and clamped–clamped P/M/P-MEE

shell respectively. In addition, linear solution is also

plotted to demonstrate the nonlinear stiffening effect

in the shell. It is evident from this figure that the results

are in excellent agreement. The backbone curves

illustrating the variation of the frequency ratio (xnl/xl)

with the non-dimensional transverse deflection (wmax/

H) to determine the magnitude of the exiting pulse

load which is responsible for causing geometrically

nonlinear deflections are plotted in Fig. 7. It may be

noticed from this figure that the paraboloid MEE shells

with R2/R1 = 10 and R1/a = 10 exhibit hardening

type nonlinearity. It is also noticed from this figure that

for causing geometrically nonlinear vibrations, the

magnitude of the exciting pulse load should be such

that the value of wmax/H be greater than 1. Hence, for

the purpose of computing the nonlinear transient

responses, the value of applied pulsed load is

arbitrarily considered as 3.2 and 3.8 kPa for P/M/P

and M/P/M sequences, respectively, such that the

amplitude ratio is more than 1 and substantial

nonlinearity is present in the uncontrolled responses.

The effect of curvature ratio (R1/a) on the response of

nonlinear frequency ratio (xnl/xl) of spherical (R2/

R1 = 1) MEE shell has been investigated in Fig. 8(a).

It is observed that the increase in the value of R1/a

increases the hardening type non linearity. Figure 8(b)

illustrates that the effect of curvature aspect ratio

(R2/R1) on the frequency ratio (xnl/xl). It is evident

from this figure that as the amplitude ratio increases,

the geometric nonlinearity in the hyperboloid shell

changes from softening type to hardening type. In

addition, the influence of the aspect ratio (a/H) on the

fundamental nonlinear frequency ratio (xnl/xl) for

paraboloid and hyperboloid shells have been tabulated

in Table 2. It may be observed from this table that the

frequency ratio increases with the increase in the

aspect ratio in case of paraboloid shells while this

trend is reverse in case of hyperboloid shells with both

P/M/P and M/P/M sequences. It may also be noted

that the frequency ratio increases with the increase in

the non-dimensional transverse deflection (wmax/H)

for both paraboloid and hyperboloid shells. The

control voltages supplied to the ACLD patches are

negatively proportional to the velocity of the point (a/

2, b/2, H/2) on the top of the substrate. The arbitrary

values of control gain are chosen for the admissible

control of nonlinear vibrations of the MEE shells

while maintaining the acceptable control voltage.

Figures 9 and 10 illustrate the nonlinear transient

responses of a simply supported MEE shells with P/

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Load (N/m2)

w/H

Nonlinear (Present)Sarangi and Ray (2011)Linear (Present)Sarangi and Ray (2011)

Fig. 5 Comparison of load

deflection curve of simply

supported MEE doubly

curved shell (with

composite properties and

R1 ? ?, R2 = R1) with

that of identical laminated

composite square plate


123

M/P and M/P/M stacking sequences, respectively,

integrated with a patch at the center or with the two

patches at the edges of the shell undergoing ACLD.

The transient responses presented in these figures are

corresponding to the cases when the patches are

passive (Kd = 0) and active (Kd = 0). In both the

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5(a)

(b)

qa4/C11

H4

w/H

Sladek et al. (2013)Present shell ( R

1= , R

2= R

1)

Linear ( R1= , R

2= R

1)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

qa4 / C11

H4

w/H

Sladek et al. (2013)Present shell ( R

1= , R

2= R

1)

Linear ( R1= , R

2= R

1)

Fig. 6 a Variation of the

central deflection of the

simply supported P/M/P-

MEE doubly curved shell

(with R1 ? ?, R2 = R1)

with the transverse load.

b Variation of the central

deflection of the clamped–

clamped P/M/P-MEE

doubly curved shell (with

R1 ? ?, R2 = R1) with the

transverse load

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.450

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency Ratio (nl

/l)

wm

ax /H

P/M/P-SSM/P/M-SSP/M/P-CCM/P/M-CC

Fig. 7 Backbone curves for

the simply supported (SS)

and clamped–clamped (CC)

MEE doubly curved shells

with P/M/P and M/P/Msequences (R1/a = 10, R2/

R1 = 10)


123

cases, the shells are set into undergoing nonlinear

vibrations with same maximum amplitude of uncon-

trolled response (Kd = 0) while the magnitudes of the

maximum control voltages are same. Figures 11 and

12 illustrate the corresponding control voltages

required for the active control of responses shown in

Fig. 9 and 10, respectively. It is observed from these

figures that the active ACLD treatment appreciably

improves the damping characteristics and has consid-

erable effect on the suppression of geometrically

nonlinear vibrations of the overall shells over the

passive damping with nominal control voltage. Also, it

may be observed from these figures that the perfor-

mance of the single ACLD patch placed at the center

of the top surface of the shell for attenuating the

nonlinear transient vibrations is considerably better

than that of the patches located at the edges.

Figures 13 and 14 illustrate the comparison of

controlled nonlinear transient responses of simply sup-

ported paraboloid shells (R2/R1 = 10) and hyperboloid

0.95 1 1.05 1.1 1.15 1.2 1.25 1.30

0.5

1

1.5

2

Frequency ratio(nl

/l)

wm

ax/H

1 1.1 1.2 1.3 1.40

0.5

1

1.5

2

Frequency ratio(nl

/l)

wm

ax/H

R1/a = 10, R

2/R

1 = -0.75

R1/a = 10, R

2/R

1 = -1

R1/a = 10, R

2/R

1 = 0.5

R1/a = 10, R

2/R

1 = 1

R1/a = 2, R

2 = R

1

R1/a = 5, R

2 = R

1

R1/a = 10, R

2 = R

1

R1/a = 20, R

2 = R

1

(a) (b)

Fig. 8 Influence of curvature ratio and curvature aspect ratio on the nonlinear frequency ratio (xnl/xl) for the simply supported P/M/P-MEE doubly curved shells (a = 200H)

Table 2 Effect of aspect ratio (a/H) of the MEE doubly curved paraboloid shell (R1/R2 = 1) and hyperboloid shell (R1/R2 = -1) on

the frequency ratio (xnl/xl) of the shells

Layer Sequence Aspect ratio (a/H) Doubly curved shell Amplitude ratio (wmax/H)

0.4 0.8 1.2 1.6

P/M/P 20 R1 = R2 1.0457 1.1250 1.2277 1.3445

R1 = -R2 1.0216 1.0823 1.1726 1.2822

50 R1 = R2 1.0646 1.1534 1.2605 1.3805

R1 = -R2 1.0195 1.0756 1.1621 1.2718

100 R1 = R2 1.0688 1.1513 1.2446 1.3462

R1 = -R2 1.0178 1.0693 1.1498 1.2534

M/P/M 20 R1 = R2 1.0415 1.1137 1.2076 1.3149

R1 = -R2 1.0194 1.0741 1.1561 1.2562

50 R1 = R2 1.0600 1.1423 1.2417 1.3535

R1 = -R2 1.0175 1.0679 1.1462 1.2461

100 R1 = R2 1.0657 1.1441 1.2326 1.3290

R1 = -R2 1.0159 1.0620 1.1345 1.2286


123

shells (R2/R1 = -10) with P/M/P and M/P/M

sequences, respectively, using the single ACLD patch

located at the center of the shell. It is evident from

these figures that the paraboloid shell possesses better

damping characteristics with both P/M/P and M/P/M

configurations than the hyperboloid MEE shell. Fig-

ure 15 illustrates the control voltages necessary for the

active control of responses for P/M/P paraboloid and

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

w/H

Uncontrolled (Kd = 0)

Center patch (Kd = 200)

Edge patches (Kd = 200)

Fig. 9 Nonlinear transient

responses of a simply

supported P/M/P-MEE

doubly curved shell

undergoing ACLD for

different locations of the

ACLD patches (R1/a = 10,

R2/R1 = 10)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

w/H

Uncontrolled (Kd = 0)





supported M/P/M-MEE

doubly curved shell

undergoing ACLD for

different locations of the

ACLD patches (R1/a = 10,

R2/R1 = 10)

0 0.05 0.1 0.15 0.2 0.25 0.3

-300

-200

-100

0

100

200

300

Time (sec)

Con

trol

vol

tage

(V

olt)



Fig. 11 Comparison of

control voltages required for

the ACLD of nonlinear

transient vibrations of the

simply supported P/M/P-MEE doubly curved shell

for the different locations of

the patches (R1/a = 10, R2/

R1 = 10)


123

hyperboloid shells. Consequently, further investiga-

tion is carried out for the paraboloid MEE shell

integrated with the single ACLD patch at the center of

the top surface of the MEE doubly curved shell only.

Figures 16 and 17 illustrate the transient responses for

the clamped–clamped MEE shells with P/M/P and M/

P/M stacking sequences, respectively using the single

patch at the center of the top surface of the shells. In

0 0.05 0.1 0.15 0.2 0.25 0.3

-200

-100

0

100

200

275

Time (sec)

Con

trol

vol

tage

(V

olt)







simply supported M/P/M-MEE doubly curved shell

for the different locations of

the patches (R1/a = 10, R2/

R1 = 10)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

w/H

Paraboloid (R2= 10R

1)

Hyperboloid (R2= -10R

1)



supported P/M/P-MEE

doubly curved shell

undergoing ACLD (R1/

a = 10, Kd = 200)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

w/H

Paraboloid (R2= 10R

1)


1)



supported M/P/M-MEE

doubly curved shell


a = 10, Kd = 150)


123

these cases also, the ACLD patch effectively attenu-

ates the nonlinear vibrations of the shells and enhances

the damping characteristics of the shells with a

nominal control voltage shown in Fig. 18. Since, the

control voltage is proportional to the velocity of the

center of the shell the depiction of the control voltage

0 0.05 0.1 0.15 0.2 0.25 0.3-300

-200

-100

0

100

200

300

Time (sec)

Con

trol

vol

tage

(V

olt)

Paraboloid (R2= 10R

1)


1)





simply supported P/M/P-


(R1/a = 10, Kd = 200)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (sec)

w/H

Kd = 0

Kd = 200


responses of a clamped–

clamped P/M/P-MEE

doubly curved shell


a = 10, R2/R1 = 10)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (sec)

w/H

Kd = 0

Kd = 150


responses of a clamped–

clamped M/P/M-MEE

doubly curved shell


a = 10, R2/R1 = 10)


123

in Fig. 18 signifies that the velocity at the concerned

point of the overall shell also reduces with time. The

phase plots shown in Fig. 19 confirm the same

demonstrating the stability of the shell. The constitu-

tive relations given by Eq. (11) reveal that the

significant improvement in damping characteristics

of the MEE shell may be attributed to the transverse

actuation by the active constraining layer of the ACLD

patch. This can be affirmed from the nonlinear

transient responses obtained by considering the

numerical value of the piezoelectric coefficient e33

as zero or nonzero. Figure 20 illustrates such transient

responses for the simply supported P/M/P MEE shell

with (e33 = 18.4107 C/m2) and without (e33 = 0)

taking into account of the induced transverse normal

stress in the active PZC constraining layer of the

ACLD patch. It may be noticed from this figure that

the active damping of the geometrically nonlinear

vibrations of the MEE shell due to the transverse

actuation (e33 = 18.4107 C/m2) by the active con-

straining layer of the ACLD patch is significantly

higher than that due to the in-plane actuation (e33 = 0)

by the same. Figure 21 illustrates the comparison

between the linear and nonlinear controlled responses

0 0.05 0.1 0.15 0.2 0.25 0.3-250

-200

-150

-100

-50

0

50

100

150

200

250

Time (sec)

Con

trol

vol

tage

(V

olt)

P/M/P, Kd = 200

M/P/M, Kd = 150

Fig. 18 Control voltages

required for the ACLD of

nonlinear transient

vibrations of the clamped–

clamped MEE doubly

curved shells (R1/a = 10,

R2/R1 = 10)

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Displacement (m)

Vel

ocit

y (m

/sec

)



0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Displacement (m)

Vel

ocit

y (m

/sec

)



SS

P/M/P M/P/M

Fig. 19 Phase plots for the simply supported MEE doubly curved shells when the ACLD patch controls the nonlinear vibrations of the

doubly curved shells (‘‘S’’ is the starting point, R1/a = 10, R2/R1 = 10)


123

of the simply supported P/M/P shell. The shell is

subjected to a transverse pulse load (3.2 kPa) that

causes substantial nonlinear deformations according

to the back-bone curve of the shell (Fig. 7). It may be

observed from Fig. 21 that the magnitude of nonlinear

responses is much smaller than that of the linear

responses and thus this figure corroborates that the

MEE shell undergoing geometrically nonlinear vibra-

tions exhibits hardening type nonlinearity. Particular

emphasis has been placed to investigate the effect of

variation of the piezoelectric fiber orientation angle

(k) in the obliquely reinforced 1–3 PZC constraining

layer on the control authority of the ACLD patch.

Accordingly, the piezoelectric fibers are considered to

be coplanar either with the vertical xz-plane or with

the vertical yz-plane, while the value of k is varied

with respect to vertical z-axis from -45� to 45�. To

quantify the performance of the ACLD patch for

controlling the large deflections of the MEE shell, a

performance index (Id) for computing the control

authority of the obliquely reinforced 1–3 PZC con-

straining layer is defined as follows:

Id ¼w a

2; b

2; H

2; 0

� � w a

2; b

2; H

2; 0:1

� w a

2; b

2; H

2; 0

� � 100 ð55Þ

In which, Id measures the percentage suppression of

the amplitude at the point (a/2, b/2, H/2) of the overall

MEE shell undergoing nonlinear transient vibrations

after 0.1 s from the initial time. For a particular value

of the mechanical load and maximum control voltage,

Figs. 22 and 23 illustrate that the control authority (i.e.

the performance index Id) of the ACLD patch varies

with the variation of the piezoelectric fiber orientation

angle (k) in the xz-plane while becoming maximum at

k = 0� for simply supported as well as clamped–

clamped MEE paraboloid shell and hyperboloid shell,

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (sec)

w/H

Linear (Kd = 200)

Nonlinear (Kd = 200)

Fig. 21 Linear and

nonlinear transient

responses of a P/M/P-MEE

doubly curved shell


a = 10, R2/R1 = 10)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

w/H

e31= -0.1902 C/m2 , e33=18.4107 C/m2

e31 = -0.1902C/m2 , e33 = 0

Fig. 20 Contributions of

in-plane and transverse

actuations by the PZC layer

in the controlled response of

the P/M/P-MEE doubly

curved shell undergoing

geometrically nonlinear

vibration (Kd = 200, R1/

a = 10, R2/R1 = 10)


123

respectively. It is observed from this figure that the

performance of the ACLD patch is independent of the

sign of the piezoelectric fiber orientation angle (k) in

the 1–3 PZC constraining layer. It may also be noted

that if the matrices [Kt/], [K//], [Ktw] and [Kww] are

set to null matrices; the nonlinear transient responses

of the MEE shell will be free of the effects of the

electro–elastic and magneto–elastic couplings. Fig-

ures 24 and 25 illustrate the effects of electro-elastic

coupling and magneto–elastic coupling on the con-

trolled responses of a simply supported MEE shell for

P/M/P and M/P/M sequences, respectively. It may be

observed from Fig. 24 that the electro-elastic and the

magneto–elastic couplings marginally affect the non-

linear transient response of the P/M/P paraboloid shell

(R2/R1 = 10). It may also be noticed from this figure

that the electro-elastic coupling is higher than the

magneto–elastic coupling for P/M/P paraboloid shell

while these effects are negligible in case of the M/P/M

paraboloid shell (Fig. 25).

6 Conclusions

A three dimensional finite element analysis has been

carried out to analyze the ACLD of geometrically

nonlinear vibrations of the MEE doubly curved shells

integrated with the patches of the ACLD treatment.

The constraining layer of the ACLD treatment is

composed of the vertically/obliquely reinforced 1–3

PZC materials. The kinematics of deformations of the

overall MEE doubly curved shell is described by a

layerwise shear deformation theory and geometric

nonlinearity is modeled by using the von Karman type

strain–displacement relations. The GHM approach is

implemented for modeling the viscoelastic layer in the

-45 -40 -30 -20 -10 0 10 20 30 40 45

10

15

20

25

30

35

Piezoelectric fiber orientation angle '

I d

P/M/P-SSP/M/P-CCM/P/M-SSM/P/M-CC

Fig. 23 Variation of the

control authority of the

ACLD patch with the

piezoelectric fiber

orientation angle (k) when

the piezoelectric fibers are

coplanar with the xz-plane

for active damping of


vibrations of the MEE

hyperboloid doubly curved

shells (R1/a = 10, R2/

R1 = -10)

-45 -40 -30 -20 -10 0 10 20 30 40 455

10

15

20

25

30

35

40

Piezoelectric fiber orientation angle '

I d

P/M/P-SS P/M/P-CC M/P/M-SS M/P/M-CC

Fig. 22 Variation of the

control authority of the

ACLD patch with the

piezoelectric fiber

orientation angle (k) when

the piezoelectric fibers are

coplanar with the xz-plane

for active damping of


vibrations of the MEE

paraboloid doubly curved

shells (R1/a = 10, R2/

R1 = 10)


123

time domain. A simple velocity feedback control law

is employed to incorporate the active damping. The

geometrically nonlinear dynamics of the MEE shells

are of hardening type. Also, the nonlinear behavior of

the MEE doubly curved shell is influenced by the

curvature ratio, the curvature aspect ratio and the

thickness aspect ratio. The numerical results suggest

that the ACLD patches significantly improve the

active damping characteristics of the MEE shells over

the passive damping for attenuating the geometrically

nonlinear vibrations of both P/M/P and M/P/M

doubly curved shells. The performance of the single

ACLD patch placed at the center of the top surface of

the MEE shell is significantly larger than that of the

two patches placed at the edges of the shells. Having

same geometrical parameters and material properties,

paraboloid shells exhibit better damping characteris-

tics than the hyperboloid shells. The contribution of

the vertical actuation by the vertically reinforced 1–3

PZC layer for attenuating the geometrically nonlinear

vibrations of the MEE shells is considerably larger

than that of the in-plane actuation by this 1–3 PZC

layer. The performance of the ACLD patch is influ-

enced by the edge boundary conditions and the

variation of the piezoelectric fiber orientation angle

(k) for suppressing the geometrically nonlinear vibra-

tions of the MEE doubly curved shells. The control

authority of the patch becomes maximum for active

damping of geometrically nonlinear vibrations if the

constraining layer of the ACLD patches is made of the

vertically reinforced 1–3 PZC (i.e. k = 0�). The

electro-elastic and the magneto–elastic couplings

marginally affect the nonlinear transient response of

the P/M/P paraboloid shell while the coupling fields

have negligible effect in case of the M/P/M parabo-

loid shell.

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

w/H

UncoupledMagneto-Elastic coupled fieldsElectro-Elastic coupled fieldsElectro-Elastic and Magneto-Elastic coupled fields


coupled and uncoupled

controlled nonlinear

transient responses of a

simply supported P/M/P-MEE doubly curved shell

integrated with a patch at the

center undergoing ACLD

(Kd = 200, R1/a = 10,

R2/R1 = 10)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

w/H

Uncoupled fieldsMagneto-elastic coupled fieldsElectro-Elastic coupled fieldsElectro-Elastic and Magneto-Elastic coupled fields


coupled and uncoupled

controlled nonlinear

transient responses of a

simply supported M/P/M-


integrated with a patch at the

center undergoing ACLD

(Kd = 150, R1/a = 10,

R2/R1 = 10)


123

Appendix 1

Transformation matrices

The transformation matrices Z1½ � � Z5½ � appearing in

Eqs. (6) and (7) are given by

Z1½ � ¼ �Z1 O::

�O

h i;

Z2½ � ¼ �Z2 hvI2 RvI2

� �;

Z3½ � ¼ �I3~O ~O �Z3

� �

Z4½ � ¼ hIr=2 I::

4

~O �Z4

h i;

Z5½ � ¼ hIr=2 hvIr�Iv

�Z5

� �

�Z1 ¼

z 0 0 zR1

z2

R1

0 z 0 zR1

z2

R1

0 0 0 1 2z

0 0 z 0 0

266664

377775;

�Z2 ¼

h2

0 0 zR1

z2

R1

0 h2

0 zR1

z2

R1

0 0 0 1 2z

0 0 h2

0 0

2666664

3777775;

�Z3 ¼z 0 z2 0

0 z 0 z2

" #

Ir ¼�1R1

0

0 �1R2

" #; �I3 ¼

1� zR1

�0

0 1� zR2

�264

375;

I::

4¼

1� z::

R1

�0

0 1� z::

R2

�264

375

�Iv ¼1� Rv

R1

�0

0 1� Rv

R2

�264

375; ~I ¼

1 0 0

0 1 0

0 0 0

264

375;

I::¼

0 0

0 0

0 1

264

375; I ¼

1 0 0

0 1 0

0 0 0

0 0 1

26664

37775; I ¼

1 0 0

0 1 0

0 0 1

264

375;

�Z4 ¼ �Z5 ¼ �Z3; Rv ¼ Z� h

2� hv; z

:: ¼ z� h

2;

I2 ¼ I6 ¼ I

Submatrices

The various submatrices Btbi½ �, Brbi½ �, Btsi½ � and Brsi½ �appearing in Eq. (25) are given by

Btbi½ � ¼

oni

ox 0 1R1

0 oni

oy1

R2

oni

oyoni

ox 0

0 0 0

266664

377775; Btsi½ � ¼

�1R1

0 oni

ox

0 �1R2

oni

oy

" #;

Brbi½ � ¼

Brbi�0 �0 �0

~O I ~O ~O

�0 �0 Brbi�0

�0 �0 �0 Brbi

26664

37775 and

Brsi½ � ¼

~I I::

~0

~0 �0 I

~O Brsi�0

T

�0T

Brsi~O

26664

37775 in which,

Brbi

� �¼

oni

ox 0

0 oni

oy

oni

oyoni

ox

2664

3775; Brsi

� �¼

oni

ox 0 0

oni

oy 0 0

" #;

where, ~0; �0; ~O; �O and O::

are the (3 9 3), (3 9 2),

(2 9 2), (4 9 2) and (3 9 4) null matrices,

respectively.

Appendix 2

The various rigidity matrices and rigidity vectors

appearing in Eqs. (30)–(32) are given by

Dtb½ � ¼X3

k¼1

Zhkþ1

hk

�Csb

� �kdz;

Dtrb½ � ¼X3

k¼1

Zhkþ1

hk

�Csb

� �kZ1½ � dz;

Drrb½ � ¼X3

k¼1

Zhkþ1

hk

Z1½ �T �Csb

� �kZ1½ � dz;

Drrs½ � ¼X3

k¼1

Zhkþ1

hk

Z3½ �T �Css

� �kZ3½ � dz;


123

Dts½ � ¼X3

k¼1

Zhkþ1

hk

�Css

� �kdz;

Dtrs½ � ¼X3

k¼1

Zhkþ1

hk

�Css

� �kZ3½ � dz;

Dptb

� �¼Zh6

h5

�Cpb

� �dz;

Dptrb

� �¼Zh6

h5

�Cpb

� �Z2½ �dz;

Dprrb

� �¼Zh6

h5

Z2½ �T �Cpb

� �Z2½ �dz;

Dpts

� �¼Zh6

h5

�Cps

� �dz;

Dptrs½ � ¼

Zh6

h5

�Cps

� �Z5½ �dz;

Dprrs

� �¼Zh6

h5

Z2½ �T �Cps

� �Z5½ �dz;

Dtsv½ � ¼Zh5

h4

�Cvs

� �dz;

Dtrsv½ � ¼Zh5

h4

�Cvs

� �Z4½ �dz;

Drrsv½ � ¼Zh5

h4

Z4½ �T �Cvs

� �Z4½ �dz;

Dptbs

� �¼Zh6

h5

�Cbs

� �dz;

Dptrbs

� �¼Zh6

h5

�Cbs

� �Z5½ �dz;

Dprtbs

� �¼Zh6

h5

Z2½ �T �Cbs

� �dz;

Dprrbs

� �¼Zh6

h5

Z2½ �T �Cbs

� �Z5½ �dz;

Dbtp

n o¼Zh6

h5

1

hpep

b

� �dz;

Dbrp

n o¼Zh6

h5

1

hpZ2½ �T ep

b

� �dz;

Dstp

n o¼Zh6

h5

1

hpep

s

� �dz;

Dsrp

n o¼Zh6

h5

1

hpZ5½ �T ep

s

� �dz;

Dtw� �

¼Zh3

h2

qsb

� � 1

hdz;

Dt/� �

¼Zh4

h3

esb

� � 1

h1 0½ �dz

þZh2

h1

esb

� � 1

h0 1½ �dz;

Drw� �

¼Zh3

h2

Z1½ �T qsb

� � 1

hdz; Dww

� �¼ 1

hl33;

Dr/� �

¼Zh4

h3

z1½ �T esb

� � 1

h1 0½ �dz

þZh2

h1

z1½ �T esb

� � 1

h0 1½ �dz;

D//� �

¼ 2s33

h

1 0

0 1

� :

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