Thin-Walled Structures 67 (2013) 63–77

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Thin-Walled Structures

0263-82

http://d

n Corr

E-m

pellif20

journal homepage: www.elsevier.com/locate/tws

Nonlinear vibrations of functionally graded cylindrical shells

Matteo Strozzi, Francesco Pellicano n

Department of Engineering ‘‘Enzo Ferrari’’, University of Modena and Reggio Emilia, Via Vignolese 905, 41125 Modena, Italy

a r t i c l e i n f o

Article history:

Received 17 February 2012

Received in revised form

8 January 2013

Accepted 24 January 2013

Keywords:

Circular cylindrical shells

Functionally graded materials

Nonlinear vibrations

31/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.tws.2013.01.009

esponding author. Tel.: þ39 059 205 6154; fa

ail addresses: francesco.pellicano@unimore.it,

00@yahoo.it (F. Pellicano).

a b s t r a c t

In this paper, the nonlinear vibrations of functionally graded (FGM) circular cylindrical shells are

analysed. The Sanders–Koiter theory is applied to model the nonlinear dynamics of the system in the

case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal,

circumferential and radial displacement fields. Simply supported, clamped and free boundary condi-

tions are considered. The displacement fields are expanded by means of a double mixed series based

on Chebyshev orthogonal polynomials for the longitudinal variable and harmonic functions for the

circumferential variable. Both driven and companion modes are considered; this allows the travelling-

wave response of the shell to be modelled. The model is validated in the linear field by means of data

retrieved from the pertinent literature. Numerical analyses are carried out in order to characterise the

nonlinear response when the shell is subjected to a harmonic external load; a convergence analysis is

carried out by considering a variety of axisymmetric and asymmetric modes. The present study is

focused on determining the nonlinear character of the shell dynamics as the geometry (thickness,

radius, length) and material properties (constituent volume fractions and configurations of the

constituent materials) vary.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded materials (FGMs) are composite materialsobtained by combining and mixing two or more different con-stituent materials, which are distributed along the thickness inaccordance with a volume fraction law. Most of the FGMs areemployed in the high-temperature environments because of theirheat shielding capacity.

The idea of FGMs was first introduced in 1984/1987 by a groupof Japanese material scientists [1]. They studied many differentphysical aspects, such as temperature and thermal stress distri-butions, static and dynamic responses.

Loy et al. [2] analysed the vibrations of the cylindrical shellsmade of FGM, considering simply supported boundary conditions.They found that the natural frequencies are affected by theconstituent volume fractions and configurations of the constitu-ent materials.

Pradhan et al. [3] studied the vibration characteristics of FGMcylindrical shells made of stainless steel and zirconia, underdifferent boundary conditions. They found that the naturalfrequencies depend on the material distributions and boundaryconditions.

ll rights reserved.

x: þ39 059 205 6126.

Arshad et al. [4] and Naeem et al. [5] analysed the effect of theboundary conditions on the frequency spectra of isotropic andFGM cylindrical shells; Darabi et al. [6] and Ng et al. [7] carriedout a nonlinear analysis of the dynamic stability under periodicaxial loading. Wu et al. [8] and Haddadpour et al. [9] studied thethermo elastic stability and the thermal effects. Iqbal et al. [10]considered vibrations by using a wave propagation approach.Shah et al. [11] analysed different exponential volume laws.

Amabili et al. [12,13] studied the nonlinear vibrations of FGMdoubly curved shallow shells; they considered the thermal effectand used a higher order shear deformation theory.

Readers interested in deepening the knowledge on shellsbehaviour are suggested to refer to the works of Leissa [14] andYamaki [15]. The first one is mainly concerned with lineardynamics of shells, exhibiting different topologies, materials andboundary conditions. The second one is focused on buckling andpost-buckling of the shells in linear and nonlinear fields. In Refs.[14,15] one can find the most important shell theories, such asDonnell, Reissner, Flugge, Sanders–Koiter, as well as solutionmethods, numerical and experimental results.

A modern treatise on the shells dynamics and stability can befound in Ref. [16], where also FGMs are considered.

Refs. [17,18] are strictly related with the present work. In Ref.[17] the effect of the geometry on the nonlinear vibrations ofisotropic shells was investigated, leading to conclusions similar tothose of the present work. The method of solution used in thepresent work was developed in Ref. [18].

h

O

x

u

w

R

L

v

�

R

zh

Fig. 1. Geometry of the functionally graded shell. (a) Complete shell; (b) cross-

section of the shell surface.

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–7764

Some other works related with the present study should bementioned. In Refs. [19–21] the nonlinear oscillations and stabi-lity of parametrically excited cylindrical shells are analysed.Amabili [22] carried out a comparison of thin shells theories forlarge-amplitude vibrations of circular cylindrical shells. Pellicano[23–26] studied the dynamic instability and sensitivity to geo-metric imperfections of circular cylindrical shells under differentexternal loads. In Refs. [27,28] the effect of the boundary condi-tions on the vibrations of circular cylindrical shells is considered.

In this paper, the nonlinear vibrations of FGM cylindrical shellsare analysed; the Sanders–Koiter theory is applied to model thenonlinear dynamics of the system in the case of finite amplitudeof vibration. The shell deformation is described in terms oflongitudinal, circumferential and radial displacement fields; thetheory considers geometric nonlinearities due to large amplitudeof vibrations. Simply supported, clamped and free boundaryconditions are considered. The FGM is made of a distributionof stainless steel and nickel, the material properties are gradedalong the thickness direction, according to a volume fractionpower-law.

The solution method consists of two steps: (1) linear analysisand eigenfunctions evaluation; (2) nonlinear analysis, using aneigenfunction based expansion.

In the linear analysis, the displacement fields are expanded bymeans of a double series based on harmonic functions for thecircumferential variable and Chebyshev polynomials for the long-itudinal variable; both driven and companion modes are consid-ered, allowing for the travelling-wave response of the shell. A Ritzbased method allows to obtain approximate natural frequencies(eigenvalues) and mode shapes (eigenfunctions).

In the nonlinear analysis, the three displacement fields arere-expanded by using approximate eigenfunctions; an energyapproach based on the Lagrange equations is considered, in orderto reduce the nonlinear partial differential equations to a set ofnonlinear ordinary differential equations.

Numerical analyses are carried out in order to characterise thenonlinear response when the shell is subjected to a harmonicexternal load. A convergence analysis is carried out to obtain thecorrect number of axisymmetric and asymmetric modes able todescribe the actual nonlinear behaviour of the shells.

The effect of the geometry on the nonlinear vibrations of theshells is analysed, i.e., the nonlinear amplitude–frequency curvesare obtained for different geometries. The influence of the con-stituent volume fractions and the configurations of the constitu-ent materials on the natural frequencies and nonlinear dynamicresponses is analysed. The present model is validated in the linearfield (natural frequencies) by means of data extracted from thepertinent literature.

2. Modelling of functionally graded materials

A general material property Pfgm of an FGM depends on thematerial properties and the volume fractions of the constituentmaterials, and it is expressed in the form [2]

Pf gmðT ,zÞ ¼Xk

i ¼ 1

~PiðTÞVf iðzÞ ð1Þ

where ~Pi and Vfi are the material property and the volume fractionof the constituent material i.

The material property ~Pi of a constituent material i can bedescribed as a function of the temperature T (K) by Touloukian’srelation [2]

~PiðTÞ ¼ P0,iðP�1,iT�1þ1þP1,iTþP2,iT

2þP3,iT

3Þ ð2Þ

where P0,i, P�1,i, P1,i, P2,i and P3,i are the coefficients of thetemperature of the constituent material i.

In the case of an FGM thin cylindrical shell with a uniformthickness h and a reference surface at its middle surface, thevolume fraction Vfi of a constituent material i can be written as [2]

Vf iðzÞ ¼zþh=2

h

� �p

ð3Þ

where the power-law exponent p is a positive real number,(0rprN), and z describes the radial distance measured fromthe middle surface of the shell, (�h/2rzrh/2), see Fig. 1.

The sum of the volume fractions of all the constituentmaterials is taken equal to unity [2]

Xk

i ¼ 1

Vf iðzÞ ¼ 1 ð4Þ

For an FGM thin cylindrical shell made of two differentconstituent materials, the volume fractions Vf1 and Vf2 can bewritten in the following form [3]:

Vf 1ðzÞ ¼ 1�zþh=2

h

� �p

, Vf 2ðzÞ ¼zþh=2

h

� �p

, Vf 1ðzÞþVf 2ðzÞ ¼ 1

ð5Þ

where the Young’s modulus E, the Poisson’s ratio n and the massdensity r are expressed as [3]

Ef gmðT,zÞ ¼ ðE2ðTÞ�E1ðTÞÞzþh=2

h

� �p

þE1ðTÞ ð6Þ

nf gmðT ,zÞ ¼ ðn2ðTÞ�n1ðTÞÞzþh=2

h

� �p

þn1ðTÞ ð7Þ

rf gmðT,zÞ ¼ ðr2ðTÞ�r1ðTÞÞzþh=2

h

� �p

þr1ðTÞ ð8Þ

3. Sanders–Koiter nonlinear theory of cylindrical shells

In Fig. 1, an FGM circular cylindrical shell having radius R,length L and thickness h is represented; a cylindrical coordinatesystem (0; x, y, z) is considered in order to take advantage from

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–77 65

the axial symmetry of the structure, the origin 0 of the referencesystem is located at the centre of one end of the shell.

Three displacement fields are represented in Fig. 1: long-itudinal u (x, y, t), circumferential v (x, y, t) and radial w (x, y, t).

3.1. Elastic strain energy, kinetic energy, virtual work of external

forces, damping forces

The Sanders–Koiter nonlinear theory of cylindrical shells is aneight-order shell theory; it is based on Love’s ‘‘first approxima-tion’’ [14]. The strain components (ex, ey, gxy) at an arbitrary pointof the shell are related to the middle surface strains (ex,0, ey,0, gxy,0)and to the changes in curvature and torsion (kx, ky, kxy) of themiddle surface of the shell by the following relationships [15]:

ex ¼ ex,0þzkx, ey ¼ ey,0þzky, gxy ¼ gxy,0þzkxy ð9Þ

where z is the distance of the arbitrary point of the cylindricalshell from the middle surface and (x, y) are the longitudinal andangular coordinates of the shell, see Fig. 1.

The middle surface strains and changes in curvature andtorsion are given by [15]

ex,0 ¼@u

L@Zþ

1

2

@w

L@Z

� �2

þ1

8

@v

L@Z�@u

R@y

� �2

þ@w

L@Z@w0

L@Z

ey,0 ¼@v

R@yþ

w

Rþ

1

2

@w

R@y�

v

R

� �2

þ1

8

@u

R@y�@v

L@Z

� �2

þ@w0

R@y@w

R@y�

v

R

� �

gxy,0 ¼@u

R@yþ@v

L@Zþ@w

L@Z@w

R@y�

v

R

� �þ@w0

L@Z@w

R@y�

v

R

� �þ@w

L@Z@w0

R@y

kx ¼�@2w

L2@Z2, ky ¼

@v

R2@y�

@2w

R2@y2, kxy ¼�2

@2w

LR@Z@yþ

1

2R3@v

L@Z�@u

R@y

� �

ð10Þ

where (Z¼x/L) is the nondimensional longitudinal coordinate andw0 describes a geometric imperfection.

In the case of FGM, the stresses are related to the strains asfollows [16]:

sx ¼EðzÞ

1�n2ðzÞðexþnðzÞeyÞ, sy ¼

EðzÞ

1�n2ðzÞðeyþnðzÞexÞ, txy ¼

EðzÞ

2ð1þnðzÞÞ gxy

ð11Þ

where E(z) is Young’s modulus and n(z) is Poisson’s ratio.The plane stress hypothesis gives sz¼0.The elastic strain energy Us of a cylindrical shell is given

by [16]

Us ¼1

2LR

Z 1

0

Z 2p

0

Z h=2

�h=2ðsxexþsyeyþtxygxyÞdZ dy dz ð12Þ

where h, R and L are the thickness, radius and length of the shell,respectively.

Using Eqs. (9), (11) and (12), the following expression of Us canbe obtained

Us ¼1

2LR

Z 1

0

Z 2p

0

Z h=2

�h=2

EðzÞ

1�n2ðzÞe2

x,0þe2y,0þ2nðzÞex,0ey,0þ

1�nðzÞ2

g2xy,0

� �dZ dy dz

þLR

Z 1

0

Z 2p

0

Z h=2

�h=2

zEðzÞ

1�n2ðzÞex,0kxþey,0kyþnðzÞðex,0kyþey,0kxÞ�

þ1�nðzÞ

2gxy,0kxy

�dZ dy dz

þ1

2LR

Z 1

0

Z 2p

0

Z h=2

�h=2

z2EðzÞ

1�n2ðzÞk2

xþk2yþ2nðzÞkxkyþ

1�nðzÞ2

k2xy

� �dZ dy dz ð13Þ

The kinetic energy Ts of a cylindrical shell (rotary inertia effectbeing neglected) is given by [16]

Ts ¼1

2LR

Z 1

0

Z 2p

0

Z h=2

�h=2rðzÞð _u2

þ _v2þ _w2

ÞdZ dy dz ð14Þ

where r (z) is the mass density of the shell.

The virtual work W done by the external forces is written as[16]

W ¼ LR

Z 1

0

Z 2p

0ðqxuþqyvþqzwÞdZ dy ð15Þ

where (qx, qy, qz) are the distributed forces per unit area acting inthe longitudinal, circumferential and radial directions, respectively.

The nonconservative damping forces are assumed to be ofviscous type and are taken into account by using Rayleigh’sdissipation function [16]

F ¼1

2cLR

Z 2p

0

Z 1

0ð _u2þ _v2þ _w2

ÞdZ dy ð16Þ

where c is the viscous damping coefficient.

4. Discretisation approach

In order to carry out the dynamic analysis of the shell, a two-step procedure is considered [18]: (i) the three displacementfields are expanded using a double series, then the Rayleigh–Ritzmethod is applied to the linearised formulation of the problem, inorder to obtain an approximation of the eigenfunctions; (ii) thedisplacement fields are re-expanded using the approximatedeigenfunctions, the Lagrange equations are considered in con-junction with the fully nonlinear expression of the potentialenergy, in order to obtain a set of nonlinear ordinary differentialequations in modal coordinates.

4.1. Linear vibration analysis

In order to carry out a linear vibration analysis only thequadratic terms are retained in Eq. (13).

A modal vibration, i.e., a synchronous motion, can be formallywritten in the form [18]

uðZ,y,tÞ ¼UðZ,yÞf ðtÞ, vðZ,y,tÞ ¼ VðZ,yÞf ðtÞ, wðZ,y,tÞ ¼WðZ,yÞf ðtÞð17Þ

where U (Z, y), V (Z, y), W (Z, y) represent the mode shape and f (t)describes the time law, which is supposed to be the same for eachdisplacement field (synchronous motion assumption).

It is worthwhile to stress that a ‘‘mode of vibration’’ isrepresented by a vector function [U, V, W]T. Now the componentsof the ‘‘mode shape’’ are expanded by means of a double mixedseries: the periodicity of deformation in the circumferentialdirection suggests that harmonic functions should be used (cosny, sin ny), whereas Chebyshev orthogonal polynomials can beconsidered in the longitudinal direction Tm

n (Z) [18]

UðZ,yÞ ¼XMu

m ¼ 0

XN

n ¼ 0

~Um,nTn

mðZÞcosny,

VðZ,yÞ ¼XMv

m ¼ 0

XN

n ¼ 0

~V m,nTn

mðZÞsinny

WðZ,yÞ ¼XMw

m ¼ 0

XN

n ¼ 0

~W m,nTn

mðZÞcosny ð18Þ

where Tn

m (Z)¼Tm (2Z�1), m is the degree of the Chebyshevpolynomials, n is the number of nodal diameters andð ~U m,n, ~V m,n, ~W m,nÞ are the generalised coordinates.

In the absence of imperfections, a linear mode shape has thefollowing simplified expression [18]:

UðZ,yÞ ¼XMu

m ¼ 0

~Um,nTn

mðZÞcosny,

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–7766

VðZ,yÞ ¼XMv

m ¼ 0

~V m,nTn

mðZÞsinny

WðZ,yÞ ¼XMw

m ¼ 0

~W m,nTn

mðZÞcosny, n¼ 0,1,2,. . . ð19Þ

It is worthwhile to stress that due to the axial symmetry andthe isotropy of the system, in the absence of imperfections and inthe case of axial symmetric boundary conditions (e.g., simply,clamped, free), the harmonic functions are orthogonal withrespect to the linear operator of the shell in the circumferentialdirection (y); i.e., the mode shapes are characterised by a specificnumber of nodal diameters (n), and the double series of Eq. (18) isreduced to the single series (19).

Note that the expansions (18) and (19) do not yet satisfy anyboundary condition.

4.2. Boundary conditions

In the present paper, simply supported, clamped and freecircular cylindrical shells are analysed; the boundary conditionsare imposed by applying constraints to the free coefficientsð ~Um,n, ~V m,n, ~W m,nÞ of expansions (18) or (19).

4.2.1. Simply supported–simply supported

Simply supported–simply supported boundary conditions aregiven by [14]

w¼ 0, v¼ 0, Mx ¼ 0, Nx ¼ 0 for Z¼ 0,1 ð20Þ

The previous conditions imply the following equations [18]:

WðZ,yÞ ¼XMw

m ¼ 0

XN

n ¼ 0

~W m,nTn

mðZÞcosny¼ 0 for Z¼ 0,1 ð21Þ

VðZ,yÞ ¼XMv

m ¼ 0

XN

n ¼ 0

~V m,nTn

mðZÞsinny¼ 0 for Z¼ 0,1 ð22Þ

W ,ZZðZ,yÞ ¼XMw

m ¼ 0

XN

n ¼ 0

~W m,nTn

m,ZZðZÞcosny¼ 0 for Z¼ 0,1 ð23Þ

U,ZðZ,yÞ ¼XMu

m ¼ 0

XN

n ¼ 0

~Um,nTn

m,ZðZÞcosny¼ 0 for Z¼ 0,1 ð24Þ

where ( � ),Z¼q( � )/qZ and ( � ),ZZ¼q2( � )/qZ

2.Such conditions are valid for any value of y and n; therefore,

Eqs. (21)–(24) can be modified as follows [18]:

XMw

m ¼ 0

~W m,nTn

mðZÞ ¼ 0, yA ½0,2p�, nA ½0,N� for Z¼ 0,1 ð25Þ

XMv

m ¼ 0

~V m,nTn

mðZÞ ¼ 0, yA ½0,2p�, nA ½0,N� for Z¼ 0,1 ð26Þ

XMw

m ¼ 0

~W m,nTn

m,ZZðZÞ ¼ 0, yA ½0,2p�, nA ½0,N� for Z¼ 0,1 ð27Þ

XMu

m ¼ 0

~Um,nTn

m,ZðZÞ ¼ 0, yA ½0,2p�, nA ½0,N� for Z¼ 0,1 ð28Þ

The linear algebraic system given by Eqs. (25)–(28) can be solvedanalytically in terms of the coefficients ð ~U 1,n, ~U2,n, ~V 0,n, ~V 1,n, ~W 0,n,~W 1,n, ~W 2,n, ~W 3,nÞ for n A [0, N], since these coefficients are linearly

dependent on the others.

4.2.2. Clamped–clamped

Clamped–clamped boundary conditions are given by [14]

u¼ 0, v¼ 0, w¼ 0, w,Z ¼ 0 for Z¼ 0,1 ð29Þ

The previous conditions imply the following equations [18]:

XMu

m ¼ 0

~Um,nTn

mðZÞ ¼ 0, yA ½0,2p�, nA ½0,N� for Z¼ 0,1 ð30Þ

XMv

m ¼ 0

~V m,nTn

mðZÞ ¼ 0, yA ½0,2p�, nA ½0,N� for Z¼ 0,1 ð31Þ

XMw

m ¼ 0

~W m,nTn

mðZÞ ¼ 0, yA ½0,2p�, nA ½0,N� for Z¼ 0,1 ð32Þ

XMw

m ¼ 0

~W m,nTn

m,ZðZÞ ¼ 0, yA ½0,2p�, nA ½0,N� for Z¼ 0,1 ð33Þ

The linear algebraic system given by Eqs. (30)–(33) can besolved analytically in terms of the coefficients ð ~U0,n, ~U 1,n, ~V 0,n,~V 1,n, ~W 0,n, ~W 1,n, ~W 2,n, ~W 3,nÞ, for n A [0, N].

4.2.3. Free–free

Free–free boundary conditions are given by [14]

Nx ¼ 0, NxyþMxy

R¼ 0, Qxþ

1

R

@Mxy

@y¼ 0, Mx ¼ 0 for Z¼ 0,1

ð34Þ

where forces and moments are given by [14]

Nx ¼Eh

1�n2ðex,0þney,0Þ, Nxy ¼

Eh

2ð1þnÞ gxy,0, Mx ¼Eh3

12ð1�n2ÞðkxþnkyÞ

Mxy ¼Eh3

24ð1þnÞ kxy, Qx ¼Eh3

12ð1�n2Þðkx,xþnky,xÞþ

Eh3

24ð1þnÞ kxy,y

ð35Þ

In this case, the complexity of the boundary conditions makesit difficult to follow the previously mentioned procedure; how-ever, in this case all the boundary conditions are ‘‘natural’’, i.e.,they involve essentially ‘‘forces’’. In such a case, it is well knownthat the Ritz procedure can be applied even if the naturalboundary conditions are not respected; therefore, here no bound-ary conditions are imposed for the free–free case. The fullexpressions of the free–free boundary conditions are reported inthe Appendix.

4.3. Linear vibration analysis: Rayleigh–Ritz procedure

The maximum number of variables needed for describing ageneral linear vibration mode can be calculated by the followingrelation (Np¼MuþMvþMwþ3�r), where Mu¼Mv¼Mw are themaximum degree of the Chebyshev polynomials and r is thenumber of equations for the boundary conditions to be respected.

For a multi-mode analysis including different nodal diametersn, the number of degrees of freedom is computed by the relation(Nmax¼Np� (Nþ1)), where N is the maximum number of thenodal diameters.

Eq. (17) are inserted into the expressions of Us and Ts (Eqs. (13)and (14)) in order to compute the Rayleigh quotient Rð ~qÞ ¼ Umax/T*, where Umax¼max (Us) is the maximum of the potential energyduring a modal vibration, T*

¼Tmax/o2, Tmax¼max(Ts) is the max-imum of the kinetic energy, o is the circular frequency of theharmonic motion and ~q ¼ ½::: ~Um,n, ~V m,n, ~W m,n:::�

T represents a vec-tor containing all the unknown variables.

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–77 67

After imposing the stationarity to the Rayleigh quotient, oneobtains the eigenvalue problem

ð�o2MþKÞ ~q ¼ 0 ð36Þ

which furnishes natural frequencies and modes of vibration(eigenvalues and eigenvectors).

The mode shape corresponding to the jth mode is given by Eq.(18), where the coefficients ð ~Um,n, ~V m:n, ~W m,nÞ are substituted withð ~UðjÞ

m,n, ~VðjÞ

m,n, ~WðjÞ

m,nÞ, which are the components of the jth eigenvec-tor ~q j obtained from Eq. (36).

The vector function

UðjÞðZ,yÞ ¼ ½UðjÞðZ,yÞ,V ðjÞðZ,yÞ,W ðjÞðZ,yÞ�T ð37Þ

describes an approximation of the jth mode of the originalproblem.

The eigenfunction vectors are normalised by imposing therelation [18]

max max½UðjÞðZ,yÞ�, max½V ðjÞðZ,yÞ�, max½W ðjÞðZ,yÞ�

h i¼ 1 ð38Þ

4.4. Nonlinear vibration analysis: Lagrange equations

In the nonlinear vibration analysis, the full expression of thepotential energy (13), containing terms up to the fourth order(cubic nonlinearity), is considered.

The displacement fields u (Z, y, t), v (Z, y, t), w (Z, y, t) areexpanded by using both the linear mode shapes U (Z, y), V (Z, y),W (Z, y) obtained in the previous linear analysis and the conjugatemode shapes Uc (Z, y), Vc (Z, y), Wc (Z, y), in the following form:

uðZ,y,tÞ ¼XNu

j ¼ 1

XN

n ¼ 1

½Uðj,nÞðZ,yÞf u,j,nðtÞþUcðj,nÞðZ,yÞf u,j,n,cðtÞ�

vðZ,y,tÞ ¼XNv

j ¼ 1

XN

n ¼ 1

½V ðj,nÞðZ,yÞf v,j,nðtÞþVcðj,nÞðZ,yÞf v,j,n,cðtÞ�

wðZ,y,tÞ ¼XNw

j ¼ 1

XN

n ¼ 1

½W ðj,nÞðZ,yÞf w,j,nðtÞþWc

ðj,nÞðZ,yÞf w,j,n,cðtÞ� ð39Þ

These expansions respect exactly the boundary conditionsexcept for the free–free case; the synchronicity is relaxed sincefor each mode j and each component (u, v, w) different time lawsare allowed.

Mode shapes U (j,n)(Z, y), V (j,n)(Z, y), W (j,n)(Z, y) are knownfunctions expressed in terms of polynomials and harmonic func-tions, see Eq. (19); the index n indicates the number of nodaldiameters, the index j is used for ordering the modes (for each n)with increasing associated natural frequency. It is interesting tonote that, in the case of simply–simply supports, j is also thenumber of longitudinal half waves (number of nodal circumfer-ences minus one), see Ref. [14].

Table 1Properties of stainless steel and nickel vs. coefficients of temperature.

Stainless steel

Coefficients E n r

P0 2.01�1011 N m�2 0.326 816

P�1 0 K 0 K 0 K

P1 3.08�10�4 K�1�2.002�10�4 K�1 0 K

P2 �6.53�10�7 K�2 3.797�10�7 K�2 0 K

P3 0 K�3 0 K�3 0 K

P (300 K) 2.08�1011 N m�2 0.318 816

Conjugate mode shapes can be straightforwardly obtainedfrom expansions (19)

Uðj,nÞc ðZ,yÞ ¼XMu

m ¼ 0

~UðjÞ

m,nTn

mðZÞsinny, V ðj,nÞc ðZ,yÞ ¼XMv

m ¼ 0

~VðjÞ

m,nTn

mðZÞcosny

W ðj,nÞc ðZ,yÞ ¼

XMw

m ¼ 0

~WðjÞ

m,nTn

mðZÞsinny n¼ 1,2,. . . ð40Þ

It is to note that axisymmetric modes do not exhibit anycompanion mode.

Expansions (39) are inserted into expressions of the strainenergy (13), kinetic energy (14), virtual work of the externalforces (15) and damping forces (16).

The Lagrange equations of motion are expressed in thefollowing form [18]

d

dt

@L

@ _qi

� ��@L

@qi

¼Qi for iA 1,Nmax½ � ðL¼ Ts�UsÞ ð41Þ

The modal coordinates are ordered in a vector q (t)¼[y, fu,j,n,fu,j,n,c, fv,j,n, fv,j,n,c, fw,j,n, fw,j,n,c,y]T and the maximum number ofdegrees of freedom Nmax depends on the number of modesconsidered in expansions (39).

The generalised forces Qi are obtained by the differentiation ofRayleigh’s dissipation function F (16) and of the virtual work doneby the external forces W (see Eq. (15)), in the form [18]

Qi ¼�@F

@ _qi

þ@W

@qi

ð42Þ

Using the Lagrange Eq. (41), a set of nonlinear ordinarydifferential equations is obtained; such system is then solved byusing numerical methods.

5. Numerical results

In this section, the linear and nonlinear vibrations of FGMcircular cylindrical shells with different geometries, materialdistributions, boundary conditions and mode shape expansionsare analysed.

The study is carried out on an FGM made of stainless steel andnickel; its properties are graded in the thickness directionaccording to a volume fraction distribution, where p is thepower-law exponent. The material properties, reported inTable 1, have been extracted from Ref. [2].

In order to validate the present method, the natural frequencies ofsimply supported FGM shells are compared with those of Loy et al.[2], see Tables 2 and 3. Two geometries are considered; both are longshells L/R¼20; the case of Table 2 represents a very thin shell, thecase of Table 3 is a shell having a ratio h/R¼0.050 that is the upperlimit for a proper use of shell theories based on the Love’s ‘‘firstapproximation’’. The comparisons show that the present methodgives results quite close to Ref. [2], the differences being less than 1%;the agreement is slightly better for the thinner shell (Table 2).

Nickel

E n r

6 kg m�3 2.24�1011 Nm�2 0.3100 8900 kg m�3

0 K 0 K 0 K�1

�2.79�10�4 K�1 0 K�1 0 K�1

�2�3.99�10�9 K�2 0 K�2 0 K�2

�3 0 K�3 0 K�3 0 K�3

6 kg m�3 2.05�1011 Nm�2 0.3100 8900 kg m�3

Table 3Comparison of natural frequencies for a simply supported FGM shell (h/R¼0.050,

L/R¼20, R¼1 m, p¼1). Polynomials of degree 11.

Natural frequency (Hz) Difference %

m n Present model Ref. [2]

1 1 13.213 13.235 0.17

1 2 32.382 32.430 0.15

1 3 90.531 90.553 0.02

1 4 173.35 173.36 0.01

1 5 280.19 280.20 0.00

1 6 410.91 410.91 0.00

1 7 565.45 565.46 0.00

1 8 743.81 743.82 0.00

1 9 945.97 945.98 0.00

1 10 1171.9 1171.9 0.00

Table 4Convergence analysis, nonlinear modelling, simply–simply supports. Modes

selected for the expansions (19). j is the number of longitudinal half waves, n is

the number of nodal diameters. Polynomials of degree 11.

(j,n) (1,6) (1,12) (1,18) (3,6) (3,12) (3,18) (1,0) (3,0) (5,0) (7,0)

6 dof u, v, w v – – – – u, w – – –

9 dof u, v, w v – – v – u, w u, w – –

12 dof u, v, w v – u, v, w v – u, w u, w – –

15 dof u, v, w v v u, v, w v – u, w u, w u, w –

18 dof u, v, w v v u, v, w v v u, w u, w u, w u, w

Table 2Comparison of natural frequencies for a simply supported FGM shell (h/R¼0.002,

L/R¼20, R¼1 m, p¼1). Polynomials of degree 11.

Natural frequency (Hz) Difference %

m n Present model Ref. [2]

1 3 4.1562 4.1569 0.02

1 2 4.4794 4.4800 0.01

1 4 7.0379 7.0384 0.01

1 5 11.241 11.241 0.00

1 1 13.211 13.211 0.00

1 6 16.455 16.455 0.00

1 7 22.635 22.635 0.00

1 8 29.771 29.771 0.00

1 9 37.862 37.862 0.00

1 10 46.905 46.905 0.00

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–7768

A specific convergence analysis is carried out for properly selectingthe degree of the Chebyshev polynomials: degree 11 has been foundsuitably accurate, details are omitted for the sake of brevity.

5.1. Nonlinear response convergence analysis

The first step of the nonlinear analysis is the convergence test onexpansions (39), which is carried out on a simply supported shellexcited with a harmonic force; the excitation frequency is close tomode (j, n), where j is the number of longitudinal half waves (halfsine) and n is the number of nodal diameters. The convergence ischecked by adding suitable modes to the resonant one, i.e.: asym-metric modes (k� j, s�n) k¼1,3, s¼1,2,3 due to quadratic and cubicnonlinearities; axisymmetric modes (k, 0) k¼1,3,5,7 due to quadraticnonlinearities; see also Table 4 and Eq. (39).

The shell is excited by an external modally distributed radialforce qz¼ f1,6 sin pZ cos 6y cos Ot; the amplitude of excitation isf1,6¼0.0012h2ro2

1,6 and the frequency of excitation O is close tothe mode (1,6), OEo1,6. The external forcing f1,6 is normalisedwith respect to the mass, acceleration and thickness; the dampingratio is equal to x1,6¼0.0005.

In the following, amplitude–frequency curves of the modalcoordinates of the shell will be presented; the modal amplitudesare normalised with respect to the thickness h of the shell andrepresented vs. the normalised frequency; for example, in repre-senting the radial amplitude of mode (1, 6), the maximumamplitude of fw,1,6 (t)/h is represented vs. O/o1,6.

In Fig. 2, amplitude–frequency curves of a simply supported FGMshell are shown (h/R¼0.002, L/R¼20, R¼1 m, p¼1, the shell is verythin and long), different expansions are compared. The 6 dof model(Table 4) gives a softening nonlinear behaviour, conversely, thehigher-order expansions converge to a hardening nonlinear beha-viour; higher order models (dof from 9 to 18) behave quite similarly;this means that the smallest expansion able to predict the dynamicswith acceptable accuracy is 9 dof model (Table 4). The main weaknessof the 6 dof expansion is the insufficient number of axisymmetricmodes, which are very important for properly modelling the circum-ferential stretching during the vibration. It is to note that the shell isvery thin, so the hardening behaviour is expected to occur (see e.g.the nonlinearity map in Ref. [17]).

In Fig. 3, a moderately thick and long shell is analysed (h/R¼0.025, L/R¼20, R¼1 m, p¼1), the amplitude–frequency curvesare obtained with the expansions of Table 4. Similar to the case ofvery thin shell of Fig. 2, again, the 6 dof model, with aninsufficient number of axisymmetric modes, is clearly inaccurate;indeed, for this kind of shell the correct behaviour is softening.

In Fig. 4, a thick shell is studied (h/R¼0.050, L/R¼20, R¼1 m,p¼1), the expected behaviour is hardening: the 6 dof model(Table 4) is still inaccurate, similar to the previous cases.

From the convergence analysis, one can claim that the 9 dof modelgives satisfactory results with the minimal computational effort;therefore, in the following the 9 dof model of Table 4 will be used.

Note that the present convergence analysis confirms andextends (to the FGMs) the results of Refs. [16–18,24,26]; in suchpapers it was found that the minimum 9 dof expansion gavegood results also for (j, n)a(1, 6); moreover, also for FGMs it isconfirmed that, in nonlinear field, the axisymmetric modes play arole of primary importance.

The additional result of the present convergence analysis is thefollowing: using an insufficient expansion leads to a wrongnonlinear behaviour; more precisely: (i) if the shell presents anactual softening behaviour, the inaccurate expansion can givehardening results; (ii) if the shell presents a hardening behaviour,an inaccurate expansion can give a softening behaviour.

It is to note that in the literature this critical behaviour ofinaccurate expansions is well known and understood; however, atour best knowledge, nobody observed yet that the inaccurateexpansion can give softening results in analysing hardeningshells; all test cases and studies present in the literature reportonly that an inaccurate expansion can give hardening behaviourswhen the shell is softening.

Our conjecture is that the present result can be probably general-ised to other kinds of circular shells (isotropic, orthotropic, etc.); thismay constitute an important novelty of the present work.

The previous considerations suggest that the following 9 dofmodel should be used for studying a generic resonant mode (j, n):

�

modes (j, n), (1, 0), (3, 0) for the longitudinal displacementfield u.

�

modes (j, n), (j, 2n), (3j, 2n) for the circumferential displace-ment field v

�

modes (j, n), (1, 0), (3, 0) for the radial displacement field w.

After selecting such modes, each expansion present in Eq. (39)is reduced to a three-term modal expansion; the resulting non-linear system has 9 dof.

0.99 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 1.010

0.2

0.4

0.6

0.8

1

1.2

1.4

Ω /ω1,6

Max

[f w,1

,6(t)

/h]

6 dof model9 dof model12 dof model15 dof model18 dof model6 dof

model18 dofmodel

Fig. 3. Convergence analysis, nonlinear amplitude–frequency curves (h/R¼0.025, L/R¼20, R¼1 m, p¼1). ‘‘ ’’, 6 dof model; ‘‘ ’’, 9 dof model; ‘‘ ’’, 12 dof

model; ‘‘–’’, 15 dof model; ‘‘ ’’, 18 dof model. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)

0.99 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 1.010

0.2

0.4

0.6

0.8

1

1.2

1.4

Ω /ω1,6

Max

[f w

,1,6

(t)/h

]

6 dof model9 dof model12 dof model15 dof model18 dof model 18 dof

model 6 dof model

Fig. 2. Convergence analysis, nonlinear amplitude–frequency curves (h/R¼0.002, L/R¼20, R¼1 m, p¼1). ‘‘ ’’, 6 dof model; ‘‘ ’’, 9 dof model; ‘‘ ’’, 12 dof

model; ‘‘–’’, 15 dof model; ‘‘ ’’, 18 dof model. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–77 69

5.2. Effect of geometry

The goal of the present section is to clarify the role of thegeometric parameters h, L, R and in particular their ratios h/R andL/R on the nonlinear response of FGM shells.

In Fig. 5, the amplitude–frequency curves of simply supportedFGM shells with different geometries are shown; the nonlinearresponse is:

�

hardening in the case (h/R¼0.002, L/R¼20, R¼1 m, p¼1), redline ( ).

�

softening in the case (h/R¼0.025, L/R¼20, R¼1 m, p¼1), blackline ( ). � hardening in the case (h/R¼0.050, L/R¼20, R¼1 m, p¼1), blue

line ( ).

The nonlinear response of the thicker shell (h/R¼0.050) ismore hardening than the thinner one (h/R¼0.002). A wideintermediate interval of thickness gives rise to a softening typebehaviour; so we can claim that the commonest type of nonlinearresponse of the thin shells is softening.

0.99 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 1.010

0.2

0.4

0.6

0.8

1

1.2

1.4

Ω /ω1,6

Max

[f w,1

,6(t)

/h]

h / R = 0.002, L / R = 20, p = 1h / R = 0.025, L / R = 20, p = 1h / R = 0.050, L / R = 20, p = 1

stainless steel on the outer surfaceand nickel on the inner surface

simply supportedboundary conditions

Fig. 5. Nonlinear amplitude–frequency curves, effect of thickness, 9 dof model, asymmetric mode (1, 6). ‘‘ ’’, (h/R¼0.002, L/R¼20, R¼1 m, p¼1); ‘‘ ’’, (h/R¼0.025,

L/R¼20, R¼1 m, p¼1); ‘‘ ’’, (h/R¼0.050, L/R¼20, R¼1 m, p¼1). (For interpretation of the references to colour in this figure caption, the reader is referred to the web

version of this article.)

0.99 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 1.010

0.2

0.4

0.6

0.8

1

1.2

1.4

Ω /ω1,6

Max

[f w

,1,6

(t)/h

]

6 dof model9 dof model12 dof model15 dof model18 dof model 6 dof

model 18 dof model

Fig. 4. Convergence analysis, nonlinear amplitude–frequency curves (h/R¼0.050, L/R¼20, R¼1 m, p¼1). ‘‘ ’’, 6 dof model; ‘‘ ’’, 9 dof model; ‘‘ ’’, 12 dof

model; ‘‘–’’, 15 dof model; ‘‘ ’’, 18 dof model. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–7770

In order to give a clear scenario regarding the influence of thegeometry on the nonlinear vibration, a parametric analysis iscarried out by varying the fundamental ratios h/R and L/R; Fig. 6represents the kind of nonlinearity, i.e., hardening or softening,depending on the aforementioned ratios. Fig. 6 presents a coarsegrid, i.e., it is obtained from few test-cases, because for each casea specific numerical model has been built; the grid is overlappedwith results of Ref. [17]; square marks describe a softening

behaviour, while circle marks describe a hardening behaviour.In Fig. 6, dashed lines are reproduced from Ref. [17], these arereferred to homogeneous shells, where such lines represent theboundaries of the hardening/softening regions. The present ana-lysis shows that FGM cylindrical shells behave similar to thehomogeneous ones: very short shells (L/Ro0.5) and very thickshells (h/R40.045) present a hardening nonlinear behaviour;conversely, a softening nonlinearity is found in a wide range of

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–77 71

shell geometries. However, for sufficiently long (L/R45) and thin(h/Ro0.005) shells, the system can be hardening.

16

16.1

16.2

16.3

16.4

16.5

16.6

16.7

16.8

16.9

17

0 2 4 6 8 10 12 14 1

ω1,

6 /2π

p

harden

natural fre

nonlinear

Fig. 7. Natural frequency o1,6 and nonlinear character NLb vs. exponent param

16

16.1

16.2

16.3

16.4

16.5

16.6

16.7

16.8

16.9

17

0 2 4 6 8 10 12 14 1

ω1,

6 /2π

p

hardeni

natural fr

nonlinear

Fig. 8. Natural frequency o1,6 and nonlinear character NLb vs. exponent param

Fig. 6. Effect of the geometry on the nonlinear response of the FGM shell. Circle

marks: hardening; square marks: softening; dashed lines: boundaries between

hardening and softening regions (from Ref. [17], homogeneous materials).

5.3. Effect of the material distribution on the nonlinear response

The effect of the material distribution on the nonlinearresponse is analysed by considering two different FGM shells:

�

6 1

ing

que

cha

ete

6

ng

equ

ch

eter

Type I FGM, nickel on the inner surface and stainless steel onthe outer surface.

� Type II FGM, stainless steel on the inner surface and nickel on

the outer surface.

In Figs. 7 and 8, the behaviour of the natural frequencyo1,6 and the nonlinear character vs. the exponent p (see Eq. (3))is shown for the following type of shell: h/R¼0.002, R¼1 m,L/R¼20.

The nonlinear character is identified by means of the followingindicator:

NLb ¼o1,6nonlin�o1,6lin

o1,6lin� 1000 ð43Þ

where o1,6 nonlin is the frequency corresponding to the maximum

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

8 20 22 24 26 28 30

NL b

ncy

racter

r p for the Type I FGM cylindrical shell (h/R¼0.002, L/R¼20, R¼1 m).

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

18 20 22 24 26 28 30

NL b

ency

aracter

p for the Type II FGM cylindrical shell (h/R¼0.002, L/R¼20, R¼1 m).

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–7772

of the amplitude–frequency diagram, see Fig. 5; the nonlinearcharacter is hardening when NLb40, softening when NLbo0.

When the stiffer material is outside (Type I FGM, Fig. 7), anincrement of the exponent p leads to: decrement of o1,6; changingthe nonlinear character from hardening (po1) to weakly hardening(p45). Increasing the exponent p physically means that the innermaterial (nickel for Type I FGM) is predominant; this justify thedecrement of the natural frequencies and the nonlinear character.

Conversely, when the stiffer material is inside (Type II FGM, Fig. 8),an increment of the exponent p leads to: increment of o1,6; changingthe nonlinear character from weakly hardening (po1) to hardening(p45). In this case, the increment of p means that the predominantmaterial is the stiffer one, as for Type II FGM the inner material isstainless steel; therefore, it is obvious that the natural frequenciesincrease as well as the nonlinear character.

Note that p-0 implies Efgm-E2 (external) and p-N impliesEfgm-E1 (internal): for Type I FGM outside is stiffer, for Type IIFGM inside is stiffer.

Figs. 9 and 10 are referred to an FGM shell having a softeningnonlinear character (h/R¼0.025, L/R¼20, R¼1 m). When the stiffermaterial is outside (Type I FGM, Fig. 9), an increment of the exponentp leads to a predominance of the material with a smaller Young’smodulus (nickel); this implies a decreasing of the natural frequenciessimilar to the case analysed in Fig. 7; also the influence on thenonlinear character is similar, i.e., an increase in the predominance ofthe weaker material produces a decrease of the nonlinearity of thesystem. In the case of FGM with stiffer material inside (Type II FGM,Fig. 10), the increment of the exponent p magnifies the presenceof stainless steel, increasing the natural frequencies and thenonlinearity.

The case of thick shells (h/R¼0.050, L/R¼20, R¼1 m) presentspictures similar to the thin shells, see Figs. 7 and 8, details are omittedfor brevity.

5.4. Effect of the boundary conditions on the nonlinear response

In this section, the effect of the boundary conditions on thenatural frequencies and the nonlinear response of the FGMcylindrical shells is analysed. The following boundary conditionsare considered: simply–simply (S–S), clamped–clamped (C–C),free–free (F–F), clamped–simply (C–S), clamped–free (C–F) and

200

201

202

203

204

205

206

207

208

209

210

211

212

0 2 4 6 8 10 12 14 16

ω1,

6/2π

p

softenin

natural fre

nonlinear

Fig. 9. Natural frequency o1,6 and nonlinear character NLb vs. exponent param

simply–free (S–F). In Fig. 11, the natural frequencies of the FGMshell (h/R¼0.002, L/R¼20, R¼1 m, p¼1) are shown. For all the sixboundary conditions, the frequencies present a non-monotonicbehaviour when n is varied (j¼1), the minimum frequency occursin between n¼2 and n¼4 (note that the minimum is alsoinfluenced by the ratios h/R and L/R); the curves of the naturalfrequencies merge asymptotically for n46, i.e., the effect of theboundary conditions on the natural frequencies is prominent forlow circumferential wavenumbers n A [1,5] and disappears forhigher circumferential wavenumbers.

In particular, for n A [1,5] the natural frequencies of the (F–F)shell are the highest ones, followed by the (C–F), (C–C), (S–F), (C–S)and (S–S).

In order to analyse the effect of the boundary conditions onthe nonlinear response, mode (1, 6) is considered; this mode isselected because for such a mode (and the higher ones) there arelimited effects of the boundary conditions on the linear beha-viour: this means that for such modes we should be able to isolatethe effect of the boundary conditions on the nonlinearbehaviour only.

In Fig. 12, amplitude–frequency curves of an FGM shell (h/R¼0.002, L/R¼20, R¼1 m, p¼1) for the driven mode (1, 6) areshown; six different boundary conditions are considered: the (S–S) and (C–S) behave similarly, the (S–F), (C–C) and (C–F) behavesimilarly, the (F–F) gives the strongest nonlinearity. It is remark-able that, even though for the mode (1, 6) all the boundaryconditions give almost the same natural frequency, the nonlinearcharacter (i.e., the value of the parameter NLb) follows thesequence of Fig. 11: the boundary condition that gives the highestlinear frequencies (F–F) presents also the strongest nonlinearcharacter, the boundary condition having the lowest frequencies(S–S) presents the weakest nonlinearity, the other boundaryconditions show a similar behaviour.

5.5. Effect of the companion mode participation on the nonlinear

response

In this section, the effect of the companion mode participation(see eq. 39) on the nonlinear response of the shells is analysed.The participation of both driven and companion modes gives apure travelling wave response, moving circumferentially around

-1.24

-1.16

-1.08

-1.00

-0.92

-0.84

-0.76

-0.68

-0.60

-0.52

-0.44

-0.36

-0.28

18 20 22 24 26 28 30

NL b

g

quency

character

eter p for the Type I FGM cylindrical shell (h/R¼0.025, L/R¼20, R¼1 m).

0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 9 10

f (H

z)

n

j = 1

F - F

C - F

C - C

S - F

C - S

S - S

Fig. 11. Variation of natural frequencies f (Hz) of the FGM shell (h/R¼0.002,

L/R¼20, R¼1 m, p¼1) with the circumferential wave number n under various

boundary conditions.

-1.16

-1.08

-1.00

-0.92

-0.84

-0.76

-0.68

-0.60

-0.52

-0.44

-0.36

-0.28

-0.20

200

201

202

203

204

205

206

207

208

209

210

211

212

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

ω1,

6 /2π

p

softening

natural frequency

nonlinear character

NL b

Fig. 10. Natural frequency o1,6 and nonlinear character NLb vs. exponent parameter p for the Type II FGM cylindrical shell (h/R¼0.025, L/R¼20, R¼1 m).

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–77 73

the shell, when the time phase shift between two conjugatemodal coordinates (e.g., fw,1,6 (t) and fw,1,6,c (t)) is p/2.

In Fig. 13(a), the amplitude–frequency curve with the compa-nion mode participation is presented (h/R¼0.025, L/R¼20, R¼1 m,p¼1, mode (1, 6)) using a 14 dof model (this expansion corre-sponds to the 9 dof model without a companion mode, i.e., thesame number of axisymmetric modes is considered). Theresponse fw,1,6 (t) with the companion mode participation, solidblue line of Fig. 13(a), is very similar to the response withoutcompanion mode participation, dashed black line, see also Fig. 5.Taking into account the companion mode, Fig. 13(b), does notproduce any variation except for a small region close to theresonance (0.9996oO/o1,6o0.9999), where the companionmode is excited by means of a 1:1 internal resonance. It isworthwhile to stress that the modal excitation does not excitedirectly the companion mode; therefore, the internal resonancemechanism induces an energy transfer between the two conju-gate (and linearly uncoupled) modes.

In Fig. 13(c–h), the amplitude–frequency curves of fv,1,12 (t),fv,3,12 (t), fw,1,0 (t), fw,3,0 (t) with the companion mode participationare shown. In particular, the amplitude of the asymmetric modes

(1, 12), (3, 12), Fig. 13(c–f), and the axisymmetric modes (1, 0), (3,0), Fig. 13(g–h), is small if compared with the amplitude of themode (1, 6), see in particular mode (3, 0) in Fig. 13(h); however,the presence of the asymmetric and axisymmetric modes in themodel is fundamental to predict the correct nonlinearity of theFGM shell, see Fig. 3.

In Fig. 14, the time histories of the driven mode (1, 6), blueline, and companion mode, red line, for O/o1,6¼0.9998 arepresented; the companion mode is initially not active, then anenergy transfer takes place, the amplitude of the driven modedecreases and eventually the companion mode is excited.

In Fig. 15, enlarged view of Fig. 14, a time phase shift betweenthe modal coordinates (conjugate modes) close to p/2 is present;therefore, a travelling wave takes place. It is worthwhile to stressthat, even though the nonlinearity of the system is not strong, theonset of a travelling wave implies that the response of the shell iscompletely different with respect to a linear model.

In Fig. 16, the spectrum of the time histories of Fig. 14 isshown: the last part of the time history is considered, i.e., thetransient dynamics are cut out. The spectrum presents fourspikes, one driven harmonic and three super of order two, threeand four, respectively: this confirms the presence and the impor-tance of quadratic and cubic nonlinearities.

6. Conclusions

In this paper, the nonlinear vibrations of FGM cylindrical shellsare analysed. The Sanders–Koiter theory is applied to model thenonlinear dynamics of the system in the case of finite amplitudeof vibration.

The functionally graded material is made of a distribution ofstainless steel and nickel, and the material properties are gradedalong the thickness direction, according to a volume fractionpower-law.

The present model is validated in the linear field by means ofcomparison with the literature.

Numerical analyses are carried out in order to characterise thenonlinear response when the shell is subjected to a harmonicexternal load.

A convergence analysis is carried out by introducing in thelongitudinal, circumferential and radial displacement fields a

0.996 0.997 0.998 0.999 1 1.001 1.002 1.003 1.0040

0.2

0.4

0.6

0.8

1

1.2

1.4

Ω /ω1,6

Max

[f w,1

,6(t)

/h]

S - SC - SS - FC - CC - FF - Fj = 1

n = 6

Fig. 12. Comparison of nonlinear amplitude–frequency curves of the FGM shell (h/R¼0.002, L/R¼20, R¼1 m, p¼1). ‘‘ ’’, simply–simply; ‘‘ ’’, clamped–

simply; ‘‘ ’’, simply–free; ‘‘ ’’, clamped–clamped; ‘‘ ’’, clamped–free; ‘‘ ’’, free–free. (For interpretation of the references to colour in this figure

caption, the reader is referred to the web version of this article.)

Max

[f w

,1,6

(t)/h

]

Max

[f w

,1,6

,c(t)

/h]

Max

[f v,

1,12

,c(t)

/h]

Max

[f v,

3,12

(t)/h

]

Max

[f v,

3,12

,c(t)

/h]

Max

[f w

,1,0

(t)/h

]

Max

[f w

,3,0

(t)/h

]

Fig. 13. Amplitude–frequency curves of the FGM shell (h/R¼0.025, L/R¼20, R¼1 m, p¼1) with the companion mode participation. Fourteen dof model. (a) ‘‘– –’’, driven

mode (1, 6) w without companion mode participation; ‘‘ ’’, driven mode (1, 6) w with companion mode participation. (b) Companion mode (1, 6) w. (c) ‘‘– –’’, driven

mode (1, 12) v without companion mode participation; ‘‘ ’’, driven mode (1, 12) v with companion mode participation. (d) Companion mode (1, 12) v. (e) ‘‘– –’’,

driven mode (3, 12) v without companion mode participation; ‘‘ ’’, driven mode (3, 12) v with companion mode participation. (f) Companion mode (3, 12) v. (g) ‘‘– –

’’, axisymmetric mode (1, 0) w without companion mode participation; ‘‘ ’’, axisymmetric mode (1, 0) w with companion mode participation. (h) ‘‘– –’’, axisymmetric

mode (3, 0) w without companion mode participation; ‘‘ ’’, axisymmetric mode (3, 0) w with companion mode participation. (For interpretation of the references to

colour in this figure caption, the reader is referred to the web version of this article.)

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–7774

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 104

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

tω1,6

f w,1

,6(t)

, fw

,1,6

,c(t)

driven modecompanion mode

Fig. 14. Time histories of the FGM shell (h/R¼0.025, L/R¼20, R¼1 m, p¼1), transient included. ‘‘ ’’, driven mode (1, 6) w with companion mode participation;

‘‘ ’’, companion mode (1, 6) w. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)

4.5 4.5005 4.501 4.5015 4.502 4.5025 4.503 4.5035 4.504 4.5045 4.505x 104

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

tω1,6

f w,1

,6(t)

, fw

,1,6

,c(t)

driven modecompanion mode

Fig. 15. Time histories of the FGM shell (h/R¼0.025, L/R¼20, R¼1 m, p¼1), steady state. ‘‘ ’’, driven mode (1, 6) w with companion mode participation; ‘‘ ’’,

companion mode (1, 6) w. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–77 75

different number of asymmetric and axisymmetric modes; thefundamental role of the axisymmetric modes is confirmed, andthe role of the higher-order asymmetric modes is clarified inorder to obtain the actual character of the shell nonlinearity.

An interesting result of the present study regards the predic-tions obtained with low-order expansions. It is well known in theliterature that small expansions could lead to hardening beha-viours when the actual shell response is softening. Here we havefound, probably for the first time, that when shells having actualhardening response are simulated with an insufficient expansion,their behaviour could appear spuriously softening.

The effect of the geometry on the nonlinear vibrations ofthe shells is analysed: very short shells and thick shells show ahardening nonlinear behaviour, conversely, a softening nonlinearityis found in a wide range of the shell geometries. For sufficientlylong and thin shells, the system behaves in a hardening nonlinearway. This confirms the results available in the literature concerninghomogeneous shells.

The effect of the material distribution on the shell nonlinearresponse is analysed by considering two different configurationsof the constituent materials: Type I FGM shell, with nickel on theinner surface and stainless steel on the outer surface, and Type II

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510-5

10-4

10-3

10-2

10-1

100

101

102

103

104

Ω /ω1,6

|FFT

(f w,1

,6(t)

)|, |F

FT(f w

,1,6

,c(t)

)|

driven modecompanion mode

Fig. 16. Spectrum of the time histories of the FGM shell (h/R¼0.025, L/R¼20, R¼1 m, p¼1), transient removed. ‘‘ ’’, driven mode (1, 6) w with companion mode

participation; ‘‘ ’’, companion mode (1, 6) w. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–7776

FGM shell, with stainless steel on the inner surface and nickel onthe outer surface.

In the case of Type I FGM, as the power-law exponentincreases, the corresponding natural frequency decreases; forType II FGM, as the power-law exponent increases, the corre-sponding natural frequency increases. In the case of a hardeningnonlinear behaviour, the nonlinearity follows increments ordecrements of natural frequency; in the case of softening, thebehaviour is opposite.

The effect of the boundary conditions on the natural frequen-cies and nonlinear responses is analysed. The effect of theboundary conditions on the natural frequencies is prominent forthe low circumferential wave numbers, and disappears for highcircumferential wave numbers. The boundary conditions stronglyinfluence the nonlinear character.

The effect of the companion mode participation on the non-linear response of the shells is analysed. Both driven and compa-nion modes are considered allowing for the travelling-waveresponse of the shell; amplitude–frequency curves with compa-nion mode participation are obtained.

It is worthwhile to stress that, even though the nonlinearity ofthe system is weak, close to the resonance of asymmetric modesthe onset of a travelling wave along the circumferential directionis possible; this is a macroscopic effect of the weak nonlinearitythat cannot be predicted with linear models.

Appendix

Free–free boundary conditions (34) imply the followingequations:

1

L

XMu

m ¼ 0

XN

n ¼ 0

~Um,nTn

m,ZðZÞcosnyþnnR

XMv

m ¼ 0

XN

n ¼ 0

~V m,nTn

mðZÞcosny

þnR

XMw

m ¼ 0

XN

n ¼ 0

~W m,nTn

mðZÞcosny¼ 0 ð44Þ

�n

R

XMu

m ¼ 0

XN

n ¼ 0

~Um,nTn

mðZÞsinnyþ1

L

XMv

m ¼ 0

XN

n ¼ 0

~V m,nTn

m,ZðZÞsinny¼ 0

ð45Þ

�1

L2

XMw

m ¼ 0

XN

n ¼ 0

~W m,nTn

m,ZZðZÞcosnyþnnR2

XMv

m ¼ 0

XN

n ¼ 0

~V m,nTn

mðZÞcosny

þn2nR2

XMw

m ¼ 0

XN

n ¼ 0

~W m,nTn

mðZÞcosny¼ 0 ð46Þ

�1

L3

1

ð1�nÞXMw

m ¼ 0

XN

n ¼ 0

~W m,nTn

m,ZZZðZÞcosnyþn

LR

nR

1

ð1�nÞþ

3

41þ

1

R

� �� �

XMv

m ¼ 0

XN

n ¼ 0

~V m,nTn

m,ZðZÞcosny

þn2

LR

nR

1

ð1�nÞþ 1þ

1

R

� �� � XMw

m ¼ 0

XN

n ¼ 0

~W m,nTn

m,ZðZÞcosny

þn2

4R21þ

1

R

� � XMu

m ¼ 0

XN

n ¼ 0

~Um,nTn

mðZÞcosny¼ 0 ð47Þ

where ( � ),Z¼q( � )/qZ, ðUÞ,ZZ ¼ @2ðUÞ=@ 2

Z and ðUÞ,ZZZ ¼ @3ðUÞ=@ 3

Z .Such conditions are valid for any real value of y and integer

value of n; therefore, Eqs. (44), (45), (46) and (47) can be modifiedas follows:

1

L

XMu

m ¼ 0

~Um,nTn

m,ZðZÞþnnR

XMv

m ¼ 0

~V m,nTn

mðZÞþnR

XMw

m ¼ 0

~W m,nTn

mðZÞ ¼ 0;

yA ½0,2p� nA ½0,N� Z¼ 0,1 ð48Þ

�n

R

XMu

m ¼ 0

~Um,nTn

mðZÞþ1

L

XMv

m ¼ 0

~V m,nTn

m,ZðZÞ ¼ 0,

yA ½0,2p� nA ½0,N� Z¼ 0,1 ð49Þ

�1

L2

XMw

m ¼ 0

~W m,nTn

m,ZZðZÞþnnR2

XMv

m ¼ 0

~V m,nTn

mðZÞ

M. Strozzi, F. Pellicano / Thin-Walled Structures 67 (2013) 63–77 77

þn2nR2

XMw

m ¼ 0

~W m,nTn

mðZÞ ¼ 0, yA 0,2p½ � nA 0,N½ � Z¼ 0,1 ð50Þ

�1

L3

1

ð1�nÞXMw

m ¼ 0

~W m,nTn

m,ZZZðZÞþn

LR

nR

1

ð1�nÞþ

3

41þ

1

R

� �� �

XMv

m ¼ 0

~V m,nTn

m,ZðZÞþn2

LR

nR

1

ð1�nÞ þ 1þ1

R

� �� �

�XMw

m ¼ 0

~W m,nTn

m,ZðZÞþn2

4R21þ

1

R

� �XMu

m ¼ 0

~Um,nTn

mðZÞ ¼ 0,

yA ½0,2p� nA ½0,N� Z¼ 0,1 ð51Þ

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