Nonlinear vibrations of clamped-free circular cylindrical shells

Journal of Sound and Vibration 330 (2011) 5363–5381

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Nonlinear vibrations of clamped-free circular cylindrical shells Ye. Kurylov a, M. Amabili b,n a b

Dipartimento di Ingegneria Industriale, Universita di Parma Viale Usberti 181/A, Parma 43100, Italy Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Canada H3A 2K6

a r t i c l e i n f o

abstract

Article history: Received 23 January 2011 Received in revised form 29 March 2011 Accepted 30 May 2011 Handling Editor: M.P. Cartmell Available online 8 July 2011

Only experimental studies are available on large-amplitude vibrations of clamped-free shells. In the present study, large-amplitude nonlinear vibrations of clamped-free circular cylindrical shell are numerically investigated for the first time. Shells with perfect and imperfect shape are studied. The Sanders–Koiter nonlinear shell theory is used to calculate the elastic strain energy. Shell displacement fields (longitudinal, circumferential and radial) are expanded by means of a double mixed series, i.e. harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. All boundary conditions are satisfied. The system is discretized by using natural modes of the shell and Lagrange equations by an energy approach, retaining damping through Rayleigh’s dissipation function. Different expansions involving from 18 to 52 generalized coordinates are used to study the convergence of the solution. The nonlinear equations of motion are numerically studied by using arclength continuation method and bifurcation analysis. Numerical responses to harmonic radial excitation in the spectral neighborhood of the lowest natural frequency are compared with experimental results available in literature. The effect of geometric imperfections and excitation amplitude are numerically investigated and fully explained. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Many efforts were made to study nonlinear vibrations of cylindrical shells due to their wide application in aerospace, mechanical and civil engineering. A great number of studies on geometrically nonlinear vibrations of circular cylindrical shells is available (see the extensive review of Amabili and Paı¨doussis [1]). The problem is also broadly discussed by Amabili in his recent monograph [2]. A good percentage of them is addressed to large-amplitude free and forced vibrations under radial harmonic excitation. In the majority of these studies the Donnell’s nonlinear shallow-shell theory is used to obtain the equations of motion; see e.g. Refs. [3–11]. Only some researchers used more refined nonlinear shell theories ¨ [12–25], as the Novozhilov, the Sanders–Koiter (also referred as Sanders) and the Flugge–Lur’e–Byrne nonlinear shell theory, or included shear deformation and rotary inertia. Numerical differences among the four most popular classical nonlinear shell theories have been numerically investigated in Ref. [12], while a discussion on differences due to small terms can be found in Ref. [26].

n

Corresponding author. Tel.: þ1 514 398 3068; fax: þ1 514 398 7365. E-mail address: [email protected] (M. Amabili). URL: http://people.mcgill.ca/marco.amabili/ (M. Amabili).

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.05.037

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Ye. Kurylov, M. Amabili / Journal of Sound and Vibration 330 (2011) 5363–5381

Many of these studies do not include geometric imperfections and some of them use a single-mode approximation to describe the shell dynamics. Most of the studies address vibrations of shells with simply supported boundary conditions, while more complicated boundary conditions, as clamped-free (i.e. cantilever) shells are still not investigated, nevertheless their wide application as storage tanks, thermal shields of rockets and engines. Therefore, a contribution toward developing a general framework that allows to study shells with different boundary conditions is particularly important. Pellicano [27] proposed the idea of using orthogonal polynomials instead of trigonometric functions to expand displacement fields in longitudinal direction. This choice allows to satisfy different boundary conditions. Kurylov and Amabili [28] followed this idea and studied nonlinear flexural vibrations of simply supported and clamped shells. The linear vibrations of clamped-free (cantilever) shells were investigated by several researchers, e.g. [29–31]. However, till now, the only available research on nonlinear vibrations of clamped-free (cantilever) shells is the experimental study performed by Chiba [32,33] on large amplitude free vibrations of polyester shells. No model has been developed yet to model nonlinear vibrations of clamped-free circular cylindrical shells since the expansions used to discretize the system take complicated expressions. In the present study, nonlinear vibrations of clamped-free circular cylindrical shells are analyzed for the first time. The Sanders–Koiter nonlinear shell theory is used to calculate the elastic strain energy. Shell displacement fields (longitudinal, circumferential and radial) are expanded by means of a double mixed series, i.e. harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. All boundary conditions are satisfied. The system is discretized by using natural modes of the shell and Lagrange equations by an energy approach, retaining damping through Rayleigh’s dissipation function. Different expansions involving from 18 to 52 generalized coordinates are used to study the convergence of the solution. The nonlinear equations of motion are numerically studied by using arclength continuation method and bifurcation analysis. Numerical responses to harmonic radial excitation in the spectral neighborhood of the lowest natural frequency are compared with experimental results available in literature. The effect of geometric imperfections and excitation amplitude are numerically investigated and fully explained.

2. Discretization and boundary conditions Fig. 1 shows a circular cylindrical shell having radius R, length L and thickness h; a cylindrical coordinate system (O; x, r,

y) is considered in order to take advantage of the axial symmetry of the structure; the origin is placed at the center of the clamped end of the shell. The displacements of arbitrary point on the middle surface are: axial u(x, y, t), circumferential v(x, y, t) and radial w(x, y, t), where t is time. Geometric imperfections are considered by means of initial radial displacements w0(x, y) associated to zero initial stress. In order to study vibrations of circular cylindrical shells, displacement fields are expanded by means of a double series: the deformation in the circumferential direction is expanded by harmonic functions due to the periodicity of the structure, which is closed on itself, while Chebyshev polynomials are used for describing the displacements in the axial direction. In particular, the axial coordinate x is rewritten by using the nondimensional variable Z ¼x/L, where L is the shell length. This transformation gives, e.g., @=@x ¼ ð1=LÞ @=@Z.

x u w

v

h

L

O

R

θ

Fig. 1. Circular cylindrical shell: coordinate system and dimensions.

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Starting with a linear analysis, let us consider a natural mode of vibration, i.e. a synchronous vibration: uðZ, y,tÞ ¼ UðZ, yÞf ðtÞ, vðZ, y,tÞ ¼ VðZ, yÞf ðtÞ, wðZ, y,tÞ ¼ WðZ, yÞf ðtÞ,

(1)

where functions UðZ, yÞ, VðZ, yÞ and WðZ, yÞ represent the mode shape, while f(t) is the time function, which is the same for all the displacements. As previously discussed, the modal shape is expanded in a double series in terms of harmonic functions and Chebyshev n polynomials Tm ðZÞ in order to satisfy the boundary conditions: MU X N X

UðZ, yÞ ¼

n Um,n,d Tm ðZÞ cosðnyÞ,

m¼0n¼0 MV X N X

VðZ, yÞ ¼

n Vm,n,d Tm ðZÞ sinðnyÞ,

m¼0n¼0 MW X N X

WðZ, yÞ ¼

n Wm,n,d Tm ðZÞ cosðnyÞ,

(2)

m¼0n¼0 n where Tm ðZÞ ¼ Tm ð2Z1Þ and Tm is the mth order Chebyshev polynomial of the first kind and n is the number of circumferential waves of the mode shape. The transformation of coordinates from Z to 2Z  1 is necessary since Chebyshev n polynomials are defined between –1 and 1, while Tm ðZÞ has been introduced in order to be defined between 0 and 1. Um,n,d , Vm,n,d and Wm,n,d are unknown coefficients depending on the integer index m and n in Eqs. (2); the last subscript d stays for driven mode. Due to the symmetry of the shell, a second mode, identical to the first one but rotated by p/2n, is always associated to the one represented by Eq. (2) for any na0. This can be represented by

MU X N X

UðZ, yÞ ¼

n Um,n,c Tm ðZÞ sinðnyÞ,

m¼0n¼1 MV X N X

VðZ, yÞ ¼

n Vm,n,c Tm ðZÞ cosðnyÞ,

m¼0n¼1

WðZ, yÞ ¼

MW X N X

n Wm,n,c Tm ðZÞ sinðnyÞ,

(3)

m¼0n¼1

where the subscript c stays for companion mode. If the mode in Eq. (2) is the one directly excited by the external excitation, this is referred as the driven mode. Then, the second mode described by Eq. (3) is referred to as the companion. Boundary conditions are considered by applying constraints to the free coefficients of expansions (2) or (3). Some of the coefficients Um,n,d ,Vm,n,d ,Wm,n,d (or Um,n,c ,Vm,n,c ,Wm,n,c ) can be suitably chosen in order to satisfy the boundary conditions. For the clamped-free (clamped at x¼0) shell the following boundary conditions are imposed [34]: u¼v¼w¼ Nx ¼ Nxy þ

@w ¼ 0 at x ¼ 0, @x

Mxy @Mxy ¼ Mx ¼ Qx þ ¼ 0 at x ¼ L, R R@y

(4a) (4b)

where Nx, Nxy, Mx, Mxy, Qx are the normal force per unit length in axial direction, shear force per unit length, bending moment per unit length in axial direction, torsion moment per unit length and force normal to the shell per unit length, respectively. It can be observed that boundary conditions (4a) are of geometric type, while natural boundary conditions are applied at the free edge of the shell, i.e. at x ¼L (or Z ¼ 1). In fact, since the Rayleigh–Ritz method is used to find the solution of the linear free vibration problem, just geometric boundary condition has to be exactly satisfied; it means that conditions (4b) do not have to be satisfied by the expansion since they will be satisfied by minimization of the system energy. Eq. (4a) can be rewritten in the following form: UðZ, yÞ ¼

MU X N X

n Um,n,d Tm ðZÞ cosðnyÞ ¼ 0 at Z ¼ 0,

(5a)

n Vm,n,d Tm ðZÞ sinðnyÞ ¼ 0 at Z ¼ 0,

(5b)

n Wm,n,d Tm ðZÞ cosðnyÞ ¼ 0 at Z ¼ 0,

(5c)

m¼0n¼0

VðZ, yÞ ¼

MV X N X m¼0n¼0

WðZ, yÞ ¼

MW X N X m¼0n¼0

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Ye. Kurylov, M. Amabili / Journal of Sound and Vibration 330 (2011) 5363–5381 MW X N X @WðZ, yÞ @T n ðZÞ ¼ Wm,n,d m cosðnyÞ ¼ 0, at Z ¼ 0: @Z @Z m¼0n¼0

(5d)

Eqs. (5a–d) are valid for any y and n; therefore they are simplified into MU X

MV X

n Um,n,d Tm ðZÞ ¼ 0,

m¼0 MW X

n Vm,n,d Tm ðZÞ ¼ 0,

m¼0 n Wm,n,d Tm ðZÞ ¼ 0,

m¼0

MW X

Wm,n,d

m¼0

n @Tm ðZÞ ¼ 0, @Z

for n ¼ 0, 1:::, at Z ¼ 0:

(6a2d)

This linear algebraic system is solved in terms of the coefficients U0,n,d ,V0,n,d ,W0,n,d ,W1,n,d , for n¼0,1y; these coefficients can be obtained exactly in terms of the remaining ones. 3. Linear vibrations The linear analysis is obtained by using the Sanders–Koiter shell theory (see Appendix A), in which only linear terms in the strain–displacement relationships are retained. Since the shell with clamped-free edges does not present a simple closed-form solution, the solution is obtained solving numerically an eigenvalue problem. Eqs. (1) and (2) or (3) are inserted into the expressions of kinetic and potential energies (see Appendix A); in particular, the nonlinear terms are neglected. Then, a set of ordinary differential equations is obtained by using the Lagrange equations. These equations can be immediately decoupled in the variable y. An intermediate step is the reordering of variables. A vector q containing all variables is built: q ¼ ½U1,0 ,U2,0 ,. . .,U1,n,d ,U2,n,d ,. . .,V1,0 ,V2,0 ,. . .,V1,n,d ,V2,n,d ,. . .,W2,0 ,W3,0 ,. . .,W2,n,d ,W3,n,d ,. . . f ðtÞ

(7)

The dimension of the vector q is Nmax. In Eq. (7), both axisymmetric and asymmetric terms are included; for axisymmetric terms the subscript d (or c in case of companion mode, which gives exactly the same frequency of the driven mode) is omitted since there is a single mode instead of the couple of driven and companion modes. For asymmetric terms, just driven modes can be considered. In fact, companion modes have exactly the same frequency and mode shapes, just exchanging sin and cos functions, as shown by comparing Eqs. (2) and (3). The number of variables needed to describe a mode with any number of circumferential waves (after decoupling in y) is MT ¼ MU þMV þ MW 1. For free vibrations, the Lagrange equations are coincident with the Rayleigh–Ritz method and are given by   d @L @L ¼ 0, i ¼ 1,2,:::,Nmax , (8)  dt @q_ i @qi where Nmax ¼ MT ðN þ1Þ and L ¼ TS US , being TS the kinetic energy and US the potential energy of the shell, which are given in Appendix A. In these energy expressions, only linear strain–displacement relationship have to be used, being the nonlinear relationships applied later in the nonlinear analysis. Using (7) and assuming harmonic motion, f ðtÞ ¼ ejot with j being the imaginary unit and o the vibration frequency, one obtains ðo2 Mþ KÞq ¼ 0

(9)

which is the classical eigenvalue problem in nonstandard   form; it gives natural frequencies o and mode shapes q. In Eq. (9) M is the mass matrix, obtained from d=dt @L=@q_ i , and K is the stiffness matrix, obtained from @L=@qi . The mode shape corresponding to the jth mode is given by Eqs. (2), where the coefficients Um,n,d , Vm,n,d , Wm,n,d are ðjÞ ðjÞ ðjÞ substituted with Um,n , Vm,n , Wm,n , which are the components of the jth eigenvector obtained from Eq. (9). Then, the following functions of the longitudinal variable Z are introduced:

U ðjÞ ðZÞ ¼

MU X N X m¼0n¼0

ðjÞ n Um,n Tm ðZÞ,

V ðjÞ ðZÞ ¼

MV X N X m¼0n¼0

ðjÞ n Vm,n Tm ðZÞ,

W ðjÞ ðZÞ ¼

MW X N X

ðjÞ n Wm,n Tm ðZÞ:

(10)

m¼0n¼0

The vector function UðjÞ ðZÞ ¼ ðU ðjÞ ðZÞ, V ðjÞ ðZÞ, W ðjÞ ðZÞÞT is the jth eigenfunction vector of the original problem. Mode shapes are normalized in order to have unit maximum displacement by U ðjÞ ðZÞ=maxðU ðjÞ ðZÞÞ, V ðjÞ ðZÞ=maxðV ðjÞ ðZÞÞ and W ðjÞ ðZÞ=maxðW ðjÞ ðZÞÞ for any Z. For simplicity, the normalized mode shapes will be indicated with U ðjÞ ðZÞ, V ðjÞ ðZÞ, W ðjÞ ðZÞ in the following part of the paper. It should be mentioned that, for accurate numerical calculations, a very high numerical accuracy is required in calculating the eigenvectors (mode shapes) and all the coefficients to be introduced in the matrices. 4. Nonlinear vibrations In order to attack the nonlinear problem, the expression of the potential shell energy (see Appendix A), obtained by using the nonlinear strain–displacement relationships, is used. The displacements uðZ, y,tÞ, vðZ, y,tÞ and wðZ, y,tÞ are

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expanded by using the linear mode shapes obtained in the previous linear analysis: uðZ, y,tÞ ¼

M X N X

U ðjÞ ðZÞ½uj,n,c ðtÞ cosðnyÞ þ uj,n,s ðtÞ sinðnyÞ,

j¼1n¼0

vðZ, y,tÞ ¼

M X N X

V ðjÞ ðZÞ½vj,n,c ðtÞ sinðnyÞ þ vj,n,s ðtÞ cosðnyÞ,

j¼1n¼1

wðZ, y,tÞ ¼

M X N X

W ðjÞ ðZÞ½wj,n,c ðtÞ cosðnyÞ þ wj,n,s ðtÞ sinðnyÞ,

(11)

j¼1n¼0

where uj,n,s ðtÞ, uj,n,c ðtÞ, vj,n,s ðtÞ, vj,n,c ðtÞ, wj,n,s ðtÞ and wj,n,c ðtÞ are the generalized coordinates. The total number of degrees of freedom in the nonlinear analysis is 2ð3M  NÞ þ 2M. In Eq. (11), the expression of the torsional displacement v starts from n ¼1 instead of n¼ 0 since the family of axisymmetric torsional modes (which is not considered in the numerical analysis since radial excitation is applied) is uncoupled from the axial and flexural displacements. This number of degrees of freedom is generally much smaller than Nmax used in the linear analysis in order to obtain a reduced-order model. The choice of the value N is linked to the number n of circumferential waves of the vibration mode considered; a good choice to keep the dimension small enough is N ¼3n. Actually not all the integers numbers between 0 (or 1) and N have to be used in the model. If no geometric imperfections are introduced in the model (this point will be discussed later in the Section 5), just modes with the following number of circumferential waves have to be considered: 0, n, 2n, 3n; any other number in between these ones will give no contribution to the model. The presence of axisymmetric modes (n ¼0) is fundamental in order to respect the physical behavior of the shell, which is flexible to bending but very stiff with respect to in-plane stretching. In Eqs. (11) both sin and cos mode shapes in y are introduced since a circular cylindrical shell is axisymmetric; therefore, both families of driven and companion modes are participating in the shell response with one-to-one internal resonances since they have exactly the same natural frequencies. Expansions (11) satisfy the boundary conditions. The normalized mode shapes U ðjÞ ðZÞ, V ðjÞ ðZÞ, W ðjÞ ðZÞ are known functions, evaluated in the previous linear analysis, and are expressed in terms of polynomials. Using expansion (11) one can select suitable shapes for each displacement separately, improving convergence and reducing the number of degrees of freedom. It is interesting to note that, due to the normalization, the generalized coordinates represent the maximum amplitude of vibration since maxðU ðjÞ ðZÞÞ, maxðV ðjÞ ðZÞÞ and maxðW ðjÞ ðZÞÞ is one. Also the trigonometric sin and cos functions have maximum value equal to one, therefore the maximum vibration amplitude is indicated by the generalized coordinates. Expansion (11) is inserted in the expressions giving strain and kinetic energies, virtual work and Rayleigh dissipation function (see Appendix A). Only a radial harmonic concentrated force excitation is assumed to act on the shell. The external radial distributed load qr applied to the shell, due to the radial concentrated force f~ , is given by ~ cosðotÞ, qr ¼ f~ dðRyRy~ ÞdðxxÞ

(12)

where o is the excitation frequency, t is the time, d is the Dirac delta function, f~ gives the radial force amplitude positive in z direction, x~ and y~ give the axial and angular positions of the point of application of the force, respectively; here, the point excitation is assumed to be located at x~ ¼ L=2, y~ ¼ 0; as a consequence of this excitation, the generalized coordinates with subscript c are directly excited (driven modes) and those with subscript s are not directly excited (companion modes). The following notation is introduced for sake of brevity: p ¼ fuj,n,c ,uj,n,s ,vj,n,c ,vj,n,s ,wj,n,c ,wj,n,s gT , j ¼ 1,. . .M, and n ¼ 0,. . .,N

(13)

where p is the time-dependent generalized coordinate vector. The generic element of the vector p is referred to as pi; the dimension of p is dofs, which is the total number of degrees of freedom used in the mode expansion (11). The generalized forces Qj are obtained by differentiation of the Rayleigh’s dissipation function and of the virtual work done by external forces Qi ¼ 

@F @W þ : @p_ i @pi

The Lagrange equations of motion for the shell are given by   d @TS @TS @US  þ ¼ Qi , i ¼ 1,. . .dofs, dt @p_ i @pi @pi

(14)

(15)

where @ TS =@ pi ¼ 0. These second-order ordinary differential equations have very long expressions containing quadratic and cubic nonlinear terms. The only nonlinear terms in Eq. (15) can be written in the form dofs dofs dofs X X X @US ¼ pk fk,i þ pj pk fj,k,i þ pj pk pl fj,k,l,i , @pi k¼1 j,k ¼ 1 j,k,l ¼ 1

(16)

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where coefficients f have long expressions that include also geometric imperfections. These terms give quadratic and cubic nonlinearities. The set of ordinary nonlinear differential equations (15) is studied by using numerical continuation methods and bifurcation analysis. 5. Numerical results The equations of motion have been obtained by using the Mathematica 6 computer software [35] in order to perform analytical surface integrals of trigonometric and Chebyshev functions. The set of ordinary nonlinear differential equations (15) is studied by using numerical continuation methods and bifurcation analysis. The Lagrange equations are rewritten in vectorial notation and pre-multiplied by the inverse of the mass matrix in order to decouple them. Then, each equation is transformed in two first-order equations. A non-dimensionalization of variables is also performed for computational convenience: the frequencies are non-dimensionalized dividing by the natural frequency of the resonant mode and the vibration amplitudes are divided by the shell thickness h. The resulting 2  dofs equations are studied by using the software AUTO [36] for continuation and bifurcation analysis of nonlinear ordinary differential equations. The software AUTO is capable of continuation of the solution, bifurcation analysis and branch switching by using arclength continuation and collocation methods. In particular, the shell response under harmonic excitation has been studied by using a numerical continuation analysis in two steps: (i) first the excitation frequency has been fixed far enough from resonance and the magnitude of the excitation has been used as bifurcation parameter; the solution has been started at zero force where the solution is the trivial undisturbed configuration of the shell and has been continued up to reach the desired force magnitude; (ii) when the desired magnitude of excitation has been reached, the solution has been continued by using the excitation frequency as bifurcation parameter. The detected pitchfork bifurcations are then followed and Neimark– Sacher bifurcations are detected on the bifurcated branch of the solution. A validation of the present linear analysis has been carried out in Table 1 by comparing the natural frequencies calculated with the present approach to those reported in Table 2.35 of Leissa [34] and originally obtained by Resnick and Dugundji [37] by using the Sanders shell theory. Numerical results in Table 1 are in good agreement and practically coincident for nZ6; the agreement improve with the circumferential wavenumber n as a possible consequence of some approximation in the in-plane inertia in Ref. [37]. 5.1. Perfect shell An isotropic circular cylindrical shell clamped at x ¼0 and free at x ¼L is considered. Calculations have been performed for a shell having the following dimensions and material properties: L¼0.48 m, R¼0.24 m, h¼0.254 mm, E¼4.65  109 Pa, r ¼ 1400 kg/m3 and n ¼0.38, which corresponds to the case of a polyester shell experimentally studied by Chiba [32,33]. Chebyshev polynomials of 15th power were used to obtain mode shapes of the problem (linear vibration study). The linear analysis shows that the fundamental mode has n¼7 circumferential waves and has natural frequency of 28.3 Hz according to the Sanders–Koiter shell theory. A nondimensional excitation in radial direction fi is introduced, which is related to the dimensional force excitation by fi ¼ f~ =mi o21,n h, where o1,n is the fundamental natural frequency in rad/s and mi is the modal mass of the shell, i.e. the ith diagonal element of the mass matrix M in Eq. (9). The modal damping ratio z ¼ 0:0005 is used for all modes. The frequency–response curve of the fundamental mode (1,n) with n ¼7, without companion modes participation and without imperfections is shown in Fig. 2 for different models. Each model has a different number of degrees of freedom (dofs). In particular, results for 18, 20, 22, 24, 26, 28, 30, 40, 43 and 47 dofs are presented. The response curves become less and less hardening increasing the number of degrees of freedom, reaching convergence for the model with 43 dofs, which shows a very mild softening behavior. In fact, the models with 43 and 47 dofs give almost coincident results, as shown in Fig. 3. The vibration amplitudes shown in the figures are the maximum amplitudes during an excitation period and they are obtained at x¼ L, i.e. at the free edge of the shell where the largest vibration amplitudes is observed. In particular, the model with 43 dofs has the following generalized coordinates: for w and u, all the generalized coordinates with subscripts (1,n,c), (1,2n,c), (1,3n,c), (2,n,c), (2,2n,c), (3,n,c), (3,2n,c) plus the first 11 axisymmetric modes, i.e. (1,0), (2,0), (3,0), y, (11,0); for v, the same as u and w but without axisymmetric modes since torsional axisymmetric modes are uncoupled Table 1 Comparison of the natural frequencies obtained by the present approach with those reported by Leissa [34] in Table 2.35 and originally obtained by Resnick and Dugundji [37] for a clamped-free aluminum circular cylindrical shell having the following dimension and material properties: L ¼0.30531 m, R ¼0.073914 m, h ¼ 0.000178 m, n ¼ 0.3; E ¼6.8258  1010 Pa, and r ¼2712.2 kg/m3. Only natural modes without internal circumferential nodes are reported. Circumferential waves number, n

2

3

4

5

6

7

8

9

10

Natural frequency (Hz), present study Natural frequency (Hz), Refs. [34,37]

456.4 489

233.9 246

175.7 181

205.3 207

279.4 280

377.6 378

494.1 494

627.1 627

776.2 776

Ye. Kurylov, M. Amabili / Journal of Sound and Vibration 330 (2011) 5363–5381

1.2

j k

h

g

f

e

d

c

b

5369

a

1.0

w1,n,c / h

0.8

0.6

0.4

0.2

0.0 1.00

1.05

1.10

1, n

Fig. 2. Frequency–response curve of the fundamental mode, without companion modes participation; no imperfections. Study of the convergence of the solution by comparison of models with different number of degrees of freedom (dofs): (a) 18 dofs; (b) 20 dofs; (c) 22 dofs; (d) 24 dofs; (e) 26 dofs; (f) 28 dofs; (g) 30 dofs; (h) 40 dofs; (j) 43 dofs; and (k) 47 dofs. Nondimensional excitation f1 ¼0.0012.

1.2

1.0

w1,n,c /h

0.8

0.6

0.4

0.2

0.0 0.996

0.998

1.000

1.002

1.004

1, n

Fig. 3. Frequency–response curve of the fundamental mode, without companion modes participation; no imperfections. Comparison of models with 43 dofs (dashed line) and 47 dofs (solid line). Nondimensional excitation f1 ¼ 0.0012.

from bending modes. The model with 47 dofs has the same generalized coordinates with the addition of two axisymmetric modes for both w and u: (12,0) and (13,0). The smallest model in Fig. 2, i.e. the one with 18 dofs, uses the following generalized coordinates: for w and u, coordinates (1,n), (1,2n), (3,n), (3,2n) plus the first 3 axisymmetric modes; for v, the same as u and w but without axisymmetric modes. In the models with 20–30 degrees of freedom, an additional axisymmetric mode is progressively added for both w and u. The 40 dofs model has the same generalized coordinates of the previous models but the first 13 axisymmetric modes are included for both w and u. All these results have been obtained by using the base of natural modes; this applies also to axisymmetric modes.

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5.2. ‘‘Artificial’’ axisymmetric modes versus natural axisymmetric modes The result of the convergence study shows that a large number of axisymmetric modes, at least 11 for each one of the displacements w and u, must be used in order to obtain accurate results. Therefore, about half of the degrees of freedom are used by models in Fig. 3 for axisymmetric generalized coordinates. It must be observed that, (i) axisymmetric modes have high natural frequencies, so that they are evaluated with less numerical accuracy in the linear eigenvalue analysis, and (ii) the generalized coordinates corresponding to axisymmetric modes have much smaller amplitudes than the other coordinates during flexural vibrations. Therefore, it is convenient to reduce the number of axisymmetric modes in the model, but at the same time it is necessary to keep the accuracy of the solution. In order to do that, the natural axisymmetric modes are replaced by ‘‘artificial’’ axisymmetric modes. These ‘‘artificial’’ axisymmetric modes are built by taking the axial shape of the asymmetric modes with n circumferential waves (n ¼7 in the present case) and making them artificially axisymmetric. The advantage of this technique is that asymmetric modes have much lower natural frequency and therefore the mode shapes can be evaluated with more accuracy. Fig. 4 shows the frequency–response curve of the fundamental mode without companion modes participation and no imperfections obtained with the model with 31 dofs using ‘‘artificial’’ axisymmetric modes versus the 47 dofs model with natural modes. The 31 dofs model with ‘‘artificial’’ axisymmetric modes has the same generalized coordinates of the 47 dofs model but uses only the first 5 axisymmetric modes for both w and u instead of 13, therefore reducing the model by 16 dofs. The two response curves are almost coincident, showing the efficiency of using ‘‘artificial’’ axisymmetric modes in reducing the order of the model. A convergence study of the models with ‘‘artificial’’ axisymmetric modes is shown in Fig. 5, where frequency–response curves obtained by using models with 22, 28, 31 and 37 degrees of freedom are compared. Excluding the model with 22 dofs, which is clearly inaccurate, the other three models present similar results. The 22 dofs model has the following generalized coordinates: for w and u the coordinates with subscript (1,n), (1,2n), (3,n), (3,2n) plus the first 5 ‘‘artificial’’ axisymmetric modes; for v, the same as u and w but without axisymmetric modes. The 28 dofs model has the same coordinates of the 22 dofs model with the addition of (2,n) and (2,2n) for all the three displacements. The 31 dofs model has the coordinates of the 28 dofs model with the addition of (1,3n) for all the three displacements. Finally, the 37 dofs model has the coordinates of the 31 dofs model with the addition of (2,3n) and (3,3n) for all the three displacements. 5.3. Companion mode participation In this section, a model with 52 dofs is used with the same generalized coordinates with ‘‘artificial’’ axisymmetric modes used in the 31 dofs model in Fig. 5. It must be observed that now both driven (subscript c) and companion (subscript s) coordinates are used for all asymmetric modes; this doubles the degrees of freedom of modes that are not axisymmetric.

1.2 1.0

w1,n,c / h

0.8 0.6 0.4 0.2 0.0 0.995

1

1.005 1, n

Fig. 4. Frequency–response curve of the fundamental mode, without companion modes participation; no imperfections. Comparison of the model with ‘‘artificial’’ axisymmetric modes, 31 dofs (solid line), and the model with ‘‘natural’’ axisymmetric modes, 47 dofs (dashed line). Nondimensional excitation f1 ¼ 0.0012.

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w1,n,c /h

1.5

1.0

0.5

0.0 0.9975

1

1.0025

1, n

Fig. 5. Frequency–response curve of the fundamental mode, without companion modes participation (using ‘‘artificial’’ axisymmetric modes); no imperfections. Study of the convergence of the solution by comparison of models with different degrees of freedom: —, 22 dofs; – –, 28 dofs;  .  , 31 dofs; ——, 37dofs. Nondimensional excitation f1 ¼0.0018.

The frequency–response curve of the fundamental mode (1,n) with n¼ 7, with companion modes participation and without imperfections, is shown in Fig. 6 for nondimensional excitation f1 ¼ 0.0025. This level of excitation is large enough to obtain a pitchfork bifurcation of the driven mode, giving rise to a second branch of the solution with companion mode participation. The appearance of two pitchfork bifurcations, giving rise to a second solution branch, is due to 1:1 internal resonance between driven and companion modes that, due to the axial symmetry of the perfect shell, have exactly the same natural frequency. The first pitchfork bifurcation on branch one is observed near the peak of the response; the second bifurcation is near the exact linear resonance. Along the second branch, the response of the shell is a combination of cosine and sine modes in circumferential direction with some phase difference, giving nodes traveling around the shell (traveling wave) instead of fixed nodes as in linear vibrations. For the axial symmetry of the shell, the traveling wave can be in clockwise or anti-clockwise direction around the shell according to initial conditions or transient history. Differently, on branch one there is no companion mode participation. If a lower excitation level is used, e.g. f1 ¼0.0015, no pitchfork bifurcations are detected and the shell response is without companion mode participation. Increasing the excitation level to f1 ¼0.0033, the frequency–response curve shown in Fig. 6 is modified into the one given in Fig. 7. The main differences are that, (i) branch 2 extends over a larger frequency range around the linear resonance (i.e. o=o1,n ¼ 1), (ii) the amplitude of the driven (w1,n,c) and companion (w1,n,s) coordinates is now almost equal on branch 2 near the linear resonance giving rise to a pure traveling wave response around the shell and, (iii) the response on branch 2 presents two Neimark–Sacker bifurcations. The shell response on branch 2 comprised between the two Neimark–Sacker bifurcations is quasi-periodic, i.e. it presents amplitude modulations. The response to harmonic excitation shown in Fig. 7 is typical of circular cylindrical shells without geometric imperfections and has been previously observed for simply supported and clamped shells, see e.g. Ref. [2]. Increasing even more the excitation level to f1 ¼0.005, the frequency–response curve becomes the one given in Fig. 8. Here a saturation phenomenon is observed on both branches. The shape of branch 1 is changed and the previous response peak is cut by an almost horizontal line appearing with a Neimark–Sacker bifurcation at vibration amplitude about 3.4h. After a second Neimark–Sacker bifurcation, the response on this horizontal line returns to be a stable periodic response without companion mode participation. Branch 2 appears with a single pitchfork bifurcation near the exact linear resonance. It presents almost the same amplitude of the driven (w1,n,c) and companion (w1,n,s) coordinates, giving rise to an almost pure traveling wave. Two Neimark–Sacker bifurcations are detected on branch 2, giving rise to a quasi-periodic response similarly to the one observed in Fig. 7. Anyway, before the low-frequency Neimark–Sacker bifurcation, the response loses stability on branch 2, differently from Fig. 7, showing an unstable almost horizontal response that can be interpreted again as a kind of saturation. Being more precise, the saturation response is at vibration level 2.4h for both the driven and companion coordinates and, while for one coordinate it has a slightly positive slope, for the other qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coordinate the slope is negative. But globally, it respects the condition w21,n,c þw21,n,s ffi3:4h, indicating again a saturation phenomenon at vibration amplitude about 3.4h.

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2.5 BP

2.0 BP 2

w1,n,c / h

1.5

1.0

0.5 1

0.0 0.995

1

1.005

1,n

1.0

0.8 2

w1,n,s /h

0.6

0.4

0.2

0.0

1

BP

BP

0.995

1

1.005

1,n Fig. 6. Frequency–response curve for the fundamental mode of the perfect shell with companion mode participation; 52 dofs model, nondimensional excitation f1 ¼0.0025. —, stable periodic solution; – –, unstable solutions; BP, pitchfork bifurcation; 1, branch 1; 2, branch 2: (a) maximum of the generalized coordinate w1,n,c and (b) maximum of the generalized coordinate w1,n,s.

In order to better understand the shell response in Fig. 8, additional generalized coordinates are shown in Fig. 9. The in-plane coordinates associated to the main driven and companion coordinates are shown in Fig. 9(a)–(c). They are slaves coordinates, reproducing the response of the main driven (w1,n,c) and companion (w1,n,s) coordinates. The coordinate w1,3n,c in Fig. 9(d) instead shows that it is activated at the saturation, where some energy is transferred to this coordinate. The first axisymmetric coordinate is shown in Fig. 9(e) showing generally a negative value, indicating that an axisymmetric contraction is necessary in order to keep the quasi-inextensibility of the shell during large-amplitude vibrations. This is the reason why, even if axisymmetric displacements are small, they must be retained in the expansion of w and u since inplane stretching would erroneously introduce a strong hardening behavior of the shell. Results in the time-domain, for the nonlinear response shown in Figs. 8 and 9, are presented in Fig. 10 for excitation frequency o ¼ 0:9986 o1,n , i.e. at a specific point in the horizontal axes of Figs. 8 and 9 corresponding to response on

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BP

3.0

TR

w1,n,c /h

2.5 2.0

TR

BP

2

1.5 1.0 0.5

1

0.0 0.994

0.997

1

1.003

1,n

TR 1.5

w1,n,s /h

2

TR

1.0

0.5

1

BP

BP

0.0 0.994

0.997

1

1.003

1,n

Fig. 7. Frequency–response curve for the fundamental mode of the perfect shell with companion mode participation; 52 dofs model, nondimensional excitation f1 ¼0.0033: —, stable periodic solution;    , stable quasi-periodic solution; – –, unstable solutions; BP, pitchfork bifurcation; TR, NeimarkSacker bifurcation; 1, branch 1; 2, branch 2: (a) maximum of the generalized coordinate w1,n,c and (b) maximum of the generalized coordinate w1,n,s.

branch 2 of the solution with companion mode participation. Fig. 10(a) shows the harmonic excitation. Comparing the time responses of the generalized coordinates to the excitation, the relative phase angle can be obtained. Also, comparing Fig. 10(b), driven mode, and Fig. 10(c), companion mode, the relative phase angle can be observed, which leads to a traveling wave around the shell. At this specific excitation frequency, not a pure traveling wave is obtained because the phase angle is different from p/2 and the amplitudes of driven and companion modes are also different. The in-plane coordinates v1,n,c and v1,n,s are clearly slave of the corresponding w1,n,c and w1,n,s, respectively, as shown in Fig. 10(d) and (e). The generalized coordinate w1,2n,c presents two oscillation periods for an excitation period, showing its double frequency response in Fig. 10(f). The same is observed for the first axisymmetric coordinate w1,0,c in Fig. 10(g), which also shows negative values representing axisymmetric double-frequency contraction. 5.4. Effect of geometric imperfections and comparison to experiments Fig. 11 shows the comparison of the frequency–response curve of the fundamental mode of the perfect shell without companion modes participation (model with 37 dofs previously used in Fig. 5) with the experimental backbone curve from Chiba [32]. The backbone curve represents the nonlinear free vibration response, and should pass through the peak of the forced vibration response. A nondimensional excitation f1 ¼0.0018 has been used in the calculation. Fig. 11 clearly shows a significant difference between the computed and experimental results by Chiba. However, the calculation are for a perfect shell. For circular cylindrical shells with different boundary conditions it has been shown that geometric imperfection can

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3.5

TR

TR

3

BP

Max (w1,n,c / h)

2.5

TR 2

2

TR

1.5 1 0.5

1

0 0.995

1

1.005 1, n

2.5

TR TR

Max (w1,n,s / h)

2

1.5 2 1

0.5

0

1 0.995

BP 1

1.005 1, n

Fig. 8. Frequency–response curve for the fundamental mode of the perfect shell with companion mode participation; 52 dofs model, nondimensional excitation f1 ¼0.005. —, stable periodic solution;    , stable quasi-periodic solution; – –, unstable solutions; BP, pitchfork bifurcation; TR, NeimarkSacker bifurcation; 1, branch 1; 2, branch 2: (a) maximum of the generalized coordinate w1,n,c and (b) maximum of the generalized coordinate w1,n,s.

change the nonlinear response. For this reason, here the effect of geometric imperfection is investigated in order to understand if the difference between numerical and experimental results can be attributed to geometric imperfections. Initially the effect of geometric imperfection with shape corresponding to the first natural mode with circumferential shape cos(2ny), where n¼7 as in previous cases, is considered. This imperfection changes the natural frequencies of the shell introducing a frequency split between the fundamental driven and companion modes, that had before exactly the same natural frequency, as shown in Fig. 12. Results presented in Fig. 12 have been obtained by using the simpler Donnell shell theory retaining in-plane inertia; this is the reason why the natural frequency of the perfect shell (for zero imperfection) is slightly higher than the 28.3 Hz previously indicated. The effect of the geometric imperfection of amplitude 2h with circumferential shape cos(2ny) on the nonlinear response is shown in Fig. 13 and is quite modest. Therefore, this geometric imperfection plays a small role. The same imperfection is increased to the amplitude 10h, which is a very large unrealistic value for this type of imperfection, in Fig. 14, but still its effect is modest. The effect of imperfections of amplitude 2h with circumferential shape cos(ny) and sin(ny) is investigated in Fig. 15. The effect of these imperfections on the nonlinear response is very small. A shell with geometric axisymmetric imperfection of maximum amplitude 10h with the shape of the first axisymmetric mode is investigated in Fig. 16. The effect of this type of imperfection is large, but it makes the shell response of hardening type. Therefore, this imperfection would increase the difference between the numerical and experimental results. The comparison of the frequency–response curves (without companion modes participation, 40 dofs model) of the fundamental mode of the shell having imperfections with the shape of the first mode with circumferential form cos(2y) and different amplitudes is shown in Fig. 17; this imperfection is an ovalization of the shell. The experimental backbone

Ye. Kurylov, M. Amabili / Journal of Sound and Vibration 330 (2011) 5363–5381

5375

0.35

0.5

0.30 0.4 Max (v1,n,s /h)

Max (v1,n,c /h)

0.25 0.3 0.2

0.20 0.15 0.10

0.1 0.05 0.0 0.995

1

0.00

1.005

0.995

1

1.005

1,n

1,n

0.10

Max (w1,3n,c /h)

Max (u1,n,c /h)

0.04

0.03

0.02

0.01

0.08 0.06 0.04 0.02 0.00

0.00 0.995

1

0.995

1.005

1

1.005 1,n

1,n

0.00

Max (w1,0,c /h)

–0.01

–0.02

–0.03

–0.04 0.995

1

1.005 1,n

Fig. 9. Frequency–response curve for perfect shell with companion mode participation; 52 dofs model, nondimensional excitation f1 ¼0.005. —, stable periodic solution;    , stable quasi-periodic solution; – –, unstable solutions: (a) maximum of generalized coordinate v1,n,c; (b) maximum of generalized coordinate v1,n,s; (c) maximum of generalized coordinate u1,n,c; (d) maximum of generalized coordinate w1,3n,c; and (e) maximum of generalized coordinate w1,0,c.

curve by Chiba [32] is also presented. The effect of maximum imperfection amplitudes 3h, 4h and 5h at the free-end of the shell are shown. Ovalization imperfections can be very large due to the manufacturing process of shells, so these values are all realistic. Fig. 17 shows that these imperfections increase significantly the softening type behavior of the shell, reducing

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0.4

1.5 1

w1,n,c (t)/h

Excitation (N)

0.2

0

–0.2

0.5 0 –0.5 –1 –1.5

–0.4

0

0.015

0.03

0

2

0.2

1

0.1

v1,n,c (t)/h

w1,n,s (t)/h

0.015

0.03

Time (s)

Time (s)

0

0 –0.1

–1

–0.2 –2 0

0.015

0

0.03

0.015

0.03

Time (s)

Time (s)

0.3

0.02

w1,2,n,c (t)/h

v1,n,s (t)/h

0.2 0.1 0

0.01

0

–0.1 –0.2

–0.01

–0.3

0

0.015 Time (s)

0

0.03

0.015 Time (s)

0.03

w1,0 (t)/h

–0.02 –0.03 –0.04 –0.05 –0.06

0

0.015 Time (s)

0.03

Fig. 10. Time response of the shell with companion mode participation showing a traveling wave response; 52 dofs model, nondimensional excitation f1 ¼ 0.005, excitation frequency o ¼ 0.9986o1,n: (a) excitation; (b) generalized coordinate w1,n,c; (c) generalized coordinate w1,n,s; (d) generalized coordinate v1,n,c; (e) generalized coordinate v1,n,s; (f) generalized coordinate w1,2n,c; and (g) generalized coordinate w1,0,c.

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5377

w1,n,c /h

1.5

1.0

0.5

0.0 0.992

0.994

0.996

0.998

1.000

1.002

1, n Fig. 11. Comparison of the frequency–response curve of the fundamental mode of the perfect shell without companion modes participation (model with 37 dofs) with the experimental backbone curve from Chiba [32]. Nondimensional excitation f1 ¼0.0018.

29.5 Frequency (HZ)

(1,n)s 29.0

28.5

(1,n)c

28.0

27.5

0.0

0.5

1.0 A1,2 n /h

1.5

Fig. 12. Fundamental natural frequencies of the shell versus imperfection with shape corresponding to the first natural mode with circumferential shape cos(2ny), where n ¼7 as in previous cases. Natural frequency of driven mode, (o1,n)c , and of companion mode, (o1,n)s.

w1,n,c /h

1.5

1.0

0.5

0.0 0.996

0.998

1.000

1.002

1.004

1, n Fig. 13. Frequency–response curve (dashed line) of the shell having imperfection with shape corresponding to the first natural mode with cos(2ny) and maximum amplitude of imperfection 2h (model with 37 dofs). Solid line, frequency–response curve of the perfect shell. Nondimensional excitation f1 ¼0.0018.

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w1,n,c /h

1.5

1.0

0.5

0.0 0.996

0.998

1.000

1.002

1.004

1, n Fig. 14. Frequency–response curve (dashed line) of the shell having imperfection with shape corresponding to the first natural mode with cos(2ny) and maximum amplitude of imperfection 10h, (model with 37 dofs). Solid line, frequency–response curve of the perfect shell. Nondimensional excitation f1 ¼ 0.0018.

1.8 1.6

w1,n,c /h

1.4 1.2 1.0 0.8 0.6 0.999

1

1.0005

1, n

Fig. 15. Frequency–response curve of the shell having imperfection of maximum amplitude 2h with shape corresponding to the fundamental mode. Solid line, perfect shell; dashed line, imperfection on mode sin(ny); dashed-dotted line, imperfection on mode cos(ny). Nondimensional excitation f1 ¼ 0.0018.

and eventually canceling, in the case of imperfection amplitude 4h, the difference between the numerical and experimental results. Here it can be observed that the experimental shell tested by Chiba [32] was obtained by a polyester sheet with lap-joined seam, which can easily lead to ovalization imperfection of the magnitude introduced. Unfortunately the actual geometry of the shell is not given in [32], so it is impossible to know the actual value of the imperfections in Chiba’s shell. The 40 dofs model used in the calculation has the same generalized coordinates of the 37 dofs model previously used, with the addition of coordinates (1,2,c) for u, v and w. The introduction of these additional coordinates is fundamental in order to obtain accurate results. Additional generalized coordinates, as (1,4,c), do not change the solution. The fundamental natural frequency of the shell is changed significantly by the introduction of the ovalization imperfection, as shown in Fig. 18. Anyway, this imperfection does not introduce any frequency split between the driven and companion modes.

6. Conclusions Nonlinear vibrations of clamped-free circular cylindrical shells have never been studied before analytically or numerically due to the difficulty of finding simple trial functions to describe the shell displacements. In fact, this specific boundary condition is important in many applications, as shown by the available experimental results.

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5379

w1,n,c /h

1.5

1.0

0.5

0.0 0.996

0.998

1.000

1.002

1.004

1, n Fig. 16. Frequency–response curve of the shell having axisymmetric imperfection of maximum amplitude 10h with the shape of the first axisymmetric mode (dashed line, 37 dofs). Solid line, frequency–response curve of the perfect shell. Nondimensional excitation f1 ¼ 0.0018.

d

c b

a

w1,n,c /h

1.5

1.0

0.5

0.0 0.992

0.994

0.996

0.998

1.000

1.002

1, n Fig. 17. Comparison of the frequency–response curves (without companion modes participation, 40 dofs) of the fundamental mode of the shell having imperfections with the shape of the first mode cos(2y) and different amplitudes. The experimental backbone curve by Chiba [32] is also shown as dashed line: (a) perfect shell; (b) imperfection with maximum amplitude 3h; (c) imperfection with maximum amplitude 4h; and (d) imperfection with maximum amplitude 5h. Nondimensional excitation f1 ¼0.0018.

55

Frequency (HZ)

50 45 40 35 30 0

1

2 A1,2 n / h

3

4

Fig. 18. Natural frequency of the fundamental mode versus imperfection with shape corresponding to the first natural mode with circumferential shape cos(2y).

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The present study has not the only purpose to study nonlinear vibrations of clamped-free circular cylindrical shells for the first time, but also to investigate the model accuracy showing the solution convergence. Then, a technique to reduce the number of degrees of freedom by introducing ‘‘artificial’’ axisymmetric modes in the expansions of the displacements is shown. The shell response with a without companion mode participation is studied with great care by investigating the qualitative and quantitative changes with the increase of the excitation amplitude. Pitchfork and Neimark–Sacker bifurcations are detected, as well as a saturation phenomenon. The effect of geometric imperfection is fully investigated to understand their effect. The conclusion is that only geometric imperfections introducing an ovalization of the shell play a significant role (and also axisymmetric imperfections); this is different from what found for circular cylindrical shells with different boundary conditions. Ovalization imperfections are actually the most frequent in manufacturing shells. Their introduction increases the softening behavior of the shell and is able to match perfectly the experimental results by Chiba [32].

Acknowledgements This work was partially supported by the NSERC Discovery Grant, Canada Research Chair and Canada Foundation of Innovation (LOF) programs and McGill University start-up grant. Appendix A: The Sanders–Koiter nonlinear shell theory and energies Strain components ex , ey and gxy at an arbitrary point of the shell are:

ex ¼ ex,0 þ zkx , ey ¼ ey,0 þ zky , gxy ¼ gxy,0 þ zkxy

(A1a2c)

where z is the distance of the arbitrary point of the shell from the middle surface. According to Sanders–Koiter nonlinear shell theory, the middle surface strain–displacement relationships and changes in the curvature and torsion for a circular cylindrical shell are given by [2]     @u 1 @w 2 1 @v @u 2 @w @w0 ex,0 ¼ þ þ  þ , (A2a) L@Z 2 L@Z 8 L@Z R@y L@Z L@Z

ey,0 ¼

      @v w 1 @w v 2 1 @u @v 2 @w0 @w v þ þ  þ  þ  R@y R 2 R@y R 8 R@y L@Z R@y R@y R

(A2b)

    @u @v @w @w v @w0 @w v @w @w0 þ þ þ þ   R@y L@Z L@Z R@y R L@Z R@y L@Z R@y R

(A2c)

@2 w , L2 @Z2

(A2d)

gxy,0 ¼

kx ¼ 

ky ¼

kxy ¼ 2

@v @2 w  , 2 R @y R2 @y2

(A2e)

  @2 w 1 @v @u 3 þ  , LR@Z@y 2R L@Z R@y

(A2f)

where Z ¼ x=L is the nondimensional longitudinal coordinate. The elastic strain energy US of a circular cylindrical shell, neglecting sz , is given by [2] Z 2p Z 1 Z h=2 1 ðsx ex þ sy ey þ txy gxy Þ dZð1þ z=RÞ dy dz, US ¼ LR 2 0 0 h=2

(A3)

where the stresses sx, sy and txy are related to the strains for homogeneous and isotropic material (sz ¼ 0, case of plane stress) by [2]

sx ¼

E ðex þ ney Þ, 1n2

sy ¼

E ðe þ nex Þ, 1n2 y

txy ¼

E g 2ð1 þ nÞ xy

where E is Young’s modulus and n is Poisson’s ratio. The kinetic energy TS of a circular cylindrical shell, by neglecting rotary inertia, is given by Z 2p Z 1 1 _ 2 Þ dZ dy, ðu_ 2 þ v_ 2 þ w TS ¼ rS hLR 2 0 0 where rS is the mass density of the shell. In Eq. (A5) the overdot denotes time derivative.

(A4a2c)

(A5)

Ye. Kurylov, M. Amabili / Journal of Sound and Vibration 330 (2011) 5363–5381

The virtual work W done by the external forces is written as Z 2p Z 1 ðqx u þqy v þqr wÞ dZ dy, W ¼ LR 0

5381

(A6)

0

where qx, qy and qr are the distributed forces per unit area acting on the shell in axial, circumferential and radial directions, respectively. The nonconservative damping forces are assumed to be of viscous type and are taken into account by using the Rayleigh’s dissipation function Z 2p Z 1 1 _ 2 Þ dZ dy F ¼ cLR ðu_ 2 þ v_ 2 þ w (A7) 2 0 0 where c is the viscous damping coefficient, which has a different value for each term of the mode expansion. References [1] M. Amabili, M.P. Paı¨doussis, Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction, Applied Mechanics Reviews 56 (2003) 349–381. [2] M. Amabili, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, New York, 2008. [3] D.A. Evensen, Nonlinear flexural vibrations of thin-walled circular cylinders, NASA TN D-4090, 1967. [4] E.H. Dowell, C.S. 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