A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Accepted Manuscript

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells E. Hasrati , R. Ansari , J. Torabi PII: DOI: Reference:

S0307-904X(17)30545-0 10.1016/j.apm.2017.08.027 APM 11939

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

13 October 2016 2 July 2017 23 August 2017

Please cite this article as: E. Hasrati , R. Ansari , J. Torabi , A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.08.027

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Highlights

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An efficient and accurate numerical solution strategy is developed for solving nonlinear free and forced vibration of cylindrical shells. The generalized differential quadrature method and periodic differential operators are used along axial and circumferential directions. The time periodic discretization technique and the pseudo-arc length continuation method is employed. The effects of variations of fundamental vibrational mode shapes on the frequency response curves are investigated.

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A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

E. Hasrati, R. Ansari, J. Torabi*

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Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran Abstract

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of

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solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von-Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and

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circumferential directions, respectively. Then, based on Hamilton’s principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained.

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Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results

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are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results

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indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.

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Keywords: Cylindrical shell; Nonlinear free and forced vibrations; Numerical variational method

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Corresponding author. E-mail address: [email protected]

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1. Introduction Cylindrical shells are widely used as essential structures in various engineering applications such as pressure vessels, pipes, storage tanks and so on. In this regard, mechanical characteristics of such structures must be studied to ensure a proper design. Cylindrical shells are often subjected to dynamic loads which may lead to vibration. In some applications, the vibration response of shells obtained based on the linear theory can be inaccurate. In other words, when the vibration amplitude becomes comparable

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to the shell thickness, the geometrically nonlinear shell theories should be considered. The nonlinear free and forced vibration analysis of circular cylindrical shells has been extensively presented by many researchers and still attracts a great deal of attentions. The first studies on the large amplitude vibration of cylindrical shells were presented by Reissner [1], Chu [2] and Evensen [3]. Later, Sinharay and Banerjee [4] presented an approach to analyze the large amplitude vibration of thin elastic shallow spherical and

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cylindrical shells. Both movable and immovable edge conditions were considered to obtain the numerical results. Nayfeh and Raouf [5] investigated nonlinear forced vibration of infinitely long circular cylindrical shells. A combination of the Galerkin procedure, the method of multiple scales and perturbation technique were used to solve the problem. In addition, by employing the finite element method the non-linear free flexural vibrations of thin circular cylindrical shells were studied by Ganapathi and Varadan [6]. The nonlinear governing equations were solved by the use of a Wilson-θ numerical integration scheme and for

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each time step, modified Newton-Raphson iterations are employed. The influence of initial stress and boundary restraints on the nonlinear vibrations of cylindrical shells were investigated by Evensen [7].

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Amabili et al. [8] studied the large amplitude vibration of circular cylindrical shells containing flowing fluid. Nonlinearity due to moderately large-amplitude deformation of the shell was taken into account

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based on Donnell's nonlinear shallow shell theory. Linear potential flow theory was also used to model the fluid-structure interaction. The response of a shell to harmonic excitation, in the spectral neighborhood of one of the lowest natural frequencies, was studied for different flow velocities.

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The effects of geometries on the nonlinear vibration analysis of cylindrical shells were reported by Pellicano et al. [9]. Nonlinear governing equations were presented on the basis of Donnell's non-linear

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shallow shell theory and the effect of viscous structural damping was considered. A discretization method based on a series expansion and Galerkin procedure was used. The influences of shell geometry such as radius, length and thickness on the non-linear vibration characteristics were analyzed. Furthermore, nonlinear vibration analysis of empty and fluid-filled cylindrical shells with movable and immovable simply-supported boundary conditions was carried out by Amabili [10]. Two different nonlinear shell theories including Donnell’s and Novozhilov’s theories were considered to obtain the elastic strain energy. In addition, a comprehensive review study was presented by Amabili and Paidoussis [11] on the geometrically nonlinear vibrations and dynamics of cylindrical shells and panels with and 3

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without fluid-structure interaction. A comparison of analytical and numerical models for nonlinear vibrations of cylindrical shells was reported by Junsen [12]. The frequency-amplitude curves for different analysis methods were compared. Kurilov and Mikhlin [13] analyzed the nonlinear vibrations of cylindrical shells with initial imperfections in a supersonic flow. The aerodynamic pressure on the shell in a supersonic flow was determined based on the piston theory. The influence of the flow and initial deflections on the nonlinear vibration behavior of the shell was further examined in the flutter range.

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Comparison of two reduced-order models including proper orthogonal decomposition and asymptotic nonlinear normal modes methods for nonlinear vibrations of fluid-filled circular cylindrical shells was performed by Amabili and Touze [14].

Goncalves et al. [15] presented low-dimensional models for the nonlinear vibration analysis of circular cylindrical shells on the basis of perturbation procedure and proper orthogonal decomposition. Geometric

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nonlinearity due to finite-amplitude shell motions were considered based on Donnell's nonlinear shallowshell theory. Forced large amplitude periodic vibrations of cylindrical shallow shells was investigated by Ribeiro [16] using -version finite element method. The governing equations were presented based on the first-order shear deformation theory and the shooting and Newton methods were used to solve the equations of motion. Kubenko et al. [17] studied the nonlinear vibrations of fluid-filled cylindrical shells subjected to longitudinal and transverse periodic excitation. Kurylov and Amabili [18] performed the

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study on the large amplitude forced vibrations of circular cylindrical shells with different boundary conditions using Polynomial and trigonometric expansions. The Sanders-Koiter nonlinear shell theory

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including in-plane inertia, was used to present the elastic strain energy. Bakhtiari-Nejad and Mousavi [19] studied the nonlinear free vibration analysis of pre-stressed circular cylindrical shells resting on Winkler and Pasternak elastic foundation. The nonlinear Sanders-Koiter shell theory was employed to derive

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strain–displacement relationships. Beam functions and the Rayleigh–Ritz procedure were applied to solve the problem in linear state. In order to determine the relationship between vibration amplitude and

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frequency in nonlinear state, the perturbation method was used. The impacts of heterogeneity and elastic foundations on the nonlinear vibration of truncated conical shells were investigated by Sofitev [20].

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The governing equations were reduced to a time dependent nonlinear differential equation and solved by the use of homotopy perturbation method. Large amplitude vibration characteristics and bifurcation of circular cylindrical shells subjected to the

traveling concentrated harmonic excitation was examined by Wang et al. [21]. The effect of viscous structure damping was taken into account and the governing equations were discretized using Galerkin's method. The method of averaging was developed to study the nonlinear traveling wave responses of the system.

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In the last decade, various studies have been performed using analytical and semi-analytical approaches on the nonlinear vibration of circular cylindrical shells made of composite materials [22-28], functionally graded materials [29-39] and functionally graded carbon nanotube reinforced composite materials [40, 41]. In the present study, a novel and effective numerical approach was developed to study the nonlinear free and forced vibrations of cylindrical shells. In order to analyze the nonlinear vibration characteristics of the structure, the matrix form of the energy functional is first presented on the basis of first-order shear

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deformation theory and von-Karman nonlinear kinematic relations. The energy functional is discretized in the space domain by the use of GDQ method in axial direction and periodic differential operators in circumferential direction. Then, according to Hamilton's principle and based on the VDQ method [42-44], the reduced forms of discretized nonlinear governing equations are obtained. By employing the mode shapes of the linear response, the number of general coordinates of the discretized system are

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considerably decreased which significantly reduces the computational cost. Finally, time periodic operators are used to discretize the governing equations in time domain, and the frequency response of cylindrical shell is obtained applying the pseudo-arc length continuation method. Note that the VDQ method is an efficient numerical method for structural analysis whose advantages can be found in [44]. The accuracy and efficiency of the proposed numerical approach are first shown, and then, comprehensive numerical results are reported to analyze the effects of important parameters on the

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nonlinear vibration behavior of cylindrical shells.

2. First-Order Shear Deformable Shell Model

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2.1. Displacements Formulation

Consider a cylindrical shell of length , radius

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coordinate system

, and constant thickness , defined in the cylindrical ⁄

⁄ . According to the first-order shear

deformation theory, in which the in-plane displacements are expanded as linear functions of thickness and the transverse deflection is constant through thickness, the displacement field is given by [

̃

*

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̃

where

] (1)

+ are the displacements of a generic point

displacement of the mid-surface, and

in the shell,

are the

denote the rotations of the transverse normal about - and -

axis, respectively. The Equation (1) can be rewritten as the following form

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̃ [

[

]

(2)

]

in which ̃ is the displacement vector and

denotes the augmented displacement vector.

2.2. Matrix form of strain-displacement relations

̃

(

)

(

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Using the Green-Lagrange strain tensor )

(3)

and neglecting nonlinear terms respective to the directions that structure is thick due to those small values, one can obtain the nonlinear von-Karman strains as following vectors ]

[

]

[

(

) (

[

]

[

]

]

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[

[

*

)

(4)

+

]

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Eq. (4) can be recast in following matrix relation

]

⁄

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[ *

in which 〈 〉

〈

are defined as

(6)

⁄

⁄

]

(7)

]

+

〉

⁄

⁄

⁄ ⁄

〈 〉

and

[

⁄ ⁄

*

The nonlinear part of Eq.(5)

〈

,

⁄

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⁄

,

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[

,

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in which denotes the Hadamard product,

(5)

+

(8)

can be represented as follow 〈

〉

〈

〉

〈

〉 (9)

〉 and accordingly Eq. (5) can be presented as 6

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(10)

2.3. Matrix form of constitutive relations The constitutive relation can be expressed as

where by considering plane-stress condition [

]

[

]

[

]

[

classical Lamé constants.

3. Hamilton’s principle Hamilton’s principle is presented as ∫ denotes the kinetic energy,

forces. In addition,

is elastic strain energy and

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where

⁄

]

and

⁄

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⁄ is the shear correction factor and

in which

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(11)

(12) (13)

are the

(14)

stands for the work of external

denotes the variation operator. In the elasticity theory, the strain energy is given as

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∫ ̃

(15)

where the strain energy per unit volume or the strain energy density ̃ is proposed in a quadratic matrix

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representation using Eqs. (4), (5), (11) and (13) as (16)

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̃

For the sake of reducing redundant number of multiplication and addition in above equation is recast as

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̃

(17)

Using Eqs. (15) and (17) the variation of strain energy on the volume ∫̃

∫(

)

can be obtained as (18)

The variation of work of external loads and kinetic energy are given as ∫

̃

(19)

∫ ̃

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( ∫ ̃̇ ̃̇ in which

∫ ̃̇

)

(20)

̃̇

indicates the force vector and

is the mass density. Additionally, the overdot symbolizes

differentiation with respect to time. Substituting Eqs. (5) and (13) in Eq. (18), the variation of total strain energy can be calculated as )

(

)

and considering that we have (

)

〈

(

〉

〈

〈

(

〉

〉

〈

〈

〉

〉

〈

〈

〉

) The integration on the volume

̅

is reduced to surface

⁄

*

⁄

+

⁄

∫

*

⁄

∫

〉)

(23)

⁄

∫

(24)

⁄

⁄

̅

+

̅

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∫

〈

(22)

by introducing following matrices

[

ED

̅

〉)

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Eq. (21) can be expanded as ∫(

(21)

)

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∫( (

]

̅

̅

⁄

(25)

∫

(

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and accordingly, Eq. (23) can be rewritten as ̅

̅

̅

̅

̅

)

(26)

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By substituting Eq. (2) into (19) and (20), one can arrive at ∫

̇

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∫

where ̅ matrix

̇

∫

̇

(27) ̇

(28)

includes the components of external transverse force and external moment, and the inertia

is given as

∫ in which

̅

∫

* and

+

(29)

are identity matrices with the size of 8

and

.

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4. Variational differential quadrature discretization In the VDQ method [42-44], after deriving the quadratic representation of energy functional, by the use of Hamilton’s principle, a special technique is employed to directly discretize it using matrix differential and integral operators. In the present study, the differential and integral matrix operators are presented based on the GDQ discretization technique and periodic operators which are described in Appendices A and B.

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Following the mathematical approach described in [42], the discretized form of Eqs. (26), (27) and (28) on the space domain can be given as (

) ̅ ̇

where the symbol

denotes the Kronecker tensor product and

(31) (32)

̅

.

(̃) and ̃ is an numerical integral operator and

Furthermore, 〈

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̇

(30)

〉

〈

〉

,

and

(35)

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(36) (37) (38)

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̅

are presented in Appendix C.

(34)

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By defining following block matrices

,

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It should be noted that the discretized matrix operators

(33)

(39)

Eqs (30)-(32) are rewritten as

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(40)

̇

(41)

̇

(42) By substituting Eqs (40)-(42) into the Hamilton’s Principle, Eq. (14), and considering integration by parts scheme in time domain one can write ∫ (

̈

)

∫

(

9

̈

)

(43)

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Then, using the fundamental lemma of the calculus of variation results in ̈

(44) which is defined using Eq. (36). The symmetric

Only un-symmetric matrix-part of above relation is is derived as 〈

〉

〈

〉

(45)

The Jacobian of stiffness parts of Eq. (44) which is called

are obtained as

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counterparts of

5. Non-dimensionalizing Considering the following dimensionless quantities (

)

̂ ̂ ̂

(

)

(

̂

)

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( ̂ ̂ ̂)

√

(46)

(47)

Where and Using these dimensionless parameters, the dimensionless form of governing equations are presented as follows ̂ ̂ ̂

̂

̂

̂ ̂ ̂

̂

̂ ̂

̂ ̂ ̂

̂

̂) ̂

(

̂

[

[

〈̂ ̂〉̂

(52) (53) (54)

and

̂

]

]

(49)

(51)

̂

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̂

AC ̂

〈 ̂ ̂〉 ̂

̂

where ̂ ̂

̂

(48)

(50)

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̂

̂ ̂ ̂

̂ ̂ ̂

̂ ̂

̂

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̂

̂

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̂ ̂̈

̂

̂

*

+ (55)

̂

̂

*

+

Furthermore, the non-dimensional discretized matrix operators ̂ , ̂ , ̂ and ̂ have been defined in Appendix D.

6. Solution procedure Grid points in ̂ direction are generated using the shifted Chebyshev–Gauss–Lobatto grid points, given by

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̂

(

)

(56)

and in ̂ direction is given by ̂

(57)

To analyze the geometrically nonlinear coupled longitudinal-transverse-rotational free and forced harmonic free vibration solution of system is assumed as ̂

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vibration characteristics, ignoring the nonlinear terms ̂ and ̂ and external force ̂ in Eq. (48), the ̃

. Substituting this harmonic solution

into Eq. (48) and applying the considered assumptions, yields the following eigenvalue problem ̂̃

̂̃

where

(58)

is the linear fundamental frequency. It should be noted that the discretized boundary conditions

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elements associated with the boundary nodes should be replaced with the typical elements in ̂ and ̂ . Solving Eq. (58) results in the linear fundamental frequencies and linear mode shapes which can be presented as ̃ and

are the

reduced generalized coordinates and a

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where

eigenvectors of system (

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matrix including the first

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[

]

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[ [

]

(59) normalized base sparse

) which can be written as

[

]

]

(60)

[

] [

[

]

]

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Assuming that cylindrical shell is subjected to a harmonic excitation transvers force with the excitation frequency of , substituting Eq. (59) into Eq. (48) and multiplying obtained relation by the based sparse function yields the following equation ̃ ̈

̃

̃

̃

̃

(61)

Eq. (61) stands for a Duffing-type in which ̃

̂

̃

̂

̃

̂

̃ 11

̂

̃

̂ (62)

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These mathematical operations result in the decrease of the general coordinates of system from

to

which can considerably reduce the computational cost and time. By considering that the system is subjected to the non-conservative external viscous damping forces, the effect of energy dissipation should be considered in the governing equation. Therefore, one can write ̃

̃

̃

̃

(63)

in which

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̃ ̇

̃ ̈

̃

̃

̃

where

and

denote damping coefficient and force amplitude, respectively. By considering the period

[

[

]

and frequency of cylindrical shell

]

⁄

, introducing

⁄

and assuming that the cylindrical

(

) ̃

)̃

(

̃

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shell is subjected to a primary resonance about the steady state region, i.e., represented as

̃

(64)

̃

, Eq. (63) is further

̃

(65)

(

)

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Due to the periodic response of Eq. (65), it can be written

(66)

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To present the discretized form of Eq. (65) on the time domain, the grid points are introduced as follows (67)

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Subsequently, Eq. (65) can be given as ̃̅

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̃̅ where ̅ and

̃̅

̅ ̅

̅ ̅

̃

(68)

are the discretized displacement vector and amplitude function on the time domain,

respectively, which are introduced as [

AC

̅

[

]

(69) ]

(70)

furthermore, by introducing the following relation |

(71)

the first and second derivative of time-dependent function

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can be presented as follows

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̅

[

]

̅

[

]

(72)

These matrices can be rewritten as ̅

̅

(73)

[

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stands for time periodic operator and can be introduced for the periodic functions as [45]

where

]

[

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{

]

{

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Also, it is worth noting that ̅ ̅ and ̅ ̅ are the recast forms of ̃

and ̃

(74)

in Eq.(65)

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which can be calculated using an input matrix, i.e.̅. Considering the relation ⌈

⌉

)⌈ ⌉, which represents the vectorized form of a relation, Eq.

(

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(68) can be rewritten in vectorized form as ̃ ) ⌈ ̅⌉

(

(

̃ ) ⌈ ̅⌉

(

̃ )⌈ ̅⌉

⌈̅ ̅ ⌉

⌈̅ ̅ ⌉

̃

(75)

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where ⌈ ⌉ stands for the vectorized form of a matrix and ⌈ ̅⌉ is the reshaped form of unknown matrix ̅ which is given as follows

[

]

AC

⌈ ̅⌉

(76)

Additionally, introducing the quantities ̿

̃

̿

̃

̿

̃

and applying to Eq. (75) leads to the following equation

13

̿

̃

(77)

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̿ ⌈ ̅⌉

̿ ⌈ ̅⌉

̿ ⌈ ̅⌉

⌈̅ ̅ ⌉

⌈̅ ̅ ⌉

̿

(78)

By using the same technique and considering the relations ⌈ ̅ ̅ ⌉

̿ ⌈ ̅ ⌉ ⌈ ̅ ⌉ and ⌈ ̅ ̅ ⌉

̿ ⌈ ̅ ⌉ ⌈ ̅ ⌉, the vectorized form of nonlinear parts of Eq. (78), i.e., ̿ and ̿ , can be obtained as ̅ ̅ ̅

̅ ̅ ̅

̅ 〈 ̅ ̅ ⌈ ̅⌉〉 ̅

̅ 〈 ̅ ̅ ⌈ ̅⌉〉 ̅

(79)

(̅ ̅ ̅ )

̿

(80)

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̿

̅

̂

̅

̂

̅

̂

̅

̂

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where

As a result, Eq. (78) can be presented as ̿

̿

̿

̿

̿ ) ⌈ ̅⌉

̿ ⌈ ̅⌉

̿

̿ ⌈ ̅⌉

(82) (83)

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(

(81)

Finally, the pseudo arc-length continuation scheme is applied to obtain the frequency-response curves of

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cylindrical shell subjected to a periodic transverse force. In addition, it is enough to neglect the damping effect ̿ and external force ̿ in Eq. (82) to find the nonlinear free vibration characteristics of system.

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7. Results and discussion

Nonlinear free and forced vibration analysis of cylindrical shells was presented. It is assumed that the

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shell structure is made of stainless steel which its material properties are given in Table 1. The accuracy of the proposed method is examined by conducting different comparison studies. In addition, the essential

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boundary conditions for clamped (C) and simply-supported (S) edges are considered as

Comparison of nonlinear to linear frequency of the simply-supported cylindrical shell is presented in Table 2. As it can be seen, the results are in good agreement. In addition, natural frequencies of cylindrical shell for various thickness-to-radius and length-to-radius ratios are presented in Tables 3 and 4, respectively. The results are presented for different boundary conditions including clamped-clamped (CC), clamped-simply supported (CS) and simply supported-simply supported (SS). In the case of SS 14

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edge condition, the comparative results are further presented which clearly show the accuracy of the present numerical approach. Furthermore, it can be seen that by decreasing length-to-radius ratio and increasing the thickness-to-radius ratio, the fundamental frequencies of the shell increase. Figs. 1-3 demonstrate the effects of thickness-to-radius ratio on the frequency response of forced and free nonlinear vibrations of the cylindrical shell for three boundary conditions of CC, CS and SS. The results indicate that by the increase of thickness-to-radius ratio of the shell from 0.005 to 0.01, the hardening

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effects decline for the CC and SS boundary conditions, however for the CS one the hardening effects increase. In this range of ⁄ ratio, the vibrational mode shapes of CC and SS cylindrical shells change and the circumferential wave number decreases from 5 to 4 for the fully clamped and form 4 to 3 for the fully simply supported, but in the case of CS boundary condition no changes are observed in the mode shape and the circumferential wave number equals to 4. By increasing

ratio from 0.01 to 0.02, the

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hardening effects decrease for CC and CS boundary conditions however, the hardening effects increase for SS one. Generally, it can be concluded that by the increase of thickness-to-radius ratio of the cylindrical shell, if the vibrational mode shape and the circumferential and longitudinal wave numbers stay constant, hardening effects increase and if the vibrational mode shape and circumferential wave numbers change, the hardening effects decrease.

The effects of length-to-radius ratio on the frequency response of forced and free nonlinear vibrations of

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cylindrical shells are investigated in Figs. 4-6. The vertical axis in this graph shows the maximum deformation and the horizontal axis shows the ratio of excitation frequency to the linear frequency. The

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results indicate that by the increase of length-to-radius ratio from 2 to 4, the circumferential wave number changes from 3 to 2 in the fully simply supported boundary condition and subsequently the hardening effects decrease in this range, but in the case of CC and CS boundary conditions the circumferential wave

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number has not changed and the hardening effects have been increased. Accordingly, it can be collectively found that when the vibrational mode shape and the circumferential and longitudinal wave

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numbers stay constant, the increase of the length-to-radius ratio leads to increase the hardening effects, but in some cases which by increasing the

ratio, the dominant vibration mode shape and the

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circumferential wave number are changed, the hardening effects decrease. It should be noted that all three boundary conditions follow the same pattern by variation of the length-to-radius ratio. Figs. 7-9 present the frequency response of the free and forced nonlinear vibrations for different values of external excitation load for three different boundary conditions. The results show that the higher amplitude of the external excitation leads to increase in the maximum amplitude of the vibration and greater deformation caused by the phenomenon of resonance happens. In addition, Figs. 10-12 illustrate the effect of damping coefficient on the frequency response of free and forced nonlinear vibrations. The

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results indicate that for all three boundary conditions, the amplitude of the deformation declines by increasing the damping coefficient. In Fig. 13, the frequency response of the free and forced nonlinear vibrations is compared for three boundary conditions including CC, CS and SS one. As it can be seen, the SS and CC boundary conditions have the highest and lowest hardening effects, respectively. In addition, it should be noted that the minimum and maximum amplitude of deflections are obtained CC and SS boundary conditions,

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respectively. Finally for more details, the first three circumferential vibrational mode shapes of cylindrical shell were presented in Fig. 14.

8. Conclusion

The nonlinear free and forced vibration analysis of cylindrical shells was presented using an efficient

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numerical solution methodology. Based on the first-order shear deformation theory and von-Karman geometric nonlinearity, the energy functional of the shell was derived and discretized applying the GDQ method and periodic differential operators. The reduced discretized nonlinear governing equations were obtained based on the VDQ method. Then, by the use of time periodic discretization technique and pseudo-arc length continuation method, the frequency response of the cylindrical shell was determined. After verifying the accuracy of the proposed numerical approach, the effects of model parameters on the

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nonlinear free and forced vibration behavior of cylindrical shells were studied. The results indicated that the decrease in length-to-radius ratio and increase in the thickness-to-radius ratio, tend the fundamental

ED

frequencies of the shell to higher values. Furthermore, it was observed that changes of fundamental vibrational mode shape play an important role in the nonlinear frequency response curves of cylindrical

PT

shells. It was generally found that by the increase of thickness-to-radius and length-to-radius ratios of the cylindrical shell, if the vibrational mode shapes remain constant, hardening effects increase, however the

CE

changes in vibrational mode shape and circumferential wave numbers, decrease the hardening effects.

Appendix A: Differential Operator

AC

a) GDQ method in axial direction

A column vector is introduced as [

]

(A-1)

Where denotes the value of at each grid point , and is the number of grid points. Also the values of r-th derivative of at each point can be defined using the following column vector

16

ACCEPTED MANUSCRIPT

|

*

|

|

+

(A-2)

Based on GDQ method one can write (A-3)

(

is evaluated at grid point values via the following formula

CR IP T

where the differentiation matrix operator [46]

) )

∑ {

In which

(A-4)

AN US

(

and is

identity matrix and

ED

M

∏

is expressed as

(A-5)

b) Periodic operator in circumferential direction

[

]

) are defined as [45]

PT

The periodic differential matrix operators ( ]

and

(A-6)

are written as

AC

CE

where the coefficients

[

(A-7)

{

17

ACCEPTED MANUSCRIPT

{

Appendix B: Integral Operators

a) Integral operator in

direction

(∑ ̃

[

)

with ̃

M

*

(∑ ̃

)

(B-1)

+

is the GDQ differential operator or the weighting coefficients matrix of the

derivative and function

ED

where

]

AN US

∫

CR IP T

(A-8)

is a column vector with

at

th order

components containing the nodal values of the arbitrary

.

direction

PT

b) Integral operator in

]

(B-3)

CE

[

(B-2)

Appendix C: Discretized matrices

AC

[

]

⁄

(C-1) ⁄

⁄

(C-2) ⁄

[

⁄

] (C-3)

18

ACCEPTED MANUSCRIPT

[

⁄ ⁄ ⁄

*

]

⁄

+

⁄ ⁄

*

(C-4)

+

in which

CR IP T

(C-5)

Appendix D: Dimensionless matrices ̂ ̂

̂

̂ ̂ ̂

̂ ̂ ̂ ⁄ ̂ ⁄ ̂

*

References:

̂ ̂

+

M

̂

+

̂

̂ ̂

* ̂

+ ⁄

(D-1)

(D-2)

(D-3)

PT

̂

*

]

ED

[

̂

̂ ̂

AN US

̂ ̂

CE

[1] Reissner, E. (1955). Non-linear effects in vibrations of cylindrical shells. Ramo-Wooldridge Corporation, Guided Missile Research Division, Aeromechanics Section.

AC

[2] Chu, H. N. (1961). Influence of large amplitudes on flexural vibrations of a thin circular cylindrical shell. Journal of the Aerospace Sciences, 28(8), 602-609. [3] Evensen D.A. (1963). Some observations on the nonlinear vibration of thin cylindrical shells, AIAA J., 1, 2857–2858. [4] Sinharay, G. C., & Banerjee, B. (1985). Large amplitude free vibrations of shallow spherical shell and cylindrical shell—a new approach. International journal of non-linear mechanics, 20(2), 69-78. [5] Nayfeh, A. H., & Raouf, R. A. (1987). Nonlinear forced response of infinitely long circular cylindrical shells. Journal of Applied Mechanics, 54(3), 571-577.

19

ACCEPTED MANUSCRIPT

[6] Ganapathi, M., & Varadan, T. K. (1996). Large amplitude vibrations of circular cylindrical shells. Journal of Sound and Vibration, 192(1), 1-14. [7] Evensen, D. A. (2000). The Influence of Initial Stresses and Boundary Restraints on the Nonlinear Vibrations of Cylindrical Shells. ASME APPLIED MECHANICS DIVISION-PUBLICATIONS-AMD, 238, 47-60.

CR IP T

[8] Amabili, M., Pellicano, F., & Paidoussis, M. P. (2000). Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid. Part IV: large-amplitude vibrations with flow. Journal of Sound and vibration, 237(4), 641-666. [9] Pellicano, F., Amabili, M., & Paıdoussis, M. P. (2002). Effect of the geometry on the non-linear vibration of circular cylindrical shells.International Journal of Non-Linear Mechanics, 37(7), 1181-1198. [10] Amabili, M. (2003). Nonlinear vibrations of circular cylindrical shells with different boundary conditions. AIAA journal, 41(6), 1119-1130.

AN US

[ ] Amabili, M., & Paıdoussis, M. P. (200 ). eview of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Applied Mechanics Reviews,56(4), 349-381. [12] Jansen, E. L. (2004). A comparison of analytical–numerical models for nonlinear vibrations of cylindrical shells. Computers & structures, 82(31), 2647-2658.

M

[13] Kurilov, E. A., & Mikhlin, Y. V. (2007). Nonlinear vibrations of cylindrical shells with initial imperfections in a supersonic flow. International Applied Mechanics, 43(9), 1000-1008.

ED

[14] Amabili, M., & Touzé, C. (2007). Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: comparison of POD and asymptotic nonlinear normal modes methods. Journal of fluids and structures, 23(6), 885-903.

PT

[15] Gonçalves, P. B., Silva, F. M. A., & Del Prado, Z. J. G. N. (2008). Low-dimensional models for the nonlinear vibration analysis of cylindrical shells based on a perturbation procedure and proper orthogonal decomposition. Journal of Sound and Vibration, 315(3), 641-663. [16] Ribeiro, P. (2008). Forced large amplitude periodic vibrations of cylindrical shallow shells. Finite Elements in Analysis and Design, 44(11), 657-674.

CE

[17] Kubenko, V. D., Koval’chuK, P. S., & Kruk, L. A. (20 0). Nonlinear vibrations of cylindrical shells filled with a fluid and subjected to longitudinal and transverse periodic excitation. International Applied Mechanics, 46(2), 186-194.

AC

[18] Kurylov, Y., & Amabili, M. (2010). Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with different boundary conditions. Journal of Sound and Vibration, 329(9), 1435-1449. [19] Bakhtiari-Nejad, F., & Bideleh, S. M. M. (2012). Nonlinear free vibration analysis of prestressed circular cylindrical shells on the Winkler/Pasternak foundation. Thin-Walled Structures, 53, 26-39. [20] Sofiyev, A. H. (2014). The combined influences of heterogeneity and elastic foundations on the nonlinear vibration of orthotropic truncated conical shells. Composites Part B: Engineering, 61, 324-339.

20

ACCEPTED MANUSCRIPT

[21] Wang, Y., Liang, L., Guo, X., Li, J., Liu, J., & Liu, P. (2013). Nonlinear vibration response and bifurcation of circular cylindrical shells under traveling concentrated harmonic excitation. Acta Mechanica Solida Sinica, 26(3), 277-291. [22] Jansen, E. L. (2008). Effect of boundary conditions on nonlinear vibration and flutter of laminated cylindrical shells. Journal of Vibration and Acoustics,130(1), 011003.

CR IP T

[23] Amabili, M. (2011). Nonlinear vibrations of laminated circular cylindrical shells: comparison of different shell theories. Composite Structures, 94(1), 207-220. [24] Li, Z. M., Chen, X. D., & Yu, H. D. (2012). Large-Amplitude Vibration Analysis of Shear Deformable Laminated Composite Cylindrical Shells with Initial Imperfections in Thermal Environments. Journal of Engineering Mechanics, 140(3), 552-565. [25] Shen, H. S. (2013). Boundary layer theory for the nonlinear vibration of anisotropic laminated cylindrical shells. Composite Structures, 97, 338-352.

AN US

[26] Wang, Y. Q. (2014). Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration. Nonlinear Dynamics, 77(4), 1693-1707. [27] Li, Z. M., & Qiao, P. (2014). Nonlinear vibration analysis of geodesically-stiffened laminated composite cylindrical shells in an elastic medium.Composite Structures, 111, 473-487. [28] Shen, H. S., & Yang, D. Q. (2014). Nonlinear vibration of anisotropic laminated cylindrical shells with piezoelectric fiber reinforced composite actuators. Ocean Engineering, 80, 36-49.

M

[29] Bich, D. H., & Nguyen, N. X. (2012). Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations.Journal of Sound and Vibration, 331(25), 5488-5501.

ED

[30] Shen, H. S. (2012). Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium. Composite Structures, 94(3), 1144-1154. [31] Sheng, G. G., & Wang, X. (2013). Nonlinear vibration control of functionally graded laminated cylindrical shells. Composites Part B: Engineering, 52, 1-10.

CE

PT

[32] Dong, L., Hao, Y. X., Wang, J. H., & Yang, L. (2013). Nonlinear vibration of functionally graded material cylindrical shell based on eddy’s third-order plates and shells theory. In Advanced Materials Research (Vol. 625, pp. 18-24). Trans Tech Publications.

AC

[33] Rafiee, M., Mohammadi, M., Aragh, B. S., & Yaghoobi, H. (2013). Nonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectric functionally graded laminated composite shells, Part I: Theory and analytical solutions. Composite Structures, 103, 179-187. [34] Du, C., Li, Y., & Jin, X. (2014). Nonlinear forced vibration of functionally graded cylindrical thin shells. Thin-Walled Structures, 78, 26-36. [35] Jafari, A. A., Khalili, S. M. R., & Tavakolian, M. (2014). Nonlinear vibration of functionally graded cylindrical shells embedded with a piezoelectric layer. Thin-Walled Structures, 79, 8-15. [36] Duc, N. D., & Thang, P. T. (2015). Nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations. Aerospace Science and Technology, 40, 115-127.

21

ACCEPTED MANUSCRIPT

[37] Sofiyev, A. H. (2016). Nonlinear free vibration of shear deformable orthotropic functionally graded cylindrical shells. Composite Structures, 142, 35-44. [38] Sofiyev, A. H. (2016). Large amplitude vibration of FGM orthotropic cylindrical shells interacting with the nonlinear Winkler elastic foundation.Composites Part B: Engineering, 98, 141-150.

CR IP T

[39] Sofiyev, A. H., Hui, D., Haciyev, V. C., Erdem, H., Yuan, G. Q., Schnack, E., & Guldal, V. (2017). The nonlinear vibration of orthotropic functionally graded cylindrical shells surrounded by an elastic foundation within first order shear deformation theory. Composites Part B: Engineering, 116, 170-185. [40] Shen, H. S., & Xiang, Y. (2012). Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments. Computer Methods in Applied Mechanics and Engineering, 213, 196-205. [41] Shen, H. S., & Yang, D. Q. (2015). Nonlinear vibration of functionally graded fiber-reinforced composite laminated cylindrical shells in hygrothermal environments. Applied Mathematical Modelling, 39(5), 1480-1499.

AN US

[42] Ansari, R., Torabi, J., & Shojaei, M. F. (2016). Vibrational analysis of functionally graded carbon nanotube-reinforced composite spherical shells resting on elastic foundation using the variational differential quadrature method. European Journal of Mechanics-A/Solids, 60, 166-182. [43] Ansari, R., Shahabodini, A., & Shojaei, M. F. (2016). Vibrational analysis of carbon nanotubereinforced composite quadrilateral plates subjected to thermal environments using a weak formulation of elasticity. Composite Structures, 139, 167-187.

M

[44] Shojaei, M. F., & Ansari, R. (2017). Variational differential quadrature: A technique to simplify numerical analysis of structures. Applied Mathematical Modelling. 49, 705-738, 2017. [45] Trefethen, L. N. (2000). Spectral methods in MATLAB (Vol. 10). Siam.

ED

[46] Shu, C. (2000). Differential quadrature and its application in engineering. Springer Science & Business Media.

PT

[47] Loy, C. T., Lam, K. Y., & Reddy, J. N. (1999). Vibration of functionally graded cylindrical shells. International Journal of Mechanical Sciences, 41(3), 309-324.

AC

CE

[48] Raju, K. K., & Rao, G. V. (1976). Large amplitude asymmetric vibrations of some thin shells of revolution. Journal of Sound and Vibration, 44(3), 327-333.

22

ACCEPTED MANUSCRIPT

Tables:

Table 1: Properties of Materials (

Stainless steel

AN US

CR IP T

Properties

)

⁄

Table 2: Comparison of frequency parameters

(

⁄

for a simply supported cylindrical shell ⁄ )

Ref.

Ref.

Present

Ref.

Ref.

Present

Ref.

Ref.

work

[48]

[30]

work

[48]

[30]

work

[48]

[30]

1

1.0010

1.0008

1.0006

1.0072

1.0060

1.0063

1.0452

1.0398

1.0414

2

1.0034

1.0029

1.0024

1.0342

1.0235

1.0250

1.1623

1.1515

1.1557

3

1.0095

1.0063

1.0053

1.0731

1.0516

1.0552

1.3325

1.3179

1.3210

AC

CE

PT

ED

M

Present

23

ACCEPTED MANUSCRIPT

Table 3: Variation of natural frequencies (Hz) against L/R ratio (h/R=0.002)

S-S

C-S

C-C

Present study

Ref. [47]

Present study

Present study

100

0. 5595

0.5595

0.8724

1.2626

50

1. 4918

1.4918

1.7156

2.0682

20

4. 2632

4.2633

4.9305

5.9623

10

8. 6034

8.6035

10.2721

12.7182

5

16. 9176

16.917

21.5437

25.1992

2

43. 3715

43.373

53.1482

62.0068

1

87. 3258

87.331

103.6168

118.9943

0.5

175.4649

175.49

199.7665

223.6622

0.2

439.2297

439.36

486.9491

543.2210

AN US

CR IP T

L/R

ED

S-S

M

Table 4: Variation of natural frequencies (Hz) against h/R ratio (L/R=20)

C-S

C-C

Present study

Ref. [47]

Present study

Present study

0.001

2.7919

2.7919

3.7308

4.4703

5.4992

5.4992

7.5429

10.2252

6.3799

6.380

8.2091

10.7280

0.01

7.9329

7.9333

9.4699

11.7237

0.02

13.5518

13.552

14.8553

16.3919

0.03

13.5566

13.557

20.1433

22.0691

0.04

13.5633

13.563

20.1534

27.6158

0.05

13.5719

13.572

20.1645

27.6329

0.005

AC

CE

0.007

PT

h/R

24

ACCEPTED MANUSCRIPT

Figures: C-C 0.7

 = 52.97 (Hz), h/R = 0.01, l

(1,4)

 = 72.59 (Hz), h/R = 0.02, l

(1,3)

 = 113.9 (Hz), h/R = 0.05, l

(1,2)

 = 120.3 (Hz), h/R = 0.07, l

(1,2)

CR IP T

0.5

0.4

f = 0.0008 c = 0.006 0.3

= 5

0.2

0.1

0 0.6

0.7

0.8

0.9

AN US

Non-dimensional maximum deflection

0.6

 = 39.08 (Hz), h/R = 0.005, (1,5) l

1

/

1.1

1.2

1.3

1.4

l

Fig.1: Non-dimensional maximum deflection of C-C cylindrical shell versus frequency ratio for different values of

AC

CE

PT

ED

M

( ) ratio

25

ACCEPTED MANUSCRIPT

C-S 0.7

 = 33.34 (Hz), h/R = 0.005, (1,4) l

0.5

 = 45.99 (Hz), h/R = 0.01, l

(1,4)

 = 60.32 (Hz), h/R = 0.02, l

(1,3)

 = 93.50 (Hz), h/R = 0.05, l

(1,2)

 = 100.5 (Hz), h/R = 0.07, l

(1,2)

0.4

0.3

CR IP T

Non-dimensional maximum deflection

0.6

f = 0.0005 c = 0.006 = 5

0.2

0 0.7

0.8

0.9

AN US

0.1

1

/

1.1

1.2

1.3

l

Fig.2: Non-dimensional maximum deflection of C-S cylindrical shell versus frequency ratio for different values of

AC

CE

PT

ED

M

( ) ratio

26

ACCEPTED MANUSCRIPT

S-S 0.7

 = 26.14 (Hz), h/R = 0.005, (1,4) l

0.5

 = 37.34 (Hz), h/R = 0.01, l

(1,3)

 = 50.19 (Hz), h/R = 0.02, l

(1,3)

 = 73.54 (Hz), h/R = 0.05, l

(1,2)

 = 81.69 (Hz), h/R = 0.07, l

(1,2)

0.4

0.3

CR IP T

Non-dimensional maximum deflection

0.6

f = 0.0004 c = 0.006 = 5

0.2

0 0.5

0.6

0.7

0.8

0.9

AN US

0.1

1

/

1.1

1.2

1.3

1.4

1.5

l

Fig.3: Non-dimensional maximum deflection of S-S cylindrical shell versus frequency ratio for different values of

AC

CE

PT

ED

M

( ) ratio

27

ACCEPTED MANUSCRIPT

C-C 0.7

(1,3)

 = 133.8 (Hz),  = 4, l

(1,3)

 = 62.22 (Hz),  = 8, l

(1,2)

 = 37.22 (Hz),  = 15, (1,2) l 0.5

 = 27.63 (Hz),  = 20, (1,1) l

0.4

CR IP T

Non-dimensional maximum deflection

0.6

 = 261.7 (Hz),  = 2, l

f = 0.004 c = 0.005

0.3

h/R = 0.05 0.2

0 0.5

0.6

0.7

0.8

0.9

AN US

0.1

1

/

1.1

1.2

1.3

1.4

1.5

l

Fig.4: Non-dimensional maximum deflection of C-C cylindrical shell versus frequency ratio for different values of

AC

CE

PT

ED

M

( ) ratio

28

ACCEPTED MANUSCRIPT

C-S 0.6

(1,3)

 = 120.8 (Hz),  = 4, l

(1,3)

 = 51.68 (Hz),  = 8, l

(1,2)

 = 33.87 (Hz),  = 15, (1,1) l  = 20.16 (Hz),  = 20, (1,1) l 0.4

CR IP T

Non-dimensional maximum deflection

0.5

 = 231.6 (Hz),  = 2, l

0.3

f = 0.003 c = 0.005 h/R = 0.05

0.2

0

0.7

0.8

0.9

AN US

0.1

1

/

1.1

1.2

1.3

l

Fig.5: Non-dimensional maximum deflection of C-S cylindrical shell versus frequency ratio for different values of

AC

CE

PT

ED

M

( ) ratio

29

ACCEPTED MANUSCRIPT

S-S 0.6

(1,3)

 = 102.4 (Hz),  = 4, l

(1,2)

 = 43.23 (Hz),  = 8, l

(1,2)

 = 23.54 (Hz),  = 15, (1,1) l  = 13.57 (Hz),  = 20, (1,1) l 0.4

0.3

CR IP T

Non-dimensional maximum deflection

0.5

 = 202.2 (Hz),  = 2, l

f = 0.002 c = 0.005 h/R = 0.05

0.2

0

0.6

0.7

0.8

0.9

AN US

0.1

1

/

1.1

1.2

1.3

1.4

l

Fig.6: Non-dimensional maximum deflection of S-S cylindrical shell versus frequency ratio for different values of

AC

CE

PT

ED

M

( ) ratio

30

ACCEPTED MANUSCRIPT

C-C 0.8

 = 62.22 (Hz), f = 0.005 l 0.7

 = 62.22 (Hz), f = 0.003 l

0.6

0.5

0.4

CR IP T

c = 0.005 h/R = 0.05 = 8

0.3

0.2

0.1

0 0.6

0.7

0.8

0.9

AN US

Non-dimensional maximum deflection

 = 62.22 (Hz), f = 0.001 l

1

/

1.1

1.2

1.3

1.4

l

AC

CE

PT

ED

M

Fig.7: Non-dimensional maximum deflection of C-C cylindrical shell versus frequency ratio for different values of force amplitude

31

ACCEPTED MANUSCRIPT

C-S 0.6

 = 51.68 (Hz), f = 0.003 l  = 51.68 (Hz), f = 0.001 l  = 51.68 (Hz), f = 0.0007 l

0.4

c = 0.005

CR IP T

Non-dimensional maximum deflection

0.5

h/R = 0.05  = 20

0.3

0.2

0

0.7

0.8

0.9

AN US

0.1

1

/

1.1

1.2

1.3

l

AC

CE

PT

ED

M

Fig.8: Non-dimensional maximum deflection of C-S cylindrical shell versus frequency ratio for different values of force amplitude

32

ACCEPTED MANUSCRIPT

S-S 0.6

 = 43.23 (Hz), f = 0.0025 l  = 43.23 (Hz), f = 0.0015 l

0.4

c = 0.005

CR IP T

Non-dimensional maximum deflection

 = 43.23 (Hz), f = 0.0008 l 0.5

h/R = 0.05 = 8

0.3

0.2

0 0.5

0.6

0.7

0.8

0.9

AN US

0.1

1

/

1.1

1.2

1.3

1.4

1.5

l

AC

CE

PT

ED

M

Fig.9: Non-dimensional maximum deflection of S-S cylindrical shell versus frequency ratio for different values of force amplitude

33

ACCEPTED MANUSCRIPT

0.4

 = 62.22 (Hz), c = 0.004 l

0.35

 = 62.22 (Hz), c = 0.006 l

 = 62.22 (Hz), c = 0.005 l

0.3

CR IP T

0.25

f = 0.002 0.2

h/R = 0.05 = 8

0.15

0.1

0.05

0 0.8

0.85

0.9

0.95

AN US

Non-dimensional maximum deflection

C-C

1

/

1.05

1.1

1.15

1.2

l

AC

CE

PT

ED

M

Fig.10: Non-dimensional maximum deflection of C-C cylindrical shell versus frequency ratio for different values of damping coefficient

34

ACCEPTED MANUSCRIPT

0.4

 = 51.68 (Hz), c = 0.004 l

0.35

 = 51.68 (Hz), c = 0.006 l

 = 51.68 (Hz), c = 0.005 l

0.3

f = 0.0015

CR IP T

0.25

h/R = 0.05 = 8 0.2

0.15

0.1

0.05

0 0.8

0.85

0.9

0.95

AN US

Non-dimensional maximum deflection

C-S

1

/

1.05

1.1

1.15

1.2

l

AC

CE

PT

ED

M

Fig.11: Non-dimensional maximum deflection of C-S cylindrical shell versus frequency ratio for different values of damping coefficient

35

ACCEPTED MANUSCRIPT

S-S 0.35

 = 43.23 (Hz), c = 0.004 l  = 43.23 (Hz), c = 0.005 l  = 43.23 (Hz), c = 0.006 l

0.25

f = 0.001

CR IP T

0.2

h/R = 0.05 = 8 0.15

0.1

0.05

0 0.8

0.85

0.9

0.95

AN US

Non-dimensional maximum deflection

0.3

1

/

1.05

1.1

1.15

1.2

l

AC

CE

PT

ED

M

Fig.12: Non-dimensional maximum deflection of S-S cylindrical shell versus frequency ratio for different values of damping coefficient

36

ACCEPTED MANUSCRIPT

 = 43.23 (Hz), (1,2), S-S 0.5

l

 = 51.68 (Hz), (1,2), C-S l l

0.3

CR IP T

0.4

f = 0.002 c = 0.005 h/R = 0.05 = 8

0.2

0.1

0.7

0.8

0.9

1

/

M

0 0.6

AN US

Non-dimensional maximum deflection

 = 62.22 (Hz), (1,2), C-C

1.1

1.2

1.3

1.4

l

AC

CE

PT

ED

Fig.13: Non-dimensional maximum deflection of cylindrical shell versus frequency ratio for different boundary conditions

37

ACCEPTED MANUSCRIPT

(m,n) = (1,2)

(m,n) = (1,1)

(m,n) = (1,3)

CR IP T

X-Y View

AN US

3-D View

AC

CE

PT

ED

M

Fig. 14: Mode shapes of the cylindrical shell for different circumferential (n) wave numbers

38