Free and forced vibration analysis of coupled conical–cylindrical shells with arbitrary boundary conditions

International Journal of Mechanical Sciences 88 (2014) 122–137

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Free and forced vibration analysis of coupled conical–cylindrical shells with arbitrary boundary conditions Xianglong Ma a,b, Guoyong Jin a,n, Yeping Xiong b, Zhigang Liu a a

College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, PR China Faculty of Engineering and the Environment, Fluid Structure Interactions Research Group, University of Southampton, Boldrewood Innovation Campus, SO16 7QF, UK

b

art ic l e i nf o

a b s t r a c t

Article history: Received 12 February 2014 Received in revised form 2 June 2014 Accepted 3 August 2014 Available online 9 August 2014

This paper presents a free and forced vibration analysis of coupled conical–cylindrical shells with arbitrary boundary conditions using a modified Fourier–Ritz method. Under the current framework, regardless of the boundary conditions, each of the displacement components of both the conical and cylindrical shells are expanded invariantly as a modified Fourier series, which is composed of a standard Fourier series and closed-form supplementary functions introduced to accelerate the convergence of the series expansion and remove all the relevant discontinuities at the boundaries and the junction between the two shell components. All the expansion coefficients are determined by using the Rayleigh–Ritz method as the generalized coordinates. By using the present method, a unified solution for the coupled conical–cylindrical shells with classical and non-classical boundary conditions can be directly derived without the need of changing either the equations of motion or the expressions of the displacements. The reliability and accuracy of the present method are validated by comparison with FEM results and those from the literature. Studies on the effects of dimensional and elastic restraint parameters on the free vibrations are also reported. Investigation on vibration of the conical–cylindrical–conical shell combination shows the extensive applicability of present method for more complex shell combinations. New numerical examples are also conducted to illustrate the forced vibration behavior of the coupled conical–cylindrical shell subjected to the excitation forces in different directions. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Vibration analysis Coupled conical–cylindrical shells Modified Fourier series Arbitrary boundary conditions

1. Introduction The coupled conical–cylindrical shell is a shell combination of great interest in practical engineering applications, such as torpedo, rocket, tower and naval hulls of submarines, owing to its excellent mechanical and physical properties. In practical designs, the shell combination is commonly used as the foundation structure subjected to the intricate environment and dynamic loads, resulting in vibration, fatigue damage and radiated noise. Since the 50s of last century, much effort has been done to study the vibration characteristics of shell structures. Many researchers, such as Donnell, Mushtari, Flügge, to name a few, have developed various shell theories based on different simplifying assumptions and approximations. These works have been well summarized by Leissa [1], Markuš [2] and Qatu [3,4]. Recently, some new approaches have been adopted to analyze the vibration behaviors of the shell of revolutions. Wu and Lee [5] used the method of differential quadrature for free vibration analysis of laminated

n

Corresponding author. Tel.: þ 86 451 82589199. E-mail address: [email protected] (G. Jin).

http://dx.doi.org/10.1016/j.ijmecsci.2014.08.002 0020-7403/& 2014 Elsevier Ltd. All rights reserved.

conical shells with variable stiffness. The equations of motion and the boundary conditions in the whole domain are replaced by a system of simultaneously algebraic equations with respect to the function values of all the sampling points. It should be noted that most of the literature concentrates on the elementary shell configurations, such as circular cylindrical, conical and spherical shells rather than the shell combinations. Compared to the elementary shell structures, different components of the shell combination find their natural description in different physical co-ordinate systems and a problem will be caused by the matching of the interface continuity conditions between the substructures, which leads directly to the difficulty of obtaining the accurate vibration solution. The finite element method (FEM) computer programs such as ANSYS, ABAQUS and NASTRAN have been well developed and applied for vibration analysis of these complex shell combinations. However, there are two main disadvantages in the computation procedure: firstly, a great number of interior points are needed if one wants to obtain the accurate solution at high frequencies, which would finally increase the computation time and storage requirement; secondly, it is hard to identify the mode shapes corresponding to the certain natural frequencies in the modal analysis. Thus, developing an accurate and efficient

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

method is of considerable technical significance to characterize the vibration of the coupled conical–cylindrical shell combinations. In the last few decades, a few but not many publications focused on the vibration analysis of the coupled conical–cylindrical shell have been reported in the literature. Kalnins [6] and Rose et al. [7] used the classic bending theory to examine rotationally symmetric shells. Hu and Raney [8] proposed a multi-segmental numerical integration technique to obtain the analytical results for joined conical–cylindrical shells. The interface continuity conditions are imposed on the segments of the shell combination in their study and good agreement is observed compared with experimental results. The transfer matrix method was used by Irie et al. [9] to investigate the vibration behavior of a conical–cylindrical shell combination. Efraim and Eisenberger [10] studied the free vibration behavior of segmented axisymmetric shells by using a power series solution and obtained relatively accurate natural frequencies. Galletly and Mistry [11] obtained the natural frequencies of cylindrical shells clamped at one end and closed at the other by different types of shells, including cones, hemispheres, ellipsoids, etc. by using variational finite differences and finite elements. Their study indicates the fact that the in-plane boundary conditions have considerable influence on the natural frequencies. A local-global B-spline finite element method was presented by Benjeddou [12] for modal analysis of the coupled shells of revolution. Caresta and Kessissoglou [13] presented a classical approach to investigate the vibration characteristics of isotropic coupled conical–cylindrical shells. In their study, a wave solution is adopted to solve the cylindrical shell equations while the conical shell equations are solved by using a power series solution. These two solutions are coupled together by the means of the continuity conditions at the junction. A three-dimensional finite element method is used by EI Damatty et al. [14] to study numerically the dynamic behavior of a joined conical–cylindrical shell. Qu et al. [15,16] proposed a modified variational approach to analyze the free and forced vibrations of ring-stiffened conical, cylindrical and spherical shell combinations. The coupled shell structure is partitioned into appropriate shell segments and all essential continuity constraints on segment interfaces are imposed by means of a modified variational principle and least-squares weighted residual method. The dynamical responses of the shell combination obtained by the method agree well with those from FEM program. Experiments and numerical simulations are adopted to investigate the plastic energy absorption behavior of expansion tubes under axial compression by conical–cylindrical die was investigated by Yang et al. [17]. Free vibration analysis of coupled cross-ply laminated conical shells was presented by Kouchakzadeh and Shakouri [18]. In their study, the cross-ply conical shell combination is considered as the general case of cylindrical–conical shells, joined cylinder-plates and cone-plates. The continuity conditions at the joining section of the cones ware achieved by the extraction of the appropriate expressions among stress resultants and deformations. Mathematically, compared to directly solve the actual problems, it is easier to obtain the solution for the shell combinations by describing the boundary value and eigenvalue problems in a variational form. This is due to the fact that expanding the solution over a set of suited admissible functions can achieve the extreme or stationary value of some kind of energy functional for a coupled conical–cylindrical shell. As the classical variational approach, the Rayleigh–Ritz method has found its efficiency in the vibration analysis of the shell combinations. Monterrubio [19] presented the Rayleigh–Ritz method and the penalty function method to solve the vibration problem of shallow shells of rectangular planform with spherical, cylindrical and hyperbolic paraboloidal geometries with classical boundary conditions. Lee et al. [20] used the Rayleigh–Ritz method to investigate the free vibration of a joined hemispherical-cylindrical shell. At the joint part of the shell

123

combination in the study, the hemispherical-cylindrical shell is assumed to have a free boundary condition while the cylindrical shell has a simply supported boundary constraints. From the review of the literature, most of previous works on the vibration analysis of the coupled conical–cylindrical shells just concentrate on the cases with classical boundary conditions rather than the general elastic boundary conditions. Even so, either theoretical formulations or the admissible functions of the displacements have to be changed if one wants to obtain the solution for the cases with different boundary restraints. Furthermore, the general boundary conditions are often encountered in practical engineering applications compared to the classical boundary conditions since the support types of practical structures are always complicated and variable in nature. Li [21] proposed originally the modified Fourier series solution for the vibration analysis of beams with general elastic restraints. The flexural displacements are expressed by an improved Fourier series, which is composed a standard Fourier series and an auxiliary polynomial function introduced to remove all the relevant discontinuities at the boundaries. Subsequently, this method was extensively adopted for the vibration analysis of rectangular plates, circular cylindrical, conical, etc. shells with classical and non-classical boundary conditions [22–27]. Ma et al. [28] investigated the active control of an elastic cylindrical shell coupled to a vibration isolation system. The cylindrical shell is simply supported at its two ends and four active control strategies are evaluated in terms of the acoustic power radiated from the supporting shell. In practice engineering applications, the foundation structures are always formulated by shell combinations with non-classical boundary conditions rather than the simply supported elementary shells. To the author's best knowledge, few publications focused on the vibration analysis of the coupled conical–cylindrical shell with general elastic boundary conditions have been reported. The main objective of this paper is to develop an alternative and unified solution for the vibration analysis of the coupled conical– cylindrical shell with general elastic boundary conditions. The Reissner's thin shell theory is used to formulate the theoretical models of the conical and cylindrical shell components. Regardless of the boundary conditions, each displacement of the two shell components is invariantly expressed by the modified Fourier series composed of a standard Fourier series and closed-form auxiliary functions. The introduction of the auxiliary functions can not only remove all the potential discontinuities at the boundaries and the junction between the two shells, but also ensure and accelerate the convergence of the series expansions. All the expansion coefficients are determined by using the Rayleigh–Ritz procedure as the generalized coordinates. The accuracy and convergence of present method are validated by comparison with FEM results and those from the literature. The effects of semi-vertex angle of the cone and the elastic restraint parameters on the free vibration behavior of the shell combination are studied. New examples are also conducted to analyze the forced vibration responses of the coupled conical–cylindrical shell subjected to the driving forces in different directions.

2. Theoretical formulation 2.1. System description The geometry and co-ordinate systems for the coupled conical– cylindrical shell are depicted in Fig. 1. The conical shell is described with the ðxc ; θc ; r c Þcoordinate system, in which xc is measured along the generator of the cone starting at its small edge, θc is the circumferential co-ordinate and rc is perpendicular to middle surface of the conical shell. The displacements of the conical shell with respect to this coordinate system are described by uc,vc and

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X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

Fig. 1. Co-ordinate systems and force and moment resultants of the conical–cylindrical shell.

wc in the xc,θc and r c directions, respectively. φ is the semi-vertex angle of the cone, R1 and R2 are respectively the radii of the cone at its small and large edges, Lc is the cone length along its generator and hc is the uniform thickness of the conical shell. The radius of the cone at any point along its length can be written as Rc ðxc Þ ¼ R1 þ xc sin φ. A cylindrical coordinate system ðxs ; θs ; r s Þ is considered for the cylindrical shell, in which xs, θs and rs, respectively, denote the axial, circumferential and radial directions. The deformations of the cylindrical shell can be defined by us,vs and ws in the xs,θs and rs directions, respectively. For both the conical and cylindrical shells, the thickness is assumed to be negligible compared to length or radii of curvature of the shell and the normals to the middle surface are considered always to be straight and normal to the middle surface. Under these assumptions, Rayleigh–Ritz energy method will be used to establish the theoretical formulation of the coupled conical–cylindrical shell based on the Reissner's thin shell theory.

2.2. Mathematical formulations of the coupling system 2.2.1. Energy functional of conical shell component According to the Reissner's thin shell theory [1], the midsurface strains, curvature and twist changes for the thin conical shell can be written as εxc 0 ¼

∂uc ∂xc

1 ∂vc uc wc þ þ εθ c 0 ¼ xc sin ðφÞ ∂θ xc xc tan ðφÞ γ xc θ c 0 ¼

∂vc 1 ∂uc vc þ  ∂xc xc sin ðφÞ ∂θc xc

kxc 0 ¼ 

∂2 wc ∂xc 2

cotðφÞ ∂vc 1 ∂2 wc 1 ∂wc  2  kθ c 0 ¼ 2 xc ∂xc xc sin ðφÞ ∂θc xc sin ðφÞ ∂θ2c τ xc θ c 0 ¼

ð1Þ

where εxc 0 , εθc 0 and γ xc θc 0 are the normal and shear strains in the middle surface of the conical shell, kxc 0 and kθc 0 are the midsurface changes in curvature and τxc θc 0 is the mid-surface twist. The strain energy and the kinetic energy for the conical shell can be written as VC ¼

Z

Ec hc 2ð1  μ2c Þ

2π

Z

Z

2π

0

0

2

Ec hc þ 24ð1  μ2c Þ

Lc

  1 μc ðγ xc θc 0 Þ2 Rc ðxc Þd xc d θc ðεxc 0 Þ2 þ ðεθc 0 Þ2 þ 2μc εxc 0 εθc 0 þ 2

Z

Lc 0

0

  1  μc ðkxc 0 Þ2 þ ðkθc 0 Þ2 þ 2μc kxc 0 kθc 0 þ ðτxc θc 0 Þ2 Rc ðxc Þd xc dθc 2

ð7Þ and TC ¼

ρc hc 2

Z

2π

Z

0

Lc

(

0

∂wc ∂t

2

 þ

∂uc ∂t

2

 þ

∂vc ∂t

2 ) Rc ðxc Þd xc d θc

ð8Þ

where, Ec, μc and ρc, respectively, are Young's modulus, Poisson ratio and density of the conical shell. 2.2.2. Energy functional of cylindrical shell component Similarly to the conical shell, the strain energy and the kinetic energy of the thin circular cylindrical shell are given as  Z 2π Z Lx ( Es hs ∂us ∂vs ws 2 VS ¼ þ þ ∂xs R2 ∂θs R2 2ð1  μ2s Þ 0 0     ) ∂us ∂vs ws ð1 μs Þ ∂vs ∂us 2 þ þ þ R2 d xs dθs  2ð1  μs Þ 2 ∂xs R2 ∂θs R2 ∂xs R2 ∂θs 3

Es hs þ 24ð1  μ2s Þ

ð2Þ

Z

2π

Z

0

Lx

0

8 !2 < ∂2 w ∂2 ws s þ : ∂x2s R22 ∂θ2s

"

 2 2 #) ∂2 ws ∂2 ws ∂ ws  R 2 d xs d θ s ∂xs ∂θs ∂x2s R22 ∂θ2s   Z 2π Z Lx ( 3 E s hs ∂vs ∂2 ws ∂vs ∂2 ws ∂vs 2 þ  2μs 2 þ 2 2 2 2 ∂θs ∂xs ∂θs R2 ∂θs R2 ∂θs 24R2 ð1 μ2s Þ 0 0

ð3Þ

 2ð1  μs Þ

ð4Þ

 2 ) ∂vs ∂2 ws ∂vs þ 2ð1 μs Þ  4ð1  μs Þ R2 d xs d θs ∂xs ∂xs ∂θs ∂xs

ð5Þ

1 ∂vc 2 2 ∂2 wc 2 ∂wc vc   þ xc tan ðφÞ ∂xc x2c tan ðφÞ xc sin ðφÞ ∂θc ∂xc x2c sin ðφÞ ∂θc ð6Þ

and TS ¼

ρs hs 2

Z 0

2π

Z

Lx 0

(

∂ws ∂t

2

 þ

∂us ∂t

2 þ

 2 ) ∂vs R2 d xs d θs ∂t

ð9Þ

ð10Þ

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

where Es, μs, and ρs respectively, denote Young's modulus, the Poisson ratio and the mass density of the cylindrical shell. 2.3. Energy functional of external force and the shell combination Point force is the typical loading case, which is frequently encountered in practice. Thoroughly understand the vibration behavior of the shell combination subjected to the point force would be helpful for the complex loading cases. The potential energy Pf caused by a point force can be written as Z 2π Z Li ðf ui ui þ f vi vi þf wi wi Þδðxi  xf ; θi  θf ÞRðxi Þdxi dθi ð11Þ Pf ¼ 0

0

where f ui , f vi and f wi are the external force in the xi, θi and ri directions, respectively. i¼c,s denote the cases of the external force acting on the conical or cylindrical shells respectively and δ is the Dirac function. When all of the energy expressions are prepared, the complete solution can be obtained by using the Rayleigh–Ritz procedure. The total energy functional of the coupled conical–cylindrical shell is taken as the sum of the energy contributions from the shell components, continuity conditions and boundary constraints. Thus, the Lagrangian energy function (L) of the shell combination can be written as L ¼ V C  T C þ V S  T S þ Pb þ Po þ Pf

ð12Þ

where Pb and Po respectively denote the potential energy caused by boundary conditions and continuity conditions, which will be described in the following sections.

125

written as ko;x ðus uc cos φ þ wc sin φÞ N x;s jxc ¼ Lc ;xs ¼ 0 ¼ 0

ð17Þ

ko;θ ðvs vc Þ  N xθ;s jxc ¼ Lc ;xs ¼ 0 ¼ 0

ð18Þ

ko;r ðws  uc sin φ  wc cos φÞ  Q x;s 

K o;r

 ∂M xθ;s  ¼0 R2 ∂θc xc ¼ Lc ;xs ¼ 0

    ∂ws ∂wc  ¼0  M x;s  ∂xs ∂xc xc ¼ Lc ;xs ¼ Ls

ð20Þ

where ko;x , ko;θ , ko;r and K o;r , respectively, denote the stiffnesses of the springs between the two shell components. As indicated in the mathematically expressions of continuity conditions and boundary conditions, all of the force and moment resultants of the conical and cylindrical shell components are restrained by employed springs. Therefore, arbitrary continuity conditions and boundary constraints can be achieved by varying the value of springs’ stiffness. Specially, the classical boundary conditions can be simulated by varying the stiffnesses of the boundary springs to be extremely large or extremely small. Thus, the potential energy stored in the boundary springs and connective springs can be described as (   ) Z 1 2π ∂wc 2 Pb ¼ kx;c u2c þ kθ;c v2c þ kr;c w2c þ K r;c jxc ¼ 0 R1 d θc 2 0 ∂xc Z

1 þ 2

2π

0

(



kx;s u2s þkθ;s v2s þ kr;s w2s þ K r;s

∂ws ∂xs

2 ) jxs ¼ Ls R2 d θs

2.4. Arbitrary boundary conditions and continuity conditions

ð21Þ

For the sake of simulating the arbitrary boundary conditions and the continuity conditions, artificial spring technique is adopted here. Specifically, four sets of stiffness-like springs are used at the un-coupled end of the conical shell with subject to the ðxc ; θc ; r c Þ coordinate system, including three sets of linear springs respectively along the xc, θc and rc directions, and one set of rotational springs around the rc directions. Similarly, with subject to the cylindrical coordinate system ðxs ; θs ; r s Þ for the cylindrical shell component, four sets of stiffness-like springs are used at uncoupled end of the cylindrical shell and another four sets of springs are used to link the two shell components. Thus, the boundary conditions for the coupled conical–cylindrical shell combination can be expressed as kx;i ui  Nx;i jxi ¼ 0 ¼ 0

ð13Þ

kθ;i vi  Nxθ;i jxi ¼ 0 ¼ 0

ð14Þ

kr;i wi  Q x;i 

K r;i

 ∂M xθ;i  ¼0 R1 ∂θi xi ¼ 0

  ∂wi þ M x;i  ¼0 ∂xi xi ¼ 0

ð19Þ

ð15Þ

ð16Þ

where Nx;i and N xθ;i is the in-plane forces, Q x;i is the transverse shear force, M x;i is the bending moment and M xθ;i is the twisting moment as shown in Fig. 1 (i ¼ c; s), ki;x , ki;θ , ki;r and K i;r respectively, denote the stiffnesses for linear springs in xi, θi and ri directions and rotational springs around ri direction Since the connective springs are designed with subject to the ðxs ; θs ; r s Þ coordinate system, the displacement components of the conical shell at the coupled end should be transferred from the ðxc ; θc ; r c Þ coordinate system into the ðxs ; θs ; r s Þ coordinate system. Thus, the continuity conditions for the two shell components are

Po ¼

1 2

Z 0

2π

fko;x ðus  uc cos φ þwc sin φÞ2 þko;θ ðvs  vc Þ2

þko;r ðws  uc sin φ  wc cos φÞ2   ) ∂ws ∂wc 2 þK o;r  jxs ¼ 0;xc ¼ Lc R2 d θs ∂xs ∂xc

ð22Þ

2.5. Unified solution for the coupling system The admissible function of the displacement is essential to achieve an accurate and convergent solution in the Rayleigh–Ritz procedure. The traditional Fourier series is a well-known form to describe the displacements of the shell structures. However, it is just applicable to some very simple boundary conditions and would lead to the discontinuities of the displacements and their derivatives at the boundaries for the cases with complex boundary conditions. Since the main purpose of the paper is to investigate the vibration behavior of the coupled conical–cylindrical shell with general boundary constraints, the admissible functions are required to satisfy not only the energy expressions, but also the continuity conditions and general elastic boundary constraints. Under the consideration mentioned above, the modified Fourier series is adopted to represent the displacements of both of the conical and cylindrical shells. The modified Fourier series is composed of a standard Fourier series and an auxiliary polynomial function introduced to remove all the potential discontinuities of the displacements and their derives at the joint and the boundaries. Thus, the displacements of the conical shell component (uc ,vc ,wc ) and the cylindrical shell component (us ,vs ,ws ) can be written in the form of the modified Fourier series with the

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X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

consideration of the symmetric modes as [24,25] 0

1

1

1

1

2

c cos ðλcm xc Þ cos ðnθc Þþ ∑ ∑ A~ ln ξl ðxc Þ cos ðnθc Þ C B ∑ ∑ C B m ¼ 0 n ¼ 0 n ¼ 0 l ¼ 1 C uc ðxc ; θc ; tÞ ¼ ejωt B C B 1 1 2 1 @ þ ∑ ∑ A c cos ðλ x Þ sin ðnθ Þþ ∑ ∑ A~ c ξ ðx Þ sin ðnθ Þ A cm c c c mn ln l c

Acmn

m ¼ 0n ¼ 1

1

1

m ¼ 0n ¼ 0

0

1

cm c

mn

c

l ¼ 1n ¼ 0

1

1

ðK  ω2 MÞD ¼ F

1

1

2

ln l

c

c

cm c

mn

m ¼ 0n ¼ 1

c

C C C C ÞA

1

4

c ~c B ∑ ∑ C mn cos ðλcm xc Þ cos ðnθc Þ þ ∑ ∑ C ln ζ l ðxc Þ cos ðnθc Þ B l ¼ 1n ¼ 0 jωt B m ¼ 0n ¼ 0 wc ðxc ; θc ; tÞ ¼ e B 1 1 4 1 @ þ ∑ ∑ C c cos ðλ x Þ sin ðnθ Þ þ ∑ ∑ C~ c ζ ðx Þ sin ðnθ l ¼ 1n ¼ 1

c

ln l

c

C C C C ÞA

ð25Þ and 0

1

1

m ¼ 0n ¼ 1

sm s

mn

s

1

1

2

s ~s B ∑ ∑ Amn cos ðλsm xs Þ cos ðnθs Þþ ∑ ∑ A ln ξl ðxs Þ cos ðnθs Þ B m ¼ 0 n ¼ 0 n ¼ 0 l ¼ 1 us ðxs ; θs ; tÞ ¼ ejωt B B 1 1 2 1 @ þ ∑ ∑ A s cos ðλ x Þ sin ðnθ Þþ ∑ ∑ A~ s ξ ðx Þ sin ðnθ l ¼ 1n ¼ 1

ln l

s

s

ð34Þ

ð24Þ

l ¼ 1n ¼ 1

c ~c B ∑ ∑ Bmn cos ðλcm xc Þ sin ðnθc Þ þ ∑ ∑ B ln ξl ðxc Þ sin ðnθc Þ B l ¼ 1n ¼ 1 jωt B m ¼ 0n ¼ 1 vc ðxc ; θc ; tÞ ¼ e B 1 1 2 1 @ þ ∑ ∑ B c cos ðλ x Þ cos ðnθ Þ þ ∑ ∑ B~ c ξ ðx Þ cos ðnθ

    Li 3 πxi L3 3πxi cos  i 3 cos 3 2Li 2Li π 3π

It can be proved that the requirement for the selection of the auxiliary polynomial function is guaranteed, since it is easy to 0 0 0″ verify that ς01 ð0Þ ¼ ς0″ 3 ð0Þ ¼ ς2 ðLÞ ¼ ς4 ðLÞ ¼ 1ξ2 ð0Þ ¼ ξ2 ðLÞ ¼ ξ2 ð0Þ ¼ 0, ξ1 ð0Þ ¼ ξ1 ðLÞ ¼ ξ01 ðLÞ ¼ 0, ξ01 ð0Þ ¼ 1,ξ02 ðLÞ ¼ 1. By Substituting Eqs. (7)–(11) and (21, 22) together with the admissible functions defined in Eqs. (23)–(34) into Eq. (12), and then setting the variation of the preceding functional with respect to the expanded and supplemented coefficients independently to zero, one obtains the equations of motion for the coupled conical– cylindrical shell combination as

ð23Þ 0

ζ 4 ðxi Þ ¼ 

C C C C ÞA

ð35Þ

where F is a column vector of the generalized forces and consists of the expression series multiplying the amplitude of corresponding force component in the order. The stiffness matrix K, the mass matrix M and the coefficient vector D can be written as " # KCC KCS K¼ ; ð36Þ T KCS KSS " M¼

MCC

0

0

MSS

# ð37Þ

ð26Þ 0

1

1

1

2

s B~ ln ξl ðxs Þ sin ðnθs Þ

1

s C B ∑ ∑ Bmn cos ðλm xs Þ sin ðnθs Þþ ∑ ∑ C B m ¼ 0n ¼ 1 l ¼ 1n ¼ 1 C B vs ðxs ; θs ; tÞ ¼ ejωt B C 1 1 2 1 @ þ ∑ ∑ B s cos ðλ x Þ cos ðnθ Þ þ ∑ ∑ B~ s ξ ðx Þ cos ðnθ Þ A m s s s s l ln mn m ¼ 0n ¼ 0

l ¼ 1n ¼ 0

ð27Þ 0

1

1

4

C smn

1

s C~ ln ζ l ðxs Þ cos ðnθs Þ

1

cos ðλsm xs Þ cos ðnθs Þ þ ∑ ∑ C B ∑ ∑ C B m ¼ 0n ¼ 0 l ¼ 1n ¼ 0 C ws ðxs ; θs ; tÞ ¼ ejωt B C B 1 1 4 1 s @ þ ∑ ∑ C s cos ðλ x Þ sin ðnθ Þ þ ∑ ∑ C~ ζ ðx Þ sin ðnθ Þ A sm s s s ln l s mn m ¼ 0n ¼ 1

l ¼ 1n ¼ 1

ð28Þ where ω is the angular frequency, t represents time, λcm ¼ mπ=Lc , c c c c c c c ~c ~c Acmn ,A~ ln ,Amn ,Aln ,Bcmn ,B~ ln ,Bmn ,B~ ln ,C cmn ,C~ ln ,C mn ,C ln and λsm ¼ mπ=Ls , s s ~s s ~s s ~s s ~ s s ~s s ~ are the expansion A ,A ,A ,A ,B ,B ,B ,B ,C ,C ,C ,C mn

ln

mn

ln

mn

ln

mn

ln

mn

ln

mn

ln

coefficients and they can be solved by the Rayleigh–Ritz procedure. ξl ðxi Þ ðl ¼ 1; 2Þ are the supplementary functions for the inplane displacements, whileςl ðxi Þðl ¼ 1; 2; 3; 4Þ are the supplementary functions for the radial displacement. Theoretically, both ξl and ςl must be closed-form functions and sufficiently smooth over the length of the conical shell in order to meet the requirements provided by the continuity conditions and boundary constraints. In this paper, these supplementary functions are specially selected as [24,25]  2 x ξ1 ðxi Þ ¼ xi i 1 ; ð29Þ Li ξ2 ðxi Þ ¼

  xi 2 xi 1 Li Li

ð30Þ

ζ 1 ðxi Þ ¼

    9Li πxi L 3πx sin  i sin 2Li 4π 2Li 12π

ð31Þ

ζ 2 ðxi Þ ¼ 

    9Li πxi L 3πxi cos  i cos 4π 2Li 12π 2Li 

ζ 3 ðxi Þ ¼

Li 3 πxi sin 2Li π3



  L3 3πxi  i 3 sin 2Li 3π

D ¼ ½AC ; BC ; CC ; AS ; BS ; CS T

ð38Þ

AC ¼ ½AC00 ; AC01 ; ⋯; ACmn ; ⋯; ACMN ; A~ 00 ; A~ 01 ; ⋯; A~ mn ; ⋯; A~ MN ; A00 ; A01 ; ⋯; C

C

C

C

C

C

C C ~C ~C ~C ~C Amn ; ⋯; AMN ; A00 ; A01 ; ⋯; Amn ; ⋯; AMN T

ð39Þ

BC ¼ ½BC00 BC01 ; ⋯; BCmn ; ⋯; BCMN ; B~ 00 ; B~ 01 ; ⋯; B~ mn ; ⋯; B~ MN ; B00 B01 ; ⋯; C

C

C

C

C

C

C C C C C C Bmn ; ⋯; BMN ; B~ 00 ; B~ 01 ; ⋯; B~ mn ; ⋯; B~ MN T

ð40Þ

C C C C C C CC ¼ ½C C00 ; C C01 ; ⋯; C Cmn ; ⋯; C CMN ; C~ 00 ; C~ 01 ; ⋯; C~ mn ; ⋯; C~ MN ; C 00 ; C 01 ; ⋯; C C ~C ~C ~C ~C C mn ; ⋯; C MN ; C 00 ; C 01 ; ⋯; C mn ; ⋯; C MN T

ð41Þ

S S S S S S AS ¼ ½AS00 ; AS01 ; ⋯; ASmn ; ⋯; ASMN ; A~ 00 ; A~ 01 ; ⋯; A~ mn ; ⋯; A~ MN ; A00 ; A01 ; ⋯; S S ~S ~S ~S ~S Amn ; ⋯; AMN ; A00 ; A01 ; ⋯; Amn ; ⋯; AMN T

ð42Þ

S S S S S S BS ¼ ½BS00 BS01 ; ⋯; BSmn ; ⋯; BSMN ; B~ 00 ; B~ 01 ; ⋯; B~ mn ; ⋯; B~ MN ; B00 B01 ; ⋯; S S S S S S Bmn ; ⋯; BMN ; B~ 00 ; B~ 01 ; ⋯; B~ mn ; ⋯; B~ MN T

ð43Þ

S S S S S S CS ¼ ½C S00 ; C S01 ; ⋯; C Smn ; ⋯; C SMN ; C~ 00 ; C~ 01 ; ⋯; C~ mn ; ⋯; C~ MN ; C 00 ; C 01 ; ⋯; S S ~S ~S ~S ~S C mn ; ⋯; C MN ; C 00 ; C 01 ; ⋯; C mn ; ⋯; C MN T

ð44Þ

The detailed expressions for the other used sub-matrices are given in Appendix A. The natural frequencies and eigenvectors of the coupled conical–cylindrical shell combination can be obtained by solving the problem of Eq. (35) when F¼0. Subsequently, the mode shapes of the shell combination can be yielded by substituting corresponding eigenvectors into the displacement expressions. For the forced vibration analysis of the shell combination, the expanded and supplemented coefficients of the displacements can be directly obtained by solve the Eq. (35).

ð32Þ 3. Numerical examples and discussions ð33Þ

In order to illustrate the convergence, accuracy and versatility of presented method, some numerical examples on vibration

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

127

Table 1 Convergence of the coupled conical–cylindrical shell with C–C, F–F, S–S boundary conditions. Boundary conditions

Mode number

MN

ANSYS

10  10

11  10

12  10

13  10

14  10

15  10

C–C

1 2 3 4 5 6

0.2244 0.2362 0.2456 0.2552 0.2668 0.2684

0.2242 0.2360 0.2456 0.2552 0.2667 0.2683

0.2241 0.2360 0.2451 0.2552 0.2667 0.2682

0.2239 0.2358 0.2448 0.2551 0.2666 0.2681

0.2236 0.2357 0.2443 0.2551 0.2666 0.2680

0.2236 0.2357 0.2443 0.2551 0.2666 0.2680

0.2238 0.2359 0.2442 0.2550 0.2664 0.2680

F–F

1 2 3 4 5 6

0.0097 0.0257 0.0302 0.0465 0.0727 0.0811

0.0097 0.0257 0.0302 0.0465 0.0727 0.0810

0.0096 0.0256 0.0299 0.0465 0.0727 0.0810

0.0096 0.0256 0.0299 0.0465 0.0727 0.0809

0.0094 0.0256 0.0291 0.0465 0.0727 0.0808

0.0094 0.0256 0.0291 0.0465 0.0727 0.0808

0.0100 0.0257 0.0291 0.0465 0.0727 0.0807

S–S

1 2 3 4 5 6

0.0812 0.2088 0.2105 0.2248 0.2267 0.2426

0.0809 0.2087 0.2105 0.2248 0.2267 0.2426

0.0807 0.2084 0.2104 0.2248 0.2266 0.2426

0.0805 0.2081 0.2102 0.2248 0.2265 0.2426

0.0804 0.2076 0.2100 0.2248 0.2264 0.2426

0.0804 0.2076 0.2100 0.2248 0.2264 0.2426

0.0809 0.2079 0.2101 0.2247 0.2264 0.2426

analysis of the coupled conical–cylindrical shell with classical boundary conditions and general elastic boundary conditions are presented in this section. For conveniently referring to the classical boundary conditions, F, S and C denote respectively free, simplysupported and clamped restraints. Unless otherwise specified, the geometric and material parameters of the coupled structure are as follows: R1 ¼ 0:4226 m, R2 ¼ Ls ¼ 1 m, hc ¼ hs ¼ h ¼ 0:01 m, φ ¼ 30o , Ec ¼ Es ¼ E ¼ 2:11  1011 Nm  2 , νc ¼ νs ¼ ν ¼ 0:3, ρc ¼ ρs ¼ ρ ¼ 7800 kg m  3 . Non-dimensional pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi frequency parameter is introduced here Ωm;n ¼ ωm;n R2 ρð1 ν2 ÞE. In addition, the zero natural frequency corresponding to the rigid-body mode of the coupled conical–cylindrical shell is neglected in all the following results. 3.1. Convergence study For the sake of validating the convergence of present method firstly, natural frequencies of the coupled conical–cylindrical shell with subject to different truncated configurations are carried out in this subsection. The first six non-dimensional frequency parameters for C–C, F–F and S–S supported conical–cylindrical shells are listed in Table 1. For different boundary conditions, there is no need to either reformulate the theoretical model or change the admissible functions of displacements, and only need to re-valuate the stiffnesses of the boundary springs. In the case of FEM program, i.e. ANSYS, 120  40 and 120  30 finite element mesh of SHELL63 elements are used respectively for the conical shell and the cylindrical shell. From the results in Table 1, it can be found that the non-dimensional frequency parameters converge rapidly with the increasing truncated number. Since the results converge well at the truncated number M¼ 14 and N ¼ 10, the following calculations will be implemented with the truncated numbers. And also, good agreement between the convergent results and those from ANSYS shows the accuracy of present method. 3.2. Validation In order to further validate the accuracy of the present method, Table 2 compares the frequency parameters of the coupled conical–cylindrical shell obtained by present method with those from ANSYS and the literature reported by Efraim and Eisenberger [10] and Caresta and Kessissoglou [13].Very good agreement is

Table 2 Non-dimensional frequency parameters Ω for the F–C supported conical– cylindrical shell. Mode order

Non-dimensional frequency parameter Ω

n

m

FEM

Efraim and Eisenberger [10]

Caresta and Kessissoglou [13] (Donnell–Mushtari)

Present

0

1 Ƭ 2 3 4 5

0.501989 0.609866 0.929602 0.953238 0.968473 1.006064

0.503779 0.609852 0.930942 0.956379 0.971634 1.012090

0.503752 0.609855 0.930916 0.956315 0.971596 1.011884

0.503792 0.609854 0.930890 0.953124 0.969493 1.009102

1

1 2 3 4 5 6

0.292689 0.633491 0.811100 0.929372 0.947084 0.983178

0.292875 0.635834 0.811454 0.931565 0.952178 0.992175

0.292908 0.635819 0.811446 0.931481 0.952189 0.991959

0.292873 0.635810 0.811231 0.930879 0.948502 0.991452

2

1 2 3 4 5 6

0.099810 0.501471 0.690708 0.857243 0.912869 0.955633

0.099968 0.502701 0.691305 0.859114 0.915870 0.960702

0.102034 0.502899 0.691479 0.858901 0.916072 0.960475

0.099915 0.502641 0.691144 0.858632 0.906351 0.960521

3

1 2 3 4 5 6

0.087406 0.390717 0.514212 0.751608 0.794909 0.915186

0.087603 0.391569 0.514478 0.753402 0.796590 0.919635

0.093771 0.392199 0.515184 0.753593 0.796983 0.919391

0.087584 0.391539 0.514379 0.750903 0.792080 0.919605

4

1 2 3 4 5 6

0.144547 0.329750 0.395380 0.645119 0.691826 0.871991

0.144619 0.330354 0.395649 0.646678 0.692805 0.871812

0.150574 0.331698 0.397604 0.647700 0.693197 0.871555

0.144599 0.330337 0.395622 0.644582 0.691144 0.871938

5

1 2 3 4 5 6

0.199367 0.295743 0.370626 0.578509 0.612690 0.815318

0.199546 0.296020 0.370901 0.579750 0.613363 0.817951

0.203896 0.296330 0.376227 0.581667 0.614222 0.819801

0.199464 0.295989 0.370866 0.578490 0.612703 0.816743

Ƭ denotes the purely torsional natural frequency

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X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

Fig. 2. Some selected mode shapes and corresponding natural frequency parameters Ωmn for the coupled conical–cylindrical shell with F–C boundary conditions.

3.3. Effects of the dimensional and the elastic restraint parameters In this section, new examples are conducted to illustrate the effects of the dimensional and elastic restraint parameters on the vibration behavior of the coupled conical–cylindrical shell. Various classical boundary conditions, including C–C, S–S and F–F, are examined here. The influence of the semi-vertex angle of the cone φ will be examined firstly. When the semi-vertex angle φ varies from 0–901, the conical shell degenerates from the annular plate to the cylindrical shell. At the meanwhile, the axial stiffness of the shell combination decreases gradually. Fig. 3 shows the lowest nature frequency parameters of the coupled conical–cylindrical shell with C–C, S–S and F–F boundary conditions as the semi-vertex angle of

0.35 C-C S-S F-F

0.3

Frequency parameter Ω

observed for not only the lower-order natural frequencies but also the higher-order frequencies. For directly illustrating the vibration behavior of the coupled conical–cylindrical shell, some selected mode shapes for the F–C supported shell structure are given in Fig. 2. It is obvious that the vibrations of the conical shell and the cylindrical shell are commonly coupled. Furthermore, it is noted that a purely torsional mode appears when n ¼0 [1], this is due to the fact that the circumferential displacement in the equations of motion for the coupled conical–cylindrical shell is uncoupled with either themeridional or radial displacement under this circumstance.

0.25 0.2 0.15 0.1 0.05 0

0

10

20

30

40

50

60

70

80

90

Semi-vertex angle φ Fig. 3. The lowest frequency parameter of the coupled conical–cylindrical shell with C–C, S–S and F–F boundary conditions as the semi-vertex angle φ increases.

the cone φ increases from 0–901. The new dimensional parameter Lc/R2 ¼0.8 is used. The lowest frequency parameter of the case with C–C boundary conditions keeps invariantly at a constant in

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

the range of φ from 0–601, and then it decreases dramatically as φ increases due to the decreasing axial stiffness. A wave-like behavior is observed for the lowest frequency curve for the case with S–S boundary conditions, which has solo peak at φ ¼601. A nearly horizontal line is presented for the case with F–F boundary conditions, which may be caused by the fact that the lowest natural frequency for the case is mainly dominated by the cylindrical shell component and thus, the variation of the semivertex angle of the conical shell component has little effect on the mode. In addition, the case with C–C boundary conditions has the largest value of the lowest natural frequency than the cases with S–S and F–F boundary conditions. One outstanding advantage of the present method is that it can be conveniently applied to deal with the cases with general elastic boundary conditions. Thus, the effects of the meridional, circumferential, radial and rotational springs’ stiffnesses on the vibration behavior of the coupled conical–cylindrical shell should be thoroughly understood. For the sake of generalizing the effects, the non-dimensional elastic restraint parameters Γ u ,Γ v ,Γ w and Γ wr are introduced here as the ratios of the meridional, circumferential, radial and rotational boundary springs’ stiffnesses to the in-plane stiffness Eh=ð1  ν2 Þ, respectively. Fig. 4 shows the first and the second frequency parameters with the variation of one group of the boundary springs' stiffnesses,

whereas the other three groups of boundary springs’ stiffnesses are assigned to be extremely large value (104). It can be found from the Fig. 4(a) that the values of the two natural frequencies increase rapidly in the range of Γ u (101–102) and keep level outside the range. In Fig. 4(b), the effect of the circumferential boundary springs’ stiffness is given. The two frequency parameters remain at the constants at the start and the end of the whole considered stiffness range. However, different from the forgoing figure, when Γ v varies from 10  4–102, there are two obvious rising stage and relatively stable intermediate values for the two curves respectively. This may be as a result of the rigid body motion of the coupled conical– cylindrical shell in the circumferential direction caused by the weakening circumferential boundary restraints. The behaviors of the two natural frequencies of the cases with the radial and rotational variable-restraints are similar to each other as shown in Fig. 4(c) and (d). The difference between the two figures is the range of the stiffness leading to the rapid variation of the two natural frequencies. For the radial springs, the range of Γ w is 10  1–101, while for the rotational springs, the range of Γ wr is from 10  4–10  2. From the four figures, it can be concluded that the meridional restraint has the greatest effect on the vibration behavior of the coupled conical– cylindrical shell than the restraints in the other three directions, and the variation of the circumferential restraint would lead to the new stable natural frequencies of the shell combinations.

0.250

0.245 1st mode 2nd mode

1st mode 2nd mode

Frequency parameter Ω

0.245

Frequency parameter Ω

129

0.240 0.235 0.230 0.225 0.220

0.240

0.235

0.230

0.225

0.215 0.210 -4 10

10

-2

0

10

10

2

10

0.220 -6 10

4

10

-4

-2

10

0

10

10

4

0.245

0.245

1st mode 2nd mode

1st mode 2nd mode 0.240

Frequency parameter Ω

Frequency parameter Ω

10

Elastic restraint parameter Гv

Elastic restraint parameter Гu

0.235

0.230

0.225

0.220 -4 10

2

10

-2

0

10

10

2

Elastic restraint parameter Г w

10

4

0.240

0.235

0.230

0.225

0.220 -6 10

10

-4

10

-2

10

Elastic restraint parameter Гwr

Fig. 4. First two frequency parameters of the coupled conical–cylindrical shell with variable stiffness of boundary springs: (a) Γ u ; (b) Γ v ; (c) Γ w ; (d) Γ wr .

0

130

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

3.4. Application of present method to complex shell combination Besides addressing the problem caused by general elastic boundary conditions, present method can be conveniently developed to the cases of complex shell combinations. In this subsection,

Table 3 Non-dimensional natural frequencies of the conical–cylindrical–conical shell combination with F–F boundary conditions. Mode no.

ANSYS

Present

Error (%)

0.5339

1.19

n

m

0

1

0.5276

1

1

0.5763

0.5838

1.30

2

1 2

0.0275 0.0315

0.0272 0.0319

1.09 1.58

3

1 2 3

0.0790 0.0821 0.3184

0.0791 0.0823 0.3208

0.13 0.24 0.75

4

1 2 3 4

0.1417 0.1447 0.2749 0.3927

0.1418 0.1446 0.2763 0.3920

0.07 0.07 0.51 0.18

5

1 2 3 4

0.1948 0.2001 0.2482 0.3661

0.1952 0.2002 0.2485 0.3654

0.21 0.05 0.12 0.19

6

1 2 3

0.2175 0.2443 0.2490

0.2183 0.2446 0.2485

0.37 0.12 0.20

the joint conical–cylindrical–conical shell combination will be taken as an example to illustrate the extensive application of present method in practice. The new conical shell is assigned at the other end of the cylindrical shell component and has the same material and geometry parameters as previous one. The whole shell combination is free supported at its two ends in the calculation. Table 3 shows the comparison of natural frequencies obtained by present method and FEM program (ANSYS). Small deviation is observed from the table, which validates the accuracy and feasibility of present method for complex shell combinations. Some selected mode shapes for the conical–cylindrical–conical shell combination with F–F boundary conditions are shown in Fig. 5. 3.5. Forced vibration analysis This section will concentrate on the forced vibration analysis of the coupled conical–cylindrical shell subject to point loads. The structural damping is essential to make the present theoretical model of practical significance. In this study, the damping is introduced by using a complex Young's modulus En ¼ Eð1 þ iηÞ [15] instead of the original one E, where η is defined as the damping of the shell combination. Four points including point A (0,0,0.4226) point B (0.5774,0,0.7113), point C (0.8660, 0, 0.8556) at the conical shell and point D (0.5, 0, 1) at the cylindrical shell are introduced in the local co-ordinate systems respectively. Fig. 6 shows the radial displacements of point C and point D for the C–C supported shell combination subjected to radial point force (f wc ¼ f wc ejωt ,f wc ¼ 1) located at point B. The results obtained by present method agree well with those from FEM program, which validate the accuracy of the present method to predict the forced vibration of the shell combination. From the comparison of the two figures, it can be found that the radial displacements of point

Fig. 5. Some selected mode shapes for the coupled conical–cylindrical–conical shell with F–F boundary conditions.

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

C and D perform identically at the first three resonant frequencies and become easy to distinguish as the frequency increases. Generally, the response at point C is larger than that at point D, which thanks to not only the relatively closer location of point C but also the fact that the vibration distribution is mainly concentrate on the conical shell rather than the cylindrical shell in the considered frequency range, although vibrations of the shell combination is usually coupled as indicated in the modal analysis. In the calculation procedure, only 2226 dofs are needed for the convergent truncated number M¼ 14 and N ¼ 10 whereas FEM program requires more than 40,000 dofs for 120  70 finite element mesh of SHELL63. Therefore, it is clear that one can save much computational cost by using the present method in favor of the computer storage requirement compared to the conventional finite element method. Furthermore, the boundary conditions at the extremities of the conical–cylindrical shell combination and the continuity conditions at the junction of the two shell components are always complicated and variable in practice. For different boundary conditions and continuity conditions, one must re-establish the computation model by using the finite element

131

program or change either the equations of motion or the admissible function of displacements with other analytical techniques. On the contrary, one can easily achieve the accurate solutions for the cases with different continuity and boundary conditions by just varying the stiffnesses of related springs with present method. The calculation procedure is significantly simplified. In general, present method offers a simply yet powerful technique to deal with the shell combinations with variable continuity and boundary conditions. An axial point force located at one end of the coupled conical– cylindrical shell is a typical loading case encountered commonly in practice. New examples are implemented to examine separately the responses of the two shell components and the F–C boundary conditions are considered here. The point load f us ¼ f us ejωt in the xs direction is set at point A in the axial direction and its amplitude is taken as f us ¼ 1. Fig. 7(a) and (b) show the responses at point B, point C and point D for the F–C supported shell combination in the meridional and radial directions respectively. For the meridional responses as shown in Fig. 7(a), although the point B is nearer to point A than point C, the displacement of point B is smaller than

-4

-5

10

10

ANSYS Present method

ANSYS -6

Radial displacement (m)

Radial displacement (m)

Present method -6

10

-8

10

-10

10

-7

10

-8

10

-9

10

-10

10

10

-11

150

200

250

300

350

10

400

150

200

Frequency (Hz)

250

300

350

400

Frequency(Hz)

Fig. 6. Radial displacements for the coupled conical–cylindrical shell subjected to the radial point force f wc at point B: (a) point C; and (b) point D.

-4

-4

10

10

Point B Point C Point D

-8

10

-10

10

150

10

-8

10

-10

10

-12

-12

10

-6

Radial displacement (m)

Axial displacement (m)

-6

10

Point B Point C Point D

200

250

300

Frequency (Hz)

350

400

10

150

200

250

300

350

400

Frequency (Hz)

Fig. 7. Responses for the coupled conical–cylindrical shell subjected to the point load f us : (a) meridional displacement; and (b) radial displacement.

132

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

-5

-4

10

10

o

o

φ=30

φ=30

o

o

φ=36

o

φ=45

Radial displacement (m)

-6

10

-8

10

-10

10

o

φ=45 -7

10

-9

10

-11

-12

10

Radial displacement (m)

φ=36

50

100

150

200

10

50

Frequency (Hz)

100

150

200

Frequency (Hz)

Fig. 8. Radial displacements for the shell components of the shell combination point load f us : (a) at point C; and (b) at point D.

that of point C when the frequency is lower than 260 Hz, and then the behavior revises, which means that the distance to the driving force is not the crucial factor for the values of response. The displacement measured at point D is smaller than those at point B and C in the whole frequency range in general. Generally, there will be none radial responses for the elementary cylindrical shell subjected to the axial load owing to the thoroughly separation of meridional and radial displacements in the energy expressions. However, when the cylindrical shell is coupled with the conical shell subjected to the axial point force, a moment around the circumferential direction will occurs at the coupled end of the cylindrical shell component as a result of force transmission through the conical shell component from the axial point force. As a result, strong radial responses are excited for the cylindrical shell component as shown in Fig. 7(b). All of three curves for point B, C and D have the almost same resonant peaks at the natural frequencies of the coupled conical–cylindrical shell. Thus, it can be concluded that the conical and cylindrical shell components are highly coupled in the radial directions. For the sake of completeness of the research, the forced vibration behaviors of the shell combination with different semi-vertex angles are numerically investigated here. Fig. 8 shows the radial responses measured at two points (C and D) for the shell combination subjected to the point load f us and considers three cases of semi-vertex angle of the cone φ(301, 361 and 451). For the radial displacement at point C as shown in Fig. 8(a), the resonant frequency increases as the angle φ increases. This provides the principle to obtain proper natural frequencies of the coupled conical–cylindrical shell combinations through changing the semi-vertex angle of the cone. However, from the numerical point of view, the responses corresponding to the relevant resonant frequencies are approximately equal. In general, the behavior of the response at point D, as shown in Fig. 8(b), is similar to that at point C in trend for all three cases of the angle, especially at the resonant frequencies, which means that the influence of the semivertex angle of the cone is consistent on both the conical and cylindrical shell components.

modified Fourier series, which is composed of a standard Fourier series and closed-form auxiliary functions introduced to accelerate the convergence of the expansion series and remove all the potential discontinuities at the boundaries and the junction between the two shell components. The boundary conditions and the continuity conditions can be achieved by the artificial spring technique, i.e. stiffnessvariable boundary springs and connective springs. Thus, the classical and non-classical boundary conditions can be simulated conveniently by just varying the stiffnesses of the corresponding artificial springs. The unknown expansion coefficients are solved by using the Rayleigh– Ritz method based on the Reissner's thin shell theory. The fast convergence and good accuracy of present method are demonstrated by comparisons of results with those from reported literature and FEM program. A number of examples are conducted to illustrate the free and forced vibration characteristics of the coupled conical–cylindrical shell combination with classical and general elastic boundary conditions. The semi-vertex angle of the cone and the boundary restraint parameters have greet influences on the vibration behavior of the coupled conical–cylindrical shell combinations. Specially, the impact due to the semi-vertex angle of the cone is consistent for both the conical and cylindrical shell components, which, however, is determined by the boundary conditions. And the variation of the circumferential restraint parameter would lead to a new stable natural frequency. Large radial responses are excited for the cylindrical shell component when the shell combination is subjected to the axial point force, which should attract much attention in practice. The accurate results obtained by present method compared with those from ANSYS for the coupled conical–cylindrical–conical shell show the applicability of present method for more complex shell combinations. The present method provides a general algorithm and successfully achieves the accurate and convergent solutions for both free and forced vibration analysis of the coupled conical–cylindrical shell. Based on this, the methodology is expected to be applicable to vibration analysis for the other shell combinations and provide accurate results for the other approaches as reference.

4. Conclusions

Acknowledgments

In this paper, a unified solution is developed for vibration analysis of the coupled conical–cylindrical shell with general elastic boundary conditions. Regardless of the boundary conditions, each displacement of the conical and cylindrical shell is invariantly expressed as the

The author would like to thank the reviewers for their constructive comments. The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China under Grant Nos.51175098 and 51279035.

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

2

Appendix A The elements of sub-matrices used in Eq. (35) are given as 2 3 Kc;uu Kc;uv Kc;uw 6 T 7 K Kc;vv Kc;vw 7; ðA1Þ Kcc ¼ 6 4 c;uv 5 KTc;uw KTc;vw Kc;ww 2

Ks;uu

6 T K Kss ¼ 6 4 s;uv KTs;uw 2

Ks;uv Ks;vv KTs;vw Kcs;uv

Ks;uw

3

7 Ks;vw 7 5 Ks;ww Kcs;uw

Kcs;uu 6 T K Kcs ¼ 6 4 cs;uv KTcs;uw

KTcs;vw

2

Mc;uu

0

0

Mc;vv

0

0

0

Mc;ww

6 Mcc ¼ 4 2

Ms;uu 6 Mss ¼ 4 0

Kcs;vv

3 ðA3Þ

3

0

7 5;

0

Ms;ww

2 0 6 6 6 6 β Kc;uv ¼ 6 Kβ 6 c;uv11 Kc;uu12 4 β Kc;uv21 Kβc;uv22 2

Kαc;uw11

7 5

ðA5Þ

Kβc;uu11

Kβc;uu12

Kβc;uu21

Kβc;uu22

Kαc;uv11 Kαc;uv21

Kαc;uv12 Kαc;uv22 0

2 0 6 6 6 6 β Kc;vw ¼ 6 Kβ 6 c;vw11 Kc;vw12 4 β Kc;vw21 Kβc;vw22

7 7 7 7; 7 7 5

ðA6Þ

3 7 7 7 7 7 7 5

ðA7Þ

0 Kβc;uw11

Kβc;uw12

Kβc;uw21

Kβc;uw22

7 7 7 7; 7 7 5

Mβc;uu22

α 6 Mα 6 c;vv21 Mc;vv22 6 6 Mc;vv ¼ 6 6 0 4

ðA12Þ

3

Mαc;vv12

0 Mβc;vv11

Mβc;vv12

Mβc;vv21

Mβc;vv22

7 7 7 7 7 7 5

ðA13Þ

3

Mαc;ww11 Mαc;ww12 α 6 Mα 6 c;ww21 Mc;ww22 6 Mc;ww ¼ 6 6 6 0 4

0 Mβc;ww11

Mβc;ww12

Mβc;ww21

Mβc;ww22

7 7 7 7 7 7 5

ðA14Þ

3

Kαcs;uu11 Kαcs;uu12 α 6 Kα 6 cs;uu21 Kcs;uu22 6 Kcs;uu ¼ 6 6 6 0 4

7 7 7 7 7 7 5

0 Kβcs;uu11

Kβcs;uu12

Kβcs;uu21

Kβcs;uu22

ðA15Þ

0 Kβc;vv11

Kβc;vv12

Kβc;vv21

Kβc;vv22

Kαc;vw11

Kαc;vw21

2

2

Kαcs;vv11

ðA10Þ

3

Kβc;ww11

Kβc;ww12

Kβc;ww21

Kβc;ww22

Kβcs;uw12

Kβcs;uw21

Kβcs;uw22

ðA17Þ

3

Kαcs;vv12

7 7 7 7 7 7 5

0 Kβcs;vv11

Kβcs;vv12

Kβcs;vv21

Kβcs;vv22

ðA18Þ

ðA19Þ Kαcs;ww11

Mαc;uu11

3

Kαcs;ww12

6 Kα α 6 cs;ww21 Kcs;ww22 6 6 Kcs;ww ¼ 6 6 0 4

2

7 7 7 7 7 7 5

Kβcs;uw11

7 7 7 7 7 7 5

Kcs;vw ¼ 0

3

0

0

ðA8Þ

ðA9Þ

Kαc;vw22 7 7 7 7 7 7 0 5

3

Kαcs;uw11 Kαcs;uw12 α 6 Kα 6 cs;uw21 Kcs;uw22 6 Kcs;uw ¼ 6 6 6 0 4

2

7 7 7 7 7 7 5

Kαc;vw12

ðA16Þ

α 6 Kα 6 cs;vv21 Kcs;vv22 6 6 Kcs;vv ¼ 6 6 0 4

3

2

Kαc;ww11 Kαc;ww12 6 Kα α 6 c;ww21 Kc;ww22 6 Kc;ww ¼ 6 6 6 0 4

Mβc;uu12

Mβc;uu21

Kcs;uv ¼ 0

3

2

Kαc;vv11 Kαc;vv12 6 Kα α 6 c;vv21 Kc;vv22 6 Kc;vv ¼ 6 6 6 0 4

Mαc;vv11

Mβc;uu11

3 0

Kαc;uw12

6 Kα α 6 c;uw21 Kc;uw22 6 Kc;uw ¼ 6 6 6 0 4

2

7 7 7 7 7 7 5

0

2

2

Kαc;uu11 Kαc;uu12 6 Kα α 6 c;uu21 Kc;uu22 6 Kc;uu ¼ 6 6 6 0 4

α 6 Mα 6 c;uu21 Mc;uu22 6 6 Mc;uu ¼ 6 6 0 4

ðA4Þ

3

0 0

3

Mαc;uu12

2

7 Kcs;vw 7; 5 Kcs;ww

0 Ms;vv

0

ðA2Þ

Mαc;uu11

133

0 Kβcs;ww11

Kβcs;ww12

Kβcs;ww21

Kβcs;ww22

0 Mβc;uu11

Mβc;uu12

Mβc;uu21

Mβc;uu22

2

ðA11Þ

Mαc;vv11 Mαc;vv12 6 Mα α 6 c;vv21 Mc;vv22 6 Mc;vv ¼ 6 6 6 0 4

ðA20Þ

3

Mαc;uu12

6 Mα α 6 c;uu21 Mc;uu22 6 6 Mc;uu ¼ 6 6 0 4

7 7 7 7 7 7 5

7 7 7 7 7 7 5

ðA21Þ

3 0 Mβc;vv11

Mβc;vv12

Mβc;vv21

Mβc;vv22

7 7 7 7 7 7 5

ðA22Þ

134

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

2

Mαc;ww11

α 6 Mα 6 c;ww21 Mc;ww22 6 6 Mc;ww ¼ 6 6 0 4

2

Kαs;uu11

3

Mαc;ww12

0 Mβc;ww11

Mβc;ww12

Mβc;ww21

Mβc;ww22

where ðA23Þ

Kαc;uu11;mn;m0 n0 ¼ K c μc sin ðϕÞδc;nn' πð  λcm Sm C m  λcm0 C m Sm0 Þ K c ð1  μc Þ δs;nn0 πnn0 C m C m0 Rc  1 þ K c δc;nn0 πC m C m0 Rc 2 þ kc;x R1 δc;nn0 π þ ðko;x R2 cos 2 φ þ ko;w R2 sin 2 φÞ

þ

δc;nn0 πð 1Þm þ m

3

Kαs;uu12

α 6 Kα 6 s;uu21 Ks;uu22 6 Ks;uu ¼ 6 6 6 0 4

7 7 7 7 7 7 5

0 Kβs;uu11 Kβs;uu21

Kβs;uu12 Kβs;uu22

Kαs;uv11

Kαs;uv12

7 7 7 7 7 7 5

ðA24Þ

0

Kαc;uu12;mn;ln0 ¼  K c δc;nn0 πλcm C m ξlð1Þ Rc þ

ðA33Þ K c ð1  μc Þ δs;nn0 πnn0 C m ξl Rc  1 2

þ K c μc sin ðϕÞδc;nn' πf λcm Sm ξl þ C m ξlð1Þ Þg 0

2 0 6 6 6 Ks;uv ¼ 6 6 Kβs;uv11 Kβs;uu12 6 4 β Ks;uv21 Kβs;uv22

Kαs;uv21

Kαs;uv22 7 7 7 7 7 7 0 5

þ ðA25Þ

2

Kαs;vv11

Kβs;uw12

Kβs;uw21

Kβs;uw22

0 Kβs;vv11

Kβs;vv12

Kβs;vv21

Kβs;vv22

2 0 6 6 6 Ks;vw ¼ 6 6 Kβs;vw11 Kβs;vw12 6 4 β Ks;vw21 Kβs;vw22

Kαs;vw11

7 7 7 7 7 7 5

Kαs;vw12

Kαs;vw21

Kαs;vw22 0

ðA27Þ

Kαs;ww11 Kαs;ww12 α 6 Kα 6 s;ww21 Ks;ww22 6 Ks;ww ¼ 6 6 6 0 4

Mαs;uu11

2 6 6 6 Ms;vv ¼ 6 6 6 4

Mαs;vv11 Mαs;vv21

2

Mαs;ww11

Kβs;ww12 Kβs;ww22

Mβs;uu12

Mβs;uu21

Mβs;uu22

0 Mβs;vv11

Mβs;vv12

Mβs;vv21

Mβs;vv22

ðA38Þ

0

0 K c ð1  μc Þ δs;nn0 πn0 ξlð1Þ ξl ð1Þ 2 0 K c ð1  μc Þ sin ðϕÞδs;nn0 πn0 ξl ξl Rc 1 þ 2

ðA39Þ

1 Kαc;uw11;mn;m0 n0 ¼ K c μc cos ðϕÞδc;nn0 πλcm0 C m Sm0 þ K c sin ð2ϕÞδc;nn0 πC m C m0 Rc 1 2

ðA40Þ

7 7 7 7 7 7 5

ðA29Þ

7 7 7 7 7 7 5

ðA30Þ

Mβs;ww12

Mβs;ww21

Mβs;ww22

0 0 1 Kαc;uw22;ln;mn0 ¼ K c μc cos ðϕÞδc;nn0 πξlð1Þ ςl þ K c sin ð2ϕÞδc;nn0 πξl ςl ð1Þ Rc 1 2 ðA43Þ

Ec ð1 þ μc Þ sin ðϕÞδs;nn0 π þ n 2 o   λcm Sm C m0  λcm0 C m Sm0 þ sin ðϕÞC m C m0 Rc  1 ðA31Þ

7 7 7 7 7 7 5

Dc ð1  μc Þ sin ðϕÞ cos 2 ðϕÞδc;nn0 π 2 fλcm λcm0 Sm Sm0 Rc 1  2λcm Sm C m0 Rc 2 þ

 2λcm0 C m Sm0 Rc 2 þ sin ðϕÞC m C m0 Rc 1 g

3

Mβs;ww11

ðA41Þ

Kαc;vv11;mn;m0 n0 ¼ K c δc;nn0 πnn0 C m C m0 Rc 1 þ Dc cos 2 ðϕÞδc;nn0 πnn0 C m C m0 Rc 3

7 7 7 7 7 7 5

0

1 Kαc;uw12;mn;ln0 ¼ K c μc cos ðϕÞδc;nn0 πλcm Sm ςl þ K c sin ð2ϕÞδc;nn0 πC m ςl Rc 1 2

1 Kαc;uw21;ln;mn0 ¼ K c μc cos ðϕÞδc;nn0 πC m ξlð1Þ þ K c sin ð2ϕÞδc;nn0 πC m ξl Rc 1 2 ðA42Þ

3

Mαs;ww12

6 Mα α 6 s;ww21 Ms;ww22 6 Ms;ww ¼ 6 6 6 0 4

ðA28Þ

3

Mβs;uu11

ðA37Þ



7 7 7 7 7 7 5

0

Mαs;vv12 Mαs;vv22 0

Kαc;uv21;ln;mn0 ¼ K c μc δc;nn0 πnC m ξlð1Þ þ K c sin ðϕÞδc;nn0 πnC m ξl Rc 1 K c ð1  μc Þ δs;nn0 πn0 λcm Sm ξl þ 2 K c ð1  μc Þ sin ðϕÞδs;nn0 πn0 C m ξl Rc 1 þ 2

3

Mαs;uu12

6 Mα α 6 s;uu21 Ms;uu22 6 6 Ms;uu ¼ 6 6 0 4

Kαc;uv12;mn;ln0 ¼  K c μc δc;nn0 πnλcm Sm ξl þ K c sin ðϕÞδc;nn0 πnC m ξl K c ð1  μc Þ K c ð1  μc Þ δs;nn0 πn0 C m ξlð1Þ þ  2 2  sin ðϕÞδs;nn0 πn0 C m ξl Rc 1

ðA36Þ

Kαc;uv22;ln;l0 n0 ¼ K c μc δc;nn0 πnξlð1Þ ξl þ K c sin ðϕÞδc;nn0 πnξl ξl Rc 1

3

0 Kβs;ww11 Kβs;ww21

0

ðA35Þ

0

2

2

ðA26Þ

3

Kαs;vv12

α 6 Kα 6 s;vv21 Ks;vv22 6 6 Ks;vv ¼ 6 6 0 4

7 7 7 7 7 7 5

0

K c ð1  μc Þ δs;nn' πnn0 ξl ξl Rc  1 2

Kαc;uv11;mn;m0 n0 ¼ K c δc;nn0 πnλcm0 C m Sm0 þ K c δc;nn0 πnC m C m0 K c ð1  μc Þ δs;nn0 πnλcm Sm C m0 þ 2

3

Kβs;uw11

0

Kαc;uu22;ln;l0 n0 ¼ K c δc;nn0 πξlð1Þ ξl ð1Þ Rc þ K c μc sin ðϕÞδc;nn0 πfξlð1Þ Þξl þ ξl ξl ð1Þ g

3

2

Kαs;uw11 Kαs;uw12 α 6 Kα 6 s;uw21 Ks;uw22 6 6 Ks;uw ¼ 6 6 0 4

ðA34Þ

þ δc;nn0 πkc;θ R1 þko;θ R1 δc;nn0 πð  1Þm þ m ðA32Þ

Kαc;vv12;mn;ln0 ¼ K c δs;nn0 πnn0 C m ξl Rc þ

0

ðA44Þ

n K c ð1 μc Þ δc;nn0 π  λcm Sm ξlð1Þ Rc  sin ðϕÞC m ξlð1Þ 2

þ sin ðϕÞλcm Sm ξl þ sin 2 ðϕÞC m ξl Rc 1

o

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

135

þDc cos 2 ðϕÞδc;nn' πnn0 C m ξl Rc 3

þn2 sin ðϕÞλcm0 C m Sm0 Rc 2 þ sin 2 ðϕÞλcm λcm0 Sm Sm0 Rc 1 g

Dc ð1  μc Þ cos 2 ðϕÞδc;nn0 π 2 f  λcm Sm ξlð1Þ Rc 1  2 sin ðϕÞC m ξlð1Þ Rc 2

þDc ð1  μc Þδs;nn0 πnn0 f sin 2 ðϕÞC m C m0 Rc 3

þ

þ sin ðϕÞλcm0 C m Sm0 Rc 2

þ2 sin 2 ðϕÞλcm Sm ξl Rc 2 þ 4 sin 2 ðϕÞC m ξl Rc 3 g

þ sin ðϕÞλcm Sm C m0 Rc 2 þ 4λcm λcm0 Sm Sm0 Rc 1 g

ðA45Þ

2

þδc;nn0 πkc;r R1 þðko;x sin φ

n

0 K c ð1  μc Þ δc;nn0 π ξlð1Þ ξl ð1Þ Rc 2 o 0 0 0  sin ðϕÞξlð1Þ ξl þ sin ðϕÞξl ξl ð1Þ þ sin 2 ðϕÞξl ξl Rc  1 0

Kαc;vv22;ln;l0 n0 ¼ K c δs;nn0 πnn0 ξl ξl Rc þ

0

þko;r cos 2 φÞR2 ð 1Þm þ m δc;nn0 π Kαc;ww12;mn;ln0 ¼ K c cos 2 ðϕÞδc;nn0 πC m ςl Rc 1  Dc δc;nn0 πλ2cm C m ςlð2Þ Rc þ Dc μc δc;nn0 πf  n2 C m ςlð2Þ Rc 1

0

þ Dc cos 2 ðϕÞδc;nn' πnn0 ξl ξl Rc 3 n 0 Dc ð1  μc Þ cos 2 ðϕÞδc;nn0 π ξlð1Þ ξl ð1Þ Rc 1 þ 2 Z Lc 0  2 sin ðϕÞ ½ξlð1Þ ðxc Þξl ðxc Þ=R2c ðxc Þdxc

 sin ðϕÞλcm Sm ςlð2Þ  n02 λ2cm C m ςl Rc 1  sin ðϕÞλ2cm C m ςlð1Þ g þ Dc δc;nn0 πfn2 n02 C m ςl Rc 3 g þ n02 sin ðϕÞλcm Sm ςl Rc 2  n2 sin ðϕÞC m ςlð1Þ Rc 2  sin ðϕÞλcm Sm ςlð1Þ Rc 1 g 2

0

 2 sin 2 ðϕÞ þ 4 sin 2 ðϕÞ

Z Z

Lc 0 Lc 0

0

½ξl ðxc Þξl ð1Þ ðxc Þ=R2c ðxc Þdxc

þ Dc ð1  μc Þδs;nn0 πnn0 f sin 2 ðϕÞC m ςl Rc 3

0

½ξl ðxc Þξl ðxc Þ=R3c ðxc Þdxc g

0

Kαc;ww22;ln;l0 n0 ¼ K c cos 2 ðϕÞδc;nn0 πςl ςl Rc 1 þ Dc δc;nn0 πςlð2Þ ςl ð2Þ 0

0

þ Dc μc δc;nn0 πf  n2 ςl ςl ð2Þ Rc 1 þ sin ðϕÞςl ςl ð2Þ

þ Dc sin ðϕÞ cos ðϕÞδs;nn0 πnλcm Sm C m0 Rc 2 Dc ð1  μc Þ cos ðϕÞδc;nn0 πn þ 2 2 f  λcm0 2 sin ðϕÞC m Sm0 Rc 2  4 sin ðϕÞC m C m0 Rc 3

þ n02 ςlð2Þ ςl Rc 1 ςlð2Þ ςl Rc 1 þ sin ðϕÞςlð2Þ ςl ð1Þ g

sin ðϕÞλcm Sm C m0 Rc 2 g

0

0

0

0

0

þ Dc δc;nn0 πfn2 n02 ςl ςl Rc 3 g  n02 sin ðϕÞςlð1Þ ςl Rc 2 0

0

 n2 sin ðϕÞςl ςl ð1Þ Rc 2 þ sin 2 ðϕÞςlð1Þ ςl ð1Þ Rc 1 g 0

0

þ Dc ð1  μc Þδs;nn0 πnn0 f sin 2 ðϕÞςl ςl Rc 3  sin ðϕÞςl ςl ð1Þ Rc 2

ðA47Þ

0

0

 sin ðϕÞςl ςl Rc 2 þ 4ςl ςl ð1Þ Rc 1 g

 K c δs;nn0 πnC m ς Rc 1  Dc μc cos ðϕÞδs;nn0 πnC m ςl Rc 1 Dc cos ðϕÞδs;nn0 πnn02 C m ςl Rc 3 l

þDc sin ðϕÞ cos ðϕÞδs;nn0 πnλcm Sm ςl Rc 2 Dc ð1  μc Þ cos ðϕÞδc;nn0 πn þ 2 2 f2 sin ðϕÞC m ςlð1Þ Rc 2 4 sin ðϕÞC m ςl Rc 3

0

0

þ δc;nn0 π½K c;w R1 ςlð1Þ ð0Þςl ð1Þ ð0Þ þ K o;r R2 ςlð1Þ ðLc Þςl ð1Þ ðLc Þ ðA53Þ Kαc;uu21 ¼ Kαc;uu21 T ;

ðA54Þ

Kαc;vv21 ¼ Kαc;vv21 T ;

ðA55Þ

Kαc;ww21 ¼ Kαc;ww21T

ðA56Þ

 K c δs;nn0 πnC m ξ Rc 1 þ Dc μc cos ðϕÞδs;nn0 πnC m ξlð2Þ Rc 1 Dc cos ðϕÞδs;nn0 πnn02 C m ξl Rc 3

Mαc;uu11;mn;m0 n0 ¼ ρc hc δc;nn0 πC m C m0 Rc ;

ðA57Þ

Dc sin ðϕÞ cos ðϕÞδs;nn0 πnC m ξlð1Þ Rc 2 Dc ð1  μc Þ cos ðϕÞδc;nn0 πn þ 2 2 f λcm 2 sin ðϕÞSm ξl Rc 2  4 sin ðϕÞC m ξl Rc 3

Mαc;uu12;mn;ln0 ¼ ρc hc δc;nn0 πC m ξl Rc

ðA58Þ

Mαc;uu22;ln;l0 n0 ¼ ρc hc δc;nn0 πξl ξl Rc ;

ðA59Þ

Mαc;ww11;mn;m0 n0 ¼ ρc hc δc;nn0 πC m C m0 Rc

ðA60Þ

Mαc;ww12;mn;ln0 ¼ ρc hc δc;nn0 πC m ςl Rc ;

ðA61Þ

þ2λcm Sm ςlð1Þ Rc 1  4 sin ðϕÞλcm Sm ςl Rc 2 g ¼

ðA52Þ

0

 Dc cos ðϕÞδs;nn0 πnn02 C m C m0 Rc 3

 2λcm λcm0 Sm Sm0 Rc 1  4

Kαc;vw21;ln;mn0

þ sin ðϕÞλcm Sm ςl Rc 2  4λcm Sm ςlð1Þ Rc 1 g þ δc;nn0 πkc;r R1 ςl ð0Þ þ δc;nn0 πko;r R2 ςl ðLc Þ

 Dc μc cos ðϕÞδs;nn0 πnλ2cm C m C m0 Rc 1

¼

 sin ðϕÞC m ςlð1Þ Rc 2

ðA46Þ

Kαc;vw11;mn;m0 n0 ¼ K c δs;nn0 πnC m C m0 Rc 1

Kαc;vw12;mn;ln0

ðA51Þ

ðA48Þ

l

0

þ2λcm Sm ξlð1Þ Rc 1 þ4 sin ðϕÞC m ξlð1Þ Rc 2 g 0

ðA49Þ 0

Kαc;vw22;ln;l0 n0 ¼  K c δs;nn0 πnξl ςl Rc 1 þK c μc cos ðϕÞδs;nn0 πnξlð2Þ ςl Rc 1

0

0

K c cos ðϕÞδs;nn0 πnn02 ξl ςl Rc 3 l l

0

Rc 2

K c sin ðϕÞ cos ðϕÞδ πnξ ς Dc ð1  μc Þ cos ðϕÞδc;nn0 πn þ 2 0 0 l l ð1Þ  2 f2 sin ðϕÞξ ς Rc  4 sin 2 ðϕÞξl ςl Rc 3 s;nn0

0

2ξlð1Þ ςl ð1Þ Rc 1 þ4

0

sin ðϕÞξlð1Þ ςl Rc 2 g

ðA50Þ

Kαc;ww11;mn;m0 n0 ¼ K c cos 2 ðϕÞδc;nn0 πC m C m0 Rc 1 þ Dc δc;nn0 πλ2cm λ2cm0 C m C m0 Rc þ Dc μc δc;nn0 πfn2 λ2cm0 C m C m Rc 1 þ sin ðϕÞλcm λ2cm0 Sm C m0  n02 λ2cm C m C m0 Rc 1 þ sin ðϕÞλ2cm λcm0 C m Sm0 g 2 02

þ Dc δc;nn0 πfn n

C m C m0 Rc 3 g þ n02

sin ðϕÞλcm Sm C m0 Rc 2

Mαc;ww22;ln;l0 n0 ¼ ρc hc δc;nn0 πςl ςl Rc

ðA62Þ

MαC;uu21 ¼ MαC;uu12 T ;

ðA63Þ

MαC;ww21 ¼ MαC;ww12T

ðA64Þ

Mαc;vv ¼ MαC;uu

ðA65Þ

Kαcs;uu11;mn;m0 n0 ¼ ko;x cos φð  1Þm δc;nn0 π

ðA66Þ

Kαcs;uu12;mn;ln0 ¼ Kαcs;uu21;mn;ln0 ¼ Kαcs;uu22;mn;ln0 ¼ 0

ðA67Þ

Kαcs;uw11;mn;m0 n0 ¼ ko;x sin φð  1Þm δc;nn0 π

ðA68Þ

Kαcs;uw12;mn;ln0 ¼ ko;x ð  1Þm ςl ð0Þδc;nn0 π

ðA69Þ

136

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

Kαcs;uw21;ln;mn0 ¼ Kαcs;uw22;ln;l0 n0 ¼ 0

ðA70Þ

Kαs;uw12;mn;ln0 ¼  K s μs δc;nn0 πλsm Sm ςl

ðA94Þ

Kαcs;vv11;mn;m0 n0 ¼  ko;θ ð  1Þm δc;nn0 π

ðA71Þ

Kαs;uw21;ln;mn0 ¼ K s μs δc;nn0 πC m ξlð1Þ

ðA95Þ

Kαcs;vv12;mn;ln0 ¼ Kαcs;vv21;mn;ln0 ¼ Kαcs;vv22;mn;ln0 ¼ 0

ðA72Þ

Kαcs;ww11mn;m0 n0 ¼  ko;r cos φð  1Þm δc;nn0 π

ðA73Þ

Kαcs;ww12;mn;ln0

¼  ko;r cos φς ð0Þδc;nn0 π

ðA74Þ

Kαcs;ww21;ln;mn0

¼  ko;r cos φδ



l

c;nn0

l

πς ðLc Þ

ðA75Þ

0

Kαs;vw11;mn;m0 n0 ¼ 

0

Ks Ds μs δs;nn0 nπC m ςl þ δs;nn0 πnC m ςlð2Þ R R Ds 2Ds ð1 μs Þ δc;nn0 πλsm n0 Sm ςlð1Þ  3 δs;nn0 πnn02 C m ςl þ R R

Kαs;vw12;mn;ln0 ¼ 

ðA76Þ Kαs;uu11;mn;m0 n0 ¼ K s δs;mm0 δc;nn0 λsm λsm0 þ ð 1Þ

m þ m0

ks;x Rδc;nn0 π þ ko;x δc;nn0 π

Kαs;uu12;mn;ln0 ¼ K s δc;nn0 πλsm Sm ξlð1Þ þ Kαs;uu22;ln;l0 n0

lð1Þ lð1Þ

¼ K s δc;nn0 πξ

Kαs;uu21 ¼ Kαs;uu12 Kαs;vv11;mn;m0 n0

πLx 1  μs πLx þ δc;mm0 δs;nn0 nn0 2 2R 2

ξ

ð1 μs ÞK s δs;nn0 πnn0 C m ξl 2R

0 ð1  μs ÞK s δs;nn0 πnn0 ξl ξl þ 2R

T

Z Lx Ks δs;nn0 πnC m ξl ðxs Þ cos ðλsm xs Þdxs R 0 2Ds ð1  μs Þ δc;nn0 πλsm n0 Sm ξlð1Þ þ R Ds μs Ds δs;nn0 πnλ2sm C m ξl  3 δs;nn0 πnn02 C m ξl  R R

Kαs;vw21;ln;mn0 ¼ 

ðA78Þ ðA79Þ ðA80Þ

  K s ð1  μs Þ 2Ds ð1  μs Þ þ δc;nn0 πλsm Sm ξlð1Þ ¼ 2 R   K s Ds þ 3 δs;nn0 πnn0 C m ξl þ R R

Kαs;vv22;ln;l0 n0 ¼

ðA77Þ

 K s ð1  μs Þ 2Ds ð1  μs Þ πLx þ δs;mm0 δc;nn0 λsm λsm0 ¼ 2 R 2   K s Ds 0 πLx þ þ δc;mm0 δs;nn0 nn R R3 2 0

Kαs;vv12;mn;ln0

ðA97Þ



 0 K s ð1  μs Þ 2Ds ð1  μs Þ þ δc;nn0 πξl ðxs Þξl ðxs Þ 2 R   0 K s Ds þ 3 δs;nn0 πnn0 ξl ðxs Þξl ðxs Þ þ R R

0 0 Ks 2Ds ð1  μs Þ δc;nn0 πn0 ξlð1Þ ςl ð1Þ δs;nn0 πnξl ςl  R R 0 0 Ds μs Ds δs;nn0 πnξl ςl ð2Þ  3 δs;nn0 πnn02 ξl ςl þ R R

T

Ks πLx K s R πLx 2 2 δc;mm0 δc;nn0 þ λ λ 0 δc;mm0 δc;nn0 kp R 2 2 sm sm K s kp πLx 2 02 K s μs kp πLx 2 02 n n þ δc;mm0 δc;nn0 λ n þ 3 δc;mm0 δc;nn0 2 R 2 sm R

ðA81Þ

2ð1  μs Þkp K s πLx 0 δs;mm0 δs;nn0 nn λsm λsm0 R 2 m þ m0 þ kr1 R2 δc;nn0 πð  1Þ þ ko;r R2 δc;nn0 π þ K o;r R2 λsm λsm0 δc;nn0 π þ

ðA82Þ

πLx 2

Ks K sR δc;nn0 πC m ςl  δc;nn0 πλ2sm C m ςlð2Þ kp R K s kp K s μs kp δc;nn0 πλ2sm n02 C m ςl þ 3 δc;nn0 πn2 n02 C m ςl þ R R 2ð1 μs Þkp K s δs;nn0 πnn0 λsm Sm ςlð1Þ  R þ ks;r R2 δc;nn0 πð  1Þm ςl ðLx Þ þ ko;r R2 δc;nn0 πςl ð0Þ ðA101Þ

ðA83Þ

ðA85Þ

Mαs;uu12;mn;m0 n0 ¼ Rρs hs δc;nn0 πC m ξl

ðA86Þ

Mαs;uu22;ln;l0 n0 ¼ Rρs hs δc;nn0 πξl ξl

ðA87Þ

Mαs;uu21 ¼ Mαs;uu12

ðA88Þ

T

0 0 Ks KsR δc;nn0 πςl ςl þ δc;nn0 πςlð2Þ ςl ð2Þ kp R 0 0 K s kp K s μs kp δc;nn0 πn02 ςlð2Þ ςl þ 3 δc;nn0 πn2 n02 ςl ςl  R R 0 2ð1 μs Þkp K s δs;nn0 πnn0 ςlð1Þ ςl ð1Þ þ R 0 0 þ ks;r R2 δc;nn0 πςl ðLx Þςl ðLx Þ þ K s;r R2 δc;nn0 πςl ðLx Þςl ðLx Þ

Kαs;ww22ln;l0 n0 ¼

0

Kαs;uv11;mn;m0 n0 ¼  K s μs δc;nn0 πn0 λm Sm C m0 þ

ðA100Þ

Kαs;ww12mn;ln0 ¼

ðA84Þ

Mαs;uu11;mn;m0 n0 ¼ Rρs hs δc;mm0 δc;nn0

ðA99Þ

Kαs;ww11;mn;m0 n0 ¼



Kαs;vv21 ¼ Kαs;vv12

ðA98Þ

Kαs;vw22;ln;l0 n0 ¼ 



þ ð  1Þm þ m ks;θ R2 δc;nn0 π þ ko;θ R2 δc;nn0 π

Ds μs πLx 2 Ds πLx 02 δc;mm0 δs;nn0 nλ 0  δc;mm0 δs;nn0 nn R 2 sm R3 2 ðA96Þ

Kαcs;ww22;ln;l0 n0 ¼  ko;r cos φςl ðLc Þςl ð0Þδc;nn0 π K o;r ςlð1Þ ðLc Þςl ð1Þ ð0Þ 

Ks nπLx 2Ds ð1  μs Þ πLx δc;mm0 δs;nn0  λsm λsm0 n0 δs;mm0 δc;nn0 R 2 2 R

0

þ ko;r R2 δc;nn0 πςl ð0Þςl ð0Þ þ K o;r R2 δc;nn0 πςlð1Þ ð0Þςl ð1Þ ð0Þ

K s ð1  μs Þ δs;nn0 πnC m Sm0 2

ðA102Þ ðA89Þ

Kαs;uv12;mn;ln0 Kαs;uv21;ln;mn0 Kαs;uv22;ln;l0 n0

K s ð1 μs Þ δs;nn0 πnC m ξlð1Þ ¼  K s μs δc;nn0 πn λm Sm ξ  2 0

l

Kαs;ww21 ¼ Kαww12 ðA90Þ

Mαs;ww11;mn;m0 n0 ¼ Rρs hs δc;mm0 δc;nn0

K s ð1 μs Þ δs;nn0 πnSm ξl þ 2

ðA91Þ

0 K s ð1  μs Þ δs;nn0 πnξlð1Þ ξl ¼ K s μs δc;nn0 πn0 ξl ξl ð1Þ  2

0

¼ K s μs δc;nn0 πn C m ξ

lð1Þ

0

Kαs;uw11;mn;m0 n0 ¼  K s μs δc;nn0 πλm Sm C m0

T

ðA103Þ πLx 2

ðA104Þ

Mαs;ww12;mn;m0 n0 ¼ Rρs hs δc;nn0 πC m ςl

ðA105Þ

ðA92Þ

Mαs;ww22;mn;m0 n0 ¼ Rρs hs δc;nn0 πςl ςl

ðA106Þ

ðA93Þ

Ms;vv ¼ Ms;uu

ðA107Þ

X. Ma et al. / International Journal of Mechanical Sciences 88 (2014) 122–137

( δs;ττ0 ¼

δc;ττ0

0

ðτ a τ0 ; τ ¼ τ0 ¼ 0Þ

1

ðτ ¼ τ0 ; τ a 0Þ

8 > <0 ¼ 1 > :2

;

ðA108Þ

ðτ a τ0 ; Þ ðτ ¼ τ0 ; τ a0Þ ðτ ¼ m; nÞ

ðA109Þ

ðτ ¼ τ0 ; τ ¼ 0Þ

Ki ¼

E i hi ; ð1  μ2i Þ

Di ¼

Ei hi ði ¼ s; cÞ 12ð1  μ2i Þ

ðA110Þ

3

Z

Lx

Sm C m0 ¼ 0

Z

Lx

C m Sm0 ¼

ðA111Þ

sin ðλim xi Þ cos ðλim xi Þdxi ;

ðA112Þ

cos ðλim xi Þ sin ðλim xi Þdxi

ðA113Þ

0

Z Sm ψ lðιÞ ¼

2π

Z

2π 0

Lx 0

0

C m ψ lðιÞ ¼

Z Z

0

Lx

sin ðλsm xs Þψ lðιÞ ðxs Þdxs ðψ ¼ ξ; ς; ι ¼ 1; 2Þ

ðA114Þ

cos ðλsm xs Þψ lðιÞ ðxs Þdxs ðψ ¼ ξ; ς; ι ¼ 1; 2Þ

ðA115Þ

The sub-matrices in Eqs. (A1–A32) with superscript β can be obtained by replacing δc;mm0 , δs;mm0 ,δc;nn0 and δs;mm0 with δs;mm0 , δc;mm0 , δs;mm0 and δc;nn0 respectively in the expressions of related submatrices with superscript α. Reference [1] Leissa AW. Vibration of shells. Washington, DC: NASA SP-288; 1973; 31–43. [2] Markuš MS. The mechanics of vibrations of cylindrical shells. Amsterdam: Elsevier; 1988. [3] Qatu M.S. Research advances in the dynamic behavior of shells. Part 1: laminated composite shells; 1989–2000. [4] Qatu, M.S. Research advances in the dynamic behavior of shells. Part 2: homogeneous shells; 1989–2000. [5] Wu CP, Lee CY. Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness. Int J Mech Sci 2001;43 (8):1853–69. [6] Kalnins A. Free vibration of rotationally symmetric shells. J Acoust Soc Am 1964;36:1355–65.

137

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