Free vibration of joined conical–cylindrical–conical shells

Acta Mech https://doi.org/10.1007/s00707-018-2133-3 O R I G I NA L PA P E R H. Bagheri · Y. Kiani · M. R. Eslami Free vibration of joined conical–cylindrical–conical shells Received: 10 December 2017 / Revised: 29 January 2018 © Springer-Verlag GmbH Austria, part of Springer Nature 2018 Abstract The present research considers the free vibration characteristics of a joined shell system that consists of three segments. The joined shell system contains two conical shells at the ends and a cylindrical shell at the middle. All shell elements are made from isotropic homogeneous material. The shell elements are unified in thickness. With the aid of the first-order shear deformation shell theory and the Donnell type of kinematic assumptions, the equations of motion of a conical shell and the associated boundary conditions are obtained. These equations are valid for each segment. The obtained equations are then discreted using the generalised differential quadratures (GDQ) method. Applying the intersection continuity conditions for displacements, rotations, forces, and moments between two adjacent shells, and also boundary conditions at the ends of the joined shell system, a set of homogeneous equations is obtained, which governs the free vibration motion of the joined shell. Comparisons are made with the available data in the open literature for the case of thin conical– cylindrical–conical shells with special types of geometry or boundary conditions. Afterwards, numerical results are provided for moderately thick shells with different geometrical and boundary conditions. 1 Introduction A joined shell that consists of different conical and cylindrical shell sections has increasing application in submarine, civil, and aerospace engineering. Close to the intersection of shells in a joined shell system, severe bending moment and shear force are developed when the shell is subjected to loading. Consequently, a fatigue phenomenon may be induced due to the vibrations by the dynamic loadings. Therefore, it is of high interest and importance to understand the vibration characteristics of joined shells to establish the fundamental requirements for a safe design. The problem of free vibration of shells of revolution with various geometries such as cylindrical, conical, and spherical shells has been the subject of many studies for a long period of time. This topic is documented in many textbooks, see, for example, [1–3]. However, the free vibration features of joined shells are less observed in the open literature. The reason is the higher number of equations and also difficulties which may arise during applying the boundary and continuity conditions. In comparison with the free vibrations of a single shell, vibrations of joined shells are less reported. An overview of the works dealing with such subject is documented below. H. Bagheri Mechanical Engineering Department, Islamic Azad University, South Tehran Branch, Tehran, Iran Y. Kiani (B) Faculty of Engineering, Shahrekord University, Shahrekord, Iran E-mail: [email protected]; [email protected] M. R. Eslami Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran H. Bagheri et al. Hu and Raney [4] performed an experimental investigation and also a theoretical examination on free vibration response of a joined shell system. The shell of this study contains cylindrical and conical parts which are unified in thickness. The case of a cantilever shell is taken into consideration. Analysis of free vibration mode shapes indicates that near the shell joints, a V-shaped notch occurs, which is the main cause of stress concentration and also fatigue phenomenon. Lashkari and Weingarten [5] conducted an experimental set-up and also a finite element procedure to analyse the free vibration behaviour of a cylinder–cone system. Results of this study show that it is possible that a segment of the shell vibrates independently. That is, one portion vibrates without exciting the other. Using a Fourier series solution, Chang and Greif [6] analysed the free vibration response of a segmented shell. Two types of cantilever shells and simply supported shells are considered. Segments of the shell system are unified in mean radius; however, they are different in thickness. Irie et al. [7] formulated the free vibration problem of joined cylindrical–conical shell system. The transfer matrix of the shell is expressed conveniently by the power series, and the frequency equations are derived for a given set of boundary conditions at the edges. As a special case, free vibration characteristics of an annular plate–cylindrical shell system are also analysed. To obtain the vibration characteristics of joined/hermetic shells, Tavakoli and Singh [8] developed a substructure synthesis method using the state space approach. In this study, different joined shell systems, such as cylinder– cylinder, cylinder–cone, sphere–sphere, and hermit capsule, are analysed. Páde approximation is established to express the matrix presentation of the governing equations associated with each shell substructure. Based on a higher-order semi-analytical finite element method, free vibration characteristics of combined cylindrical– conical shell systems are investigated by Sivadas and Ganesan [9]. Patel et al. [10] analysed a three-segment shell, i.e. a cone–cylinder–cone shell in the free vibration regime using the first-order Mindlin shear deformation shell theory. A semi-analytical finite element formulation is developed for joined shells made of composite laminated materials. The established eigenvalue problem based on the finite elements formulation may be used to extract the natural frequencies of the shell and also the associated mode shapes. Lee et al. [11] performed an investigation on the free vibration characteristics of free–free, simply supported–free, and clamped–free cylindrical–spherical joined shell systems. The Flügge shell theory and the Rayleigh technique are applied in order to analyse the free vibration characteristics of the joined shell structure and individual shell components. In this research, also a modal test and a finite element simulation are performed. It is shown that the vibrational response of the joined spherical–cylindrical shell structure is independent of the shallowness of a hemispherical shell. In particular, in the case of the joined spherical–cylindrical shell with the free–free boundary condition, the shallowness is not effective at all. Efraim and Eisenberger [12] analysed the free vibration and mode sequence of segmented thin Reissner–Naghdi shells. In this research, the exact dynamic stiffness matrix for each segment is derived and used in the assembly of the complete structure dynamic stiffness matrix. Caresta and Kessissoglou [13] obtained the free vibration characteristics of a joined conical–cylindrical shell system according to two different numerical schemes. A trigonometric expansion solution is used to describe the displacements of the cylindrical shell, while the displacements of the conical sections are obtained using a power series solution. Both thin classes of Donnell–Mushtari and Flügge shells are analysed. Free vibration characteristics of the shell are given for different combinations of boundary conditions. Unlike most of the available works, which are developed according to the flexural shell theories, Kang [14] analysed the free vibration of a joined cylindrical–conical shells within the framework of three-dimensional elasticity theory. Thickness variation of the shell system is also considered in this research. The developed solution method is based on the multi-term Ritz method, where the shape functions are constructed according to the simple polynomials. With implementation of a variational approach, Qu et al. [15] analysed the free vibration of a joined cylindrical–conical shell system with classical or non-classical boundary conditions. The thin shell assumptions of Reissner–Naghdi theory are used as the fundamental theoretical assumptions. The interface continuity and geometric boundary conditions are approximately enforced by means of a modified variational principle and least-squares weighted residual method. The above-mentioned solution method is also used to analyse the free vibration behaviour of other complex shell geometries. For instance, ring-stiffened joined conical–cylindrical shell systems [16], joined conical–cylindrical–spherical shell systems [17], joined cylindrical–spherical shell with elastic support boundary conditions [18], and spherical–cylindrical–spherical shells [19] are also analysed by Qu and his co-authors. Following the hybrid numerical method of Caresta and Kessissoglou [13], which is mainly based on the power series solution, Kouchakzadeh and Shakouri [20] developed the method of vibration analysis of conical–conical shell systems made of cross-ply composite laminated shells. Shakouri and Kouchakzadeh [21] also analysed the free vibration of a thin conical–conical shell system using the power series solution system. Various combinations of shell–shell systems and shell– Free vibration of joined conical shells plate systems may be extracted as the special cases of a conical–conical shell system. This research also contains an experimental study for the special case of a joined conical–conical shell with both ends free. Free and forced vibration characteristics of joined shells are investigated by Ma et al. [22] using a Fourier–Ritz formulation. The developed solution method is applicable for arbitrary combinations of boundary conditions. Two- and three-segment shells comprising of cylindrical and conical shells are analysed in this research. In the mentioned work, each of the displacement components is 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. Chen et al. [23] analysed the free and forced vibration of a ring-stiffened joined conical–cylindrical shell with arbitrary boundary conditions. The combined shell is initially divided into substructures according to the junctions of shell–shell and shell-plate, and/or the location of driving point. Unlike many other available works which use the smeared technique to model the stiffeners, in this research the stiffeners with rectangular cross section are treated as discrete members. Wang and Guo [24] analysed the free vibration characteristics of compound shells submerged in fluid. To analyse the effects of angular rotation on the free vibration characteristics of a joined shell, Sarkheil and Saadat Foumani [25] developed a power series solution similar to the investigations of Shakouri and Kouchakzadeh [21] and Kouchakzadeh and Shakouri [20]. In the most general case, Sarkheil et al. [26] investigated the free vibration behaviour of a joined shell system composed of n conical shells. This study also proposed an experimental study for the special case of a joined conical–conical–conical shell with both ends free. The solution technique of this research is according to the power series expansion of the displacement field. Also, Sarkheil et al. [27] formulated the free vibration characteristics of a joined shell system with arbitrary number of shells under the effect of rotation, using the method of power series solution. The free vibration characteristics of a cylindrical shell attached to a spherical dome at the end are investigated by Kang [28] using the Ritz formation and the three-dimensional elasticity theory. Recently, Bagheri et al. [29] analysed the applicability of the generalised differential quadratures method to analyse the free vibration response of a joined conical–conical shell using the shear deformable shell model. Also, Bagheri et al. [30] investigated the free vibration of a conical shell which is ring-stiffened in an arbitrary position. In this work, the conical shell is divided into two different cones, and the intersection continuity conditions are applied directly in the ring position. These two works of Bagheri et al. [29,30] may be applied for arbitrary combinations of boundary conditions and thin to moderately thick shells. The above literature survey covers almost the main works reported so far on the vibration characteristics of the joined shell systems. As the above survey reveals, most of them are mainly focused on the two-segment shells such as cylindrical–conical shells and conical–conical shells. On the other hand, the problem of free vibration of joined conical–cylindrical–conical shells has been the subject of a few studies which are limited to a class of thin of shells. This study aims to analyse the free vibration response of a joined shell system using a shear deformable shell model. A joined shell system consists of two cones at the ends, and a cylindrical shell at the middle is considered for the analysis. The geometry of the shell is assumed to be symmetric with respect to the shell middle length. However, different combinations of boundary conditions are considered for the analysis. The governing equations for each segment are obtained using the first-order shell model and the Donnell kinematic assumptions of the shallow shells. All shells are unified in thickness. Using the wave expansion through the circumferential direction and generalised differential quadrature expansion along the shell length, the governing equations of the joined shell, boundary conditions, and the continuity conditions are discretized to provide an eigenvalue problem. The resulting system of equations is solved to obtain the frequencies of the shell and the associated mode numbers. After validating the proposed solution method via some comparison studies, a series of parametric studies is performed to examine the influences of cone angles, shell thickness, cone radii, and boundary conditions. 2 Governing equations Consider a joined circular conical–cylindrical–conical shell made of an isotropic homogeneous material of uniform thickness h. The system may be considered as three different shells. The first one is conical, the second one is cylindrical, and the third one again is conical. For the ith shell, the end radii are assumed as Ri and Ri+1 . Due to the symmetry of the shell geometry, R1 = R4 and R2 = R3 , as shown in Fig. 1. Also, the semi-vertex angle of the ith cone is denoted by αi . As a result, α2 = 0 and α1 = −α3 = α. The length of each shell is denoted by L i , where again due to symmetry L 1 = L 3 . Meridional, circumferential, and normal directions of H. Bagheri et al. Fig. 1 Geometric parameters and coordinate system sign of a joined conical–cylindrical–conical closed shell each conical shell are denoted by 0 ≤ x i ≤ L i , i = 1, 2, 3, 0 ≤ θ ≤ 2π, and −h/2 ≤ z ≤ +h/2, respectively. The adopted coordinate system (x i , θ, z), geometric characteristics, and sign convention of the joined shell are depicted in Fig. 1. As a suitable theory for thin to moderately thick shells, the first-order shear deformation (FSDT) shell theory of Mindlin is used in the present research to estimate the displacement components across the thickness of the shell. According to the FSDT, for the ith segment of the shell, the components of the displacement on a generic point may be represented by       u i x i , θ, z, t = u i0 x i , θ, t + zϕxi x i , θ, t ,       v i x i , θ, z, t = v0i x i , θ, t + zϕθi x i , θ, t ,     wi x i , θ, z, t = w0i x i , θ, t (1) Free vibration of joined conical shells where u, v, and w are, respectively, the meridional, circumferential, and through-the-thickness displacements. A subscript 0 indicates the characteristics of the mid-surface. Besides, ϕθ and ϕx are, respectively, the transverse normal rotations about the x- and θ -axes. Furthermore, herein and all the rest, superscript i takes the values of 1,2, and 3, and is associated with the ith shell segment. According to the first-order shear deformation shell theory, the in-surface components of strains are linear functions of the through-the-thickness coordinate, while the transverse shear strains are assumed to be constant [31], ⎧ i ⎫ ⎧ i ⎫ ⎧ i ⎫ ⎪ εx x ⎪ ⎪ ⎪ κx x ⎪ ⎪ ⎪ εx x0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i i ⎪ i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ε κ ε ⎨ θθ ⎬ ⎨ θθ0 ⎬ ⎨ θθ ⎪ ⎬ i i i . (2) = γxθ 0 + z κxθ γxθ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i i i ⎪ γx z ⎪ ⎪ κx z ⎪ ⎪ γx z0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ i ⎪ ⎩ i ⎪ ⎩ i ⎪ ⎭ ⎭ ⎭ ⎪ κθ z γθ z γθ z0 The components of the strain field on the mid-surface of the shell according to the first-order shear deformation theory and compatible with the Donnell type of kinematic assumptions are [31] ⎧ ⎫ u i0,x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ i ⎪ ⎪ ⎪ vi ⎪ ⎪ εx x0 ⎪ ⎪ ⎪ ⎪ ⎪ ) ) sin(α cos(α ⎪ ⎪ ⎪ ⎪ i i 0,θ i i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ ⎪ + + w u ⎪ 0 0⎪ i i i ε ⎪ ⎪ ⎪ ⎪ r (x ) r (x ) r (x ) θ θ 0 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎬ ⎨ i u i sin(α ) i 0,θ i i , (3) γxθ 0 = + v v − ⎪ ⎪ ⎪ 0,x i ⎪ ⎪ r (x i ) r (x i ) 0 ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i i γx z0 ⎪ ⎪ ⎪ w0,x ⎪ ⎪ ⎪ ⎪ ⎪ i + ϕx ⎪ ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ i γθ z0 ⎪ ⎪ w ⎪ ⎪ cos (αi ) i 0,θ ⎪ ⎪ i ⎪ ⎪ + ϕ − v ⎩ ⎭ θ 0 i i r x r x and the components of change in curvature in the Donnell sense compatible with the FSDT are [31] ⎧ ⎫ i ϕx,x ⎪ ⎪ i ⎪ ⎪ ⎧ ⎪ ⎫ ⎪ ⎪ ⎪ i ⎪ ⎪ κ ⎪ ⎪ ⎪ ⎪ x x ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ ⎪ ⎪ ϕ sin(α ) ⎪ ⎪ ⎪ ⎪ i θ,θ i ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ ⎪ ⎪ + ϕ κ ⎪ ⎪ ⎪ ⎪ x ⎨ θθ ⎬ ⎨ ⎬ r (x i ) r (x i ) i = i κ xθ ⎪ ϕx i ,θ ⎪ sin(αi ) i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ κxi z ⎪ ⎪ ⎪ ⎪ ⎪ + ϕ − ϕθ ⎪ i ⎪ ⎪ ⎪ ⎪ i i θ,x ⎪ ⎪ ⎪ ⎪ r (x ) r (x ) ⎪ ⎪ ⎩ i ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ 0 κθ z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 (4) where in the above equations (),x i and (),θ denote the derivatives with respect to the meridian and circumferential directions of the shell, respectively. Furthermore, r (x i ) = Ri + x i sin(αi ) stands for the radius of the joined shell at each point along the length. For the case when material properties of the shell are linearly elastic, the components of stress in terms of strains are evaluated as ⎧ i ⎫ ⎡ ⎤⎧ i ⎫ σx x ⎪ ⎪ Q 11 Q 12 0 0 0 ⎪ εx x ⎪ ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ ⎪ ⎨ σθi θ ⎪ ⎬ ⎢ Q 12 Q 22 0 0 0 ⎥⎪ ⎨ εθ θ ⎪ ⎬ ⎢ ⎥ (5) τθi z = ⎢ 0 0 Q 44 0 0 ⎥ γθi z ⎪ ⎪ ⎣ 0 0 0 Q ⎪ ⎦⎪ ⎪ ⎪ 55 0 ⎪ τxi z ⎪ ⎪ ⎪ γxi z ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ ⎩ i i ⎭ 0 0 0 0 Q 66 τxθ γxθ where Q i j ’s (i, j = 1, 2, 4, 5, 6) are the reduced material stiffness coefficients and are obtained as follows: Q 11 = Q 22 = E , 1 − ν2 Q 12 = νE , 1 − ν2 Q 44 = Q 55 = Q 66 = E . 2(1 + ν) (6) H. Bagheri et al. The components of the stress resultants are obtained using the components of stress field as [32] ⎧ i ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ i ⎫ ⎨ Mx x ⎬  +h/2 ⎨ σxi x ⎬ ⎨ N x x ⎬  +h/2 ⎨ σxi x ⎬ = z σθi θ dz, Ni σθi θ dz, Mθi θ = ⎩ ⎩ i ⎭ ⎭ ⎩ ⎭ ⎩ θi θ ⎭ i i −h/2 −h/2 N xθ τxθ Mxθ τxθ  i   +h/2  i  Qxz σx z κ = dz. Q iθ z σθi z −h/2 (7) In the above equation, κ is the shear correction factor of FSDT. In this research, the shear correction factor is set equal to κ = 5/6. Substitution of Eqs. (4) into (7) with the simultaneous aid of Eqs. (2), (3), and (4) generates the stress resultants in terms of the mid-surface characteristics of the shell as ⎧ i ⎫ ⎡ ⎤ ⎧ εi ⎫ ⎪ ⎪ A11 A12 0 0 0 0 0 0 ⎪ ⎪ N xi x ⎪ ⎪ ⎪ xi x0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ N ⎢ ⎪ ⎪ ⎪ ⎥ A A 0 0 0 0 0 0 θ θ ⎪ i ⎪ ⎪ ⎪ ⎢ 12 22 ⎪ εθiθ 0 ⎪ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ γxθ 0 ⎪ ⎪ ⎪ ⎪ ⎢ 0 0 A66 0 0 0 0 0 ⎥⎪ ⎪ ⎪ N xθ ⎪ ⎪ ⎨ ⎬ ⎬ ⎢ ⎨ ⎥ Mxi x 0 0 ⎥ κxi x ⎢ 0 0 0 D11 D12 0 . (8) = ⎢ 0 0 0 D D 0 0 ⎥ κθi θ ⎪ Mθi θ ⎪ ⎪ ⎢ ⎥⎪ 12 22 0 ⎪ ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ ⎪ ⎢ 0 0 0 0 0 D ⎪ i ⎪ ⎪ 0 0 ⎥ 66 Mxθ ⎪ ⎪ ⎪ κxθ ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 0 0 0 0 0 κ A44 0 ⎦ ⎪ i i ⎪ ⎪ ⎪ γθ z0 ⎪ ⎪ ⎪ ⎪ Qθ z ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ i ⎭ ⎩ ⎭ 0 0 0 0 0 0 0 κ A i 55 γx z0 Qxz In the above equation, the constant coefficients Ai j and Di j indicate the stretching and bending stiffness, respectively, which are calculated by  +0.5h (Ai j , Di j ) = (Q i j , z 2 Q i j )dz. (9) −0.5h The complete set of the equations of motion and boundary conditions of a joined shell system may be obtained based on the generalised Hamilton’s principle [32]. Hamilton’s principle reads  t2    δ K i − U i + V i dt = 0 t1 at t = t1 , t2 : δu i0 = δv0i = δw0i = δϕxi = δϕθi = 0, (10) where in the above equation δ K i is the virtual kinetic energy of the ith shell element which is equal to  L i  2π  +h/2   δKi = ρ(z) u̇ i δ u̇ i + v̇ i δ v̇ i + ẇi δ ẇi r (x i )dzdθ dx i . (11) 0 0 −h/2 Here, a (˙) indicates the derivative with respect to t. Besides, δU i is the virtual strain energy of the ith shell element which may be calculated as  L i  2π  +h/2   i i σxi x δεxi x + σθi θ δεθi θ + τxθ δU i = + κτxi z δγxi z + κτθi z δγθi z r (x i )dzdθ dx i . δγxθ 0 0 (12) −h/2 And δV i is the virtual potential energy of the external loads which is absent for the free vibration problem. Integrating the above expressions with respect to the z coordinate and performing the Green–Gauss theorem to relieve the virtual displacement gradients results in the expressions for the linear equations of motion of the ith conical shell as N xi x,x i + Nθi θ,θ r (x i ) sin(αi ) i N xθ,θ + (N x x − Nθi θ ) = I1 ü i0 , r (x i ) r (x i ) + N xθ,x i + 2 sin(αi ) i cos (αi ) i Q θ z = I1 v̈0i , N xθ + i r (x ) r xi Free vibration of joined conical shells 1 sin (αi ) i cos (αi ) i Q iθ z,θ + Qxz − Nθ θ = I1 ẅ0i , i i r x r x r xi  sin (αi )  i 1 i i i i M + M − M + xx xθ,θ θ θ − Q x z = I3 ϕ̈x , r xi r xi 2 sin (αi ) i 1 Mθi θ,θ + Mxθ − Q iθ z = I3 ϕ̈θi + i r x r xi Q ix z,x i + Mxi x,x i i Mxθ,x i (13) where the following definitions apply: (I1 , I3 ) =  +h/2 ρ(1, z 2 )dz. (14) −h/2 The complete set of the boundary conditions are revealed through the process of virtual displacement relieving. For the closed ends of the shell, i.e. x 1 = 0 and x 3 = L 3 , the boundary conditions are as follows: i i N xi x δu i0 = N xθ δv0i = Q ix z δw0i = Mxi x δϕxi = Mxθ δϕθi = 0. (15) 3 Boundary and matching conditions For the two ends of the joined shell system, various types of boundary conditions may be defined. Each of the edges x 1 = 0 and x 3 = L 3 may be clamped (C), simply supported (S), or free (F). The mathematical expression of edge supports on the curved edges takes the form C : u 0 = v0 = w0 = ϕx = ϕθ = 0, S : N x x = v0 = w0 = Mx x = ϕθ = 0, F : N x x = N xθ = Q x z = Mx x = Mxθ = 0. (16) At the intersection of the shell system, the continuity of displacement components as well as the force and moment resultants should be satisfied. The compatibility of the displacements at the intersection reads i+1 u i0 cos(αi ) − w0i sin(αi ) = u i+1 0 cos(αi+1 ) − w0 sin(αi+1 ), i+1 u i0 sin(αi ) + w0i cos(αi ) = u i+1 0 sin(αi+1 ) + w0 cos(αi+1 ), v0i = v0i+1 , ϕxi = ϕxi+1 , ϕθi = ϕθi+1 , (17) and similarly, the compatibility of the stress resultants at the intersection results in i+1 N xi x cos(αi ) − Q ix z sin(αi ) = N xi+1 x cos(αi+1 ) − Q x z sin(αi+1 ), i+1 N xi x sin(αi ) + Q ix z cos(αi ) = N xi+1 x sin(αi+1 ) + Q x z cos(αi+1 ), Mxi x = Mxi+1 x , i+1 i N xθ = N xθ , i+1 i Mxθ = Mxθ where superscript i takes the values 1, 2, and 3 and is associated with the ith shell segment. (18) H. Bagheri et al. 4 Solution procedure Upon substitution of the stress resultants in terms of the mid-surface characteristics from Eq. (8) into the equations of motion (13), boundary condition (16), and matching conditions (17) and (18), the complete set of equations of motion, boundary, and matching conditions may be evaluated in terms of the displacements and curvatures. Since the shell is in the form of a shell of revolution, displacement components and rotations are periodic functions in terms of the circumferential coordinate. Considering the derivative order of these functions in the governing equations, and following the separation of variables technique, the displacement components and rotations may be written as ⎧ ⎫ u i0 (x i , θ, t) ⎪ ⎪ ⎤⎧ i i ⎫ ⎡ ⎪ ⎪ U (x ) ⎪ ⎪ ⎪ ⎪ sin(nθ ) 0 0 0 0 ⎪ ⎪ ⎪ i i ⎪ i i ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V (x ) ⎪ ⎬ ⎢ 0 ⎨ v0 (x , θ, t) ⎪ ⎬ cos(nθ ) 0 0 0 ⎥⎪ ⎥ ⎢ i i i 0 sin(nθ ) 0 0 ⎥ W (x ) (19) w0 (x i , θ, t) = cos(ωt + ψ) ⎢ 0 ⎪ ⎪ ⎦⎪ ⎣ 0 i (x i ) ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 sin(nθ ) 0 i i  ⎪ x ⎪ ⎪ ϕ (x , θ, t) ⎪ ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ 0 0 0 0 cos(nθ ) ⎩ i (x i ) ⎭ ⎩ i i ⎭ θ ϕθ (x , θ, t) where in the above equation n is the wave number through the circumferential direction. The time dependency in the solution of (19) is a simple harmonic function which is chosen to overcome the periodicity condition of the field variables in time domain due to the free vibration motion. In this function ω is the natural frequency of the joined shell. Substitution of the above equation into the equations of motion (13) results in new 15 coupled ordinary differential equations, where each five of them belong to a shell. These equations are in terms of the unknown through-the-meridian functions U i (x i ), V i (x i ), W i (x i ), ix (x i ), and iθ (x i ). The transformed equations and the associated boundary conditions for the ith segment are not given here and are presented in “Appendix A”. As expected, the presented equations in Eqs. (A.1)–(A.5) for the ith shell along with a proper choice of boundary and matching conditions result in a system of homogeneous equations. To solve the system of equations as an eigenvalue problem, the generalised differential quadratures method is implemented to transform the ordinary differential equations (A.1)–(A.5) into new linear algebraic equations. The GDQ method is quite well known, and its details are not repeated herein. Meanwhile, one may refer to [33] for more details. Also the distribution of grid points for each section is according to the Chebyshev–Gauss–Lobatto method [34,35]. Different techniques may be used to apply the boundary and matching conditions to the joined shell system. In this research, the boundary and matching conditions are applied directly to the equations. Based on this technique, one should apply the GDQ method to both equations of motion and boundary conditions. The equations of motion after discretising, applying the matching and boundary conditions and global assembling, take the form K = ω2 M (20) where in the above equation M is the generalised mass matrix, K is the generalised stiffness matrix, and  is the unknown displacement vector. The natural frequencies of the structure may be obtained by solving the standard eigenvalue problem (20). In this research, the solution procedure by means of the GDQ technique has been implemented in a MATLAB code. 5 Numerical results and discussion The developed formulation and solution method in the previous Section of the present study are used in here to obtain the natural frequencies of a shear deformable joined conical–cylindrical–conical shell system. The shells are made from an isotropic homogeneous material and are unified in thickness. In the numerical results of this study, for the sake of generality, the dimensionless frequency parameter is defined, which is equal to   = ω R2 ρ 1 − ν 2 /E. (21) For all of the numerical results, Poisson’s ratio is set equal to ν = 0.3. In the subsequent results, two comparison studies are given to justify the validity and accuracy of the developed formulation in the present study. Afterwards, parametric studies are given to show the importance Free vibration of joined conical shells of different parameters on the frequencies of the shells. Six different combinations of boundary conditions may be considered, since each end may be clamped, simply supported, or free, and the shell geometry is symmetric. It also should be mentioned that each segment of the shell is divided into 31 nodal points in the GDQ method after examination of the convergence of the frequencies up to the desired accuracy. 5.1 Comparison studies In this Section two comparison studies are provided. The first comparison study of this research aims to compare the natural frequencies of a cylindrical shell which is reinforced with two annular plates at the ends. This compound shell structure is analysed by Irie et al. [7]. It should be mentioned that the results of our study may be reduced easily to those of Irie et al. [7] when the semi-vertex angle of the cone is set equal to α = 90◦ . To model the shell of Irie et al. [7], other geometrical parameters are as follows: R2 / h = 100, R1 /R2 = 0.5, L 2 /R2 = 3. The two ends of the compound shell are assumed to be clamped, which means that the inner radius of the annular plates is set to be clamped. In the analysis of Irie et al. [7], which is according to the thin shell formulation, the governing equations are discretized by means of the transformed matrix technique. Their results provide the first six frequency parameters of the shell for each of the circumferential mode numbers. Comparison is provided in Table 1. It is observed that our results are in excellent agreement with those given by Irie et al. [7] for both symmetric (n = 0) and asymmetric (n > 0) frequencies. The second comparison study is dedicated to the free vibration of a conical–cylindrical–conical shell system. In this study, both ends of the shell system are assumed to be free. Numerical results of this study are compared with the numerical results of Ma et al. [22]. Geometric characteristics of the shell are R2 / h = 100, R1 /R2 = 0.4226, L 2 /R2 = 1, and α = 30◦ . Results of Ma et al. [22] provide the frequencies of the shell for some selective circumferential mode numbers, and for each mode number, the first one, two, or three frequencies are provided. It is again observed that the frequencies obtained by our study are in close agreement with those given by Ma et al. [22], which again verifies the correctness of the developed formulation and solution method (Table 2). 5.2 Parametric studies After validating the proposed solution method and formulation via performing comparison studies in the previous Section, in the subsequent results, the free vibration response of a shear deformable conical–cylindrical– conical shell system is analysed. Figure 2 provides the minimum frequency of a joined shell system for different circumferential mode numbers. In this Figure, four different types of boundary conditions are considered, and geometric characteristics of the shell are h/R2 = 0.01, L 2 /R2 = 1, and R1 /R2 = 0.4226. It is verified that for a joined shell system, variation of minimum frequency for a prescribed circumferential mode number is not monotonic with respect to the semi-vertex angle of the cone. Furthermore, in the joined shell system, similar to a conical shell system, the minimum frequency of the shell belongs to the higher circumferential mode numbers. Further observation of the effect of boundary conditions on the frequencies of the compound shell reveals that, for a prescribed Table 1 First six natural frequency parameters for each circumferential mode number n in the case of a C–C joined annular plate–cylindrical shell–annular plate m 1 Source n=0 Present 0.01751 0.01753 Irie et al. [7] 2 Present 0.2277 Irie et al. [7] 0.2283 3 Present 0.2341 0.2347 Irie et al. [7] 4 Present 0.6255 0.6290 Irie et al. [7] 5 Present 0.6398 0.6435 Irie et al. [7] 6 Present 0.8507 0.8510 Irie et al. [7] Geometric characteristics of the system are R2 / h n=1 n=2 n=3 0.2259 0.1660 0.09840 0.2262 0.1661 0.09868 0.2287 0.2363 0.2497 0.2291 0.2368 0.2502 0.2748 0.2374 0.2524 0.2750 0.2380 0.2530 0.4054 0.3938 0.2822 0.4056 0.3941 0.2826 0.6218 0.5647 0.4468 0.6248 0.5652 0.4471 0.6321 0.6302 0.5741 0.6338 0.6329 0.5745 = 100, R1 /R2 = 0.5, L 2 /R2 = 3, and α = 90◦ n=4 n=5 0.07437 0.07512 0.1994 0.1999 0.2752 0.2760 0.2775 0.2784 0.3474 0.3479 0.4750 0.4755 0.08149 0.08269 0.1595 0.1603 0.2728 0.2735 0.3100 0.3111 0.3143 0.3154 0.3958 0.3964 H. Bagheri et al. Table 2 Frequency parameters for each circumferential mode number n in the case of an F–F conical–cylindrical–conical system n Ma et al. [22] m Present 2 1 0.0272 2 0.0319 3 1 0.0791 2 0.0823 3 0.3208 4 1 0.1418 2 0.1446 3 0.2763 4 0.3920 5 1 0.1952 2 0.2002 3 0.2485 4 0.3654 6 1 0.2183 2 0.2446 3 0.2485 Geometric characteristics of the shell are R2 / h = 100, R1 /R2 = 0.4226, L 2 /R2 = 1, and α = 30◦ 0.4 0.25 n n n n n 0.35 0.3 = = = = = 1 2 3 4 5 n n n n n 0.2 0.25 = = = = = 1 2 3 4 5 0.15 0.2 f f 0.0277 0.0322 0.0790 0.0822 0.3202 0.1416 0.1445 0.2760 0.3668 0.1951 0.2002 0.2486 0.3656 0.2181 0.2446 0.2490 0.1 0.15 0.1 0.05 0.05 (a) 0 0 10 20 30 40 50 60 70 80 (b) 0 90 0 10 20 30 α[°] 50 60 70 60 70 90 0.4 n n n n n 0.35 0.3 = = = = = 1 2 3 4 5 n n n n n 0.35 0.3 0.25 0.2 0.2 f 0.25 0.15 0.15 0.1 0.1 0.05 = = = = = 1 2 3 4 5 0.05 (c) 0 80 α[°] 0.4 f 40 0 10 20 30 40 α[°] 50 60 70 80 (d) 90 0 0 10 20 30 40 50 80 90 α[°] Fig. 2 Influence of the semi-vertex angle of the cone and boundary conditions on the minimum frequency of the joined shell for different circumferential mode numbers. Geometric characteristics of the shell are h/R2 = 0.01, L 2 /R2 = 1, and R1 /R2 = 0.4226. a C–C shell, b C–F shell, c C–S shell, d S–S shell Free vibration of joined conical shells Table 3 First six natural frequency parameters for each circumferential mode number n in various boundary conditions of a joined conical–cylindrical–conical shell B.Cs. n m=1 m=2 m=3 m=4 C–C 1 0.2962 0.6275 0.7102 0.8250 2 0.2782 0.4212 0.5939 0.7953 3 0.2466 0.3011 0.4142 0.6477 4 0.2193 0.2405 0.3128 0.5467 5 0.2140 0.2271 0.2592 0.4876 6 0.2185 0.2501 0.2544 0.4705 C–S 1 0.2923 0.6194 0.6791 0.8084 2 0.2731 0.4158 0.5910 0.7933 3 0.2376 0.2912 0.4095 0.6418 4 0.2064 0.2330 0.3093 0.5353 5 0.2068 0.2222 0.2569 0.4719 6 0.2183 0.2468 0.2531 0.4580 S–S 1 0.2889 0.6050 0.6623 0.7853 2 0.2682 0.4102 0.5879 0.7914 3 0.2314 0.2790 0.4044 0.6371 4 0.2019 0.2174 0.3054 0.5297 5 0.2044 0.2130 0.2542 0.4682 6 0.2182 0.2454 0.2499 0.4561 C–F 1 0.0645 0.3992 0.6766 0.7646 2 0.0314 0.3206 0.5158 0.6697 3 0.0805 0.2643 0.3696 0.4920 4 0.1430 0.2278 0.2948 0.3812 5 0.1971 0.2203 0.2546 0.3684 6 0.2183 0.2461 0.2530 0.4238 S–F 1 0.0110 0.3982 0.6598 0.7448 2 0.0297 0.3130 0.5111 0.6686 3 0.0805 0.2489 0.3624 0.4908 4 0.1429 0.2089 0.2902 0.3808 5 0.1968 0.2094 0.2515 0.3684 6 0.2182 0.2450 0.2495 0.4238 Geometric characteristics of the shell are R2 / h = 100, R1 /R2 = 0.4226, L 2 /R2 = 1, and α = 30◦ m=5 m=6 0.9093 0.8314 0.6881 0.5702 0.4986 0.4766 0.8914 0.8299 0.6834 0.5640 0.4943 0.4740 0.8790 0.8284 0.6776 0.5521 0.4779 0.4608 0.8133 0.8095 0.6651 0.5571 0.4927 0.4734 0.7999 0.8067 0.6556 0.5401 0.4729 0.4585 0.9516 0.8505 0.7530 0.6539 0.5750 0.5183 0.9362 0.8503 0.7519 0.6525 0.5739 0.5175 0.9210 0.8500 0.7507 0.6510 0.5727 0.5166 0.9116 0.8455 0.7373 0.6324 0.5627 0.5156 0.8891 0.8448 0.7354 0.6308 0.5615 0.5147 mode number, the frequency of a C–C shell is higher than of a C–S shell, and the latter case has a higher frequency than an S–S shell. For the case of a thin joined conical–cylindrical conical shell system, Table 3 presents the first six frequency parameters of combined shells with different boundary conditions as functions of different circumferential mode numbers. Results of this Table are given for a compound shell with geometrical characteristics Table 4 First six lowest asymmetric natural frequency parameters for various boundary conditions and different thickness ratios of an isotropic homogeneous joined conical–cylindrical–conical shell B.Cs. C–C h/R2 1 2 3 4 5 0.02 0.2773(4) 0.2778(3) 0.2935(5) 0.3011(2) 0.3117(1) 0.05 0.3310(1) 0.3436(2) 0.3709(3) 0.4275(4) 0.5058(5) 0.1 0.3471(1) 0.3969(2) 0.4910(3) 0.6317(4) 0.8358(5) C–F 0.02 0.0595(2) 0.0655(1) 0.1552(3) 0.2512(4) 0.2931(5) 0.05 0.0674(1) 0.1403(2) 0.3294(3) 0.4267(4) 0.5058(5) 0.1 0.0703(1) 0.2517(2) 0.4833(3) 0.6314(4) 0.8358(5) C–S 0.02 0.2679(3) 0.2702(4) 0.2932(5) 0.2957(2) 0.3094(1) 0.05 0.3291(1) 0.3364(2) 0.3639(3) 0.4270(4) 0.5058(5) 0.1 0.3436(1) 0.3882(2) 0.4885(3) 0.6315(4) 0.8358(5) S–S 0.02 0.2614(3) 0.2663(4) 0.2907(2) 0.2930(5) 0.3073(1) 0.05 0.3272(1) 0.3300(2) 0.3587(3) 0.4266(4) 0.5058(5) 0.1 0.3402(1) 0.3806(2) 0.4861(3) 0.6314(4) 0.8358(5) F–F 0.02 0.0000(1) 0.0547(2) 0.1518(3) 0.2472(4) 0.2928(5) 0.05 0.0000(1) 0.1305(2) 0.3197(3) 0.4261(4) 0.5058(5) 0.1 0.0000(1) 0.2338(2) 0.47703() 0.6312(4) 0.8357(5) Number in parentheses indicates the circumferential mode number. Geometric characteristics of the shell are R1 /R2 L 2 /R2 = 1, and α = 30◦ 6 0.3131(6) 0.6312(6) 1.1006(6) 0.3131(6) 0.6312(6) 1.1006(6) 0.3131(6) 0.6312(6) 1.1006(6) 0.3131(6) 0.6312(6) 1.1006(5) 0.3131(6) 0.6312(6) 1.1006(6) = 0.4226, H. Bagheri et al. R2 / h = 100, R1 /R2 = 0.4226, L 2 /R2 = 1, and α = 30◦ . It is seen that lower frequencies may appear in higher circumferential mode numbers for all combinations of boundary conditions. Table 4 yields the first six frequency parameters of a joined conical–cylindrical–conical shell system for various circumferential mode numbers and different combinations of boundary conditions. Geometric characteristics of the shell are R1 /R2 = 0.4226, L 2 /R2 = 1, and α = 30◦ . In this Table, three different thickness-to-radius ratios are considered, which are h/R2 = 0.02, 0.05, and 0.1. It is observed that for a compound shell, which is free at both ends, the minimum frequency with circumferential mode number n = 1 is associated with a zero-valued frequency, which is due to the rigid body motion of the shell. 6 Conclusions In the present research, free vibration characteristics of a joined shell system composed of two conical shells at the ends and a cylindrical one at the middle are analysed. First-order shear deformation shell theory suitable for moderately thick shells and the Donnell type of kinematic assumptions are used as the basic assumptions. The governing equations of motion are established for the conical shell system. The governing equations for each shell segment are discretized by means of the generalised differential quadratures. Applying the boundary conditions for the two ends of the shells and also matching conditions at the intersection of the two adjacent shells, a system of eigenvalue problem is achieved, which may be used for free vibration analysis of compound shells. Comparison studies are provided to assure the accuracy of the developed formulation. Afterwards, by parametric studies the frequencies of a joined conical–cylindrical–conical shell system are obtained. Funding This article has received no funding. Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest. 7 Appendix After applying Eq. (19) to the equations of motion (13), the following system of equations is extracted. For the sake of simplicity, the superscript i is dropped out. A11 U,x x + A11 A12 sin (α) U,x + −nV,x + cos (α) W,x r (x) r (x) A66 −n 2 U + n sin (α) V − nr (x) V,x r 2 (x) A22 sin (α) − (sin (α) U − nV + cos (α) W ) + I1 ω2 U = 0, r 2 (x) A12 n A22 U,x + 2 n sin (α) U − n 2 V + n cos (α) W r (x) r (x) A66 sin (α) A66 + nU,x + r (x) V,x x nU − sin (α) V + r (x) V,x + 2 r (x) r (x) κ A44 cos (α) (− cos (α) V + r (x) θ + nW ) + I1 ω2 V = 0, r 2 (x) A22 cos (α) A12 cos (α) U,x − − (sin (α) U − nV + cos (α) W ) r (x) r 2 (x) κ A55 sin (α) x + W,x + κ A55 x,x + W,x x + r (x) κ A44 + 2 n cos (α) V − nr (x) θ − n 2 W + I1 ω2 W = 0, r (x) D12 sin (α) D12 n θ,x − D11 ,x x − (sin (α) x − nθ ) r (x) r 2 (x) (A.1) (A.2) (A.3) Free vibration of joined conical shells D66 D11 sin (α) x,x −n 2 x + n sin (α) θ − nr (x) θ,x + 2 r (x) r (x) (D12 − D11 ) sin (α) + (sin (α) x − nθ ) − κ A55 x + W,x + I3 ω2 x = 0, r 2 (x) D12 D22 D66 nx,x + 2 nx,x + r (x) θ,x x n sin (α) x − n 2 θ + r (x) r (x) r (x) D66 sin (α) nx − sin (α) θ + r (x) θ,x + r 2 (x) κ A44 + (− cos (α) V + r (x) θ + nW ) + I3 ω2 θ = 0. r (x) + (A.4) (A.5) Similarly, one should interpret the boundary conditions (15) with the aid of variable change (19). 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