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172 ALEKSANDR KORJAKIN ET AL. shells is discussed, for example, in References [2,3,8,15,22,23,33,35,43,44,48]. Though the damping characteristics of laminated composite structures have been studied extensively, studies of sandwich composite structures are limited. The analysis of sandwich beams and constrained layer damping in beams has been performed in Reference [9]. A three-layered beam theory was employed where the continuity of the displacements and the transverse shear stresses is satisfied at the interfaces. The energy method is used to derive the governing equations of motion for transverse vibrations of curved sandwich beams [46]. Flexural vibrations of viscoelastic damped sandwich plates has been analyzed in Reference [17]. A finite element analysis associated with an asymptotic solution method for the harmonic flexural vibrations was proposed. The finite element method was employed for the analysis of the harmonic response of damped three-layer plates in Reference [20]. General non-linear equations of motion of viscoelastic damped sandwich plates and cylindrical panels were derived with the help of the principle of virtual work in Reference [49]. A special finite element approach was developed in Reference [36] to study the vibration and damping characteristics of three-layered conical shells with a viscoelastic core. A finite element based on the discrete layer theory and taking into account the shear deformations was used for analysis of spherical shells with a viscoelastic core in Reference [15]. Damping properties of three-layered shallow spherical shells have been studied in Reference [33]. Expressing the in-plane displacements in terms of auxiliary functions, the general solution of the equations of motion for non-axisymmetric modes was given in terms of Bessel’s functions. Different shell and plate theories can be used to analyze the sandwich structures. Good results can be obtained employing the so called zig-zag theories. In Reference [10] a multilayered plate element is proposed which meets computational requirements and includes both the zig-zag distribution along the thickness coordinate of the in-plane displacements and the interlaminar continuity for the transverse shear stresses. A simple layerwise higher-order zig-zag model is proposed for the bending of laminated composite plates. The model accounts for cubic variation of the transverse shear stresses across the laminate with zero values at the free surface [27]. A beam finite element based on a discrete layer laminated beam theory with the sublaminate first-order zig-zag kinematic assumptions is presented and assessed for thick and thin laminated beams [50]. Authors have used a modified form of DiSciuva’s linear zig-zag laminate kinematics, in which continuity interfacial transverse shear stresses are satisfied identically. A computationally efficient finite-element formulation for linear and non-linear analysis of flat sandwich panels is presented in Reference [11]. Mechanical accuracy is acquired by allowing a zig-zag in-plane displacement field in the thickness direction and by fulfilling interlaminate equilibrium at the interface between the core and skins for the transverse shear stress components. A high accuracy for prediction of stresses can be obtained employing a higher-order zig-zag theory [6].
Free Damped Vibrations of Sandwich Shells of Revolution 173 The vibration and damping analysis of sandwich shells has not been attempted so far and it is a subject of the present investigation. The natural frequencies and the loss factors of cylindrical, conical and spherical shells are calculated. A sandwich shell finite element based on the zig-zag model and described in Reference [12] is used for the dynamic analysis with damping. The energy method (EM) is employed to model the damping. The background of the EM is in finding the energy dissipated in a natural mode by adding up the contributions from the individual components of deformation. Such approach has been used by several authors, for example, in References [1,8,29,51]. The EM has been used for damping analysis of laminated plates [40]. The present paper is an extension of a similar investigation in the case of laminated shells presented in Reference [26]. Governing Equations THEORETICAL BACKGROUND Considering a composite sandwich shell of uniform moderate thickness h with three anisotropic laminae, each of them may be arbitrarily oriented. The influence of the interphases between the laminae is neglected that means rigid connection between the layers. In addition, in the numerical analysis the typical sandwich assumptions (soft core, the thickness of the outer layers is much thinner in comparison with the core thickness) are taken into account. Displacements The kinematic equations for the first-order shear deformation theory (FOSDT) including the drilling rotation of the normal can be obtained from three-dimensional equations of the theory of elasticity using the Taylor series approximation of the vector of position function in the deformed state with respect to the coordinate x 3 (see, for example, References [7,39,41]). Let us introduce curvilinear coordinates x i ={x α ,x 3 } on the midsurface of the shell in the initial state which is defined by the triad {a α , a 3 }. The use of Greek indices means the values 1, 2. The unit vector a 3 denotes the normal to the middle surface of the shell. Curvilinear coordinates X i ={X α ,X 3 } are related to the deformed state which is defined by the triad {A α ,A 3 }. In the deformed state the vector A 3 may not be perpendicular to the midsurface of the shell. The representation of the displacement vector u of an arbitrary point of the shell with respect to the first-order approximation is α α α 3 = + 3 u( x , x ) v( x ) x ( x ) (1) Here v is the displacement vector of the midsurface and denotes the vector of
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