Dynamic stability of cylindrical orthotropic shells with an elastic core under a longitudinal periodic load

The dynamic instability of a cylindrical orthotropic shell with an elastic core subjected to a longitudinal periodic load is considered. Equations are obtained for determining the regions of dynamic instability for different core models.     

Effect of an elastic core on the dynamic stability of an orthotropic cylindrical shell

A method of determining the regions of dynamic instability of an orthotropic cylindrical shell "bonded" to an elastic cylinder is proposed. An expression for the core reaction is obtained from the coupling conditions for the forces normal to the lateral surface and the radial displacements of the shell and the core at the contact surface. When the reaction is substituted in the system of equations of motion of the shell, the part corresponding to the free vibrations of the cylinder is discarded. The system of equations of motion of the shell is reduced to an equation of Mathieu type, from which transcendental equations for determining the boundaries of the regions of dynamic instability are obtained. These regions are analyzed for various modes of loss of stability and different values of the core modulus of elasticity.     

Effect of transverse shear on the stability of an orthotropic cylindrical shell with an elastic core in axial compression

The effect of transverse shear on the stability of a composite shell in axial compression has been investigated. The core is treated as an isotropic elastic cylinder bonded to the inner surface of the shell. The effect of the tangential shearing forces between the shell and the core is taken into account.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 267–273, March–April, 1973.     

Parametric vibrations of cylindrical shells with a viscoelastic core

The method of calculating the axisymmetric and nonaxisymmetric parametric vibrations of a cylindrical shell bonded to an elastic core [2] is extended to the case of hollow and solid viscoelastic cores by substituting for the material moduli in the equations of motion of the core integral operators with kernels in the form of an exponential and a sum of exponentials. Expressions are given for the reaction of the viscoelastic core, together with the equation of the boundaries of the spectrum of principal regions of dynamic instability. The effect of relaxation time and the long-term modulus of elasticity of the core on the shape and location of the regions of dynamic instability is analyzed.     

Dynamic stability of an elastic orthotropic cylindrical shell with allowance for transverse shears

The resolvents for the dynamic stability of an elastic orthotropic cylindrical shell are obtained in accordance with the Ambartsumyan and Timoshenko-type refined theories. The regions of instability given by the classical and refined theories are compared. The dependence of the refinements on the shell parameters, the shear moduli of the material, and the buckling modes are investigated.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 312–320, March–April, 1973.     

Stability of composite shells with an elastic core

The stability in axial compression of orthotropic cylindrical shells with an elastic core is investigated with allowance for the following factors: transverse shear strains in the shell material, precritical state of stress of the core, and the moments due to the surface forces exerted on the shell by the core. The numerical results obtained are compared with the results of approximate methods in which the above-mentioned factors are disregarded.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 3, pp. 544–547, May–June, 1976.     

Stability of shear-compliant cylindrical shells connected with an elastic foundation in a geometrically nonlinear formulation

The effect of transverse shear strains on the stability "in the large" of a cylindrical transversal-isotropic shell with an elastic filler under the effect of axial compression is investigated. The length of the cylindrical shell is assumed to be greater than the diameter. The approximate solution is obtained by the Bubnov-Galerkin method. Such a problem for an isotropic shell was considered earlier in [2, 4] on the basis of the equations of the classical Kirchhoff-Love theory.L'vov Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR. Ternopol Branch, L'vov Polytechnic Institute. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 113–117, January–February, 1972.     

夹层圆柱壳在移动内压作用下的临界速度研究

基于夹层壳理论和三维弹性动力学理论,研究了无限长夹层圆柱壳在移动内压作用下的临界速度．首先,基于夹层壳理论,考虑夹芯的压缩和剪切变形以及面板的剪切变形,研究了轴对称简谐波在无限长夹层圆柱壳中的传播问题;其次,基于三维弹性动力学理论,将位移变量用Legendre正交多项式系表示,同时引入位置相关函数,将求解导波问题化为简单的特征值问题．利用这两种方法得到了最低模态的频散曲线,最小相速便是内压移动的临界速度．最后,用算例和数值模拟来验证方法的有效性．结果表明,两种理论得到临界速度吻合得较好;当波数较小时,两种理论得到的频散曲线吻合得很好,当k→∞时,夹层壳理论和弹性动力学理论得到的极限相速分别趋于面板和夹芯的剪切波波速．波数较小时,两种理论分析夹层圆柱壳的导波问题是有效的．数值模拟预测的临界速度与理论分析的结果吻合得很好．     

The effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk

It is well known that either the asymmetric disk or transverse crack brings parametric inertia (or stiffness) excitation to the rotor-bearing system. When both of them appear in a rotor system, the parametric instability behaviors have not gained sufficient attentions. Thus, the effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk is studied. First, the finite element equations of motion are established for the asymmetric rotor system. Both the open and breathing transverse cracks are taken into account in the model. Then, the discrete state transition matrix (DSTM) method is introduced for numerically acquiring the instability regions. Based upon these, some computations for a practical asymmetric rotor system with open or breathing transverse crack are conducted, respectively. Variations of the primary and combination instability regions induced by the asymmetric disk with the crack depth are observed, and the effect of the orientation angle between the crack and asymmetric disk on various instability regions are discussed in detail. It is shown that for the asymmetric angle around 0, the existence of transverse (either open or breathing) crack has attenuation effect upon the instability regions. Under certain crack depth, the instability regions could be vanished by the transverse crack. When the asymmetric angle is around π/2, increasing the crack depth would enhance the instability regions.     

Stress and strain-inertia gradient elasticity in free vibration analysis of single walled carbon nanotubes with first order shear deformation shell theory

A gradient-enriched shell formulation is introduced in the present study based on the first order shear deformation shell model and the stress gradient and strain-inertia gradient elasticity theories are used for dynamic analysis of single walled carbon nanotubes. It provides extensions of the first order shear deformation shell formulation with additional higher-order spatial derivatives of strains and stresses. The higher-order terms are introduced in the formulation by using the Laplacian of the corresponding lower-order terms. The proposed shell formulation includes two length scale size parameters related to the strain gradients and inertia gradients. The effects of the transverse shear, aspect ratio, circumferential and half-axial wave numbers and length scale parameters on different vibration modes of the single-walled carbon nanotubes are elucidated. The results are also compared with those obtained from a classical shell theory with Sanders–Koiter strain-displacement relationships.     

Parametric resonance of a FG cylindrical thin shell with periodic rotating angular speeds in thermal environment

Parametric resonance of a functionally graded (FG) cylindrical thin shell with periodic rotating angular speeds subjected to thermal environment is studied in this paper. Taking account of the temperature-dependent properties of the shell, the dynamic equations of a rotating FG cylindrical thin shell based upon Love's thin shell theory are built by Hamilton's principle. The multiple scales method is utilized to obtain the instability boundaries of the problem with the consideration of time-varying rotating angular speeds. It is shown that only the combination instability regions exist for a rotating FG cylindrical thin shell. Moreover, some numerical examples are employed to systematically analyze the effects of constant rotating angular speed, material heterogeneity and thermal effects on vibration characteristics, instability regions and critical rotating speeds of the shell. Of great interest in the process is the combined effect of constant rotating angular speed and temperature on instability regions.     

Natural and parametric vibrations of a cylindrical shell-filler system with tangential interaction taken into account

Conclusions 1. The spectrum of the eigenfrequencies and dynamical instability regions of a shell-filler system breaks up into two infinite spectra for each mode of wave formation; the first of them is determined by the shear modulus of the filler material, and the second — by the volume deformation modulus. The second spectrum is absent for incompressible fillers.2. It has been shown that taking tangential interactions into account has a strong effect on the arrangement and width of the dynamical instability regions belonging to the first spectrum and hardly changes the arrangement and widths of the regions of the second spectrum.3. As a result of the limiting transition as the frequency of the driving force tends to zero, expressions are obtained from the formulas of this paper for calculating the static stability of a shell with an incompressible filler. The numerical results obtained for this case correspond to results given in the literature.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. P. Stuchki Latvian State University, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 503–509, May–June, 1977.     

Stability of transversally isotropic shells coupled with an elastic foundation

The stability of shells coupled with an elastic Winkler foundation is investigated. It is assumed that the shell is made of a material (glass-reinforced plastic) with low resistance to shear, as a result of which generalized theories that take transverse shear strains into account [1–4] must be used in the stability calculations. The solution obtained is compared with the corresponding solution obtained on the basis of the classical Kirchhoff-Love theory [8].Lvov Polytechnic Institute. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 669–673, July–August, 1969.     

Dynamic stability of a viscoelastic orthotropic cylindrical shell

A method based on the use of Laplace transforms has been developed for reducing the system of equations of motion of a viscoelastic orthotropic cylindrical shell to a single integro-differential equation. The effect of the viscous components on the regions of dynamic instability is investigated (creep due to the action of the shear stresses is taken into account).Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 714–721, July–August, 1973.     

Identifying the elastic moduli of composite plates by using high-order theories. 2. Theoretical-experimental approach

The identification of elastic properties of laminated composite plates from measured eigenfrequencies has been performed. The elastic moduli of the laminates were determined by using a multilevel modeling and a two-step identification procedure. At the first step, based on a genetic algorithm, the Young’s and shear moduli were found, but at the second one, by minimizing an error function, the values of transverse moduli were refined. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 207–216, March–April, 2008.     

Nonlinear flexural vibrations of composite shallow open shells

This paper addresses geometrically nonlinear flexural vibrations of open doubly curved shallow shells composed of three thick isotropic layers. The layers are perfectly bonded, and thickness and linear elastic properties of the outer layers are symmetrically arranged with respect to the middle surface. The outer layers and the central layer may exhibit extremely different elastic moduli with a common Poisson's ratio ν. The considered shell structures of polygonal planform are hard hinged supported with the edges fully restraint against displacements in any direction. The kinematic field equations are formulated by layerwise application of a first order shear deformation theory. A modification of Berger's theory is employed to model the nonlinear characteristics of the structural response. The continuity of the transverse shear stress across the interfaces is specified according to Hooke's law, and subsequently the equations of motion of this higher order problem can be derived in analogy to a homogeneous single-layer shear deformable shallow shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear response of shear deformable FGM curved panels resting on elastic foundations and subjected to mechanical and thermal loading conditions

This paper presents an analytical investigation on the nonlinear response of thick functionally graded doubly curved shallow panels resting on elastic foundations and subjected to some conditions of mechanical, thermal, and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation shell theory taking into account geometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, explicit relations of load-deflection curves for simply supported curved panels are determined. Effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the panels are analyzed and discussed. The novelty of this study results from accounting for higher order transverse shear deformation and panel-foundation interaction in analyzing nonlinear stability of thick functionally graded cylindrical and spherical panels.     

Numerical stability for a dynamic Naghdi shell

We consider an elastic model for a shell incorporating shear, membrane, bending and dynamic effects. We make use of the theory proposed by Arnold and Brezzi [1] based on a locking free non-standard mixed variational formulation. This method is implemented in terms of the displacement and rotation variables as the minimization of an altered energy functional. We extend this theory to the shell vibrations problem and establish optimal error estimates independent of the thickness, thereby proving that shear and membrane locking is avoided. We study the numerical stability both in static and dynamic regimes. The approximation schemes are tested on specific examples and the numerical results confirm the estimates obtained from theory.     

Axisymmetric buckling of laminated thick annular spherical cap

Axisymmetric buckling analysis is presented for moderately thick laminated shallow annular spherical cap under transverse load. Buckling under central ring load and uniformly distributed transverse load, applied statically or as a step function load is considered. The central circular opening is either free or plugged by a rigid central mass or reinforced by a rigid ring. Annular spherical caps have been analysed for clamped and simple supports with movable and immovable inplane edge conditions. The governing equations of the Marguerre-type, first order shear deformation shallow shell theory (FSDT), formulated in terms of transverse deflection w, the rotation ψ of the normal to the midsurface and the stress function Φ, are solved by the orthogonal point collocation method. Typical numerical results for static and dynamic buckling loads for FSDT are compared with the classical lamination theory and the dependence of the effect of the shear deformation on the thickness parameter for various boundary conditions is investigated.     

Shear buckling modes of cylindrical sandwich shells under external pressure, tension-compression of bearing layers with unequal forces, and inhomogeneous across-the-thickness temperature

Equations are set up for describing, in a correct statement and with an accuracy sufficient in actual practice, the shear buckling modes (BMs) of cylindrical sandwich shells with a transversely soft core of arbitrary thickness. Based on them, solutions are obtained to a number of problems on the buckling instability according to shear modes under some force and thermal loadings. It is found that the BMs occur in the shell along the circumferential and axial directions if, in the precritical state, a normal compressive stress arises in the transverse direction. It is shown that this condition is fulfilled in the following cases: in axial tension of the shell with unequal forces applied to the end faces of bearing layers (the parameter of critical load is maximum if the tensile forces are equal); under external (internal) pressure; on cooling the outer and heating the inner layers. The results obtained are presented in the form of simple analytical formulas for determining the corresponding critical parameters of the force and thermal actions.Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 37–48, January–February, 2005.