Natural vibrations of a conical orthotropic shell with small curvatures of the generatrix

The Ostrogradskii-Hamilton principle is used with the variational-difference approach to solving eigenvalue problems as the basis for analyzing the influence of small distortions of the generatrix on the natural frequencies of vibration of free conical shells and conical shells precompressed in the axial direction. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 64–68, March 1999.     

Dependence of the deformation of flexible cylindrical shells on their geometry and the orthotropy and nonlinear properties of the composite materials

A study is made of geometrically and physically nonlinear inverse problems concerning the axisymmetric deformation of cylindrical shells into conical shells. Results obtained from the numerical solution of the problems are used to determine the laws of distribution of the surface loads, stresses, strains, and displacements in relation to the initial parameters and nonlinearities of the shells. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 86–91, June, 1999.     

Nonlinear problems of the dynamics of elastic shells partialiy filled with a liquid

Theoretical and experimental investigations of the nonlinear vibrations and dynamic stability of thin shells partially filled with a liquid are reviewed. The paper deals with the basic laws governing the dynamic high-deflection deformation of carrying shell structures and the considerable vibrations of the free liquid surface due to the natural, forced, and parametrically excited vibrations of the combined system and also due to impulse loads acting on the carrying object. The nonlinear dynamic interaction of shells with a liquid filler is analyzed with allowance for the wave motions of the free liquid surface. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 3–34, April, 2000.     

Stability of conical shells with small curvatures of the generatrix

Numerical solutions are used as the basis for a comparative analysis of the qualitative and quantitative effect of small initial curvatures of the generatrices of conical and cylindrical shells on the critical axial pressures of the shells S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 2, pp. 36–40, February, 1999.     

Stability of the tubular layer of a deformed material in a rotating horizontal cylinder

The stability conditions for the steady-state motion of the tubular layer of a treated deformable material in a rotating horizontal cylinder are determined analytically. With allowance for the accepted similarity criteria, universal diagrams of the boundaries of transition of modes of motion of liquid and loose materials in the cylinder are obtained on the basis of experimental data. Analysis of the diagrams shows the identity of the stability conditions for a liquid layer and a loose medium, which can be regarded as a Newtonian liquid upon fast relative motions. It is shown also that the analytical stability conditions for the liquid layer correspond to the experimental data for large Reynolds numbers when the mode hysteresis occurs and do not correspond to these data for small Reynolds numbers when secondary circulating flows form. Rovno State Pedagogical Institute, Rovno 266000, Ukraine. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 120–127, January–February, 2000.     

Stability of multilayered shells of revolution with allowance for geometric nonlinearity of the subcritical state

The finite-difference method and the Trefftz-Reissner variational principle are used to obtain a system of equations in mixed from to describe the stability and geometric nonlinearity of composite shells of revolution. Methods are developed and an algorithm is proposed to calculate the components of the geometrically nonlinear subcritical stress-strain state and to use those components to determine the “upper” critical values for shells with zero Gaussian curvature loaded by uniform external pressure, an axisymmetric load, or a combination of these loads. The stability of cylindrical, conical, and compound shells under uniform pressure is examined for different support conditions. Linear and nonlinear methods of determining the subcritical stress-strain state are compared and their effect on the critical loads is estimated. Ukrainian Transportation Institute and the Ukrainian Academy of Water Management, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 60–66, June, 1999.     

Three-dimensional free vibration analysis of rotating laminated conical shells: layerwise differential quadrature (LW-DQ) method

This paper focuses on the free vibration analysis of thick, rotating laminated composite conical shells with different boundary conditions based on the three-dimensional theory, using the layerwise differential quadrature method (LW-DQM). The equations of motion are derived applying the Hamilton’s principle. In order to accurately account for the thickness effects, the layerwise theory is used to discretize the equations of motion and the related boundary conditions through the thickness of the shells. Then, the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equation applying the DQM in the meridional direction. This study demonstrates the applicability, accuracy, stability and the fast rate of convergence of the present method, for free vibration analyses of rotating thick laminated conical shells. The presented results are compared with those of other shell theories obtained using conventional methods and a special case where the angle of the conical shell approaches zero, that is, a cylindrical shell and excellent agreements are achieved.     

Elastoplastic state of nonshallow flexible ellipsoidal shells under concentric loading

The elastoplastic stress-strain state of flexible ellipsoidal shells under concentric loading is investigated by the solution of nonlinear boundary-value problems. Numerical results are presented for open shells in the case of a free (unsupported) edge and a shell with a central opening (the periphery of the opening is supported by a thin ring). Solutions of the problems in linear and nonlinear statements are analyzed. Numerical results for ellipsoidal and spherical shells are compared. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 4, pp. 54–60, April, 1999.     

Two-dimensional problem of the impact of a vertical wall on a layer of a partially aerated liquid

The two-dimensional unsteady problem of the impact of a vertical wall on a layer of a liquid which is mixed with air near the wall and does not contain air bubbles away from the wall is solved in a linear approximation. The gas-liquid mixture is modeled by a homogeneous, ideal, and weakly compressible medium with a reduced sound velocity dependent on the air concentration in the gas-liquid mixture. Outside the gas-liquid layer, the liquid is considered ideal and incompressible. During the initial stage of the impact, the liquid flow and the hydrodynamic pressure are determined using the linear theory of the potential motion of an inhomogeneous liquid. The dependence of the amplitude of the impact pressure along the wall on the air concentration in the gas-liquid layer and on the thickness of this layer is investigated. For a small relative thickness of the layer, the thin-layer approximation is used. It is shown that the solution of the original problem tends to the approximate solution as the thickness of the layer decreases. It is shown that the presence of the gas-liquid layer leads to wall pressure oscillations. Estimates are obtained for the pressure amplitude and the oscillation period. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 34–46, September–October, 2006.     

Magnetoelastic nonlinear deformation of a conical shell of variable stiffness

A complete system of equations of nonlinear magnetoelasticity of shells is presented. A resolving system of equations of a conical shell of variable thickness is constructed for the axisymmetric case. A numerical example of a flexible shell illustrates the possibility of using the proposed technique to solve this class of problems. T. G. Shevchenko National University, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 34–39, November, 1999.     

On the theory and solution of dynamic problems of the nonstationary behavior of cylindrical shells with discrete variable stiffeners

The formulation and development of a numerical algorithm for solving shell theory problems concerning cylindrical shells with variable-stiffness reinforcements are considered. The Hamilton-Ostrogradskii variational principle is used to obtain the equation of motion. The algorithm is based on finite-difference approximation of the initial variational functional. A numerical example is presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 34, No. 6, pp. 53–59, June, 1999.     

The dynamics of a compressible viscous liquid (review). II

This article is the second part of a review of the dynamics of rigid and elastic bodies in a compressible viscous liquid in a linearized formulation. The following processes are investigated: the forced harmonic vibrations of rigid bodies in moving and resting compressible viscous liquids, the nonstationary motion of rigid bodies in a compressible viscous liquid at rest, the movement of rigid bodies in a resting compressible viscous liquid under the action of radiation forces that are due to the interaction with propagating acoustic harmonic waves, the propagation of harmonic waves in thin-walled cylindrical elastic shells containing a compressible viscous liquid, and the propagation of harmonic waves in hydroelastic systems consisting of a resting compressible viscous liquid and elastic compressible or incompressible bodies with initial stresses. Publications concerning the above problems are analyzed. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 3–30, March, 2000.     

Evaluation of loads critical to stability loss of ribbed conical and spherical shells

A new formula has been proposed for estimation of critical stresses in ribbed conical and spherical shells. Numerical values of the unknown coefficients that enter into the formula have been defined by the supposition that the curve constructed in accordance with the proposed formula has to envelop from below all experimental values of the critical stresses known to the authors. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 47–50, September, 1999.     

On the vibration of laminated nonhomogeneous orthotropic shells

In this paper, an analytical procedure is given to study the free vibration of the laminated homogeneous and non-homogeneous orthotropic conical shells with freely supported edges. The basic relations, the modified Donnell type motion and compatibility equations have been derived for laminated orthotropic truncated conical shells with variable Young’s moduli and densities in the thickness direction of the layers. By applying the Galerkin method, to the basic equations, the expressions for the dimensionless frequency parameter of the laminated homogeneous and non-homogeneous orthotropic truncated conical shells are obtained. The appropriate formulas for the single-layer and laminated complete conical and cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the influences of the non-homogeneity, the number and ordering of layers and the variations of the conical shell characteristics on the dimensionless frequency parameter are investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

One approach to the construction of the basic equations of the theory of nonthin shells with variable thickness

A method for determining the principal curvatures of medium-thickness shells through the vector representation of local bases is proposed based on the parametrization of the base median surface of a shell and its front surfaces. Repeated partial derivatives of vector functions with respect to variable Gaussian coordinates, which is a standard practice in theory of shells when calculating the curvatures as coefficients of the second quadratic form, are not used. Such an approach allows us to accurately account for the geometry of the front surfaces of the shell in determining its stress-strain state. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 58–65, September, 1998.     

Analysis of the stress state of a flattened conical shell of moderate thickness with a structurally weakening hole

Conclusions We determined the relationship between the nature of the stress distribution on the hole surface in a flexible plate as a function of thickness. We observed a great difference between the stress densities in flattened, thin and moderate-thickness conical shells and the stress concentrations near holes in thin cylindrical shells and thin, almost cylindrical, conical shells. The stress distribution near the hole in flattened conical shells of moderate thickness is similar to the stress distribution near the holes in flexible, thick plates. During loading of conical shells by an axial force, the lowest stress concentration factor near the holes is obtained when the axis of the hole is parallel to the shell axis. As the thickness of the shell is increased, the stress concentration factor near the holes increases.Kiev University. Ukrainian Institute of Water Management Engineers, Rovno. Translated from Prikladnaya Mekhanika, Vol. 24, No. 9, pp. 65–70, September, 1988.     

Nonsteady operating conditions of a multimode cylindrical radiator, taking account of processes in the cable channel

A multimode cylindrical piezotransformer submerged in liquid is considered, in the case of excitation by nonsteady pulses traveling through the extended cable channel of the electroded sections (the other section is short-circuited). The accompanying transient process is modeled on the basis of the theory of thin electroelastic shells, the acoustic approximation, and the telegraph equations. Numerical results are given for an electric signal whose profile is specified by the Heaviside function. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 35–43, August, 1999.     

Nonstationary oscillations of three-layer cylindrical shells under axisymmetric loading

The nonstationary behavior of three-layer cylindrical shells under an axisymmetric loading is considered with the application of hypotheses to each layer. Independent postulations are proposed for the approximation of displacements and transverse strains across the thickness of each layer. Reissner's variational principle for dynamic processes is used to derive the motion equations. The problem of the dynamic deformation of three-layer cylindrical shells under a nonstationary loading is considered in the case where the ends of the shells are rigidly fixed. The values obtained were compared with those predicted from hypotheses relating to the whole packet of the structure (the Timoshenko-type theory of multilayered shells). S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 3–9, August, 1999.     

Vibration effects of the dynamic interaction between a gas-liquid medium and elastic shells

The paper presents the results of experimental investigations of regimes of the nonlinear deformation of elastic shells filled with a liquid. These regimes are due to the interaction of local clusters of gas bubbles formed in a vibrating liquid with the vibration modes of an elastic shell excited in the circumferential and longitudinal directions. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 67–73, July, 2000.     

Effect of local loads on the stress-strain state of nonthin orthotropic conical shells

The paper analyzes the stress-strain state of nonthin orthotropic conical shells under local loads. The distribution of deflections and stresses in conical shells under local loads of some types is given as an example Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 103–113, August 2008.