Stability of Axially Compressed Noncircular Cylindrical Shells Consisting of Panels of Constant Curvature

The stability problem is solved for an axially compressed cylindrical shell. Its cross section is formed by circular arcs of radius r with ends supported on a closed circle of radius R. The solution is based on the Flügge equations of the classic theory of deep cylindrical shells. It is shown that the critical axial load for shells of medium length and appropriately chosen cross-sectional profile can be increased by a factor of R/r approximately, compared with the circular shell. The shells length affects considerably the efficiency of noncircular shells of this type. This design model allows us to find out how the local properties of the shell and its stiffness are related     

The Problem of Forced Nonlinear Vibrations of Cylindrical Shells Completely Filled with Liquid

The forced nonlinear vibrations of a thin cylindrical shell completely filled with a liquid are studied. A refined mathematical model is used. The model takes into account the nonlinear terms up to the fifth power of the generalized displacement of the shell. The Bogolyubov’Mitropolsky averaging method is used to plot amplitude’frequency response curves for steady-state vibrations. The steady-state vibrations at the frequency of principal harmonic resonance are analyzed for stability__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 52–59, February 2005.     

A closed-form solution procedure for circular cylindrical shell vibrations

A systematic procedure for obtaining the closed-form eigensolution for thin circular cylindrical shell vibrations is presented, which utilizes the computational power of existing commercial software packages. For cylindrical shells, the longitudinal, radial, and circumferential displacements are all coupled with each other due to Poissons ratio and the curvature of the shell. For beam and plate vibrations, the eigensolution can often be found without knowledge of absolute dimensions or material properties. For cylindrical shell vibrations, however, one must know the relative ratios between shell radius, length, and thickness, as well as Poissons ratio of the material. The mode shapes and natural frequencies can be determined analytically to within numerically determined coefficients for a wide variety of boundary conditions, including elastic and rigid ring stiffeners at the boundaries. Excellent agreement is obtained when the computed natural frequencies are compared with known experimental results.     

Nonlinear azimuthal waves in a centrifuge

Azimuthal wave motions in a liquid which partially fills a cylinder (centrifuge) rapidly rotating about a horizontal axis are discussed in this paper. Under the action of centrifugal force the liquid is pressed to the wall of the cylinder and moves together with it about the central air core. The vibrations of the free surface which arise are called centrifugal waves [1]. The difficulties of their theoretical investigation are related to the nonlinearity both of the basic equations and also of the boundary condition for the pressure on the free surface; therefore they have previously been studied only by linear methods [1, 2]. Nonlinear azimuthal waves in a centrifuge with an infinite radius of the rotating cylinder are analytically described below. The waves found are an analog of Gerstner trochoidal waves on a cylindrical surface. An approximate solution for a centrifuge with a finite outer radius is constructed by matching the waves obtained to the known linear ones.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 86–89, May–June, 1984.In conclusion the author expresses his gratitude to E. I. Yakubovich for useful discussion.     

Influence of general rotation on the transfer of torsional vibrations through axisymmetric layers of viscous liquid

A study is made of the motion of a viscous incompressible liquid in a gap between a cylinder of finite length and a jacket that contains it. The motion is due to small torsional vibrations of the jacket, whereas the complete system rotates uniformly around the common symmetry axis. The equation is linearized under the assumption that the Rossby number is small. Three problems are considered: the one-dimensional problem of vibrational motion in the cylindrical gap, the self-similar problem for the flow in the end interdisk gaps, and the two-dimensional problem that describes the flow in the corner region. It is established that the superposition of the general rotation makes the damping properties of the liquid layers between the jacket and the ends of the cylinder much less good. The influence of boundary effects is clarified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 26–31, May–June, 1991.     

Electrization of dielectric liquids flowing in tubes

The methods of the mechanics of continuous media [1] are used to consider the problem of electrization of dielectric liquids flowing in tubes [2–6]. According to modern ideas [2–6], there is always dissolved in such liquids a slight admixture of an electrolyte, whose molecules in such a dilute solution dissociate to a certain extent into positively and negatively charged ions. On the walls, oxidizing and reducing reactions take place, as a result of which the negative and positive ions, respectively, give up to the wall surplus electrons or take missing electrons from it. Thus, a positive (respectively, negative) total electric charge is induced in the liquid by the flow. We consider in this paper the electrization of a dielectric liquid in laminar flow in a circular cylindrical tube. We find the distribution of the electric charge in the liquid, the maximal electric current, and the dependence of the length over which the distribution of the electric charge in the tube is established on the tube radius, the Debye radius of the liquid, and the Péclet diffusion number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 41–47, November–December, 1979.We thank V. V. Gogosov for helpful comments made in a discussion of thwe work.     

Maximum stability formulas for reinforced cylindrical shells under external pressure

The problem of determining the reinforcement structure that maximizes the stability of a cylindrical shell subjected to external pressure was formulated in [1], where a numerical solution was obtained for a particular class of structures on the basis of a formula for the stability limit of a hinged anisotropic circular cylindrical shell of medium length in the membrane state. In the present article the stability limit is determined more accurately, without any constraint on the length of the shell, and the optimization is carried out over a broader class of structures.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 5, pp. 159–167, September–October, 1977.     

Free vibrations of two capillary liquids rotating in a cylindrical vessel

The problem of the small natural vibrations of two coaxially disposed ideal liquids rotating in a cylindrical vessel under conditions of complete weightlessness is considered. The set of normal vibrations of the system consists of internal wave motions and surface waves. Asymptotic formulas are derived for the vibration frequencies of the surface waves. The results of computer calculations are presented in the form of graphs and tables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 97–104, September–October, 1976.     

Nonlinear Vibrations of a Cylindrical Shell Containing a Flowing Fluid

The Bogolyubov-Mitropolsky method is used to find approximate periodic solutions to the system of nonlinear equations that describes the large-amplitude vibrations of cylindrical shells interacting with a fluid flow. Three quantitatively different cases are studied: (i) the shell is subject to hydrodynamic pressure and external periodical loading, (ii) the shell executes parametric vibrations due to the pulsation of the fluid velocity, and (iii) the shell experiences both forced and parametric vibrations. For each of these cases, the first-order amplitude-frequency characteristic is derived and stability criteria for stationary vibrations are established__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 75–84, April 2005.     

Electroviscous effects in steady fully developed flow of a power-law liquid through a cylindrical microchannel

Electroviscous effects in steady, fully developed, pressure-driven flow of power-law liquids through a uniform cylindrical microchannel have been investigated numerically by solving the Poisson–Boltzmann and the momentum equations using a finite difference method. The pipe wall is considered to have uniform surface charge density and the liquid is assumed to be a symmetric 1:1 electrolyte solution. Electroviscous resistance reduces the velocity adjacent to the wall, relative to the velocity on the axis. The effect is shown to be greater when the liquid is shear-thinning, and less when it is shear-thickening, than it is for Newtonian flow. For overlapping electrical double layers and elevated surface charge density, the electroviscous reduction in the near-wall velocity can form an almost stationary (zero shear) layer there when the liquid is shear-thinning. In that case, the liquid behaves approximately as if it is flowing through a channel of reduced diameter. The induced axial electrical field shows only a weak dependence on the power-law index with the dependence being greatest for shear-thinning liquids. This field exhibits a local maximum as surface charge density increases from zero, even though the corresponding electrokinetic resistance increases monotonically. The magnitude of the electroviscous effect on the apparent viscosity, as measured by the ratio of the apparent and physical consistency indices, decreases monotonically as the power-law index increases. Thus, overall, the electroviscous effect is stronger in shear-thinning, and weaker in shear-thickening liquids, than it is when the liquid is Newtonian.     

Vibrations and Dissipative Heating of a Cylindrical Panel under a Periodic Moving Load

An analytic solution is obtained to describe the vibrations and dissipative heating of a simply supported infinite cylindrical panel under periodic normal loads moving over its surface with a constant velocity. Special attention is focused on resonant vibrations, which result in the most intensive dissipative heating. It is additionally assumed that the material of the panel is viscoelastic, its properties are independent of temperature, and Poisson’s ratio is real. The influence of thickness, radius of curvature, load velocity, and viscoelastic properties on the thermal state of the panel is analysed against the thermal state of the plate__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 100–109, April 2005.     

Free torsional vibrations of an elastic cylinder with laminated periodic structure

The theory of torsional vibrations of a circular cylinder, with a periodic variation of elastic constants and density normal to the axis of the cylinder, is developed in terms of Floquet waves. Floquet waves are quasi-periodic waves, whose amplitude profile has the same periodicity as that of the material and repeats with the periodicity of the cell. Using Floquet's theory, the dispersion spectrum is obtained for time-harmonic waves propagating in a laminated cylinder with periodic structure. It is shown that the dispersion spectrum has a band structure, consisting of passing bands and stopping bands. Motion in the case of grazing incidence, and motion at the end of the zones is discussed. It is also shown that as the radius of the cylinder tends to infinity, the torsional waves in a circular cylinder degenerate to SH-waves in laminated plates.     

Wave Deformation Modes of Fluid-Containing Cylindrical Shells Under Periodic Force

The forced nonlinear vibrations of a cylindrical shell fully filled with a fluid in instability zones of single-mode deformation are analyzed. It is shown that the deformation mode of the shell in these zones is traveling bending waves propagating in the circumference direction. The effect of the fluid on the characteristics of these waves is studied__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 5, pp. 76–82, May 2005.     

Oscillating flows in capillaries filled with immiscible liquids

Hydrodynamic flows generated by mechanical vibrations of a capillary filled with immiscible liquids are investigated. Air bubbles are contained at the hermetically sealed ends of the capillary. Equations for the change in the volumes of the air bubbles as functions of time and velocity distribution in the liquids are obtained for the case when the radius of the capillary is much less than the lengths of the liquid columns. Results of numerical calculations are given for a capillary filled with two liquids: water and mercury. Amplitude-frequency dependences of the change in volumes of the air bubbles are constructed which have a resonance nature. Graphs of the dependence of the velocity of the water and the mercury on the radial coordinate at different times are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 13–18, September–October, 1987.     

Shape of an axisymmetric liquid film on the surface of a rotating cylinder

The steady axisymmetric motion of a viscous film together with a cylinder is investigated. The shape of an axisymmetric film of constant mass depends not only on the physical properties of the liquid, the rate of rotation and the radius of the cylinder but also on the pressure difference between the liquid and the ambient medium. By calculation and by means of a qualitative investigation of the first integral of the basic equation it is shown that for different values of the parameters the free surface of the film may be cylindrical or wavy, intersect itself or consist of periodically distributed isolated annular layers. The calculation results correspond with the experimental data the more closely the thinner the film and the greater its transverse velocity. This is attributable to the absence of gravitational acceleration from the model of the steady axisymmetric motion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 23–27, July–August, 1989.     

Optimization of cylindrical shells with compliant cores

Thin-walled, cylindrical structures are found extensively in both engineering components and in nature. The weight to load bearing ratio is a critical element of design of such structures in a variety of engineering applications, including space shuttle fuel tanks, aircraft fuselages, and offshore oil platforms. In nature, thin-walled cylindrical structures are often supported by a honeycomb- or foam-like cellular core, as for example, in plant stems, porcupine quills, or hedgehog spines. Previous studies have suggested that a compliant core increases the buckling resistance of a cylindrical shell over that of a hollow cylinder of the same weight. In this paper, we extend the linear-elastic buckling theory by coupling it with basic plasticity theory to provide a more comprehensive analysis of isotropic, cylindrical shells with compliant cores. We examine the optimal design of a thin-walled cylinder with a compliant core, of given radius and specified materials, for a prescribed load bearing capacity in axial compression. The analysis gives the values of the shell thickness, the core thickness, and the core density that maximize the load bearing capacity of the shell with a compliant core over an equivalent weight hollow shell. The analysis also identifies the optimum ratio of the core modulus to the shell modulus and is supported by a Lagrangian optimization technique. The analysis further discusses the selection of materials in the design of a cylinder with a compliant core, identifying the most suitable material combinations. The performance of a cylinder with a compliant core is compared with competing designs (optimized hat-stiffened shell and optimized sandwich-wall shell). Finally, the challenges associated with achieving the optimal design in practice are discussed, and the potential for practical implementation is explored.     

Bifurcation Analyses in the Parametrically Excited Vibrations of Cylindrical Panels

We consider parametrically excited vibrations of shallow cylindrical panels. The governing system of two coupled nonlinear partial differential equations is discretized by using the Bubnov–Galerkin method. The computations are simplified significantly by the application of computer algebra, and as a result low dimensional models of shell vibrations are readily obtained. After applying numerical continuation techniques and ideas from dynamical systems theory, complete bifurcation diagrams are constructed. Our principal aim is to investigate the interaction between different modes of shell vibrations under parametric excitation. Results for system models with four of the lowest modes are reported. We essentially investigate periodic solutions, their stability and bifurcations within the range of excitation frequency that corresponds to the parametric resonances at the lowest mode of vibration.     

Thermocapillary instability of the interface between two immiscible liquids in the presence of volume heat sources

The problem of the stability of the interface between two infinite layers of different immiscible liquids is considered. It is assumed that within the liquid a distributed volume heat source, simulating Joule heating, is given. The stability of the rest state with respect to small unsteady disturbances is investigated. The investigation is carried out using the real boundary conditions at the interface between the two liquids rather than the model boundary conditions usually employed in such problems [5]. The problem considered is related to the practical question of the stability of electrolyzer processes. In the present case a possible threshold mechanism of development of oscillations of the electrolyte-aluminum interface is examined. A numerical example with liquid parameters that coincide with those of the electrolyte and aluminum shows that the thermocapillary instability mechanism can, in fact, be the source of surface waves at the electrolyte-aluminum interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 156–160, September–October, 1990.     

Numerical simulation of shock wave interaction with a water column

The paper describes a scheme which is based on the CIP scheme and modified to properly describe a gas/liquid interface without smearing the density jump across the interface. This was achieved by calculating the density separately for each phase. The density at each grid point was determined by using a density function in a similar fashion as CIP. As a result a sharp density gradient was obtainable throughout the flow field and the scheme could handle properly gas/liquid interfaces having a large density ratio. Shock wave interaction with a cylindrical water column was simulated. The numerical results were compared with appropriate interferograms. Good agreement was found between the two. The results obtained for the cylindrical water column were compared with a similar solid cylinder case. The comparison reveled that even 40 s after shock impingement some differences were found between a liquid column and the solid cylinder. Received 11 July 2000 / Accepted 28 March 2001     

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.