Vibrations of a viscoelastic cylinder with an elastic shell

The nonlinear vibrations of a viscoelastic cylinder with an elastic shell subjected to two harmonic forces are investigated using the averaging scheme described in [3, 4]. Nonresonance, resonance, and subharmonic vibrations are examined. It is shown that the presence of viscosity in the system leads to a single stationary equilibrium position for which the stability conditions are given.V. I. Lenin Tashkent State University. Translated from Mekhanika Polimerov, No. 4, pp. 691–697, July–August, 1973.     

Geometrically nonlinear equations of motion of an elastic sandwich shell

A variational function of the dynamics of a thin shallow composite shell of regular structure is derived on the basis of a mixed variational principle of geometrically nonlinear elasticity theory. The shell consists of alternating elastic bonding layers and binder layers. The condition of stationarity of the functional permits obtaining the fundamental dynamics equations for a sandwich shell.     

Deformation of a glass-reinforced plastic cylindrical shell encased in a thick-walled elastic cylinder

A solution is given for the axisymmetric problem of the deformation of an orthotropic glass-reinforced plastic shell of finite length encased in a soft transversely isotropic cylinder under internal pressure.Moscow Institute of Heat Engineering. Translated from Mekhanika Polimerov, No. 1, pp. 164–168, January–February, 1973.     

Dent patterns on the surface of a longitudinally compressed,non-linear elastic cylindrical shell

A novel version of reductive perturbation theory is proposed for analysing the dynamics of bends in a longitudinally compressed, non-linear elastic cylindrical shell near the stability threshold given by the linear theory. Soliton-like annular folds and patterns of diamond-shaped dents on the shell surface are predicted and analytically described. Similar formations, which are both stress concentrators and precursors of plastic flow of the material, contain information on the precritical stress state of the shell. It is shown that a shell with dents supports an external load, which is tens of percent less than the upper critical load in the linear theory of shells. The conditions for the formation of and explicit expressions for solitary waves that propagate along the generatrix of the shell on a background of arrays of folds and dents are found.     

Axisymmetric interaction of an acoustic wave with a thin elastic spherical shell stiffened by an elastic rib

In nonstationary formulation in the context of the classical theories of shells and curvilinear rods we solve the axisymmetric problem of scattering of sound on a spherical shell stiffened by a rib. We analyze the effect of the rib on the spectral density of the echo-signal. We give the results of computing the reflected pulse and classify the sequence of pulses of peripheral waves in the echo-signal. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 142–147.     

The Reissner-Sagoci type problem for a non-homogeneous elastic cylinder embedded in an elastic non-homogeneous half-space

The problem considered is that of the torsion of a non-homogeneouselastic cylinder, which is embedded in a non-homogeneous elastichalf-space (matrix) of different rigidity modulus. A rigid discis bonded to the flat surface of the cylinder and torque isapplied to the cylinder through a rigid disc. It is assumedthat there is perfect bonding at the common cylindrical surface.Using integral transformation techniques the solution of theproblem is reduced to dual integral equations. Later on thesolution of the dual integral equations is transformed intothe solution of a Fredholm integral equation of the second kind.Solving the Fredholm integral equation numerically the numericalresults for torque and shear stress inside the cylinder areobtained and displayed graphically to demonstrate the effectof non-homogeneity of the elastic material on the torque andshear stress.     

Scattering of an incident acoustic wave by an infinite elastic cylinder in a shallow sea

The scattering of sound waves by an isotropic elastic circular cylinder of infinite extent in a water of finite depth is investigated taking into account the shear waves that can exist in addition to compressional waves in scatterers of solid material. The axis of the cylinder is parallel to the water level. The reflected and transmitted energies are plotted for the various values of the radius of the cylinder, and the farfield scattered pressure is obtained for various depth and presented in a graphical form. Copyright © 2008 John Wiley & Sons, Ltd.     

The scattering of acoustic waves by a cylinder with a non-uniform elastic coating

The problem of the scattering of a plane acoustic wave by a solid cylinder with a radially non-uniform elastic coating is considered. An analytical expression describing the scattered acoustic field is obtained. The equations of motion of the non-uniform elastic cylindrical layer are reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by the power-series method. The results of calculations of the directional pattern of the scattered field are presented.     

Diffraction of impulsive elastic waves by a fluid cylinder

We consider the diffraction of impulsive SV waves by a fluid circular cylinder. The cylinder is embedded in an unbounded isotropic homogeneous elastic medium and it is filled with some acoustic fluid. The line source, generating the incident pulse, is situated outside the cylinder parallel to its axis. We investigate the problem by the method of dual integral transformation as developed by Friedlander. The resulting integrals are evaluated approximately to obtain the short-time estimate of the motion near the wave front in the shadow zone of the elastic medium. We also interpret the approximate solution in terms of Keller’s geometrical theory of diffraction.     

Edge resonance in an elastic semi-infinite cylinder

We study the three-dimensional elasticity operator in a semi-infinite circular cylinder subject to free boundary conditions, in the case of zero Poisson ratio. We prove, adapting the method from [15], i.e., by first finding an invariant subspace for the elasticity operator such that the essential spectrum has a strictly positive lower bound and then finding a test function in this space for which the variational quotient takes a value below the bottom of the essential spectrum, that there is an eigenvalue embedded in the continuous spectrum. Physically, an eigenvalue corresponds to a "trapped mode", that is, a harmonic oscillation localized near the edge. This effect, known in mechanics as the "edge resonance" has been extensively studied numerically and experimentally. Our paper extends the mathematical justification of such phenomena provided by [15] to a three-dimensional setting     

On steady motion of a drop in an infinite liquid medium

A rigorous proof is given of solvability of the free boundary problem for the Navier-Stokes equations governing a steady fall (or uprising) of a drop in an infinite liquid medium. It is assumed that the densities of both liquids are close. The solution is obtained as a perturbation of the rest state. However, in comparison with other problems of this type, the proof requires much more delicate arguments since the Frechet derivative of the corresponding operator computed at the rest state is not invertible. Bibliography: 8 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 233–254.     

The stability of generalized steady motion

The influence of dissipative forces on the stability of the generalized steady motion of a mechanical system with time-dependent constraints is investigated. As a preliminary, the problem of the limiting behaviour of solutions of a non-autonomous system is solved on the assumption that m first integrals and a function that decreases along every solution of the system are known. As an example, the motion of a gyroscope in gimbals.     

Elastic buckling of an axially compressed open circular cylindrical shell

The paper is devoted to stability problem of an axially compressed open elastic circular cylindrical shell. Two curvilinear edges of the open cylindrical shell are simply supported while two straight edges are free. Two load cases of the shell are assumed. The first load case ‐ was the invariable axially normal force intensity, and the second load case ‐ the linearly varying axially normal force intensity. Critical loads of the shell for both load cases are determined. These critical loads are compared with a classical load for closed circular cylindrical shell. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress state of an isotropic shell with an elastic inclusion

Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 76–80, 1990.     

Flexure of a micropolar elastic circular cylinder

Flexure problem in the linear theory of micropolar elasticity is solved for a circular cylinder by Iesan’s semi-inverse method. Expressions for the macrodisplacement vector and microrotation vector are obtained.     

Electromagnetic interaction of a conducting cylinder with a magnetic dipole caused by steady rotation

The motion of a conductor in a magnetic field induces eddy currents whose interaction with the field produces Lorentz forces opposing the motion. One can determine the velocity of the conductor from the force on the magnet system since the latter is equal but opposite to the Lorentz force on the conductor. This contactless method is known as Lorentz force velocimetry (LFV). We study an idealized configuration of LFV, i.e. a rotating solid cylinder interacting with a point dipole. The understanding of parameter influences in this setup can be helpful for more realistic configurations. We use a purely kinematic approach appropriate for low magnetic Reynolds numbers. Numerical results for small and large distances between dipole and cylinder have been obtained with the commercial software COMSOL Multiphysics. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Torsional stability of a glass-reinforced plastic cylindrical shell with an elastic core

The problem of the stability of a glass-reinforced plastic cylindrical shell with an elastic core subjected to twisting moments applied to the edges of the shell is considered. As in various other studies [4–6], the glass-reinforced plastic is treated as an elastically orthotropic material. The core is treated as an isotropic elastic cylinder, whose outer surface is bonded to the shell. Expressions for the critical stresses are obtained for an infinitely long shell and a shell of finite length.Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1082–1086, November–December, 1970.