The motion of a rigid spherical inclusion in an elastic medium under the action of plane waves

Two problems on the motion of a rigid spherical inclusion in an elastic medium under the action of a nonstationary, longitudinal, plane, compressional wave and a harmonic shear wave are considered. Using the method of vector eigenfunctions and the integral Laplace transform with respect to time their precise solutions are constructed. Numerical results are given which illustrate the dependence of characteristics of motion of the inclusion on parameters of the falling wave.Translated from Dinamicheskie Sistemy, No. 9, pp. 37–47, 1990.     

On the motion of chaplygin''s sphere on a horizontal plane

The motion of a heavy sphere on a fixed horizontal plane is considered. It is assumed that the centre of mass of the sphere is at its geometric centre, while the principal central moments are different (Chaplygin's sphere). Using the method of averaging, the motion of the sphere is investigated under slip conditions when there is low viscous and also low dry friction. It is shown that when the sphere moves with viscous friction it tends, for the majority of initial data, to rotate about the longest of the axes of the principal central moments of inertia. The motion of the sphere centre tends to become uniform so that the slip velocity approaches zero exponentially. A system of averaged equations, which is fully integrable, is obtained in the case of almost equal moments of inertia, when the friction is dry. The solutions are analyzed.     

The motion of a cylindrical body in a stratified fluid under the action of a radiation force

The free motion of a thin cylindrical body is investigated based on a previously derived expression for the radiation force acting on moving point sources in a stratified fluid. The fundamental equations of motion are derived, the limits of applicability of the approximation used are indicated and the results of calculations of typical trajectories of a body which begins to move with a specified velocity from a position of neutral buoyancy at an angle to the horizon are presented. Calculations of the trajectory of motion of a thin cylindrical body in a stratified fluid when the total radiation force is taken into account show that the effect of the lateral component of this force is considerable and leads not only to quantitative corrections but also to qualitative effects (for example, to an increase in the oscillations of the body and a change in its direction of motion). The results obtained pertain both to the motion of solids in fluids and to the translational motion of vortex dipoles in weakly stratified media.     

The motion of a body supported at three points on a rough plane

The problem of the motion of a heavy rigid body, supported on a rough horizontal plane at three of its points, is considered. The contacts at the support points are assumed to be unilateral and subject to the law of dry (Coulomb) friction. The dynamics of possible motions of such a body under the action of gravity forces and dry friction is investigated. In the case of a plane body, it is possible to obtain particular integrals of the equations of motion.     

The motion of a submerged sphere in a liquid under an ice sheet

The solution of the linear steady problem of the flow of an inviscid, incompressible and infinitely deep liquid around a sphere under an ice sheet, which is modelled by a thin elastic stressed plate of constant thickness is constructed. Special cases of this problem are the motion of a submerged sphere under broken ice, a membrane, and also under the free surface both in the presence and absence of capillary effects. The method of multipole expansions is used in the framework of the linear potential wave theory. The hydrodynamic loads (the wave drag and the buoyancy) acting on the body and also the distribution of the deflections of the ice sheet are calculated as a function of the body velocity, the ice thickness and the value of the compressing or stretching forces. It is shown that all the flow characteristics depend considerably on the ratio of the body velocity and the critical velocity of flexural-gravitational waves.     

Terminal motion of a thin elliptical plate over a horizontal plane with orthotropic friction

The problem on the terminal motion of a thin elliptic plate over a horizontal plane taking into account orthotropic friction forces has been considered. Differential equations of the movement of the plate have been derived. The system of equations has been numerically solved under various initial conditions. It has been shown that sliding and spinning cease simultaneously. It has been found that the limiting behavior of the plate is governed not only by the ratio of the moment of inertia to the coefficient of friction, but also by the orientation of the plate. A comparison of the behaviors of the elliptic and circular plates has been carried out. The results of the calculations can be used to describe the phenomena that occur in a rail–wheel contact in more detail.     

Distributing points uniformly on the unit sphere under a mirror reflection symmetry constraint

Uniformly distributed point sets on the unit sphere with and without symmetry constraints have been found useful in many scientific and engineering applications. Here, a novel variant of the Thomson problem is proposed and formulated as an unconstrained optimization problem. While the goal of the Thomson problem is to find the minimum energy configuration of N electrons constrained on the surface of the unit sphere, this novel variant imposes a new symmetry constraint – mirror reflection symmetry with the x–y plane as the plane of symmetry. Qualitative features of the two-dimensional projection of the optimal configurations are briefly mentioned and compared to the ground-state configurations of the two dimensional system of charged particles laterally confined by a parabolic potential well.     

The interconnection of sliding and rolling in the problem of the motion of a homogeneous sphere on a rough horizontal plane

The motion of a homogeneous sphere on a rough horizontal plane when the angular velocities of the twisting and spinning of the sphere are equal to zero at the initial instant is considered. It is proved that, for any initial conditions, the angular velocity of the rolling of the sphere and the sliding velocity vanish after the same finite time. It is shown that the sliding and rolling are interconnected and, in particular, that the rolling of a sphere without sliding is impossible.     

A fiberwise analogue of the Borsuk-Ulam theorem for sphere bundles over a 2-cell complex

For a vector bundle α, let indα denote the largest integer m for which there exists a Z/2-map from Sm−1 to S(α). We prove that the equality indα=dimα holds for every vector bundle α over the complex Sn−1k_∪en, where n?2 and k≠0, if and only if either k is even and n≠2,3,4,8 or k is odd.     

Stokes flow for the axisymmetric motion of several spherical particles perpendicular to a plane wall

The creeping flow around several spherical particles moving on a line perpendicular to a plane wall is calculated numerically using the boundary integral method. The locations of the point forces on the surfaces of the spheres are chosen so as to describe precisely the lubrication regions when the surfaces are close to one another. Earlier results are recovered for the cases of a single sphere and a wall and of two equal spheres far from a wall. New results are presented for two (equal or unequal) spheres close to a plane wall and several equal spheres far from a wall.     

The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere

In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.     

A fiberwise analogue of the Borsuk-Ulam theorem for sphere bundles over a 2-cell complex II

We describe a finite complex B as I-trivial if there does not exist a Z₂-map from Si−1 to S(α) for any vector bundle α over B and any integer i with i>dimα. We prove that the m-fold suspension of projective plane FP² is I-trivial if and only if m≠0,2,4 for F=C, m≠0,4 for F=H. In the case where F is the Cayley algebra, the m-fold suspension is shown to be I-trivial for every m>0.     

On the motion of a heavy body with a circular base on a horizontal plane and riddles of curling

In this paper we discuss the dynamics of an axisymmetric rigid body whose circular area moves upon a horizontal rough surface. We investigate the interaction between the character of the law of friction and the curvature of the body’s trajectory. For the case of a curling stone it is shown that the observed effects can only be explained using the dependence of the friction coefficient on the Gümbel number. The procedure for constructing the law of friction based on experimental data is developed. It is shown that the available data can only be substantiated by means of anisotropic friction. The simplest model of such friction is constructed which provides quantitative coincidence with the experiment.     

Slow motion of a sphere moving normal to two infinite parallel plane walls in a micropolar fluid

In the present work, we consider the slow steady motion of a rigid sphere moving normal to two parallel plane walls in a micropolar fluid. Non-dimensional variables are introduced. A combined analytical-numerical technique based on the superposition principle and a numerical method, namely the collocation method, is used. The drag force and the wall correction factor are evaluated. Numerical results are obtained and represented graphically.     

Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface

The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on S ² are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.     

The virtual mass of a sphere in a suspension of spherical particles

The problem of the virtual mass of a sphere, moving in an ideal incompressible fluid when there are other identical spherical particles of arbitrary mass present is considered. A solution is constructed for the velocity potential of the fluid in the form of the superposition of perturbation fields, introduced into the flow by each of the particles. The perturbation fields are obtained in the form of functional series, the coefficients of which are mutually consistent by a defined system of equations. An explicit expression is obtained for the hydrodynamic force acting on the sphere in the form of a function of the coordinates of all the particles. A simple analytical dependence of the mean value of the force and the virtual mass of the sphere on the particle-to-fluid density ratio in a first approximation of the volume fraction of the dispersed phase is obtained for a statistically uniform distribution of the dispersed particles in the suspension, using the procedure of averaging over their different possible configurations in space.     

The analytic torsion of a cone over a sphere

We compute the analytic torsion of a cone over a sphere of dimensions 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.