A. A. Zobova

Ball on a Viscoelastic Plane

We consider dynamical problems arising in connection with the interaction of an absolutely rigid ball and a viscoelastic support plane. The support is a relatively stiff viscoelastic Kelvin-Voigt medium that coincides with the horizontal plane in the undeformed state. We also assume that under the deformation the support induces dry friction forces that are locally governed by the Coulomb law. We study the impact appearing when a ball falls on the plane. Another problem of our interest is the motion of a ball “along the plane.” A detailed analysis of various stages of the motion is presented. We also compare this model with classical models of interaction of solid bodies.

Various friction models in two-sphere top dynamics

The dynamics of a two-sphere tippe top on a rough horizontal plane is considered. The top is bounded by a nonconvex surface consisting of two spherical segments of distinct radii and a cylinder; the cylinder axis coincides with the common symmetry axis of the segments. If the top is initially placed so that its center of mass is almost in the lowest position, the symmetry axis is almost vertical, and a high angular velocity of rotation about the vertical symmetry axis is imparted to the top, then it turns upside down from its base to the leg and starts to rotate on the leg. Then the top gradually returns to the stable equilibrium. The problem of the top motion is often used to demonstrate the efficiency of various proposed friction models [1–3].The effects arising in the two-sphere top dynamics with various dry friction models are compared in the present paper. Both analytic methods based on the theory of stability and bifurcations and numerical calculations are used. The numerical study is performed under the assumption that the supporting plane is deformable, which permits one to describe transient processes accompaniedwith impacts by using a single system of equations.

Free and controlled motions of an omniwheel vehicle

The dynamics of a vehicle whose three omniwheels are symmetrically arranged is considered in the case when the vehicle moves on a horizontal plane. Two wheels are parallel to each other, whereas the third one is perpendicular to them; the centers of the wheels are located at the vertices of an isosceles triangle. A phase portrait is constructed under the assumption that there are no external actions (except for gravity). The stability conditions for uniform rectilinear motions are compared with the Chaplygin sleigh model. The stability and bifurcation of steady motions are discussed in the case of constant control.

Tippe-top on visco-elastic plane: steady-state motions, generalized smale diagrams and overturns

We consider the dynamics of a tippe-top on a visco-elastic plane with dry friction. Tippetop is a rigid dynamically symmetric sphere with the center of mass that lies on the dynamical axis of symmetry but does not coincide with the center of the sphere. It was shown earlier [1] that the full mechanical energy conserves only on the steady-state motions of the tippe-top while the linear on the pseudovelocities Jellet integral remains constant for a quite general model of friction, for example, viscous friction. It allows studying the stability of the steady-state motions and constructing the generalized Smale diagrams [1]. Here we consider another model of friction—so-called distributed dry friction proposed in [2] that is more suitable for the convex bodies on a rough plane. In this case, both energy and Jellet function change their value on solutions. The earlier investigated case is considered as generating. Steady state motions transform in pseudo-steady motions with slow changing parameters. Calculations illustrate the analytical investigation.

Different Models of Friction in Double-Spherical Tippe-Top Dynamics

Dynamics of a two-spherical tippe-top on a rough horizontal plane is considered. The tippe-top is bounded by a non-convex surface that consists of two-spheres and a cylinder; the axis of cylinder coincides with the common spheres’ axes of symmetry. Being fast spun around its axis of symmetry the tippe-top overturns from the bottom (the big sphere) to the leg (that is modeled by a small sphere) and some time it returns back to stable equilibrium position. Different models of friction are examined analytically and numerically. Numerical investigation is done in assumption that supporting plane is slightly deformable, that allows one to describe rolling and impacts by the same system of dynamical equations. Results of numerical investigation coincide with analytical conclusions.

On conjugation of solutions to two integrable problems: Rolling of pointed body in the plane

In this paper, the problem of motion of heavy rigid body in the rough plane is studied. The surface of body is such that the axis of rotation has two “points,” where the plane tangent to the surface is not specified. The motion is described by two systems of equations; each of them is true for its side of the phase space. One of them is a system of equations that describes rolling of body in the plane without slippage (the body is tangent to the plane by its convex part); the second, is a system of equations that describes motion of body with the fixed point for the Lagrange case (the body is basing on the plane by the point). Questions of existence of global first integrals and potential of the cited one-dimensional system are studied.

Dynamics of Rolling with a Microslip for an Elastic Cylinder on an Elastic Half-Space

For the first time, the dynamic problem of plane-parallel rolling with the slip of an elastic cylinder along an elastic base of the same material is analyzed. The distribution of the normal and tangential stresses in the contact-interaction region consisting of relative slip and stick subregions corresponds to the solution of the quasi-static problem of the theory of elasticity. The solution obtained is compared with the solution of the problem of sliding of an absolutely rigid cylinder along an absolutely rigid plane with Amonton−Coulomb dry friction.