Rolling of a non-homogeneous ball over a sphere without slipping and twisting

Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.      

Theoretical model of laminar flow in tubes in rolling motion

The theoretical model of laminar flow in tubes in rolling motion is established. The velocity and temperature correlations are derived, and the frictional resistance coefficient and Nusselt number are also obtained. The oscillation of parameters is induced by the tangential force due to rolling motion. The effect of centrifugal and Coriolis forces on the flow is negligible. The tangential force does not effect on the average parameters. The oscillating amplitude of Nusselt number increase with the Prandtl number increasing. Both the oscillating amplitudes of frictional resistance coefficient and Nusselt number increase with the rolling frequency increasing.     

Relation of the reinforcing action of fillers with molecular motion in polymers

In the case of filled systems based on butadiene-styrene copolymers, it has been shown that the attainment of mobility of kinetic segments in definite temperature regions leads in amorphous polymers to an increase in their strength and reinforcement effect, and also to a change in the morphology of the breakdown surface in these temperature regions. The established regularity has been examined from the point of view of the adhesion theory of reinforcement.Moscow Institute of Meat and Milk Industry. Central Scientific-Research Laboratory of Chemical Packaging Materials, Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 909–911, September–October, 1972.     

Local compatibility and the adhesion of polymers

The role of the thermodynamic factor in creating an adhesion bond between partly or wholly compatible polymers is discussed. The importance of this factor depends on the fact that polymers brought into contact will diffuse into each other only if they are wholly or partly mutually soluble. This mutual solubility is confined to a surface layer and is determined by the local diffusion. In considering the mechanism of formation of an adhesion bond between two polymers it is necessary to take their surface properties into account. In the diffusion theory of adhesion diffusion is regarded as the motive force in the creation of an adhesion bond.Moscow Lomonosov Institute of Fine Chemical Technology. Translated from Mekhanika Polimerov, No. 3, pp. 455–459, May–June, 1969.     

Increasing the adhesion of polymers to metals by adding substances with conjugation systems

The electron-donor properties of polymers with conjugation systems (PCS) and their capability of forming -complexes with metals is of interest in relation to adhesion. Small amounts (0.01–0.1%) of substances with polyconjugation systems (polydiphenylbutadiene, polyphenylacetylene, and copolymers of naphthalene and anthracene with benzene) increase the adhesion between aluminum and polymers used as electrical insulation coatings (polyesters, polyimides, and their derivatives). With optimum PCS dosage, the resistance to peeling rises by 15–38%.Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 365–367, March–April, 1974.     

Rubber rolling over a sphere

“Rubber” coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied by “marble” coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2–3–5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group G ₂). The 2–3–5 nonholonomic geometries are classified in a companion paper [2] via Cartan’s equivalence method [3]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [4–8] with SO(3) symmetry group, total space Q = SO(3) × S ² and base S ², that can be reduced to an almost Hamiltonian system in T*S ² with a non-closed 2-form ω_NH. In this paper we present some basic results on the sphere-sphere problem: a dynamically asymmetric but balanced sphere of radius b (unequal moments of inertia I _j but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ω_NH is conformally symplectic [9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1/2(b/a − 1). Using sphero-conical coordinates we verify the result by Borisov and Mamaev [10] that the system is integrable for p = −1/2 (ball over a plane). They have found another integrable case [11] corresponding to p = −3/2 (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates separates the Hamiltonian in this case. No other integrable cases with different I _j are known.      

Effect of sliding velocity on area of real contact and on force of friction in high-elastic materials

The velocity dependence of the force of friction and the area of real contact of SKN-18, SKN-26, and SKN-40 rubbers has been simultaneously determined on an optical instrument. In the region of small sliding velocities the area of real contact is virtually independent of the sliding velocity, while the force of friction increases in proportion to the logarithm of the sliding velocity, in accordance with the molecular-kinetic theory of friction. At high sliding velocities a deviation in the velocity dependences of both quantities is observed. The constant of friction has been determined over the entire velocity range and the resulting velocity dependence is examined in molecular terms.Mekhanika Polimerov, Vol. 3, No. 2, pp. 309–311, 1967     

Nature and mechanism of friction of rubberlike polymers in different physical states

We have investigated the frictional properties of crosslinked butadiene-nitrile and butadiene-styrene copolymers and natural rubber in friction against polished steel under vacuum conditions in the temperature interval from –200 to +150° C, which embraces the glassy and high-elastic states, as well as the transition region between them. The temperature dependence of polymer friction is characterized by two maxima, a principal and a low-temperature maximum. The principal maximum, observed in the glass transition region, is not associated with the mechanical loss maximum observed in the polymers themselves. The temperature dependence of the force of friction is composed of three parts. In the high-elastic region there is an increase in the force of friction with fall in temperature, in accordance with the molecular-kinetic theory of friction of rubberlike polymers. In this region the nature of friction is associated with mechanical losses in the surface layer of polymer. The mechanical losses inside the polymer itself are unimportant. The deviation from the theoretical curve and the fall in the force of friction below a certain temperature in the transition region are chiefly associated with a decrease in the actual area of contact as the polymer passes into the glassy state. In the glassy region the friction is significantly determined by the mechanical losses in the polymer itself associated with the repeated elastic and forced-elastic deformation of the asperities in the layer of polymer in contact with the rigid surface. Therefore the low-temperature maximum is closely related to the mechanical loss maximum observed in the same temperature region in dynamic tests. Apart from this, the friction maximum is also associated with the increase in the forces of adhesion and the reduction of the actual area of contact at temperatures at which a forced-elastic mechanism of compression of the polymer asperities is not realized.Mekhanika Polimerov, Vol. 3, No. 1, pp. 123–135, 1967     

A study of turbulent flow of solutions of a polymer and a surface-active substance

We give the results of laser doppler anemometer measurements of the velocity profiles and mean-square values of the longitudinal velocity pulses in the flow of solutions of a polymer (polyacrylamides) or a surface-active substance (undustrial “metaupon” paste) that lower hydrodynamic frictional resistance and mixtures of such substances. It is shown that a solution of a mixture has a significantly greater effect on the turbulence of the flow than the polymers or surface-active substances do individually. Two figures. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 142–146.     

Lamellar structure and plasticity micromechanisms in bulk crystalline polymers. Communication 1

It is shown by electron microscopy that the lamellar structure typical of bulk crystalline polymers is identical in its structural-morphological characteristics with the typical products of martensite transformations in metal systems. It is also established that the polymer crystallization process has the typical characteristics of transformations governed by the shear (martensite) mechanism. These conclusions are used as a basis for an examination of the principal factors controlling the formation of the real structure of bulk polymers. The thermodynamic conditions under which bulk polymers crystallize require that the lamellar-spherulitic structure be formed in accordance with a self-consistent shear mechanism. In accordance with the new model, the spherulites represent an organization of the martensite lamellae in which the structural stress fields are mutually compensated. The proposed model underlines the fact that polycrystalline polymers and metal systems in the martensitic state are structurally similar materials.E. O. Paton Electrowelding Institute, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 3, pp. 387–394, May–June, 1976.     

On a geometrically exact theory for contact interactions

The focus of the contribution is on the development of the unified geometrical formulation of contact algorithms in a covariant form for various geometrical situations of contacting bodies leading to contact pairs: surface-to-surface, line-to-surface, point-to-surface, line-to-line, point-to-line, point-to-point. The computational contact algorithm will be considered in accordance with the geometry of contact bodies in a covariant form. This combination forms a geometrically exact theory of contact interaction, see [1]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adhesion and wetting in urea fiberglass plastics

Results of a study of the adhesion of urea polymers to fiberglass and of the phenomenon of wetting of the glass surface by oligomers are presented; the introduction of linear high-molecular-weight compounds with active functional groups into the make-up of the binder aids in an improvement of adhesion and wetting. It has been shown that an interrelationship exists between adhesion, wetting, and the physicomechanical properties of fiberglass plastics.     

Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with coulomb friction

In this paper, we study the problem of predicting the acceleration of a set of rigid, 3-dimensional bodies in contact with Coulomb friction. The nonlinearity of Coulomb's law leads to a nonlinear complementarity formulation of the system model. This model is used in conjunction with the theory of quasi-variational inequalities to prove for the first time that multi-rigid-body systems with all contacts rolling always has a solution under a feasibility-type condition. The analysis of the more general problem with sliding and rolling contacts presents difficulties that motivate our consideration of a relaxed friction law. The corresponding complementarity formulations of the multi-rigid-body contact problem are derived and existence of solutions of these models is established. The research of this author was based on work supported by the National, Science Foundation under grants DDM-9104078 and CCR-9213739. The research of this author was partially supported by the National Science Foundation under grant IRI-9304734, by the Texas Advanced Research Program grant 999903-078, and by the Texas Advanced Technology Program under grant 999903-095.     

A. Yu. Ishlinskii's researches in the field of rolling friction and their development

The results of the research by A. Yu. Ishlinskii and others on the theory of rolling contact of deformable bodies, taking into account their imperfect elasticity and partial slip in the area contact region, are presented.     

Thermal analysis of hot strip rolling using finite element and upper bound methods

Thermal analysis of hot rolling process has been studied in this work. A finite element method has been coupled with an upper bound solution assuming, triangular velocity field, to predict temperature field during hot strip rolling operation. To do so, an Upwind Petrov–Galerkin scheme together with isoparametric quadrilateral elements has been employed to solve the steady-state heat transfer equation. A comparison has been made between the published and the model predictions and a good agreement was observed showing the accuracy of the proposed model.     

Autohesion of commercial elastomer systems

The effect of the content of various plasticizers on the autohesive strength of polymers SKN-40 and P-200 is studied. It is established that an increase in the content of compatible plasticizer is accompanied by a monotonic increase in autohesion. The addition to the same polymers of increasing quantities of incompatible plasticizers causes a continuous fall in autohesion. The obtained results agree with the diffusion theory of autohesion phenomena.Mekhanika Polimerov, Vol. 1, No. 3, pp. 77–80, 1965     

Rolling of a homogeneous ball over a dynamically asymmetric sphere

We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of “clandestine” linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.     

Examples of non-uniqueness for the linear problem of potential flow around semi-submerged bodies

The plane Neumann-Kelvin problem, which uses a linear approximation of the theory of waves of small amplitude to describe the steady, vortex-free motion of semi-submerged cylindrical bodies in an ideal, incompressible, heavy liquid with a free surface, is considered. For each fixed value of the free-stream velocity and a convoy of two or more bodies it is shown that the geometry of the bodies can be so chosen that the homogeneous Neumann-Kelvin problem will have a non-trivial solution. A family of potentials is constructed that provide such examples of non-uniqueness. The corresponding configurations can be obtained by choosing some of the streamlines of the solutions as the body contours. Examples are given.