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2013
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    Alexander Ivanov

    Alexander Ivanov
    Inststitutskii per. 9, Dolgoprudnyi 141700, Russia
    Moscow Institute of Physics and Technology

    Publications:

    Ivanov A. P.
    On the control of a robot-ball using two omni-wheels
    2015, Vol. 11, No. 2, pp.  319-327
    Abstract
    We discuss the dynamics of balanced body of spherical shape on a rough plane, managed by the movement of the built-in shell. These two shells are set in relative motion due to rotation of the two symmetrical omni-wheels. It is shown that the ball can be moved to any point on the plane along a straight or (in the case of the initial degeneration) polygonal line. Moreover, any prescribed curvilinear trajectory of the ball center can be followed by appropriate control strategy as far as the diameter, connecting both wheels, is non-vertical.
    Keywords: robot-ball, omni-wheel, control of motion
    Citation: Ivanov A. P.,  On the control of a robot-ball using two omni-wheels, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 2, pp. 319-327
    DOI:10.20537/nd1502007
    Ivanov A. P.
    Comments on the H.Beghin papers
    2013, Vol. 9, No. 4, pp.  765-766
    Abstract
    Citation: Ivanov A. P.,  Comments on the H.Beghin papers, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 765-766
    DOI:10.20537/nd1304012
    Ivanov A. P.
    Abstract
    We discuss the basic problem of dynamics of mechanical systems with constraints-finding acceleration as a function of the phase variables. It is shown that in the case of Coulomb friction, this problem is equivalent to solving a variational inequality. The general conditions for the existence and uniqueness of solutions are obtained. A number of examples is considered.

    For systems with ideal constraints discussed problem has been solved by Lagrange in his «Analytical Dynamics» (1788), which became a turning point in the mathematization of mechanics. In 1829, Gauss gave his principle, which allows to obtain the solution as the minimum of a quadratic function of acceleration, called «constraint». In 1872 Jellett gaves examples of nonuniqueness of solutions in systems with static friction, and in 1895 Painlev´e showed that in the presence of friction, together with the non-uniqueness of solutions is possible. Such situations were a serious obstacle to the development of theories, mathematical models and practical use of systems with dry friction. An unexpected and beautiful promotion was work by Pozharitskii, where the author extended the principle of Gauss on the special case where the normal reaction can be determined from the dynamic equations regardless of the values of the coefficients of friction. However, for systems with Coulomb friction, where the normal reaction is a priori unknown, there are still only partial results on the existence and uniqueness of solutions.

    The approach proposed here is based on a combination of the Gauss principle in the form of reactions with the representation of the nonlinear algebraic system of equations for the normal reactions in the form of a variational inequality. The theory of such inequalities includes the results of existence and uniqueness, as well as the developed methods of solution.
    Keywords: principle of least constraint, dry friction, Painlevé paradoxes
    Citation: Ivanov A. P.,  On the variational formulation of dynamics of systems with friction, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 3, pp. 479-498
    DOI:10.20537/nd1303006
    Ivanov A. P.
    Abstract
    Citation: Ivanov A. P.,  Comments on the P.Painlevé paper “Sur les lois du frottement de glissement”, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 5, pp. 981-984
    DOI:10.20537/nd1205009
    Ivanov A. P., Sakharov  A. V.
    Abstract
    We consider a rigid body which moves upon a rough plane by means of displacements of internal masses. To make turns, we change the angular momentum of the rotor. This leads to asymmetry in normal stresses and appearance of vertical momentum of friction forces.
    Keywords: dry friction, mobile devices without external drivers
    Citation: Ivanov A. P., Sakharov  A. V.,  Dynamics of rigid body, carrying moving masses and rotor, on a rough plane, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 4, pp. 763-772
    DOI:10.20537/nd1204006
    Ivanov A. P., Erdakova N. N.
    On a mechanical lens
    2012, Vol. 8, No. 4, pp.  773-781
    Abstract
    The problem of dynamics of heavy uniform ball moving on the fixed rough plane under its own inertia and forces of dry friction is considered. Assuming that friction coefficient is variable, the switching curve for change the value of friction coefficient is constructed. Using this curve to change the value of friction coefficient we have shown that the bundle of equal balls starting from one interval with equal linear and angular velocities should gather at one point.
    Keywords: dry friction, variable friction coefficient, ball’s dynamics
    Citation: Ivanov A. P., Erdakova N. N.,  On a mechanical lens, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 4, pp. 773-781
    DOI:10.20537/nd1204007
    Ivanov A. P.
    Abstract
    Dynamical systems with discontinuous right-hand sides are considered. It is well known that the trajectories of such systems are nonsmooth and the fundamental solution matrix is discontinuous. This implies the presence of the so-called discontinuous bifurcations, resulting in a discontinuous change in the multipliers. A method of stepwise smoothing is proposed allowing the reduction of discontinuous bifurcations to a sequence of typical bifurcations: saddle-node, period doubling and Hopf bifurcations. The results obtained are applied to the analysis of the well-known system with friction a block on the moving belt, which serves as a popular model for the description of selfexcited frictional oscillations of a brake shoe. Numerical techniques used in previous investigations of this model did not allow general conclusions to be drawn as to the presence of self-excited oscillations. The new method makes it possible to carry out a complete qualitative investigation of possible types of discontinuous bifurcations in this system and to point out the regions of parameters which correspond to stable periodic regimes.
    Keywords: non-smooth dynamical systems, discontinuous bifurcations, oscillator with dry friction
    Citation: Ivanov A. P.,  Analysis of discontinuous bifurcations in nonsmooth dynamical systems, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 2, pp. 231-247
    DOI:10.20537/nd1202003
    Ivanov A. P., Shuvalov N. D.
    Abstract
    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 Gumbel 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.
    Keywords: mixed friction, Stribeck curve, curling rock dynamics
    Citation: Ivanov A. P., Shuvalov N. D.,  On the motion of a heavy body with a circular base on a horizontal plane and riddles of curling, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 521-530
    DOI:10.20537/nd1103008
    Ivanov A. P.
    Abstract
    Comparative analysis of the dynamics of a homogeneous ball on a plane with dry friction is conducted for two conjectures: 1) single contact point (non-holonomic statement); 2) the normal load is distributed in the circle spot of contact with radius $ε$. It is assumed that for given active forces and coefficient of friction the non-slip motion is possible. The expression for load distribution function $φ$ at the contact spot (second statement) is arbitrary, with general mild restrictions, which ensure correctness of the passage to the limit. It is shown that for $ε → 0$ the trajectory of the ball with contact spot approaches the trajectory of the ball with single contact point.

    Previously similar result was obtained by Fufaev [1] in the case $φ = \rm{const}$. The possibility of approximation of reactions of non-holonomic constraints by means of forces of viscous friction was proved [2,3], as well as by means of forces of dry friction with infinitely large coefficient of friction [4].
    Keywords: systems with rolling motion, dry friction
    Citation: Ivanov A. P.,  Comparative analysis of friction models in dynamics of a ball on a plane, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 4, pp. 907-912
    DOI:10.20537/nd1004015
    Ivanov A. P.
    Abstract
    Citation: Ivanov A. P.,  On the mathematical treatment of the impact in billiards, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 439-447
    DOI:10.20537/nd1002015
    Ivanov A. P.
    Abstract
    Examples of irregular behavior of dynamical systems with dry friction are discussed. A classification of frictional contacts with respect to their dimensionality, associativity, and the possibility of interruptions is proposed and basic models showing typical features are stated. In particular, bifurcation conditions for equilibrium families are obtained and formulas for the monodromy matrix for systems with friction are constructed. It is shown that systems with non-associated contacts possess singularities that lead to the nonexistence or nonuniqueness of phase trajectories; these results generalize the paradoxes of Painlev´e and Jellett. Owing to such behavior, a number of earlier results, including the problem on the motion of a rigid body on a rough plane, require an improvement.
    Keywords: non-smooth dynamical systems, dry friction, discontinuous bifurcation
    Citation: Ivanov A. P.,  Bifurcations in systems with friction: basic models and methods, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 479-498
    DOI:10.20537/nd0904003
    Ivanov A. P.
    Abstract
    The classical mechanical problem about the motion of a heavy rigid body on a horizontal plane is considered within the framework of theory of systems with unilateral constraints. Under general assumptions about the character of friction, we examine the question on the possibility of detachment of the body fromthe plane under the action of reaction of the plane and forces of inertia. For systems with rolling, we find new scenarios of the appearing of motions with jumps and impacts. The results obtained are applied to the study of stationary motions of a disk.We have showed the following.
    1) In the absence of friction, the detachment conditions on stationary motions do not hold. However, if the angle $θ$ between the symmetry axis and the vertical decreases to zero, motions close to stationary motions are necessarily accompanied by detachments.
    2) The same conclusion holds for a thin disk that rolls on the support without sliding.
    3)For a disk of nonzero thickness in the absence of sliding, the detachment conditions hold on stationary motions in some domain in the space of parameters; in this case, the angle $θ$ is not less than 49 degrees. For small values of $θ$ the contact between the body and the support does not break in a neighborhood of stationary motions.
    Keywords: unilateral constraint, friction, Painlevé paradoxes
    Citation: Ivanov A. P.,  On Detachment Conditions in the Problem on the Motion of a Rigid Body on a Rough Plane, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 3, pp. 287-302
    DOI:10.20537/nd0803003
    Ivanov A. P.
    Abstract
    Mechanical systems with unilateral constraints that can be represented in the contact mode on the phase plane are considered. On the phase plane we construct domains that satisfy the following conditions 1) a detachment from the constraint is impossible; 2) the sign of the constraint reaction corresponds to its unilateral character. These conditions are equivalent for an ideal constraint [1, 2], but they can differ in the presence of friction [3]. Trajectories without detachments belong to intersections of these domains. A circular disc moving on a horizontal support with viscous friction and a disc with the sharp edge moving on an icy surface [4, 5] are considered as examples.

    Usually for the control of contact conservation one uses only the second condition from above, which can lead to invalid qualitative conclusions.
    Keywords: unilateral constraint, friction, detachment conditions
    Citation: Ivanov A. P.,  Geometric Representation of Detachment Conditions in Systems with Unilateral Constraints, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 3, pp. 303-312
    DOI:10.20537/nd0803004

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