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2013
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# Alexander Ivanov

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

## Publications:

 Ivanov A. P. On Final Motions of a Chaplygin Ball on a Rough Plane 2016, vol. 21, no. 7-8, pp.  804-810 Abstract A heavy balanced nonhomogeneous ball moving on a rough horizontal plane is considered. The classical model of a “marble” body means a single point of contact, where sliding is impossible. We suggest that the contact forces be described by Coulomb’s law and show that in the final motion there is no sliding. Another, relatively new, contact model is the “rubber” ball: there is no sliding and no spinning. We treat this situation by applying a local Coulomb law within a small contact area. It is proved that the final motion of a ball with such friction is the motion of the “rubber” ball. Keywords: Coulomb friction, Chaplygin ball, asymptotic dynamics Citation: Ivanov A. P.,  On Final Motions of a Chaplygin Ball on a Rough Plane, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 804-810 DOI:10.1134/S1560354716070030
 Ivanov A. P. On the Control of a Robot Ball Using Two Omniwheels 2015, vol. 20, no. 4, pp.  441-448 Abstract We discuss the dynamics of a balanced body of spherical shape on a rough plane, controlled by the movement of a built-in shell. These two shells are set in relative motion due to rotation of the two symmetrical omniwheels. 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 an appropriate control strategy as far as the diameter connecting both wheels is nonvertical. Keywords: robot ball, omniwheel, control Citation: Ivanov A. P.,  On the Control of a Robot Ball Using Two Omniwheels, Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, pp. 441-448 DOI:10.1134/S1560354715040036
 Ivanov A. P. On the Impulsive Dynamics of M-blocks 2014, vol. 19, no. 2, pp.  214-225 Abstract This paper is concerned with the motion of a cubic rigid body (cube) with a rotor, caused by a sudden brake of the rotor, which imparts its angular momentum to the body. This produces an impulsive reaction of the support, leading to a jump or rolling from one face to another. Such dynamics was demonstrated by researchers from Massachusetts Institute of Technology at the IEEE/RSJ International Conference on Intelligent Robots and Systems in Tokio in November 2013. The robot, called by them M-block, is 4 cm in size and uses an internal flywheel mechanism rotating at 20 000 rev/min. Initially the cube rests on a horizontal plane. When the brake is set, the relative rotation slows down, and its energy is imparted to the case. The subsequent motion is illustrated in a clip [13]. Here the general approach to the analysis of dynamics of M-cube is proposed, including equations of impulsive motion and methods of their solution. Some particular cases are studied in details. Keywords: gyrostat, impulsive dynamics, friction, robotic dynamics Citation: Ivanov A. P.,  On the Impulsive Dynamics of M-blocks, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 214-225 DOI:10.1134/S1560354714020051
 Ivanov A. P. On the Variational Formulation of the Dynamics of Systems with Friction 2014, vol. 19, no. 1, pp.  100-115 Abstract We discuss the basic problem of the dynamics of mechanical systems with constraints, namely, the problem of finding accelerations 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 are considered. For systems with ideal constraints the problem under discussion was 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 one to obtain the solution as the minimum of a quadratic function of acceleration, called the constraint. In 1872 Jellett gave examples of non-uniqueness of solutions in systems with static friction, and in 1895 Painlevé showed that in the presence of friction, the absence of solutions is possible along with the nonuniqueness. Such situations were a serious obstacle to the development of theories, mathematical models and the practical use of systems with dry friction. An elegant, and unexpected, advance can be found in the work [1] by Pozharitskii, where the author extended the Gauss principle to 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 [2–4]. 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 [5] includes results on the 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 the Dynamics of Systems with Friction, Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 100-115 DOI:10.1134/S1560354714010079
 Ivanov A. P. Analysis of Discontinuous Bifurcations in Nonsmooth Dynamical Systems 2012, vol. 17, no. 3-4, pp.  293-306 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: saddlenode, period doubling and Hopf bifurcations. The results obtained are applied to the analysis of the well-known dry friction oscillator, which serves as a popular model for the description of self-excited frictional oscillations of a braking system. 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: nonsmooth dynamical systems, discontinuous bifurcations, oscillators with dry friction Citation: Ivanov A. P.,  Analysis of Discontinuous Bifurcations in Nonsmooth Dynamical Systems, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 293-306 DOI:10.1134/S1560354712030069
 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 2012, vol. 17, no. 1, pp.  97-104 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 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. Keywords: mixed friction, Stribeck’s curve, dynamics of curling stones 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, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 97-104 DOI:10.1134/S156035471201008X
 Ivanov A. P. Bifurcations in systems with friction: basic models and methods 2009, vol. 14, no. 6, pp.  656-672 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 Painlevé 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, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 656-672 DOI:10.1134/S1560354709060045
 Ivanov A. P. Geometric Representation of Detachment Conditions in Systems with Unilateral Constraint 2008, vol. 13, no. 5, pp.  435-442 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, detachment conditions Citation: Ivanov A. P.,  Geometric Representation of Detachment Conditions in Systems with Unilateral Constraint, Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 435-442 DOI:10.1134/S1560354708050067
 Ivanov A. P. On Detachment Conditions in the Problem on the Motion of a Rigid Body on a Rough Plane 2008, vol. 13, no. 4, pp.  355-368 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 from the 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 $\theta$ 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 $\theta$ is not less than 49 degrees. For small values of $\theta$, 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, Regular and Chaotic Dynamics, 2008, vol. 13, no. 4, pp. 355-368 DOI:10.1134/S1560354708040096