# Dmitry Treschev

Gubkina 8, Moscow, 119991 Russia
Steklov Mathematical Institute, Russian Academy of Sciences

## Publications:

 Borisov A. V., Mamaev I. S., Treschev D. V. Rolling of a rigid body without slipping and spinning: kinematics and dynamics 2012, Vol. 8, No. 4, pp.  783-797 Abstract In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form. Keywords: rolling without slipping, nonholonomic constraint, Chaplygin system, conformally Hamiltonian system Citation: Borisov A. V., Mamaev I. S., Treschev D. V.,  Rolling of a rigid body without slipping and spinning: kinematics and dynamics, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 4, pp.  783-797 DOI:10.20537/nd1204008
 Ramodanov S. M., Tenenev V. A., Treschev D. V. Self-propulsion of a body with rigid surface and variable coefficient of lift in a perfect fluid 2012, Vol. 8, No. 4, pp.  799-813 Abstract We study the system of a 2D rigid body moving in an unbounded volume of incompressible, vortex-free perfect fluid which is at rest at infinity. The body is equipped with a gyrostat and a so-called Flettner rotor. Due to the latter the body is subject to a lifting force (Magnus effect). The rotational velocities of the gyrostat and the rotor are assumed to be known functions of time (control inputs). The equations of motion are represented in the form of the Kirchhoff equations. In the case of piecewise continuous controls, the integrals of motion are indicated. Using these integrals we obtain a (reduced) system of first-order differential equations on the configuration space. An optimal control problem for several types of the inputs is then solved using genetic algorithms. Keywords: perfect fluid, self-propulsion, Flettner rotor Citation: Ramodanov S. M., Tenenev V. A., Treschev D. V.,  Self-propulsion of a body with rigid surface and variable coefficient of lift in a perfect fluid, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 4, pp.  799-813 DOI:10.20537/nd1204009
 Treschev D. V., Erdakova N. N., Ivanova T. B. On the final motion of cylindrical solids on a rough plane 2012, Vol. 8, No. 3, pp.  585-603 Abstract The problem of a uniform straight cylinder (disc) sliding on a horizontal plane under the action of dry friction forces is considered. The contact patch between the cylinder and the plane coincides with the base of the cylinder. We consider axisymmetric discs, i.e. we assume that the base of the cylinder is symmetric with respect to the axis lying in the plane of the base. The focus is on the qualitative properties of the dynamics of discs whose circular base, triangular base and three points are in contact with a rough plane. Keywords: Amontons–Coulomb law, dry friction, disc, final dynamics, stability Citation: Treschev D. V., Erdakova N. N., Ivanova T. B.,  On the final motion of cylindrical solids on a rough plane, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp.  585-603 DOI:10.20537/nd1203012
 Salnikova T. V., Treschev D. V., Gallyamov S. R. On the motion of free disc on the rough horisontal plane 2012, Vol. 8, No. 1, pp.  83-101 Abstract We consider the problem of a disk sliding on a horizontal plane under the action of dry friction forces. The model is based on three hypotheses. The law of interaction of a small element of the disk’s surface with the plane is the Amonton–Coulomb law, the pressure distribution over the contact patch is a linear (generally speaking, time-dependent) function of Cartesian coordinates, the height of the disk is not high. The equations of motion possess a rich group of symmetry, which enables a detailed qualitative analysis of the problem. Keywords: dry friction, Amontons–Coulomb law Citation: Salnikova T. V., Treschev D. V., Gallyamov S. R.,  On the motion of free disc on the rough horisontal plane, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp.  83-101 DOI:10.20537/nd1201006
 Kuleshov A. S., Treschev D. V., Ivanova T. B., Naymushina O. S. A rigid cylinder on a viscoelastic plane 2011, Vol. 7, No. 3, pp.  601-625 Abstract The paper considers two two-dimensional dynamical problems for an absolutely rigid cylinder interacting with a deformable flat base (the motion of an absolutely rigid disk on a base which in non-deformed condition is a straight line). The base is a sufficiently stiff viscoelastic medium that creates a normal pressure $p(x) = kY(x)+ν\dot{Y}(x)$, where $x$ is a coordinate on the straight line, $Y(x)$ is a normal displacement of the point $x$, and $k$ and $ν$ are elasticity and viscosity coefficients (the Kelvin—Voigt medium). We are also of the opinion that during deformation the base generates friction forces, which are subject to Coulomb’s law. We consider the phenomenon of impact that arises during an arbitrary fall of the disk onto the straight line and investigate the disk’s motion «along the straight line» including the stages of sliding and rolling. Keywords: Kelvin–Voight medium, impact, viscoelasticity, friction Citation: Kuleshov A. S., Treschev D. V., Ivanova T. B., Naymushina O. S.,  A rigid cylinder on a viscoelastic plane, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp.  601-625 DOI:10.20537/nd1103014
 Borisov A. V., Bolotin S. V., Kilin A. A., Mamaev I. S., Treschev D. V. Valery Vasilievich Kozlov. On his 60th birthday 2010, Vol. 6, No. 3, pp.  461-488 Abstract Citation: Borisov A. V., Bolotin S. V., Kilin A. A., Mamaev I. S., Treschev D. V.,  Valery Vasilievich Kozlov. On his 60th birthday, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 3, pp.  461-488 DOI:10.20537/nd1003001