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# Volume 14, Number 6, 2009

 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. Superintegrable system on a sphere with the integral of higher degree Abstract We consider the motion of a material point on the surface of a sphere in the field of $2n + 1$ identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [1], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional $N$-particle system discussed in the recent paper [2] and show that for the latter system an analogous superintegral can be constructed. Keywords: superintegrable systems, systems with a potential, Hooke center Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Superintegrable system on a sphere with the integral of higher degree, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 615-620 DOI:10.1134/S156035470906001X
 Kharlamov M. P. Separation of variables in the generalized 4th Appelrot class. II. Real solutions Abstract We continue the analytical solution of the integrable system with two degrees of freedom arising as the generalization of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields [Kharlamov, RCD, vol. 10, no. 4]. The separated variables found in [Kharlamov, RCD, vol. 12, no. 3] are complex in the most part of the integral constants plane. Here we present the real separating variables and obtain the algebraic expressions for the initial Euler–Poisson variables. The finite algorithm of establishing the topology of regular integral manifolds is described. The article straightforwardly refers to some formulas from [Kharlamov, RCD, vol. 12, no. 3]. Keywords: Kowalevski top, double field, Appelrot classes, separation of variables Citation: Kharlamov M. P., Separation of variables in the generalized 4th Appelrot class. II. Real solutions, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 621-634 DOI:10.1134/S1560354709060021
 Fernandez O. E.,  Mestdag T.,  Bloch A. M. A generalization of Chaplygin’s Reducibility Theorem Abstract In this paper we study Chaplygin’s Reducibility Theorem and extend its applicability to nonholonomic systems with symmetry described by the Hamilton–Poincare–d’Alembert equations in arbitrary degrees of freedom. As special cases we extract the extension of the Theorem to nonholonomic Chaplygin systems with nonabelian symmetry groups as well as Euler–Poincaré–Suslov systems in arbitrary degrees of freedom. In the latter case, we also extend the Hamiltonization Theorem to nonholonomic systems which do not possess an invariant measure. Lastly, we extend previous work on conditionally variational systems using the results above. We illustrate the results through various examples of well-known nonholonomic systems. Keywords: Hamiltonization, nonholonomic systems, reducing multiplier Citation: Fernandez O. E.,  Mestdag T.,  Bloch A. M., A generalization of Chaplygin’s Reducibility Theorem, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 635-655 DOI:10.1134/S1560354709060033
 Ivanov A. P. Bifurcations in systems with friction: basic models and methods 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
 Jirout M.,  Mack W.,  Lugner P. Non-smooth dynamics of a magnetic track brake Abstract During the first stages of braking with a magnetic track brake in low suspension impacts occur between the force transmitting components, giving rise to a non-smooth system behaviour. Initially, when the electric current is switched on, the magnet moves down until it impinges on the rail. Thereafter, it is decelerated by the friction force, and subsequently there occurs a first impact of the transmission link of the magnet on the transmission link of the bogie frame. Due to the elasticity of the components there follows a high-frequency series of impacts with decreasing intensity until the velocity of the magnet relative to the bogie frame vanishes. Of course, the occurring forces are multiples of the steady-state ones, and this must be taken into account at the design of the force transmitting components. Keywords: non-smooth dynamics, impact, friction, magnetic track brake Citation: Jirout M.,  Mack W.,  Lugner P., Non-smooth dynamics of a magnetic track brake, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 673-681 DOI:10.1134/S1560354709060057
 Edelmann J.,  Plochl M. Handling characteristics and stability of the steady-state powerslide motion of an automobile Abstract Powerslide of an automobile may be defined as a steady-state cornering motion at a large side slip angle of the vehicle, considerably large traction forces and a large negative steering angle of the handwheel. In this case the front wheels direct towards the outside of the turn. As this extrem driving condition, which can be seen e.g. in Rallye sports, is hardly addressed in literature so far, this paper investigates the respective handling characteristics. Therefore, a nonlinear four-wheel vehicle model is applied including nonlinear tyre characteristics, the load transfer between inner and outer wheels and the influence of the traction forces on the lateral tyre forces. A basic stability analysis reveals the unstable nature of the steady-state owerslide motion of a certain test vehicle. To approve the numerical findings, measurements have been performed with a sports utility vehicle with rear-wheel drive at various speeds on a wet circular test track. Keywords: vehicle dynamics, handling behaviour, steady-state cornering, powerslide Citation: Edelmann J.,  Plochl M., Handling characteristics and stability of the steady-state powerslide motion of an automobile, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 682-692 DOI:10.1134/S1560354709060069
 Vil'ke V. G.,  Kosenko I.,  Aleksandrov E. B. On computer implementation of the Hertz elastic contact model and its simplifications Abstract The paper is concerned with an elastic contact model of rigid bodies which is developed in the framework of the Hertz model. For this new model, we suggest more effective algorithms with reduced computational time. We also present an algorithm for representation of the geometry of the contacting surfaces in the local contact coordinate system. This coordinate system tracks permanently the surfaces of the bodies, which are able to contact. An approach to computation of the normal elastic force is presented. It is based on the reduction to a single transcendental scalar equation that includes the complete elliptic integrals of the first and second kinds. The computational time in the Hertz-model simulation was considerably reduced due to the use of the differential technique for computation of the complete elliptic integrals and due to the replacement of the implicit transcendental equation by a differential one. Using the classic solution of the Hertz contact problem, we then present an invariant form for the force function, which depends on the geometric properties of the intersection of the undeformed volumes occupied with the rigid bodies (so-called volumetric model). The reduced expression for the force function obtained is shown to be different from that accepted in the classic contact theory hypotheses. Our expression has been tested in several examples dealing with bodies that contact elastically including Hertz’s classical model. In the context of the Hertz contact problem, an approximate model for computation the resulting wrench of the dry friction tangent forces is set up. The wrench consists of the total friction force and the drilling friction torque. The approach under consideration naturally extends the contact model constructed earlier. The dry friction forces and torque are integrated over the contact elliptic spot. Generally an analytic computation of the integrals mentioned is bulk and cumbersome leading to decades of terms that include rational functions depending on complete elliptic integrals. To be able to implement a fast computer model of elastically contacting bodies, one should first set up an approximate model in the way initially proposed by Contensou. To verify the model developed, we have used results obtained by several authors. First we test our method on the Tippe-Top dynamic model. Simulations show that the top’s evolution can be verified with a high quality compared with the use of the theory of set-valued functions. In addition, the ball bearing dynamic model has been also used for a detailed verification of different approaches to the computation of tangent forces. Then the friction model of the regularized Coulomb type and the approximate Contensou one, each embedded into the whole bearing dynamic model, were thoroughly tested and compared. It turned out that the simplified Contensou approach provides a computer model that runs even faster compared with the case of the point contact. In addition, the volumetric model demonstrated a reliable behavior and an acceptable accuracy. Keywords: Hertz contact model, theorem of existence and uniqueness, volumetric contact model, Contensou–Erismann model, Tippe-Top model, ball bearing model Citation: Vil'ke V. G.,  Kosenko I.,  Aleksandrov E. B., On computer implementation of the Hertz elastic contact model and its simplifications, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 693-714 DOI:10.1134/S1560354709060070

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