The rolling of a wheel with a reinforced tyre along a plane without slip

A model of a wheel with a reinforced tyre, whose surface is simulated by a flexible strip (tread) attached to parts of two tori (the sidewalls of the tyre) is considered. The disk of the wheel (a rigid body) has six degrees of freedom and is in contact with the plane along part of the tread. Based on several assumptions, the potential energy functional of the deformed wheel is found as a function of the deformations of the centre line of the tread. On the assumption that the wheel is rolling without slip in the region of contact of the tread with the plane along a previously unknown section of the tread, the complete system of equations of motion is obtained. The equilibrium of the wheel and the steady state of rolling in a straight line with given swivel and tilt are investigated, and all characteristics of the motion are found (the contact region, the tyre deformation, and the forces and torques applied to the wheel disk).     

The rolling of a wheel with a pneumatic tyre on a planet

A model of a pneumatic tyre as a system with an infinite number of degrees of freedom is proposed, when its surface is represented by the deformed surface of a torus. Using a number of hypotheses a functional of the potential energy of the deformations of the tyre is obtained as a function of the deformations of its tread. A complete system of equations of motion is obtained, assuming that the wheel rolls without slipping in the area of contact of the tread with the plane, with respect to the previously unknown part of the tread. In two special cases of the rolling of a wheel with breakaway and on a banking, all the characteristics of the motion (the contact area, the tyre deformation, and the forces and moments applied to the disc of the wheel) are obtained.     

Controlled rolling of a disc along a plane curve

The plane-parallel rolling of a disc along a fairly smooth curve under the action of perturbing and controlling forces and moments of forces is investigated. Models of the controlled motion corresponding to an explicit or implicit coordinate specification of the curve are constructed. The requirements imposed on the values of the sliding friction force and the normal pressure force, which ensure that the disc rolls without slipping and without loss of contact are determined. The problem of bringing the disc into the required state of motion in a time-optimal way by the action of the force of gravity and a limited axial moment of forces are considered. The situation when the reference curve contains a linear component (inclined upwards or downwards) and a periodic (wavy) component is investigated. Problems of optimal synthesis for a relatively large and relatively small controlling moment are considered.Investigations of the rolling of bodies of different geometrical shape on surfaces, both flat and curved (see 1, 2, 3, 4, etc.), are important for applications in modern machine construction, transport, mobile robot systems,5, 6, 7 etc. However, there are no results of systematic investigations of the problems of the dynamics and control of rolling of bodies on curved surfaces, including comparatively simple bodies (cylinders, discs and spheres). From the applied point of view, the problem of controlled rolling of a disc under gravitational forces on a curve containing a linear incline and periodic smooth changes in the slope is of interest as a model of the motion of a wheel along a path with a curvilinear profile. The solution of problems of the control of the motion of a disc along a closed curve, for example, inside or outside a circle, surmounting obstacles or “rolling out” from hollows of different shapes etc. is of fundamental importance.     

Control of the rolling of a wheel by changing its imbalance

The motion of a wheel (rolling without slipping) in a vertical (longitudinal) plane is considered. The load, which is modelled as a point mass, can be moved within the wheel in a hollow which is positioned along its diameter. By moving the load from one end of the hollow to the other in a specific way, it is possible to manage the rolling of the wheel. An algorithm is constructed for control of the position of the load using which the load is periodically moved from one end of the hollow to the opposite end while the wheel rolls (non-uniformly) over the surface.     

Influence of variable heat flux on natural convection along a corrugated wall in porous media

In this work the coupled non-linear partial differential equations, governing the free convection from a wavy vertical wall under a power law heat flux condition, are solved numerically. For both Darcy and Forchheimer extended non-Darcy models, a wavy to flat surface transformation is applied and the governing equations are reduced to boundary layer equations. A finite difference scheme based on the Keller Box approach has been used in conjunction with a block tri-diagonal solver for obtaining the solution. Detailed simulations are carried out to investigate the effect of varying parameters such as power law heat flux exponent m, wavelength–amplitude ratio a and the transformed Grashof number Gr′. Both surface undulations and inertial forces increase the temperature of the vertical surface while increasing m reduces it. The wavy pattern observed in surface temperature plots, become more prominent with increasing m or a but reduces as Gr′ increases.     

Parametric resonance in the problem of a heavy rigid body rolling along a straight line on a plane

The problem of the stability of a heavy rigid body, bounded by the surface of an ellipsoid and with a cavity in the form of a coaxial ellipsoid, rolling along a straight line on a horizontal rough plane is investigated. It is shown that in the case of a body that is close to being dynamically symmetrical, parametric resonance always occurs leading to instability of the rolling. Each ellipsoid has its own “individual” resonance angular velocity. In the general case, regions in which the necessary stability conditions are satisfied can be distinguished in parameter space. The problem of calculating the resonance coefficient corresponding to instability for parametric resonance in a reversible third-order system is solved.     

Micro‐ and nanoscale modelling of dry friction with application to the wheel/rail contact

Friction is a phenomenon involving elastic interactions, plastic deformation and failure processes at different length scales. A model of dry friction is established based on the method of Movable Cellular Automata (MCA). The influence of material and loading parameters has been investigated within a large number of numerical simulations. The new friction law is applied to the calculation of stresses, deformations and tractive forces in wheel/rail contact with rough surfaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Contact simulation for the calculation of the kinematic excitation of vibrations from surface roughness of wheel and rail

A simulation method has been developed to predict the sound emission of the railway wheelsets due to the excitation of wheelset vibrations by surface roughness of wheel and rail. This model treats the contact in a linearized manner and is therefore not capable of modelling the local effects of the roughness in the wheel-rail contact zone. These effects are very important since the contact acts as a low pass filter which is calculated within a contact simulation. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The steady rolling of a disc on a rough plane

The well-known problem of the rolling without slipping of a heavy circular disc along a horizontal plane is considered. The steady motions of a disc for which the angle between the plane of the disc and the supporting plane (the angle of nutation) is constant are investigated. The problem of the range of variation of the angle of nutation within which the given motions are stable, irrespective of the values of the constants of the two linear first integrals or, in other variables, irrespective of the angular velocities of precession and proper rotation, is investigated. It is shown that this range is wider than was established earlier in [1].     

The problem of a viscoelastic cylinder rolling on a rigid half-space

The problem of a viscoelastic cylinder rolling on a rigid base, propelled by a line force acting at its centre, is solved in the noninertial approximation. The method used is based on a decomposition of hereditary integrals developed by the authors in previous work, and on the viscoelastic Kolosov-Muskhelishvili equations which are used to generate a Hilbert problem. In this formulation, the problem reduces to a nonsingular integral equation in space and time, which simplifies under steady-state conditions and for exponential decay materials, to algebraic form. There are also two subsidiary conditions.In the case of a standard linear model, explicit analytic results and numerical examples are given for the pressure function, for surface displacements, and also for hysteretic friction.     

The motion of a point mass along a string

An equation for the trajectory of a point mass (a particle) when it moves (slides) in a plane by inertia along a weightless elastic thread (string), stretched between two fixed points is obtained. The time dependence of the trajectory parameter is established. An equation of the trajectory of the particle when it suddenly decelerates is obtained. Forced motion of the particle along a straight string (as a model of the swinging of a lift on an elastic tether in zero gravity) is considered.     

The motion of a point mass along a string

As a supplement to results obtained earlier [1], the general integral of motion of a point mass along a string is determined, and the influence of friction is evaluated.     

The motion of a three-link system along a plane

Controllable motions of a three-link system along a horizontal plane when there is dry friction are considered. Previously obtained results are generalized and refined.     

The rigidity of “corrugated” surfaces of revolution

The rigidity is proven of certain surfaces of revolution with infinite alternation of portions of positive and negative curvature.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 517–522, October, 1973.     

The nearly horizontally rolling of a thick disk on a rough plane

In the paper the analytical solution of the partially linearized equations of motion of nearly horizontally rolling of a thick rigid disk on a perfectly rough horizontal plane under the action of gravity is given in terms of Whittaker functions. The solution is used to obtain the asymptotic solutions for a very small inclination angle, the study of unilateral contact between a disk and a plane and the study of disk colliding motion.      

The amalgamated duplication of a ring along a multiplicative-canonical ideal

After recalling briefly the main properties of the amalgamated duplication of a ring R along an ideal I, denoted by , see M. D’Anna and M. Fontana, to appear in J. Algebra Appl., we restrict our attention to the study of the properties of , when I is a multiplicative canonical ideal of R, see W. J. Heinzer, J. A. Huckaba and I. J. Papick, Comm. Algebra. In particular, we study when every regular fractional ideal of is divisorial.     

On the euclidean dimension of a wheel

Following Erdös, Harary, and Tutte, the euclidean dimension of a graphG is the minimumn such thatG can be embedded in euclideann-spaceR ⁿ so that each edge ofG has length 1. We present constructive proofs which give the euclidean dimension of a wheel and of a complete tripartite graph. We also define the generalized wheelW _m,n as the join and determine the euclidean dimension of all generalized wheels.     

A computer simulation of a rail network

A computer simulation of a rail segment is presented. The goal is to provide a capability for scheduling and routing with respect to predetermined objectives. The simulation is founded on a decomposition of the given line segment into fundamental units representing node to node subsegments with each node being an interlocking of the real system. A decision subroutine is activated every time a train reaches a node; all feasible options are then examined with respect to the current configuration of the system. Ultimately, it is hoped the simulation will have on-line monitoring capabilities.