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.     

A model of a reinforced tyre with a rod protector

A tyre design consisting of a steel-cord-reinforced rigid bond with sides connected to the wheel disc and a protector(tread) in contact with the road is examined. The tread is in the form of a set of rods connected by one end to the band, with the other end either free or in contact with the road. The rod end in contact with the road is acted upon by a force applied from the road, represented by a force normal to the road plane and a shear force due to dry friction. If the modulus of the shear force does not exceed the magnitude of the normal force multiplied by the dry friction coefficient, there is no slip at the contact point. In the opposite case, the rod end will be displaced along the road by an amount sufficient to distribute the normal and shear forces. The dynamics of longitudinal and transverse strains of the rods in contact with the road is analysed using the motion separation method in the quasi-static approximation. The behaviour of the tread rods as a function of the vertical displacement of the wheel centre is investigated, the contact area is found and the conditions are determined under which the contact area is divided into parts in which either slip of the rod ends occurs or does not occur, depending on the magnitude of the longitudinal displacement of the wheel centre or its turning relative to the horizontal axis. An analogue of a continuous model of a rod-like tread is considered, and the magnitudes of the forces and moments are found as a function of the wheel disc displacements. The equations of wheel rolling are obtained, and the conditions under which steady motions exist are found.     

Cornerning radial tyres

An angular difference between direction of wheel motion and wheel plane produced a lateral force on a rolling tyre. A simple mechanism of shear stress generation in the contact area of tread is used. The coefficient of friction plays a limiting role. The resulting lateral force and self-aligning torque are computed by means of the belt model of a radial tyre. Finally, some numerical results are given to illustrate the theory.     

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 motion of a railway wheelset

The rolling of a railway wheelset along rails without slipping is investigated taking the creep hypothesis into account. The wheelset is represented by two cones that have a common base, and the rails are represented by two circular cylinders with parallel axes. The kinematic characteristics of the unperturbed rolling motion of the wheelset, which occurs when the centre of mass moves along a straight line, and of the perturbed motion, which occurs when the centre of mass of the wheelset describes a sinusoidal trajectory, are determined. The constraint reactions are found for the motions investigated up to small second-order values of the perturbed variables. When the elastic properties of the material in the contact area are taken into account, the creep hypothesis is used, averaging over the fast variables is employed, and the value of the critical speed, above which the rectilinear rolling of the wheelset becomes unstable, is found using averaged equations. In the latter case a periodic mode with two time intervals when the wheel flanges come into contact with the rails is investigated. The reaction force, the work of the dry friction force, and the moment of the active forces needed to maintain the periodic mode are found at the flange/rail contact point within the dry friction model. The boundaries of the stability regions, the parameters of the periodic mode and the moment of the resistance forces as functions of the problem parameters are determined from the formulae obtained by analytical methods.     

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.     

An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane

This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.     

Modelling and control of the motion of a disk rolling on a sloping plane

This work deals with the stabilization and control of the motion of a disk rolling on a sloping plane. It is assumed here that the motion of the disk is controlled by a tilting moment, a directional moment, and a pedalling moment. By using a kind of an inverse control transformation a control strategy is proposed under which the motion of the disk is stabilized and is able asymptotically to track any smooth trajectory which is located on the sloping plane.     

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.     

Efficient Calculation of Tread Block Vibrations

A model is presented that allows to calculate the dynamics of tyre tread blocks. The numerical efficiency is achieved by the combination of static and important modal ansatz functions, also known as Hurty/Craig/Bampton transformation. Nonlinear single point contacts provide the contact forces and consider the surface roughness. The non-smooth friction characteristic is approximated an arctan function and a parameter study of the slope parameter is conducted. Typical stickslip limit cycles of the tread block are shown. These phenomena can have a part in the tyre/road noise. Experimental results qualitatively agree with the modelled tread block vibrations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of steady motions of a mobile robot with roller-carrying wheels and a displaced centre of mass

The motion of a mobile three-wheel robotic vehicle on a horizontal surface is investigated. Passive rollers are fastened along the rim of each wheel, enabling each wheel not only to roll in the usual manner, but also to move perpendicular to its plane. Each of these wheels, as well as the ordinary wheels, is equipped with one drive, which rotates the wheel about its axis. The vehicle equipped with roller-carrying wheels can move in any direction with any orientation. The motion of the robot on a horizontal surface is studied in the case where the centre of mass of the robot deviates from the geometric centre of the triangular platform, and there is no slip at the points of contact of the rollers with the supporting surface. In the case of free motion of the robot, an additional first integral is pointed out and the exact solution found is analysed. An equation for specifying steady motions, under which a constant voltage is supplied to the DC motors that drive the wheels, is constructed. The stability of the rectilinear motion of the robot is investigated.     

Stabilization and control of the motion of a rolling disk

This work deals with the stabilization and control of the motion of a disk rolling on the horizontal plane. It is assumed here that the motion of the disk is controlled by a tilting moment, a directional moment, and a pedalling moment. By using a kind of an inverse control transformation, a control strategy is proposed under which the motion of the disk is stabilized and is able asymptotically to track any given smooth ground trajectory.     

Dynamics-Based Motion Planning for a Pendulum-Actuated Spherical Rolling Robot

This paper deals with the dynamics and motion planning for a spherical rolling robot with a pendulum actuated by two motors. First, kinematic and dynamic models for the rolling robot are introduced. In general, not all feasible kinematic trajectories of the rolling carrier are dynamically realizable. A notable exception is when the contact trajectories on the sphere and on the plane are geodesic lines. Based on this consideration, a motion planning strategy for complete reconfiguration of the rolling robot is proposed. The strategy consists of two trivial movements and a nontrivial maneuver that is based on tracing multiple spherical triangles. To compute the sizes and the number of triangles, a reachability diagram is constructed. To define the control torques realizing the rest-to-rest motion along the geodesic lines, a geometric phase-based approach has been employed and tested under simulation. Compared with the minimum effort optimal control, the proposed technique is less computationally expensive while providing similar system performance, and thus it is more suitable for real-time applications.     

The rolling of a wheel along a corrugated rail

The rolling without detachment of a rigid massive wheel, carrying a static load, along a rail with undulations on the running surface, which arises as a result of non-uniform wear, is investigated. The rail is supported by an elastoviscous base. Because of the inertia of the wheel and the carriage the horizontal component of the velocity of the wheel centre differs only slightly from a constant quantity, and hence the motion of the wheel along the rail is assumed to be uniform. Steady vertical vibration of the wheel is considered. The vertical coordinate of the wheel centre, and also the difference between the longitudinal coordinates of the wheel centre and the point of contact of the wheel and the rail, are periodic and, correspondingly, even and odd functions of the longitudinal coordinate of the wheel centre, and their period is equal to the wave length on the rail surface. The periodic force of interaction of the wheel and the rail is given in the form of a Fourier series. Short waves, the amplitude of which is much less than their length, are often observed on the rail surface, and this length is much less than the wheel radius. In this case the coefficients of the Fourier series are expressed in terms of Bessel functions of the first kind of integer order. Observations show that the depth of the short wave on the rail surface increases until the radius of curvature in the rail trough approximates to the wheel radius, and hence it is assumed that these radii are close to or equal to one another. In this case the trajectory of the wheel centre differs considerably from the wave on the rail surface.     

A qualitative analysis of the dynamics of a disc on an inclined plane with friction

The problem of the motion of a disc on an inclined plane with dry friction is investigated. It is shown that, if the friction coefficient is greater than the slope of the plane, the disk will come to rest after a certain finite time, and its sliding and rotation will cease simultaneously. The limit position of the instantaneous centre of velocities is indicated. The limit motions of the disc in the case when the ratio of the friction coefficient to the slope of the plane is equal to or less than unity: uniform sliding (in the case of a general position) and equiaccelerated sliding (always) of the disc along the line of greatest slope of the plane, respectively, are obtained. The case when the friction coefficient is equal to the slope, while the initial sliding velocity is directed upwards along the line of greatest slope, is an exception. In this case, the disc comes to rest after a finite time, and the sliding velocity and the angular velocity of the disc vanish simultaneously.     

The vibrations of a free and loaded tyre

Using the model of a wheel with a reinforced tyre [Vil’ke VG, Kozhevnikov IF. A model of a wheel with a reinforced tyre. Vestnik MGU. Ser.1. Matematika Mekhanika 2004;4:37–45], the natural frequencies and natural forms of vibrations of a free or loaded tyre in the neighbourhood of the equilibrium position are determined. The spectrum of natural frequencies and natural forms of vibration are found analytically for a free (unloaded) tyre with a fixed disc. A similar problem is solved numerically in the case of a loaded tyre. The results of this analysis can be used to estimate the level of noise which occurs when a vehicle moves on an uneven surface.     

The optimum rectilinear motion of a two-mass system

The forward rectilinear motion of a system of two rigid bodies along a horizontal plane is considered. Forces of dry friction act between the bodies and the plane, and the motion is controlled by internal forces of interaction between the bodies. A periodic motion in which the system moves along a straight line is constructed. The optimum parameters of the system and a control law are found corresponding to the maximum mean velocity of motion of the system as a whole.     

The brachistochrone problem for a disc

The motion of a vertical disc along a curve under the influence of gravity is investigated. On the assumption of regular rolling without slip and separation of contact points, the problem of plotting the curve of most rapid motion of the disc centre from the origin of coordinates to an arbitrary fixed point of the lower half-plane is solved. As usual, the velocity at the initial instant of time is zero, and at the final instant of time it is not fixed. In explicit parametric form, the classical brachistochrone for contact points of the disc is plotted and investigated. The response time, trajectory and kinematic and dynamic characteristics of motion are calculated analytically. Previously unknown qualitative properties of regular rolling are established. In particular, it is shown that the disc centre moves along a cycloid connecting specified points. The envelopes of the boundary points of the disc, produced as its centre moves along the cycloid, are brachistochrones. The feasibility of mechanical coupling of the disc and the curve by reaction forces at the contact point (the normal pressure and dry friction) is investigated.