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Register a new userOn the Normal Force and Static Friction Acting on a Rolling Ball Actuated by Internal Point Masses
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The goal of this paper is to investigate the normal and tangential forces acting at the point of contact between a horizontal surface and a rolling ball actuated by internal point masses moving in the balls frame of reference. The normal force and static friction are derived from the equations of motion for a rolling ball actuated by internal point masses that move inside the balls frame of reference, and, as a special case, a rolling disk actuated by internal point masses. The masses may move along one-dimensional trajectories fixed in the balls and disks frame. The dynamics of a ball and disk actuated by masses moving along one-dimensional trajectories are simulated numerically and the minimum coefficients of static friction required to prevent slippage are computed.
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The motion of a rolling ball actuated by internal point masses that move inside the balls frame of reference is considered. The equations of motion are derived by applying Euler-Poincares symmetry reduction method in concert with Lagrange-dAlemberts principle, which accounts for the presence of the nonholonomic rolling constraint. As a particular example, we consider the case when the masses move along internal rails, or trajectories, of arbitrary shape and fixed within the balls frame of reference. Our system of equations can treat most possible methods of actuating the rolling ball with internal moving masses encountered in the literature, such as circular motion of the masses mimicking swinging pendula or straight line motion of the masses mimicking magnets sliding inside linear tubes embedded within a solenoid. Moreover, our method can model arbitrary rail shapes and an arbitrary number of rails such as several ellipses and/or figure eights, which may be important for future designs of rolling ball robots. For further analytical study, we also reduce the system to a single differential equation when the motion is planar, that is, considering the motion of the rolling disk actuated by internal point masses, in which case we show that the results obtained from the variational derivation coincide with those obtained from Newtons second law. Finally, the equations of motion are solved numerically, illustrating a wealth of complex behaviors exhibited by the systems dynamics. Our results are relevant to the dynamics of nonholonomic systems containing internal degrees of freedom and to further studies of control of such systems actuated by internal masses.
The controlled motion of a rolling ball actuated by internal point masses that move along arbitrarily-shaped rails fixed within the ball is considered. Application of the variational Pontryagins minimum principle yields the balls controlled equations of motion, a solution of which obeys the balls uncontrolled equations of motion, satisfies prescribed initial and final conditions, and minimizes a prescribed performance index.
The controlled motion of a rolling ball actuated by internal point masses that move along arbitrarily-shaped rails fixed within the ball is considered. The controlled equations of motion are solved numerically using a predictor-corrector continuation method, starting from an initial solution obtained via a direct method, to realize trajectory tracking and obstacle avoidance maneuvers.
In rubber friction studies it is often observed that the kinetic friction coefficient {mu} depends on the nominal contact pressure p. We discuss several possible origins of the pressure dependency of {mu}: (a) saturation of the contact area (and friction force) due to high nominal squeezing pressure, (b) non-linear viscoelasticity, (c) non-randomness in the surface topography, in particular the influence of the skewness of the surface roughness profile, (d) adhesion, and (e) frictional heating. We show that in most cases the non-linearity in the {mu}(p) relation is mainly due to process (e) (frictional heating), which softens the rubber, increases the area of contact, and (in most cases) reduces the viscoelastic contribution to the friction. In fact, since the temperature distribution in the rubber at time t depends on on the sliding history (i.e., on the earlier time t0 < t), the friction coefficient at time t will also depend on the sliding history, i.e. it is, strictly speaking, a time integral operator. The energy dissipation in the contact regions between solids in sliding contact can result in high local temperatures which may strongly affect the area of real contact and the friction force (and the wear-rate). This is the case for rubber sliding on road surfaces at speeds above 1 mm/s. In Ref. [14] we have derived equations which describe the frictional heating for solids with arbitrary thermal properties. In this paper the theory is applied to rubber friction on road surfaces. Numerical results are presented and compared to experimental data. We observe good agreement between the calculated and measured temperature increase.
In this work we propose an extension to the analytical one-dimensional model proposed by E. Gnecco (Phys. Rev. Lett. 84:1172) to describe friction. Our model includes normal forces and the dependence with the angular direction of movement in which the object is dragged over a surface. The presence of the normal force in the model allow us to define judiciously the friction coefficient, instead of introducing it as an {sl a posteriori} concept. We compare the analytical results with molecular dynamics simulations. The simulated model corresponds to a tip sliding over a surface. The tip is simulated as a single particle interacting with a surface through a Lennard-Jones $(6-12)$ potential. The surface is considered as consisting of a regular BCC(001) arrangement of particles interacting with each other through a Lennard-Jones $(6-12)$ potential. We investigate the system under several conditions of velocity, temperature and normal forces. Our analytical results are in very good agreement with those obtained by the simulations and with experimental results from E. Riedo (Phys. Rev. Lett. 91:084502) and Eui-Sung Yoon (Wear 259:1424-1431) as well.
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