Expression For Friction Force Research Articles

Shaping the trajectory of the billiard ball with approximations of the resultant contact forces

This paper presents mathematical models of the contact pressure distribution on a circular contact area and the corresponding rolling resistance. Hertzian pressure distribution is distorted in a special way in order to move rolling centre outside the geometrical centre of the contact area. With the assumption of fully developed sliding and classical Coulomb friction law on each element of the contact, integral models of the total friction force and moment reduced to the contact centre are given. In order to improve the convenience of use of the contact models in numerical simulations of rigid body dynamics and decrease their computational cost, special approximations of the integral models of friction force and moment are proposed. Moreover, special modifications of the corresponding expressions for friction forces and rolling resistance are proposed, which allows avoiding their singularities for vanishing relative motion of the contacting bodies. The application of the proposed contact models in mathematical modelling of a rigid ball rolling and sliding over a deformable table is presented. Furthermore, possibilities of use of the developed simulation models in shaping the billiard ball's trajectory are presented.

QUASI-STATIONARY TWO-DIMENSIONAL WAVE FILTRATIONAL FIELDS OF PRESSURE FOR THE GIVEN HARMONIC DISTURBANCES ON THE BOUNDARY

In this paper the use of average asymptotic method developed by the authors before to study filtration- wave fields in the oil and gas reservoirs is illustrated. To prove it, the problem of wave field in the half space presented a three-layer medium with different filtration properties has been discussed in the paper. The central region is a semi-infinite layer bounded by two parallel half-planes. Pressure perturbations that are the source of seepage waves in a half space are taken on the boundary of a layer, and on the edges of environments perturbations these are absent. Each of the three media is homogeneous and isotropic. At the boundaries of the contact areas the equality of pressures and the flow of fluid are chosen. The main task is determining the field of pressure in each of the media. This can be achieved by solving the conjugation problem for the wave equation with damping. The equation is derived on the basis of the laws of conservation of mass and momentum, presented in the form of the equation of continuity and motion, in which the effect of friction forces has been taken into consideration. The expression for friction force is obtained on the basis of Darcy's law. To find the pressure wave fields the desired functions are presented in the form of asymptotic formulas by which the original problem is reduced to the interface simpler for the coefficients of the asymptotic expansion. However, the problems for the coefficients contain traces of derivatives for external areas nonlocal boundary conditions, and therefore they are non-classical. The simple analytical dependences for calculation fields in the formations for zeroth and first coefficients have been found. It is shown that a zero expansion coefficient equivalent describes the plane wave, for the range of the central reservoir pressure does not depend on the vertical position; with the first factor provides the main part of the amendment, specifying the geometry of the wave front. The accuracy of this method has been justified by comparison of the asymptotic solutions with the expansion coefficients of the exact solution of the problem in the space of a parameterized Fourier sine transform in the Maclaurin series for the formal parameter.

Nernst--Planck analog equations and stationary state membrane electric potentials.

A set of coupled nonlinear differential equations, determining the concentration profiles and electric potentials valid for isothermal transport of ions and molecules across a diffusion barrier are formulated, using a correction to the limiting expression for chemical potential gradients and the molecular expression for frictional force. These differential equations are similar to Nernst-Planck equations and reduce to these under appropriate approximations. Solutions of these equations valid under specified conditions are presented. Expressions for permeability, concentration profiles of many ion systems are included.