Mathematical Model of Interaction of Free Rolling Flat Disk With Soil

Abstract

When constructing a mathematical model for the interaction of a free-rotating plane disk with soil, it is necessary to take into account that the magnitude of its kinematic parameter, equal to the ratio of its circumferential velocity to the speed of translational movement of the disk, is not a given quantity, but a definite quantity. With uniform rotation of the disk and its translational movement at a constant speed, the kinematic parameter is determined from the equilibrium equation of external forces, applied to the disk. The generalized mathematical model of disk-soil interaction, proposed earlier, was taken into account, but in view of relative complexity it was not widely used. The aim of the study is to construct a simpler, but adequate mathematical model for the interaction of a free-spinning disc with soil. The model is constructed under the assumptions of the constancy of the translational velocity of the disk, the permanence of its penetration and the possibility of replacing the pressure on the disc’s lateral surfaces with its mean value and replacing the force per unit length of the blade with its mean value. Since the distribution of the elementary forces of soil reactions is ultimately determined by the distribution of the absolute velocities of the points of the disk in contact with the soil, the resultant reactions of the soil and their total moment are functions of the kinematic parameter of the disk and its relative burial. These functions are given by integrals, that are not expressed in terms of elementary functions by a finite number of operations. However, the proximity of the kinematic parameter of the free-spinning disk to unity makes it possible, with the help of an estimate of these integrals, to obtain approximate expressions in terms of elementary functions for the resultant reactions of the soil and their total angular momentum. It is shown that the accuracy of the approximations obtained is sufficient for engineering practice.

An approximating mathematical model of interaction between a freely rotating disk and soil

“…This model has not found wide application because of its relative complexity: power characteristics of the disk are determined in it by means of elliptic integrals. A simplified model of FRDK interaction with soil was built relatively recently [25]. It has made it possible to obtain approximate expressions for elliptic integrals under condition that l » 1.…”

“…As noted in [8], the moment of friction forces in the disk bearing is negligible. Therefore, the kinematic parameter, l, is determined from the condition of equality to zero of the sum of the resultant moments of the soil reaction forces acting on the disk blade and its side faces which takes the following form [12,25] when taking into account 23:…”

“…The moment m O * is expressed through these parameters with the help of (20) and (6) through «non-taken» integrals that are in equalities (8) and (3). Therefore, this equation (25) has to be solved numerically for different values of the dimensionless x and n parameters with respect to the l parameter. As a result of numerical solution of this equation, the function l * = l * (x, n) is found.…”

“…It follows from equalities (27), (25) and 24that the kinematic parameter of the FRDK can be found as the extremum point of the power consumptions necessary for overcoming the friction forces acting on its side face and the forces of soil crushing with the DK blade. Since it follows from equality (27) that:…”

“…Therefore, the kinematic parameter of the FRDK is easily determined from equation ∂ ∂ = W * l 0 depending on the relative penetration depth and the dimensionless dynamic coefficient n which reflects soil properties. The computational experiment consisted in determining the kinematic coefficient, l, using numerical solution of equation (25). The moment M O * was expressed in terms of parameters l and x with the help of equalities (20) and (17).…”