An Approximating Mathematical Model of Interaction Between a Freely Rotating Disk and Soil

A generalized mathematical model of disk interaction with soil was built under general assumptions regarding the mode of the disk knife motion in soil, namely, in a mode of slippage, skidding or rolling without slippage and skidding. Previously constructed models follow from it as particular cases at certain values of parameters. However, because of computational complexity of this model for the case of a freely rotating disk knife consisting in the need for a preliminary numerical solution of a transcendental equation to determine the mode of disk motion, the generalized mathematical model has not found wide application. Therefore, an analytical two-dimensional approximation of a generalized model of disk interaction with soil which is a new model of approximation type was constructed on the basis of a computer experiment using the least squares method.An explicit expression was obtained for the kinematic parameter of a freely rotating disk knife which determines its mode of motion. It was established that this parameter is a rational function of relative depth of the disk penetration and the dimensionless dynamic coefficient characterizing soil properties. Also, explicit expressions were obtained for the projections of the resultant soil reaction forces acting on the blade of the disk knife and its side faces depending on the data of dimensionless parameters. It has been established that the horizontal component of the reaction which determines tractive resistance of the disk is also a rational function of the relative penetration depth and the dimensionless dynamic coefficient. It was established that the magnitude of the kinematic parameter significantly affects the magnitude and direction of the resultant soil reactions to the disk. The expressions obtained make it possible to significantly simplify experiments to determine the resultant soil reaction forces to a freely rotating disk knife and reduce their required number. These expressions make it possible to carry out strength calculations of soil-cultivating working tools with disks and determine their optimal parameters according to the strength criteria and the minimum specific energy consumption with accuracy sufficient for engineering practice. Adequacy of the obtained expressions was confirmed by comparison with experimental data of the disk knife dynamometr