Use of Euler equations in research into three-dimensional oscillations of sugar beet root during its vibration-assisted lifting
¹National University of Life and Environmental Sciences of Ukraine, 15, Heroyiv Oborony Str., 03041 Kyiv, Ukraine
²National Scientific Centre ‘Institute for Agricultural Engineering and Electrification’, 11, Vokzalna Str., Glevaкha-1, Vasylkiv District, 08631 Kiev Region, Ukraine
³Estonian University of Life Sciences, Institute of Technology, Kreutzwaldi 56, EE51014 Tartu, Estonia; *Correspondence: email@example.com
⁴Slovak University of Agriculture in Nitra, 2, Trieda Andreja Hlinku, 949 76 Nitra, Slovakia
Following the results of the research into the physical process of the vibratory interaction between the digging tool and the beet root, it has been found that the latter, while standing in soil, i.e. amid an elastic medium, has strong attachment to the soil in its lower (the densest and driest) part, which virtually implies one conventional fixed point. This finding provides the basis for examination of the three-dimensional motion of the beet root’s body during its lifting from the ground in case of its asymmetric interaction with one of the shares of the vibrating digging tool. We have studied the gyration of the beet root’s body about a point initiated by its interaction with the inclined face of the vibrating digging tool share that makes oscillatory movements in the longitudinal vertical plane. The aim of the study is to establish the values of the angular displacements of the root’s body at the moment of its getting in asymmetric contact with the vibrating digging tool followed by the breaking of its bonds with the surrounding elastic medium, i.e. to develop a new mathematical model of the vibration-assisted digging of a beet root out of the soil. Basing on the use of the original equations of Euler, a new differential equation system has been obtained, which facilitates the analytical treatment of the mentioned work process. That system of differential equations for the three-dimensional oscillations of the root caused by the action of a perturbing force comprises three dynamic and three kinematic equations. It is a determined system, which makes possible its solution, i.e. the numerical modelling of the process of root lifting from the ground under different digging conditions, because it includes all necessary parameters of the vibrating digging tool, the sugar beet root and the soil surrounding it.