Theory of vertical oscillations and dynamic stability of combined tractor-implement unit
¹National University of Life and Environmental Sciences of Ukraine, 15, Heroyiv Oborony Str., UK 03041 Kyiv, Ukraine
²National Scientific Centre, Institute for Agricultural Engineering and Electrification, 11, Vokzalna Str., Glevaкha-1, Vasylkiv District, UK 08631 Kiev Region, Ukraine
³Estonian University of Life Sciences, Kreutzwaldi 56, EE51014 Tartu, Estonia
⁴Tavria State Agrotechnological University of Ukraine, Khmelnytskoho pr. 18, Melitopol, UK 72312 Zaporozhye region, Ukraine
Currently, throughout the world quite extensive use is made of combined tractor-implement units, which are capable of performing several process operations in the same pass. At the same time, the state-of-the-art ploughing and general-purpose tractors that can carry as front- so rear-mounted implements and accordingly feature both the front and rear PTOs, also able to travel efficiently as forward so in reverse gear, are most suited for the performance of such operations. Authors developed and successfully tested a combined tractor-implement unit on the basis of a wheeled ploughing and general-purpose tractor, which can in one pass efficiently chop the after harvesting crop residues with a front-mounted rotary chopper and simultaneously perform tillage with a rear-mounted plough. The aim of this study is the elaboration of the theoretical basis for the process of vertical oscillation of the combined ploughing and chopping tractor-implement unit and the validation of its dynamic stability in the longitudinal and vertical plane. The research has been performed with the use of the methods of designing the analytical mathematical models of functioning of agricultural machines and machine assembly units based on the theory of tractor, the vibration theory, the theory of automatic control and dynamic stability and the methods of computer programme construction and PC-assisted numerical computation. The dynamics of the said unit have been studied basing on the analysis of the amplitude frequency characteristics of the unit as a dynamic system responding to external perturbations appearing in the form of soil surface irregularities. Following the results of the undertaken analytical study, first the equivalent schematic model of the discussed combined tractor-implement unit in the longitudinal and vertical plane was developed, the unit’s characteristic points were defined, the linear and angular displacements specified and acting forces applied. Each pneumatic-tyre wheel of the unit represented by its elastically damping model had point contacts with the soil surface irregularities defined by the respective elevations. Using the original dynamic equations in the form of the Lagrange equations of the second kind, first we defined the generalised coordinates and the formulae for the kinetic and potential energy, dissipation functions and generalised forces, then, after performing the necessary transformations, we set up the system of four differential equations, which described the motion of the dynamic system under consideration. Further, we applied the Laplace transformations to the obtained differential equation system, which provided for obtaining the system of equations in the operator form and preparing them for the representation suitable for PC-assisted numerical calculations with the use of the developed computer programme. In accordance with the numerical computation results, graphs were plotted for the amplitude and phase frequency response characteristics of the tractor’s vertical oscillations at different stiffness coefficients of its steering wheels, the amplitude frequency response characteristics of the chopper’s oscillations depending on its mass and its support wheel tyres’ stiffness coefficient as well as the characteristics of the plough’s oscillations at different stiffness coefficients of its pneumatic-tyre ground support wheel.