MATHEMATICAL MODELLING OF OPERATIONAL STABILITY OF SOWING MACHINES' MECHANICAL SYSTEMS

This study addresses the construction of mathematical models for the operational stability of mechanical systems in agricultural seeders. The objects of research are the coulter systems of seeders with a support-packer roller and direct sowing planters, which are the disc working bodies of most sowing machines. The complexity in mathematical modeling of systems stability is due to a significant number of factors whose values are of variable and random character. To model them, it is proposed to explore the stability of systems based on their control parameters: lengths and angles of inclination, installation height of nodes and parts of a seeder, etc. The endogenous and exogenous parameters that include the following: step at sowing, soil surface depth and its properties, dimensions and weight of seeds, etc., are fixed at preset limits, in accordance with agricultural conditions. Difficulties in solving such systems of differential equations and obtaining the analytical solutions are explained by the condition for an inverse problem: the forces that act on a system must by in a constant equilibrium. The proposed modeling of system stability is based on the Lyapunov second method, which implies the construction and investigation of functions of perturbed motion at variable control parameters. We have established expressions for determining the stage of asymptotic stability of the system, which are characterized by the magnitude of time and distance that are required to return it to the unperturbed state. The obtained resulting mathematical expressions allowed us to establish significant factors: the length of a hitch, the rigidity of a spring, the inclination angle of a hitch, distance to the point of fastening a spring rod, which define the perturbed path of the coulter system. The result of modeling is the obtained damping character of the perturbed motion of the examined systems, as well as the established dependences of perturbed path of coulter systems on the above-specified parameters. To test the adequacy of the derived mathematical expressions for determining the stability of sowing machines, to refine the ranges in the variation of significant parameters, we carried out an experimental research. We employed the procedure for a multi-factorial experiment. As a result, with the help of the developed modelling method and based on the experiment conducted, we determined the rational parameters for working bodies of sowing machines of different types, which could be applied for efficient operation and in the design process of similar machine