Theoretical investigation in the stability of motion of the asymmetric top-gathering machine

Abstract

Goal. To determine based on numerical computer modeling of the obtained analytical dependences the parameters of stable plane-parallel motion of an asymmetric top-gathering machine (TGM). Methods. Methods of higher mathematics for the transformation of systems of differential equations, the theory of stability of motion, automatic control, the study of the behavior of tracking systems, a compilation of computer programs, numerical calculations, construction of graphical dependencies and their analysis are used. Results. According to the results of the analytical study, the values are gained of the amplitude and phase-frequency characteristics of the oscillations of the course angle of the tractor during the reproduction of the perturbing unit in the form of oscillations of the angle of rotation of the hitch in the horizontal plane. Based on mathematical modeling data the necessary graphic dependences are constructed. Their analysis made it possible to establish the influence of the working speed of the test unit, the values of the input coefficients of the tires of the front and rear wheels of the tractor, as well as the layout of the power tool (universal or integrated) on the stability of the test unit. Conclusions. Based on the received system of differential equations the stability of movement of the asymmetric TGM during the performance of the technological process is investigated. Quantitative indicators of this stability are the values of the amplitude and phase-frequency characteristics of the oscillations of the course angle of the tractor during the development by the unit of external perturbation in the form of oscillations of the angle of rotation of the TGM. Calculations have shown that reducing the value of the input coefficient of the tires of the tractor wheels increases its sensitivity to the effects of perturbation. This sensitivity is greater the closer the wheels of the power tool are located to the point of attachment of the trailed TGM. From tractors of classical and integral arrangement at the assembly of the asymmetric TGM, the preference should be given to the first one. According to the results of mathematical modeling, the qualitative increase in the speed of the tractor based on tractors of both schemes from 1.5 to 2.5 m/s leads to an undesirable increase in amplitude-frequency response and the desired increase in phase-frequency response during their reproduction of external disturbance in the form of oscillations of the angle of rotation of the TGM. At the same time, in quantitative terms, the sensitivity of the unit based on a classic tractor to the effect of perturbing on the amplitude is 11–12% less, and the phase shift — almost the same, which is the best option in terms of stability of the tractor in the horizontal plane.