Building a mathematical model of the oscillatory movement of a frontal installed machine for the beet’s greens harvesting

Abstract

Purpose. It is improving the front-mounted stubble harvester work quality due to the reasonable selection of its parameters by modeling the minimization of the amplitude of the angular oscillations when moving along the unevenness of the field surface. Methods. Theoretical studies were carried out using the basic principles of higher mathematics and theoretical mechanics, particularly the dynamics of a mechanical system. Numerical calculations and graphical dependences were obtained using a PC using PTC Mathcad 15.0 and Grapher 20.1 software environments. Experimental studies and processing of the obtained laboratory and field data were carried out according to standard methods. Results. Based on the developed equivalent scheme of the unit for the beet’s greens harvesting, the differential equation of the angular oscillations of the beet’s greens harvesting machine in the form of a LaGranzh of the second kind was drawn up relative to the centre of instant rotation of the hinged mechanism of the aggregating tractor. Due to the numerical solution of this equation, it was established that, with increasing the speed of the translational movement of the unit from 1.5 to 2.5 m∙c-1 amplitude of oscillations of the angle of rotation of the frontal harvesting machine increases from 0.88° at a speed 1.5 m∙c-1 to 1,18° at speed of movement 2.5 m∙c-1. However, with the unit’s direction for the beet’s greens harvesting at a speed of 1.5 m∙c-1 the length of the transition process is 0.22 c. With the unit’s movement at a speed of 2.5 m∙c-1, the value of the estimated indicator is desirable to the mark of 0.14 c, i.e. by 36%. This means that in the range of motion speeds of the machine for the beet’s greens harvesting 1.5–2.5 m∙c-1, the preference should be given to more values of this parameter. To reduce the amplitude of the angular oscillations of the machine for the beet’s greens harvesting in the longitudinally-vertical plane, its musculoskeletal wheels must be equipped with stringent tires and the most distant from the instantaneous center of rotation of the frontal tingling mechanism of the tractor. The central thrust of this mechanism should be installed with as much angle of inclination to the vertical. It is proved that the greater the period of fluctuations of the longitudinal profile of the field with the roots of sugar beet, the more stable the movement of the front-frontically aggregated machine for the beet’s greens harvesting in the longitudinal-vertical plane. Experimental studies show that the amplitude of oscillations of the lower ends of the knives of the rotor cutting apparatus up to 5 cm of the loss of bite is slight and increases significantly if the amplitude of such oscillations exceeds the specified value. Conclusions 1. In the range of motion speeds of the machine for the beet’s greens harvesting 1.5–2.5 m∙c-1, the preference should be given to a larger value of this parameter. This will ensure good productivity of its work with a slight increase in the amplitude of angular vibrations and small (in absolute dimension), but the desired decrease in the duration of the racing and acting of the output of the dynamic system on a stable mode of technological functioning. 2. Reducing the value of the coefficient of the rigidity of the copiers of the copies of the machine for the beet’s greens harvesting causes angular vibrations with slightly larger positive and much less negative amplitude. 3. The more rigid the pneumatic tires of the machine copiers for the beet’s greens harvesting, the smaller the angle (in its positive measurement) at it deviates from its static position. At the same time, due to the relative vigour of this deviation, it is possible to state the invariance of the machine of the harvesting of the beet’s greens to change the values of the coefficient μ of the decrement of pneumatic tires of its copy wheels. 4. To reduce the amplitude of the angular oscillations of the machine for the beet’s greens harvesting in the longitudinal-vertical plane, its copy wheels should be, as far as possible, from the instantaneous center of rotation of the frontal hinge mechanism of the tractor. The central draft of this mechanism should be installed with as much angle of inclination to the vertical. 5. It is established that the greater the period of oscillations of the longitudinal profile of the soil surface in the rows of sugar beets roots, the more stable the movement in the longitudinal-vertical plane is the frontally installed on the aggregating tractor of the machine for the beet’s greens harvesting. 6. For the amplitude of oscillations of the lower ends of the rotor machine knives for the beet’s greens harvesting up to 5 cm, the loss of the beet’s greens is insignificant. However, with the increase in the amplitude of these fluctuations, such as 7 cm, these losses will already be 12%. Keywords: sugar beets, beet's greens harvesting, mathematical model, differential equation, amplitude, frequency, parameters.