Engineering, Energy, Transport AIC

Andrii BABII – Doctor of Technical Sciences, Professor, Head of Engineering Mechanics and Agricultural Machines Department Ternopil Ivan Puluj National Technical University (вул. Руська, 56, м. Тернопіль, Україна, 46001, e-mail: avbabiytntu@gmail.com, https://orcid.org/0000-0001-6198-0100). 
Ivan VOVK – Postgraduate of Engineering Mechanics and Agricultural Machines Department Ternopil Ivan Puluj National Technical University (Ruska str., 56, Ternopil, Ukraine, 46001, e-mail: ivan_5479@ukr.net, https://orcid.org/0009-0000-5305-3550).

The paper proposes a mathematical model of contact interaction between the blade as the working body of a rotary shredder of plant residues and plant stems removed from the field surface. The method of removing such vegetation involves shredding it and scattering it over the field surface. The practical purpose of implementing the mathematical model is to determine the minimum circular speed of the working body when it interacts with the stems.
The mathematical model is based on the assumption that a plant stem can be modelled using Euler-Bernoulli beam theory. To theoretically reproduce the impact pulse interaction of the blade on the stem, a load approximation was used with delta functions having a narrow peak section. The model allows the deformation and oscillatory processes that occur in the stem after impact by the blade of the rotating working body to be evaluated. A solution to the Euler-Bernoulli equation for impulse impact has been constructed. The obtained expressions for the transverse displacements of the beam axis point have a modal decomposition for impulse impact. Since real stems have actual energy dissipation, modal damping, expressed by the corresponding coefficient, has been introduced into the model. For the numerical solution of the problem, a truncated series is considered: 10 modes for smooth pulses and 50 modes for very short pulses. This number of modes already gives a good approximation of the displacement and depends on the geometry and parameters of the interacting bodies.
The result of the numerical experiment is to obtain the values of the maximum bending moments that cause bending stress and compare them with the critical stresses that cause stem failure. For the reliability of the results obtained, the safety factor for bending has been increased to 1.2, which will ensure guaranteed stem failure. Based on this, the minimum speed of interaction between the blade and the stem when the breaking condition is met was found. This speed is 8-10 m/s, at which the stems will only be broken, but not crushed. Imposing the condition of shredding the stems into particles with a length of approximately 0.03 m, a series of peripheral circumferential speeds of the rotary working element's blade was obtained for various ground speeds of the machine. For example, at a forward speed of the unit of 3.0 km/h (0.83 m/s), the circular speed of the blade should be 12-14 m/s; for a forward speed of the unit of 4.0 km/h (1.11 m/s) – 16-18 m/s; for 5.0 km/h (1.39 m/s) – 20-22 m/s; for 6.0 km/h (1.67 m/s) – 24-28 m/s. Thus, the implementation of the mathematical model of the interaction between the blade of the rotary working element and the stems of the plants being removed, based on the Euler–Bernoulli beam model, allowed us to substantiate the minimum circumferential speed of the working element for various forward speeds of the unit and the required length of the shredded material.

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