Mathematical model of the cross-section of wheat grain

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Abstract

BACKGROUND: When studying the optimal length of the holes in the lattice bottom of the inclined chamber of a combine harvester, which ensures preliminary separation of the combed grain heap, the cross section of wheat grain was modeled in the shape of a separate ball or a cut cylinder. This is due to the fact that the description of the technological process is significantly simplified with this shape of grain. However, such models of the grain cross-section are very far from the real shape of the object, since the dorsal side of the grains is convex, and there is a longitudinal groove on the ventral side. The kind of surface closest to the real shape of the grain is the Pascal’s snail mathematical model. For this model, the centroid coordinates are determined, and equations are obtained for calculating its cross-sectional area and moments of inertia for each coordinate axis. Verification of the obtained equations in the KOMPAS-3D software showed that the discrepancy between the real and theoretically predicted values of the centroid coordinates is about 13%, which reduces the adequacy of the calculations and requires their refinement.

AIM: Refinement of the mathematical model of the cross-section of wheat grain shaped as the Pascal’s snail.

METHODS: The object of the study is a cross section of wheat grain shaped as the Pascal’s snail. When determining the centroid coordinates, methods of theoretical mechanics were used, and the resulting expressions were verified in the KOMPAS-3D three-dimensional modeling software.

RESULTS: Mathematical expressions for analytical calculation of the centroid coordinates are obtained for different versions of the Pascal’s snail: a = b (cardioid), a < b (the Pascal’s snail without an internal loop), a > b (the Pascal’s snail with an internal loop). Verification of the obtained expressions proves their adequacy, since the convergence of theoretical and experimental data is 100%.

CONCLUSIONS: The use of refined mathematical models of the cross-section of wheat grain can significantly simplify the modeling of the separation process of combed heaps, as well as to increase the accuracy of calculations. To simplify the description of this process, it is advisable to use the KOMPAS-3D three-dimensional modeling software.

About the authors

Victor V. Nikitin

Bryansk State Agrarian University

Author for correspondence.
Email: viktor.nike@yandex.ru
ORCID iD: 0000-0003-1393-2731
SPIN-code: 5246-6938
Scopus Author ID: 57201686117

Associate Professor, Dr. Sci. (Engineering), Head of the Technical Service Department

Russian Federation, 2a Sovetskaya street, 243365 Kokino, Vygonichsky District of Bryansk Oblast

Victor N. Ozhereliev

Bryansk State Agrarian University

Email: vicoz@bk.ru
ORCID iD: 0000-0002-2121-3481
SPIN-code: 3423-0991
Scopus Author ID: 57195608281

Professor, Dr. Sci. (Agriculture), Professor of the Technical Systems in Agrobusiness, Environmental Management and Road Construction Department

Russian Federation, 2a Sovetskaya street, 243365 Kokino, Vygonichsky District of Bryansk Oblast

Natalia V. Sinyaya

Bryansk State Agrarian University

Email: sinzea@yandex.ru
ORCID iD: 0000-0002-1794-1347
SPIN-code: 9225-4347

Cand. Sci. (Engineering), Associate Professor of the Technical Service Department

Russian Federation, 2a Sovetskaya street, 243365 Kokino, Vygonichsky District of Bryansk Oblast

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2. Fig. 1. Special cases of the cross-section of wheat grain shaped as the Pascal’s snail: a) a = b (cardioid); b) a < b (the Pascal’s snail without an internal loop); c) a > b (the Pascal’s snail with an internal loop).

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3. Fig. 2. Screenshot of the working window of the KOMPAS-3D software when determining the centroid of the cardioid (a = b = 20 mm).

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4. Fig. 3. Screenshot of the working window of the KOMPAS-3D software when determining the centroid of the Pascal’s snail at a = 20 mm and b = 30 mm.

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5. Fig. 4. The Pascal’s snail with an internal loop.

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6. Fig. 5. Screenshot of the working window of the KOMPAS-3D software when determining the centroid of the Pascal’s snail at a = 20 mm and b = 15 mm.

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