Mathematical model of the cross-section of wheat grain

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.

Keywords

cross-section of wheat grain / Pascal’s snail / cardioid / centroid coordinates / cross-sectional area of grain

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