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 the wheat grain was modeled in the form of a separate ball or a cut cylinder. This is due to the fact that with this form of grain the description of the technological process is significantly simplified. 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 closest surface to the real shape of the grain is a mathematical model, which is Pascal's snail. For this model, the coordinates of the figures center of gravity are determined, and equations are obtained for calculating its cross-sectional area and moments of inertia for each coordinate axis. Checking the obtained equations in the KOMPAS-3D program showed that the discrepancy between the real and theoretically predicted values of the coordinates of the figures center of gravity is about 13%, which reduces the adequacy of the calculations and requires their clarification.
 METHODS: The object of the study is a cross section of a wheat grain, made in the form of Pascal's snail. When determining the coordinates of the figures center of gravity, methods of theoretical mechanics were used, and the resulting expressions were verified in the three-dimensional modeling system KOMPAS-3D.
 RESULTS: Mathematical expressions are obtained for analytically finding the coordinates of the centers of gravity for different versions of Pascal's snail: a = b (cardioid), a b (Pascal's snail without an internal loop), a b (Pascal's snail contains an internal loop). Checking the obtained expressions indicates their adequacy, since the convergence of theoretical and experimental data is 100%.
 CONCLUSION: 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 increase the accuracy of calculations. To simplify the description of this process, it is advisable to use the KOMPAS-3D three-dimensional modeling system.