Mathematical model of the cross-section of wheat grain

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BACKGROUND

The requirement for the implementation of new progressive technologies in the harvesting process in the Russian Federation is driven by the constant increase in the production of grain crops in the country. For example, the Ministry of Agriculture predicts that Russia will maintain its leadership in wheat production in 2024 and increase exports to 50 million tons of grain. However, addressing the challenges of reducing labor intensity and energy costs and improving the quality of grain and seeds to the required parameters is currently impossible using traditional combine technologies [1, 2]. The high consumption of fuels and lubricants (up to 7 liters per ton of harvested grain), the rising cost of production (up to 12,000 rubles per ton of grain), and direct and indirect grain losses of up to 30%–50% [3, 4] of the entire harvest require the adoption of modern farming methods.

Numerous studies and production experiences have indicated the possibility of increasing the productivity and energy efficiency of combine harvesters and reducing direct and indirect grain losses through combine combing. A comparative study indicated that using a stripper header instead of a straight-flow header resulted in notable fuel savings (up to 40%) per hectare and an increase in productivity by a factor of 1.3 [5]. However, one of the challenges of this harvesting technology is that the combed heap contains up to 85% free grain, which, when entering the threshing gap, leads to increased crushing (up to 8%) by the working parts of the threshing drum and a decrease in quality indicators [6].

To avoid this issue, it is advisable to perform preliminary separation of the combed heap to remove free grain and directly send the latter for cleaning, bypassing the threshing device. With the current design of a combine harvester, an additional separating device can be placed in the inclined chamber [7, 8]. Theoretical studies exploring the optimal length of the holes in the lattice bottom of a combine harvester’s inclined chamber, designed to aid in the preliminary separation of the combed grain heap, modeled the cross section of the wheat grain either as a separate ball or a truncated cylinder [9–11]. This representation of caryopsides considerably simplifies the description of the separation process. However, these models are quite far from the real shape of the grain, as the dorsal side of the caryopsides is convex, and there is a longitudinal groove on the ventral side. Consequently, the difference in the rate of separation of free grain between theoretical and experimental data was found as ~30% [12].

The surface closest to the actual cross-sectional shape of the grain is represented by the mathematical model developed by I.A. Mayatskaya, which is Pascal’s snail (Fig. 1) [13, 14]. For these models, the author determined the coordinates of the center of gravity of the figure and derived equations for calculating its cross-sectional area and moments of inertia for each coordinate axis.

Models of the grain cross section were constructed in the KOMPAS-3D program, and the discrepancy between the real and theoretically predicted values of the coordinates of the figure center of gravity was ~13% [12], which reduces the accuracy of the calculations and necessitates further refinement.

STUDY AIM

The study aimed to present the mathematical model of the cross section of a wheat grain, represented in the form of Pascal’s snail.

MATERIALS AND METHODS

The study focused on a cross section of a wheat grain represented in the form of Pascal’s snail. The coordinates of the figure center of gravity were determined using methods from theoretical mechanics, and the resulting equations were validated in the three-dimensional modeling system KOMPAS-3D.

RESULTS AND DISCUSSION

Запишем уравнение улитки Паскаля в прямоугольной системе координатах в общем виде [7]

${(x^2+y^2-a\cdot x)}^2-b^2\cdot (x^2+y^2)=\mathrm{0,}$                                                                (1)

where a and b are the geometric parameters of the figure (Fig. 1), and x and y are Cartesian coordinates.

 

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).

Рис. 1. Частные случаи поперечного сечения зерна пшеницы выполненной в виде улитки Паскаля: a) a = b (кардиоида); b) a < b (улитка Паскаля без внутренней петли); c) a > b (улитка Паскаля содержит внутреннюю петлю).

 

If a = b, then Pascal’s snail becomes a cardioid. The Pascal’s snail equation in polar coordinates (φ; r) (-π ≤ φ ≤ π) can be represented as follows:

$r=a\cdot (1+\cos \phi )$,                                                                                       (2)

where φ is the polar angle of the radius vector of the current point of the curve..

The center of gravity of the figure bounded by the cardioids, denoted as point С (х_с; у_с) (см. рис. 1, а). ), is determined (Fig. 1a). Owing to the symmetry of the cardioid, у_с = 0, only х_с as the abscissa of point С needs to be determined:

$x_c=\frac{M_y}{S}$,                                                                                                  (3)

where S is the area bounded by the cardioid, and M_y is the static moment of the body bounded by the cardioid relative to the у-axis.

The square S of the figure bounded by the cardioid is determined as follows:

$S=\frac{1}{2}{\int }_0^{2\cdot \pi }{r}^2\left(\phi \right)d\phi =\frac{1}{2}{\int }_0^{2\cdot \pi }{\left[a\cdot (1+\cos \phi )\right]}^{2}d\phi =\frac{3\cdot \pi \cdot a^2}{2}$                       (4)

According to its definition, the static moment of a figure is as follows [15]:

$M_y=\underset{S}{\iint }xdxdy$.                                                                                         (5)

The static moment is calculated by transitioning to polar coordinates for the points (x; y) of the indicated figure

$\begin{array}{c}x=\rho \cos \phi ;y=\rho \sin \phi ;dxdy=\rho d\rho d\phi ;\\ -\pi \le \phi \le \pi ;0\le \rho \le a(1+\cos \phi ).\end{array}$                                                 (6)

Then

$\begin{array}{l}{M}_y=\underset{S}{\iint }xdxdy=\underset{S}{\iint }\rho \cos \phi \cdot \rho d\rho d\phi =\underset{-\pi }{\overset{\pi }{\int }}\cos \phi d\phi \underset{0}{\overset{a(1+\cos \phi )}{\int }}{\rho }^{2}d\rho =\underset{-\pi }{\overset{\pi }{\int }}\cos \phi d\phi \cdot \frac{1}{3}{\rho }^3\left|\begin{array}{c}{}^{a(1+\cos \phi )}\\ {}_{\begin{array}{cc}\begin{array}{ccc}0& & \end{array}& \end{array}}\end{array}\right.=\\ =\frac{a^3}{3}\underset{-\pi }{\overset{\pi }{\int }}{(1+\cos \phi )}^3\cos \phi d\phi =\frac{5\pi }{4}{a}^3.\end{array}$(7)

According to Eq. (4) and Eq. (7), from Eq. (3) we obtain:

$x_c=\frac{5}{6}a$.                                                                                                 (8)

Consider an example in which the diameter of the initial circle a is 20 mm. Then, according to Eq. (8), the center of gravity of the cardioid is located at a distance x_c of 16.67 mm from the origin.

 

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

Рис. 2. Скриншот рабочего окна программы «КОМПАС-3D» при определении центра тяжести кардиоиды a = b = 20 мм.

 

To validate the theoretical studies, a cardioid was constructed using the KOMPAS-3D program (Fig. 2), and the abscissa of the center of gravity of the figure was determined as x_c = 16.67 mm, with its cross-sectional area S = 1884.95 mm². The construction results demonstrate the accuracy of the derived equations (4) and (8), as there is 100% agreement between theoretical and experimental data.

By analogy with the cardioid, we determine the position of the center of gravity of the figure bounded by Pascal’s snail for the case when 0 < a < b (Fig. 1b).

The Pascal’s snail equation in polar coordinates (φ; r) (-π ≤ φ ≤ π) has the following form [7]:

$r=a\cdot \cos \phi +b$.                                                                                        (9)

The area S bounded by Pascal’s snail is as follows:

$S=\frac{1}{2}{\int }_0^{2\cdot \pi }{r}^2\left(\phi \right)d\phi =\frac{1}{2}{\int }_0^{2\cdot \pi }{(a\cdot \cos \phi +b)}^{2}d\phi =\frac{\pi \cdot a^2}{2}+\pi \cdot b^2$.              (10)

The static moment of the figure is calculated by transitioning to polar coordinates for the points (x; y) of the specified figure.

$\begin{array}{c}x=\rho \cos \phi ;y=\rho \sin \phi ;dxdy=\rho d\rho d\phi ;\\ -\pi \le \phi \le \pi ;0\le \rho \le a\cdot \cos \phi +b.\end{array}$                                               (11)

Then

$\begin{array}{l}{M}_y=\underset{S}{\iint }xdxdy=\underset{S}{\iint }\rho \cos \phi \cdot \rho d\rho d\phi =\underset{-\pi }{\overset{\pi }{\int }}\cos \phi d\phi \underset{0}{\overset{a\cdot \cos \phi +b}{\int }}{\rho }^{2}d\rho =\underset{-\pi }{\overset{\pi }{\int }}\cos \phi d\phi \cdot \frac{1}{3}{\rho }^3\left|\begin{array}{c}{}^{a\cdot \cos \phi +b}\\ {}_{\begin{array}{cc}\begin{array}{ccc}0& & \end{array}& \end{array}}\end{array}\right.=\\ =\frac{1}{3}\underset{-\pi }{\overset{\pi }{\int }}{(a\cdot \cos \phi +b)}^3\cos \phi d\phi =\frac{\pi \cdot a\cdot (a^2+4\cdot b^2)}{4}.\end{array}$ (12)

According to Eq. (10) and Eq. (11), from Eq. (3) we obtain the following:

$x_c=\frac{a\cdot (a^2+4\cdot b^2)}{2\cdot (a^2+2\cdot b^2)}.$                                                                               (13)

Consider an example in which the studied version of Pascal’s snail has the initial parameters of a = 20 mm, b = 30 mm. Then, according to Eq. (13), the center of gravity of the figure is located at a distance from the origin of coordinates x_c = 18.18 mm.

With the same data, Pascal’s snail was constructed using the KOMPAS-3D program, and the actual coordinate of the figure center of gravity was determined as x_c = 18.18 mm (Fig. 3). The construction results affirm the accuracy of the derived equation (13), with 100% agreement between theoretical and experimental data.

Consider the special case 3, where 0 < b < a, and Pascal’s snail contains an internal loop as illustrated in Fig. 4.

The Pascal’s snail equation is presented in polar coordinates (φ; r) [7]:

$x=r\cdot \cos \phi ,y=r\cdot \sin \phi .$                                                                      (14)

Then Eq. (1) takes the form:

${(r^2-a\cdot r\cdot \cos \phi )}^2-b^2\cdot r^2=\mathrm{0,}\Rightarrow r=a\cdot \cos \phi \pm b.$                                    (15)

This equation is analyzed for different signs in front of b.

With the (−) sign we obtain the following:

$r=a\cdot \cos \phi -b.$                                                                                        (16)

With the condition r ≥ 0, we obtain the following:

$a\cdot \cos \phi -b\ge 0\Rightarrow \cos \phi \ge \frac{b}{a}\Rightarrow -\alpha \le \phi \le \alpha ,\text{\hspace{1em}где\hspace{1em}}\alpha \text{=arccos}\frac{b}{a}.$   (17)

This results in the following:

$r=a\cdot \cos \phi -b\left(-\alpha \le \phi \le \alpha ;\alpha \text{=arccos}\frac{b}{a}\right).$                                          (18)

The equation of the loop in polar coordinates (i.e., Eq. 18) corresponds to Fig. 4.

 

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

Рис. 4. Улитка Паскаля с внутренней петлей.

 

With the (+) sign in Eq. (15), we obtain (again, subject to r ≥ 0):

$r=a\cdot \cos \phi +b\left(-(\pi -\alpha )\le \phi \le \pi -\alpha \right).$                                                  (19)

Equation (19) represents the external line of Pascal’s snail in polar coordinates.

According to Eqs. (18) and (19), we determine the center of gravity of Pascal’s snail. For this purpose, we calculate the area bounded by the figure in the absence of an internal loop:

$\begin{array}{l}S={\int }_{-(\pi -\alpha )}^{\pi -\alpha }d\phi {\int }_0^{a\cdot \cos \phi +b}\rho d\rho =\frac{1}{2}{\int }_{-(\pi -\alpha )}^{\pi -\alpha }{(a\cdot \cos \phi +b)}^{2}d\phi ={\int }_0^{\pi -\alpha }{(a\cdot \cos \phi +b)}^{2}d\phi =\left|\alpha =\arccos \frac{b}{a}\right|=\\ =\left(\frac{a^2}{2}+b^2\right)\cdot (\pi -\alpha )+\frac{3}{2}\cdot b\cdot \sqrt{a^2-b^2}.\end{array}$ (20)

The static moment is calculated by converting to polar coordinates for the points (x; y) of the given figure:

$\begin{array}{c}x=\rho \cos \phi ;y=\rho \sin \phi ;dxdy=\rho d\rho d\phi ;\\ -\pi \le \phi \le \pi ;0\le \rho \le a\cdot \cos \phi +b.\end{array}$                                               (21)

Then,

$\begin{array}{l}{M}_y=\underset{S}{\iint }xdxdy=\underset{S}{\iint }\rho \cos \phi \cdot \rho d\rho d\phi =\underset{-(\pi -\alpha )}{\overset{\pi -\alpha }{\int }}\cos \phi d\phi \underset{0}{\overset{a\cdot \cos \phi +b}{\int }}{\rho }^{2}d\rho =\underset{-(\pi -\alpha )}{\overset{\pi -\alpha }{\int }}\cos \phi d\phi \cdot \frac{1}{3}{\rho }^3\left|\begin{array}{c}{}^{a\cdot \cos \phi +b}\\ {}_{\begin{array}{cc}\begin{array}{ccc}0& & \end{array}& \end{array}}\end{array}\right.=\\ =\frac{2}{3}\underset{0}{\overset{\pi -\alpha }{\int }}{(a\cdot \cos \phi +b)}^3\cos \phi d\phi =\left|\alpha =\arccos \frac{b}{a}\right|=\frac{1}{4}\cdot a\left(a^2+4\cdot b^2\right)\left(\pi -\alpha \right)+\frac{1}{12}\cdot \frac{\sqrt{a^2-b^2}}{a}\cdot b\cdot \left(13\cdot a^2+2\cdot b^2\right).\end{array}$ (22)

According to Eq. (20) and Eq. (22), from Eq. (3) we obtain:

$x_c=\frac{\frac{1}{4}\cdot a\left(a^2+4\cdot b^2\right)\left(\pi -\alpha \right)+\frac{1}{12}\cdot \frac{\sqrt{a^2-b^2}}{a}\cdot b\cdot \left(13\cdot a^2+2\cdot b^2\right)}{\left(\frac{a^2}{2}+b^2\right)\cdot (\pi -\alpha )+\frac{3}{2}\cdot b\cdot \sqrt{a^2-b^2}}.$                (23)

Consider an example in which the studied version of Pascal’s snail has the initial parameters of a = 20 mm, b = 15 mm. According to Equation (23), the center of gravity of the figure is positioned at a distance of х_с = 15,38 mm from the origin of the coordinates.

The same version of Pascal’s snail is constructed. Using the application package in the KOMPAS-3D three-dimensional modeling system, the actual value of the coordinate of the figure center of gravity was determined as х_с = 15.38 mm (Fig. 5). The construction results indicate the accuracy of the derived Eq. (23), with 100% agreement between theoretical and experimental data.

 

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.

Рис. 5. Скриншот рабочего окна программы «КОМПАС-3D» при определении центра тяжести улитки Паскаля при a = 20 мм и b = 15 мм.

 

Thus, the utilization of refined mathematical models of the cross section of wheat grain enables increased accuracy in calculating the process of separating combed heaps on the lattice bottom of the inclined chamber of a combine harvester. Moreover, to simplify the description of this process, employing the KOMPAS-3D three-dimensional modeling system is recommended.

CONCLUSIONS

1. During theoretical studies on separation, it is advisable to model the cross section of a wheat grain in the form of Pascal’s snail.
2. Mathematical equations are derived for analytically determining the coordinates of the center of gravity for different versions of Pascal’s snail: a = b (cardioid), a < b (Pascal’s snail without an internal loop), and a > b (Pascal’s snail contains an internal loop). The accuracy of the derived equations is demonstrated by the 100% agreement between theoretical and experimental data.
3. The utilization of refined mathematical models of the cross section of wheat grain enables increased accuracy in calculating the separation process of the combed heap on the lattice bottom of a grain combine harvester’s inclined chamber. The KOMPAS-3D three-dimensional modeling system should be employed to simplify the description of this process.

ADDITIONAL INFORMATION

Authors’ contribution. V.V. Nikitin — analysis of publications on the research topic, writing the text of the manuscript; V.N. Ozhereliev — expert opinion, editing the text of the manuscript, approval of the final version of the manuscript; N.V. Sinyaya — creating figures, editing the text of the manuscript. All authors made a substantial contribution to the conception of the work, acquisition, analysis, interpretation of data for the work, drafting and revising the work, final approval of the version to be published and agree to be accountable for all aspects of the work.

Competing interests. The authors declare that they have no competing interests.

Funding source. This study was not supported by any external sources of funding.

ДОПОЛНИТЕЛЬНАЯ ИНФОРМАЦИЯ

Вклад авторов. В.В. Никитин — поиск публикаций по теме статьи, написание текста рукописи; В.Н. Ожерельев — экспертная оценка, редактирование текста рукописи, утверждение финальной версии; Н.В. Синяя — создание изображений, редактирование текста рукописи. Авторы подтверждают соответствие своего авторства международным критериям ICMJE (все авторы внесли существенный вклад в разработку концепции, проведение исследования и подготовку статьи, прочли и одобрили финальную версию перед публикацией).

Конфликт интересов. Авторы декларируют отсутствие явных и потенциальных конфликтов интересов, связанных с публикацией настоящей статьи.

Источник финансирования. Авторы заявляют об отсутствии внешнего финансирования при проведении исследования.

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

References

Supplementary files