Theoretical Foundations of Placing Onion Seeds in the Seed Box of Planting Machine

Akramjon Umurzakov; Utkirbek Khakimov; Gulomjon Makhkamov

doi:10.20944/preprints202503.1996.v1

Submitted:

26 March 2025

Posted:

26 March 2025

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Abstract

Studies on how many onion seeds are placed in the wheel slot of the planting machine were carried out taking into account the center of gravity. The result is a system of equations that allows you to determine the probable number of placements in a slot. Based on the numerical solution, it was found that onion seeds are more likely to fall with a large surface area.

Keywords:

seed

;

seed box

;

wheel

;

planting machine

;

probability

;

center of gravity

;

equation

;

expression

;

side

Subject:

Engineering - Industrial and Manufacturing Engineering

The quality of planting onions seeds with planting machine depends on the probability of how many seeds fall into the cells of the wheel. The fact that the seeds fall into the wheel cells can be considered a coincidence. Therefore, we are investigating how many onion seeds are likely to fall into the disc slots.

The fact that how many seeds fall into the wheel slots of the planting machine and their position depends on their shape. According to ([1] p.7-8), [2,3], seeds can be in the following forms: spherical, elliptical, pyramidal, truncated pyramidal, oval, and bean-shaped (Table 1).

Based on the above information, based on the above, we can assume that the shape of the onion seed will be a ABCD pyramid whose base consists of an equilateral ABS triangle. Here, let

A C = a

,

A B = B C = b

, be the center of gravity of the base M₀ (the point of intersection of the medians). For convenience, let the AC side lie on the x axis and have point B₁ in its center. Assuming point B₁ as the coordinate head, we align the positive direction of that axis with the light B₁B (Figure 1). In this case, the coordinates of points A, B and C are as follows

A (- \frac{a}{2}; 0; 0); B (0; \sqrt{b^{2} - \frac{a^{2}}{4}}; 0); C (\frac{a}{2}; 0; 0) .

The coordinate of point М₀ is the arithmetic mean of the coordinates of A, B, C points, i.e. М₁

\equiv \frac{A + B + C}{3}

. If we consider the coordinates of A, B, C points, the coordinates of point М₀ will be

M_{1} (0; \frac{1}{3} \sqrt{b^{2} - \frac{a^{2}}{4}}; 0)

(1)

Using Figure 1, we write the coordinates of point D through the height h drawn from that point as follows

D (0; \frac{1}{3} \sqrt{b^{2} - \frac{a^{2}}{4}}; h),

(2)

here h is the height from point D to the center of gravity

We write the coordinate М₀ of the center of gravity of the onion seed, considering the point D, as in the following

M 1 \equiv \frac{A + B + C + D}{4}

(3)

Given the coordinates of points A, B, C, D, the coordinates М₁, the center of gravity of the onion seed as in the following

M_{1} (0; \frac{1}{3} \sqrt{b^{2} - \frac{a^{2}}{4}}; \frac{h}{4})

(4)

Whichever side the center of gravity of the onion seed is closer, the more likely it is to fall with that side. It follows that onion seeds are more likely to fall with the side where their center of gravity is. We therefore calculate the distances from the center of gravity to all sides.

d (M; A B C) = \frac{h}{3}

(5)

Considering the ADC and BCD edges of the onion seed as a triangular flatness, we form the equation of plane of these triangles in accordance with ([4] p.43-44), [5,6,7].

The following is the equation of plane for ADC side of an onion seed:

A D C : |\begin{matrix} x + \frac{a}{2} & y & z \\ a & 0 & 0 \\ \frac{a}{2} & \frac{1}{3} \sqrt{b^{2} - \frac{a^{2}}{4}} & h \end{matrix}| = 0

(6)

Simplifying the determinant, we obtain the following expression for the ADC side of an onion seed:

a {(- 1)}^{2 + 1} (h y - z \frac{1}{3} \sqrt{b^{2} - \frac{a^{2}}{4}}) = 0, 6 h y - \sqrt{4 b^{2} - a^{2}} z = 0

(7)

The following is the equation of plane for the BCD side of an onion seed:

B C D : |\begin{matrix} x - \frac{a}{2} & y & z \\ - \frac{a}{2} & \sqrt{b^{2} - \frac{a^{2}}{4}} & 0 \\ - \frac{a}{2} & \frac{1}{3} \sqrt{b^{2} - \frac{a^{2}}{4}} & h \end{matrix}| = 0

(8)

If we simplify the determinant, the following expression is formed for the BCD side of an onion seed:

\begin{array}{l} (x - \frac{a}{2}) h \sqrt{b^{2} - \frac{a^{2}}{4}} - y (- \frac{a}{2}) h + z (- \frac{a}{6} \sqrt{b^{2} - \frac{a^{2}}{4}} + \frac{a}{2} \sqrt{b^{2} - \frac{a^{2}}{4}}) = x \frac{h}{2} \sqrt{4 b^{2} - a^{2}} + y \frac{a h}{2} + \\ + \frac{a}{3} \sqrt{b^{2} - \frac{a^{2}}{4}} z - \frac{a}{2} h \sqrt{b^{2} - \frac{a^{2}}{4}} = 0 \end{array} 6 h \sqrt{4 b^{2} - a^{2}} x + 6 a h y + 2 a \sqrt{4 b^{2} - a^{2}} z - 3 a h \sqrt{4 b^{2} - a^{2}} = 0

(9)

Considering the distances between the sides of an onion seed and the center of gravity М₁ as plane, we determine the distance from the point to the plane. According to ([4] p. 47-48), [9], the distance from the point to the plane is expressed in the following:

M (x_{0}; y_{0}; z_{0}), α : A x + B y + C z + D = 0 d (M; α) = \frac{|A x_{0} + B y_{0} + C z_{0} + D|}{\sqrt{A^{2} + B^{2} + C^{2}}}

(20)

Using expression (10), we determine the distances between the sides of an onion seed and the center of gravity М₁

d (M_{1}; A B C) = \frac{h}{3}

(13)

\begin{array}{l} d (M_{1}; A D C) = \frac{|6 h \frac{1}{3} \sqrt{b^{2} - \frac{a^{2}}{4}} - \sqrt{4 b^{2} - a^{2}} \frac{h}{4}|}{\sqrt{36 h^{2} + 4 b^{2} - a^{2}}} = \frac{1}{4} \frac{|4 h \sqrt{4 b^{2} - a^{2}} - h \sqrt{4 b^{2} - a^{2}}|}{\sqrt{36 h^{2} + 4 b^{2} - a^{2}}} = \frac{3}{4} h \frac{\sqrt{4 b^{2} - a^{2}}}{\sqrt{36 h^{2} + 4 b^{2} - a^{2}}} \end{array} \begin{array}{l} d (M_{1}; A D C) = \frac{3}{4} h \frac{\sqrt{4 b^{2} - a^{2}}}{\sqrt{36 h^{2} + 4 b^{2} - a^{2}}} \end{array}

(42)

\begin{array}{l} d (M_{1}; A D B) = d (M_{1}; D C B) = \frac{|6 h \sqrt{4 b^{2} - a^{2}} 0 + 6 a h \frac{1}{3} \sqrt{b^{2} - \frac{a^{2}}{4}} + 2 a \sqrt{4 b^{2} - a^{2}} \frac{h}{4} - 3 a h \sqrt{4 b^{2} - a^{2}}|}{\sqrt{36 h^{2} (4 b^{2} - a^{2}) + 36 a^{2} h^{2} + 4 a^{2} (4 b^{2} - a^{2})}} = \\ = \frac{|a h \sqrt{4 b^{2} - a^{2}} + \frac{a h}{2} \sqrt{4 b^{2} - a^{2}} - 3 a h \sqrt{4 b^{2} - a^{2}}|}{\sqrt{144 h^{2} b^{2} - 36 a^{2} h^{2} + 36 a^{2} h^{2} + 16 a^{2} b^{2} - 4 a^{4}}} = \frac{\frac{3}{2} a h \sqrt{4 b^{2} - a^{2}}}{2 \sqrt{36 h^{2} b^{2} + 4 a^{2} b^{2} - a^{4}}} = \frac{3 a h \sqrt{4 b^{2} - a^{2}}}{4 \sqrt{36 h^{2} b^{2} + 4 a^{2} b^{2} - a^{4}}} \end{array} d (M_{1}; A D B) = d (M_{1}; D C B) = \frac{3 a h \sqrt{4 b^{2} - a^{2}}}{4 \sqrt{36 h^{2} b^{2} + 4 a^{2} b^{2} - a^{4}}}

(53)

The longer the distance is from the center of gravity to the side, the smaller the probability of falling with this side. If we consider that the ratios of the distances from the point to the sides are equal to the inverse of the probabilities of falling with that point:

d_{1} : d_{2} : d_{3} : d_{4} = \frac{1}{P_{1}} : \frac{1}{P_{2}} : \frac{1}{P_{3}} : \frac{1}{P_{4}}, P_{1} : P_{2} : P_{3} : P_{4} = \frac{1}{d_{1}} : \frac{1}{d_{2}} : \frac{1}{d_{3}} : \frac{1}{d_{4}}

(64)

here, d₁ is the distance from the center of gravity to the side ABC, mm; d₂ is the distance from the center of gravity to the side ACD, mm; d₃ is the distance from the center of gravity to the side ABD, mm; and d₄ is the distance from the center of gravity to the side BCD, mm.

What can be concluded from the above equations is that since the onion seed is considered to be pyramidal, the sum of the probabilities of falling into the slot with each side is equal to one, i.e. [9,10,11,12].

P_{1} + P_{2} + P_{3} + P_{4} = 1

(75)

Given the expressions (14) and (15), we obtain the following:

\{\begin{cases} P_{1} (А В С) = \frac{\frac{1}{d_{1}}}{\frac{1}{d_{1}} + \frac{1}{d_{2}} + \frac{1}{d_{3}} + \frac{1}{d_{4}}} = \frac{1}{d_{1} (\frac{1}{d_{1}} + \frac{1}{d_{2}} + \frac{1}{d_{3}} + \frac{1}{d_{4}})} \\ P_{2} (А С D) = \frac{\frac{1}{d_{2}}}{\frac{1}{d_{1}} + \frac{1}{d_{2}} + \frac{1}{d_{3}} + \frac{1}{d_{4}}} = \frac{1}{d_{2} (\frac{1}{d_{1}} + \frac{1}{d_{2}} + \frac{1}{d_{3}} + \frac{1}{d_{4}})} \\ P_{3} (А B D) = P_{4} (B D C) = \frac{\frac{1}{d_{3}}}{\frac{1}{d_{1}} + \frac{1}{d_{2}} + \frac{1}{d_{3}} + \frac{1}{d_{4}}} = \frac{1}{d_{3} (\frac{1}{d_{1}} + \frac{1}{d_{2}} + \frac{1}{d_{3}} + \frac{1}{d_{4}})} \end{cases}

(86)

Based on the expression (13), we obtain the following

\begin{array}{l} \frac{1}{d_{1}} + \frac{1}{d_{2}} + \frac{1}{d_{3}} + \frac{1}{d_{4}} = \frac{3}{h} + \frac{4 \sqrt{36 h^{2} + 4 b^{2} - a^{2}}}{3 h \sqrt{4 b^{2} - a^{2}}} + \frac{8 \sqrt{36 h^{2} b^{2} + 4 a^{2} b^{2} - a^{4}}}{3 a h \sqrt{4 b^{2} - a^{2}}} = \\ = \frac{3 a \sqrt{4 b^{2} - a^{2}} + 4 a \sqrt{36 h^{2} + 4 b^{2} - a^{2}} + 8 \sqrt{36 h^{2} b^{2} + 4 a^{2} b^{2} - a^{4}}}{3 a h \sqrt{4 b^{2} - a^{2}}} \end{array}

(97)

\begin{array}{l} \frac{1}{\frac{1}{d_{1}} + \frac{1}{d_{2}} + \frac{1}{d_{3}} + \frac{1}{d_{4}}} = \frac{3 a h \sqrt{4 b^{2} - a^{2}}}{3 a \sqrt{4 b^{2} - a^{2}} + 4 a \sqrt{36 h^{2} + 4 b^{2} - a^{2}} + 8 \sqrt{36 h^{2} b^{2} + 4 a^{2} b^{2} - a^{4}}} \end{array}

(108)

In order to simplify the operations on expressions, we introduce the following notation:

\begin{array}{l} E = \frac{3 a h \sqrt{4 b^{2} - a^{2}}}{3 a \sqrt{4 b^{2} - a^{2}} + 4 a \sqrt{36 h^{2} + 4 b^{2} - a^{2}} + 8 \sqrt{36 h^{2} b^{2} + 4 a^{2} b^{2} - a^{4}}} \end{array}

(119)

Using expressions (17), (18) and (19), we obtain the following results:

P_{1} (A B C) = \frac{3 E}{h} P_{2} (A C D) = \frac{4 \sqrt{36 h^{2} + 4 b^{2} - a^{2}}}{3 h \sqrt{4 b^{2} - a^{2}}} E P_{3} (A B D) = P_{4} (B C D) = \frac{4 \sqrt{36 h^{2} b^{2} + 4 a^{2} b^{2} - a^{4}}}{3 a h \sqrt{4 b^{2} - a^{2}}} E

(20)

To determine the probability of the side that the onion seed is likely to fall, we perform a numerical solution of the system of equations (20).

We perform the numerical solution of the system of equations (20) in the following dimensions of the onion seed: a=2,54, мм; b=3,54, мм; h=2,17, мм.

E = \frac{3 * 1, 69 * 2, 17 \sqrt{4 * 3, 54^{2} - 1, 69^{2}}}{3 * 1, 69 \sqrt{4 * 3, 54^{2} - 1, 69^{2}} + 4 * 1, 69 \sqrt{36 * 2, 17^{2} + 4 * 3, 54^{2} - 1, 69^{2}} + 8 \sqrt{36 * 2, 17^{2} * 3, 54^{2} + 4 * 1, 69^{2} * 3, 54^{2} - 1, 69^{4}}} = 0, 1294

P_{1} (A B C) = \frac{3 * 0, 1294}{2, 17} = 0, 18

P_{2} (A C D) = \frac{4 \sqrt{36 * 2, 17^{2} + 4 * 3, 54^{2} - 1, 69^{2}}}{3 * 2, 17 \sqrt{4 * 3, 54^{2} - 1, 69^{2}}} * 0, 1294 = 0, 17

P_{3} (A B D) = P_{4} (B C D) = \frac{4 \sqrt{36 * 2, 17^{2} * 3, 54^{2} + 4 * 1, 69^{2} * 3, 54^{2} - 1, 69^{4}}}{3 * 1, 69 * 2, 17 \sqrt{4 * 3, 54^{2} - 1, 69^{2}}} * 0, 1294 = 0, 325

P_{1} + P_{2} + P_{3} + P_{4} = 0, 18 + 0, 17 + 0, 325 + 0, 325 = 1

According to the results of numerical solutions of the system of equations (20), the probability of onion seeds falling into the wheel slot with a large surface side is 65%, the probability of falling with a medium surface side is 18% and the probability of falling with a small surface side is 17%.

Conclusions

Studies on how many onion seeds are placed in the wheel slot of the planting machine were carried out taking into account the center of gravity. The result is a system of equations that allows you to determine the probable number of placements in a slot. Based on the numerical solution, it was found that onion seeds are more likely to fall with a large surface area.

References

N.P. Kryuchin. Seeding machines. Design features and development trends: Textbook. - Samara: RIC SGSKhA 2009, p.175.
Ya. Narmanov. Analyst geometry. - T .: National society of Uzbekistan Philosophers Press, 2008. – p.176.
V.E. Gmurman. Probability Theory and a Handbook for solving problems from mathematical statistics. - T .: O’qituvchi, 1980. – p. 322.
Umurzakov, A.K., Turdaliev, V.M. & Khakimov, U.A. Low-Power Hydraulic Motor for Mobile Micropower Stations and Pumps. Russ. Engin. Res. 42, 791–793 (2022). [CrossRef]
Makhkamov, G.U., Khakimov, U.A. Experimental Study of Micro-Hydropower Plants. Calculation of Water Wheel Efficiency. Russ. Engin. Res. 43, 1524–1527 (2023). [CrossRef]
Khakimov, U. A. (2025). Issues of Creation of Water Engines for Mobile Micro-HPP and Pumps.
Hakimov, UA (2025). Mobil mikro GES va nasoslar uchun suv dvigatellarini yaratish masalalari. Oldindan chop etish. [CrossRef]
Khakimovich, U. A.; Akramovich, K. U. Creation of Water Engines for Mobile Micro Hydropower. Preprints 2025, 2025031277. [CrossRef]
A X Umurzakov; M X Imomov; F R Maxmudov; S X Mamasoliyeva. The influence of the front section teeth lengths on the agrotechnical and energy performance of a two-stage vibratory gear hardware for land. 2023, 1284, 012025 . [CrossRef]
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Figure 1. Conditionally accepted form of onion seed. Here a – the width of an onion seed, mm; b – its length, mm; h – height, mm.

Table 1. The classification of seed forms.

as per N.N.Ulrix	as per V.N.Dobrokhotov	as per A.N.Semenov
Spherical (a=b=l)

Elliptical (a<b<l, a=b<l)

Pyramidal (a<b<l)

Truncated pyramidal (a<b<l)

Oval (a<b=l)

Bean-shaped (a<b<l, a<b=l)

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Theoretical Foundations of Placing Onion Seeds in the Seed Box of Planting Machine

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