## 2 Materials and Methods
### Theoretical Notions for Corn Seed Calibration on Cylindrical Sieves
#### 2.4.2 Phase of Relative Repose  
Considering this, at the initial moment, \(t=t_{0}=0\), the seed is at point \(M_{0}\) of the surface (Figure 5) and has the same movement as the surface; in other words, it is in the phase of relative repose. An analysis is conducted to see under which conditions the particle is at relative repose if the cylinder; after a certain time interval, it rotates with the angle \(\alpha=\omega t\), and without the seed slipping, it moves with the surface from position \(M_{0}\) to position \(M\). The seed will remain in relative repose if the forces acting of it will be in balance. The following forces act on the seed:  
1. the force of gravity, \(G=\) mg, directed downwards;
2. reaction \(N\) of the surface, normal in point \(M\) and directed towards the centre \(O\);
3. the friction force \(F=f_{N}\), directed on the tangent in the direction of revolution;
4. the inertial force in the transport movement mR\(\omega^{2}\), directed in the direction of the radius outwards. A mobile system of coordinate axes \(\xi O\eta\) is chosen, with its center in \(O\), rotating with the cylinder with the angular speed \(\omega=\) const. Figure 4: Corn seed dimensions: a—length; b—width; c—thickness. The forces mentioned above are projected on the coordinate axes. For the seed to be at relative repose, the sum of the projections on the coordinate axes must be equal to zero. \[mR\omega^{2}-N+mgcos\omega t=0 \tag{3}\]  
\[F-mgsin\omega t=0 \tag{4}\]  
From relation (3), the normal is determined:  
\[N=m\left(R\omega^{2}+g\ cos\omega t\right)=mg\left(\frac{R\omega^{2}}{g}+cos \omega t\right) \tag{5}\]  
Relation (4) includes the friction force \(F\), which can have any value that is not higher than the limit value \(F_{lim}=fN\), where \(f\) is the friction coefficient that will be denoted by \(tg\varphi\) (\(\varphi\) being the friction angle). The maximum value of the friction force is limited by the friction coefficient and the normal force, which is equal to the weight of the seed. Therefore, relation (4) will be written as  
\[fN-mgsin\omega t\geq 0 \tag{6}\]  
The inequality states that the maximum force of friction, \(fN\), must be greater than or equal to the component of the gravitational force acting tangentially to the circular path of the seed, \(mgsin(\omega t)\), in order for the seed to remain in circular motion without slipping. This condition ensures that the object remains in contact with the surface and does not slide or slip. Coefficient \(f\) is calculated depending on the material of the sieve, the revolution speed, the inclination angle, and the characteristics of the seeds. Or, taking note of relation (5), will be  
\[tg\varphi\left(r\omega^{2}+g\ cos\ \omega t\right)\geq g\ sin\ \omega t \tag{7}\]  
After some simple transformations, the relation becomes  
\[k\ sin\ \varphi\geq sin\ (\omega t-\varphi) \tag{8}\]  
Figure 5: The forces acting on a seed. where \(k=\frac{\tau_{0}e^{2}}{g}\) is the index characterizing the movement regime of the cylinder. Relations (5) and (8) determine the conditions that ensure the phase of relative repose of the seed on the inner surface of the moving cylinder. For this, it is necessary that the reaction \(N\) of the surface be a positive quantity, namely,  
\[N=mg\ (k+cos\ \omega t)>0 \tag{9}\]  
Only in these conditions will the seed be found on the surface of the cylinder. Relation (8) determines the condition for the seed, located on the surface of the sieve, not to slide off it, meaning that it has an identical movement to that of the sieve. The same relation contains a variable quantity, namely, time \(t\); the other quantities \(k\), \(r\) and \(\varphi\), for a given cylinder and seed, are constant. Under these conditions, the left side of relation (8) is constant and the right side increases with time; and at a given moment when \(t=t_{1}\), both sides become equal.