## 2 Materials and Methods
### Model of Grain Motion in the System of a Piezolectric Sensor  
Sowing crops and cultivated plants requires appropriate dosing of grains to be positioned in rows, which ensures utilization of an optimal amount of grains at an appropriate distance between them (appropriate grain distribution in the soil). For this purpose, it is  
Figure 1: The framework of the conducted research. necessary to control the material feeding into the colter. A control chamber with a piezoelectric sensor PZT (ceramic) was proposed to count the grains. By means of the chamber, grains that fall down a delivery tube are counted. Figure 2 shows the scheme of the control chamber equipped with a PZT (ceramic) piezoelectric sensor. Selection of an appropriate shape and dimensions as well as the distance between the sensor and its chamber walls is of key importance in the presented solution. They need to be matched so that a falling seed will contact the sensor surface only once. It is important to provide grains with a suitable tilt angle as this conditions the grain motion direction and the impact range. A small impact range causes a double hit against the sensor surface which results in a double count of the same grain and the control system indication error. In terms of usability, this is not a desired situation, and it would disrupt the regularity of the control system performance. To optimize the structural features of the sensor chamber and its dimensions it is necessary to take into account the grain motion in the chamber and its characteristic extreme situations. Dimensions of the sensor and the chamber must be correlated with each other in such a way that after exiting the output tube a grain hits the sensor only once. Grain movement was considered for a flat sensor in the form of a plate (Figure 2). Grain motion can be divided into four stages:  
1. Motion in the seed delivery tube,
2. Motion after leaving the seed delivery tube until hitting against the seed drill surface,
3. Impact with the sensor surface, with partial energy loss,
4. Grain motion after collision with velocity \(v_{k}\), different (smaller due to impact) from its velocity before a collision with the sensor. An equation of grain motion in a chamber results from [PERSON]'s law of dynamics. Grain can be transported gravitationally or pneumatically in the form of a grain-air mixture. Gravitational transport, that is, falling under the impact of \(G\) gravitation force and Magnus  
Figure 2: Scheme of a piezoelectric sensor chamber, d—grain exit hole diameter, h—distance between the highest point of the sensor mounting place, H—height of the chamber with sensor, l—length of piezoelectric sensor, c—length between the most protruding point of the sensor and the casing wall, \(\alpha\)—tilt angle with the horizontal, a—width of the chamber with sensor, k-seed. \(M\)[32] force was considered first. It should be assumed that a falling grain performs translational motion and a rotational motion around its own axis. During collision with the sensor surface a grain is affected by the force of contact \(F_{C}\) with normal component \(N\) (connected with the impact pressure and damping) and tangential component \(T\) resulting from grain sliding friction (Figure 3). The equation of grain translational motion is expressed by:  
\[m\frac{d\overset{\rightarrow}{v}}{dt}=\overset{\rightarrow}{G}+\overset{ \rightarrow}{F}_{c}+\overset{\rightarrow}{M} \tag{1}\]  
and rotational motion:  
\[I\frac{d\overset{\rightarrow}{\omega}}{dt}=\overset{\rightarrow}{M}_{ \omega}+\overset{\rightarrow}{M}_{T} \tag{2}\]  
where: \(m\)--grain mass, \(\overset{\rightarrow}{v}\)--translational motion velocity, \(\overset{\rightarrow}{G}\)--gravitational force, \(\overset{\rightarrow}{F}_{c}\)--force resulting from the seed contact with the sensor surface, \(M\)--Magnus force, \(I\)--inertia moment, \(\overset{\rightarrow}{\omega}\)--speed angular velocity, \(\overset{\rightarrow}{M}_{\omega}\)--torque, \(\overset{\rightarrow}{M}_{T}\)--rolling resistance moment.