## 2 Materials and Methods
### Model of Grain Motion in the System of a Piezolectric Sensor
Both the force of contact and the moment of rolling resistance against the surface occur when a particle hits a surface or another particle. In the remaining cases they are equal to 0. Gravitational force is defined as follows:  
\[\overset{\rightarrow}{G}=m\overset{\rightarrow}{g} \tag{3}\]  
Contact force \(\overset{\rightarrow}{F}_{c}\) is defined as a resultant of the surface reaction forces upon impact \(\overset{\rightarrow}{N}_{c}\) and tangential forces \(\overset{\rightarrow}{T}_{c}\)[32]:  
\[\overset{\rightarrow}{F}_{c}=\sum_{i=1}^{n}\biggl{(}\overset{\rightarrow}{ N}_{c}+\overset{\rightarrow}{T}_{c}\biggr{)} \tag{4}\]  
When modeling contacts, i.e., an impact between two bodies, a few available models of normal and tangential forces can be used [33]. Normal force \(\overset{\rightarrow}{N}_{c}\) that occurs during the impact can be determined according to Hertz theory [32; 33], whereas determination of tan  
Figure 3: Scheme of forces affecting grains upon impact with the sensor surface, N–normal force, T–tangential force, G–gravitation force, \(\alpha\)–tilt angle of sensor surface, x–the distance the grain collides with the sensor, l–the sensor length. gential forces \(\overrightarrow{T}_{c}\) was carried out using a model proposed by [PERSON] and [PERSON] [34] for stick-slip behavior during contact. Both normal forces \(\overrightarrow{N}_{c}\) and tangential forces \(\overrightarrow{T}_{c}\) are the effect of stiffness components \(\overrightarrow{N}_{si}\) and \(\overrightarrow{T}_{si}\) and damping \(\overrightarrow{N}_{Ti}\) or as \(\overrightarrow{T}_{Ti}\), hence contact force can be expressed as:  
\[\overrightarrow{F}_{c}=\sum_{i=1}^{n}\left(\overrightarrow{N}_{c}+ \overrightarrow{T}_{c}\right)=\sum_{i=1}^{n}\left(\overrightarrow{N}_{si}+ \overrightarrow{N}_{Ti}+\overrightarrow{T}_{si}+\overrightarrow{T}_{Ti}\right) \tag{5}\]  
Normal force results from two forces--elastic force \(\overrightarrow{N}_{si}\) and damping force \(\overrightarrow{T}_{Ti}\):  
\[\overrightarrow{N}_{c}=\overrightarrow{N}_{si}+\overrightarrow{N}_{Ti} \tag{6}\]  
Elastic force according to [PERSON] theory [32; 33] is defined by dependency:  
\[N_{si}=\frac{4}{3}E^{*}\sqrt{R^{*}}|h_{o}|^{\frac{3}{2}} \tag{7}\]  
Figure 2:0  
Figure 2:1  
Figure 2:2  
where: \(E^{*}\)--reduced Young modulus (Equation (8)), \(R^{*}\)--effective radius of contact with a flat surface (Equation (9)), \(h_{o}\)--particle overlap (Equation (10)), \(E_{1}\), \(E_{2}\)--Young moduli of two bodies at contact, \(v_{1},v_{2}\)--Poisson coefficient of bodies at contact, \(R\)--radius of grain curvature at the contact point, \(d\)--length between the grain gravity center and the contact surface. The damping force dependent on the restitution coefficient is expressed as:  
Figure 2:3  
where: \(c_{nH}\)--damping coefficient of normal forces in Hertz theory, \(m^{*}\)--effective mass (relative), \(v_{n}^{\prime nl}\)--relative velocity of normal component. The damping coefficient of normal forces in Hertz theory is connected with the linear damping coefficient \(c_{n}\) according to equation:  
Figure 2:4  
whereas \(c_{n}\) is determined based on the value of the restitution coefficient \(\varepsilon\). The restitution coefficient and damping coefficient are connected by the following dependency:  
\[\varepsilon=\left\{ \begin{array}{cc}\exp\left[-\frac{c_{n}}{\sqrt{1-c_{n}^{ 2}}}\left(\pi-\arctan\frac{2c_{n}\sqrt{1-c_{n}^{2}}}{1-2c_{n}^{2}}\right) \right]&if\ \ 0\leq c_{n}<\frac{1}{\sqrt{2}}\\ \exp\left[-\frac{c_{n}}{\sqrt{1-c_{n}^{2}}}\arctan\frac{2c_{n}\sqrt{1-c_{n}^{2 }}}{2c_{n}^{2}-1}\right]&if\ \ \frac{1}{\sqrt{2}}\leq c_{n}\leq 1\\ \exp\left(-\frac{c_{n}}{\sqrt{c_{n}^{2}-1}}\ln\frac{c_{n}+\sqrt{c_{n}^{2}-1}}{ c_{n}-\sqrt{c_{n}^{2}-1}}\right)&if\ \ \ c_{n}>1\end{array} \right.