## Materials and Methods
### Mathematical Contact Model  
The mass motion equations in this model can be recorded using the force equilibrium condition and the second law of motion as two ordinary second-order equations, which can be presented in the form of four first order equations suitable for numerical calculations (Equations (1)-(4)):  
\[\frac{dv_{1}}{dt}\ =\ \frac{1}{m_{1}}S_{1} \tag{1}\]  
\[\frac{dv_{2}}{dt}\ =\ \frac{1}{m_{2}}(S_{1}-S_{2}) \tag{2}\]  
\[v_{1}\ =\ \frac{dx_{1}}{dt} \tag{3}\]  
\[v_{2}\ =\ \frac{dx_{2}}{dt} \tag{4}\]  
where:  
\(m_{1}\), \(m_{2}\)--particle mass (kg);  
\(S_{1}\), \(S_{2}\)--contact force (N);  
\(t\)--time (s);  
\(v_{1}\), \(v_{2}\)--velocity (m s\({}^{-1}\)). Forces \(S_{1}\) and \(S_{2}\) for the viscoelastic model result from the relationship of (5) and (6):  
\[S_{1}\ =\ k_{1}\cdot x_{1}+c_{1}v_{1} \tag{5}\]  
\[S_{2}\ =\ k_{2}(x_{1}-x_{2})+c_{2}(v_{1}-v_{2}) \tag{6}\]  
where:  
\(c_{1}\), \(c_{2}\)--damping parameter (kg s\({}^{-1}\));  
\(k_{1}\), \(k_{2}\)--contact stiffness (N m\({}^{-1}\));  
Figure 3: Diagram of a piezoelectric force sensor (**a**) and physical model of the grain sensor system at the moment of impact (**b**): (1) grain mass m\({}_{2}\), (2) grain contact stiffness k\({}_{2}\), (3) grain damping parameter c\({}_{2}\), ( 4) parasitic force sensor mass m\({}_{1}\), (5) stiffness of the measuring element of the force transducer k\({}_{1}\), (6) damping of the measuring element of the force transducer c\({}_{1}\), (7) sensor dial, (8) piezoelectric crystals, and (9) electrode. \(x_{1}\), \(x_{2}\)--overlap (m). For the perfectly plastic impact model, it was assumed that the contact force had a constant value selected so that the velocity of the grain fell to zero during the impact (zero value of the velocity restitution parameter [18; 26; 27]). This condition was met when the momentum of this force was equal to the momentum of the grain during the impact. This force value can be calculated from this condition assuming the contact time \(t_{k}\) in (7) and (8):  
\[S_{2}(t)\ =\ m_{2}\frac{v_{2}}{t_{k}},\ \text{if}\ t\ <\ t_{k} \tag{7}\]  
\[S_{2}(t)\ =\ 0,\ \text{if}\ t\ >\ t_{k} \tag{8}\]  
where:  
\(t_{k}\)--contact time (s). To best develop a model that reflects reality, in the calculation, a trapezoidal course was introduced instead of a rectangular one, which is physically impossible to develop, with short sections of this force rising and falling. Any relationships with the contact force can be introduced to the model, given, among others, in [20]; but only the two presented above as the extreme cases were used in the calculations. The system of nonlinear ordinary equations described above was integrated with a simple Euler algorithm, using a very short integration step. The calculations were performed using the proprietary computer software, coded in the Rad Delphi Embaracdelo 2010 system, with the working name GKWK. The software included the main calculation module and procedures for data input, as well as for graphical presentation of results. The software also contained a procedure for automatic analysis of the resultant oscillograms of \(S_{1}\) and \(S_{2}\) forces. By changing the value of the integration step by one row, it was possible to check whether the solution was subject to the length of this step. The repeated results indicated that the adopted integration step dt = \(1\times 10^{-10}\) s was sufficiently short. Figure 4 shows the proprietary software interface. The grains used in the simulation were standard spherical grains with the following constant parameters:  
* Poisson parameter--0.2 (-);
* Kirchhoff's module--9.0 (MPa);
* Particle friction parameter--0.25 (-);
* Damping parameter--1.0 (Ns-m\({}^{-1}\));
* Grain diameter--5.0 (mm);  
Figure 4: GKWK, Poznan, Poland, 2017, software interface view. * Grain velocity--2.0 (m\(\cdot\)s\({}^{-1}\));
* Grain density--765 (kg\(\cdot\)m\({}^{-3}\)). It was also assumed that the force sensor had the following constant parameters:  
* Sensor diameter--10.0 (mm);
* Disc thickness--2.0 (mm).