## 4 Numerical Results  
Numerical simulations aimed at exploring the dynamics of vibration conveying of cylindrical parts beyond the limits of the analytical model of Section 2. In particular, numerical simulations made it possible to analyze the effect of the impacts of the cylindrical part on the stop plate. A realistic reference case was defined, which is characterized by the following parameters of the vibratory conveyor: \(x_{0}\)= 0.0002 m, \(\omega=314\) rad/s (frequency 50 Hz), \(\theta=3^{\circ}\), \(\psi=0^{\circ}\). The maximum modulus of conveyor acceleration is 19.72 m/s\({}^{2}\) (\(a/g=2\)). The rolling part has a radius \(r=0.007\) m and mass \(m=0.096\) kg. The friction parameters are \(f_{v}=0.005\), \(\mu_{c}=0.3\), and \(\mu_{s}=0.6\). The value of the rolling resistance is more than twice that of the typical rolling elements of bearings [16] and takes into account that the cylindrical part may be rough or dirty. The kinetic Coulomb friction coefficient value is inside the typical range of values found in vibratory conveyors [1; 6; 9]. The static Coulomb friction coefficient was set to be twice the kinetic coefficient [17]. Finally, the impact stiffness was set to \(k=1,000,000\) N/m; this value made it possible to obtain values of contact forces similar to the measured ones (about \(60\div 80\) N). The first parametric simulations aimed at analyzing the effect of angle \(\psi\); Figure 6 shows numerical results in terms of the absolute displacement of the cylinder center of the mass. In the same figure, for comparison, the displacement of the conveyor is represented (along the direction tilted by \(\psi\) with respect to the \(x\) axis). In the plots of Figure 6, two different phases can be identified. The first phase (phase 1), which ends at about 0.25 s and includes 10 \(\div\)12 vibration periods, is the initial motion of the cylindrical part without impacts. As foreseen by theoretical analysis, the cylindrical parts begins its motion at \(t=0\) owing to the initial velocity given by the conveyor, reaches the maximum displacement, and eventually impacts the stop plate. The maximum displacement is rather small (about 0.003 m), and this value is in agreement with the theoretical predictions (Equation (21)). Figure 5: Logical flow of the simulation. In this phase, the increase in the angle \(\psi\) up to \(20^{\circ}\) leads to an increase in the traveled distance, a further increase leads to a reduction in the traveled distance. This particular behavior is due by the presence of two opposite effects. The first effect can be understood looking at the sliding velocity (\(v_{sl}\)) of the cylindrical part:  
\[v_{sl}=v_{x}-V_{x}+R\dot{\varphi} \tag{27}\]  
which is plotted in Figure 7 with conveyor acceleration. If \(\psi=0\), the cylindrical part slides both when the conveyor acceleration is negative and positive (forwards and backwards sliding, respectively). Backwards sliding causes a reduction in the traveled distance. This phenomenon is in agreement with the analytical results of Figure 4, which show that for \(\psi=0\), the static friction coefficient needed to guarantee the pure rolling motion is larger than \(\mu_{s}=0.6\) both for negative (-2) and for positive (+2) values of non-dimensional acceleration. Conversely, Figure 7 shows that if \(\psi\) increases, there is only forwards sliding, since the static friction coefficient is able to prevent backwards sliding when the conveyor acceleration becomes positive, and the cylindrical part behaves similar to a block. It is worth noticing that this behavior is in agreement with the analytical results of Figure 4.  
Figure 6: Numerical results, displacement of the cylindrical part along the track, \(\theta=3^{\circ}\), \(f_{\rm{F}}=0.005\), \(\mu_{\rm{c}}=0.3,\ \mu_{\rm{s}}=0.6\), and parametric values of \(\psi\). Figure 7: Numerical results, sliding velocity of the cylindrical part along the track, \(t<0.1\) s, \(\theta=3^{\circ}\), \(f_{\rm{F}}=0.005\), \(\mu_{\rm{c}}=0.3,\ \mu_{\rm{s}}=0.6\), and parametric values of \(\psi\).