## 4 Numerical Results
After the first impact (phase 2), the variation in the rolling friction coefficient changes the instants in which the most energetic impacts take place, but the traveled distance does not significantly change. Figure 13 shows the effect of contact stiffness on the conveying of cylindrical parts. The contact stiffness does not affect phase 1, and it does not have a clear effect on the traveled distance in phase 2. This phenomenon agrees with physical intuition, because the contact stiffness is another factor that contributes to randomly varying the relative velocity at the impact, which is the main factor influencing the rebound of the cylindrical part. Figure 10: Numerical results, sliding velocity of the cylindrical part along the track, \(\theta=3^{\circ}\), \(f_{v}=0.005\), \(\mu_{c}=0.3\), \(\mu_{s}=0.6\), and parametric values of \(\psi\). Figure 11: Numerical results, displacement of the cylindrical part along the track, \(\theta=3^{\circ}\), \(\psi=0^{\circ}\), \(f_{v}=0.005\), \(\mu_{c}=0.3,\ \mu_{s}=0.6\), and parametric values of frequency. \[v_{sl}=v_{x}-V_{x}+R\dot{\varphi} \tag{27}\]0 Numerical results, displacement of the cylindrical part along the track, \(\theta=3^{\circ}\), \(\psi=0\), \(\mu_{c}=0.3,\ \mu_{s}=0.6\), and parametric values of \(f_{v}\). \[v_{sl}=v_{x}-V_{x}+R\dot{\varphi} \tag{27}\]1 Numerical results, displacement of the cylindrical part along the track, \(\theta=3^{\circ}\), \(\psi=0\), \(f_{v}=0.005\), \(\mu_{c}=0.3,\ \mu_{s}=0.6\), and parametric values of contact stiffness. Figure 14 shows that \(\mu_{s}\) has a small effect on phase 1: if the static friction coefficient increases above 0.6, the traveled distance increases. This phenomenon can be explained looking at Figure 15, which shows the sliding speed for the same values of \(\mu_{s}\). With \(\mu_{s}=1\) during phase 1, the cylinder does not slide at all, and the traveled distance increases, since the pure rolling motion dissipates less energy than the rolling with sliding motion. It is worth noticing that this result agrees with the analytical results of \[v_{sl}=v_{x}-V_{x}+R\dot{\varphi} \tag{27}\]2  
After the first rebound, the impacts take place at random instants and, again, the relative velocity at the impact has the largest effect. The effect of a proportional variation of both friction coefficients (\(\mu_{c}\) and \(\mu_{s}\)) is presented in \[v_{sl}=v_{x}-V_{x}+R\dot{\varphi} \tag{27}\]3 In phase 1, the cylinder with the highest \(\mu_{s}\) reaches the largest distance because it rolls without sliding, as shown in \[v_{sl}=v_{x}-V_{x}+R\dot{\varphi} \tag{27}\]4 After the first impact (phase 2), the motion is dominated by the relative velocity at the impact; nevertheless, a decrease in \(\mu_{c}\) leads to an average increase in the traveled distance. With \(\mu_{c}=0.2\), there are 4 rebounds with a traveled distance larger than 0.008 m, whereas with \(\mu_{c}=0.4\), there is no rebound with a traveled distance larger than 0.008 m. This phenomenon occurs because for every value of \(\mu_{c}\) and \(\mu_{s}\), there are large sliding velocities (Figure 17) after the impacts, and a reduction in \(\mu_{c}\) leads to a reduction in energy dissipation. \[v_{sl}=v_{x}-V_{x}+R\dot{\varphi} \tag{27}\]5 Numerical results, displacement of the cylindrical part along the track, \(\theta=3^{\circ}\), \(\psi=0\), \(f_{v}=0.005\), \(\mu_{c}=0.3\), and parametric values of static friction coefficient \(\mu_{s}\). \[v_{sl}=v_{x}-V_{x}+R\dot{\varphi} \tag{27}\]6 Numerical results, sliding velocity of the cylindrical part along the track, \(\theta=3^{\circ}\), \(\psi=0\), \(f_{v}=0.005\), \(\mu_{c}=0.3\), and parametric values of static friction coefficient \(\mu_{s}\). The numerical results presented in this section clearly highlight that only the decrease in the vibration frequency can significantly increase the distance traveled by the cylindrical part.