## 4 Numerical Results
The second effect is the decrease in the initial velocity of the cylindrical part when \(\psi\) increases; see Equation (20). In summary, for small values of \(\psi\), the first effect is dominant, and the traveled distance increases, whereas for large values of \(\psi\), the second effect reduces the traveled distance. The second phase of the motion (phase 2, \(t>0.25\) s) is dominated by the impacts of the cylindrical part on the stop plate. The cylindrical part reaches the maximum displacement in this phase, and this value chiefly depends on impact mechanics. If the relative velocity (Equation (22)) is large, because the cylindrical part and the conveyor move in opposition, the conveyor transfers a lot of energy to the cylindrical part, which has a large rebound after the impact. This phenomenon occurs at \(t=2.04\) for \(\psi=0^{\circ}\). Figure 8, which is a zoom of Figure 6 of about \(t=2.04\) s, shows that the impact occurs when the conveyor has the maximum positive velocity, whereas the velocity of the cylindrical part is negative. Conversely, if the relative velocity is small, because the cylindrical part and the conveyor move in phase, the conveyor transfers less energy to the cylindrical part, and a smaller rebound occurs. This phenomenon takes place at \(t=0.49\) s for \(\psi=0^{\circ}\), since Figure 9 shows that the cylindrical part impacts the conveyor when it has a negative velocity. Figure 8: Numerical results, displacement of the cylindrical part along the track, impact with large relative velocity at \(t=2.04\), \(\psi=0^{\circ}\), \(\theta=3^{\circ}\), \(f_{v}=0.005\), \(\mu_{c}=0.3\), \(\mu_{s}=0.6\). Figure 9: Numerical results, displacement of the cylindrical part along the track, impact with small relative velocity at \(t=0.49\), \(\psi=0^{\circ}\), \(\theta=3^{\circ}\), \(f_{v}=0.005\), \(\mu_{c}=0.3\), \(\mu_{s}=0.6\). The angle of conveyor acceleration (\(\psi\)) has a negligible effect on the second phase (impact dominated) of the motion of the cylindrical part on the conveyor, since the relative velocity at the impact depends much more on the previous motion than on conveyor maximum velocity, which slightly decreases if \(\psi\) increases. Actually, after some impacts, the relative velocity at the impact changes randomly. Thus, the motion of the cylindrical part on the conveyor is similar to the motion of a bouncing ball that impacts a vibrating table, which may exhibit a chaotic behavior [18; 19]. Figure 10 shows for the same cases of Figure 6 the sliding velocity of the cylindrical part and highlights that, owing to the large value of non-dimensional acceleration (\(a/g\ =2\)), very often the cylinder slides on the conveyor track. The largest sliding velocities take place just after the impacts. The second parametric simulation aimed at analyzing the effect of conveyor vibration frequency. Conveyor acceleration was kept equal to the previous value (\(a/g\ =2\)); therefore, an increase in frequency led to a decrease in velocity according to this equation:  
\[v=\frac{a}{\omega}. \tag{28}\]  
Figure 11 clearly shows that the increase in the frequency has a negative effect on the motion of the cylindrical part both before and after the first impact. The first effect takes place because the initial velocity is smaller, while the second effect takes place because the relative velocity at the impact is smaller. If the vibration frequency decreases to 25 Hz, the traveled distance increases up to 0.043 m. It is worth noticing that even if the dynamics of conveying of block-shaped parts and cylindrical parts are very different, the decrease in vibration frequency has a positive effect in both cases [1]. Then, the effect of rolling friction coefficient was investigated by means of numerical simulations. The results, which are presented in Figure 12, show that realistic variations in \(f_{v}\) have a negligible effect on the conveying of cylindrical parts. Before the first impact (phase 1), the decrease in \(f_{v}\) causes a very small increase in the traveled distance.