## 2 Theoretical Basis
The angular momentum equation about the center of mass of the cylindrical part is:  
\[I_{G}\ddot{\varphi}=r\ R_{x}-u_{rs}R_{y}sgn(\dot{\varphi}). \tag{8}\]  
Reaction force \(R_{n}\) does not appear in Equation (8), because it passes through the center of mass of the cylinder. \(I_{G}\) is the moment of inertia about the center of mass, \(r\) is the cylinder radius, \(u_{rs}\) is the coefficient of rolling resistance in static condition, \(\ddot{\varphi}\) is the angular acceleration, \(\dot{\varphi}\) is the angular velocity, and \(sgn\) is the signum function. Since the beginning of the rolling motion is considered, \(\ddot{\varphi}=0\) and  
Figure 1: Block (**a**) and cylindrical part; (**b**) on the vibrating conveyor. \(sgn\left(\dot{\varphi}\right)=-1\), because after the beginning of relative motion, a negative rolling velocity will take place: \(\dot{\varphi}=-v_{r}/R\). Tangential reaction force \(R_{x}\) can be calculated from the angular momentum equation:  
\[R_{x}=\ -\frac{u_{rs}}{R}R_{y}. \tag{9}\]  
The term \(\frac{u_{rs}}{R}\) is the non-dimensional rolling friction coefficient in static condition, which is named \(f_{\text{rs}}\). The reaction force that the stop plate exerts on the cylinder (\(R_{n}\)) can be calculated from Equations (5) and (9). The contact between the cylinder and the stop plate is unilateral (\(R_{n}>0\)). Hence, the condition \(R_{n}\leq 0\) in conjunction with the adoption of the maximum rolling friction coefficient in Equation (9) defines the beginning of the relative motion between the cylinder and the conveyor track:  
\[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]0  
If Equation (6) is introduced into Equation (10), the following condition for the beginning of rolling motion is obtained:  
\[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]1  
Figure 2 depicts the values of the static rolling friction coefficient (\(f_{\text{rs}}\)) that prevents the rolling motion against non-dimensional acceleration (\(a/g\)) for parametric values of \(\psi\). Only negative values of acceleration are considered, since a negative acceleration is required to begin the rolling motion; the maximum value of non-dimensional acceleration is \(a/g=-sin(\theta)/cos(\psi)\), since for larger values, the resultant of the gravity and inertia forces pushes the cylinder against the stop plate. Figure 2 shows that very large and unrealistic values [16] of static rolling friction coefficient (\(f_{\text{rs}}\)) are needed to prevent the beginning of the rolling motion. When \(a/g\) is lower than \(-0.35\), the increase in angle \(\psi\) increases the value of \(f_{\text{rs}}\). Hypothetically, the cylindrical part excited by conveyor vibrations could begin a sliding motion as well. The following condition for the beginning of the sliding motion can be derived:  
\[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]2  
It is worth noticing that this condition is the same that holds true for non-cylindrical parts [1]. Equation (12) is similar to Equation (11), but \(f_{\text{rs}}\) is substituted by \(\mu_{s}\), which is the static Coulomb friction coefficient. For most of the materials used in industrial applications [16], the static Coulomb friction coefficient is much larger than the non-dimensional rolling friction coefficient in static condition. \[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]3 Static rolling friction coefficient that prevents the rolling motion, \(\theta=3^{\circ}\) and parametric values of \(\psi\). Therefore, the negative acceleration needed to begin the sliding motion is much larger (in modulus) than the one needed to begin the rolling motion. The evolution of the rolling motion is now analyzed.