## 2 Theoretical Basis  
There are very important differences between the vibration conveying of parts that can roll on the conveyor track and parts that cannot roll. A 2D model with the plane of motion perpendicular to the axis of the cylinder is enough to point out these differences. Vibration conveying of non-rolling parts is a dynamic phenomenon dominated by dry friction. Figure 1a shows a small block on the track of the conveyor, which is tilted by angle \(\theta\) with respect to the horizontal direction. A fixed reference frame with axis \(x\) parallel to the track and axis \(y\) perpendicular to track is introduced. \(R_{x}\) and \(R_{y}\) are the components of the reaction force exerted by the track surface on the block (the positive direction is represented); they are related by the static Coulomb friction coefficient \(\mu_{s}\) and by the kinetic Coulomb friction coefficient \(\mu_{c}\). Acceleration \((\overrightarrow{a})\) of the conveyor is tilted by angle \(\psi\) with respect to the conveyor track. If the motion of the conveyor is harmonic with amplitude \(x_{0}\) and angular frequency \(\omega\), the components of the acceleration are:  
\[A_{x}=acos(\psi)=-x_{0}\omega^{2}sin(\omega t)cos(\psi) \tag{1}\]  
\[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]  
Angle \(\psi\) is essential for the vibration conveying of non-rolling parts. When acceleration has the direction shown in Figure 1a, the inertia force adds to the weight of the block, the reaction force \(R_{y}\) is large, the modulus of the demanded tangential reaction force \(|R_{x}|\) is smaller than the maximum static friction force \((\mu_{s}|R_{y}|)\), and the block cannot slide backwards. Conversely, when acceleration \(\overrightarrow{a}\) changes its direction, the inertia force tends to cancel the weight force, \(R_{y}\) decreases, and the modulus of the demanded tangential reaction force \(|R_{x}|\) may become larger than the maximum static friction force;hence, the block moves, and forwards sliding begins. This condition repeats at every period of the harmonic motion and eventually the block moves forwards with respect to the track. The scenario drastically changes when the part has a cylindrical shape, as depicted in Figure 1b. A stop-plate is introduced, and \(R_{n}\) is the reaction force exerted by the stop plate on the cylinder (the positive direction is represented), \(M_{r}\) is the rolling resistance torque (the positive direction is represented), \(m\) is the cylinder mass, and \(mg\) is the gravity force (the actual direction is represented). First, the cylindrical part is assumed to be steady on the vibrating conveyor, and the dynamic conditions that make possible the beginning of the relative motion are analyzed. The equations of motion in the \(x\) and \(y\) directions are:  
\[ma_{x}=-mgsin(\theta)+\ R_{x}+R_{n} \tag{3}\]  
\[ma_{y}=-mgcos(\theta)+\ R_{y}. \tag{4}\]  
In Equations (3) and (4), \(a_{x}\) and \(a_{y}\) are the two components of the acceleration of the center of mass of the cylindrical part. When there is no relative motion, \(a_{x}\) and \(a_{y}\) coincide with the components of acceleration of the conveyor (\(A_{x}\) and \(A_{y}\)); therefore, the equations of motion become:  
\[macos(\psi)=-mgsin(\theta)+\ R_{x}+R_{n} \tag{5}\]  
\[ma\ sin(\psi)=-mgcos(\theta)+\ R_{y}. \tag{6}\]  
The perpendicular reaction force can be calculated from Equation (6). \[R_{y}=\ ma\ sin(\psi)+mgcos(\theta) \tag{7}\]  
The contact between the cylinder and the track is unilateral; hence, \(R_{y}\geq 0\). When \(R_{y}\) tends to zero, the cylinder separates from the track, and the hopping motion begins. The hopping motion can be avoided decreasing the vibration acceleration, and this solution leads to a noise reduction [9]. Thus, this kind of motion is not considered in the framework of this research.