## 2 Theoretical Basis
The equations of motion are:  
\[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]4  
\[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]5  
\[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]6  
Moreover, during pure rolling, the following kinematic condition has to be fulfilled:  
\[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]7  
\(R_{y}\) can be calculated from Equation (14); then, Equations (13), (15), and (16) can be solved to calculate \(R_{x}\), \(a_{x}\), and \(\ddot{\varphi}\). Taking into account that for a cylinder \(I_{G}=0.5 mR^{2}\), the following results are obtained:  
\[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]8  
\[A_{y}=asin(\psi)=-x_{0}\omega^{2}sin(\omega t)sin(\psi). \tag{2}\]9  
\[ma_{x}=-mgsin(\theta)+\ R_{x}+R_{n} \tag{3}\]0  
It is worth noticing that when the cylinder rolls forwards (\(\dot{\varphi}<0\)), the rolling friction term in Equation (19) adds to the gravity term. The motion of the cylinder on the conveyor track can be studied integrating Equations (18) and (19); the input is \(acos(\psi)\). Generally speaking, a numerical integration is needed (see Section 3) owing to the \(sgn\) function. Nevertheless, \(f_{y}<0.01\) in most industrial processes, and an approximation of the integrals of Equations (18) and (19) can be obtained assuming \(f_{y}=0\). According to Equation (11), the forward motion begins when \(a/g\leq-sin(\theta)/cos(\psi)\). Since in normal conveying applications angle \(\theta\) is rather small, another simplification can be made assuming that the forward rolling motion begins when \(acos(\psi)=-x_{0}\alpha^{2}sin(\omega t)cos(\psi)\) becomes negative. This assumption simplifies the initial conditions, because in the harmonic motion vibration, the acceleration changes its sign when the displacement is zero and the velocity has the maximum positive value (\(x_{0}\alpha\)). The integration of Equation (18) with the above-mentioned simplifications and initial conditions leads to the following equations for the absolute velocity (\(v_{x}\)) and displacement (\(x\)) of the center of mass of the cylinder:  
\[ma_{x}=-mgsin(\theta)+\ R_{x}+R_{n} \tag{3}\]1  
\[ma_{x}=-mgsin(\theta)+\ R_{x}+R_{n} \tag{3}\]2  
From the point of view of conveying, the most important parameter is the relative velocity \(v_{r}\) between the part and the track of the conveyor:  
\[ma_{x}=-mgsin(\theta)+\ R_{x}+R_{n} \tag{3}\]3  
\[ma_{x}=-mgsin(\theta)+\ R_{x}+R_{n} \tag{3}\]4The relative velocity includes a positive and constant term due to initial velocity (\(x_{0}\omega\)); a harmonic term due to the vibration of the conveyor plate, whose integral over a period (\(T=2\pi/\omega\)) is zero; and a negative term due to gravity that increases (in modulus) with time. If \(\theta=0\) (horizontal conveying), the average conveying velocity in the period (\(\overline{v_{r}}\)) is constant and equal to \(\frac{2}{3}\omega x_{0}cos(\psi)\). If \(\theta>0\) the average conveying velocity decreases, and at the nth period it assumes the value:  
\[ma_{x}=-mgsin(\theta)+\ R_{x}+R_{n} \tag{3}\]5  
When \(\overline{v_{r}}\) becomes negative, the cylindrical part rolls backwards and eventually impacts the stop plate. Figure 3 depicts the ratio between \(\overline{v_{r}}\) and the maximum vibration speed (\(x_{0}\omega\)) versus period number (\(n\)) for \(\theta=3^{\circ}\). In Figure 3a, the ratio between the vibration acceleration and gravity acceleration (\(a/g\)) is 1 and the parametric values of angle \(\psi\) are considered; the increase in \(\psi\) has a negative effect on the conveying speed. In Figure 3b, \(\psi\) is set to zero, and the parametric values of the ratio \(a/g\) are considered; the increase in vibration acceleration has a positive effect on vibration conveying. Equation (24) holds true with the assumption \(f_{v}=0\). If rolling resistance is present, it contributes to decreasing the average conveying speed in the period.