## 1 Introduction  
The automation of industrial processes very often requires the handling of small [1; 2; 3] and micro items [4], which include mechanical and electrical components used in the assembly processes, finite products that have to be packaged, and row materials that have to be processed. In many industrial scenarios, these small items have to be transported, oriented, and sorted. Nowadays, vibratory systems are very common, because they allow handling small items in a versatile and efficient way. They have a simple and sturdy construction and are suited to handle dusty and hot materials as well. For these reasons, some researchers have studied the kinematics and dynamics of vibratory conveying, orienting, and sorting. In 1963, [PERSON] et al. [5] carried out a pioneering study that highlighted the role of friction and clearly stated the differences between the gliding and hopping motions of a machine part on a vibrating conveyor, in which the machine part was considered a point mass. The influence of directional friction characteristics was analyzed in [6], assuming a point mass model of the vibrating part. In [7], the mechanical part was modeled as a rigid body with six degrees-of-freedom (DOFs) and the contact mechanics were deeply analyzed; however, the rolling motion was neglected, since a block-shaped component was studied. The dynamics of a block moving along the spiral track of a vibratory bowl feeder were studied in [8], and the block was modeled as a point mass. Some recent studies dealt with the design of the conveyor [9; 10; 11]; also in these studies, the part was modeled as a point mass, and no rolling motion was considered. Cylindrical parts, similar to pins and plugs, are very common in electrical and mechanical industry; also, a screw is better represented by a cylinder than by a block. Other manufacturing processes require the handling of cylindrical and quasi-cylindrical parts, e.g., the pharmaceutical industry has to convey cylindrical pills and phials. Cylindrical parts are processed by classical conveyors, if their axes of rotation form a small angle with the direction of conveying [12]. When this condition is not satisfied, they begin to roll down the track. Most of the scientific literature dealing with conveyors neglects this rolling motion [1], because sometimes, this phenomenon does not have a significant effect on the process. For example, in orientation processes, often a cylindrical part that can roll also has a wrong orientation and it has to be rejected both if it slides and if it rolls. Nevertheless, in some applications, the cylindrical part has to travel a larger distance, and the rolling motion strongly influences the process. Thus, this paper focuses on the dynamics of cylindrical parts in a vibrating conveyor, with the aim of explaining the basic differences between the vibration conveying of a block and of a rolling cylinder. The scientific literature in this field is rather scarce. Some papers [13; 14] analyzed the probabilities of natural resting aspects of cylinders and parts with curved surfaces in vibratory bowl feeders. In [15], a preliminary numerical analysis of the motion of a cylindrical part on a conveyor was carried out, and the possibility of rolling, sliding, and hopping motions was highlighted. This paper is an extension of the previous research [15] that includes analytical studies, a more detailed mathematical model that considers impacts, and a wide parametric analysis. The paper is organized as follows. First, the dynamic equations governing the rolling motion of a cylindrical part are analytically derived, the differences between vibratory conveying of rolling and non-rolling parts are highlighted, and the parameters with the largest influence on the rolling motion are found. Then, a two-dimensional (2D) non-linear numerical model is developed that takes into account the transition between pure rolling and rolling with sliding and the impacts of the cylindrical parts with the edge of the conveyor.