Abstract. Generalized differential equations of relative particle motion over the internal surface of an inclined cylinder, which performs an oscillating motion, have been formulated. All the points of a cylinder trace ellipses in vertical planes. Such cases of cylinder oscillation, when either semi-axes or one of them is equal to zero, that is to say, a cylinder performs reciprocating motion, have been considered. The equations have been solved applying numerical methods and trajectories of relative particle motion over the surface of a cylinder have been built. Graphs of other kinematic characteristics as a timevarying function have been presented. Certain cases, when a cylinder axis is located horizontally or at an angle to a horizontal plane, have been considered.