Dynamics of a Particle on a Movable Wavy Surface

The relative motion of a particle on a linear wavy surface with a sinusoid cross-section is considered in the article. The rectilinear generatrices of the surface are parallel to each other and parallel to the horizontal plane. The surface oscillates in such a way that all its points describe circles in horizontal planes. Numerical methods solve the differential equations of the relative displacement of a particle. The trajectories of sliding of a particle on a surface and graphs of its reaction were obtained. The relative trajectory of the particle after movement stabilization can be closed or periodic spatial curves. When the diameter of the circle during oscillations is equal to the period of the sinusoid, the trajectory of the relative motion of the particle can be a periodic curve. In this case, the particle moves in a direction, close to the transverse, overcoming depressions and ridge surfaces. In other cases, the trajectory of sliding is a closed spatial curve whose horizontal projection is close to the circle. A partial example of a surface is a plane, and the trajectory of relative motion (the sliding of a particle) is a circle. An analytical expression of its radius was found.