SLIDING OF THE PARTICLE ALONG THE MOVABLE HORIZONTAL PLANE

Abstract

A lot of engineering problems relate to the interaction of working bodies of machines with particles of technological material and require analytical dependencies of particle movement on a moving rough plane. When a particle hits a moving horizontal rough plane, which remains horizontal during movement, the particle begins to slide along it. The character of the movement of the plane determines the shape of the particle's sliding trajectories. The reciprocating oscillations of the horizontal plane, as well as translational oscillations, when all points of the plane describe circles, are sufficiently researched. However, there is no rotation of the plane in these cases. However, numerous parts of machines and mechanisms carry out just such a movement. If the plane moves translationally, the particle moves along a trajectory, which is similar to the curve described by the plane. During the rotational movement of the plane, the particle moves along a spiral of the correct shape. In the case of a combination of these two movements, at the initial stage the relative motion of the particle is chaotic, but over time it acquires the shape of a spiral regardless of point of particle incidence to the plane. The article deals with the relative motion of a particle on a horizontal rough plane, which carries out complex oscillations, which are the result of moving a point of the plane in a circle with a constant angular velocity relative to its center and simultaneous rotation of the plane around this point with the same angular velocity in the opposite direction. The partial case of oscillations, when the lengths of the crank and slider are equal to zero, is considered. Analytical dependencies of particle motion were found by methods of differential geometry. The obtained regularities make it possible to significantly expand the theory of particle movement on the surface. In addition, they can be applied to the geometric design of mechanisms of the crank-slider type, in which the length of the slider is equal to the length of the crank.

Key words: relative motion, horizontal plane, oscillations with rotational motion, particle, differential equations.