MATHEMATICAL MODEL OF A PARTICLE SLIDING ON THE SURFACE OF A ROTATING VERTICAL RIGHT HELICOID

Abstract

A right helicoid can be either open or closed depending on the method of its formation. If a straight horizontal line segment undergoes helical motion while intersecting the vertical axis, the resulting helicoid is closed and is technically known as an auger (or screw). If the segment does not intersect the axis and moves at a certain distance from it, the resulting helicoid is open. Screw conveyors typically utilize closed helicoids as standard technical helical surfaces. However, other helical surfaces may be used, which requires the creation of a corresponding model of particle motion.

The distinctive feature of this study is the development of a generalized mathematical model that describes particle motion along both types of helicoids. This differentiates it from existing works focused on a specific surface. The obtained second-order differential equations describe the sliding trajectory of a particle on the surface. Depending on the design parameters, such a surface can be an open or closed helicoid, or a special case — the rotation of a horizontal flat disk. This approach allowed for obtaining particle motion parameters on various surfaces and comparing the results.

By solving the differential equations using numerical methods, sliding trajectories of a particle on the surfaces of closed and open vertical helicoids rotating around their own axes were constructed. It was established that there is no fundamental difference between the sliding trajectories. Simultaneously, such sliding results in the upward movement of the particle. Initially, on a closed helicoid, the particle moves downward before ascending, causing its ascent velocity to be slightly lower compared to an open helicoid. If the helicoids are stationary, the particle begins to descend along the surface, moving away from the axis, and eventually stops. The particle sliding trajectories are constructed both within the boundaries of the surface segment and under the condition that it is not constrained by a cylinder.

Keywords: right and open helicoids, trajectory, angular velocity, compound motion, differential equations.