Designing a gravity chute based on the given trajectory of cargo movement

This study's object is the process of cargo movement along the helical surface of an oblique open helicoid under the action of its natural weight. Such movement takes place in gravity chutes where the cargo descends under the action of its natural weight. Gravity (screw) chutes are used for transportation, separation, and enrichment of material. For a given surface, the problem is solved by composing differential equations of motion of a mathematical point, which is conditionally replaced by cargo, in projections onto the axis of the spatial coordinate system. If the surface is helical, then after stabilization of the motion, it is possible to find the parameters of the helical line – the trajectory of cargo movement. The task implies solving the inverse problem – constructing a helical surface along a given trajectory of cargo descent, which is a helical line. The results are attributed to the use of two accompanying trihedra of the trajectory with a common vertex and tangent orts to the trajectory, which coincide. One of them is a Frenet trihedron whose position is determined by the differential characteristics of the curve, and the second is a Darboux trihedron, the position of which depends on the point of the trajectory on the surface. In addition to the two coincident orts, the remaining four orts are located in the plane normal to the trajectory. The use of these two orts makes it possible to compose differential equations of motion of the load in projections onto a moving Darboux trihedron, one of the planes of which is tangent to the surface. A feature of the solution to the problem is that the trajectory of the load, i.e., the helix, is given by radius r of the cylinder on which it is located and velocity V of the load. Using these data, angle β of its ascent is determined. For example, at r = 0.5 m, V = 2.5 m/s, the angle of elevation is β = 20.7°. Then, a helical linear surface is constructed that passes through the given trajectory