Construction of a generalized mathematical model for particle sliding on the surface of a rotating vertical straight helicoid

The object of this study is the complex motion of a particle on the surface of a vertical straight helicoid rotating around its own axis. In screw conveyors, closed helicoids are used as well-known technical helical surfaces. An issue is not the disadvantages of using classical closed helicoids but the limitations of existing mathematical models of particle motion, which essentially reduce engineering research only to this type of surfaces. The lack of a generalized model for other helical surfaces makes their analysis and practical application impossible. The proposed approach expands the class of helicoids under consideration and creates the prerequisites for finding new design solutions. The derived second-order differential equations describe the trajectory of particle sliding on the surface. Depending on the structural parameters, such a surface can be an open or closed helicoid, as well as a special case of rotation of a horizontal flat disk. That has made it possible to define the parameters of particle motion on different surfaces and compare the results. In particular, the particle sliding trajectories along closed and open helicoids rotating with angular velocity ω = 10 s–1 and ω = 20 s–1 were constructed. In this case, the friction coefficient f = 0.3 and the lift angle β = 15° of the outer edge of the surface were assumed at a radius of R = 0.1 m of the limiting cylinder. The particle sliding trajectories were constructed within the surface compartment, as well as under the condition that it is not limited by the cylinder. The practical significance of the results is the possibility of using the model built for designing energy-efficient screw conveyors without an external casing. This makes it possible to reduce the metal content of structures by 15–20% and prevent jamming during the transportation of fractional materials. The resulting analytical dependences make it possible to calculate the optimal screw pitch and shaft radius to ensure a given material movement trajectory.