Calculation of a gravitational screw chute with an axial cross-section curve defined by an explicit equation

Screw gravity chutes are used for energy-free transportation of cargo and gravitational separation of ores, which necessitates an optimal combination of design parameters of the working surface, speed of movement, and compactness of the trajectory. The aim of the work was to analytically describe the movement of a cargo along a screw surface, given by the curve of its axial cross-section, under the action of its own weight, using the example of a material particle. Methods of classical mechanics, differential surface theory and numerical methods were used to solve the problem. The main results of the study were based on the fact that after stabilising its motion, the material particle begins to slide along the surface at a constant speed and a constant distance from the axis of the helical surface, taking into account the shape of the curve of its axial cross-section. It was established that the equation of this curve may include constant values that affect its shape, i.e., the kinematic parameters of the particle. This made it possible to find the required values of the constants to ensure the specified parameters of particle sliding. The differential equations of motion of a particle sliding along a helical surface were compiled in projections on the axes of a stationary coordinate system. A parabola was considered as the curve of the axial cross-section of the surface, the equation of which includes a constant value. The obtained analytical dependencies made it possible to determine the optimal values of the constants in the equation of the axial cross-section curve, which ensured the required sliding speed of the particle and the distance from the axis of the helical surface. This opened up opportunities for designing screw chutes taking into account specific technological requirements, in particular for gravitational separation or energy-free transportation of bulk materials. The practical application of the proposed calculations is demonstrated by the example of a parabolic cross-section, which confirms the effectiveness of the method for optimising the kinematic parameters of motion. As a result of the study, expressions for the design of screw gravity chutes were obtained and the influence of the introduced constant on the kinematic parameters of particle sliding was determined