Study of the motion of a material particle on the internal surface of a stationary inclined cylinder

Cylindrical surfaces as working bodies of agricultural machines have quite wide application. The motion of a material particle on cylindrical surfaces is considered in the works of academicians of the Ukrainian Academy of Sciences P. M. Vasylenko and P. M. Zaika. However, P. M. Vasylenko failed to integrate the differential equations of motion and he considers approximate solutions. A similar problem was solved by P. M. Zaika, considering the motion of a particle on the inner surface of an inclined cylinder rotating around its own axis. The purpose of the study is to find the kinematic parameters of the motion of a material particle on the inner surface of a stationary inclined cylinder under the action of its own weight under different initial conditions. The article investigates the motion of a material particle on the inner surface of an inclined cylinder under the action of its own weight. Cases are considered when the particle moves accelerated, decelerated and at a constant speed. The system of differential equations is solved by numerical methods. The results were visualized. It was established that the motion of a material particle along the inner surface of an inclined cylinder under the action of its own weight in the presence of friction can be divided into three cases: 1) the angle of inclination of the generating cylinder to the horizon is greater than the angle of friction; 2) the angle of inclination of the generating cylinder to the horizon is equal to the angle of friction; 3) the angle of inclination of the generating cylinder to the horizon is less than the angle of friction. If we exclude the rectilinear motion along the lower generating cylinder, then all three cases are characterized by oscillatory motion, which over time acquires certain signs of stability, in particular, by decreasing the amplitude. In this case, in the first case, the particle velocity increases over time, in the second - it stabilizes and becomes constant, in the third - it decreases until the particle stops completely. In the absence of friction, in all three cases, the oscillations will continue indefinitely with an increase in their period.