Research of the movement of a material particle on the internal surface of a vertical cylinder that performs planetary motion

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Planetary motion of a cylinder is understood as its motion when it is in two rotational motions at the same time: it rotates around its own vertical axis with a constant angular velocity, and the cylinder axis itself rotates with a constant angular velocity around a vertical fixed axis. The movement of the particle will be complex and will consist of its relative movement along the inner surface of the cylinder and the translational movement of the cylinder itself. Such a drive scheme is used in cylindrical sieves for sorting seeds of agricultural crops. Problems on the complex motion of a particle can be successfully solved using the trihedron and Frenet's formulas. The purpose of the study is to establish the complex movement of a material particle along the inner surface of a cylindrical sieve using a trihedron and Frenet's formulas at the same and different angular velocities of transfer and relative rotation of the sieve. A characteristic feature of the application of the trihedron and Frenet formulas is that the independent variable in them is not time t, as is generally accepted in problems of kinematics and dynamics of a point, but the length of the arc s of the directional curve (in our case, a circle of radius R), so the relationship was established the connection between rotational movements through this parameter. The system of differential equations is integrated by numerical methods. An exact analytical solution was found in the case when the motion of the particle stabilizes and its speed becomes constant. The obtained results were visualized. Some regularities of the relative and absolute motion of a particle in a cylindrical sieve were established when the angular velocity of rotation of the cylinder around its own axis is zero and is not equal to zero. In the first case, it was found that the particle on the surface of the cylinder occupies a position at which it is as far as possible from the axis of rotation of the cylinder around a vertical line and then it moves down the plane of the cylinder uniformly accelerated, uniformly or uniformly decelerated until it "sticks" depending on the value angular velocity. In the second case, the particle behaves similarly: it remains at the maximum distance after the motion is stabilized. At the same time, it slides along the surface of the cylinder with a constant relative speed along a helical line. The direction of the rise of the helical line changes to the opposite when the direction of the angular velocity of the cylinder changes. Such movement is possible in a certain range of angular velocities of the particle and the cylinder. As the angular velocity of the particle increases, there comes a moment when it cannot maintain the described state of sliding and begins to move along the surface of the cylinder with stops and is prone to "sticking". This state occurs sooner when the angular velocities of the particle and the cylinder have the same direction, and later when they are directed in opposite directions.