Differential equations for relative yarn movement in the end sections of the channel in the winding drum

The spinning and rewinding of the yarn is accompanied by the movement of the yarn. The specificity of these technological processes leads to the appearance of a curved section of the thread between the thread guide and the slider. This section has a double curvature. In this case, the thread participates in two movements: a figurative movement around the axis and a relative one along its quasi-static form of equilibrium. Air resistance forces, gravity force, portable and Coriolis forces of inertia will act on the thread. Within the framework of this problem, we will consider the relative linear velocity and the translational angular velocity to be constant. The introduction of a spherical coordinate system makes it possible to simplify the original system of differential equations. These coordinates must satisfy a nonlinear nonholonomic relationship. When integrating the system of differential equations, the fourth-order Runge-Kutta method was used. The calculations were carried out for cotton yarn