Дослідження динамічних процесів у неголономній моделі кульового млина

Annotation. Problem. Creation of a universal mathematical model of dynamic processes in a shockless ball mill model to determine the technological qualities of the device and select its rational parameters. Goal. A nonholonomic mathematical model of dynamic processes in a ball mill has been created. Differential equations of motion of a ball in a spherical shell that performs circular translational motion are obtained in the form of the Appel equations. When writing the acceleration function, quasi-velocities were used. An algorithm for converting the Appel equations into first-order differential equations in the Cauchy form is proposed. The computational studies carried out made it possible to establish important regularities in the operation of the device. Methodology. A mathematical model of dynamic processes in a ball mill has been created. The equations of motion of a ball in a spherical shell, which makes a circular translational motion, have been obtained in the form of the Appel equations. An algorithm for their transformation into ordinary first-order differential equations in the Cauchy form has been proposed. Computational studies have been carried out using the developed model. Important regularities of the influence of parameters on the technological indicators of the device have been established. Results. A mathematical model of dynamic processes in a ball mill has been created. The equations of motion of a ball in a spherical shell, which makes a circular translational motion, have been obtained in the form of the Appel equations. An algorithm for their transformation into ordinary first-order differential equations in the Cauchy form has been proposed. Computational studies have been carried out using the developed model. Important regularities of the influence of parameters on the technological indicators of the device have been established. Originality. For the first time, a non-holonomic mathematical model of dynamic processes in a ball mill is presented in the form of the Appel equations. It is shown that the calculation of the acceleration energy and the formulation of the equations are significantly simplified when quasi-accelerations are used instead of generalized accelerations. Practical significance. The created model allows you to calculate the trajectories of the point of contact of the ball with the spherical chamber, the pressure force on the shell, which is important from a technological point of view for choosing rational parameters of the device.