MATHEMATICAL MODELING OF THE ROTATION PHASE OF A SOLID BODY MOVEMENT WHEN DESCENDING FROM THE INCLINED RAMP
Ключові слова:plane-parallel motion, rod, inclined ramp, Lagrange equation of the second kind, ordinary differential equations, Cauchy problem, numerical simulation, Runge-Kutta method
Known mathematical models of solids coming off a conveyor belt or sloping ramp are based on the construction of three differential equation systems in a rectangular Cartesian system of coordinates. However, there are no results of calculations of motion parameters using these models and their comparison with other calculation and experimental data for specific objects in open print.The aim of the work is to build and test a new adequate mathematical model of the phase of rotation of solid movement when coming down from the sloping ramp to study the parameters of its movement at the beginning of free fall.The task of the solid body rotation phase (a convergence from the supporting surface with a growing angle of inclination from the moment when the center of the body mass is above the edge of the support, to the detachment from the support of its rear end) is considered, which is simulated by a straight homogeneous rod, when descending from the inclined ramp in the polar coordinate system.Differential equations of the rod movement are made with the help of Lagrange equations of the second kind. The generalized coordinates are the distance from the edge of the support to the center of the rod masses and the polar angle between the horizontal axis and the axis of the rod. For the received non-linear resolution system of two ordinary differential equations of the second order, the corresponding task of Koshi is formulated, which is solved numerically by the Runge-Kutt method of the fourth order of accuracy. On the basis of the proposed approach, numerical experiments were conducted, the results of which are presented in the form of graphs dependencies time the rotation phase, angle of the turn and the angular speed and linear velocity of center of mass at the end of the rotation phase from the initial velocity of the center of mass of the rod for rods of 5, 10 and 15 m by an angle of ramp 35°.It has been established that the increase in the initial speed of the center of the mass leads to an increase in its final speed, as well as a decrease in the time of the rotation phase, the angle of the turn and the angular speed of the rod. As the length of the rod increases, the rotational phase and the final speed of the center of the mass increase, and the turning angles and angular speeds of the rod decrease.
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