Maple-моделі руху частинки по нерухомих шорстких поверхнях 2-го порядку

Анотація

УДК 514.18

MAPLE MODEL THE MOTION OF THE PARTICLE IN A STATIONARY ROUGH SURFACES OF THE 2ND ORDER

1. A.   Nevidomin

Analytical output law of motion of a particle on the rough surface is reduced to drawing up a system of differential equations of 2nd order desired dependency which is the trajectory of the particle, its velocity, acceleration, length of the path, the force of normal reaction, the move to its stop and other trajectory-kinematic characteristics . The sequence of analytical output of differential equations and ways of its solution is quite time consuming.

In contrast, computer simulation of a particle on the surface allows to remove cumbersome analytic transformation and provide a convenient interactive mode scientist for the necessary computational experiments on the analysis of a particle at different baseline throwing it on any rough surface. But the development of computer models of a particle on the surface needs to address a number of theoretical and practical nature which make up the relevance of research. First, develop a general algorithm for automatic withdrawal of differential equations law of motion of particles on any surface that is randomly located in space; trajectory analysis, kinematic characteristics of a particle not only in time, but depending on the position of the particles on the surface and the direction of its movement on the surface; illustrate the results of research in the form of numerical data, graphics and motion simulations particles on the surface.

The purpose of research - development-Maple models of a particle on all surfaces 2nd order.

Materials and methods of research. For cylinder arbitrary rotation position developed three models of a particle, which allow it to explore the trajectory-kinematic properties according to the time, location and direction of movement. Derived laws of motion of particles in the cylinder.

Results. The shape of the trajectories of particles on the vertical cylinder essentially depends on the angle 0 °, 45 °, 90 °, 120 ° throwing them over time t all the particles gain speed free fall. For horizontal cylinder ξ = 90 ° to the motion of a particle to its full stop significantly depends on its initial position on the surface, which affects the possible lead particles from the surface. Thus, the particle to the original position 0.9π and throwing angle 60 °, 90 ° break away from horizontal surface of the cylinder at the time t≈0.24s - will free fall. At this point the normal reaction force is zero, and graphs showing speed velocity of separation from the surface. Particles with an angle throwing 0 °, 30 ° not  break away from the surface of the horizontal cylinder. The nature of a particle on the inner surface of the cylinder includes older properties as the vertical cylinder and the horizontal, what significantly affects the angle deviation from its vertical position. For horizontal cylinders and older probable movement of particles on its outer surface, where the centrifugal force will not pin it to the surface, but rather break away it at the top of the cylinder.

We derive laws of motion of the particles on rough inner surface of the vertical cylinder

The trajectories of particle velocity and graphics on rough surface vertical cylinder that particles thrown at an angle of 30 ° minimum speed is 2.8 m / s at a time when the particle was at a distance of 4.9 rectilinear generators. Particles thrown straight generators perpendicular to the cylinder (90 °), first reduce their speed to a certain value of 5.8 m / s, and then gaining it. The greater the friction, the lower the value of the minimum speed of the particles and the faster it falls. With decreasing initial velocity particles shortens the time before its fall.

Based on the proposed method of forming laws of motion of particles on rough surfaces in the interior of the coordinates in the projection of ort accompanying trihedron trajectory was set maple-set models of a particle in all ruled rough surfaces 2nd order. In order to evaluate each of these surfaces to make the separation or movement of loose material held complex computational experiments on trajectory analysis, kinematic properties of a particle at different initial conditions. In particular, the trajectory of the vertical rotation hyperboloid not cross his neck, as particles break away. Bulk material thrown on top of the slope hyperboloid first will accumulate around the neck, which after passing trajectories of particles will be closer to each other.

Automatic execution of analytical transformations in the formation law of motion of a particle on the rough surface of the sphere, ellipsoid, paraboloid [4] and hyperboloid of revolution arbitrary position is so cumbersome that they illustrate is impractical - the user sets only the initial conditions of the experiment and selects appearance presenting the results (Fig. 4).

Conclusions

A set of computer models of automatic generation of a particle of the law as a system of two differential equations of 2nd order for any rough surface randomly located in space. It was developed through computer tools has become possible to perform online research trajectory-kinematic characteristics of a particle on all surfaces 2nd order, cylindrical surfaces with orthogonal sections as transcendental curves of helical surfaces.