Machinery & Energetics

The article considers the construction of the surface of rotation, which is came down to finding its meridian by given condition. Such condition is the nature of the movement of the particle on the inner surface during its rotation around the vertical axis. The absolute movement of a particle is formed from the ratio of the rotational movement of the surface and the relative movement (sliding) of the particle on the surface. Classic example of such movement is the particle movement inside a vertical cone that rotates at a constant angular velocity around its axis, and the partial case when the angle of inclination of the generatrices of the cone is equal to zero and it turns into a horizontal disk. The meridian curve can be set by explicit equations or parametrical equations as a function of an independent variable. The article considers the case when the meridian of the surface of rotation is given by parametrical equations in the function of time. It allows us to make a differential equation of movement of a particle, in which all dependencies are functions of time. These dependences must be found from the composite differential equation of particle movement. When a particle hits the surface, it begins to slide on it, describing a curvilinear trajectory. Taking into account the rotational movement of the surface, there is an absolute trajectory. The first derivative of its length in time gives an absolute velocity, and the second – absolute acceleration, the expression of which includes unknown functions describing the meridian. The differential equation of movement is compiled in projections on three axes of the Cartesian coordinate system. The system of three equations includes four unknown functions: two equations that define the meridian, the dependence of the angular velocity of the particle and the reaction of the surface. To solve the equation, the number of unknown functions should be reduced to three. One dependence was set for this. This method leads to partial cases, one of which is the movement of a particle on a horizontal disk rotating around a vertical axis. A specific example is considered and the meridian curve is constructed as a result of numerical solution of the equations, provided that the particle rises upwards inside the surface at a constant given velocity.

surface of rotation, meridian, angular velocity, particle, sliding, trajectory, differential equations