EXTERNAL ROLLING OF A POLYGON ON CLOSED CURVILINEAR PROFILE

The rolling of a flat figure in the form of an equilateral polygon on a curvilinear profile is considered. The profile is periodic. It is formed by a series connection of an arc of a symmetrical curve. The ends of the arc rely on a circle of a given radius. The equation of the curve, from which the curvilinear profile is constructed, is found. This is done provided that the centre of the polygon, when it rolls in profile, must also move in a circle. Rolling occurs in the absence of sliding. Therefore, the length of the arc of the curve is equal to the length of the side of the polygon. To find the equations of the curve of the profile, a first-order differential equation is constructed. Its analytical solution is obtained. The parametric equations of the curve are obtained in the polar coordinate system. The limits of the change of an angular parameter for the construction of a profile element are found. It is a part of the arc of the curve. According to the obtained equations, curvilinear profiles with different numbers of their elements are constructed

Rolling of a single-cavity hyperboloid of rotation on a helicoid on which it bends

The bending of a single-cavity hyperboloid with rotation while maintaining the rectilinear generator is considered. The resulting bending surfaces are a plural of open skew helicoids, including partial cases of oblique closed and open ordinary helicoid s. The parametric equations for the continuous bending of these surfaces are established by changing the angle between the straight line and its axis. The possibility of pure unrolling of a hyperboloid along a helicoid from the set of its bends with linear contact along a common rectilinear generator of both surfaces is shown. Using the obtained equations, the surfaces are constructed and the images of hyperboloid and helicoid with the common rectilinear generator of their contact are shown