Geodesic lines on the cyclic helical surfaces

Abstract

At present, wind turbines with a vertical axis are widely used in engineering. Due to the twist of the blades, the rotation of the helicoid rotor is more uniform, which significantly reduces the dynamic load on the support nodes and, thus, increases their service life. But in the process of wind load there are cracks in the surface of blades, the development of which may eventually lead to their partial destruction or the entire structure, so their detection and elimination are an important part of design and manufacturing. It is known that trajectories of cracks coincide with families of geodesic surface lines, which ultimately leads to the idea that the voltages along the geodesics are larger in comparison with the stresses along the lines of curvature. The term “geodesic” was first used by P. Laplas in relation to the “shortest lines” on the earth’s surface. For a long time, the geodesic line existed only theoretically in differential geometry as a line of internal surface geometry. But at the present time families of geodetic lines find practical application in engineering and production. For example, in the formation of surfaces for laying out of fiberglass or strengthening the cylinders of composite materials with reinforced threads, in the calculation of the trajectory of mobile robot. The obstacle in finding geodesic surface lines is the problem of a computational nature. Only for a limited list of known surfaces (cylinder, cone, pseudosphere) can one find the equation of geodesic lines in explicit form. For all others, including the sphere, their retrieval is reduced to the integration of differential equations by numerical methods.In this paper, the equation of the geodesic lines on the cyclic helical surface is obtained by minimizing the curve length. Since all the formulas are rather cumbersome for calculations, we used Simpson’s formulas to calculate certain integrals. Several geodesics were built on such surfaces in a graphic editor to confirm the results.At present, wind turbines with a vertical axis are widely used in engineering. Due to the twist of the blades, the rotation of the helicoid rotor is more uniform, which significantly reduces the dynamic load on the support nodes and, thus, increases their service life. But in the process of wind load there are cracks in the surface of blades, the development of which may eventually lead to their partial destruction or the entire structure, so their detection and elimination are an important part of design and manufacturing. It is known that trajectories of cracks coincide with families of geodesic surface lines, which ultimately leads to the idea that the voltages along the geodesics are larger in comparison with the stresses along the lines of curvature. The term “geodesic” was first used by P. Laplas in relation to the “shortest lines” on the earth’s surface. For a long time, the geodesic line existed only theoretically in differential geometry as a line of internal surface geometry. But at the present ...