Devising a technique for constructing approximate sweeps of helicoids based on the theory of surface bending
DOI:
https://doi.org/10.15587/1729-4061.2026.362946Keywords:
surface pitch, flat ring, flat workpiece, screw turn, straight helicoidAbstract
This study investigates the process of constructing approximate helicoid sweeps based on the classical theory of their continuous bending in the surface of revolution. Straight helicoids are non-sweep surfaces; therefore, a flat workpiece for fabricating them can only be an approximate sweep. Such an approximate sweep is a flat ring bounded by the inner and outer arcs of circles whose radii are tabular data. The dimensions of the ring must be such as to ensure a minimum of plastic deformations when forming them into a helicoid coil.
To find the dimensions of the ring, the classical theory of bending of non-sweep surfaces has been applied. According to the theorem of differential geometry, any helical surface can be bent into a surface of revolution. Such bending is carried out by reducing the pitch of the surface to zero: that makes it possible to visually observe the deformation of the surface. The resulting surface of revolution can be approximated by a truncated cone. The exact sweep of the truncated cone will be an approximate sweep of the helicoid turn. This approach is based not on experimental data but on theoretical approaches to the bending process. Depending on the type of helicoid, the surface of revolution can be a catenoid or a single-cavity hyperboloid of revolution. This makes it possible to choose such sections of the surface of revolution for approximation by a truncated cone where it most closely fits it. This will correspond to the minimum of plastic deformations when forming the sweep of the cone into a helicoid turn.
In this work, approximate sweeps have been constructed for straight closed and open helicoids with the same design data: surface pitch, H = 100 linear units; radii of the cylinders bounding the surface, r = 20 linear units; and R = 60 linear units. The results are attributed to a new approach to finding approximate sweeps using the theory of bending of non-sweeping surfaces
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Copyright (c) 2026 Andrii Nesvidomin, Serhii Pylypaka, Tetiana Volina, Tetiana Kresan, Oleksandr Savoiskyi, Oksana Yurchenko, Oleksandr Savchenko, Serhii Borodai, Olha Zalevska, Olena Nalobina
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