BENDING OF A SURFACE OF REVOLUTION INTO A HELICAL CONOID

Abstract

The article explores the process of bending a catenoid into a helical conoid, which is significant for manufacturing of screws. A helical conoid (a straight closed helicoid) is formed by the helical motion of a line segment around an axis, with the segment intersecting the axis at a right-angle during motion. Such a surface cannot be bent into a plane; however, by gradually reducing the pitch, it can be transformed into a surface of revolution – a catenoid. During this deformation, the lengths of the lines and the area of the turn as a whole remain unchanged, meaning the deformation occurs similarly to developable surfaces. This deformation is based on the theory of surface bending. Any helical surface can be bent into a surface of revolution and vice versa.

The bending of the non-developable surface of a helical conoid into a catenoid is a classic example of differential geometry. Any surface of revolution can be bent into a helical surface. A surface relates to two families of coordinate lines, and a point on the surface is defined by the values of two curvilinear coordinates. If the meridian of the surface of revolution is given by an explicit equation, it can be described by parametric equations. This approach allows finding an approximate flat blank for manufacturing a screw turn.

It is proposed to approximate the obtained catenoid with a truncated cone. The development of the truncated cone will be the approximate development of the screw turn. These are the features of finding the approximate development, which in engineering practice is calculated using different formulas. The article presents parametric equations describing a one-parameter set of intermediate surfaces during the bending of a helical conoid by gradually reducing the pitch of the surface to zero.

Keywords: surface pitch, catenoid, truncated cone, approximation, approximate development.