Geometric Transformation For Double Helical Wire Rods

Abstract

The true shape of a double-helical geometry in a rope cross-section is needed for the design of ropes to predict the interstitial voids that need to be filled with water-blocking material. Commercially-available software such as AutoCAD, Inventor, Ideas, Mechanical Desktop, etc. are unable to develop this geometry. Thus, the objective of this study is to develop the parametric equations needed to produce scaled rope cross-section plots. Two fundamental mathematical tools are used to describe the cross-section of helical wire rods. The premier tool is the equation of the "Pencil of Spheres" where coordinates of the second helix centerline in a global coordinate system is found. To establish the centerline, coordinates of the helical path geometric transformations are developed. The equation of the "Pencil of Spheres" for the second helix is then intersected with a plane perpendicular to the rope axis. This results in the true shape of the helical round rods. The objective of this research is to derive a mathematical model that describes the exact geometry of double-helical round rods in a plane perpendicular to the rope axis. Using this model, a computer program is developed to plot the rope cross-section to a true scale.