Mathematical model and closure criteria for wire-wound toroidal helical structures

This study establishes a mathematical model for steel wire-wound toroidal helical structures and defines a precise closure criterion. On the basis of the Frenet–Serret frame, mathematical models for single- to triple-order toroidal helices are derived. An optimized method to calculate the Frenet–Serret normal vector is proposed to substantially reduce derivation intricacy. This study proposes for the first time the mathematical modeling of single-thread winding patterns and rigorously investigates their closure criterion. Results demonstrate that the winding mechanism shares an identical criterion with multithread winding patterns. 3D software modeling by using the proposed methodology reveals enhanced accuracy in helical structure analysis. The findings provide valuable insights into the modeling of complex helical geometries.