Geometric analysis on rotary rolling process of integral helical high finned tubes

Spatial analytic geometry is introduced to analyze the essentially geometric characters of integral helical finned tubes and the spatial relationship between rollers and rolled pieces in the rotary rolling process. A mathematical model is constructed for helical finned tubes based on the concept of helicoidal surfaces and cylindrical helixes. The binormal vector and tangential vector of any point on the helix, corresponding to the roller's axial vector on the point of the tube surface, are achieved by an expression with the main structural parameters of helical finned tubes. In order to research the spatial relationship between rollers and rolled pieces, two coordinate systems are established to set rollers and rolled pieces respectively, and a coordinate transformation expression between the two coordinate systems is deduced. Based on coordinate transformation, the process is introduced to deduce the value of roller radius and the coordinate values of contact points between rollers and rolled pieces.