A note on phase characteristic among rollers before thread or spline rolling

A certain difference (i.e., phi) of phase among rollers (i.e., rolling dies) before rolling process of complex profile (such as thread, spline, screw) is demanded to make the threaded or splined profile forming by different rollers connect well. The mathematical models of phase requirement between rollers before thread or spline rolling were established by Zhang et al. (Int J Adv Manuf Technol 81:513-528, 2015). Then, considering the staggered phase by roller itself, the adjusting the difference phi for rollers before thread or spline rolling was investigated. However, the rotated direction of roller for adjusting the difference phi before rolling is closely involved in rotating direction of workpiece in the forming process, especially for difference phi of adjustment before rolling forming process with three or more rollers. But the influence of rotation of workpiece on the difference phi was not considered in pervious study. Thus, in this note, the rotation of workpiece was considered in modified models by theoretical analysis. Some analyses for thread or spline rolling with three rollers were different from pervious study, and numerical simulations of rolling forming processes have been used to verify the modified models. The phase characteristic among rollers is close to the remainder from dividing the sum of starts or teeth between workpiece and roller by number of rollers. Some analyses for thread or spline rolling with three or more rollers were corrected by the modified models. The modified models are suitable for all situations of thread or spline rolling forming process.