Phase characteristic between dies before rolling for thread and spline synchronous rolling process

The thread and spline synchronous rolling process can form the external threaded and external splined tooth profiles on different parts of component by only one rolling process. However, it is a coupled rolling process where the thread rolling coupling with the spline rolling, and both phase adjusting requirements of dies before rolling for thread rolling and spline rolling should be satisfied in the synchronous rolling process. Thus, the phase adjustment only by rotating die may be difficult to ensure the thread and spline rolling by different dies well connects respectively. In this paper, requirements of phase difference adjustment for dies before thread rolling and before spline rolling have been studied systematically, and the mathematical expressions of phase difference adjustment have been established, and then the requirements of phase difference under different starts of threaded workpiece and die (or teeth of splined workpiece and die) for two and three rolling dies before thread rolling (or spline rolling) have been investigated. Based on these, the ratio S of phase difference between corresponding threaded sections of rolling dies to that between corresponding splined sections of rolling dies was introduced, and the phase characteristic between dies and the method of phase difference adjustment under different parameters in thread and spline synchronous rolling process were systematically researched. The results indicated that: (1) for thread (or spline) rolling process, if the remainder from dividing the starts nw of thread (or teeth Zw of spline) of workpiece by number N of rolling dies is equal to the remainder from dividing the starts nd of thread (or teeth Zd of spline) of rolling die by N, i.e., nw≡nd(mod N) (or Zw≡Zd(mod N)), then the phases of dies before rolling are the same, else there is a phase difference between dies; (2) for thread and spline synchronous rolling process, if the phase difference of splined section between rolling die j (j = 2,…, N) and rolling die 1 is φsj1=hNθsh=1,…,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varphi}_{{\mathrm{s}}_{j1}}=\frac{h}{N}{\theta}_s\left(h=1,\dots, N\right) $$\end{document} and ratio of phase difference is Sj1=Nhk+1k=0,1,2,3,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {S}_{j1}=\frac{N}{h}k+1\left(k=0,1,2,3,\dots \right) $$\end{document}, then the die structures of two dies are the same, i.e., the relative phase between threaded and splined sections for two dies are the same, else the die structures of rolling dies j and 1 are different and the phase difference for threaded sections of rolling dies and the phase difference for splined sections of rolling dies are assured by die structure itself.