Research on the profile modification of power skiving tool for internal gears

Abstract

Power skiving provides an effective solution and considerable machining efficiency for the machining of internal gears. The tool profile design and the reusability after resharpening is crucial in gear manufacturing. In this paper, a novel method of tool profile correction based on the inverse error-complement of involute profile is proposed. Firstly, the mathematical model of involute cutter with rake angle and relief angle is established. The profile error relative to the target gear is calculated by the mathematical model. Then, the distribution of the gear profile errors is fitted by a fifth-order multinomial. The fitted multinomial function is attached to the cutter profile. The maximum theoretical error of the target gear profile is in 10e-7 mm order of magnitude through the calculation of fewer iterations. Finally, the distribution of the multinomial coefficients along the resharpening direction is obtained by linear programming. The result shows that the cutter designed by the proposed method possess almost negligible error to the gear profile and good repeatability of resharpening.

Appendices

Appendix 1. Nomenclature

m	normal module of gear
z c	teeth number of cutter
z 1	teeth number of internal gear
α	normal pressure angle of cutter
α t	Transverse pressure angle of cutter
β c	helical angle of cutter
β	helical angle of gear
α f	cone angle of cutter
x 0	Initial tooth profile shift coefficient of cutter
x	Tooth profile shift coefficient of gear
α k	Rake angle of cutter
S a	Cutting edge mark(+ 1 L, − 1 R)
S c	hand of spiral mark of cutter(+ 1 R, − 1 L)
r b	base radius of cutter
r 0	radius of reference circle of cutter
R m	radius of reference circle of gear
R max	radius of internal gear for dedendum
R min	radius of internal gear for addendum
d g	resharpening length
E t	error tolerance
γ	offset of cutter

Appendix 2

$${\boldsymbol M}_{21}=\begin{bmatrix}\cos\theta&S_a\sin\theta&0&0\\-S_a\sin\theta&\cos\theta&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}{\;\boldsymbol M}_{32}=\begin{bmatrix}\cos\theta_\mu&-S_c\sin\theta_\mu&0&0\\S_c\sin\theta_\mu&\cos\theta_\mu&0&0\\0&0&\mu&0\\0&0&0&1\end{bmatrix}$$

$${\boldsymbol M}_{43}=\begin{bmatrix}\cos\varphi_c&\sin\varphi_c&0&0\\-\sin\varphi_c&\cos\varphi_c&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}{\;\boldsymbol M}_{54}=\begin{bmatrix}1&0&0&0\\0&\cos\Sigma&-S_c\sin\Sigma&0\\0&S_c\sin\Sigma&\cos\Sigma&0\\0&0&0&1\end{bmatrix}$$

$${\boldsymbol M}_{65}=\begin{bmatrix}1&0&0&a\\0&1&0&-\gamma\\0&0&1&-d_s\\0&0&0&1\end{bmatrix}{\;\boldsymbol M}_{76}=\begin{bmatrix}\cos\varphi_1&-\sin\varphi_1&0&0\\\sin\varphi_1&\cos\varphi_1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}$$

$${\boldsymbol T}_{ba}=\begin{bmatrix}\cos\alpha_k&0&\sin\alpha_k\\0&1&0\\-\sin\alpha_k&0&\cos\alpha_k\end{bmatrix}{\;\boldsymbol T}_{3b}=\begin{bmatrix}1&0&0\\0&\cos\beta_c&S_c\sin\beta_c\\0&-S_c\sin\beta_c&\cos\beta_c\end{bmatrix}$$