Gear shaving remains a critical finishing process in gear manufacturing due to its high efficiency and cost-effectiveness. However, it frequently induces tooth profile concave errors near the pitch circle, significantly impacting gear meshing performance. Traditional research focuses on design modifications of shaving cutters or overlap ratios, often overlooking fundamental gear shaving mechanics. We address this gap by establishing a cutting force model that quantitatively links process parameters to cutting forces and subsequent profile errors.
The cutting force model integrates metal-cutting theory with elastic contact mechanics. Starting from the fundamental metal-cutting equation:
$$F = \tau_s a_{p} f (1.4\xi + C)$$
where \( \tau_s \) is the material shear yield point, \( \xi \) is the deformation coefficient, \( C \) depends on the tool rake angle, \( a_p \) is the depth of cut, and \( f \) is the axial feed. In gear shaving, \( a_p \) relates to radial feed \( f_r \) through elastic deformation \( \delta_c \) and material removal:
$$a_p = \Delta_{f_r} + \delta_c \quad \text{where} \quad \delta_c = e \left( \frac{3\pi \lambda F_{nc}}{2 \sum_{i=1}^{m} \frac{k_i}{C_i^{3/2}} \right)^{2/3}$$
Here, \( \Delta_{f_r} \) is material removed per radial feed increment, \( \lambda \) combines Poisson’s ratios and elastic moduli of gear and cutter, and \( k_i \), \( C_i \) are curvature terms. Substituting radial feed \( f_r = \Delta / (2 \sin \alpha) \) and experimental radial force \( F_r = P e^{-Q/S} \) yields the final gear shaving cutting force \( F_c \):
$$F_c = \tau_s f (1.4\xi + C) \left[ \frac{2i \sin \alpha f_r}{n} + e \left( \frac{3\pi \lambda F_{nc}}{2 \sum_{i=1}^{m} \frac{k_i}{C_i^{3/2}} \right)^{2/3} \right]$$
where \( n \) is spindle speed and \( i \) is the transmission ratio.
Spindle Speed Influence
Higher spindle speeds reduce cutting forces but exhibit diminishing returns. At \( f = 60 \text{mm/min} \) and \( f_r = 0.045 \text{mm} \):
$$\begin{array}{c|c}
n \, (\text{r/min}) & F_c \, \text{at Pitch Circle (N)} \\
\hline
140 & 34.76 \\
170 & 30.32 \\
200 & 27.63 \\
230 & 26.46 \\
\end{array}$$
The reduction occurs due to increased shear angles and thermal softening at elevated speeds. Force concentrations consistently peak near the pitch circle across all speeds, correlating with the onset of gear shaving concave errors.
Axial Feed Impact
Increasing axial feed linearly amplifies cutting forces. At \( n = 170 \text{r/min} \) and \( f_r = 0.045 \text{mm} \):
$$\begin{array}{c|c}
f \, (\text{mm/min}) & F_c \, \text{at Pitch Circle (N)} \\
\hline
30 & 14.52 \\
36 & 18.64 \\
51 & 24.87 \\
60 & 30.32 \\
\end{array}$$
Larger feeds increase material removal rates, intensifying plastic deformation. Force maxima consistently localize near the pitch circle during gear shaving, explaining error accumulation in this region.
Radial Feed Dominance
Radial feed exerts the strongest influence on cutting forces. At \( n = 170 \text{r/min} \) and \( f = 60 \text{mm/min} \):
$$\begin{array}{c|c}
f_r \, (\text{mm}) & F_c \, \text{at Pitch Circle (N)} \\
\hline
0.033 & 18.57 \\
0.039 & 25.82 \\
0.045 & 30.32 \\
0.058 & 36.61 \\
\end{array}$$
Partial derivatives quantify parametric sensitivity:
$$\left| \frac{\partial F_c}{\partial f_r} \right| \approx 450.9 \quad \left| \frac{\partial F_c}{\partial f} \right| \approx 31.6 \quad \left| \frac{\partial F_c}{\partial n} \right| \approx 1.85$$
Radial feed’s order-of-magnitude higher sensitivity directly links aggressive radial increments to exacerbated concave errors during gear shaving.
Finite Element Validation
A five-tooth FEM model simulated cutting forces under varied parameters. Material behavior followed Johnson-Cook criteria with explicit dynamic analysis. Key results:
$$\begin{array}{c|c|c|c|c}
\text{Condition} & n \, (\text{r/min}) & f \, (\text{mm/min}) & f_r \, (\text{mm}) & \text{Simulated } F_c \, (\text{N}) \\
\hline
1 & 170 & 60 & 0.045 & 29.73 \\
2 & 200 & 60 & 0.045 & 27.63 \\
3 & 170 & 36 & 0.045 & 18.64 \\
4 & 170 & 60 & 0.033 & 25.82 \\
\end{array}$$
Force distribution patterns matched theoretical predictions, with pitch-circle force concentrations confirming concave-error formation mechanisms in gear shaving.
Experimental Verification
Tests on a YX4230CNC5 shaving machine used 45 steel gears and measured profile deviations:
$$\begin{array}{c|c|c|c|c}
\text{Run} & n \, (\text{r/min}) & f \, (\text{mm/min}) & f_r \, (\text{mm}) & \text{Profile Error (μm)} \\
\hline
1 & 100 & 120 & 0.055 & 21.8 \\
2 & 160 & 120 & 0.055 & 24.6 \\
3 & 200 & 120 & 0.055 & 26.0 \\
4 & 160 & 90 & 0.055 & 23.8 \\
5 & 160 & 120 & 0.050 & 18.6 \\
6 & 160 & 120 & 0.045 & 17.4 \\
\end{array}$$
Radial feed reduction decreased errors by 29.3% (Run 2 vs. Run 6), validating its dominant role in gear shaving accuracy.
Conclusions
1. Cutting forces during gear shaving correlate positively with radial/axial feeds and negatively with spindle speed. Radial feed exhibits 14× greater influence than axial feed and 244× greater than spindle speed.
2. Force concentrations at the pitch circle directly cause localized over-cutting, initiating tooth profile concave errors.
3. Optimizing radial feed minimizes cutting forces and subsequent errors without compromising efficiency. This model enables parameter selection for precision gear shaving.