In the modern machinery industry, the demand for high-speed, heavy-load, and low-noise transmission systems has led to the widespread application of helical gears and herringbone gears. As a result, the manufacturing precision requirements for these gears have become increasingly stringent. The Sachs helical slotting cutter, which is specifically designed for machining herringbone gears with small or no undercut grooves, has attracted significant attention due to its absence of theoretical manufacturing errors. It effectively improves the tooth profile accuracy of the processed gears while achieving favorable tool rake and clearance angles. However, due to the lack of research on the grinding method for herringbone gear slotting cutters (Sachs helical slotting cutters), the application of this tool has been limited in our country. Therefore, we conducted a comprehensive study on the grinding principle, grinding method, and grinding device for herringbone gear slotting cutters, thereby providing the necessary conditions for the popularization and application of this type of tool in our country.
1. The Grinding Principle of Herringbone Gear Slotting Cutters
The method for generating the rake angle of the herringbone gear slotting cutter involves grinding a chip-breaking groove along the involute cutting edge on the blunt side of the tooth, while grinding a facet along the involute cutting edge on the sharp side. In this study, we focus on the grinding principle of the chip-breaking groove on the blunt side, as the grinding of the facet on the sharp side is analogous. To simplify the grinding process and improve efficiency, we use a single base circle plate for grinding all slotting cutters with the same nominal pitch circle diameter. As illustrated in the figure, the principle of grinding is based on the generating motion between the grinding wheel and the cutter, where the contact line approximates the desired involute profile.
In the coordinate system Oxy, the center of the slotting cutter to be ground is O, the tip circle radius is \(r_a\), the base circle radius is \(r_b\), and the involute segment of the tooth is AB. In the coordinate system O1x1y1, the substitute base circle plate of the grinding machine has a radius \(r_{b0}\), and its center is O1. The involute generated by this plate is CO2D, starting from point C at the intersection of the y1 axis and the base circle rb0. In the coordinate system O2x2y2, the grinding wheel surface is conical, with a base circle diameter d and a chip-breaking groove depth t. The x2 axis is tangent to the substitute base circle plate. The center of the grinding wheel base circle is located at a distance (d/2 – t) from the O2x2y2 plane along the positive z2 axis. During the grinding of the chip-breaking groove, the intersection line between the conical surface of the grinding wheel and the end face of the slotting cutter is a quasi-involute GH. Since the curvature radius of the involute AB increases outward, the line GH has the same characteristic. By sliding the line GH along AB, we can always find a position where GH is closest to AB. In this optimal position, the coordinate system O1x1y1 is shifted and rotated relative to Oxy by a horizontal shift a, a vertical shift b, and a rotation angle β.
To establish the optimization model, we digitize the involute AB into an array of coordinates (xi, yi) in the Oxy system:
\[
\begin{aligned}
x_i &= r_b [\sin(\phi + \omega_0) – \phi \cos(\phi + \omega_0)] \\
y_i &= r_b [\cos(\phi + \omega_0) + \phi \sin(\phi + \omega_0)] \quad (i=1,2,\ldots,k)
\end{aligned}
\]
Here, \(\phi\) is the involute roll angle, \(\omega_0\) is the half-angle of the tooth space at the base circle of the slotting cutter, and k is the number of points. Similarly, the intersection line (x2 < 0) between the grinding wheel and the end face is digitized into coordinates (x2i, y2i) in the O2x2y2 system:
\[
y_{2i}^2 + \left(\frac{d}{2} – t\right)^2 – \left(\frac{d}{2} + x_{2i} \tan \gamma \right)^2 = 0 \quad (i=1,2,\ldots,m)
\]
where γ is the half-cone angle of the grinding wheel, and m is the number of points. The coordinate transformation matrix from O1x1y1 to Oxy is:
\[
M_{01} = \begin{pmatrix}
\cos\beta & -\sin\beta & a \\
\sin\beta & \cos\beta & b \\
0 & 0 & 1
\end{pmatrix}
\]
The transformation matrix from O2x2y2 to O1x1y1 is:
\[
M_{12} = \begin{pmatrix}
\cos\phi & \sin\phi & r_{b0}(\sin\phi – \phi \cos\phi) \\
-\sin\phi & \cos\phi & r_{b0}(\cos\phi + \phi \sin\phi) \\
0 & 0 & 1
\end{pmatrix}
\]
Thus, the combined transformation from O2x2y2 to Oxy is:
\[
M_{02} = M_{01} M_{12} = \begin{pmatrix}
\cos(\phi-\beta) & \sin(\phi-\beta) & r_{b0}[\sin(\phi-\beta) – \phi \cos(\phi-\beta)] + a \\
-\sin(\phi-\beta) & \cos(\phi-\beta) & r_{b0}[\cos(\phi-\beta) + \phi \sin(\phi-\beta)] + b \\
0 & 0 & 1
\end{pmatrix}
\]
The coordinates (x2i, y2i) in the Oxy system are then:
\[
\begin{aligned}
x_i’ &= (x_{2i} – r_{b0}\phi) \cos(\phi – \beta) + (y_{2i} + r_{b0}) \sin(\phi – \beta) + a \\
y_i’ &= -(x_{2i} – r_{b0}\phi) \sin(\phi – \beta) + (y_{2i} + r_{b0}) \cos(\phi – \beta) + b \quad (i=1,2,\ldots,m)
\end{aligned}
\]
Similarly, the coordinates (xi, yi) in the O2x2y2 system are:
\[
\begin{aligned}
x_{2i}’ &= (x_i – a) \cos(\beta – \phi) + (y_i – b) \sin(\beta – \phi) + r_{b0}\phi \\
y_{2i}’ &= -(x_i – a) \sin(\beta – \phi) + (y_i – b) \cos(\beta – \phi) – r_{b0} \quad (i=1,2,\ldots,k)
\end{aligned}
\]
For a given set of a, b, β, and a given φ value, we obtain m coordinate pairs (xi’, yi’). When n values of φ are considered, we have m×n pairs. The distance between each point (xi, yi) and all points (xi’, yi’) is computed as:
\[
\begin{pmatrix}
\varepsilon_{i1} \\ \varepsilon_{i2} \\ \vdots \\ \varepsilon_{i,m\times n}
\end{pmatrix}
=
\begin{pmatrix}
\sqrt{(x_i – x_1′)^2 + (y_i – y_1′)^2} \\
\sqrt{(x_i – x_2′)^2 + (y_i – y_2′)^2} \\
\vdots \\
\sqrt{(x_i – x_{m\times n}’)^2 + (y_i – y_{m\times n}’)^2}
\end{pmatrix}
\]
Let εimin be the minimum among these distances. Applying this to all (xi, yi) points yields k minimum distances: ε1min, ε2min, …, εkmin. The optimization model is then:
\[
\begin{aligned}
&\varepsilon = \sum_{i=1}^k \varepsilon_{i\min} \to \min \\
&\text{subject to:} \quad (y_{2i}’)^2 – (y_{2i})^2 \geq 0 \\
&\qquad\qquad\,\, x_{2i}’ – x_{2i} < 0 \\
&\qquad\qquad\,\, \varepsilon_{i\min} < 0.025 \\
&\qquad\qquad\,\, \varepsilon_{i\min} > 0
\end{aligned}
\]
When the values of a, b, and β are optimized, the total error ε is minimized. At this point, the envelope GH of the intersection line between the grinding wheel and the end face of the slotting cutter is closest to the desired involute AB.
2. Parameter Selection and Calculation
2.1 Grinding Wheel Diameter and Rake Angle of the Slotting Cutter
During the grinding of the chip-breaking groove, it is essential to ensure that the envelope GH does not interfere with the involute cutting edge AB of the slotting cutter. In the vicinity of y2 = 0, the curvature radius ρint of the intersection line between the grinding wheel and the end face must be smaller than the curvature radius ρstart of the involute AB at the starting point S of the effective working section. The condition is:
\[
\rho_{\text{int}} < \rho_{\text{start}}
\]
where:
\[
\rho_{\text{int}} = \frac{\left[ y_2^2 + \left(\frac{d}{2} + x_2 \tan \gamma \right)^2 \tan^2 \gamma \right]^{3/2}}{y_2^2 \tan^2 \gamma – \left(\frac{d}{2} + x_2 \tan \gamma\right)^2 \tan^2 \gamma}
\]
\[
\rho_{\text{start}} = a \sin \alpha_{\text{mesh}} – \sqrt{r_{a\_gear}^2 – r_{b\_gear}^2}
\]
Here, a is the center distance between the slotting cutter and the gear being cut, αmesh is the mesh angle, ra\_gear is the tip circle radius of the gear, and rb\_gear is the base circle radius of the gear. From the expression for ρint, it is evident that a smaller grinding wheel base diameter d and a smaller half-cone angle γ lead to a smaller ρint and thus a smaller error ε. However, if the grinding wheel peripheral speed is too low, the surface roughness of the rake face increases, affecting the surface quality of the processed gear and increasing cutting forces and heat. Therefore, we recommend d = 60–70 mm.
2.2 Rake Angle of the Slotting Cutter and Half-Cone Angle of the Grinding Wheel
The relationship between the grinding wheel half-cone angle and the slotting cutter rake angle is given by:
\[
\gamma’ = \beta_{b0} – \gamma_{c1}
\]
\[
\gamma” = \beta_{b0} + \gamma_{c2}
\]
where γ’ is the grinding wheel half-cone angle for grinding the facet on the sharp side, γ” is the grinding wheel half-cone angle for grinding the chip-breaking groove on the blunt side, βb0 is the nominal base circle helix angle of the slotting cutter, γc1 is the side rake angle on the sharp side, and γc2 is the side rake angle on the blunt side. For gears made from structural steel that has been normalized or quenched and tempered, we recommend γc1 = γc2 = 5°–7°.
2.3 Facet on the Sharp Side
To ensure that the herringbone gear slotting cutter can correctly cut the ridge of the herringbone gear, the cutting edge on the blunt side should be higher than that on the sharp side by a value Δk:
\[
\Delta k = B_1 \cdot \sin(\beta_{b0} – \gamma’_{c1})
\]
where B1 is the facet width, which should be selected as:
\[
B_1 = (0.25 \sim 0.4) S_{aot}
\]
and Saot is the tooth tip width of the slotting cutter in the end face. The parameter Δl in the adjustment calculation is given by Δl = rb0·β.
2.4 Adjustment Parameters and Calculated Errors
Table 1 shows the parameter selection and adjustment calculations for grinding a series of herringbone gear slotting cutters. The data include the module, tooth count, helix angle, substitute base circle radius, grinding wheel diameter, rake angle, horizontal shift a, vertical shift b, extension Δl, and the calculated error ε.
| Module m (mm) | Number of Teeth z | Helix Angle β (° ‘ “) | Substitute Base Circle Radius rb0 (mm) | Grinding Wheel Diameter d (mm) | Rake Angle γ’ = γ” (°) | Horizontal Shift a (mm) | Vertical Shift b (mm) | Extension Δl (mm) | Calculated Error ε (mm) |
|---|---|---|---|---|---|---|---|---|---|
| 4 | 31 | 23°23’05” | 63 | 65 | 6 | 5.113 | 3.281 | 1.275 | 0.121 |
| 4.5 | 28 | 23°23’05” | 63 | 65 | 6 | 7.041 | 4.216 | 2.147 | 0.413 |
| 5 | 25 | 23°23’05” | 63 | 65 | 6 | 6.910 | 3.604 | 1.156 | 0.152 |
| 5.5 | 23 | 23°23’05” | 63 | 65 | 6 | 8.114 | 4.336 | 1.401 | 0.138 |
| 6 | 21 | 23°23’05” | 63 | 65 | 6 | 9.474 | 4.051 | 1.843 | 0.157 |
3. Grinding Machine Structure and Experimental Verification
The grinding machine we designed is based on the grinding principle described above. The main structural components include an L-shaped column with a lifting table, a transverse sliding plate, a substitute base circle plate supported by annular rolling guides, two layers of small sliding plates (one longitudinal, one transverse) above the base circle plate, a dividing head for clamping the slotting cutter, and a grinding wheel driven by a motor via a belt for high-speed rotation. Steel bands are fixed to the substitute base circle plate and stretched horizontally; the axis of the grinding wheel lies in the same vertical plane as the centerline of the straight section of the steel band. When the handle is turned to move the transverse sliding plate left or right, the relative motion between the grinding wheel axis and the substitute base circle plate generates the required involute motion (i.e., the grinding wheel axis acts as the generating line of the involute). The adjustments of a, b, and β are realized by rotating specific handles, and the depth of the chip-breaking groove is controlled by another handle.
Experiments on the cutting performance and the resulting gear accuracy of the herringbone gear slotting cutter ground using the above principle were conducted on an imported Sachs gear shaping machine. Table 2 shows the experimental data for cutting performance and machining accuracy.
| Cutting Speed V (m/min) | Circular Feed s (mm) | Workpiece Material | Heat Treatment State | Tool Material |
|---|---|---|---|---|
| 39 | 0.2 | 45 Steel | Quenched and Tempered | W18Cr4V |
| Workpiece Teeth Number z | Workpiece Module m (mm) | Wear Criterion VB (mm) | Tool Life T (min) | Tooth Profile Error Δff (μm) |
|---|---|---|---|---|
| 50 | 6 | 0.3 | 492 | 9 |
The experimental results demonstrate that the ground herringbone gear slotting cutters exhibit excellent cutting performance and produce gears with high tooth profile accuracy. The tool life reached 492 minutes before reaching the wear criterion of VB = 0.3 mm, and the tooth profile error was only 9 μm, which satisfies the requirements for high-precision herringbone gear manufacturing.
4. Conclusion
Based on our research and experimental verification, the following conclusions can be drawn:
- The grinding method for herringbone gear slotting cutters has been validated through experiments, confirming the correctness of the grinding principle, calculation formulas, and computer programs.
- After proper grinding using the proposed method and device, herringbone gear slotting cutters exhibit significantly improved tool life and produce gears with higher tooth profile accuracy. The tool life reached 492 minutes, and the profile error was reduced to 9 μm, which is superior to conventional grinding methods.
- The use of a single substitute base circle plate for different slotting cutters with the same nominal pitch circle diameter greatly simplifies the grinding process and improves efficiency. The optimization model effectively determines the optimal parameters (a, b, β) to minimize the approximation error between the ground surface and the theoretical involute.
- The grinding device we developed is practical and reliable, enabling the precise grinding of chip-breaking grooves and facets for herringbone gear slotting cutters. This device plays a key role in promoting the application of herringbone gear slotting cutters in domestic manufacturing.
In summary, our study provides a comprehensive solution for the grinding of herringbone gear slotting cutters. The method and device have been successfully applied in practice, contributing to the advancement of herringbone gear manufacturing technology. Future work will focus on further optimizing the grinding parameters and exploring the grinding of other types of advanced gear cutters for herringbone gears.