The forming method is a primary machining technique for the ring gear (large wheel) of hyperboloid gear pairs, prized for its advantages of fewer adjustment parameters and high processing efficiency. Consequently, research into the machining calculations and tooth profile correction for this method holds significant practical importance. Hyperboloid gears are critical components in the transmission systems of automobiles, ships, and various engineering and port machinery. The quality of the tooth flank directly impacts overall machine performance. While extensive research exists on the machining and correction of spiral bevel and hyperboloid gears, most studies focus on generating methods. There is a notable gap in research concerning profile correction techniques specific to the forming method. This article addresses this gap by proposing an equivalent correction strategy for modifying the tooth profile angle during the forming process of hyperboloid gears.
The forming process for hyperboloid gears typically involves parameters such as cutter radial location (S1), cutter rotational angle or swivel (q1), axial workpiece offset (XG2), and workpiece tilt or installation angle (γ1). The workpiece is tilted according to its root angle, the tooth profile angle is generally equal to the cutter blade angle, and the cutter axis is parallel to the horizontal plane (considering a horizontal machine model). The root apex of the gear coincides with the machine center, which is also the cutting crossing point. Inaccuracies stemming from machine tool errors, setup errors, and adjustment errors can cause the actual tooth profile angle to deviate from its theoretical specification, necessitating a correction procedure.
This paper introduces a method to correct the tooth profile angle by conceptually rotating the gear blank. The core principle involves an initial rotation of the blank around the tangential direction at the reference point on the tooth flank. This rotation changes the effective pressure angle. However, this rotated position is not physically realizable on a standard machine. Therefore, an equivalent transformation is performed: the entire setup (cutter and blank) is rotated back around the cutter axis until the workpiece axis is returned to a horizontal orientation. The changes in relative position caused by this sequence of rotations are compensated for by recalculating and adjusting the machine settings—namely, the radial and angular cutter positions, and the workpiece installation angle. This method effectively achieves the desired profile angle modification and is suitable for both fine-tuning and more substantial adjustments.
To establish a foundation for this correction, a mathematical model for the forming method machining of a hyperboloid gear must first be constructed. The coordinate systems are defined as follows: $S_m (O_m – X_m, Y_m, Z_m)$ is the machine coordinate system, with its origin $O_m$ at the machine center and the $X_mO_mY_m$ plane coinciding with the cutter tip plane. $S_c (O_c – X_c, Y_c, Z_c)$ is the cutter coordinate system, with its origin $O_c$ at the center of the cutter tip plane. $S_p (O_p – X_p, Y_p, Z_p)$ is the reference point coordinate system, with its origin $O_p$ at the cutting reference point $P$ on the tooth surface. The axis $X_p$ is tangent to the tooth lengthwise direction at $P$, $Y_p$ is perpendicular to $Y_c$, and the $X_pO_pY_p$ plane is parallel to $X_mO_mY_m$. The root apex of the gear, $O_r$, coincides with the machine center $O_m$. $O_2$ is the designed crossing point of the gear. Parameters $S_1$ and $q_1$ are the radial and angular cutter positions, $r_c$ is the cutter radius, $\beta_0$ is the spiral angle at the reference point, $\gamma_1$ is the workpiece installation angle, $X_{G2}$ is the axial offset from the crossing point (also the axial workpiece correction), $R_{02}$ is the generating gear cone distance, $L_1$ is the workpiece axis direction vector, and $h_r$ is the dedendum.
The cutter blade surface is represented in the cutter coordinate system as $\mathbf{r}_c(u_c, \theta_c)$, where $u_c$ and $\theta_c$ are surface parameters. The equation of the generating gear surface in the machine coordinate system is:
$$\mathbf{r}_m = \mathbf{M}_{mc}(S_1, q_1) \cdot \mathbf{r}_c(u_c, \theta_c)$$
where $\mathbf{M}_{mc}$ is the transformation matrix from $S_c$ to $S_m$. For the specific reference point $P$ with corresponding cutter coordinates $(u_p, \theta_p)$, the generating gear cone distance $R_{02}$ is the magnitude:
$$R_{02} = |\mathbf{r}_m(S_1, q_1, u_p, \theta_p)|$$
The vector from point $P$ to the root apex $O_r$ expressed in $S_p$ is:
$$\overrightarrow{PO_r} = [-R_{02}\cos\beta_0,\quad R_{02}\sin\beta_0,\quad h_r]^T$$
The direction vector of the workpiece axis $L_1$, pointing towards the mounting base, expressed in $S_p$ is:
$$\mathbf{L}_1 = [\cos\gamma_1 \cos\beta_0,\quad -\cos\gamma_1 \sin\beta_0,\quad \sin\gamma_1]^T$$
When a discrepancy $\Delta\alpha$ exists between the required gear tooth profile angle and the cutter blade angle (or when a deliberate correction $\Delta\alpha$ is needed), the equivalent correction method is applied. The gear blank is first conceptually rotated by $\Delta\alpha$ around the $X_p$ axis at point $P$. After this rotation, the new coordinate system at $P$ is denoted as $S_{p2}(O_{p2}-X_{p2}, Y_{p2}, Z_{p2})$. The vector from $P$ to the new (virtual) root apex $O_{r2}$ and the new workpiece axis direction $\mathbf{L}_{20}$ in $S_{p2}$ are:
$$\overrightarrow{PO_{r2}} = \mathbf{T}(\Delta\alpha) \cdot \overrightarrow{PO_r}$$
$$\mathbf{L}_{20} = \mathbf{T}(\Delta\alpha) \cdot \mathbf{L}_1$$
where $\mathbf{T}(\Delta\alpha)$ is the 3×3 rotation matrix about the $X_{p2}$ axis. The cutter axis direction in $S_{p2}$ is $\mathbf{L}_c = [0, 0, 1]^T$. The angle between the rotated workpiece axis $\mathbf{L}_{20}$ and the machine plane (now defined by the new orientation) becomes the new installation angle $\gamma_2$:
$$\gamma_2 = \frac{\pi}{2} – \arccos(\mathbf{L}_c \cdot \mathbf{L}_{20})$$
Let $\mathbf{L}_{Ym2}$ be the unit direction vector of the new machine $Y_m$ axis expressed in $S_{p2}$. It is perpendicular to $\mathbf{L}_{20}$, leading to the condition $\mathbf{L}_{Ym2} \cdot \mathbf{L}_{20} = 0$. This relationship determines the new effective spiral angle parameter $\beta_2$ at the reference point, which is a function of the original parameters and $\Delta\alpha$: $\beta_2 = f(\gamma_1, \beta_0, \Delta\alpha)$. Expressing $\mathbf{L}_{20}$ in the new machine coordinate system $S_{m2}$ yields $\mathbf{L}_2 = (X_{L2}, Y_{L2}, Z_{L2}) = \mathbf{L}(\beta_2) \cdot \mathbf{L}_{20}$. To make the workpiece axis horizontal again, the entire system must be rotated back around the cutter axis by an angle $\Delta\theta$:
$$\Delta\theta = \arctan\left(\frac{Y_{L2}}{X_{L2}}\right)$$
Consequently, the corrected angular cutter position $q_2$ is:
$$q_2 = q_1 + \Delta\theta$$
Next, the corrected radial cutter location $S_2$ and axial workpiece offset $X_{G2}^{(new)}$ are calculated. In $S_{p2}$, the vector from $P$ to the cutter center $O_c$ is $\overrightarrow{PO_c} = [0, r_c, h_r]^T$. The vector from the virtual root apex $O_{r2}$ to $O_c$ is:
$$\overrightarrow{O_{r2}O_c} = (X_{cr2}, Y_{cr2}, Z_{cr2}) = \overrightarrow{PO_c} – \overrightarrow{PO_{r2}}$$
The vector from $O_{r2}$ to the new machine center $O_{m2}$ in $S_{p2}$ is:
$$\overrightarrow{O_{r2}O_{m2}} = [0, Z_{cr2} \tan \gamma_2, Z_{cr2}]$$
Therefore, the magnitude of the vector from $O_c$ to $O_{m2}$, which is the corrected radial distance $S_2$, is:
$$S_2 = |\overrightarrow{O_cO_{m2}}| = |\overrightarrow{O_{r2}O_{m2}} – \overrightarrow{O_{r2}O_c}|$$
Finally, the corrected axial workpiece offset $X_{G2}^{(new)}$, representing the distance from the design crossing point $O_2$ to the new machine center $O_{m2}$, is:
$$X_{G2}^{(new)} = X_{G2}^{(0)} – \frac{Z_{cr2}}{\sin \gamma_2}$$
where $X_{G2}^{(0)}$ is the original axial offset value. The set of parameters $(\gamma_2, q_2, S_2, X_{G2}^{(new)})$ constitutes the complete equivalent correction for modifying the tooth profile angle by $\Delta\alpha$ in the forming process of a hyperboloid gear.
To validate the correctness of the derived equivalent correction theory, a numerical example of a hyperboloid gear pair is analyzed. The basic geometric parameters of the gear set are summarized in the following table:
| Parameter Name | Pinion | Gear (Hyperboloid Gear) |
|---|---|---|
| Number of Teeth | 6 | 37 |
| Transverse Module | 11.732 | |
| Shaft Angle | 90° | |
| Offset Distance | 35 mm | |
| Outer Diameter | 113.39 mm | 434.8 mm |
| Spiral Angle at Ref. Point | 45° (LH) | 34.4° (RH) |
| Face Width | 67.63 mm | 62 mm |
| Pressure Angle | Gear Convex / Pinion Concave: 22° Gear Concave / Pinion Convex: 23° |
|
| Outer Addendum | 14.89 mm | 1.84 mm |
| Outer Whole Depth | 19.14 mm | 18.97 mm |
For this analysis, a correction of $\Delta\alpha = +0.5°$ is applied to the gear’s convex side and $\Delta\alpha = -0.5°$ to the concave side, aiming for final pressure angles of 22° and 23° respectively, using a standard cutter with a blade angle of 22.5°. The machining parameters before and after the equivalent correction are calculated and compared below:
| Parameter Name | Before Correction | After Correction |
|---|---|---|
| Cutter Radius (rc) | 152.4 mm | 152.4 mm |
| Point Width | 5.2 mm | 5.2 mm |
| Cutter Blade Angle | 22.5° | 22.5° |
| Radial Setting (S) | 163.19 mm | 162.48 mm |
| Cutter Swivel (q) | 50.4° | 52.27° |
| Axial Workpiece Offset (XG2) | -0.78 mm | 0.18 mm |
| Workpiece Installation Angle (γ) | 74.93° | 74.62° |
Based on these parameters, tooth surfaces are computed for both the uncorrected and corrected cases. The deviations of these computed surfaces from the theoretical target surface (with the desired 22°/23° pressure angles) are analyzed. For the convex side before correction, deviations at the toe-top, heel-top, toe-root, and heel-root were +0.0725 mm, +0.0860 mm, -0.0640 mm, and -0.0873 mm respectively, indicating a pressure angle larger than the target (cumulative deviation of ~0.173 mm along the profile). For the concave side before correction, the deviations showed the opposite trend, indicating a pressure angle smaller than the target.
After applying the equivalent correction parameters, the calculated deviations reduced dramatically. For the corrected convex side, deviations were on the order of ±0.003 mm, with a cumulative profile deviation of only about 0.0037 mm at the toe and 0.0007 mm at the heel. Similarly, for the corrected concave side, deviations were within ±0.002 mm, with cumulative deviations of 0.0035 mm and 0.0006 mm. This confirms that the corrected machining parameters successfully generate a hyperboloid gear tooth surface with the intended profile angle, validating the mathematical model and correction algorithm.
To transition from theory to practical application, a cutting test was conducted. The corrected parameters were used to machine a hyperboloid gear ring on a domestic YK2260X gear milling machine. The machined gear was then measured on a Gleason P65 Coordinate Measuring Machine (CMM), using the theoretical corrected surface as the reference. The measured deviations closely matched the calculated predictions. For the convex flank, measured deviations at key points were within ±0.008 mm, with a profile form error of less than 0.002 mm. For the concave flank, deviations were within ±0.006 mm with a similarly small form error. The close agreement between the measured results and the theoretical calculations provides strong experimental validation for the feasibility and accuracy of the proposed equivalent tooth profile angle correction method for hyperboloid gears machined via the forming process.
In conclusion, this article has presented a novel and effective method for correcting the tooth profile angle during the forming process of hyperboloid gears. The method is based on an equivalent transformation principle involving a conceptual rotation of the gear blank followed by a compensatory rotation of the entire setup, ultimately realized by recalculating the standard machine settings. Detailed mathematical models were derived to calculate the corrected radial and angular cutter positions, axial offset, and workpiece installation angle. A numerical case study demonstrated the effectiveness of the correction, reducing tooth profile deviations to negligible levels. Finally, practical machining and measurement tests on a real hyperboloid gear confirmed the theory’s validity. This method provides a robust and straightforward approach to controlling first-order tooth profile geometry in formed hyperboloid gears, offering valuable theoretical support for improving manufacturing accuracy and consistency in industrial applications. Unlike complex conversions between different machine types, this method operates within the framework of a standard forming machine, requiring only parameter recalculation based on measurement feedback, thus offering a practical tool for quality control and refinement in the production of hyperboloid gears.