In modern mechanical transmission systems, helical gears play a critical role due to their high load capacity and smooth operation. However, achieving high precision in tooth profile modification, such as lead crowning, remains challenging due to inherent machining errors in CNC forming gear grinding processes. This research focuses on analyzing and compensating for tooth surface errors in internal helical gears during lead modification. I developed a comprehensive error model by establishing the spatial meshing relationship between the grinding wheel and the gear, and proposed a compensation method that balances errors on both left and right tooth surfaces. Experimental validation confirmed the effectiveness of this approach in improving gear accuracy.
The foundation of this work lies in the kinematics of CNC forming gear grinding machines. These machines typically feature six axes: three linear (x, y, z) and three rotational (A, B, C). The grinding wheel is mounted on a swivel arm that rotates about the A-axis, while the workpiece rotates about the B-axis. The coordinated motion of the B-axis and z-axis generates the helical path along the tooth flank. To model the process, I defined coordinate systems attached to the gear (s1) and the grinding wheel (s_t). The transformation between these systems accounts for the gear rotation angle φ1, wheel installation angle γm, center distance E_tp(0), and axial displacement L_t. The tooth surface in the gear coordinate system is represented as a parametric equation:
$$ r_1(u, \theta) = [x_1, y_1, z_1, 1]^T $$
where u and θ are surface parameters. Using homogeneous transformation matrices, the surface equation in the wheel coordinate system becomes:
$$ r_t(u, \theta, \phi_1) = M_{tc} M_{ca} M_{a1} r_1(u, \theta) $$
Here, M_a1 represents the rotation of the gear, M_ca the translation between centers, and M_tc the wheel installation. The meshing condition requires that the wheel surface and tooth surface maintain continuous contact, described by the equation:
$$ f_t = n_t(u_1, \theta_1, \phi_1) \cdot [y_t \times r_t(u_1, \theta_1, \phi_1)] = 0 $$
where n_t is the unit normal vector of the tooth surface. The contact line projected onto the wheel’s axial plane yields the wheel profile, calculated as:
$$ x_w(u_1, \theta_1, \phi_1) = \sqrt{x_t^2 + z_t^2}, \quad y_w(u_1, \theta_1, \phi_1) = y_t $$
For lead modification, I introduced additional radial motion (x-axis) and rotational motion (B-axis) to the grinding path. The modification profile follows a parabolic curve, with the radial offset ΔE given by:
$$ \Delta E = \begin{cases}
\frac{a_{ml}}{2} (\theta – \theta_b)^2 & \theta_a \leq \theta \leq \theta_b \\
0 & \theta_b \leq \theta \leq \theta_c \\
\frac{a_{ml}}{2} (\theta – \theta_c)^2 & \theta_c \leq \theta \leq \theta_d
\end{cases} $$
where a_ml is the lead modification coefficient, and θ_a to θ_d define the modification zones along the gear width. This approach enables precise control over the tooth profile, but it introduces errors due to multi-axis additional motions.
The impact of multi-axis additional motions on tooth surface errors is significant. The x-axis additional motion causes a radial displacement Δx, which varies along the tooth profile due to changing pressure angles from root to tip. This results in profile slope deviation f_H. For a point on the tooth surface, the displacement at the tip Δx_a and root Δx_f are:
$$ \Delta x_a = \Delta x \sin \lambda_a, \quad \Delta x_f = \Delta x \sin \lambda_f, \quad f_H = \Delta x_f – \Delta x_a $$
where λ_a and λ_f are the pressure angles at tip and root, respectively. The B-axis additional rotation Δb causes a tangential displacement, leading to profile slope deviation calculated as:
$$ \Delta b_a = r_a \Delta b, \quad \Delta b_f = r_f \Delta b, \quad f_H = \Delta b_f – \Delta b_a $$
Here, r_a and r_f are the tip and root radii. The helical motion is generated by interpolating the z-axis movement with the workpiece rotation, related by:
$$ C_B = Z_B \times \frac{\tan \beta}{r} $$
where Z_B is the axial displacement, β is the helix angle, and r is the pitch radius. These relationships highlight the coupling between machine motions and tooth errors.
Furthermore, the grinding wheel installation angle γm influences the contact line morphology. Deviating from the standard setting Σ = β can optimize the contact line shape. For right-hand helical gears, reducing γm shortens the contact line and increases its curvature, while increasing γm lengthens it. This adjustment, combined with workpiece parameters, helps mitigate errors. The following table summarizes key parameters for a typical internal helical gear used in this study:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Normal module m_n (mm) | 2 | Face width d (mm) | 65 |
| Number of teeth z | 79 | Normal shift coefficient x_n | 0.4987 |
| Pressure angle α_n (°) | 20 | Lead modification (μm) | 5 ± 4 |
| Helix angle β (°) | 15 | Evaluation start (mm) | 4.5 |
| Hand of spiral | Right | Evaluation end (mm) | 40.5 |
Based on the kinematics and error analysis, I developed a tooth surface error model. The theoretical tooth surface is represented as a point cloud, and the grinding contact lines are simulated. The error is defined as the deviation between the actual ground surface and the theoretical model. For lead modification, the ideal surface is symmetric, but multi-axis errors cause twisting—over-correction on one flank and under-correction on the other. The error model quantifies these deviations, enabling compensation.
To compensate for errors, I proposed a method that balances the left and right tooth surfaces. Instead of correcting one side aggressively, this approach slightly adjusts both sides to achieve an overall improvement. The compensation parameters include the spiral angle adjustment β_g and the swing angle adjustment C_B_g, calculated as:
$$ \beta_g = \arctan\left( \frac{(f_{H\beta L} – f_{H\beta R}) / 2}{D_1} \right), \quad C_{B_g} = \frac{D_1 \tan \beta_g}{r} $$
where f_HβL and f_HβR are the lead slope deviations for left and right flanks, and D_1 is the evaluation length. I implemented this in a MATLAB-based software tool that computes compensation values based on measurement data. The interface inputs include measurement positions and slope deviations, and outputs the required machine adjustments.
Experimental validation was conducted on a YK7350 CNC grinding machine with a Siemens control system. The gear parameters matched those in the table above. After initial grinding, teeth were measured on a Gleason 650GMS analyzer. The lead deviations for left and right flanks were recorded. Before compensation, the left flank had an average total lead deviation F_β of 8.6 μm and slope deviation f_Hβ of -2.3 μm, corresponding to grade 7 accuracy. The right flank had F_β of 5.4 μm and f_Hβ of -4.7 μm, grade 6. After applying compensation, the left flank improved to F_β of 4.3 μm and f_Hβ of -2.6 μm, while the right flank had F_β of 10.0 μm and f_Hβ of -6.0 μm, both achieving grade 6. The results demonstrate the compensation’s effectiveness in enhancing accuracy for helical gears.
| Parameter | Flank | Tooth 1 | Tooth 20 | Tooth 40 | Tooth 60 |
|---|---|---|---|---|---|
| F_β (μm) Before | Left | 6.8 | 10.0 | 6.2 | 11.5 |
| Right | 6.6 | 4.0 | 8.0 | 2.9 | |
| f_Hβ (μm) Before | Left | 6.9 | -10.2 | 5.4 | -11.4 |
| Right | -6.2 | -3.7 | -8.3 | -0.4 | |
| F_β (μm) After | Left | 5.1 | 6.2 | 2.9 | 3.1 |
| Right | 5.4 | 9.0 | 14.0 | 11.5 | |
| f_Hβ (μm) After | Left | -4.6 | -5.2 | 0.5 | -1.0 |
| Right | -2.4 | -4.4 | -9.6 | -7.4 |
The mathematical modeling of helical gears involves complex geometries. The tooth surface of an involute helical gear can be derived from the base helix parameters. The transverse pressure angle α_t relates to the normal pressure angle α_n by:
$$ \tan \alpha_t = \frac{\tan \alpha_n}{\cos \beta} $$
The base helix angle β_b is given by:
$$ \sin \beta_b = \sin \beta \cos \alpha_n $$
These parameters are essential for defining the tooth flank equations. In grinding, the wheel profile must match the gear geometry to avoid interference. The condition for non-interference is ensured by the continuous tangency between the wheel and tooth surfaces, expressed through the equation of meshing.
In summary, this research provides a systematic approach to compensating tooth profile errors in helical gears during CNC forming grinding. By analyzing multi-axis motions and their impact on lead modification, I developed an error model and a balanced compensation method. The experimental results confirm that the proposed technique improves gear accuracy, making it suitable for high-precision applications involving helical gears. Future work could explore real-time compensation algorithms and extend the method to other gear types.