In modern mechanical transmission systems, helical gears play a critical role due to their ability to provide smooth and efficient power transfer. However, the increasing demands for high-performance gear systems in advanced equipment necessitate superior accuracy in gear manufacturing. One common technique to enhance gear performance is tooth profile modification, which aims to optimize load distribution and reduce stress concentrations. Specifically, for internal helical gears, achieving precise tooth profile modification through CNC form grinding presents unique challenges, including inherent machining errors caused by multi-axis additional motions. These errors can lead to tooth surface distortions, adversely affecting the overall gear quality. In this study, I focus on analyzing the mechanisms behind these errors and developing effective compensation strategies to improve the grinding accuracy of internal helical gears. The research involves establishing a mathematical model for the grinding process, investigating the impact of machine tool motions, and proposing a compensation method that balances errors on both tooth flanks. Experimental validation is conducted to verify the effectiveness of the proposed approach, demonstrating significant improvements in tooth direction accuracy.
The foundation of this research lies in the CNC form grinding process for internal helical gears, which involves complex interactions between the grinding wheel and the gear tooth surface. To model this process, I first establish a spatial meshing coordinate system that defines the relative positions and motions of the grinding wheel and the internal helical gear. This system accounts for multiple axes of movement, including linear motions along the x, y, and z directions and rotational motions around the A, B, and C axes. The coordinate transformation between the gear and grinding wheel is essential for deriving the tooth surface equations and the contact conditions. Let me denote the coordinate systems as follows: the gear-fixed coordinate system \( s_1 \) with axes \( x_1, y_1, z_1 \), and the grinding wheel-fixed coordinate system \( s_t \) with axes \( x_t, y_t, z_t \). The transformation from \( s_1 \) to \( s_t \) can be expressed using homogeneous transformation matrices. For instance, the rotation of the gear by an angle \( \phi_1 \) around the \( z_1 \)-axis is represented by the matrix \( M_{a1} \):
$$ M_{a1} = \begin{bmatrix} \cos\phi_1 & \sin\phi_1 & 0 & 0 \\ -\sin\phi_1 & \cos\phi_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Similarly, the translation between the gear and grinding wheel, accounting for the center distance \( E_{tp}^{(0)} \) and axial movement \( L_t \), is given by \( M_{ca} \):
$$ M_{ca} = \begin{bmatrix} 1 & 0 & 0 & E_{tp}^{(0)} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & L_t \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
The installation angle of the grinding wheel, denoted as \( \gamma_m \), is incorporated through the matrix \( M_{tc} \):
$$ M_{tc} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\gamma_m & -\sin\gamma_m & 0 \\ 0 & \sin\gamma_m & \cos\gamma_m & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Using these transformations, the tooth surface equation in the grinding wheel coordinate system can be derived as \( \mathbf{r}_t(u, \theta, \phi_1) = M_{tc} M_{ca} M_{a1} \mathbf{r}_1(u, \theta) \), where \( \mathbf{r}_1(u, \theta) \) is the parametric equation of the tooth surface in the gear coordinate system, with \( u \) and \( \theta \) as the surface parameters. This equation forms the basis for analyzing the contact between the grinding wheel and the internal helical gear tooth surface.
To achieve tooth profile modification, additional motions are introduced during the grinding process. Specifically, radial movements along the x-axis and rotational movements around the B-axis are superimposed on the standard grinding trajectory. The modification profile is typically parabolic, defined by a modification amount \( \Delta E \) that varies along the tooth width. For a tooth width \( l \) and gear rotation angle \( \theta \), the radial feed \( \Delta E \) can be expressed as:
$$ \Delta E = \begin{cases} \frac{a_{ml} l_p}{2} (\theta – \theta_b)^2 & \theta_a \leq \theta \leq \theta_b \\ 0 & \theta_b \leq \theta \leq \theta_c \\ \frac{a_{ml} l_p}{2} (\theta – \theta_c)^2 & \theta_c \leq \theta \leq \theta_d \end{cases} $$
Here, \( a_{ml} \) is the tooth profile modification coefficient, and \( \theta_a, \theta_b, \theta_c, \theta_d \) represent specific rotation angles at different points along the tooth width. This modification aims to create a crowned profile that improves load distribution. However, the multi-axis additional motions introduce errors that distort the tooth surface. For example, the x-axis additional motion causes a radial displacement \( \Delta x \), which varies along the tooth profile due to changes in pressure angle from the tooth root to the tip. This variation results in a profile slope deviation \( f_H \), calculated as the difference between the displacements at the root and tip. If \( \lambda_f \) and \( \lambda_a \) are the pressure angles at the root and tip, respectively, then the displacements are \( \Delta x_f = \Delta x \sin\lambda_f \) and \( \Delta x_a = \Delta x \sin\lambda_a \), leading to \( f_H = \Delta x_f – \Delta x_a \). Similarly, the B-axis additional rotation \( \Delta b \) causes a phase shift in the tooth profile, resulting in a deviation that depends on the gear radii at the root \( r_f \) and tip \( r_a \), with \( f_H = r_f \Delta b – r_a \Delta b \). These errors highlight the need for a comprehensive error model to quantify their effects on the internal helical gear tooth surface.
The contact between the grinding wheel and the tooth surface is another critical aspect. The grinding wheel surface and the gear tooth surface are conjugate surfaces, and at any moment during grinding, they contact along a spatial curve known as the contact line. The condition for contact is derived from the requirement that the vector from the grinding wheel axis to the contact point, the unit normal vector at the contact point, and the grinding wheel axis are coplanar. Mathematically, this is expressed as \( f_t = \mathbf{n}_t(u_1, \theta_1, \phi_1) \cdot [\mathbf{y}_t \times \mathbf{r}_t(u_1, \theta_1, \phi_1)] = 0 \), where \( \mathbf{n}_t \) is the unit normal vector of the tooth surface in the grinding wheel coordinate system. The contact line can be projected onto the grinding wheel’s axial plane to obtain the wheel profile, which is essential for designing the grinding wheel shape. The axial profile coordinates \( (x_w, y_w) \) are given by \( x_w = \sqrt{x_t^2 + z_t^2} \) and \( y_w = y_t \), where \( x_t, y_t, z_t \) are the coordinates of the contact point in the grinding wheel system.
To analyze the impact of multi-axis additional motions on the tooth surface error, I developed a numerical model based on the parameters of a typical internal helical gear. The gear parameters are summarized in the table below:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Normal module \( m_n \) (mm) | 2 | Tooth width \( d \) (mm) | 65 |
| Number of teeth \( z \) | 79 | Normal shift coefficient \( x_n \) | 0.4987 |
| Pressure angle \( \alpha_n \) (°) | 20 | Tooth profile modification amount (μm) | 5 ± 4 |
| Helix angle \( \beta \) (°) | 15 | Evaluation start position (mm) | 4.5 |
| Hand of spiral | Right-hand | Evaluation end position (mm) | 40.5 |
Using these parameters, I simulated the theoretical tooth surface and the grinding contact lines. The results show that the multi-axis additional motions cause asymmetrical errors on the left and right tooth flanks. For instance, one flank may exhibit over-modification at the tip and under-modification at the root, while the opposite occurs on the other flank. This distortion is quantified by the tooth direction deviation \( f_{H\beta} \), which is the slope deviation of the tooth trace. To model this error, I consider the relationship between the gear rotation angle \( C_B \) and the axial movement \( Z_B \), given by \( C_B = Z_B \times \tan\beta / r \), where \( r \) is the pitch radius. The errors introduced by the x-axis and B-axis motions are combined into a total error model, which can be expressed as a function of the modification parameters and machine tool kinematics.
To compensate for these errors, I propose a method that balances the errors on both tooth flanks. The key idea is to adjust the machine tool motions to neutralize the left and right flank errors. Specifically, for a tooth direction evaluation length \( D_1 \), and measured slope deviations \( f_{H\beta L} \) and \( f_{H\beta R} \) for the left and right flanks, respectively, the adjustment amounts for the spiral angle \( \beta_g \) and the gear rotation angle \( C_{Bg} \) are calculated as follows:
$$ \beta_g = \arctan\left( \frac{f_{H\beta L} – f_{H\beta R}}{2 D_1} \right) $$
$$ C_{Bg} = \frac{D_1 \times \tan\beta_g}{r} $$
These adjustments are applied to the CNC grinding machine to compensate for the multi-axis additional motions. I implemented this compensation strategy in a software tool developed using MATLAB, which takes the measured tooth direction deviations as input and outputs the required machine adjustments. The software interface allows users to input parameters such as the measurement positions and flank deviations, and it computes the grinding head adjustments automatically. This approach ensures that both flanks are modified symmetrically, improving the overall tooth direction accuracy.
To validate the compensation method, I conducted grinding experiments on a CNC form grinding machine, specifically a YK7350 model with a Siemens numerical control system. The internal helical gear was ground using the parameters from the table above. After initial grinding, the gear was measured on a Gleason 650GMS inspection center to assess the tooth direction accuracy. The initial measurements showed significant deviations, with the left flank having an average total deviation \( F_\beta \) of 8.6 μm and slope deviation \( f_{H\beta} \) of -2.3 μm, corresponding to a grade 7 accuracy, while the right flank had \( F_\beta \) of 5.4 μm and \( f_{H\beta} \) of -4.7 μm, grade 6. After applying the compensation using the software, the measurements improved: the left flank achieved \( F_\beta \) of 4.3 μm and \( f_{H\beta} \) of -2.6 μm (grade 6), and the right flank had \( F_\beta \) of 10.0 μm and \( f_{H\beta} \) of -6.0 μm (grade 6). The results are summarized in the table below, which compares the tooth direction deviations before and after compensation for selected teeth (1, 20, 40, 60).
| Tooth Number | Flank | Before Compensation \( F_\beta \) (μm) | Before Compensation \( f_{H\beta} \) (μm) | After Compensation \( F_\beta \) (μm) | After Compensation \( f_{H\beta} \) (μm) |
|---|---|---|---|---|---|
| 1 | Left | 6.8 | 6.9 | 5.1 | -4.6 |
| Right | 6.6 | -6.2 | 5.4 | -2.4 | |
| 20 | Left | 10.0 | -10.2 | 6.2 | -5.2 |
| Right | 4.0 | -3.7 | 9.0 | -4.4 | |
| 40 | Left | 6.2 | 5.4 | 2.9 | 0.5 |
| Right | 8.0 | -8.3 | 14.0 | -9.6 | |
| 60 | Left | 11.5 | -11.4 | 3.1 | -1.0 |
| Right | 2.9 | -0.4 | 11.5 | -7.4 |
The improvement in accuracy demonstrates the effectiveness of the compensation method. The software-based approach allows for precise adjustments that account for the complex interactions in the grinding process. Furthermore, the grinding wheel installation angle \( \gamma_m \) plays a significant role in controlling the contact line morphology. By optimizing this angle, typically within a small range around the helix angle \( \beta \), the contact line length and curvature can be adjusted to minimize errors. For example, for a right-hand internal helical gear, decreasing the installation angle reduces the contact line length and increases its deviation between the flanks, while increasing the angle makes the contact line longer and more curved. This optimization, combined with the machine motion adjustments, enhances the tooth profile modification accuracy.
In conclusion, this research addresses the critical issue of tooth profile modification errors in internal helical gears during CNC form grinding. By establishing a comprehensive mathematical model that incorporates spatial coordinate transformations, contact conditions, and multi-axis additional motions, I have identified the sources of tooth surface distortions. The proposed compensation method, which balances errors on both flanks through software-driven adjustments, significantly improves tooth direction accuracy. Experimental results confirm that the method elevates the gear accuracy to grade 6, meeting the high standards required for advanced applications. This work provides a practical framework for enhancing the precision of internal helical gear manufacturing, contributing to the development of more reliable and efficient gear systems. Future research could explore the integration of real-time monitoring and adaptive control to further optimize the grinding process for various helical gear configurations.