We address the challenge of reduced tooth surface accuracy in form grinding of lead-modified internal helical gears caused by machine tool errors. In this study, we propose a systematic method for tooth surface deviation correction. First, we construct a processing motion model for internal helical gears, establish the tooth profile modification coordinate equation, and derive the form grinding wheel axial cross-sectional profile based on spatial meshing theory. Then, we develop a mathematical model for the CNC form grinding machine and analyze the quantitative influence of machine tool position errors on tooth surface deviations. By identifying the coupling relationship between machine tool errors and surface deviations, we implement a correction algorithm and validate it through experiments on an actual grinding machine. The results demonstrate that the corrected tooth surface accuracy improves from Grade 7 to Grade 6, confirming the effectiveness of the proposed approach.
1. Introduction
Helical gears are widely used in high-precision transmission systems due to their smooth meshing and high load capacity. For internal helical gears, form grinding is an efficient finishing process, especially when tooth profile modifications are required to improve load distribution and reduce noise. However, during form grinding, machine tool geometric errors, installation errors, and thermal effects can cause significant deviations in the tooth surface, particularly for left and right flanks. These deviations often manifest as pressure angle errors, helix angle errors, and flank twist. Traditional compensation methods rely on empirical adjustments, which are time-consuming and less accurate. Therefore, we aim to establish a theoretical error model and develop a corrective approach that can precisely adjust the machine tool parameters to minimize tooth surface deviations. In this paper, we focus on three primary error sources: the grinding wheel frame installation error Δy, the grinding wheel installation angular error Δφ, and the workpiece radial machining error Δx. We quantitatively analyze their effects on the tooth surface and propose a coupled correction strategy. The methodology is integrated into a dedicated software tool and experimentally verified.
2. Mathematical Model of Modified Internal Helical Gears
2.1 Tooth Profile Modification Principle
Tooth profile modification is represented by a second-order parabolic function superimposed on the standard involute. In the cross‑sectional coordinate system {S₁}, the X‑axis is along the tooth depth direction and coincides with the tooth space symmetry line. The base circle radius is rb, and the base circle tooth space half‑angle is σ₀. For any point M on the involute with roll angle u, the modification amount ΔE is given by:
$$ \Delta E = \begin{cases} a_{mp}(u – u_c)^2 & u_d \le u \le u_c \\ 0 & u_c < u < u_b \\ a_{mp}(u – u_b)^2 & u_b \le u \le u_a \end{cases} $$
where amp is the modification coefficient, and ua, ub, uc, ud are the roll angles at the start and end of the tip and root modification zones. The modified tooth profile vector in the end section is:
$$ \mathbf{r}(u) = \begin{pmatrix} r_b \cos(\sigma_0 + u) + (r_b u + \Delta E) \sin(\sigma_0 + u) \\ r_b \sin(\sigma_0 + u) – (r_b u + \Delta E) \cos(\sigma_0 + u) \end{pmatrix} $$
The half‑angle σ₀ is determined from the pitch circle tooth space half‑angle: σ₀ = π/(2z) – (tan α – α), where z is the number of teeth and α is the pressure angle.
2.2 Tooth Surface Equation of Modified Internal Helical Gears
We establish the generating coordinate system for internal helical gears, as shown in Figure 2 of the original work. The transformation from the auxiliary coordinate system {S₂} to the moving gear coordinate system {S₃} is:
$$ \mathbf{M}_{3,2} = \begin{pmatrix} \cos\varphi & -\sin\varphi & 0 & 0 \\ \sin\varphi & \cos\varphi & 0 & 0 \\ 0 & 0 & 1 & h \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
The relation between the helical motion rotation angle φ and the axial displacement h is:
$$ \varphi = \frac{2h \sin\beta}{m z} $$
where β is the helix angle, m is the module, and z is the number of teeth. The complete tooth surface equation of the modified internal helical gear is:
$$ \mathbf{r}_3(u, h) = \mathbf{M}_{3,2} \cdot \mathbf{r}_2(u) = \begin{pmatrix} r_b \cos(\sigma_0 + u + \varphi) + (r_b u + \Delta E) \sin(\sigma_0 + u + \varphi) \\ r_b \sin(\sigma_0 + u + \varphi) – (r_b u + \Delta E) \cos(\sigma_0 + u + \varphi) \\ h \\ 1 \end{pmatrix} $$
The unit normal vector is:
$$ \mathbf{n}_3 = \frac{ \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial h} }{ \left| \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial h} \right| } $$
2.3 Form Grinding Wheel Profile Solution
During the grinding process, the form grinding wheel and the gear tooth surface are conjugate surfaces. The contact line exists on both surfaces. By projecting this contact line onto the wheel’s transverse plane, we obtain the axial cross‑sectional profile of the grinding wheel. The transformation matrices between the gear coordinate system {S₃} and the grinding wheel coordinate system {Ss} are:
$$ \mathbf{M}_{s,3} = \begin{pmatrix} 1 & 0 & 0 & -a \\ 0 & \cos\beta & -\sin\beta & 0 \\ 0 & \sin\beta & \cos\beta & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
$$ \mathbf{M}_{4,s} = \begin{pmatrix} \cos\Sigma & 0 & -\sin\Sigma & 0 \\ 0 & 1 & 0 & 0 \\ \sin\Sigma & 0 & \cos\Sigma & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
The wheel cross‑sectional profile is given by:
$$ \mathbf{r}_4(u, h, \Sigma) = \mathbf{M}_{4,s} \cdot \mathbf{M}_{s,3} \cdot \mathbf{r}_3(u, h) $$
and the normal vector transformation:
$$ \mathbf{n}_4 = \mathbf{M}_{4,s} \cdot \mathbf{M}_{s,3} \cdot \mathbf{n}_3 $$
Here, a is the center distance between the wheel and gear, Σ is the rotation angle required to project the contact line into the transversal plane, and β is the helix angle. By solving the meshing condition (n₄ · v = 0), we can determine the wheel profile points.
3. CNC Form Grinding Machine Motion Model for Internal Helical Gears
3.1 Machine Tool Structure
The CNC internal gear form grinding machine has six motion axes: three linear axes (X, Y, Z) and three rotary axes (A, B, C). The grinding wheel frame moves along the Z and Y axes. The grinding wheel arm is mounted on the frame and rotates about the A axis. The form grinding wheel rotates about the C axis. The workpiece is mounted on a rotary table (B axis) and can also move along the X axis.
3.2 Mathematical Model of the Grinding Process
We establish the kinematic coordinate system of the CNC machine tool. The gear coordinate system {Sg} is fixed to the center of the rotary table. The grinding wheel frame initial coordinate system is {Sa}, and the grinding wheel initial coordinate system is {Ss}. The transformation matrices are defined as follows:
From the gear initial coordinate system {Sg} to the gear moving coordinate system {Sm}:
$$ \mathbf{M}_{m,g} = \begin{pmatrix} \cos\varphi_g & \sin\varphi_g & 0 & 0 \\ -\sin\varphi_g & \cos\varphi_g & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
where φg is the rotation angle of the gear about its axis.
From the wheel frame coordinate system {Sa} to the swing arm moving coordinate system {Sn}:
$$ \mathbf{M}_{n,a} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\varphi_a & -\sin\varphi_a & 0 \\ 0 & \sin\varphi_a & \cos\varphi_a & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
where φa is the swing arm rotation angle about the xa axis.
From the wheel coordinate system {Ss} to the wheel moving coordinate system {Sf}:
$$ \mathbf{S}_{f,s} = \begin{pmatrix} \cos\varphi_s & 0 & -\sin\varphi_s & 0 \\ 0 & 1 & 0 & 0 \\ \sin\varphi_s & 0 & \cos\varphi_s & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
where φs is the grinding wheel rotation angle.
In the machine zero coordinate system, cy and cz denote the translations of the wheel frame along Y₀ and Z₀ axes, and cx denotes the translation of the gear origin along X₀. The translation matrices are:
$$ \mathbf{M}_{0,n} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -c_y \\ 0 & 0 & 1 & c_z \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
$$ \mathbf{M}_{0,g} = \begin{pmatrix} 1 & 0 & 0 & -c_x \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
The wheel tooth surface is transformed into the gear moving coordinate system {Sm} as:
$$ \mathbf{r}_m(u, \sigma, \varphi) = \mathbf{M}_{m,g} \mathbf{M}_{0,g} \mathbf{M}_{0,n} \mathbf{M}_{n,a} \mathbf{S}_{f,s} \mathbf{r}_s(u, \sigma) $$
This complete kinematic chain allows us to simulate the theoretical tooth surface.
4. Machine Tool Position Error Analysis
4.1 Error Factors
We consider three main error sources: Δy (grinding wheel frame installation error in the Y direction), Δφ (angular error of the grinding wheel installation), and Δx (radial machining error of the workpiece along the X axis). These errors are incorporated into the transformation matrices as follows:
$$ \mathbf{M}’_{0,n} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -c_y + \Delta y \\ 0 & 0 & 1 & c_z \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
$$ \mathbf{M}’_{n,a} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos(\varphi_a + \Delta\varphi) & -\sin(\varphi_a + \Delta\varphi) & 0 \\ 0 & \sin(\varphi_a + \Delta\varphi) & \cos(\varphi_a + \Delta\varphi) & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
$$ \mathbf{M}’_{0,g} = \begin{pmatrix} 1 & 0 & 0 & -c_x + \Delta x \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
The actual tooth surface with errors is then:
$$ \mathbf{r}_m'(u, \sigma, \varphi) = \mathbf{M}_{m,g} \mathbf{M}’_{0,g} \mathbf{M}’_{0,n} \mathbf{M}’_{n,a} \mathbf{S}_{f,s} \mathbf{r}_s(u, \sigma) $$
4.2 Quantitative Analysis of Error Effects
We analyze the influence of each error on the tooth surface deviation using the gear parameters listed in Table 1.
| Parameter | Value |
|---|---|
| Number of teeth z | 79 |
| Normal module mn (mm) | 2 |
| Pressure angle αn (°) | 20 |
| Helix angle β (°) | 15 |
| Profile shift coefficient xn | 0.4987 |
| Face width B (mm) | 65 |
| Tip modification amount (μm) | 5 ± 4 |
| Root modification amount (μm) | 5 ± 4 |
| Grinding wheel radius (mm) | 65 |
Effect of Δy: Δy mainly causes pressure angle changes. Increasing Δy reduces the pressure angle on the left flank and increases it on the right flank. Decreasing Δy has the opposite effect. The deviation pattern is symmetrical for both flanks but with opposite signs.
Effect of Δφ: Δφ primarily induces helix angle changes. A positive Δφ makes the tooth direction shift to the right (right-handed trend), while a negative Δφ shifts to the left (left-handed trend). This error affects both flanks in the same direction.
Effect of Δx: Δx introduces pressure angle changes on both flanks in the same direction. Increasing Δx increases the pressure angle on both flanks, while decreasing Δx reduces it.
The quantitative results are summarized in Table 2 for simulated errors of ±0.15 mm or ±0.15°.
| Error Source | Error Value | Left Flank Effect | Right Flank Effect |
|---|---|---|---|
| Δy = +0.15 mm | Pressure angle decrease | Pressure angle increase | |
| Δy = –0.15 mm | Pressure angle increase | Pressure angle decrease | |
| Δφ = +0.15° | Rightward helix shift | Rightward helix shift | |
| Δφ = –0.15° | Leftward helix shift | Leftward helix shift | |
| Δx = +0.15 mm | Pressure angle increase | Pressure angle increase | |
| Δx = –0.15 mm | Pressure angle decrease | Pressure angle decrease |
5. Correction Method for Tooth Surface Deviation
5.1 Deviation Quantification
We define the tooth profile slope deviation fHα and the helix slope deviation fHβ as the two key indicators. The measured tooth surface points are compared with the theoretical ones. The rotation angle θ of the actual profile relative to the theoretical profile is computed using the dot product:
$$ \theta = \arccos\left( \frac{x_1 x_\theta + y_1 y_\theta}{r^2} \right) $$
The rotation matrix is:
$$ \mathbf{M}_{\theta1} = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix} $$
The actual tooth profile after including machining errors is:
$$ \mathbf{r}_1(\theta) = \begin{pmatrix} r_b\cos(u+\sigma_0) + r_b u\sin(u+\sigma_0) + \Delta x \\ r_b\sin(u+\sigma_0) – r_b u\cos(u+\sigma_0) + \Delta y \end{pmatrix} $$
The rotated profile is:
$$ \mathbf{r}_\theta = \mathbf{M}_{\theta1} \mathbf{r}_1(\theta) $$
The normal deviation ej at each point j is:
$$ e_j = \sqrt{(x_{1j} – x_{\theta j})^2 + (y_{1j} – y_{\theta j})^2} \quad \text{or} \quad -\sqrt{(x_{1j} – x_{\theta j})^2 + (y_{1j} – y_{\theta j})^2} $$
depending on whether yθj ≤ y1j. The profile slope deviation is then en – e1.
5.2 Correction Strategy
From the measured left and right flank profile slope deviations fHαL and fHαR, we compute the required corrections in the X and Y axes:
$$ f_{H\alpha x} = \frac{f_{H\alpha L} + f_{H\alpha R}}{2}, \quad f_{H\alpha y} = \frac{f_{H\alpha L} – f_{H\alpha R}}{2} $$
For helix slope correction, we adjust the interpolation between the Z‑axis movement and the gear rotation. The standard relationship is:
$$ B_g = \frac{Z_g \tan\beta}{r} $$
where r is the pitch radius. After measuring the left and right helix slope deviations fHβL and fHβR, the corrected helix angle βe and the required additional gear rotation Bge are:
$$ \beta_e = \arctan\left( \frac{f_{H\beta L} – f_{H\beta R}}{2H} \right), \quad B_{ge} = \frac{H \tan\beta_e}{r} $$
where H is the tooth direction measurement length.
5.3 Software Implementation
We developed a dedicated tooth surface deviation correction software using MATLAB, as shown in Figure 12 of the original work. The interface allows the user to input basic gear parameters, modification parameters, and the measured left and right flank deviations. The software automatically computes the required machine tool adjustments (Δx, Δy, Δφ) and outputs the corrected tool path parameters.
6. Experimental Verification
6.1 Grinding Experiment
We performed form grinding of internal helical gears on a YK7350 CNC form grinding machine. The gear parameters are given in Table 1. After initial trial grinding, the tooth surfaces were measured using a Gleason 650GMS gear inspection center. The first measurement results are shown in Figure 15 of the original reference. The profile slope deviations and helix slope deviations were recorded.
6.2 Correction and Second Grinding
Based on the measured deviations, we applied the correction method described in Section 5. The machine tool parameters were adjusted accordingly: Δx, Δy, and the swing arm angle φa were fine-tuned. A second grinding was performed, and the gear was re-measured. The results are shown in Figure 16 of the original reference. The quantitative comparison is listed in Table 3.
| Parameter | Before Correction (μm) | After Correction (μm) |
|---|---|---|
| fHαL (Left flank profile slope) | –8.5 | 4.6 |
| fHαR (Right flank profile slope) | 1.9 | 4.6 |
| fHβL (Left flank helix slope) | 12.2 | 1.2 |
| fHβR (Right flank helix slope) | –4.1 | –0.9 |
The tooth surface accuracy improved from Grade 7 to Grade 6 according to the ISO standard. The correction method successfully reduced the deviations, confirming its validity and practicality.
7. Conclusion
We have presented a comprehensive method for correcting tooth surface deviations in form grinding of modified internal helical gears. By establishing an accurate mathematical model of the gear and machine tool, and quantitatively analyzing the influence of three key machine tool position errors (Δy, Δφ, Δx), we derived a coupled correction strategy. The method was implemented in a software tool and experimentally validated on a real CNC form grinding machine. The results demonstrate that the proposed approach effectively improves the tooth surface accuracy from Grade 7 to Grade 6, providing a reliable solution for high-precision manufacturing of helical gears.
