Spiral bevel gears are critical components in power transmission systems for intersecting axes, widely used in automotive, marine, and aerospace industries due to their high load capacity, smooth operation, and excellent degree of coincidence. However, machining accuracy of spiral bevel gears is significantly affected by various errors, among which thermal errors in machine tools play a pivotal role. Thermal deformation during machining alters the relative position between the tool and workpiece, leading to deviations in the tooth surface profile. This study focuses on developing a comprehensive model to predict tooth surface errors in spiral bevel gears caused by thermal effects in five-axis CNC gear milling machines. By integrating gear meshing theory, multi-body system dynamics, and experimental data, we establish a mapping relationship between machine tool thermal errors and tooth surface inaccuracies. The proposed methodology not only enhances the understanding of thermal impacts on gear quality but also provides a foundation for error compensation strategies.
The manufacturing of spiral bevel gears typically follows the “imaginary crown gear” principle, where a cutting tool simulates the generation motion to form the tooth surface. In traditional mechanical cradle-style machines, the tool rotates and oscillates to mimic the engagement between an imaginary generating gear (the crown gear) and the workpiece. The mathematical model of the tooth surface is derived from the tool surface equations, considering spatial relationships and relative motions. For a spiral bevel gear, the tooth surface can be represented using parametric equations based on the tool geometry and kinematic conditions. The double-sided cutter, comprising alternating inner and outer blades, simultaneously machines both sides of the tooth slot. The tool surface is described in the tool coordinate system $O_tx_ty_tz_t$ with the radial vector $\mathbf{r}^t$ and unit normal vector $\mathbf{n}^t$ given by:
$$ \mathbf{r}^t(u, \theta) = \begin{bmatrix} (r \pm u \sin \alpha) \cos \theta \\ (r \pm u \sin \alpha) \sin \theta \\ -u \cos \alpha \\ 1 \end{bmatrix} $$
and
$$ \mathbf{n}^t(\theta) = \begin{bmatrix} \cos \alpha \cos \theta \\ \cos \alpha \sin \theta \\ \pm \sin \alpha \end{bmatrix} $$
where $u$ and $\theta$ are tool parameters, $\alpha$ is the tool profile angle, $r$ is the distance from the tool tip to the origin, and the signs correspond to outer and inner blade surfaces. The generation process involves coordinate transformations from the tool to the workpiece system. Using homogeneous transformation matrices, the position vectors and normal vectors in the workpiece coordinate system $O_wx_wy_wz_w$ are derived as:
$$ \mathbf{r}^w = \mathbf{M}_{wd} \cdot \mathbf{M}_{db} \cdot \mathbf{M}_{ba} \cdot \mathbf{M}_{a0} \cdot \mathbf{M}_{0c} \cdot \mathbf{M}_{ct} \cdot \mathbf{r}^t $$
and
$$ \mathbf{n}^w = \mathbf{H}_{wd} \cdot \mathbf{H}_{db} \cdot \mathbf{H}_{ba} \cdot \mathbf{H}_{a0} \cdot \mathbf{H}_{0c} \cdot \mathbf{H}_{ct} \cdot \mathbf{n}^t $$
where $\mathbf{M}_{ij}$ and $\mathbf{H}_{ij}$ are transformation matrices between coordinate systems. The meshing condition between the generating gear and the workpiece requires that the relative velocity at the contact point is perpendicular to the common normal vector, expressed as:
$$ \mathbf{V}^{0w} \cdot \mathbf{n}^0 = 0 $$
with $\mathbf{V}^{0w} = \mathbf{V}^0 – \mathbf{V}^w = \boldsymbol{\omega}^0 \times \mathbf{r}^0 – \boldsymbol{\omega}^w \times (\mathbf{r}^w – \mathbf{r}^e)$, where $\boldsymbol{\omega}^0$ and $\boldsymbol{\omega}^w$ are angular velocities, and $\mathbf{r}^e$ is the vector between origins. Solving these equations yields the tooth surface equation for the spiral bevel gear.
To discretize the tooth surface, we employ the rotational projection method, which constraints grid points on the tooth surface via axial cross-section projections. The tooth surface is divided into 7 rows along the tooth height and 9 columns along the tooth width, resulting in 63 discrete points. The coordinates of these points in the workpiece system are obtained by solving nonlinear equations derived from the meshing condition and projection relationships. For each grid point $(i,j)$, the following equations hold:
$$ z^w(u, \theta, \phi_0) = x_{ij}^p – L $$
and
$$ [x^w(u, \theta, \phi_0)]^2 + [y^w(u, \theta, \phi_0)]^2 = (y_{ij}^p)^2 $$
where $(x_{ij}^p, y_{ij}^p)$ are coordinates in the projection plane, and $\phi_0$ is the generating gear angle. Iterative methods are used to solve for parameters $u$, $\theta$, and $\phi_0$, which are then substituted into the tooth surface equation to obtain discrete point coordinates.
In modern five-axis CNC gear milling machines, such as the YKH2235 model, the traditional cradle mechanism is replaced by interpolated motions of linear and rotary axes. The machine structure includes a bed, column, spindle head, workpiece box, and rotary table. For generating machining of the spiral bevel gear, the workpiece rotation (S-axis) and interpolated motions of X and Y axes simulate the generation motion, while Z-axis controls depth and B-axis sets the root angle. The motion equivalence between the CNC machine and traditional machine is established by equating transformation matrices. The CNC machine transformation from tool to workpiece is:
$$ \mathbf{M}_{w2t} = \mathbf{M}_{w2h} \cdot \mathbf{M}_{hk} \cdot \mathbf{M}_{kg} \cdot \mathbf{M}_{g0} \cdot \mathbf{M}_{0e} \cdot \mathbf{M}_{et} $$
Setting $\mathbf{M}_{w2t}$ equal to the traditional machine matrix $\mathbf{M}_{wt}$ yields the relationship between CNC axes positions and traditional parameters:
$$ x = (L + X_w) \cos \delta_M + S \cos q $$
$$ y = S \sin q + E $$
$$ z = (L + X_w) \sin \delta_M – (X_b + X_w) $$
where $S$ is radial cutter position, $q$ is angular cutter position, $\delta_M$ is machine root angle, $X_b$ is bed position, $X_w$ is axial workpiece position, $E$ is vertical offset, and $L$ is distance from rotary center to crossing point.
Thermal errors in the machine tool arise from internal and external heat sources, causing structural deformations that affect the relative tool-workpiece position. Using multi-body system theory, we model the thermal errors by considering the machine’s topological structure and low-order body array. The ideal and actual positions of the tool tip in the workpiece coordinate system are given by:
$$ \mathbf{P}^{wi} = \prod_{u=1}^{n} \left[ \mathbf{T}_{L(u)}^{p} \cdot \mathbf{T}_{L(u)}^{s} \right] \cdot \mathbf{C}^t $$
and
$$ \mathbf{P}^{w} = \prod_{u=1}^{n} \mathbf{E T}_{L(u)} \cdot \mathbf{C}^t $$
where $\mathbf{E T}_{ij} = \Delta \mathbf{T}_{ij}^{p} \cdot \mathbf{T}_{ij}^{p} \cdot \Delta \mathbf{T}_{ij}^{s} \cdot \mathbf{T}_{ij}^{s}$, and $\Delta \mathbf{T}_{ij}^{p}$ and $\Delta \mathbf{T}_{ij}^{s}$ are error matrices due to thermal deformations. The relative error at the tool tip is:
$$ \mathbf{e}_p = \mathbf{P}^{w} – \mathbf{P}^{wi} = [\Delta e_x, \Delta e_y, \Delta e_z, 1]^T $$
The components of $\mathbf{e}_p$ are derived as functions of linear and angular thermal errors of each axis. For example, the error in X-direction can be expressed as:
$$ \Delta e_x = B \sin \delta_M (\Delta \beta_{xz} – 0.688 \Delta \gamma_z + 0.535 \Delta \alpha_x + \Delta \beta_x) + \text{…} $$
These errors are mapped to the tooth surface by modifying the ideal tooth surface points. The actual tooth surface point with errors is:
$$ \mathbf{r}^{we}(u, \theta, \phi_0, \Delta e) = \mathbf{r}^{w}(u, \theta, \phi_0) + \mathbf{d}_k $$
where $\mathbf{d}_k$ is the error vector at the k-th grid point. The tooth surface error in the normal direction is the dot product between $\mathbf{d}_k$ and the unit normal vector $\mathbf{n}^w_k$:
$$ h_k = \mathbf{d}_k \cdot \mathbf{n}^w_k $$
This represents the deviation due to thermal effects at each discrete point on the spiral bevel gear tooth surface.
To validate the model, we conducted thermal characterization experiments on the five-axis CNC gear milling machine. Temperature sensors were installed near heat sources, such as screw nuts, guide rails, and motors for linear axes, and on the spindle housing for the spindle system. The thermal errors were measured using laser interferometers and a ballbar system. For instance, the Z-axis thermal positioning error and straightness errors were recorded over multiple cycles. The spindle thermal errors were measured using the five-point method. The experimental results showed significant thermal drifts, which were incorporated into the error model.
| Parameter | Value |
|---|---|
| Number of Teeth | 38 |
| Hand of Spiral | Right |
| Face Width (mm) | 32.18 |
| Module (mm) | 5.391 |
| Normal Pressure Angle (°) | 20 |
| Shaft Angle (°) | 90 |
| Outer Cone Distance (mm) | 106.61 |
| Whole Depth (mm) | 10.40 |
| Addendum (mm) | 1.57 |
| Dedendum (mm) | 8.82 |
| Pitch Angle (°) | 73.54 |
| Face Angle (°) | 75.02 |
| Root Angle (°) | 68.21 |
The cutting experiments involved machining two spiral bevel gears: one under cold-start conditions and another after thermal warm-up. The same tooling and setup were used to minimize extraneous errors. The tooth surface deviations were measured on a gear measuring center. The difference between the warm and cold gear deviations represents the thermal-induced error. The predicted tooth surface errors based on the model showed a root mean square error of 2.57 µm compared to measured values, with similar trends in deviation patterns. The following table summarizes the experimental conditions for thermal error measurement:
| Component | Temperature Sensor Locations | Motion Parameters |
|---|---|---|
| Linear Axes | Motors, bearing seats, sliders, nuts | Velocity: 3 m/min for X, Y, Z |
| Spindle | Spindle housing top, sides, front, rear | Spindle Speed: 300 rpm |
The results demonstrate that the proposed model effectively predicts thermal-induced tooth surface errors in spiral bevel gears. The integration of gear geometry, machine kinematics, and thermal error modeling provides a comprehensive approach for accuracy enhancement in gear manufacturing. Future work could focus on real-time error compensation and optimization of cutting parameters to further improve the quality of spiral bevel gears.
In conclusion, this study presents a detailed methodology for modeling and validating thermally induced tooth surface errors in spiral bevel gears. By combining theoretical analysis with experimental data, we have established a robust framework that can be utilized in industrial applications to reduce errors and enhance the performance of spiral bevel gear transmissions. The insights gained from this research contribute to the advancement of precision machining processes for complex gear systems.