In precision manufacturing, thermal errors significantly impact the accuracy of CNC gear hobbing machines, accounting for a substantial portion of machining inaccuracies. This study focuses on developing a robust thermal error model for a CNC gear hobbing machine by integrating finite element analysis and Bayesian network theory. The approach involves simulating thermal-structural behavior under operational conditions, optimizing temperature variables through fuzzy clustering, and constructing a predictive model to compensate for thermal displacements. By reducing the number of required sensors and enhancing prediction accuracy, this research aims to improve the performance of gear hobbing processes in industrial applications.
The CNC gear hobbing machine comprises key components such as the column, direct-drive hob spindle system, worktable, feed system, and bed. Thermal sources primarily include cutting heat from the gear hobbing process, motor heating, and bearing friction. Accurate modeling of these heat sources is crucial for effective error compensation. The finite element method is employed to simulate temperature distribution and thermal deformation, while Bayesian networks provide a probabilistic framework for error prediction. This integrated methodology ensures reliable thermal error management in gear hobbing operations.
Heat generation in a gear hobbing machine arises from multiple sources. Cutting heat during gear hobbing is calculated based on the cutting force and spindle speed. A portion of this heat, typically 5%, transfers to the hob. The heat power \( Q \) is given by:
$$ Q = P \nu $$
where \( P \) is the cutting force and \( \nu \) is the spindle speed. The maximum torque \( M_{\text{max}} \) for the hob is derived as:
$$ M_{\text{max}} = 9.1 m^{1.75} S^{0.65} T^{0.81} \nu_0^{-0.26} z^{0.27} K_{\text{material}} K_{\text{hardness}} K_{\text{helix}} $$
Here, \( m \) is the normal module, \( S \) is the axial feed rate, \( T \) is the depth of cut, \( \nu_0 \) is the spindle speed, \( z \) is the number of gear teeth, and \( K_{\text{material}} \), \( K_{\text{hardness}} \), \( K_{\text{helix}} \) are correction factors for material, hardness, and helix angle, respectively. The cutting force \( P \) is then:
$$ P = \frac{2 M_{\text{max}}}{d} $$
with \( d \) as the hob outer diameter. Bearing friction heat, particularly from sliding bearings, is another critical source. The frictional heat \( Q_1 \) is expressed as:
$$ Q_1 = f p v $$
where \( f \) is the friction coefficient, \( p \) is the bearing load, and \( v \) is the journal circumferential speed. For rolling bearings like thrust ball bearings, the heat generation power \( q \) is:
$$ q = 1.047 \times 10^{-4} M \cdot n $$
with \( M \) as the bearing frictional torque and \( n \) as the spindle speed. The total frictional torque \( M \) combines load-dependent torque \( M_0 \) and viscous torque \( M_1 \):
$$ M = M_0 + M_1 $$
where:
$$ M_0 = \begin{cases}
10^{-7} \cdot f_0 \cdot (\nu n)^{2/3} \cdot d_m^3, & \nu n \geq 2000 \\
160 \times 10^{-7} \cdot f_0 \cdot d_m^3, & \nu n < 2000
\end{cases} $$
and:
$$ M_1 = f_1 p_1 d_m $$
Here, \( f_0 \) and \( f_1 \) are constants related to bearing type and lubrication, \( \nu \) is the lubricant kinematic viscosity, \( p_1 \) is the load for moment calculation, and \( d_m \) is the bearing pitch diameter. The direct-drive motor heat \( Q_d \) is calculated from input power and efficiency:
$$ Q_d = P_1 (1 – \eta) $$
with \( P_1 \) as the motor input power and \( \eta \) as the motor efficiency.
Boundary conditions for thermal analysis include convective heat transfer. For forced convection around rotating components like the hob spindle and motor rotor, the heat transfer coefficient \( \alpha \) is:
$$ \alpha = \frac{N_u \lambda}{d_s} $$
The Nusselt number \( N_u \) is given by:
$$ N_u = 0.133 R_e^{2/3} P_r^{1/3} $$
and the Reynolds number \( R_e \) is:
$$ R_e = \frac{\omega d_s^2}{\nu_f} $$
where \( \lambda \) is the fluid thermal conductivity, \( d_s \) is the equivalent diameter, \( \omega \) is the angular velocity, and \( \nu_f \) is the fluid kinematic viscosity. For water cooling in the motor, the convection coefficient \( h_1 \) is:
$$ h_1 = 0.023 R_e^{0.8} P_r^{0.4} \frac{\lambda}{D} $$
with \( D \) as the characteristic diameter and \( u \) as the fluid velocity.
Finite element analysis is conducted to simulate steady-state and transient thermal behavior. The model incorporates material properties, as summarized in Table 1, and heat source intensities from Table 2. The mesh consists of 397,046 elements and 632,875 nodes, with refinement around the hob spindle for accuracy.
| Component | Material | Density (kg/m³) | Elastic Modulus (Pa) | Poisson’s Ratio | Specific Heat (J/(kg·K)) | Thermal Conductivity (W/(m·K)) | Thermal Expansion Coefficient (1/K) |
|---|---|---|---|---|---|---|---|
| Hob Spindle | 40Cr | 7870 | 2.11E+11 | 0.269 | 460 | 44 | 1.20E-05 |
| Motor Shaft | 45 Steel | 7890 | 2.10E+11 | 0.278 | 450 | 48 | 1.17E-05 |
| Bed | HT200 | 7200 | 1.48E+11 | 0.31 | 510 | 45 | 1.10E-05 |
| Sliding Bearing | CuAl10Fe3 | 7400 | 1.10E+11 | 0.3 | 380 | 56 | 1.70E-05 |
| Rolling Bearing | GCr15 | 7830 | 2.19E+11 | 0.3 | 460 | 81 | 1.25E-05 |
| Hob | W18Cr4V | 8260 | 2.25E+11 | 0.25 | 420 | 27.2 | 1.21E-05 |
| Source | Intensity (W) |
|---|---|
| Hob Cutting Heat | 238.5 |
| Tailstock Bearing Heat | 46.5 |
| Motor Heat | 502 |
| Sliding Bearing II | 40.6 |
| Rolling Bearing | 1.6 |
Steady-state thermal analysis reveals temperature distribution, with the motor rotor reaching 41.6°C and the tailstock bearing at 33°C. Thermal displacement results show maximum deformation in the motor region (92 μm) and smaller displacements in the hob spindle (35 μm). Transient analysis tracks temperature and displacement over time, indicating gradual increases aligned with heat accumulation. The hob spindle exhibits significant displacements in the X and Y directions, with minimal Z-direction movement.
To optimize temperature variables, fuzzy clustering is applied to simulation data from seven initial points. The similarity matrix \( R \) is constructed using Pearson correlation coefficients \( r_{ij} \):
$$ r_{xy} = \frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sqrt{\sum (x_i – \bar{x})^2 \sum (y_i – \bar{y})^2}} $$
The transitive closure \( \hat{R} \) is computed iteratively until \( R^{2k} = R^{2(k+1)} \). Setting a threshold \( \lambda = 0.998 \), temperature variables are clustered into four groups. Correlation with thermal displacement identifies key points, reducing sensors to three: the gear hobbing machine’s tool post, tailstock bearing, and sliding bearing II. Table 3 summarizes the clustering and correlation results.
| Temperature Variable | Cluster | Correlation with Displacement |
|---|---|---|
| T1, T2 | 1 | 0.943, 0.941 |
| T3, T4 | 2 | 0.982, 0.990 |
| T5, T6 | 3 | 0.995, 0.989 |
| T7 | 4 | 0.924 |
Bayesian network modeling leverages these thermal key points. The network structure \( S \) is a directed acyclic graph with nodes representing temperature variables and thermal error. The joint probability distribution is:
$$ P(X_1, X_2, \ldots, X_n) = \prod_{i=1}^n P(X_i | \text{Parents}(X_i)) $$
For the gear hobbing machine, nodes include tailstock bearing temperature \( \Delta T_1 \), tool post temperature \( \Delta T_4 \), sliding bearing II temperature \( \Delta T_5 \), and thermal error \( \Delta E \). Conditional probabilities are derived from experimental data, discretized into ten states based on value ranges. For example, \( \Delta T_1 \) intervals are [0, 0.04], [0.04, 0.08], …, [0.36, 0.40] °C, and \( \Delta E \) intervals are [-1.1, -0.79], [-0.79, -0.48], …, [1.69, 2.0] μm. Table 4 shows the independent distribution probabilities for \( \Delta T_1 \) and \( \Delta E \).
| State | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| \( \Delta T_1 \) | 0.179 | 0 | 0.339 | 0.143 | 0.107 | 0.071 | 0.036 | 0.036 | 0.036 | 0.036 |
| \( \Delta E \) | 0.036 | 0 | 0.071 | 0.107 | 0.143 | 0.179 | 0.143 | 0.107 | 0.071 | 0.071 |
Conditional probability tables, such as \( P(\Delta T_1 | \Delta E) \), are populated from data. Prediction involves finding the state \( j \) that maximizes \( P(\Delta E_j | \Delta T_1, \Delta T_4, \Delta T_5) \), and the error increment \( \Delta E_k \) is taken as the midpoint of the predicted state. The cumulative error is updated as:
$$ E_{k+1} = E_k + \Delta E_k $$
Experimental validation involves collecting temperature and displacement data from the gear hobbing machine under no-load conditions at 160 rpm. Temperature sensors (DS18B20) monitor the key points, and displacement sensors track hob spindle movement. The Bayesian network model predictions are compared with measured data over 7 hours, showing a maximum residual error of 3.5 μm. This demonstrates the model’s effectiveness in predicting thermal errors for gear hobbing applications.
The integration of finite element analysis and Bayesian networks provides a comprehensive framework for thermal error modeling in CNC gear hobbing machines. By optimizing temperature variables and leveraging probabilistic reasoning, the model reduces sensor requirements and enhances prediction accuracy. Future work could explore real-time compensation systems to further improve gear hobbing precision in dynamic environments.