To accurately calculate the time-varying meshing stiffness of spur gear pairs under real meshing conditions, this study proposes a methodology combining Tooth Contact Analysis (TCA) and Load Tooth Contact Analysis (LTCA). The method accounts for tooth profile modifications and installation errors while leveraging mathematical programming and finite element analysis to determine load distribution factors and transmission errors. Cubic Hermite interpolation is applied to derive time-varying meshing stiffness over a full engagement cycle.
1. Theoretical Framework
1.1 Definition of Time-Varying Meshing Stiffness
The time-varying meshing stiffness \( K \) of spur gears is defined as the ratio of instantaneous normal force \( P \) to total elastic deformation \( \delta \):
$$ K = \frac{P}{\delta} $$
1.2 Coordinate Transformation for TCA
The coordinate systems for spur gear meshing analysis are established as:
$$ \mathbf{r}_{f1}(u_1, \theta_1, \phi_1) = [M_{f1}(\phi_1)]\mathbf{r}_1(u_1, \theta_1) $$
$$ \mathbf{r}_{f2}(u_2, \theta_2, \phi_2) = [M_{f2}(\phi_2)]\mathbf{r}_2(u_2, \theta_2) $$
Where transformation matrices are:
$$ M_{f1} = \begin{bmatrix}
\cos\phi_1 & \sin\phi_1 & 0 & 0 \\
-\sin\phi_1 & \cos\phi_1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} $$
1.3 Normal Clearance Calculation
The normal clearance \( b_k \) between spur gear tooth surfaces is determined through discrete point analysis:
$$ b_k = \sqrt{(x_{M2}-x_{M1})^2 + (y_{M2}-y_{M1})^2 + (z_{M2}-z_{M1})^2} $$
2. LTCA Implementation
2.1 Flexibility Matrix Construction
The normal flexibility matrix \( [F] \) for spur gear teeth is formulated as:
$$ [F] = \begin{bmatrix}
f_{11} & \cdots & f_{1n} \\
\vdots & \ddots & \vdots \\
f_{n1} & \cdots & f_{nn}
\end{bmatrix} $$
where \( f_{ij} \) represents the deformation at node \( j \) caused by unit load at node \( i \).
2.2 Load Distribution Model
The deformation compatibility equation for spur gear pairs under load is:
$$ [F]_k[p]_k + [w]_k = [\delta] + [d]_k $$
With force equilibrium constraint:
$$ \sum_{j=1}^n (p_j^I + p_j^{II}) = P $$
3. Numerical Results
3.1 Meshing Characteristics
Key parameters for spur gear analysis:
| Parameter | Pinion | Gear |
|---|---|---|
| Teeth | 42 | 43 |
| Module (mm) | 3.5 | |
| Pressure Angle | 20° | |
| Young’s Modulus (GPa) | 240 | |
3.2 Stiffness Comparison
Comparison between TCA/LTCA and FEM results:
| Method | Avg. Stiffness (N/m) | Error |
|---|---|---|
| TCA/LTCA | 6.103×10⁸ | – |
| FEM | 6.294×10⁸ | 3.1% |
3.3 Modification Effects
Impact of profile modification on spur gear meshing stiffness:
$$ \Delta K = K_0 – K_m = \alpha \delta_g + \beta h_g $$
Where \( \delta_g \) = modification depth and \( h_g \) = modification length.
| Modification Length (mm) | Avg. Stiffness (N/m) | Amplitude (N/m) |
|---|---|---|
| 1.7 | 7.138×10⁸ | 2.056×10⁸ |
| 2.3 | 6.896×10⁸ | 1.649×10⁸ |
4. Conclusion
The proposed TCA/LTCA method effectively calculates time-varying meshing stiffness for spur gears, demonstrating:
- 3.1% deviation from FEM results validates computational accuracy
- Tooth profile modifications reduce stiffness amplitude by 19.8%
- Nonlinear relationship between modification parameters and stiffness reduction
This methodology provides critical insights for spur gear design optimization, particularly in vibration reduction and load capacity improvement.