We investigate the inverse algorithm for determining pinion machining parameters in spiral bevel gears. The tooth surface equation for spiral bevel gears generated by the modified-roll method is derived, and a least squares optimization model is established to achieve the predesigned target surface. Three distinct algorithms are implemented and compared: the generalized inverse matrix method, truncated singular value decomposition, and the Levenberg-Marquardt (L-M) iterative algorithm with trust region strategy.
Theoretical Tooth Surface Formulation
The cutting cone surface Σp in coordinate system Sp is expressed as:
$$ \mathbf{r_p} = \begin{bmatrix} (R_p + u \sin \alpha_1) \cos \theta \\ (R_p + u \sin \alpha_1) \sin \theta \\ -u \cos \alpha_1 \end{bmatrix} $$
where \(u\) and \(\theta\) are surface parameters, \(R_p\) is cutter point radius, and \(\alpha_1\) is tool pressure angle. The coordinate transformation from machine tool to workpiece system is governed by:
$$ \mathbf{r_1}(u,\theta,\phi_p) = \mathbf{M_{1b}}(\phi_1) \mathbf{M_{bh}} \mathbf{M_{hm}} \mathbf{M_{mc}} \mathbf{M_{cp}}(\phi_p) \mathbf{r_p}(u,\theta) $$
The meshing equation follows from differential geometry principles:
$$ f(u, \theta, \phi_p) = (\mathbf{r_{1,u}}, \mathbf{r_{1,\theta}}, \mathbf{r_{1,\phi_p}}) $$
where subscripts denote partial derivatives. For modified roll machining, the pinion rotation angle \(\phi_1\) relates to cradle rotation \(\phi_p\) through:
$$ \phi_1 = m_{1p} (\phi_p – C\phi_p^2 – D\phi_p^3) $$
with \(m_{1p}\) as roll ratio, and \(C\), \(D\) as modification coefficients.
Deviation Modeling and Optimization
Discrete tooth surface points for machining parameters \(\mathbf{d} = [\alpha_1, R_p, S_r, q_1, \gamma_1, X_B, E_m, X_G, m_{1p}, C, D]^T\) are defined by position and normal vectors:
$$ \mathbf{p_i} = \mathbf{p_i}(\varepsilon_i, \mathbf{d}), \quad \mathbf{n_i} = \mathbf{n_i}(\varepsilon_i, \mathbf{d}) \quad (i=1,\dots,m) $$
The deviation vector \(\mathbf{h(d)} = [h_1(\mathbf{d}), \dots, h_m(\mathbf{d})]^T\) between target surface \(\mathbf{p^*_i}\) and initial surface satisfies:
$$ \mathbf{p^*_i} = \mathbf{p_i} + \mathbf{n_i} h_i $$
The nonlinear system for surface parameters is solved via:
$$ \begin{cases}
[\mathbf{p^*_i} – \mathbf{p_i}(\varepsilon_i, \mathbf{d})] \cdot \mathbf{p_{i,\theta_i}}(\varepsilon_i, \mathbf{d}) = 0 \\
[\mathbf{p^*_i} – \mathbf{p_i}(\varepsilon_i, \mathbf{d})] \cdot \mathbf{p_{i,u_i}}(\varepsilon_i, \mathbf{d}) = 0 \\
f(\varepsilon_i, \mathbf{d}) = 0
\end{cases} $$
yielding the residual error:
$$ h_i = [\mathbf{p^*_i} – \mathbf{p_i}(\varepsilon_i, \mathbf{d})] \cdot \mathbf{n_i}(\varepsilon_i, \mathbf{d}) $$
The optimization minimizes sum of squared residuals:
$$ \min f(\mathbf{d}) = \frac{1}{2} \mathbf{h}^T(\mathbf{d}) \mathbf{h}(\mathbf{d}) $$
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 23 | 65 |
| Module (mm) | 3.9 | |
| Spiral angle (°) | 25 | |
| Hand of spiral | RH | LH |
| Shaft angle (°) | 90 | |
| Pitch angle (°) | 19.4861 | 70.5139 |
Parameter Identification Algorithms
The Jacobian matrix \(\mathbf{J}\) of residual vector \(\mathbf{h}\) contains sensitivity coefficients:
$$ \mathbf{J} = \frac{\partial \mathbf{h}}{\partial \mathbf{d}} \quad (\in \mathbb{R}^{m \times n}), \quad J_{ij} = -\frac{\partial \mathbf{p_i}}{\partial d_j} \cdot \mathbf{n_i} $$
Generalized Inverse Method
Parameter adjustment solves \(\mathbf{S} \Delta \mathbf{d} = \mathbf{h}\) via pseudoinverse:
$$ \Delta \mathbf{d} = (\mathbf{S}^T\mathbf{S})^{-1} \mathbf{S}^T \mathbf{h} $$
Truncated SVD Method
Singular value decomposition \(\mathbf{S} = \mathbf{U\Sigma V}^T\) filters small singular values to stabilize solutions.
Levenberg-Marquardt Algorithm
The trust-region based L-M iteration step is:
$$ \mathbf{d_{k+1}} = \mathbf{d_k} – (\mathbf{G_k} + \mu_k \mathbf{I})^{-1} \mathbf{g_k} $$
where \(\mathbf{g_k} = \mathbf{J_k^T h_k}\), \(\mathbf{G_k} = \mathbf{J_k^T J_k}\), and \(\mu_k\) is dynamically adjusted using trust region ratio \(\eta_k\):
$$ \eta_k = \frac{f(\mathbf{d_k}) – f(\mathbf{d_k} + \Delta \mathbf{d_k})}{q(\mathbf{0}) – q(\Delta \mathbf{d_k})} $$
with \(q\) as quadratic approximation model. The damping parameter updates as:
$$ \mu_{k+1} = \begin{cases}
\mu_k / c & \eta_k \geq 0.75 \\
\mu_k & 0.25 \leq \eta_k < 0.75 \\
c \mu_k & \eta_k < 0.25
\end{cases} \quad (c \in \mathbb{N}) $$
| Parameter | Value |
|---|---|
| Pressure angle (°) | 22.5 |
| Cutter radius (mm) | 92.8943 |
| Radial distance (mm) | 110.7413 |
| Angular position (°) | 50.0484 |
| Roll ratio | 0.3484 |
Numerical Validation
A spiral bevel gear pair with design contact ratio 1.44 was modified to target surface (contact ratio 1.75). Initial deviation sum of squares was \(f_0\) = 0.0845 mm². The identified parameters and residual errors are compared below:
| Parameter | Generalized Inverse | Truncated SVD | L-M Algorithm |
|---|---|---|---|
| Cutter radius (mm) | 90.5924 | 92.1837 | 92.2411 |
| Radial distance (mm) | 111.8262 | 110.2901 | 109.6558 |
| Angular position (°) | 49.8765 | 49.5114 | 49.5837 |
| Roll ratio | 0.3410 | 0.3463 | 0.3483 |
| Algorithm | Max error (μm) | Sum of squared errors (mm²) |
|---|---|---|
| Generalized inverse | 21 | 1.4723×10⁻³ |
| Truncated SVD | 17 | 8.2969×10⁻⁴ |
| L-M with trust region | 2 | 1.4993×10⁻⁵ |
The L-M algorithm reduces surface deviation by 98.2% compared to truncated SVD, demonstrating superior reverse gear capability for precision tooth surface approximation. Contact pattern analysis confirms the reverse gear solution achieves the target contact ratio of 1.74 (0.57% deviation).
Conclusions
We establish a complete mathematical framework for reverse gear parameter identification in spiral bevel gear machining. The trust-region enhanced L-M algorithm demonstrates:
- 98.2% lower residual error versus truncated SVD method
- Precise control of contact ratio (0.57% deviation)
- Stable convergence with physically realizable parameters
This reverse gear methodology enables active modification design and deviation correction for high-performance spiral bevel gears. Future work will extend the algorithm to account for machine kinematics constraints during parameter optimization.