In engineering applications, helical gears are critical components in transmission systems due to their high efficiency, stability, and durability. However, wear during meshing can lead to noise, vibration, and eventual failure, posing significant challenges to system reliability. Traditional probabilistic reliability methods often require extensive data to define accurate probability distributions, which may not be available in practice. To address this, we propose a non-probabilistic interval model to handle uncertainties in geometric parameters, material properties, and loading conditions for helical gears. This approach enables a robust wear reliability optimization design, ensuring safety and performance under data-scarce scenarios.
The geometric modeling of helical gears involves key parameters such as the number of teeth, normal module, helix angle, face width, pressure angle, addendum coefficient, and clearance coefficient. We developed a parametric 3D model using UG software, which allows for precise control over gear geometry. The assembly of helical gear pairs is crucial for accurate meshing simulation, as misalignment can lead to interference and incorrect wear predictions. Our model ensures proper engagement by defining both driving and driven gears within the same coordinate system, facilitating realistic finite element analysis.
Finite element analysis (FEA) is employed to simulate the meshing process and evaluate contact stresses and wear in helical gears. Using ABAQUS, we model the gear pair with refined meshing at the contact surfaces to capture high-stress regions accurately. The material properties for the gears, made of 20CrMnTi, are defined with elastic modulus, Poisson’s ratio, and density as interval variables to account for uncertainties. The contact between teeth is governed by a penalty function with a friction coefficient of 0.1. Load application involves three steps: initial contact establishment, torque application on the driven gear, and rotational motion of the driving gear to simulate operational conditions. The maximum contact stress from FEA is compared with Hertz theory to validate the model, showing a deviation of less than 3%, which confirms the reliability of our approach.
The wear analysis integrates the Archard wear model with adaptive meshing techniques in ABAQUS. The wear depth is computed based on contact pressure and sliding distance, updated through a user-defined subroutine. For helical gears, wear distribution varies along the tooth width due to the helix angle, with higher wear observed at the root and tip regions. Our results indicate that the maximum wear depth occurs at the root, and it increases from the front to the rear of the tooth face. This non-uniform wear pattern underscores the importance of considering geometric and operational uncertainties in design.
To handle uncertainties, we adopt a non-probabilistic interval model, where variables are defined by their lower and upper bounds instead of probability distributions. The key uncertain parameters include geometric dimensions (normal module, helix angle, face width, number of teeth), material properties (elastic modulus, Poisson’s ratio), and loading conditions (rotational speed, torque). The functional requirements, such as allowable contact stress and wear depth, are also treated as interval variables. The reliability of helical gears is assessed using non-probabilistic reliability indices for contact strength and wear, derived from the interval analysis of performance functions.
The contact strength function is defined as: $$g_1(\mathbf{x}) = \delta_{HS} – \delta_{\text{max}}(m_n, \beta, B, E, \nu, n, T)$$ where $\delta_{HS}$ is the allowable contact stress, and $\delta_{\text{max}}$ is the maximum contact stress from FEA. The wear function is: $$g_2(\mathbf{x}) = W_s – W_h(m_n, \beta, B, E, \nu, n, T)$$ where $W_s$ is the allowable wear depth, and $W_h$ is the computed wear depth. The non-probabilistic reliability index $\eta$ for each function is calculated as: $$\eta = \frac{g^c}{g^r}$$ where $g^c$ is the midpoint and $g^r$ is the radius of the interval response. A reliability index greater than 1 indicates a safe design.
We use Kriging surrogate models to approximate the nonlinear relationships between input variables and output responses, as explicit functions are not available. The Kriging models are trained on sample points from design of experiments (DOE), and their accuracy is verified with R² values above 0.95. The interval bounds of the performance functions are determined using sequential quadratic programming (SQP), and the reliability indices are computed accordingly. Our analysis shows that both contact strength and wear reliability indices exceed 1, confirming the safety of the helical gear design under uncertainties.
For optimization, we formulate a reliability-based design optimization (RBDO) problem with the objective of minimizing the total volume of the helical gear pair. The design variables are the normal module, face width, number of teeth on the driving gear, and helix angle. The constraints ensure that the non-probabilistic reliability indices for contact strength and wear are above specified thresholds. The optimization model is: $$\begin{aligned} \text{find} \quad & \mathbf{X} \\ \text{min} \quad & f(\mathbf{X}) = V(x_1, x_2, x_3, x_4) \\ \text{s.t.} \quad & \eta_1 \geq 1 \\ & \eta_2 \geq 1 \\ & \mathbf{X}^L \leq \mathbf{X} \leq \mathbf{X}^U \\ & \mathbf{Y}^L \leq \mathbf{Y} \leq \mathbf{Y}^U \end{aligned}$$ where $\mathbf{X}$ are design variables, $\mathbf{Y}$ are interval variables, and $V$ is the volume. The inner loop uses SQP to compute reliability indices, while the outer loop employs a genetic algorithm to optimize the design variables.
| Category | Variable | Mean | Lower Bound | Upper Bound |
|---|---|---|---|---|
| Geometry | Normal module $m_n$ (mm) | 4 | 3.98 | 4.02 |
| Geometry | Helix angle $\beta$ (°) | 13 | 12.935 | 13.065 |
| Geometry | Face width $B$ (mm) | 40 | 39.8 | 40.2 |
| Material | Elastic modulus $E$ (GPa) | 210 | 199.5 | 220.5 |
| Material | Poisson’s ratio $\nu$ | 0.3 | 0.285 | 0.315 |
| Loading | Rotational speed $n$ (r/min) | 140 | 135 | 145 |
| Loading | Torque $T$ (N·m) | 300 | 285 | 315 |
The optimization results demonstrate a significant reduction in gear pair volume while maintaining reliability. After rounding the design variables to practical values, the volume decreases by 25.11%, and the maximum wear depth is reduced by 16.09%. The contact stress increases slightly but remains within allowable limits. This highlights the effectiveness of our non-probabilistic approach in achieving lightweight and reliable helical gear designs.
| Variable | Mean | Lower Bound | Upper Bound |
|---|---|---|---|
| Allowable contact stress $\delta_{HS}$ (MPa) | 1992.375 | 1892.76 | 2092 |
| Allowable wear depth $W_s$ (μm) | 25 | 23.75 | 26.25 |
In conclusion, our study presents a comprehensive framework for wear reliability optimization of helical gears using non-probabilistic models. By addressing uncertainties through interval analysis and surrogate modeling, we ensure robust performance under data-scarce conditions. The optimized helical gears exhibit improved wear resistance and reduced volume, validating the proposed method. Future work could explore dynamic loading conditions and thermal effects on wear in helical gears.
| Design Variable | Normal Module $m_n$ (mm) | Face Width $B$ (mm) | Number of Teeth $z_1$ | Helix Angle $\beta$ (°) | $\eta_1$ | $\eta_2$ | Gear Pair Volume $V$ (10^6 mm³) |
|---|---|---|---|---|---|---|---|
| Initial Value | 4 | 40 | 26 | 13 | 3.7286 | 1.2332 | 3.178 |
| Optimized Value | 3.4188 | 39.01 | 25.7 | 13.415 | 2.144 | 2.8446 | 2.2195 |
| Rounded Value | 3.5 | 39 | 26 | 13.4 | – | – | 2.38 |
The non-probabilistic reliability analysis for helical gears involves computing the interval responses of the performance functions. For contact strength, the function $g_1(\mathbf{x})$ has a midpoint $g_1^c = 1281.1$ MPa and radius $g_1^r = 343.6$ MPa, giving a reliability index $\eta_1 = 3.7286$. For wear, $g_2(\mathbf{x})$ has a midpoint $g_2^c = 12.234$ μm and radius $g_2^r = 9.922$ μm, resulting in $\eta_2 = 1.2332$. Both indices exceed 1, confirming the reliability of the helical gears. The Kriging models effectively capture the nonlinear behavior, with R² values of 0.98061 for contact stress and 0.95374 for wear, ensuring accurate predictions.
In the optimization process, the design variables are iteratively adjusted to minimize volume while satisfying reliability constraints. The genetic algorithm explores the design space globally, and SQP ensures precise computation of reliability indices. The final optimized design reduces the gear pair volume from 3.178×10^6 mm³ to 2.38×10^6 mm³, a 25.11% reduction, and the wear depth decreases from 6.306 μm to 5.2912 μm, a 16.09% improvement. These results demonstrate the practical benefits of integrating non-probabilistic reliability into the design of helical gears, leading to more efficient and durable transmission systems.
Further analysis of helical gears under varying operational conditions could enhance the model’s applicability. For instance, incorporating thermal effects on material properties and lubrication regimes would provide a more comprehensive wear assessment. Additionally, extending the non-probabilistic framework to multi-objective optimization could balance trade-offs between weight, cost, and reliability in helical gears. The proposed method serves as a foundation for advanced reliability-based design in mechanical engineering, particularly for components like helical gears where uncertainty is prevalent.