In mechanical transmission systems, helical gears are widely used due to their superior meshing performance, high overlap ratio, and compact structure compared to spur gears. However, traditional design approaches often focus solely on meeting the overall strength requirements of the gear pair, neglecting the significant difference in bending strength between the pinion and the gear. This oversight can lead to premature failure of the weaker gear, resulting in increased maintenance costs and reduced system reliability. To address this issue, we propose a multi-objective optimization framework that minimizes both the difference in maximum root bending stress between the helical gears and the contact stress on the tooth surface. This approach ensures balanced bending strength and enhanced surface load capacity, thereby improving the overall lifespan and performance of helical gear transmissions.
The optimization model incorporates key geometric parameters of helical gears as design variables, including the pinion tooth number, normal module, helix angle, and normal shift coefficients for both gears. Constraints are applied to ensure the gear pair meets strength, overlap ratio, and tooth tip thickness requirements. We employ the Multi-Objective Particle Swarm Optimization (MOPPSO) algorithm to solve this model, leveraging its simplicity, convergence efficiency, and ability to handle real-valued optimization problems. The optimized results are validated through simulation using MASTA software, demonstrating significant improvements in both bending strength equality and contact stress reduction. This study provides a comprehensive reference for the optimal design of helical gear macroscopic geometric parameters, contributing to more reliable and efficient gear systems.
Helical gears are critical components in various industrial applications, such as reducers, automotive transmissions, and heavy machinery. Their inclined teeth allow for smoother and quieter operation compared to spur gears, but they also introduce complexities in stress distribution and load sharing. The geometric parameters of helical gears, such as the helix angle, module, and shift coefficients, directly influence the root bending stress and surface contact stress. Inadequate design can lead to issues like tooth breakage due to excessive bending stress or pitting caused by high contact stress. Therefore, optimizing these parameters is essential to achieve a balanced performance where both bending and contact strengths are maximized, and the difference in strength between the mating gears is minimized.
To formulate the optimization problem, we define two objective functions. The first objective aims to minimize the contact stress on the tooth surface, which is a primary factor affecting the surface durability of helical gears. The contact stress for helical gears can be expressed using the following formula based on the Hertzian contact theory:
$$ \min f_1 = \sigma_H = Z_H Z_E \sqrt{ \frac{2000 T \cos^3 \beta}{m_n^2 z_1^2 b \epsilon_\alpha} \cdot \frac{u + 1}{u} \cdot K_1 } $$
Here, $$ \sigma_H $$ represents the contact stress in MPa, $$ Z_H $$ is the zone factor, $$ Z_E $$ is the elasticity factor in MPa, $$ T $$ is the torque in N·m, $$ \beta $$ is the helix angle in radians, $$ m_n $$ is the normal module in mm, $$ z_1 $$ is the pinion tooth number, $$ b $$ is the face width in mm, $$ \epsilon_\alpha $$ is the transverse contact ratio, $$ u $$ is the gear ratio, and $$ K_1 $$ is the load factor for the pinion. Minimizing this stress helps prevent surface failures such as pitting and wear, which are common in heavily loaded helical gears.
The second objective focuses on minimizing the difference in the maximum root bending stress between the pinion and the gear. This ensures that both gears have nearly equal bending strength, reducing the risk of premature failure in the weaker gear. The bending stress for each gear is calculated as follows:
$$ \min f_2 = \left| \sigma_{F1} – \sigma_{F2} \right| = \left| \frac{2000 T K_1 Y_{F1} Y_{S1} Y_\beta \cos \beta}{m_n^2 z_1 b} – \frac{2000 T K_2 Y_{F2} Y_{S2} Y_\beta \cos \beta}{m_n^2 z_1 b} \right| $$
In this equation, $$ \sigma_{F1} $$ and $$ \sigma_{F2} $$ are the maximum root bending stresses for the pinion and gear, respectively, in MPa. $$ Y_{F1} $$ and $$ Y_{F2} $$ are the form factors, $$ Y_{S1} $$ and $$ Y_{S2} $$ are the stress correction factors, $$ Y_\beta $$ is the helix angle factor, and $$ K_2 $$ is the load factor for the gear. The form factor and stress correction factor depend on the tooth geometry and are given by:
$$ Y_F = \frac{6 h_{Fe} / m_n \cos \alpha_{Fen} }{ (s_{Fn} / m_n)^2 \cos \alpha_n } $$
and
$$ Y_S = (1.2 + 0.13 L) q_s^{1 / (1.21 + 2.3 / L)} $$
where $$ s_{Fn} $$ is the chordal thickness at the critical section in mm, $$ h_{Fe} $$ is the bending moment arm in mm, $$ \alpha_{Fen} $$ is the load angle in radians, $$ \alpha_n $$ is the normal pressure angle in radians, $$ q_s = s_{Fn} / (2 \rho_F) $$ is the root fillet parameter, $$ \rho_F $$ is the radius of curvature at the critical point in mm, and $$ L = s_{Fn} / h_{Fe} $$. The helix angle factor is defined as:
$$ Y_\beta = 1 – \frac{\beta}{120^\circ} $$
These formulas capture the intricate relationships between the geometric parameters and the stress states in helical gears, enabling a precise optimization.
The design variables for the optimization are selected based on their significant impact on the objective functions. We choose the pinion tooth number $$ z_1 $$, normal module $$ m_n $$, helix angle $$ \beta $$, and normal shift coefficients for the pinion and gear, denoted as $$ \chi_{n1} $$ and $$ \chi_{n2} $$, respectively. Thus, the design vector is:
$$ X = [ z_1 \quad m_n \quad \beta \quad \chi_{n1} \quad \chi_{n2} ]^T = [x_1 \quad x_2 \quad x_3 \quad x_4 \quad x_5]^T $$
To ensure the practicality and feasibility of the optimized helical gears, several constraints are imposed. First, the contact fatigue strength must be within allowable limits for both gears:
$$ g_1 = \sigma_{H1} – [\sigma_{HP}]_1 \leq 0 $$
$$ g_2 = \sigma_{H2} – [\sigma_{HP}]_2 \leq 0 $$
where $$ [\sigma_{HP}]_1 $$ and $$ [\sigma_{HP}]_2 $$ are the allowable contact stresses for the pinion and gear, respectively. Similarly, the bending fatigue strength constraints are:
$$ g_3 = \sigma_{F1} – [\sigma_{FP}]_1 \leq 0 $$
$$ g_4 = \sigma_{F2} – [\sigma_{FP}]_2 \leq 0 $$
with $$ [\sigma_{FP}]_1 $$ and $$ [\sigma_{FP}]_2 $$ being the allowable bending stresses. The center distance must match the design value with a tolerance of 0.01 mm:
$$ g_5 = | a’ – a | – 0.01 \leq 0 $$
where $$ a’ $$ is the actual center distance and $$ a $$ is the design center distance. The total contact ratio $$ \epsilon_\gamma $$ should be at least 2 to ensure smooth meshing:
$$ g_6 = 2 – \epsilon_\gamma \leq 0 $$
Additionally, the tooth tip thickness must be sufficient to prevent sharp edges and ensure durability:
$$ g_7 = 0.4 m_n – d_{a1} \left( \frac{\pi + 4 \chi_{n1} \tan \alpha_n}{2 z_1} + \text{inv} \alpha_t – \text{inv} \alpha_{a1} \right) \leq 0 $$
$$ g_8 = 0.4 m_n – d_{a2} \left( \frac{\pi + 4 \chi_{n2} \tan \alpha_n}{2 z_2} + \text{inv} \alpha_t – \text{inv} \alpha_{a2} \right) \leq 0 $$
Here, $$ d_{a1} $$ and $$ d_{a2} $$ are the tip diameters, $$ \alpha_t $$ is the transverse pressure angle, $$ z_2 $$ is the gear tooth number, and $$ \alpha_{a1} $$ and $$ \alpha_{a2} $$ are the tip pressure angles for the pinion and gear, respectively. The shift coefficients are constrained to avoid undercutting and excessive thinning:
$$ g_9 = \frac{h_a – z_1 \sin^2 \alpha_t / (2 \cos \beta) – \chi_{n1} \leq 0 $$
$$ g_{10} = \frac{h_a – z_2 \sin^2 \alpha_t / (2 \cos \beta) – \chi_{n2} \leq 0 $$
where $$ h_a $$ is the addendum coefficient. Finally, boundary constraints are applied to the design variables:
$$ 17 \leq z_1 \leq 30 $$
$$ 2 \leq m_n \leq 4 $$
$$ 8^\circ \leq \beta \leq 20^\circ $$
$$ 0.2 \leq \chi_{n1} \leq 0.5 $$
$$ -0.5 \leq \chi_{n2} \leq 0.5 $$
The MOPPSO algorithm is employed to solve this multi-objective optimization problem due to its efficiency in handling non-linear constraints and generating Pareto-optimal solutions. The algorithm initializes a population of particles, each representing a potential solution in the design space. The velocity and position of each particle are updated iteratively based on its personal best and the global best from the archive. The update equations are:
$$ V_{t+1} = w V_t + c_1 r_1 (p_{\text{best}} – x_t) + c_2 r_2 (g_{\text{best}} – x_t) $$
$$ x_{t+1} = x_t + V_{t+1} $$
where $$ V_t $$ and $$ x_t $$ are the velocity and position at iteration $$ t $$, $$ w $$ is the inertia weight, $$ c_1 $$ and $$ c_2 $$ are learning factors, $$ r_1 $$ and $$ r_2 $$ are random numbers in [0,1], $$ p_{\text{best}} $$ is the personal best position, and $$ g_{\text{best}} $$ is the global best position selected from the external archive based on crowding distance and Pareto dominance. The archive maintains non-dominated solutions, and its size is controlled to ensure diversity.
To demonstrate the application of this optimization framework, we consider a case study of a single-stage reducer with helical gears. The initial design parameters are as follows: pinion tooth number $$ z_1 = 18 $$, gear tooth number $$ z_2 = 67 $$, normal module $$ m_n = 4 $$ mm, helix angle $$ \beta = 17.27^\circ $$, pinion normal shift coefficient $$ \chi_{n1} = 0.2 $$, gear normal shift coefficient $$ \chi_{n2} = 0.3115 $$, center distance $$ a = 180 $$ mm, face width $$ b = 60 $$ mm. The operating conditions include an input power of 45 kW, input speed of 1500 rpm, and a service life of 10 years. The gear material is 20CrMnTi with a density of 7800 kg/m³. The allowable stresses are set to $$ [\sigma_{FP}]_1 = [\sigma_{FP}]_2 = 450 $$ MPa for bending and $$ [\sigma_{HP}]_1 = [\sigma_{HP}]_2 = 1300 $$ MPa for contact.
Using the MOPPSO algorithm with a population size of 200, archive size of 30, inertia weight of 0.8, learning factors $$ c_1 = c_2 = 2 $$, and 800 iterations, we obtain a set of Pareto-optimal solutions. The optimization results show significant improvements in both objectives. For instance, one optimal solution yields $$ z_1 = 21 $$, $$ m_n = 3.5 $$ mm, $$ \beta = 20^\circ $$, $$ \chi_{n1} = 0.213 $$, $$ \chi_{n2} = 0.145 $$, with objective values $$ f_1 = 687.209 $$ MPa and $$ f_2 = 2.7393 $$ MPa. Compared to the initial design, where $$ f_1 = 720.9034 $$ MPa and $$ f_2 = 21.3577 $$ MPa, the contact stress is reduced by 33.6944 MPa, and the bending stress difference is minimized by 18.6184 MPa.
The following table summarizes the comparison between the initial and optimized geometric parameters for the helical gears:
| Parameter | Initial Design | Optimized Design |
|---|---|---|
| Pinion tooth number $$ z_1 $$ | 18 | 21 |
| Gear tooth number $$ z_2 $$ | 67 | 75 |
| Normal module $$ m_n $$ (mm) | 4 | 3.5 |
| Helix angle $$ \beta $$ | 17.27° | 20° |
| Pinion shift coefficient $$ \chi_{n1} $$ | 0.2 | 0.213 |
| Gear shift coefficient $$ \chi_{n2} $$ | 0.3115 | 0.145 |
| Contact stress $$ f_1 $$ (MPa) | 720.9034 | 687.209 |
| Bending stress difference $$ f_2 $$ (MPa) | 21.3577 | 2.7393 |
To validate the optimization results, we perform static simulation using MASTA software. Three-dimensional models of the initial and optimized helical gear pairs are created, and a torque of 286.47 N·m is applied to the pinion while constraining the gear. The mesh is refined with 5 nodes along the tooth height and 9 nodes along the face width to ensure accuracy. The simulation results confirm the theoretical calculations: the initial design has a maximum contact stress of 755.8734 MPa and bending stresses of 91.1735 MPa for the pinion and 117.3698 MPa for the gear, yielding a difference of 26.1963 MPa. In contrast, the optimized design shows a maximum contact stress of 722.8305 MPa and bending stresses of 103.6059 MPa for the pinion and 108.0811 MPa for the gear, with a difference of 4.4752 MPa. The errors between simulation and theory are within acceptable engineering limits, verifying the effectiveness of the optimization.
In conclusion, the multi-objective optimization of helical gear geometric parameters using the MOPPSO algorithm successfully achieves a balance between minimizing contact stress and equalizing bending strength. The optimized helical gears exhibit improved performance, with reduced risk of premature failure and enhanced load capacity. This approach provides a robust framework for designing helical gears in various applications, ensuring reliability and efficiency. Future work could explore the inclusion of dynamic effects and manufacturing constraints to further refine the optimization process.