Spur gears are widely used in mechanical transmission systems due to their high efficiency, reliability, and compact structure. However, conventional design methods for spur gear transmissions often involve complex calculations, reliance on engineering experience, and repetitive verification processes, leading to suboptimal results. To address these limitations, this study focuses on optimizing a two-stage spur gear transmission system with the objective of minimizing the overall volume. An improved particle swarm optimization (PSO) algorithm is employed to enhance the design efficiency and performance. The optimization model incorporates geometric constraints, contact fatigue strength, and bending fatigue strength to ensure the reliability of the spur gear system. Results demonstrate that the proposed approach significantly reduces the volume of the spur gear transmission compared to conventional methods, highlighting its effectiveness in practical applications.
Spur gears are essential components in various industrial applications, including automotive and automation systems, owing to their ability to transmit power smoothly and efficiently. The design of spur gear transmissions typically involves multiple factors such as material selection, load conditions, gear dimensions, and strength requirements. Traditional approaches rely on iterative calculations and empirical data, which can be time-consuming and may not yield optimal solutions. In this context, intelligent optimization algorithms offer a promising alternative. The particle swarm algorithm, known for its simplicity and efficiency, is particularly suitable for such problems. This paper introduces an improved PSO algorithm that incorporates a linear decreasing inertia weight mechanism and simulated annealing principles to optimize the design of two-stage spur gear systems. The primary goal is to minimize the volume of the spur gear transmission while adhering to all design constraints, thereby reducing material usage and costs.
The mathematical model for the two-stage spur gear transmission is established to formulate the optimization problem. The system consists of a high-speed stage (gear 1 and gear 2) and a low-speed stage (gear 3 and gear 4), as illustrated in the structure diagram. The objective function aims to minimize the total volume of the spur gears, which is critical for achieving compactness and cost-effectiveness. The volume \( v \) is expressed as a function of the gear parameters:
$$ v = \frac{\pi z_1^3 m_1^3}{4} (1 + i_1^2) \phi_{d1} + \frac{\pi z_3^3 m_2^3}{4} (1 + i_2^2) \phi_{d2} $$
where \( z_1 \) and \( z_3 \) are the number of teeth for the high-speed and low-speed stage pinions, respectively; \( m_1 \) and \( m_2 \) are the modules; \( i_1 \) and \( i_2 \) are the transmission ratios; and \( \phi_{d1} \) and \( \phi_{d2} \) are the face width coefficients. The design variables are selected as follows:
$$ \mathbf{x} = [z_1, z_3, m_1, m_2, i_1]^T = [x_1, x_2, x_3, x_4, x_5]^T $$
Constraints are imposed to ensure the spur gears meet geometric and strength requirements. Geometric constraints include minimum module and tooth number limits to prevent issues like tooth breakage and interference. For instance, the modules must satisfy \( m_1 \geq 2 \) and \( m_2 \geq 2 \), while the pinion teeth must be at least 17 to avoid undercutting and ensure stability. Additionally, to prevent interference between the high-speed gears and the output shaft, the following condition is applied:
$$ \frac{(z_3 m_2 + z_3 m_2 i_2) – z_1 m_1 i_1}{2} \geq Q $$
where \( Q \) represents the minimum distance required. The contact fatigue strength constraints for both stages are given by:
$$ \sigma_{H1} = Z_E Z_H \sqrt{\frac{2 K_1 T_1}{\phi_{d1} m_1^3 z_1^3} \cdot \frac{u_1 + 1}{u_1}} \leq [\sigma_H] $$
$$ \sigma_{H2} = Z_E Z_H \sqrt{\frac{2 K_2 T_2}{\phi_{d2} m_2^3 z_3^3} \cdot \frac{u_2 + 1}{u_2}} \leq [\sigma_H] $$
Here, \( \sigma_H \) is the contact stress, \( [\sigma_H] \) is the allowable contact stress, \( Z_E \) and \( Z_H \) are the elastic and zone coefficients, \( K_1 \) and \( K_2 \) are load factors, \( T_1 \) and \( T_2 \) are torques, and \( u_1 \) and \( u_2 \) are gear ratios. The bending fatigue strength constraints for all spur gears are defined as:
$$ \sigma_{F1} = \frac{2 K_1 T_1 Y_{Fa1} Y_{Sa1}}{\phi_{d1} m_1^3 z_1^2} \leq [\sigma_{F1}] $$
$$ \sigma_{F2} = \frac{2 K_1 T_1 Y_{Fa2} Y_{Sa2}}{\phi_{d1} m_1^3 z_1^2} \leq [\sigma_{F2}] $$
$$ \sigma_{F3} = \frac{2 K_2 T_2 Y_{Fa3} Y_{Sa3}}{\phi_{d2} m_2^3 z_3^2} \leq [\sigma_{F3}] $$
$$ \sigma_{F4} = \frac{2 K_2 T_2 Y_{Fa4} Y_{Sa4}}{\phi_{d2} m_2^3 z_3^2} \leq [\sigma_{F4}] $$
where \( \sigma_F \) is the bending stress, \( [\sigma_F] \) is the allowable bending stress, and \( Y_{Fa} \) and \( Y_{Sa} \) are the form factor and stress correction factor for each spur gear.
The improved particle swarm algorithm is designed to handle this constrained optimization problem efficiently. The basic PSO algorithm simulates the social behavior of birds, where each particle represents a potential solution. The position and velocity of the \( i \)-th particle in a \( D \)-dimensional space are updated as follows:
$$ \mathbf{x}_i(k+1) = \mathbf{x}_i(k) + \mathbf{v}_i(k+1) $$
$$ \mathbf{v}_i(k+1) = w \mathbf{v}_i(k) + c_1 r_1 [\mathbf{p}_i(k) – \mathbf{x}_i(k)] + c_2 r_2 [\mathbf{p}_g(k) – \mathbf{x}_i(k)] $$
Here, \( w \) is the inertia weight, \( c_1 \) and \( c_2 \) are learning factors, \( r_1 \) and \( r_2 \) are random numbers between 0 and 1, \( \mathbf{p}_i \) is the personal best position, and \( \mathbf{p}_g \) is the global best position. To enhance performance, a linear decreasing inertia weight strategy is adopted:
$$ w = w_{\text{max}} – \frac{k}{T} (w_{\text{max}} – w_{\text{min}}) $$
where \( w_{\text{max}} \) and \( w_{\text{min}} \) are the maximum and minimum inertia weights, \( k \) is the current iteration, and \( T \) is the maximum number of iterations. Additionally, simulated annealing is incorporated to avoid local optima by using the Metropolis criterion:
$$ p = \begin{cases}
1 & \text{if } f(\mathbf{x}_i’) < f(\mathbf{x}_i) \\
\exp\left(-\frac{f(\mathbf{x}_i’) – f(\mathbf{x}_i)}{a T}\right) & \text{otherwise}
\end{cases} $$
where \( f(\mathbf{x}_i’) \) and \( f(\mathbf{x}_i) \) are the fitness values of the new and current solutions, respectively, and \( a \) is the temperature decay coefficient. The steps of the improved PSO algorithm are as follows: initialize particle positions and velocities; evaluate fitness; update personal and global bests using simulated annealing; adjust inertia weight; update velocities and positions; handle boundaries and constraints; and repeat until convergence.
An optimization example is presented to validate the approach. The two-stage spur gear transmission has an input power of 10 kW, a total transmission ratio of 15, and an input speed of 1450 rpm. The design variables are bounded as: \( 17 \leq x_1 \leq 35 \), \( 17 \leq x_2 \leq 35 \), \( 2 \leq x_3 \leq 5 \), \( 2 \leq x_4 \leq 5 \), and \( 3.7 \leq x_5 \leq 4.5 \). Other parameters, such as \( Z_H = 2.5 \), \( Z_E = 189.8 \sqrt{\text{MPa}} \), \( \phi_{d1} = \phi_{d2} = 0.8 \), and \( Q = 50 \), are determined from standard tables. The improved PSO parameters are set as: population size \( N = 50 \), iterations \( T = 500 \), \( c_1 = c_2 = 2 \), \( w_{\text{max}} = 0.9 \), \( w_{\text{min}} = 0.4 \), and \( a = 0.9 \). Constraints are handled using a penalty function method, where infeasible solutions are assigned high fitness values.
The optimization results are compared between the improved PSO and standard PSO algorithms. The average convergence curves over 10 independent runs show that the improved PSO achieves better optimization accuracy. After rounding the optimized parameters to practical values, the results are summarized in the following table:
| Parameter | Improved PSO | Conventional Method |
|---|---|---|
| High-speed pinion teeth \( z_1 \) | 23 | 21 |
| High-speed gear teeth \( z_2 \) | 101 | 84 |
| Low-speed pinion teeth \( z_3 \) | 27 | 24 |
| Low-speed gear teeth \( z_4 \) | 92 | 90 |
| High-speed module \( m_1 \) | 2.5 | 3 |
| Low-speed module \( m_2 \) | 3.5 | 4 |
| Objective function value (mm³) | 9.1095 × 10⁶ | 1.1044 × 10⁷ |
The table clearly demonstrates that the improved PSO reduces the volume of the spur gear transmission by approximately 17.5% compared to the conventional method, while satisfying all constraints. This significant reduction highlights the effectiveness of the algorithm in optimizing spur gear systems. The use of spur gears in this context ensures efficient power transmission, and the optimization process enhances their design for compactness and cost savings.
In conclusion, the improved particle swarm algorithm offers a robust approach for optimizing two-stage spur gear transmissions. By integrating linear decreasing inertia weight and simulated annealing, the algorithm effectively minimizes the volume of spur gears under various constraints. The optimization results confirm that the proposed method outperforms conventional design techniques, leading to more efficient and economical spur gear systems. Future work could explore multi-objective optimization for spur gears, considering factors like noise reduction and thermal performance, to further advance their application in industrial settings.