In the context of coal mining, thin seam extraction has gained prominence due to the depletion of medium-thick coal reserves. As a foundational energy source, coal necessitates efficient mining methods, particularly in low-height seams where operational conditions are challenging. Thin seam shearers must achieve high power density to maintain productivity, which drives the need for compact and robust transmission systems. The planetary gear transmission, commonly used in heavy machinery, offers advantages such as high load capacity, compact structure, and efficiency. However, in thin seam shearers, the radial and axial dimensions of the planetary gear system are constrained by the drum diameter and cutting depth, limiting the power density. This study focuses on optimizing the planetary transmission gear in the rocker arm of a thin seam shearer to minimize volume while ensuring reliability under complex loads. By establishing a mathematical model based on optimization theory, we apply nonlinear programming and genetic algorithms in MATLAB to solve the problem. The results demonstrate significant volume reduction, enhancing the shearer’s adaptability to harsh mining conditions.
The planetary gear system in the rocker arm, specifically an NGW-type configuration, consists of a sun gear, planet gears, an internal ring gear, and a carrier. This system transmits high torque from the motor to the drum through a series of gears, including idlers and a final spur gear. The NGW-type planetary reducer, with four planet gears, is designed to handle large power and complex loads while maintaining compactness. The sun gear employs a floating structure to ensure even load distribution and self-centering, while the planet gears rotate around their axes and drive the carrier. The carrier then outputs torque to the drum via a spline connection. Key materials include high-strength alloy steel for the sun and planet gears, and nitrided steel for the internal ring gear, to enhance fatigue resistance. The traditional design, based on empirical data, often leads to oversized components with excessive safety margins. This approach is inefficient for high-power-density thin seam shearers, necessitating modern optimization techniques.
To address these challenges, we formulate an optimization model with the objective of minimizing the total volume of the planetary transmission gears. The design variables include the number of teeth on the sun gear, internal ring gear, and planet gears, as well as the gear width and module. Let the design vector be defined as \( \mathbf{x} = [x_1, x_2, x_3, x_4, x_5]^T = [Z_A, b, m_n, Z_B, Z_C]^T \), where \( Z_A \) is the sun gear teeth, \( b \) is the face width, \( m_n \) is the module, \( Z_B \) is the internal ring gear teeth, and \( Z_C \) is the planet gear teeth. The objective function, representing the total volume \( V \), is derived from the sum of the volumes of the sun gear, internal ring gear, and the planet gears. For an NGW-type planetary system with \( C_S = 4 \) planet gears, the volume is approximated as:
$$ V = \frac{\pi}{4} b \left( Z_A^2 m_n^2 + Z_B^2 m_n^2 + C_S Z_C^2 m_n^2 \right) $$
Simplifying, we obtain the target function:
$$ f(\mathbf{x}) = \frac{\pi}{4} x_2 x_3^2 \left( x_1^2 + x_4^2 + 4 x_5^2 \right) $$
The constraints are categorized into boundary conditions and planetary gear matching conditions. Boundary constraints ensure the gears meet design standards and strength requirements. First, to avoid undercutting in the sun gear during generation cutting, with a pressure angle of \( 20^\circ \) and addendum coefficient of 1, the constraint is:
$$ g_1(\mathbf{x}) = x_1 – 17 \geq 0 $$
Second, the face width coefficient \( \psi_d = b / (m_n Z_A) \) must lie between 0.7 and 1.5 to balance load distribution and compactness:
$$ g_2(\mathbf{x}) = 1.5 – \frac{x_2}{x_1 x_3} \geq 0 $$
$$ g_3(\mathbf{x}) = \frac{x_2}{x_1 x_3} – 0.7 \geq 0 $$
Third, the module should be at least 2 mm, following standard series:
$$ g_4(\mathbf{x}) = x_3 – 2 \geq 0 $$
Fourth, the face width must be at least 150 mm for structural integrity:
$$ g_5(\mathbf{x}) = x_2 – 150 \geq 0 $$
Fifth, the transmission ratio \( i \) should be at least 5.2 to meet speed reduction requirements:
$$ g_6(\mathbf{x}) = \left(1 + \frac{x_4}{x_1}\right) – 5.2 \geq 0 $$
Sixth, the contact fatigue strength constraint for the sun gear, based on the Hertzian contact stress formula, is derived as:
$$ g_7(\mathbf{x}) = 13758.72 – x_1 x_2 x_3^2 (x_1 + x_5) \geq 0 $$
where parameters include input torque \( T_A = 15.92 \times 10^3 \, \text{Nm} \), application factor \( K_A = 1.80 \), load distribution factor \( K_\beta = 1.967 \), and contact fatigue limit \( \sigma_{H \lim} = 1350 \, \text{MPa} \). Seventh, the bending fatigue strength constraint, using the Lewis formula, is:
$$ g_8(\mathbf{x}) = 4.63 \times 10^5 – x_1 x_2 x_3^5 \geq 0 $$
with bending fatigue limit \( \sigma_{F \lim} = 420 \, \text{MPa} \) and form factor \( Y_F = 3.10 \).
The planetary gear matching constraints ensure proper assembly and operation. The concentricity condition requires the center distances between meshing gears to be equal:
$$ g_9(\mathbf{x}) = x_1 + x_5 – x_4 = 0 $$
The assembly condition mandates that the sum of sun and ring gear teeth be an integer multiple of the number of planet gears:
$$ g_{10}(\mathbf{x}) = \text{rem}(x_1 + x_4, C_S) = 0 $$
The adjacency condition prevents interference between adjacent planet gears by ensuring the sum of tip circle radii is less than the center distance:
$$ g_{11}(\mathbf{x}) = (x_1 + x_5) – 2 \left( x_5 + 2 \right) > 0 $$
This results in a complex optimization model with five variables, seven boundary constraints, and three equality/inequality constraints.
For optimization, we employ two methods: nonlinear programming using the fmincon function in MATLAB and a genetic algorithm (GA). The fmincon function is suitable for constrained nonlinear problems and uses a gradient-based approach. We set the initial point as the traditional design: \( \mathbf{x}_0 = [17, 175, 9, 83, 33]^T \). The function call includes the objective function, constraints, and bounds. The algorithm iteratively searches for a local minimum, but the solution may depend on the initial point. In contrast, the genetic algorithm is a global optimization technique inspired by natural selection. It operates on a population of solutions, applying selection, crossover, and mutation to evolve toward better solutions. We configure the GA with a population size of 500, crossover probability of 0.8, mutation probability of 0.1, and a maximum of 500 generations. This stochastic approach is less likely to be trapped in local minima and is robust for complex problems.
The optimization process involves solving the model with both methods, then rounding the continuous variables to integers as per gear design standards. For example, if the optimal solution is \( \mathbf{x}^* = [X_1^*, X_2^*, X_3^*, X_4^*, X_5^*] \), we consider integer candidates around these values and evaluate the objective function to find the best integer solution. This post-processing ensures practical feasibility.
The results from both optimization methods are compared with the traditional design in the table below. The table summarizes key parameters, including gear teeth numbers, face width, module, transmission ratio, face width coefficient, total volume, and volume reduction percentage.
| Parameter | Traditional Design | Nonlinear Programming (fmincon) | Genetic Algorithm (GA) |
|---|---|---|---|
| Sun Gear Teeth \( Z_A \) | 17 | 19 | 17 |
| Planet Gear Teeth \( Z_C \) | 33 | 31 | 31 |
| Internal Ring Gear Teeth \( Z_B \) | 83 | 81 | 79 |
| Face Width \( b \) (mm) | 175 | 171 | 172 |
| Module \( m_n \) (mm) | 9 | 8 | 8 |
| Transmission Ratio \( i \) | 5.88 | 5.26 | 5.65 |
| Face Width Coefficient \( \psi_d \) | 1.14 | 1.12 | 1.26 |
| Total Volume \( V \) (mm³) | 1.184 × 10⁸ | 9.254 × 10⁷ | 8.969 × 10⁷ |
| Volume Reduction | 0% | 21.8% | 24.3% |
As shown, both optimization methods reduce the volume significantly—by 21.8% with nonlinear programming and 24.3% with genetic algorithm—compared to the traditional design. The genetic algorithm achieves a slightly better reduction due to its global search capability, which avoids local optima. The adjustments in gear teeth and module maintain the transmission ratio within acceptable limits while improving compactness. The face width coefficient remains in the optimal range, ensuring load distribution without excessive unevenness. The planet gears’ configuration satisfies all matching constraints, confirming the feasibility of the optimized designs.
In conclusion, the optimization of planetary transmission gears in thin seam shearers is crucial for achieving high power density in constrained environments. By formulating a comprehensive mathematical model and applying advanced optimization techniques, we demonstrate substantial improvements in volume reduction. The genetic algorithm outperforms nonlinear programming in finding global optima, highlighting its value for complex engineering problems. This approach not only enhances the efficiency of thin seam shearers but also provides a framework for optimizing other planetary gear systems in heavy machinery. Future work could explore multi-objective optimization, including factors like noise reduction and thermal performance, to further advance planetary gear design.
The integration of these optimization methods into the design process enables more precise and reliable planetary gear systems, contributing to sustainable mining practices. The use of planet gears in such applications underscores their importance in transmitting high torques efficiently. As mining conditions become more challenging, continued refinement of these techniques will be essential for meeting operational demands.