Internal gear pumps are widely used in hydraulic systems due to their superior performance in reducing trapped oil, noise, and flow pulsation compared to external gear pumps. As an internal gear manufacturer, optimizing the tooth profile parameters is crucial for enhancing pump efficiency and minimizing flow pulsation. This article explores the use of a hybrid genetic algorithm to optimize parameters such as pressure angle, modification coefficient, and addendum coefficient for internal gears, aiming to achieve the lowest possible flow pulsation rate. The methodology integrates mathematical modeling, algorithm improvements, and simulation results to demonstrate significant reductions in pulsation.
The flow pulsation in internal gear pumps arises from the periodic variation in the meshing point radius during operation, leading to discontinuous fluid discharge. This phenomenon is quantified by the flow pulsation rate, defined as the ratio of the difference between maximum and minimum instantaneous flow to the average flow. For a PGH-type internal gear pump, the instantaneous flow rate can be expressed as:
$$Q = \frac{B \omega_1}{2} \left[ R_1’^2 + (h_1 + h_2)^2 – (i h_2 – h_1)^2 – m^2 \right]$$
where \( B \) is the tooth width, \( \omega_1 \) is the angular velocity of the internal gear, \( R_1′ \) is the pitch circle radius, \( h_1 \) and \( h_2 \) are the addendum heights of the ring gear and internal gear, respectively, \( i \) is the transmission ratio, and \( m \) is the distance between the meshing point and the node. The maximum instantaneous flow occurs when the meshing point coincides with the node (\( m = 0 \)):
$$Q_{\text{max}} = \frac{B \omega_1}{2} \left[ R_1’^2 + (h_1 + h_2)^2 – (i h_2 – h_1)^2 \right]$$
The minimum instantaneous flow occurs at the extreme meshing position (\( m = \pm t_j \)), where \( t_j \) is the base pitch:
$$Q_{\text{min}} = \frac{B \omega_1}{2} \left[ R_1’^2 + (h_1 + h_2)^2 – (i h_2 – h_1)^2 – t_j^2 \right]$$
Thus, the flow pulsation rate \( \sigma \) is derived as:
$$\sigma = \frac{Q_{\text{max}} – Q_{\text{min}}}{Q_{\text{tB}}} = \frac{B \omega_1}{n q} \left(1 – \frac{t_j^2}{4}\right)$$
where \( n \) is the pump speed and \( q \) is the displacement. The optimization objective is to minimize \( \sigma \), which depends on the base pitch \( t_j \), a function of tooth profile parameters. For internal gears, these parameters include the number of teeth \( z \), modification coefficient \( x \), pressure angle \( \alpha \), addendum coefficient \( h_a^* \), and module \( m \). The design variables are selected as:
$$X = [x_1, x_2, z_2, \alpha, h_a^*, m]^T$$
where \( x_1 \) and \( x_2 \) are the modification coefficients of the ring gear and internal gear, respectively, \( z_2 \) is the number of teeth of the internal gear, and other variables are as defined. The constraints ensure structural integrity, proper meshing, and avoidance of interference. For instance, the base circle must not exceed the addendum circle, tooth thickness must suffice for strength, and overlap coefficients must lie between 1.03 and 1.05. Additional constraints prevent sharp teeth and ensure positive modification.
The hybrid genetic algorithm (NIGA) combines an improved genetic algorithm (IGA) with a niche genetic algorithm (NGA) to enhance local search capabilities and prevent premature convergence. Traditional genetic algorithms (SGA) use selection, crossover, and mutation operators but often suffer from local optima. In IGA, the mutation rate is adaptive, calculated as:
$$P_m = \frac{P_{m1} + P_{m2}}{2} = \frac{P_{m0} – (P_{m0} – P_{\text{min}}) \cdot \frac{\sigma – \sigma_{\text{min}}}{\sigma}}{2}$$
where \( P_{m0} \) is the initial mutation rate, \( P_{\text{min}} \) is the minimum allowed rate, \( \sigma \) is the average population evaluation, and \( \sigma_{\text{min}} \) is the maximum evaluation. This adaptivity balances exploration and exploitation. NGA employs a sharing function to maintain diversity:
$$S(d_{ij}) = \begin{cases}
1 – \frac{d_1(x_i, x_j)}{\sigma_1} & \text{if } d_1(x_i, x_j) < \sigma_1 \\
1 – \frac{d_2(x_i, x_j)}{\sigma_2} & \text{if } d_2(x_i, x_j) < \sigma_2 \\
1 – \frac{d_1(x_i, x_j) d_2(x_i, x_j)}{\sigma_1 \sigma_2} & \text{if } d_1(x_i, x_j) < \sigma_1 \text{ and } d_2(x_i, x_j) < \sigma_2 \\
0 & \text{otherwise}
\end{cases}$$
Here, \( d_1 \) and \( d_2 \) represent evaluation and Hamming distances, respectively, and \( \sigma_1 \), \( \sigma_2 \) are niche radii. The shared fitness is:
$$F'(x_i) = \frac{F(x_i)}{\sum_{j=1}^M S(d_{ij})}$$
where \( F(x_i) \) is the original fitness and \( M \) is the generation number. NIGA integrates these features to improve convergence speed and solution quality.
For optimization, a PGH-type internal gear pump with initial parameters is considered: displacement \( q = 3.18 \, \text{mL/rad} \), pressure \( p = 20 \, \text{MPa} \), speed \( n = 12,560 \, \text{rad/min} \), ring gear teeth \( z_1 = 19 \), internal gear teeth \( z_2 = 14 \), and tooth width \( B = 20 \, \text{mm} \). The optimization process in MATLAB minimizes \( \sigma \) subject to 28 constraints, including geometric and meshing conditions. Results before and after optimization are summarized in the table below.
| Parameter | Before Optimization | After Optimization | Change (%) |
|---|---|---|---|
| Ring Gear Teeth (\( z_1 \)) | 19 | 17 | -10.53 |
| Ring Gear Modification Coefficient (\( x_1 \)) | 0.53 mm | 0.74 mm | 39.62 |
| Internal Gear Teeth (\( z_2 \)) | 14 | 13 | -7.14 |
| Internal Gear Modification Coefficient (\( x_2 \)) | 0.62 mm | 0.58 mm | -6.45 |
| Addendum Coefficient (\( h_a^* \)) | 0.80 mm | 0.82 mm | 2.50 |
| Pressure Angle (\( \alpha \)) | 20° | 25° | 25.00 |
| Module (\( m \)) | 2.12 mm | 3.50 mm | 65.09 |
| Flow Pulsation Rate (\( \sigma \)) | 3.44% | 3.19% | -7.27 |
The optimization reduces the flow pulsation rate by 7.27%, demonstrating the effectiveness of NIGA. Key changes include a decrease in tooth numbers, an increase in the ring gear modification coefficient, and adjustments to pressure angle and module. These alterations enhance the pump’s structural efficiency and operational stability. For internal gear manufacturers, such optimizations are vital for producing high-performance components that meet industrial demands for low pulsation and noise.
Further analysis shows that the base pitch \( t_j \) is influenced by the pressure angle and module, as \( t_j = \pi m \cos \alpha \). Substituting into the pulsation rate formula:
$$\sigma = \frac{B \omega_1}{n q} \left(1 – \frac{(\pi m \cos \alpha)^2}{4}\right)$$
This equation highlights the inverse relationship between \( \sigma \) and \( m \cos \alpha \), justifying the parameter adjustments. The constraints ensure feasible designs; for example, the overlap coefficient \( \epsilon \) is maintained within 1.03 to 1.05 using:
$$\epsilon = \frac{z_1}{2\pi} (\tan \alpha_{a1} – \tan \alpha’) + \frac{z_2}{2\pi} (\tan \alpha’ – \tan \alpha_{a2})$$
where \( \alpha_{a1} \) and \( \alpha_{a2} \) are the addendum pressure angles, and \( \alpha’ \) is the operating pressure angle. The hybrid genetic algorithm efficiently handles these nonlinear constraints through penalty functions and niche operations.
In practice, internal gears must be manufactured with precision to avoid interference. The internal gear manufacturer must ensure that the ring gear’s transition curve does not interfere with the internal gear’s profile, enforced by constraints such as:
$$g_{16}(X) = z_2 \tan \alpha – z_1 \tan \alpha’ – \frac{4(h_a^* – x_1)}{\sin 2\alpha} \geq 0$$
This prevents undercutting and ensures smooth meshing. The optimization process iteratively refines parameters until the minimum \( \sigma \) is achieved, typically requiring hundreds of generations in NIGA due to its improved convergence.
The table below summarizes the constraint functions used in the optimization model, highlighting their roles in ensuring feasible designs for internal gears.
| Constraint Number | Function | Description |
|---|---|---|
| \( g_1 \) to \( g_{12} \) | Bounds on variables | Ensure parameters within practical ranges |
| \( g_{13} \) | \( 1 – \frac{h_a^* – x_2}{z_2 (1 – \cos \alpha)} \leq 0 \) | Base circle ≤ addendum circle |
| \( g_{14}, g_{15} \) | Tooth thickness ≥ 0.4 | Prevent sharp teeth |
| \( g_{16} \) to \( g_{19} \) | Interference avoidance | Ensure no meshing conflicts |
| \( g_{20}, g_{21} \) | \( 1.03 \leq \epsilon \leq 1.05 \) | Overlap coefficient range |
| \( g_{22} \) | \( x_2 – x_1 \geq 0 \) | Positive modification |
| \( g_{23} \) to \( g_{26} \) | Cosine function bounds | Maintain geometric validity |
| \( g_{27}, g_{28} \) | Meshing point position | Ensure pulsation calculation accuracy |
The hybrid genetic algorithm’s performance is superior to traditional methods in solving this multi-constraint problem. For internal gear manufacturers, implementing such algorithms in design software can lead to more efficient pumps with reduced flow pulsation, enhancing system reliability. Future work could explore real-time optimization or integration with computational fluid dynamics for broader applications.
In conclusion, the hybrid genetic algorithm effectively optimizes internal gear pump tooth profile parameters, reducing flow pulsation by 7.27% while maintaining structural integrity. This approach underscores the importance of advanced algorithms in hydraulic system design, offering significant benefits for internal gear manufacturers striving for high-performance products.