Abstract
Gear hobbing is a crucial manufacturing process in the production of various mechanical components. The optimization of process parameters in CNC gear hobbing is challenging due to the numerous and intricately coupled parameters. This paper proposes a multi-objective optimization method for CNC gear hobbing process parameters, aiming to minimize energy consumption, processing time, and quality errors. By employing the HDBSCAN algorithm, GMOMPA, and an improved TOPSIS, we efficiently address the multi-objective optimization problem. The results demonstrate that the proposed algorithm can significantly reduce energy consumption, enhance processing efficiency, and stabilize processing quality.
1. Introduction
Gears are fundamental and critical components in various mechanical equipment. CNC gear hobbing is a widely used method for gear manufacturing, capable of producing straight-tooth cylindrical gears, helical cylindrical gears, bevel gears, crowned gears, worm gears, and other types of gears. Despite the maturity of CNC gear hobbing technology, there are still challenges in practical applications due to the complexity and variability of the hobbing process. The selection of process parameters directly affects the accuracy and efficiency of gear manufacturing, making their optimization a key issue.
Previous research on the process optimization of CNC gear hobbing has primarily focused on using different multi-objective optimization algorithms to obtain suitable process parameter solutions. However, these methods still have limitations in mathematical model construction and algorithm application. Therefore, this paper proposes a multi-objective optimization method for CNC gear hobbing process parameters based on the Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN), Guided Multi-Objective Marine Predators Algorithm (GMOMPA), and improved Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS).
2. Problem Description of Multi-Objective Optimization for CNC Gear Hobbing Process Parameters
The three essential elements of a multi-objective optimization problem are decision variables, constraint conditions, and objective functions. The multi-objective optimization problem for CNC gear hobbing process parameters can be represented as ,h h h Q p C, where =h p p u represents the problem to be solved, and 12,, h h h C c c denotes the collection of past processing cases. In this context, u represents the descriptive parameters of the gear hobbing process given by the user, while e represents the parameters to be optimized, which are the decision variables in the multi-objective optimization problem.
Table 1: Decision Variables in Gear Hobbing Process
| Decision Variables | Description |
|---|---|
| 0z | Number of hobbing cutter heads |
| 0a d | Outer diameter of hobbing cutter |
| 0n | Spindle speed |
| f | Axial feed rate |
2.1 Constraint Conditions
To achieve multi-objective optimization of the CNC gear hobbing process parameters, several types of constraints are constructed:
- Boundary Constraints: Each parameter to be optimized should fall within a specific range.
- Accuracy Constraints: According to precision requirements, 0z and 0n should be integers, while 0a d and f should be retained to two decimal places.
- Hobbing Cutter Set Constraints: Defined as 12,,H h h, representing the collection of hobbing cutters.
- Cutting Force Constraint: The cutting force during gear hobbing should not exceed the maximum cutting force.
- Surface Roughness Constraint: The gear surface roughness requirement must be met.
2.2 Objective Functions
The optimization of a complete gear hobbing process considers objectives including energy consumption E, processing time T, and quality error Q. The energy consumption E consists of standby energy consumption sE, air-cutting energy consumption aE, and cutting energy consumption cE. The processing time T is composed of standby time sT, air-cutting time aT, and cutting time cT. The quality error Q includes tooth profile error and tooth direction error.
3. Methodology
3.1 Determination of Process Parameter Range Based on HDBSCAN Clustering
Before starting the multi-objective optimization, it is necessary to find a subset of samples in the past processing case sample set hC that have a high similarity to the problem to be solved hp. The upper and lower limits of all ce in this subset are used as the value range of the parameters to be optimized pe.
3.2 Multi-Objective Optimization of Process Parameters Based on GMOMPA
The Guided Multi-Objective Marine Predators Algorithm (GMOMPA) is used for multi-objective optimization of process parameters. The framework of the GMOMPA algorithm.
In GMOMPA, the prey population is initialized with a uniform distribution within the search space ul and ll. Each prey solution represents a set of process parameters to be optimized. The predator population, composed of elite solutions, guides the movement of the prey population.
3.3 Optimal Solution Set Ranking Based on Improved TOPSIS
The obtained Archive set from multi-objective optimization contains multiple Pareto optimal solutions. Each Pareto optimal solution has different advantages and disadvantages in various objectives. Therefore, optimal solution set ranking is necessary to provide valuable information and a reference for decision-makers. In this section, the Analytic Hierarchy Process (AHP) is used to obtain the weights of each indicator, and the improved TOPSIS method is employed for optimal solution set ranking.
4. Algorithm Feasibility Verification and Comparative Analysis
4.1 Algorithm Feasibility Verification
A certain CNC gear hobbing machine tool manufacturing enterprise has accumulated a certain number of cylindrical gear processing case samples. A cylindrical gear workpiece is ready for processing on a six-axis five-linkage high-speed dry and wet cutting vertical CNC gear hobbing machine (model YK3118B-CNC) equipped with the Siemens SINUMERIK 828D CNC system. The descriptive parameters pu are given, and the hobbing method is specified as the axial hobbing method with secondary cyclical clockwise and counterclockwise cutting. The optimization objectives are energy consumption E, processing time T, and quality error Q during the hobbing process.
Table 2: Description Parameters of the Workpiece to Be Processed
| Material | Modulus m (mm) | Number of Teeth 1z | Pressure Angle (rad) | Helix Angle (rad) | Outer Diameter 1e d (mm) | Full Tooth Width B (mm) | Cutting Depth pa (mm) |
|---|---|---|---|---|---|---|---|
| 45# Steel | 4 | 16 | 0.349 | 0.524 | 82.63 | 42 | 9.22 |
4.1.1 Determination of Process Parameter Range
A total of 30 processing cases are read. The HDBSCAN algorithm is used to cluster the processing case samples, and 2k and 3mcs are set. Four clusters are obtained, and the closest cluster to hp is selected. Based on the processing information of all samples in this cluster, the value range of the parameters to be optimized pe is determined.
4.1.2 Multi-Objective Optimization of Process Parameters
The GMOMPA is used for multi-objective optimization, with a population size of 100N, a maximum archive size of _100max arch, and a maximum number of iterations of _300max iter. During each iteration, the minimum values of the three objectives in the Archive set are recorded, and the iteration process is observed. The values of the objective functions gradually decrease as the number of iterations increases, indicating that the algorithm is continuously optimizing and is effective in terms of convergence of the objective functions. After iteration, GMOMPA outputs the Archive set containing 99 solutions, which are mutually non-dominated in the three objectives and can be used for subsequent analysis and decision-making.
4.2 Comparative Analysis
To further validate the efficacy of the presented approach, we employed two prominent multi-objective optimization algorithms, namely MOPSO and NSGA-II, to tackle the problem using the predefined range of parameters to be optimized. A detailed comparison of the parameter configurations for these three algorithms is outlined in Table 3.
Table 3: Parameter Settings of Three Multi-Objective Optimization Algorithms in Comparative Analysis
| Algorithm | Population Size (N) | Maximum Archive Size (_max_arch) | Maximum Iterations (_max_iter) | Additional Parameters |
|---|---|---|---|---|
| GMOMPA | 100 | 100 | 300 | FADs effect probability = 0.2 |
| MOPSO | 100 | 100 | 300 | Local speed factor = 1.2c |
| NSGA-II | 100 | 100 | 300 | (Standard parameters) |
By leveraging the aforementioned parameter settings, we conducted a comparative analysis to assess the performance of GMOMPA against MOPSO and NSGA-II. The results of this comparison are depicted through various metrics and visualizations.
Firstly, the Pareto frontier diagram for the three target fitness values obtained by each optimization method. This diagram provides a clear visualization of the trade-offs between the different objectives and showcases the distribution of optimal solutions identified by each algorithm.
It is evident that the solutions obtained by MOPSO and NSGA-II exhibit uneven distribution on the Pareto frontier, with noticeable clustering in certain regions. This indicates that these algorithms tend to get trapped in local optima more frequently compared to GMOMPA. Conversely, the solutions found by GMOMPA are more evenly spread across the Pareto frontier, demonstrating a broader exploration of the solution space and a lower tendency to converge prematurely.
Furthermore, Table 3 (continued below for clarity) presents a quantitative comparison of the optimal solution results achieved by each method across the three objectives: energy consumption (E), processing time (T), and quality error (Q).
Table 3 (Continued): Comparison of Optimal Solution Results Among Different Optimization Methods
| Objective | Method | Optimized Parameters (z, a_d /mm, n /r/min, f /mm/r) | Objective Value |
|---|---|---|---|
| E (kWh) | GMOMPA | (2, 76.50, 1092, 0.80) | 0.5741 |
| MOPSO | (2, 83.00, 1053, 0.75) | 0.6014 | |
| NSGA-II | (2, 78.00, 1067, 0.80) | 0.5766 | |
| T (s) | GMOMPA | (Same as above for E) | 393.69 |
| MOPSO | (Same as above for E) | 409.15 | |
| NSGA-II | (2, 78.00, 1086, 0.80) | 395.19 | |
| Q (mm) | GMOMPA | (2, 92.00, 1096, 0.50) | 0.00158 |
| MOPSO | (2, 86.50, 1007, 0.56) | 0.00161 | |
| NSGA-II | (2, 88.50, 1081, 0.51) | 0.00158 |
Analyzing these results, it is apparent that GMOMPA outperforms both MOPSO and NSGA-II in terms of energy consumption and processing time. Specifically, GMOMPA achieves a 4.54% reduction in optimal energy consumption compared to MOPSO and a 0.43% reduction compared to NSGA-II. Similarly, for processing time, GMOMPA delivers a 3.78% improvement over MOPSO and a 0.38% enhancement compared to NSGA-II. Regarding quality error, while GMOMPA and NSGA-II yield comparable results, GMOMPA still demonstrates a slight edge with a 1.86% lower optimal quality error than MOPSO.
In summary, the comparative analysis confirms the superiority of the proposed GMOMPA algorithm in addressing the multi-objective optimization problem of CNC gear hobbing process parameters. By efficiently balancing energy consumption, processing time, and quality error, GMOMPA demonstrates its capacity to identify high-quality solutions that are both diverse and well-distributed across the Pareto frontier. These findings underscore the practical implications of the GMOMPA algorithm in enhancing the efficiency and sustainability of CNC gear hobbing operations.