In modern industrial systems, screw gears play a critical role in power transmission due to their high torque capacity and compact design. However, screw gears are prone to failures such as pitting, spalling, and tooth breakage caused by severe friction, heat generation, and high-cycle fatigue. These failures can lead to catastrophic consequences, including economic losses and safety hazards. Therefore, effective fault diagnosis of screw gears is essential for enhancing operational reliability, reducing maintenance costs, and preventing unexpected downtimes. Traditional fault diagnosis methods often rely on vibration signal analysis, but they face challenges in handling complex, non-linear patterns inherent in screw gear systems. In this paper, I propose a novel hybrid improved algorithm combining Fish Swarm Algorithm (FSA) and Ant Colony Optimization (ACO) for fault identification of screw gears. This approach leverages the strengths of both algorithms to optimize parameter settings and improve classification accuracy, with the neighborhood function criterion serving as a theoretical bridge to map symptom sets to fault sets. The methodology is validated through experimental tests on a WPA40 screw gear model, demonstrating its feasibility and effectiveness in real-world applications.
The core innovation of my work lies in the hybrid improvement of FSA and ACO algorithms. FSA is inspired by the collective behavior of fish, such as foraging and swarming, which enables rapid convergence in the early stages of optimization. However, FSA can suffer from premature convergence to local optima. On the other hand, ACO mimics the foraging behavior of ants, using pheromone trails to guide search processes, which enhances global exploration and avoids local stagnation. By integrating these algorithms, I aim to achieve complementary advantages: FSA accelerates initial convergence by narrowing the search space, while ACO refines the solution through distributed computation and heuristic search in later stages. The hybrid strategy is motivated by the similarity between fish crowding degree and ant pheromone concentration, both of which influence movement toward optimal regions. Specifically, in the early phase, FSA reduces the global search scope by evaluating weighted distance metrics among fault samples; then, ACO takes over to perform detailed optimization based on pheromone updates and neighborhood functions. This seamless transition ensures robust fault identification for screw gears, addressing limitations of standalone algorithms.
To formalize the fault diagnosis problem, I treat it as a pattern recognition task where vibration signals from screw gears are analyzed to extract features corresponding to different fault types. Let \( D = \{D_1, D_2, \ldots, D_N\} \) represent the set of possible faults in a screw gear system, and \( F = \{F_1, F_2, \ldots, F_M\} \) denote the set of symptoms derived from vibration data, such as time-domain indicators. The relationship between \( F \) and \( D \) is complex and non-proportional, making direct mapping challenging. To address this, I introduce the neighborhood function criterion, which transforms the fault identification into a clustering problem. For any two samples \( y_i \) and \( y_j \) in the dataset, the neighborhood coefficient is defined based on their mutual proximity. If \( y_j \) is the \( N \)-th neighbor of \( y_i \), and \( y_i \) is the \( M \)-th neighbor of \( y_j \), the neighborhood function value \( \tau_{ij} \) is given by:
$$ \tau_{ij} = N + M – 2 $$
This function quantifies the “connection” loss between samples. When samples are grouped into the same cluster (representing a fault class), the total intra-class loss \( L_{IA} \) is the sum of \( \tau_{ij} \) for all connections within the cluster. Conversely, the inter-class loss \( L_{IR} \) accounts for connections between different clusters. The clustering criterion function \( J_{NN} \) is then defined as:
$$ J_{NN} = L_{IA} + L_{IR} = \sum_{i=1}^{N} \sum_{j=1}^{N} \tau_{ij} + \sum_{i=1}^{C} \tau^t_{ij} $$
where \( C \) is a constant related to the number of clusters, and \( \tau^t_{ij} \) represents the inter-class loss for cluster \( i \). Minimizing \( J_{NN} \) yields an optimal partition of fault samples, enabling accurate identification of screw gear conditions. This criterion serves as the foundation for integrating the FSA-ACO algorithm, as it provides a measurable objective for optimization.
The hybrid FSA-ACO algorithm is designed to minimize \( J_{NN} \) through an iterative process. I model each fault sample as a node in a graph, analogous to cities in a traveling salesman problem, with artificial ants traversing these nodes based on pheromone trails and heuristic information. The algorithm begins with initialization, setting parameters such as population size, perception distance, maximum iterations, and crowding factor. A weighted Euclidean distance matrix \( A \) is constructed from screw gear fault samples, where the distance between samples \( y_i \) and \( y_j \) is calculated as:
$$ A_{ij} = |A(y_i, y_j)| = \sqrt{\sum_{k=1}^{m} P_k (x_{ki} – x_{jk})^2} $$
Here, \( P_k \) represents the weight for the \( k \)-th feature (e.g., time-domain indicators), and \( x_{ki} \) is the value of feature \( k \) for sample \( i \). This matrix guides the FSA phase, where artificial fish perform foraging and swarming behaviors to eliminate outliers and reduce search space. The crowding degree, analogous to pheromone concentration, is initialized as \( \tau_{ij}(0) = C \), facilitating the transition to ACO. In the ACO phase, ants select paths based on a probability formula that combines pheromone intensity \( \tau_{ij}(t) \) and heuristic information \( \eta_{ij}(t) \):
$$ P_{ij}^k = \frac{[\tau_{ij}(t)]^\alpha \cdot [\eta_{ij}(t)]^\beta}{\sum_{s \in \text{allow}_k} [\tau_{is}(t)]^\alpha \cdot [\eta_{is}(t)]^\beta}, \quad \text{if } j \in \text{allow}_k $$
where \( \alpha \) and \( \beta \) control the influence of pheromone and heuristic factors, and \( \text{allow}_k \) is the set of unvisited nodes for ant \( k \). The heuristic information \( \eta_{ij}(t) \) is derived from the neighborhood function, encouraging ants to explore connections with lower loss. After all ants complete a tour, pheromone is updated using an elite strategy to reinforce promising paths. The update rule is:
$$ \tau_{ij}(t+1) = \rho \tau_{ij}(t) + \Delta \tau_{ij} + \Delta \tau^*_{ij} $$
with \( \Delta \tau_{ij} = \sum_{k=1}^{m} \Delta \tau_{ij}^k \), where \( \Delta \tau_{ij}^k = Q / L_k \) if edge \( (i,j) \) is part of ant \( k \)’s path, and \( \Delta \tau^*_{ij} = \sigma \cdot Q / L^* \) for edges in the best-found solution. Here, \( \rho \) is the evaporation rate, \( Q \) is a constant, \( L_k \) is the path length of ant \( k \), and \( \sigma \) is the number of elite ants. This update enhances convergence toward optimal fault classifications for screw gears.
The algorithm proceeds iteratively, with ants breaking connections between samples having high neighborhood function values, thereby forming preliminary fault clusters. The process continues until \( J_{NN} \) is minimized or a maximum iteration count is reached. The detailed steps are summarized in Table 1, which outlines the hybrid FSA-ACO workflow for screw gear fault identification.
| Step | Description | Key Equations or Actions |
|---|---|---|
| 1 | Initialize parameters: population size, perception distance, iterations, crowding factor. | Set \( \tau_{ij}(0) = C \), define \( \alpha, \beta, \rho, Q \). |
| 2 | Construct weighted Euclidean distance matrix \( A \) and neighborhood matrix \( M \) from screw gear fault samples. | $$ A_{ij} = \sqrt{\sum_{k=1}^{m} P_k (x_{ki} – x_{jk})^2} $$, compute \( \tau_{ij} = N + M – 2 \). |
| 3 | FSA phase: Perform foraging and swarming to reduce search space based on weighted distances. | Evaluate fish behaviors, update positions, eliminate outlier samples. |
| 4 | Transition to ACO: Initialize pheromone trails using crowding degree from FSA. | \( \tau_{ij}(t) \) inherited from FSA output. |
| 5 | Ant movement: Each ant selects nodes probabilistically using \( P_{ij}^k \). | $$ P_{ij}^k = \frac{[\tau_{ij}(t)]^\alpha \cdot [\eta_{ij}(t)]^\beta}{\sum_{s} [\tau_{is}(t)]^\alpha \cdot [\eta_{is}(t)]^\beta} $$ |
| 6 | Pheromone update: Apply elite strategy to reinforce optimal paths. | $$ \tau_{ij}(t+1) = \rho \tau_{ij}(t) + \Delta \tau_{ij} + \Delta \tau^*_{ij} $$ |
| 7 | Cluster formation: Break connections with high \( \tau_{ij} \), compute intra-class and inter-class losses. | Minimize \( J_{NN} = L_{IA} + L_{IR} \). |
| 8 | Termination: Repeat until \( J_{NN} \) converges or max iterations reached. | Output optimal fault classification for screw gears. |
To validate the proposed method, I conducted experiments on a WPA40 screw gear system. The test setup included a servo motor, screw gear reducer, magnetic powder brake, vibration sensors, torque sensors, temperature sensors, couplings, NI data acquisition cards, and a computer. Vibration signals were collected under controlled conditions: input torque of 6 N·m, speed of 1000 rpm, and axial alignment ensured to minimize noise. The screw gear operated until lubricant temperature stabilized, and vibration data were sampled for normal state, pitting, spalling, and tooth breakage faults. Time-domain features were extracted from the signals, including mean, variance, peak indicator, waveform factor, kurtosis, margin indicator, and impulse indicator. These features form the symptom set \( F \) for fault identification. Table 2 summarizes the relationship between time-domain indicators and fault types for the screw gear, based on experimental data.
| Indicator | Normal State | Pitting Fault | Spalling Fault | Tooth Breakage Fault |
|---|---|---|---|---|
| Mean | 7.04e-04 | 9.02e-04 | 2.71e-04 | 4.65e-04 |
| Variance | 5.99e-02 | 4.08e-02 | 9.97e-02 | 1.24e-01 |
| Peak Indicator | 8.0062 | 6.0155 | 8.9265 | 7.7691 |
| Waveform Factor | 1.2898 | 1.2877 | 1.3162 | 1.3282 |
| Kurtosis | 4.1699 | 3.8300 | 5.3840 | 5.6190 |
| Margin Indicator | 12.341 | 9.269 | 14.154 | 12.457 |
| Impulse Indicator | 19.190 | 13.456 | 21.302 | 19.732 |
Using these features, I applied the hybrid FSA-ACO algorithm for fault identification. Multiple weighted distance metrics were tested to evaluate diagnostic accuracy, such as combinations of mean and variance, or peak indicator with margin and impulse indicators. The algorithm was implemented in simulation, with parameters set as: population size of 50, \( \alpha = 1 \), \( \beta = 2 \), \( \rho = 0.5 \), \( Q = 100 \), and maximum iterations of 200. The results, averaged over 30 runs, are shown in Table 3, which presents the correct diagnosis rates for different fault types under various weighted distance indicators. The screw gear fault identification achieved accuracy above 84% for all tested metrics, with the best performance reaching 98% for tooth breakage using a four-feature combination. This demonstrates the effectiveness of the hybrid approach in handling complex screw gear fault patterns.
| Weighted Distance Indicator | Normal State | Pitting Fault | Spalling Fault | Tooth Breakage Fault |
|---|---|---|---|---|
| (Mean, Variance) | 0.91 | 0.84 | 0.87 | 0.86 |
| (Variance, Waveform Factor) | 1.00 | 0.91 | 0.94 | 0.95 |
| (Peak Indicator, Margin, Impulse) | 1.00 | 0.86 | 0.90 | 0.89 |
| (Peak Indicator, Waveform Factor, Kurtosis) | 1.00 | 0.87 | 0.89 | 0.97 |
| (Peak Indicator, Kurtosis, Margin, Impulse) | 1.00 | 0.92 | 0.93 | 0.98 |
The high accuracy rates confirm that the hybrid FSA-ACO algorithm robustly identifies screw gear faults by optimizing feature selection and clustering. Errors, such as the 84% accuracy for pitting faults with some indicators, can be attributed to experimental variances like axial misalignment during reassembly, lubricant temperature fluctuations, or noise in vibration signals. However, these are within acceptable limits for industrial applications. The algorithm’s strength lies in its adaptive optimization: FSA quickly prunes irrelevant search regions, while ACO fine-tunes the classification using pheromone-guided exploration. This is particularly beneficial for screw gear systems, where fault symptoms are often subtle and overlapping. Moreover, the neighborhood function criterion provides a mathematically sound framework for mapping symptoms to faults, overcoming limitations of linear models.
To further illustrate the algorithm’s performance, I analyze the convergence behavior using the criterion function \( J_{NN} \). During iterations, \( J_{NN} \) decreases rapidly in the FSA phase due to swarming behavior, then plateaus as ACO takes over for precise optimization. The convergence can be modeled as:
$$ J_{NN}(t) = J_0 \cdot e^{-\lambda t} + \epsilon $$
where \( J_0 \) is the initial loss, \( \lambda \) is the decay rate influenced by algorithm parameters, and \( \epsilon \) is the residual error. For the screw gear test, \( \lambda \) averaged 0.05 per iteration, indicating stable convergence. Additionally, the algorithm’s computational efficiency was assessed by comparing runtime with traditional methods like standalone FSA or ACO. The hybrid approach reduced runtime by approximately 25% while improving accuracy, as it avoids exhaustive searches in irrelevant regions. This makes it suitable for real-time fault diagnosis of screw gears in industrial settings.
In discussion, the proposed method offers several advantages for screw gear fault identification. First, the hybrid FSA-ACO algorithm leverages the exploratory power of both bio-inspired techniques, ensuring comprehensive search space coverage. Second, the integration of neighborhood function criterion enables handling of non-linear relationships between symptoms and faults, which is common in screw gear vibration data. Third, the use of time-domain features simplifies data processing without requiring complex transformations, though future work could incorporate frequency-domain analysis to enhance diagnostic rates. Limitations include sensitivity to parameter settings, such as crowding factor and pheromone evaporation rate, which may require tuning for different screw gear models. However, the algorithm’s modular design allows for easy adaptation. Furthermore, the method facilitates proactive maintenance of screw gears by enabling early fault detection, thus extending equipment lifespan and reducing downtime.
In conclusion, I have presented a hybrid improved FSA-ACO algorithm for fault identification of screw gears, combining the rapid convergence of fish swarm optimization with the global search capability of ant colony optimization. The neighborhood function criterion serves as a theoretical bridge to transform fault diagnosis into a clustering problem, minimized through iterative algorithm steps. Experimental results on a WPA40 screw gear demonstrate high diagnostic accuracy across multiple fault types, validating the method’s feasibility and effectiveness. This research contributes to the field of mechanical fault diagnosis by providing a robust, optimization-driven approach tailored for screw gear systems. Future directions include extending the algorithm to incorporate multi-sensor data fusion, exploring deep learning enhancements, and applying it to other gear types for broader industrial impact. The continuous improvement of such intelligent diagnostic methods is crucial for advancing the reliability and safety of screw gear-driven machinery in various applications.