Spiral bevel gears are widely used in equipment such as ship power systems, helicopters, aviation engines, and automotive transmission systems due to their advantages such as smooth operation, large transmission ratio, high torque transmission, reliable operation, and compact structure. Under long-term heavy and variable load operating conditions, if damage to the tooth surface of spiral bevel gears occurs, it can cause equipment vibration, reduce transmission performance, and even equipment damage and even casualties. Therefore, Accurately identifying the degree of damage caused by faults in spiral bevel gears, timely grasping the development status of damage faults, providing effective support for condition based maintenance, is crucial for ensuring the safe and stable operation of spiral bevel gears.
The meshing process of spiral bevel gears is complex and variable, and the working environment is also complex. Due to the strong background noise, strong nonlinearity, and non-stationary characteristics of fault vibration signals, it is difficult to identify their fault characteristics. Modulation Signal Bispectrum (MSB) has strong demodulation ability, which can effectively process nonlinear data, It can also suppress Gaussian noise and non Gaussian noise, and can obtain fault characteristics of rotating machinery in strong noise environments. Li Jiawei et al. proposed a fault diagnosis method for spiral bevel gears based on spectrum editing and MSB, which can effectively suppress the interference of harmonic signals and white noise; Guo Junchao et al. [3] combined Weighted Average Ensemble Empirical Mode Decomposition (WAEEMD) with MSB to extract fault features of rolling bearings; Zhu Danchen et al. combined second-order total variation denoising with MSB to diagnose bearings; TIAN et al. improved the MSB to make it more stable, and the improved MSB is suitable for classifying bearing faults.
The current research on fault diagnosis of spiral bevel gears mainly extracts features from vibration signals to analyze faults, such as using BP (BackPropagation) neural networks to identify the wear status of spiral bevel gears; BUZZONI et al. used Bayesian classifiers to identify the wear and pitting status of spur bevel gears. However, traditional fault pattern recognition methods are difficult to effectively construct complex mapping relationships between faults and signals. With the development of Convolutional Neural Network (CNN), the accuracy of pattern recognition is increasing, The ability of data mining is also becoming stronger. In recent years, CNN has been preliminarily studied and applied in fault diagnosis of rotating machinery. CNN can improve the shortcomings of traditional fault pattern recognition algorithms, but how to construct fault samples of spiral bevel gears based on vibration signals is the key to training CNN models for identifying damage degree, Traditional methods for constructing time-domain and frequency-domain feature samples are difficult to extract weak fault features. However, MSB is a high-order spectrum that can effectively reflect the weak features of signals, and research has shown that it is suitable for fault diagnosis of rotating machinery
ZHY Gear combines MSB with CNN and uses modulation signal bispectrum of spiral bevel gear vibration signals to construct input samples for CNN. A method for identifying the degree of damage to spiral bevel gears based on modulation signal bispectrum and convolutional neural networks is proposed, and the effectiveness of the method is verified through experiments
1. MSB Theory
YANG et al.’s research shows that MSB has strong demodulation ability and can also obtain fault characteristics of rotating machinery in strong noise environments. MSB is an improved bispectral method, which is also derived from second-order power spectrum. The Fourier transform X (f) of the discrete-time vibration signal x (t) is:
The MSB of x (t) is:
In the formula: BMS (fc, fx) is the modulation signal bispectrum of signal x (t); X * (fc) is the conjugate complex of X (fc); E [] is the mathematical expectation; Fc is the carrier frequency; Fx is the modulation frequency; (fc+fx) is the upper sideband frequency; (fc fx) is the upper sideband frequency. The amplitude AMS (fc, fx) and phase of MSB φ MS (fc. Fx) is:
The calculation formula for MSB sideband BSMES (fc, fx) is shown in equation (5):
2. Identification of Damage Degree of Spiral Bevel Gears Based on Modulation Signal Bispectrum and CNN
The process of identifying the degree of damage to spiral bevel gears based on modulation signal bispectrum and CNN is as follows:
Step 1: Conduct a vibration test for the fault of the spiral bevel gears, use an accelerometer to collect the vibration signal of the spiral bevel gears, set an appropriate sampling frequency, and collect sufficient vibration signal fragments
Step 2: Divide the collected vibration signal segments into several equal length data segments and perform preprocessing and normalization. Perform modulation signal bispectral analysis on them, extract the top view image of the modulation signal bispectrum, construct a feature map sample set, and divide it into training and testing sets
Step 3: Initialize the learning rate k, pooling layer sampling size s, minimum training amount n, convolution kernel size o, and iteration number m of the convolutional neural network, and construct these parameters into a parameter set P. Set the learning rate, iteration number, minimum training amount, and convolution kernel size separately. Set the iteration turn threshold for pooling layer sampling size to x1~x5, and adjust the step size to y1~y5
Step 4: Input the training set into the CNN model for model training, optimize the selection of key parameters such as iteration number and learning rate, input the model parameter with the lowest error rate to construct the parameter set P, and complete the CNN model training
Step 5: Input the test sample set into the trained CNN model for recognition, verify the effectiveness of the model, and thus achieve the identification of the degree of damage to spiral bevel gears
3. Damage fault test of spiral bevel gears
3.1 Experimental data collection
In order to verify the effectiveness of the method proposed in this article, vibration simulation tests were conducted on a comprehensive fault simulation platform for spiral bevel gear systems. The spiral bevel gear reducer is shown in Figure 1. According to the degree of damage, normal spiral bevel gears and two different degrees of damage fault spiral bevel gears were set, and the three states of spiral bevel gears are shown in Figure 2. Due to the fault spiral bevel gears being set on the input shaft, The acceleration sensor is installed on the input shaft. The BK testing system is used to collect the vibration signals of spiral bevel gears in normal and different degrees of damage states. The vibration signals of normal spiral bevel gears, slightly damaged spiral bevel gears, and moderately damaged spiral bevel gears are collected, with a sampling frequency of 3.2kHz and a speed of 900r/min
Divide the vibration signals collected in each state into several data segments with a length of 1024, perform modulation signal bispectral analysis on each data segment, and obtain the modulation signal bispectral map under the default equal viewing angle as shown in Figure 3. In order to observe all feature components and avoid feature loss caused by partial occlusion, adjust Figure 3 to the top-down angle state to obtain the modulation signal bispectral map under the top-down angle (Figure 4) And convert it into RGB images as subsequent model training samples
3.2 CNN Model Construction and Parameter Selection
The convolutional neural network structure used includes 1 input layer, 3 convolutional layers, 2 max pooling layers, 1 flattening layer, 2 fully connected layers, and 1 output layer. The process of the convolutional neural network is as follows:
(1) Replace the activation function with ReLU;
(2) Adjust the feature image size of the model input to 128 x 128 x 1;
(3) Using maximum pooling method to accelerate network training speed;
(4) Add BatchNorm operation after the convolutional layer to accelerate the convergence speed and stability of the network;
(5) The Dropout operation is added to the fully connected layer to effectively prevent overfitting.
The architecture of the convolutional neural network is shown in Figure 5, and specific parameters are shown in Table 1
| Model parameters | Number of feature maps | Convolutional kernel (sampling) size | Step size | Activation function |
| Input layer | 1 | — | — | — |
| Convolutional layer 1 | 8 | 3 x 3 | 1 x 1 | ReLU |
| Pooling layer 1 | 8 | 2 × 2 | 1 × 1 | — |
| Convolutional layer 2 | 16 | 3 x 3 | 1 x 1 | ReLU |
| Pooling layer 2 | 16 | 2 × 2 | 1 × 1 | — |
| Convolutional layer 3 | 32 | 3 × 3 | 1 × 1 | ReLU |
| Flattening layer | 1 | — | — | — |
| Fully connected layer 1 | 1 | — | — | ReLU |
| Fully connected layer 2 | 1 | — | — | ReLU |
| Output layer | 1 | — | — | Softmax |
In order to obtain a high-precision CNN diagnostic model, it is important to choose appropriate parameters. In this paper, the hierarchical optimization method is used to select the main parameters of CNN: 90 iterations, 0.0001 learning rate, a minimum training amount of 50 per training, and a convolutional kernel size of 3 × 3, The sampling size of the pooling layer is 2 × 2. The modulation signal bispectral maps of three different states of spiral bevel gears obtained are used as CNN input samples to construct an MSB-CNN fault diagnosis system. Each fault state contains 500 samples, with 150 randomly selected as training samples and the remaining 350 as test samples
3.3 Fault identification results and analysis of spiral bevel gears
To verify the advantages and effectiveness of modulation signal bispectrum, the vibration signal+CNN and modulation signal bispectrum+CNN methods were selected for comparison in the experiment. The classification confusion matrices for different input samples are shown in Figure 5, where 0, 1, and 2 respectively represent normal spiral bevel gears, lightly damaged spiral bevel gears, and moderately damaged spiral bevel gears
In order to eliminate the problem of differences in single diagnosis, each group of experiments was repeated 100 times, and the fault state recognition results were taken as the average of the repeated experiments. The recognition results and model training time of the two methods are shown in Table 2. From Table 2, it can be seen that the average recognition accuracy of the modulation signal bispectrum+CNN method is 99.91%, The average recognition accuracy of the vibration signal+CNN method is lower than that of the method proposed in this paper
| Fault identification method | Number of test samples | Average recognition accuracy/% | Training time/s |
| Modulation signal bispectrum+CNN | 1050 | 99.91 | 67 |
| Vibration signal+CNN | 1050 | 93.53 | 107 |
As an input sample, the vibration signal is one-dimensional data, which contains a large amount of noise, so the average recognition accuracy has decreased. MSB can not only effectively process the nonlinear components of the signal, but also suppress various noises. Using MSB as a sample input to CNN, the extracted features are more accurate, so the average recognition accuracy is higher
To further verify the advantages and effectiveness of the method proposed in this article compared to other intelligent fault recognition methods, the recognition results of modulation signal bispectrum+SVM and modulation signal bispectrum+BP were selected for comparison in the experiment. The SVM used RBF kernel function, and the parameter selection was kernel function parameter selection δ ＝ 3. The penalty factor C=4, the number of hidden layer nodes in the BP neural network is 32, and the node activation function is ReLU. The classification confusion matrix obtained from the last test set is shown in Figure 6, where 0, 1, and 2 represent spiral bevel gears, lightly spiral bevel gears, and moderately spiral bevel gears, respectively. The diagonal in Figure 7 represents the specific prediction results of each type of signal
The recognition results and model training time are shown in Table 3. According to Table 3, the average recognition accuracy of modulated signal bispectrum+CNN is 99.91%, while the average recognition accuracy of the other two intelligent spiral bevel gears fault recognition methods is lower than that of the method proposed in this paper; The training time of the method proposed in this paper is 97 seconds, which is much better than the other two intelligent methods for identifying faults in spiral bevel gears, and significantly improves the efficiency of fault identification by sampling modulation signal bispectrum as the model input sample
| Fault identification method | Number of test samples | Average recognition accuracy/% | Training time/s |
| Modulation signal bispectrum+CNN | 1050 | 99.91 | 97 |
| Modulation signal bispectrum+SVM | 1050 | 99.42 | 2374 |
| Modulation signal bispectrum+BP | 1050 | 89.37 | 621 |
When dealing with multi classification problems, SVM has a complex model and slow training speed due to the construction of the classifier. Moreover, SVM classifiers only train sample data with the same label, resulting in slow training and testing of SVM classification. By using modulation signal bispectrum as sample input, SVM extracts more accurate features, Therefore, the average recognition accuracy is relatively high. However, the BP neural network has a shallow structure and limited ability to handle nonlinear problems, which makes it difficult to recognize some features of the image and limits its ability to identify fault information of spiral bevel gears. Convolutional neural networks have a deep structure and strong non-linear processing ability, and the weight sharing mechanism of the convolutional layers in the convolutional neural network reduces the number of trainable parameters in the network, improving training efficiency, The average recognition accuracy of the model is higher
4. Conclusion
1) The constructed convolutional network model and modulation signal bispectral sample construction method combine feature automatic learning with spiral bevel gears fault classification, compensating for the shortcomings of traditional fault recognition methods that require manual feature extraction and simplifying the diagnostic process
2) Compared with vibration signals, the modulation signal bispectrum as the input sample has a higher average recognition accuracy
3) Compared with traditional intelligent spiral bevel gears fault recognition methods, the proposed recognition method has advantages in average recognition accuracy and model training time.