In order to accurately diagnose the faulty parts of the rolling bearing under different operating conditions, the KJADE (Kernel Function Joint Approximate Diagonalization of Eigenmatrices) algorithm is proposed to reduce the dimensionality of the high-dimensional feature data. Then, the VNWOA (Von Neumann Topology Whale Optimization Algorithm) is used to optimize the LSSVM (Least Squares Support Vector Machine) method to diagnose the fault type of the rolling bearing. The VNWOA algorithm is used to optimize the regularization parameters and kernel parameters of LSSVM. The low-dimensional nonlinear features contained in the multidomain feature set are extracted by KJADE and compared with the results of PCA, LDA, KPCA, and JADE methods. Finally, VNWOA-LSSVM is used to identify bearing faults and compare them with LSSVM, GA-LSSVM, PSO-LSSVM, and WOA-LSSVM. The results show that the recognition rate of fault diagnosis is up to 98.67% by using VNWOA-LSSVM. The method based on KJADE and VNWOA-LSSVM can diagnose and identify fault signals more effectively and has higher classification accuracy.
Rolling bearings are widely used in modern machinery and are one of the most vulnerable mechanical parts in rotating machinery [
The traditional bearing fault diagnosis process based on feature extraction mainly includes four steps: vibration signal acquisition, signal preprocessing, feature extraction, and fault identification. In the traditional bearing fault diagnosis method, the three steps of signal preprocessing, feature extraction, and fault identification are closely connected [
In the aspect of mechanical fault diagnosis based on empirical mode decomposition, Ali et al. [
Past research has made significant progress, but there are some limitations. The fault diagnosis is essentially a pattern recognition problem, and deep learning can be used for pattern recognition by extracting the features layer by layer, which can be largely rid of relying on various advanced signal processing techniques and heavy artificial feature extraction. Therefore, the fault diagnosis method improved by the conventional method has achieved a large room for improvement and the effect is also good. Therefore, the study of reasonable methods and optimization algorithms is particularly important for the realization of intelligent and efficient diagnosis of rolling bearings.
The main purpose of this study is to extract the nonlinear characteristic parameters of normal bearing, inner ring fault, outer ring fault, rolling element fault, and three fault levels (a total of ten bearing states) under various fault modes. Then, the KJADE algorithm is used to feature the extracted original high-dimensional feature matrix to extract the effective features that are more sensitive to the bearing state. Finally, a fault diagnosis method for rolling bearing based on the improved whale algorithm is proposed to effectively identify bearing faults. The diagnostic results of LSSVM, GA-LSSVM, PSO-LSSVM, WOA-LSSVM, and VNWOA-LSSVM are compared, respectively, to verify the effect of the improved whale algorithm on fault diagnosis accuracy and diagnostic recognition rate of rolling bearings. Three contributions have been made.
This study used Case Western Reserve University’s bearing test data as the verification object. The experimental data of the model SKF6205 deep groove ball bearing were used as the simulation data. The experimental data are processed into different points of failure at different positions of the bearing by electric spark. The vibration signal of the bearing is recorded by the acceleration sensor at a sampling frequency of 48 kHz. The torque sensor is mainly used to monitor the speed of the bearing at a speed of 1750 rpm. The load motor provides a load to the bearing. The bearing failure damage diameters are 0.007 inches (0.1778 mm), 0.014 inches (0.3556 mm), and 0.021 inches (0.5334 mm), respectively. By taking the bearing test data as the original input signal, the bearing normal, inner ring fault, outer ring fault, and rolling element fault are extracted. Finally, the fault diagnosis research and analysis of the nonlinear characteristic parameters of three fault levels (a total of ten bearing states) under different fault modes are carried out.
Vibration signal feature extraction is an important step in the research of rolling bearing fault diagnosis. It is essential to extract the most useful features from the vibration signal. The extracted feature is a high-dimensional feature matrix, but there will still be some useless signals in it. Therefore, how to eliminate the redundancy between features and reduce the complexity of decision making is very important. In terms of fault diagnosis, due to some problems such as the convergence and accuracy of the optimization algorithm, the final fault diagnosis accuracy and fault recognition rate are not very high. This paper attempts to extract features from the time domain, frequency domain, and time-frequency domain of the vibration signal. Then, the algorithm is used to reduce the dimension of the feature. Finally, the Von Neumann topology is used to improve the WOA and optimize the LSSVM regularization parameters and the parameters of the nuclear parameters to diagnose the rolling bearing fault.
A method based on KJADE and VNWOA-LSSVM is proposed to diagnose the rolling bearing-fault signal. Firstly, the multidomain feature set of nonlinear feature parameters is obtained by extracting time-frequency domain features from time domain, frequency domain, and VMD envelope spectrum. The feature fusion of the high-dimensional feature matrix in the feature set is then performed by KJADE. Finally, the Von Neumann topology is used to improve the WOA and optimize the LSSVM regularization parameters and kernel parameters, which effectively improves the accuracy and convergence speed of the LSSVM. At the same time, it maintains a good overall performance, which improves the ability to diagnose rolling bearing faults.
The remainder of this paper is organized as follows. In Section
Figure
Schematic framework diagram.
The original signal is decomposed by VMD (Variational Mode Decomposition) to obtain corresponding components, wherein the
The Hilbert transform is performed on the
Construct the parsing signal:
Corresponding amplitude function
The resulting Hilbert spectral expression is
From the entire process of HHT analysis, the frequency and amplitude of each IMF (Intrinsic Mode Function) are a function of time, so
KJADE is a new feature fusion method based on the JADE (Joint Approximative Diagonalization of Eigenmatrix) algorithm, and it has greatly improved the handling of nonlinear problems and the improvement of the robustness of nonlinear results. The core idea of this algorithm is to perform high-dimensional mapping of the observed data
Nonlinear mapping.
The specific steps of the KJADE algorithm are summarized as follows: The sample point data in the observed data Calculating the covariance matrix in the feature space For the abovementioned kernel matrix Selecting the matrix group Calculating the rotation matrix Finally, the mixed matrix
When the bearing fails, the extracted characteristic distribution of each bearing has a good class separability. Therefore, the difference between the vibration signals of each bearing and each part can be calculated by constructing two classes of model, which can be used as the basis for evaluating the performance degradation of the bearing. In the classification measurement of the sample, the distance between between-class and within-class has been successfully applied to the class separability metric [
The two classes of model are shown in Figure
The diagrammatic sketch of the two classes of model.
Then, the between-class scatter matrix is
The within-class scatter matrix is
SVM is a relatively new machine learning method based on statistical theory as basic research. In the case, where the sample is very limited, it can be used to find the optimal solution. LSSVM introduces the least squares based on the traditional SVM. Based on Vapnik’s support vector machine, the optimization goal is defined by a quadratic loss function with a square term. Finally, the inequality constraint in the original quadratic optimization problem becomes the equality constraint problem for solving linear equations, which can reduce the complexity of the overall calculation and increase the operation speed [
The optimization objective function using the LSSVM algorithm is
The final optimization problem becomes
In order to solve the problem of optimization well, the Lagrange multiplier
Then, using the constraints on the KKT condition. The relevant parameters of the Lagrange function are separately subjected to a partial derivative operation at the extreme points sought, and the result is zero. The resulting linear matrix expression is as follows:
There are some relationships in the above expression:
In the LSSVM algorithm, the resulting optimal classification function expression is
In the above formula,
Mirjalili and Lewis studied the predatory behavior of humpback whales [
Bubble-net prey behavior of whales.
In the model surrounding the prey, assuming that the target prey position or the position closest to the target prey is equated with the current best candidate solution, after the defined best search agent is subsequently defined, the other agents will update their position and gradually nearest to the best search agent. Its location update expressions are as follows:
In the above equations (
In the model of hunting prey, the model consists of two parts.
The first part is narrowing the range of prey, that is, the value of
The second part is position update with spiral features:
Humpback whales contain both the first and second parts described above when hunting prey, and the system defines a 50% probability as a threshold to apply to the position of the humpback whale. The specific expression is as follows:
In the model of searching for prey, for the individual whales, they are randomly supplemented with prey within a certain range. When
In the above formulas,
In view of the shortcomings of the WOA in optimizing some complex problems, the accuracy is not high and the convergence speed is slow. Therefore, the WOA is improved by using the Von Neumann topology. As shown in Figure
Von Neumann topology.
Studying the abovementioned whale position update formula, it shows that the change has a great relationship with the global optimal solution and will change with the update of the global optimal solution. The midpoints of the local and global optimal positions are selected and adopted so that the whales are affected by the global and local optimal solutions while performing position updating. It can exchange sufficient information with other local whales to enhance the local search ability of the algorithm. As the number of iterations increases, the local optimum and the global optimal will appear approximately coincident and the position of the whale will be updated to return to the original formula [
In the above formula,
The VNWOA uses the Von Neumann topology. Within each neighborhood, the optimal solution found by the humpback whale at each central location affects only the other four whales in the same neighborhood. The full exchange of whales in the neighborhood can maintain the multidirectionality and diversity of the population. While multiple neighborhoods maintain the diversity of fish stocks, avoiding the fact that a whale finds a local optimal solution prematurely, the whole population falls into local optimum, and it is improved in terms of maintaining globality and convergence speed. By improving the whale position update formula in the algorithm, the midpoint of the local optimal position and the global optimal position is selected. In this way, the whale can not only be guided by the global optimum when the location is updated, but also enhance the whale communication ability in the local area. The update of its location is also affected by a part of the local optimum, thus enhancing the local search ability of the algorithm. As the iteration progresses, the local optimal value gradually coincides with the global optimal value and the whale’s position update eventually returns to the original formula. The ability of the algorithm to achieve convergence and precision is enhanced.
Step 1 (data preprocessing): The time domain, frequency domain, and time-frequency domain characteristics are obtained for the original vibration signals and normalized. The feature reduction is performed by KJADE, and the training set and test set are divided by 7 : 3. Step 2: Initializing the whale position and setting the population to Step 3: Determining the network topology of the VNWOA and the range of values for initializing the LSSVM, that is, the range of values of Step 4: Calculating the corresponding fitness of each whale, and determining the initial optimal individual and initial optimal fitness according to the order of fitness. Step 5: Using the Von Neumann topology to perform neighborhood search, exchanging information in the neighborhood, finding the best whale in the neighborhood, and then following the formulas ( Step 6: The whale swims in a spiral shape to the prey while shrinking the encirclement. The position of the other whales is updated according to the selected whale position, forcing the whale to deviate from the prey thus obtaining the best training accuracy in the LSSVM. Step 7: Repeating steps 5 and 6. Until the maximum number of iterations
Vibration signal preprocessing: by selecting the normal state of the bearing and three types of faults (corresponding to three fault levels for each of the three fault types), a total of ten bearing states are used as verification objects. Then, the time domain and frequency domain of ten bearing states and the method of using VMD envelope spectrum are used to extract the time-frequency domain features of the rolling bearing so as to obtain the multidomain feature set of nonlinear characteristic parameters. Feature fusion: KJADE is used for feature fusion in the high-dimensional feature matrix of the feature set, and the evaluation factor SS composed of between-class and within-class scatters is employed to depict the clustering performance of the proposed new features quantitatively. It is then compared to the PCA, LDA, KPCA, and JADE clustering effects. Selection of fault signal for training and testing: a total of 1000 fault signal samples are randomly selected from ten types of bearing states. Among them, 700 samples were used as training samples and 300 samples were used as test samples and the data length of each sample was 15. Fault diagnosis: the method used for fault diagnosis mainly uses the Von Neumann topology to improve the WOA and optimize the LSSVM regularization parameters and the parameters of the nuclear parameters. In the experiment, the maximum number of iterations is set to 100, and then the 700 sets of samples in step 3 are input to the whale position with the best fitness as the parameters of the LSSVM for training. Finally, the remaining 300 sets of test data are diagnosed.
The rolling bearing fault data used in this experiment were obtained from the Case Western Reserve University Bearing Data Center [
Bearing fault diagnosis test equipment.
This experiment uses an acceleration sensor adsorbed on a magnetic body to obtain the vibration data. At the drive and blast ends of the motor frame, the sensor is placed at 12 o'clock. The vibration signal was recorded with a 16-channel digital recording recorder and saved as a Matlab (
“Normal” indicates bearing under normal conditions; “IR” indicates inner ring failure; “OR” indicates outer ring failure; and “B” indicates rolling element failure. The ten bearing data obtained are numbered and indicated by I-X. The specific bearing data are shown in Table
Bearing data.
Bearing status | Degree of failure (inches) | Abbreviation | Category labeling |
---|---|---|---|
Normal | 0 | Normal | I |
|
|||
Inner ring failure | 0.007 | IR07 | II |
0.014 | IR14 | III | |
0.027 | IR21 | IV | |
|
|||
Outer ring fault | 0.007 | OR07 | V |
0.014 | OR14 | VI | |
0.027 | OR21 | VII | |
|
|||
Rolling element failure | 0.007 | B07 | VIII |
0.014 | B14 | IX | |
0.027 | B21 | X |
When the motor speed is 1797 rpm (the frequency conversion is also considered to be
Time domain waveform of 10 kinds of bearings.
A time-domain waveform diagram of the bearing in a healthy state is shown in Figure
Four characteristic indexes of the center frequency, frequency standard deviation, root mean square frequency, and frequency concentration of the bearing signal are extracted as frequency domain features, and the time-frequency domain characteristic signals are extracted by using the VMD envelope spectrum method to form a multidimensional feature set. Since the corresponding spectrum and the envelope spectrum have the same range of values on the abscissa, image blending is employed and normalized.
The two characteristics of the time-frequency spectrum and the envelope spectrum when the bearing is in normal operation is shown in Figure
Fault-free normalized waveform.
B07 normalized waveform.
B14 normalized waveform.
B21 normalized waveform.
IR07 normalized waveform.
IR14 normalized waveform.
IR21 normalized waveform.
OR07 normalized waveform.
OR14 normalized waveform.
OR21 normalized waveform.
From the original bearing vibration information, the number of samples is 1000 and the multidomain feature set is extracted from the time domain, the frequency domain, and the time-frequency domain, thereby obtaining a feature matrix of
It can be clearly seen from Figures
LDA feature distribution map.
PCA feature distribution map.
KPCA feature distribution map.
JADE feature distribution map.
KJADE feature distribution map.
SS of different methods.
Method | LDA | PCA | KPCA | JADE | KJADE |
---|---|---|---|---|---|
ss | 16.7 | 9.5 | 27.6 | 48.2 | 202.7 |
From the total sample of bearing vibration data, 700 samples were randomly selected as training samples and 300 samples were used as test samples, and the data length of each sample was 15. The characteristics of the training set are identified by the category and then input to the VNWOA-LSSVM training and compared with the LSSVM, GA-LSSVM, PSO-LSSVM, and WOA-LSSVM.
It can be seen from the fitness curve of Figure
LSSVM optimal fitness value.
In Figures
LSSVM troubleshooting results.
GA-LSSVM troubleshooting results.
PSO-LSSVM troubleshooting results.
WOA-LSSVM troubleshooting results.
VNWOA-LSSVM troubleshooting results.
Algorithm performance comparison.
Algorithm type | Fitness value | Training accuracy (%) | Diagnostic accuracy (%) | Training mean square error (%) |
---|---|---|---|---|
LSSVM | — | 87.46 | 85.67 |
|
GA-LSSVM | 0.8996 | 89.38 | 88.00 |
|
PSO-LSSVM | 0.9187 | 90.96 | 90.67 |
|
WOA-LSSVM | 0.9583 | 94.32 | 94.00 |
|
VNWOA-LSSVM | 0.9784 | 98.79 | 98.67 |
|
Figure
Algorithm time complexity comparison chart.
Through the normal, inner ring fault, outer ring fault, rolling element fault and three fault levels in various fault modes (a total of ten bearing states), the time domain, the frequency domain, and the method using the VMD envelope spectrum, the time-frequency domain features are extracted to obtain the multidomain feature set of nonlinear feature parameters. Then, the KJADE is used to feature the high-dimensional feature matrix in the feature set. The extracted low-dimensional features have very good clustering in the feature space. The effect and the effective characteristics that are more sensitive to the bearing state can be obtained. At the same time, in the fault diagnosis, the Von Neumann topology is used to improve the WOA and optimize the LSSVM regularization parameters and the parameters of the nuclear parameters. The accuracy and convergence speed of LSSVM are improved and a good overall globality is maintained, thus improving the ability of fault diagnosis of rolling bearings. It shows that the proposed method for fault diagnosis of rolling bearings has a good diagnostic effect and also provides a diagnosis method for rolling bearing fault diagnosis.
Convolutional neural network
Deep belief network
Autoregression
Back propagation neural network
Particle swarm optimization-support vector machine
Hilbert–Huang transform
Intrinsic mode function
Empirical mode decomposition
Electrical discharge machining
Variational mode decomposition
Support vector machine
Least squares support vector machine
Principal component analysis
Linear discriminant analysis
Kernel principal component analysis
Joint approximative diagonalization of eigenmatrix
Kernel function joint approximate diagonalization of eigenmatrices
Whale optimization algorithm
Genetic algorithm-least squares support vector machine
Particle swarm optimization-least squares support vector machine
Whale optimization algorithm-least squares support vector machine
Von Neumann topology whale optimization algorithm-least squares support vector machine
The data used to support the findings of this study are available from the specific operational procedures upon request. Specific operational procedures, the bearing data used to support the findings of this study, have been deposited in the “
The authors declare that they have no conflicts of interest.
This article was supported by Shanghai Polytechnic University Graduate Program Fund [EGD18YJ0003].