Bearing is an important mechanical component that easily fails in a bad working environment. Support vector machines can be used to diagnose bearing faults; however, the recognition ability of the model is greatly affected by the kernel function and its parameters. Unfortunately, optimal parameters are difficult to select. To address these limitations, an escape mechanism and adaptive convergence conditions were introduced to the ALO algorithm. As a result, the EALO method was proposed and has been applied to the more accurate selection of SVM model parameters. To assess the model, the vibration acceleration signals of normal, inner ring fault, outer ring fault, and ball fault bearings were collected at different rotation speeds (1500 r/min, 1800 r/min, 2100 r/min, and 2400 r/min). The vibration signals were decomposed using the variational mode decomposition (VMD) method. The features were extracted through the kernel function to fuse the energy value of each VMD component. In these experiments, the two most important parameters for the support vector machine—the Gaussian kernel parameter
The rolling bearing is a core mechanical component that is widely used in wind turbines, aeroengines, ships, automobiles, and other important mechanical equipment [
Currently, the vibration monitoring method is the most commonly adopted method to monitor bearing conditions [
The support vector machine (SVM) [
In 2015, a new bionic intelligent algorithm termed “Ant Lion Optimizer” (ALO) was devised by Mirjalili [
At present, there are few studies regarding ALO application in bearing fault diagnosis. As a new bionic optimization algorithm, there are some ant lion individuals in the ALO algorithm with relatively poor fitness in the iteration process. If the ants select poor fitness ant lions for walking, the probability of falling into a local extremum increases. In addition, resource waste will result if poor fitness ant lions search around the local extremum and partially affect the optimization performance and convergence efficiency of the ALO algorithm.
Given the aforementioned problems, this paper uses the rolling bearing as its test object and to improve the ant lion algorithm. When combined with SVM, it also sought to diagnose bearing faults. This work has both great theoretical significance and practical value to improve the accuracy of fault diagnosis in rolling bearings, thereby ensuring the safety and stability of functional rolling bearings.
The ALO algorithm was modeled on the hunting behavior of ant larvae in nature. As constructed, the optimization algorithm mimics the walking of random ants, constructs traps, lures the ants into the trap, captures the ants, and reconstructs the traps. The ALO algorithm conducts a global search by walking around randomly selected ant lions, and local refinement optimization is achieved by adaptive boundary of the ant lion trap.
The total number of ants and ant lions is defined by
The fitness vectors
The iterative process of the ALO algorithm is to continuously update the position according to the interaction between the ants and the ant lions; after this update, it then reselects the elite ant lions. The ALO algorithm primarily includes random ant walks, trapping in an ant lion’s pit, building traps, sliding ants towards the ant lion, catching prey, rebuilding the pit, and elitism.
When searching for food in nature, ants move stochastically; as such, a random walk
In order to ensure the random walks of ants in the search space, the position of the ants is normalized by
According to its fitness, an individual ant lion is selected from the ant lion population of the previous generation through a roulette operation, defined as follows:
As
This process is described as the ants walking around the “trap.” The boundary of the walking area is affected by the position of the elite, which can be defined by the following formula:
As soon as the ant starts sliding towards the trap, the ant lions realize the ant is in the trap and shoot sand to the center of the pit to prevent it from escaping. The process can be described as an adaptive decrease in the radius of a given ant’s random walk hypersphere:
The ant lion kills the ant and eats its body. If a prey ant’s fitness is higher than the elite’s fitness, the elite ant lions update their position to the prey ants. In other words, the elite ant lions will build new traps for the next prey. Given this scenario, the following equation is proposed:
The elite ant lion affects the movements of all ants, with each ant randomly walking around a selected ant lion; this walking is done according to both the roulette wheel and the elite, simultaneously. This behavior is modeled according to the following function:
In the 1950s, Vapnik [
For the two types of classes to be classified, the sample dataset is defined as follows:
The establishment of an SVM classification model is done to find an optimal classification surface
By solving the following dual problem, the optimal solution
This allows the optimal hyperplane normal vector in equation (
The corresponding samples to
The decision function yielded is as follows:
The above SVM model has been established for linear sample classification; for nonlinear classification, the nonlinear transform
As shown in formula (
Common kernel function expressions and parameters.
Kernel functions | Expressions | Parameters |
---|---|---|
Linear kernel |
|
None |
Polynomial kernel |
|
|
Gauss kernel |
|
|
Sigmoid kernel |
|
|
As shown in Table
The standard SVM solves the problem of two-class classification; in reality, encountering a multiclass problem is more common than a two-class problem. Therefore, the study of a multiclass SVM problem is of great significance. At present, researchers have proposed a handful of effective multiclass SVM construction methods. These approaches can be divided into two categories, with the first being the direct construction method. This method improves the discriminant function of a two-class SVM model to construct a multiclass model. This method uses only one SVM discriminant function to achieve a multiclass output. The discriminant function of this algorithm is very complex, and its classification accuracy is not good. The second method is to realize the construction of a multiclass SVM classifier by combining multiple two-class SVMs. In practice, this method is more widely used and includes one-against-one, one-against-all, direct acyclic graph, and binary tree approaches [
In the ALO algorithm, ants randomly walk around the elite ant lion and the roulette wheel-selected ant lion; these ants gradually fall into the trap set by the ant lion. As the number of iterations increases, the walk range of the ants becomes increasingly smaller. In turn, this means the range of the search optimization solution becomes increasingly smaller as well. If the elite ant lion is located at the local extremum value, the risk of falling into the local extremum is increased. This reduces the optimization performance of the ALO algorithm. In nature, when an ant lion builds an ant trap, it is not always successful in catching the ants that fall into the trap. If the ants find that there is an ant lion nearby, they will avoid it to escape being eaten.
Here—and based on the aforementioned considerations—the ant escape mechanism was introduced into the ALO algorithm. This introduction resulted in an improved ALO algorithm, termed here as the EALO algorithm. By introducing the ant escape mechanism, the possibility of the algorithm falling into a local extremum value is reduced, thereby improving the optimization ability of the algorithm.
The optimization performance of the ALO algorithm includes primarily precision and time consumption. In the ALO algorithm, algorithm convergence is controlled by setting the maximum number of iterations
Assuming that
Assuming that
Flow diagram of the EALO-SVM algorithm.
Bearing fault vibration tests were conducted to verify the performance of the EALO-SVM algorithm. These tests were performed on a machinery fault simulator (SpectraQuest Inc., Hermitage Road, Richmond, Virginia, USA) and are shown in Figure
Bearing fault simulator and experimental setup.
Time waveform for sensor 1 (1500 r/min).
As shown in Figure
Variational mode decomposition (VMD) is a new signal decomposition technology that was proposed by Dragomiretskiy [
Assuming that
Introducing Gaussian kernel function results in
Define vector
Finally, a feature sample
In this experiment,
Figure
Feature vectors at different speeds.
Fifty groups of samples were selected from each fault sample dataset (normal, inner ring, outer ring, and ball fault bearings) to construct the training sample matrix with dimensions of 8 × 200; the remaining samples of each fault bearing dataset were used to construct the testing sample matrix with dimensions of 8 × 200. The parameters Gaussian kernel function
To verify the effectiveness of the EALO method proposed here, the ALO, genetic algorithm (GA), and particle swarm optimization (PSO) methods with different parameters were selected for comparison. All algorithms were executed on a computer running Windows 10 ×64 operating system with an Intel® Core™ i7-8700k CPU@3.70 GHz and with a memory capacity of 64 GB. To prevent the algorithms from iterating indefinitely, the maximum number of iterations
Details of different algorithm parameters.
Algorithm | Searching range | Convergence condition | Parameter | Initial value |
---|---|---|---|---|
EALO |
|
|
Ant population | 50 |
Ant lion population | 25 | |||
Escape ant population | 10 | |||
Ant escape probability | 0.7 | |||
|
||||
ALO |
|
|
Ant population | 50 |
Ant lion population | 25 | |||
|
||||
GA-1 |
|
|
Population | 50 |
Crossover probability | 0.1 | |||
Mutation probability | 0.01 | |||
|
||||
GA-2 |
|
|
Population | 50 |
Crossover probability | 0.4 | |||
Mutation probability | 0.01 | |||
|
||||
GA-3 |
|
|
Population | 50 |
Crossover probability | 0.1 | |||
Mutation probability | 0.1 | |||
|
||||
GA-4 |
|
|
Population | 50 |
Crossover probability | 0.4 | |||
Mutation probability | 0.1 | |||
|
||||
GA-5 |
|
|
Population | 50 |
Crossover probability | 0.7 | |||
Mutation probability | 0.5 | |||
|
||||
PSO-1 |
|
|
Particle size | 50 |
Acceleration factor |
0.1 | |||
Acceleration factor |
0.1 | |||
Weight |
0.5 | |||
|
||||
PSO-2 |
|
|
Particle size | 50 |
Acceleration factor |
0.5 | |||
Acceleration factor |
0.1 | |||
Weight |
0.8 | |||
|
||||
PSO-3 |
|
|
Particle size | 50 |
1.0 | ||||
Acceleration factor |
0.5 | |||
Acceleration factor |
1.0 | |||
Weight |
100 | |||
|
||||
PSO-4 |
|
|
Particle size | 50 |
Acceleration factor |
1.5 | |||
Acceleration factor |
1.7 | |||
Weight |
1.0 | |||
|
||||
PSO-5 |
|
|
Particle size | 50 |
Acceleration factor |
2.0 | |||
Acceleration factor |
2.0 | |||
Weight |
1.5 |
Using the rotation speed of 1500 r/min as an example, the distributional status of the ant and ant lion populations in the SVM parameter optimization processed by EALO is shown in Figure
Ant and ant lion distribution status during the EALO-SVM model optimization process. (a) Initial status. (b) 3rd iteration. (c) 5th iteration. (d) 7th iteration. (e) 9th iteration. (f) 13th iteration.
When considering the escape mechanism, there is a given probability that some ants will escape the trap of an ant lion. As shown in Figure
Using bearing fault diagnosis at a speed of 1500 r/min as an example, the convergence curves for EALO, ALO, GA, and PSO are all shown in Figure
Convergence curves for different optimization algorithms (1500 r/min). (a) EALO. (b) ALO. (c) GA. (d) PSO.
Using different parameters, the convergence performance of the GA method is different. Using inappropriate parameters resulted in GA performance deterioration; as shown in GA-5, the performance threshold
It has been reported that the binary tree model is a more suitable approach to classify bearing faults than many other classification models [
Model optimization and bearing fault diagnostic results at 1500 r/min.
Methods |
|
|
Time (s) | Iteration number | SV number | Testing dataset | Training dataset | ||
---|---|---|---|---|---|---|---|---|---|
Faults | Recognition rate (%) | Faults | Recognition rate (%) | ||||||
EALO | 19.5487 | 6.9794 | 0.9065 | 13 | 62 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 100 | ||||||
Average | 99.5 | Average | 99 | ||||||
Total average | 99.25 | ||||||||
|
|||||||||
ALO | 18.5233 | 8.3764 | 4.6771 | 100 | 59 | Normal | 100 | Normal | 100 |
Inner | 94 | Inner | 94 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 98 | ||||||
Average | 98 | Average | 98 | ||||||
Total average | 98 | ||||||||
|
|||||||||
GA-1 | 27.9121 | 2.4963 | 9.7142 | 100 | 43 | Normal | 92 | Normal | 100 |
Inner | 90 | Inner | 88 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 98 | ||||||
Average | 95 | Average | 96.5 | ||||||
Total average | 95.75 | ||||||||
|
|||||||||
GA-2 | 18.2271 | 9.5530 | 9.6264 | 100 | 48 | Normal | 100 | Normal | 100 |
Inner | 94 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 98 | ||||||
Average | 98 | Average | 98.5 | ||||||
Total average | 98.25 | ||||||||
|
|||||||||
GA-3 | 20.0440 | 9.1353 | 9.9713 | 100 | 34 | Normal | 100 | Normal | 100 |
Inner | 94 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 98.5 | Average | 98.5 | ||||||
Total average | 98.5 | ||||||||
|
|||||||||
GA-4 | 17.7436 | 4.6214 | 10.9412 | 100 | 163 | Normal | 100 | Normal | 100 |
Inner | 94 | Inner | 90 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 94 | Ball | 86 | ||||||
Average | 97 | Average | 94 | ||||||
Total average | 95.5 | ||||||||
|
|||||||||
GA-5 | 10.6886 | -2.4792 | 10.3611 | 100 | 200 | Normal | 100 | Normal | 100 |
Inner | 92 | Inner | 88 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 0 | Ball | 4 | ||||||
Average | 73 | Average | 73 | ||||||
Total average | 73 | ||||||||
|
|||||||||
PSO-1 | 22.3069 | 5.9467 | 10.6698 | 100 | 41 | Normal | 100 | Normal | 100 |
Inner | 92 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 98 | ||||||
Average | 97.5 | Average | 98.5 | ||||||
Total average | 98 | ||||||||
|
|||||||||
PSO-2 | 30.0000 | 2.5238 | 9.1010 | 100 | 47 | Normal | 52 | Normal | 68 |
Inner | 92 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 98 | ||||||
Average | 85.5 | Average | 91 | ||||||
Total average | 88.25 | ||||||||
|
|||||||||
PSO-3 | 30.0000 | 9.8034 | 8.9303 | 100 | 171 | Normal | 96 | Normal | 98 |
Inner | 72 | Inner | 78 | ||||||
Outer | 98 | Outer | 100 | ||||||
Ball | 82 | Ball | 70 | ||||||
Average | 87 | Average | 86.5 | ||||||
Total average | 86.75 | ||||||||
|
|||||||||
PSO-4 | 19.0039 | 8.0955 | 8.3374 | 100 | 56 | Normal | 100 | Normal | 100 |
Inner | 94 | Inner | 9 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 98 | ||||||
Average | 98 | Average | 98 | ||||||
Total average | 98 | ||||||||
|
|||||||||
PSO-5 | 10.8694 | -1.9647 | 8.8360 | 100 | 200 | Normal | 96 | Normal | 98 |
Inner | 72 | Inner | 78 | ||||||
Outer | 98 | Outer | 100 | ||||||
Ball | 50 | Ball | 46 | ||||||
Average | 79 | Average | 80.5 | ||||||
Total average | 79.75 |
Model optimization and bearing fault diagnostic results at 1800 r/min.
Methods |
|
|
Time (s) | Iteration number | SV number | Testing dataset | Training dataset | ||
---|---|---|---|---|---|---|---|---|---|
Faults | Recognition rate (%) | Faults | Recognition rate (%) | ||||||
EALO | 23.7849 | 3.8057 | 0.9349 | 15 | 20 | Normal | 100 | Normal | 98 |
Inner | 100 | Inner | 100 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 100 | ||||||
Average | 99.5 | Average | 99.5 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
ALO | 19.8388 | 8.68205 | 4.3750 | 100 | 33 | Normal | 100 | Normal | 98 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 100 | ||||||
Average | 99.5 | Average | 98.5 | ||||||
Total average | 99 | ||||||||
|
|||||||||
GA-1 | 22.8205 | 5.7719 | 9.1252 | 100 | 29 | Normal | 100 | Normal | 98 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 100 | ||||||
Average | 99.5 | Average | 99 | ||||||
Total average | 99.25 | ||||||||
|
|||||||||
GA-2 | 23.9121 | 7.6404 | 10.4754 | 100 | 18 | Normal | 100 | Normal | 98 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 98.5 | ||||||
Total average | 99 | ||||||||
|
|||||||||
GA-3 | 27.8681 | 6.7904 | 10.6654 | 100 | 21 | Normal | 100 | Normal | 98 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 98.5 | ||||||
Total average | 99.25 | ||||||||
|
|||||||||
GA-4 | 19.9341 | -4.3258 | 9.4085 | 100 | 200 | Normal | 100 | Normal | 100 |
Inner | 92 | Inner | 88 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 44 | Ball | 68 | ||||||
Average | 84 | Average | 89 | ||||||
Total average | 86.5 | ||||||||
|
|||||||||
GA-5 | 19.5385 | 9.28187 | 10.6452 | 100 | 30 | Normal | 100 | Normal | 98 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 100 | ||||||
Average | 99.5 | Average | 98.5 | ||||||
Total average | 99 | ||||||||
|
|||||||||
PSO-1 | 20.0281 | 8.3309 | 8.4335 | 100 | 32 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 100 | ||||||
Average | 99.5 | Average | 99 | ||||||
Total average | 99.25 | ||||||||
|
|||||||||
PSO-2 | 18.5136 | 10.0000 | 8.9782 | 100 | 31 | Normal | 100 | Normal | 98 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 100 | ||||||
Average | 99.5 | Average | 98.5 | ||||||
Total average | 99 | ||||||||
|
|||||||||
PSO-3 | 28.0514 | 7.4172 | 8.0311 | 100 | 22 | Normal | 100 | Normal | 98 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
PSO-4 | 18.6287 | 10.0000 | 8.0532 | 100 | 31 | Normal | 100 | Normal | 98 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 100 | ||||||
Average | 99.5 | Average | 98.5 | ||||||
Total average | 99 | ||||||||
|
|||||||||
PSO-5 | 18.4047 | 10.0000 | 7.5552 | 100 | 31 | Normal | 100 | Normal | 98 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 98 | Ball | 100 | ||||||
Average | 99.5 | Average | 98.5 | ||||||
Total average | 99 |
Model optimization and bearing fault diagnostic results at 2100 r/min.
Methods |
|
|
Time (s) | Iteration number | SV number | Testing dataset | Training dataset | ||
---|---|---|---|---|---|---|---|---|---|
Faults | Recognition rate (%) | Faults | Recognition rate (%) | ||||||
EALO | 23.2934 | 7.6761 | 0.2366 | 3 | 19 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
ALO | 25.7334 | 7.2869 | 3.3147 | 100 | 20 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
GA-1 | 25.4725 | 5.1417 | 7.4716 | 100 | 22 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
GA-2 | 29.2674 | 4.5188 | 11.8305 | 100 | 17 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
GA-3 | 28.7179 | 9.4871 | 9.7435 | 100 | 20 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
GA-4 | 20.1026 | 9.2599 | 8.0058 | 100 | 19 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
GA-5 | 13.0549 | 8.0581 | 8.5892 | 100 | 139 | Normal | 100 | Normal | 98 |
Inner | 98 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 99.5 | Average | 98 | ||||||
Total average | 99 | ||||||||
|
|||||||||
PSO-1 | 27.0491 | 5.2417 | 8.8048 | 100 | 19 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
PSO-2 | 15.4353 | 10.0000 | 8.2438 | 100 | 32 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 98.5 | ||||||
Total average | 99.25 | ||||||||
|
|||||||||
PSO-3 | 18.4043 | 9.0937 | 7.7971 | 100 | 22 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
PSO-4 | 20.3913 | 10.0000 | 7.7289 | 100 | 20 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
PSO-5 | 30.0000 | 9.8561 | 5.8882 | 100 | 19 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 98 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 |
Model optimization and bearing fault diagnostic results at 2400 r/min.
Methods |
|
|
Time (s) | Iteration number | SV number | Testing dataset | Training dataset | ||
---|---|---|---|---|---|---|---|---|---|
Faults | Recognition rate (%) | Faults | Recognition rate (%) | ||||||
EALO | 26.5027 | 6.4664 | 0.2656 | 3 | 22 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 99.5 | ||||||
Total average | 99.75 | ||||||||
|
|||||||||
ALO | 30.0000 | 1.7908 | 3.4479 | 100 | 27 | Normal | 94 | Normal | 88 |
Inner | 100 | Inner | 100 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 98.5 | Average | 97 | ||||||
Total average | 97.75 | ||||||||
|
|||||||||
GA-1 | 24.4322 | 5.4787 | 9.9083 | 100 | 21 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 99.5 | ||||||
Total average | 99.75 | ||||||||
|
|||||||||
GA-2 | 23.2381 | 7.7137 | 10.2449 | 100 | 22 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 99.5 | ||||||
Total average | 99.75 | ||||||||
|
|||||||||
GA-3 | 21.7289 | 8.6957 | 9.1041 | 100 | 23 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 99.5 | ||||||
Total average | 99.75 | ||||||||
|
|||||||||
GA-4 | 22.4615 | 6.4680 | 8.4840 | 100 | 22 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 99.5 | ||||||
Total average | 99.75 | ||||||||
|
|||||||||
GA-5 | 24.9011 | 9.3258 | 10.8618 | 100 | 24 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 99.5 | ||||||
Total average | 99.75 | ||||||||
|
|||||||||
PSO-1 | 30.0000 | 1.8006 | 6.9233 | 100 | 26 | Normal | 88 | Normal | 84 |
Inner | 100 | Inner | 100 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 97 | Average | 96 | ||||||
Total average | 96.5 | ||||||||
|
|||||||||
PSO-2 | 30.0000 | 1.7888 | 8.8493 | 100 | 27 | Normal | 92 | Normal | 88 |
Inner | 100 | Inner | 100 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 98 | Average | 97 | ||||||
Total average | 97.5 | ||||||||
|
|||||||||
PSO-3 | 30.0000 | 1.88131 | 7.6882 | 100 | 29 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 96 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 99 | ||||||
Total average | 99.5 | ||||||||
|
|||||||||
PSO-4 | 30.0000 | 7.4530 | 6.5468 | 100 | 24 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 99.5 | ||||||
Total average | 99.75 | ||||||||
|
|||||||||
PSO-5 | 20.5845 | 9.0898 | 7.2608 | 100 | 23 | Normal | 100 | Normal | 100 |
Inner | 100 | Inner | 98 | ||||||
Outer | 100 | Outer | 100 | ||||||
Ball | 100 | Ball | 100 | ||||||
Average | 100 | Average | 99.5 | ||||||
Total average | 99.75 |
As shown in Table
Table
SV number of the SVM model optimized by different methods at different speeds.
Methods | SV number | Average SV number | Total average SV number | |||
---|---|---|---|---|---|---|
1500 r/min | 1800 r/min | 2100 r/min | 2400 r/min | |||
EALO | 31 | 20 | 19 | 22 | — | 23.00 |
ALO | 59 | 33 | 20 | 27 | — | 34.75 |
GA-1 | 43 | 29 | 22 | 21 | 28.75 | 55.75 |
GA-2 | 48 | 18 | 17 | 22 | 26.25 | |
GA-3 | 34 | 21 | 20 | 23 | 24.5 | |
GA-4 | 163 | 200 | 19 | 22 | 101 | |
GA-5 | 200 | 30 | 139 | 24 | 98.25 | |
PSO-1 | 41 | 32 | 19 | 26 | 29.5 | 45.15 |
PSO-2 | 47 | 31 | 32 | 27 | 34.25 | |
PSO-3 | 171 | 22 | 22 | 29 | 61 | |
PSO-4 | 56 | 31 | 20 | 24 | 32.75 | |
PSO-5 | 200 | 31 | 19 | 23 | 68.25 |
The average bearing fault recognition rates using different optimization methods (e.g., EALO, ALO, GA, and PSO) at different speeds are shown in Table
Average recognition rate of bearing faults using four optimization methods at different speeds.
Methods | Recognition rate (%) | Average recognition rate (%) | Total average recognition rate (%) | |||
---|---|---|---|---|---|---|
1500 r/min | 1800 r/min | 2100 r/min | 2400 r/min | |||
EALO | 99.25 | 99.50 | 99.50 | 99.75 | — | 99.50 |
ALO | 98.00 | 99.00 | 99.50 | 97.75 | — | 98.56 |
GA-1 | 95.75 | 99.25 | 99.50 | 99.75 | 98.56 | 96.99 |
GA-2 | 98.25 | 99.00 | 99.50 | 99.75 | 99.13 | |
GA-3 | 98.50 | 99.25 | 99.50 | 99.75 | 99.25 | |
GA-4 | 95.50 | 86.50 | 99.50 | 99.75 | 95.31 | |
GA-5 | 73.00 | 99.00 | 99.00 | 99.75 | 92.69 | |
PSO-1 | 98.00 | 99.25 | 99.50 | 96.50 | 98.31 | 96.84 |
PSO-2 | 88.25 | 99.00 | 99.25 | 97.50 | 96.00 | |
PSO-3 | 86.75 | 99.50 | 99.50 | 99.50 | 96.31 | |
PSO-4 | 98.00 | 99.00 | 99.50 | 99.75 | 99.06 | |
PSO-5 | 79.75 | 99.00 | 99.50 | 99.75 | 94.50 |
Based on the classical ALO algorithm, the EALO algorithm was proposed by introducing an escape mechanism and adaptive iterative convergence conditions. This algorithm was then applied to the diagnosis of bearing faults. Comparing with more traditional methods (e.g., ALO, GA, and PSO), the following conclusions can be drawn: The escape mechanism was effective and reduced the possibility that the classical ALO algorithm would fall into a local extremum value. This improved the global optimization performance. The proposed adaptive convergence conditions effectively reduced the iteration number, saving optimization time and improving the optimization performance of the EALO algorithm. The proposed EALO algorithm was suitable for SVM parameter optimization. When compared with the classical ALO, GA, and PSO approaches, the EALO algorithm had the best performance. The feature extraction method based on the VMD and kernel function was effective and provides a new reference point for bearing fault diagnosis.
The data used to support the findings of this study are included in the supplementary file “Datasets.zip.”
The authors declare that they have no conflicts of interest.
This study was supported by the National Natural Science Foundation of China (Grant Nos. 11702091 and 51575178), the Natural Science Foundation of Hunan Province of China (Grant Nos. 2018JJ3140, 2019JJ50156, and 2018JJ4084), Hunan Provincial Key Research and Development Program (Grant No. 2018GK2044), and Open Funded Projects of Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment (Grant No. 201605).
The supplementary file “Datasets.zip” is the original bearing faults data were used in experiments.