The stochastic resonance (SR) method is widely applied to fault feature extraction of rotary machines, which is capable of improving the weak fault detection performance by energy transformation through the potential well function. The potential well functions are mostly set fixed to reduce computational complexity, and the SR methods with fixed potential well parameters have better performances in stable working conditions. When the fault frequency changes in variable working conditions, the signal processing effect becomes different with fixed parameters, leading to errors in fault detection. In this paper, an underdamped second-order adaptive general variable-scale stochastic resonance (USAGVSR) method with potential well parameters’ optimization is put forward. For input signals with different fault frequencies, the potential well parameters related to the barrier height are figured out and optimized through the ant colony algorithm. On this basis, further optimization is carried out on undamped factor and step size for better fault detection performance. Cases with diverse fault types and in different working conditions are studied, and the performance of the proposed method is validated through experiments. The results testify that this method has better performances of weak fault feature extraction and can accurately identify different fault types in the input signals. The method proves to be effective in the weak fault extraction and classification and has a good application prospect in rolling bearings’ fault feature recognition.
Rotary machines are generally applied to modern industrial production, and rolling bearings play a key role in rotary machines [
The information on the running states of rolling bearings is contained in vibration signals. In recent years, many scholars have used the vibration signals of rolling bearings for fault diagnosis [
In recent years, the SR method has been widely applied in fault feature extraction and recognition of bearings [
In order to simplify the complexity of the system and increase computational efficiency, most of the existing SR methods fix the parameters as specific values. The potential barrier is crucial in the output of the SR system. The optimal potential barrier corresponds to the optimal output of the system. However, different input signals have different dominant barriers, and fixed SR barrier parameters limit the system to achieving the optimal output. Optimizing the barrier height is the basis for the system to achieve the optimal output. Therefore, an underdamped second-order adaptive general variable-scale stochastic resonance method with the optimization of potential well parameters is introduced for bearing fault detection. According to different input signals, the optimal parameters of the system are adaptively matched by optimization algorithm, and optimal barriers corresponding to different input signals are obtained, respectively. On the basis of the most dominant barrier, the optimal matching of noise, the input signal, and the nonlinear system are realized, and the weak fault features of bearings are recognized.
The following research contents are as follows: in the second part, the USAGVSR diagnosis method is introduced. The third part carries on the simulation. In the fourth part, cases of diverse fault types in rolling bearings are studied. In the fifth part, the rolling bearing faults under different working conditions are studied, and the accuracy of the proposed method is verified. The sixth part is the result and discussion of this paper. The seventh part draws the conclusion of this paper.
The classical model of SR is a bistable system. The dynamic equation of bistable SR is [
The output effect of SR depends on the transition of the Brownian particle in the potential well. The motion diagram of the Brownian particle in the SR system is shown in Figure
The motion diagram of the Brownian particle.
As shown in Figure
In practical engineering applications, underdamping phenomenon is common, and the damping factor of the SR system will affect the output response of the system. Therefore, considering the damping factor of the system, the model of SR is expressed as
Classical SR is suitable for small parameters, but it is a high-frequency signal in engineering practice, so the general variable-scale stochastic resonance model is used to make it suitable for the high-frequency weak signal, and substitution variables are introduced [
Make
Equation (
The output of the USAGVSR system is complex and difficult to be analyzed by the analytical method. Therefore, the fourth-order Runge–Kutta is applied to numerical analysis. The expression of iterative algorithms is as follows:
For equation (
Make
The output SNR is [
In SR systems, the barrier height determines the output of SR. For different input signals, the corresponding most advantageous barrier is different. Therefore, it is very important to determine the most advantageous barriers for different input signals in practical engineering applications. In this paper, the bistable adaptive general variable-scale stochastic resonance model and ant colony optimization algorithm are applied to find the most advantageous well parameters
The steps of ant colony algorithm are as follows [ Initialize system parameters, including the number of ants Place Find the maximum probability and judge whether Output the maximum SNR and optimal parameters
For different input signals, the ant colony algorithm is applied to match the corresponding most advantageous barrier adaptively by calculating SNR, and the optimal parameters
The USAGVSR method with potential well parameters’ optimization.
The potential barrier plays an important role in the output of the SR system, and the optimal potential barrier corresponds to the optimal output of the system. For different input signals, the corresponding most advantageous barrier is different. Optimization of potential well parameters is the basis of SR system processing. The steps of the USAGVSR method with potential well parameters’ optimization are as follows: vibration signals of rolling bearings are collected by vibration sensors, and the collected vibration signals are preprocessed. The underdamping factor in equation (
To confirm the effectiveness of the proposed method, three analog signals are introduced to analysis parameters. The analog signal is
Barriers for different well parameters.
Figure
For three different analog signals, the optimal SNR is searched by the ant colony algorithm, respectively, the most dominant barriers corresponding to the analog signals are obtained, and the best potential well parameters
SNR variation of analog signal
max1 represents the maximum SNR corresponding to the analog signal in the case of the optimal potential well parameters, and max2 represents the maximum SNR corresponding to the analog signal in the case of the fixed potential well parameters. Similarly, for analog signal
SNR variation of (a) analog signal
The figures show that the most dominant barriers corresponding to different analog signals are different, and the most dominant well parameters are also different. The maximum SNR values corresponding to the most dominant well parameters of the three analog signals are all larger than the SNR values of the fixed potential well parameters. Therefore, for different input signals, it is crucial to determine the most dominant wells to obtain the best output of the system.
After determining the most dominant well parameters of the three analog signals, the SNR changes with the change of step size and underdamping factor are shown in Figure
The trend of SNR with (a) step size and (b) underdamping factor.
Figure
A machinery fault simulator is used for case analysis in this paper, as shown in Figure
The machinery fault simulator.
The testing bearings are installed on the test bench, and the bearing information is shown in Table
Information of the tested bearings.
Number of rolling elements | Rolling element diameter (inch) | Pitch diameter (inch) | Contact angle |
---|---|---|---|
8 | 0.3125 | 1.319 | 0 |
According to the relevant parameters of bearings, Table
Fault characteristic frequencies of rolling bearings.
Speed of the motor (r/min) | |||
---|---|---|---|
1200 | 61.05 | 98.95 | 39.84 |
2400 | 122.09 | 197.91 | 79.68 |
4800 | 244.19 | 395.82 | 159.36 |
The outer ring fault is used as a case, which is set in the outer raceway of the bearing. The speed of the motor is 2400 r/min. Figure
Original vibration signal.
As can be seen from Figure
At present, most of the existing SR methods fix the potential well parameters as specific values to simplify the complexity of the system and improve the efficiency of calculation. The potential well parameters are fixed as
Signal processed by SR.
As can be seen from Figure
The potential well parameters are fixed as
Signal processed by TFSR.
As can be seen from Figure
In the same way, the potential well parameters are fixed as
Signal processed by USSR.
Figure
On the basis of the USSR, the adaptive general variable-scale stochastic resonance is introduced, and the optimal barrier parameters are first made sure according to input signals. The collected vibration signal is preprocessed and then input into the adaptive general variable-scale stochastic resonance system. Set the scale coefficient
Signals processed by USAGVSR.
Figure
The fault is set in the inner raceway of the bearing. The speed of the motor is 2400 r/min. The original vibration signal time and frequency domain graphs are shown in Figure
Original vibration signal.
The potential well parameters are fixed as
Signal processed by SR.
Figure
The potential well parameters are fixed as
Signal processed by TFSR.
Figure
Similarly, the parameters
Signal processed by USSR.
Figure
Similarly, the collected vibration signal is input into the adaptive general variable-scale stochastic resonance system, and the corresponding most dominant barrier parameters are found by ant colony algorithm, which are
Signals processed by USAGVSR.
Figure
In addition, the fault is set in the rolling element of the bearing. The speed of the motor is 2400 r/min. Figure
Original vibration signal.
The potential well parameters are fixed as
Signal processed by SR.
Figure
The potential well parameters are fixed as
Signal processed by TFSR.
Figure
The parameters
Signal processed by USSR.
Figure
Similarly, the collected vibration signal is input into the adaptive general variable-scale stochastic resonance system, and the corresponding most dominant barrier parameters are found by ant colony algorithm, which are
Signals processed by USAGVSR.
Figure
The outer ring fault is used as a case, the rotational speed of the shaft is set to 1200 r/min, 2400 r/min, and 4800 r/min, respectively, and the rotation speed of 2400 r/min is analyzed as above. The following is an analysis of the rotational speeds of 1200 r/min and 4800 r/min.
The original vibration signal time and frequency domain graphs are shown in Figure
Original vibration signal.
As can be seen from Figure
The parameters
Signal processed by USSR.
Figure
Similarly, the collected vibration signal is input into the adaptive general variable-scale stochastic resonance system, and the corresponding most dominant barrier parameters are found by ant colony algorithm, which are
Signals processed by USAGVSR.
Figure
The original vibration signal time and frequency domain graphs are shown in Figure
Original vibration signal.
As can be seen from Figure
The parameters
Signal processed by USSR.
Figure
Similarly, the collected vibration signal is input into the adaptive general variable-scale stochastic resonance system, and the corresponding most dominant barrier parameters are found by ant colony algorithm, which are
Signals processed by USAGVSR.
Figure
The USAGVSR method proposed in this paper can effectively extract the weak fault feature information of rolling bearings. For different input signals, the adaptive general variable-scale stochastic resonance system is used to match the optimal potential barriers, and the optimal potential well parameters
Output SNR comparison of different fault types.
Output SNR of SR (dB) | Output SNR of TFSR (dB) | Output SNR of USSR (dB) | Output SNR of USAGVSR (dB) | |
---|---|---|---|---|
−0.5710 | −0.5821 | −0.5330 | −0.0328 | |
−0.7629 | −0.7751 | −0.7650 | −0.4319 | |
−0.6003 | −0.5969 | −0.5924 | −0.1355 |
Output SNR comparison of different rotation speeds.
Output SNR of USSR (dB) | Output SNR of USAGVSR (dB) | SNR increment (dB) | |
---|---|---|---|
1200 r/min | −−4.0449 | −3.2370 | 0.8079 |
2400 r/min | −0.5330 | −0.032 | 0.5002 |
4800 r/min | −1.2260 | −0.4633 | 0.7627 |
According to different fault types of rolling bearings and the faults under different working conditions, compared with the other methods, the SNR of the USAGVSR method is significantly improved. For different input signals, the USAGVSR method makes the transition frequency of the Brownian particle and the frequency of the input signal match best and determines the corresponding most advantageous barrier, which is the basis of subsequent SR processing. On the basis of the most dominant barrier, the optimal output of the SR is achieved. For different fault types and faults under different working conditions, the signals processed by this method have an obvious peak value at
The potential well parameters’ optimization on USAGVSR is conducted for weak fault detection of rolling bearings. The main findings are as follows: This method is compared with traditional methods with fixed potential well parameters, and the SNR is significantly improved, the fault characteristics of the output waveform are more obvious, and the bearing fault types are easier to identify According to different input signals, the corresponding barriers are adaptively matched, which lays a foundation for SR processing, and it is easier to obtain the best system output The method extracts the weak fault features of rolling bearings, which is beneficial to fault identification and accurate determination of fault types
The USAGVSR method has a good application prospect in rolling bearing fault diagnosis. In future research, we will combine the advantages of the proposed method with other fault diagnosis methods to carry out weak fault feature recognition of rolling bearings.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Science Foundation of China (nos. 52075348, 51905357, 52005352, and 51805337), Key Laboratory of Vibration and Control of Aero-Propulsion System, Ministry of Education, Northeastern University (VCAME202001), and Natural Science Foundation of Liaoning Province (no. 2019-ZD-0654).