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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 (_{1} is changed to 1/^{2} of the original signal, it can meet the requirements of classical SR for small parameters.

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 [_{S}(_{N}(

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 _{1} and _{1} corresponding to different input signals, and the optimal potential barriers are found. The SNR is applied to evaluate the output of the system, and SNR is expressed as_{k}) represents the discrete Fourier transform of the corresponding signal sequence _{1}(_{2}(

The steps of ant colony algorithm are as follows [

Initialize system parameters, including the number of ants _{1}, pheromone concentration

Place _{1} ants randomly on the initial node, constantly modify the information of nodes with ants, calculate SNR, and update pheromone based on the SNR.

Find the maximum probability and judge whether

Output the maximum SNR and optimal parameters _{1} and _{1} of the system.

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 _{1} and _{1} are determined. On this basis, the signals are processed by the underdamped second-order stochastic resonance (USSR) system, and then ant colony optimization algorithm is applied to search for optimal underdamped factor and step size, so as to achieve the optimal system response. The USAGVSR method with potential well parameters’ optimization is shown in Figure

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 (_{1} and _{1} are output. Then, the vibration signals are input to the underdamped second-order stochastic resonance system and take advantage of the corresponding most advantageous barriers, and the system parameters are initialized. The optimum SNR is searched by the optimization algorithm, and then the optimal output parameters are obtained. The corresponding frequency domain graphs of vibration signals are obtained, the weak fault features are extracted, and the fault types are identified.

To confirm the effectiveness of the proposed method, three analog signals are introduced to analysis parameters. The analog signal is _{1} = 0.1; _{2} = 0.5; _{3} = 1, frequency _{1} = 50 Hz; _{2} = 100 Hz; and _{3} = 150 Hz, and Gaussian white noise is added to the analog signals, respectively. For given different potential well parameters _{1} and _{1}, Figure

Barriers for different well parameters.

Figure _{1} and _{1} are different, the height and width of the barrier corresponding to the system are also different, and the Brownian particle also has different transition frequencies. Therefore, for different input signals, it is important to determine the most dominant barriers and the best potential well parameters in order to achieve the optimal system output.

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 _{1} and _{1} are obtained. For analog signal _{1}, the optimal barrier height _{1} = 0.241 and _{1} = 0.852. Set fixed potential well parameters _{2} = 1 and _{2} = 1. For the optimal and fixed potential well parameters, the SNR change of analog signal _{1} with the increase of

SNR variation of analog signal _{1.}

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 _{2}, the optimal barrier height _{1} = 0.076 and _{1} = 0.754. For analog signal _{3}, the optimal barrier height _{1} = 0.364 and _{1} = 0.958. Figure _{2} and _{3} with the increase of

SNR variation of (a) analog signal _{2} and (b) analog signal _{3}.

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 _{cf}) of rolling bearings are as follows:_{cf} of the outer ring, inner ring, and rolling element, respectively, _{1} represent the diameter and number of rolling elements, respectively, _{1} is the pitch diameter, _{1} is the rotational speed of the shaft.

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 _{c}) and double rotational frequency (2_{c}) are more obvious, there are a large number of mixed peaks, and the weak fault features of rolling bearings are not easy to extract, and they can easily lead to misdiagnosis. The proposed USAGVSR method is used to process the vibration signal and compared with the stochastic resonance, time-delayed feedback stochastic resonance (TFSR), and underdamped second-order stochastic resonance methods as follows.

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 _{2} = 1 and _{2} = 0.2, the collected vibration signal is preprocessed, and the carrier frequency _{c1} = 100 Hz is set for modulation processing. Then, the SR system deals with the processed vibration signal, and the step size is optimized by ant colony algorithm. When SNR = −0.5710 dB, the best result is obtained,

Signal processed by SR.

As can be seen from Figure _{c} and 2_{c} are more obvious. Moreover, there is an obvious peak value at the inner ring fault characteristic frequency, which easily leads to the misidentification of bearing fault types.

The potential well parameters are fixed as _{2} = 1 and _{2} = 0.2. The TFSR system is used to process the vibration signal, and ant colony algorithm is used to optimize the step size

Signal processed by TFSR.

As can be seen from Figure _{c} and 2_{c} are obvious, which is not conducive to the extraction of rolling bearing weak fault features.

In the same way, the potential well parameters are fixed as _{2} = 1 and _{2} = 0.2. The USSR system is used to process the vibration signal, and the underdamped factor and step size are optimized by ant colony algorithm. When SNR = −0.533 dB,

Signal processed by USSR.

Figure _{c} is obvious, and the corresponding peak value at _{cf} is also evident.

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 _{1} and _{1} are found by ant colony algorithm. When SNR is the largest, the optimal barrier and well parameters _{1} and _{1} are obtained, which are _{1} = 0.0315, and _{1} = 0.0609, respectively. After the optimal barrier is determined, the USSR system processes the vibration signal, and the underdamped factor and step size are optimized by ant colony algorithm. The optimal system result is obtained when SNR = −0.0328 dB, at which

Signals processed by USAGVSR.

Figure _{c}, and compared with the above systems, the peak value at _{cf} is more obvious, and the SNR is also significantly improved. Therefore, the USAGVSR system has better effects, more weak fault features are extracted, and the fault types are accurately recognized.

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 _{c} is obvious, the fault features are weak, and they are not easy to extract.

Original vibration signal.

The potential well parameters are fixed as _{2} = 1 and _{2} = 0.2. The SR system is used to process the vibration signal, and ant colony algorithm is used to optimize the step size. When SNR = −0.7629 dB and

Signal processed by SR.

Figure _{c} is obvious, the fault features are weak, and there are some chaotic peaks, which are not conducive to the extraction of fault information.

The potential well parameters are fixed as _{2} = 1 and _{2} = 0.2. The TFSR system is used to process the vibration signal, and ant colony algorithm is used to optimize the step size and feedback strength. When SNR = −0.7751 dB,

Signal processed by TFSR.

Figure

Similarly, the parameters _{2} = 1 and _{2} = 0.2 are set, and the vibration signal is processed by the USSR system. When SNR = −0.765 dB, the optimal system output is obtained, at which

Signal processed by USSR.

Figure _{cf}.

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 _{1} = 0.175, _{1} = 0.0625, and

Signals processed by USAGVSR.

Figure _{cf} is relatively evident, and compared with the processing results of the above systems, SNR is also significantly improved.

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 _{2} = 1 and _{2} = 0.2. The SR system is used to process the vibration signal, and ant colony algorithm is used to optimize the step size. When SNR = −0.6003 dB and

Signal processed by SR.

Figure _{c} is relatively obvious, the fault feature is weak, and the fault information of the rolling bearing cannot be extracted.

The potential well parameters are fixed as _{2} = 1 and _{2} = 0.2. The TFSR system is used to process the vibration signal, and ant colony algorithm is used to optimize the step size and feedback strength. When SNR = −0.5969 dB,

Signal processed by TFSR.

Figure _{c}, and the fault feature is not obvious. In addition, there are a large number of hybrid peaks at _{cf}, and the fault identification of the rolling bearing cannot be carried out.

The parameters _{2} = 1 and _{2} = 0.2 are set, and the vibration signal is processed by the USSR system. When SNR = −0.5924 dB, the optimal system output is obtained, at which

Signal processed by USSR.

Figure _{cf}, which can diagnose the existence of the rolling element fault, but the corresponding amplitude at _{c} is relatively high.

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 _{1} = 0.0875, _{1} = 0.0625, and

Signals processed by USAGVSR.

Figure _{c} is significantly decreased, the amplitude at _{cf} is obviously increased, and SNR is also significantly improved. Therefore, the USAGVSR method has a better effect and can precisely identify the fault types of bearings.

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 _{cf} of the original vibration signal is obvious, but there are some hybrid peaks, which are not conducive to the weak fault feature extraction of rolling bearings. The original vibration signal is processed by USSR and the proposed USAGVSR method, and the processing results are as follows.

The parameters _{2} = 1 and _{2} = 0.2 are set, and the vibration signal is processed by the USSR system. When SNR = −4.0449 dB, the optimal system output is obtained, at which

Signal processed by USSR.

Figure _{cf}, but there is still a high hybrid peak frequency, which is easy to cause misdiagnosis.

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 _{1} = 0.42, _{1} = 0.094, and

Signals processed by USAGVSR.

Figure _{cf} of the signal processed by the USAGVSR system is obvious, the hybrid peak value is weakened, and the SNR is improved obviously.

The original vibration signal time and frequency domain graphs are shown in Figure

Original vibration signal.

As can be seen from Figure _{c} amplitude of the original vibration signal is obvious, while the _{cf} amplitude is the lower, which is not conducive to the weak fault feature extraction of rolling bearings. The original vibration signal is processed by USSR and the proposed USAGVSR method, and the processing results are as follows.

The parameters _{2} = 1 and _{2} = 0.2 are set, and the vibration signal is processed by the USSR system. When SNR = −1.226 dB, the optimal system output is obtained, at which

Signal processed by USSR.

Figure _{c} and 2_{c}, but _{cf} cannot be extracted, which is not conducive to the fault diagnosis of rolling bearings.

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 _{1} = 0.467, _{1} = 0.018, and

Signals processed by USAGVSR.

Figure _{c} and _{cf} of the signal processed by the USAGVSR system are obvious, and the SNR is improved obviously, which is beneficial to extracting the fault characteristic information of rolling bearings. Therefore, the USAGVSR method has a better effect and can precisely identify the fault types of bearings.

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 _{1} and _{1} are determined. Then, the signal is processed by the USSR system, and the parameters are adjusted by ant colony algorithm to get the optimal system output. When the fault types are different, the comparison between the proposed method and the results of the above three systems is shown in Table

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 _{cf}, which can better extract weak fault information, obtain the optimal output results, and accurately recognize the fault types.

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).