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Since the weak fault characteristics of mechanical equipment are often difficult to extract in strong background noise, stochastic resonance (SR) is widely used to extract the weak fault characteristics, which is able to utilize the noise to amplify weak fault characteristics. Although classical bistable stochastic resonance (CBSR) can enhance the weak characteristics by adjusting the parameters of potential model, when potential barrier height is adjusted potential well width is also changed and vice versa. The simultaneous change of both potential well width and barrier height is difficult to obtain a suitable potential model for better weak fault characteristic extraction and further fault diagnosis of machinery. For this reason, the output signal-to-noise ratio (SNR) of CBSR is greatly reduced, and the corresponding enhancement ability of weak fault characteristics is limited. In order to avoid the shortcomings, a new SR method is proposed to extract weak fault characteristics and further diagnose the faults of rotating machinery, where the classical bistable potential is replaced with a bistable confining potential to get the optimal SR. The bistable confining potential model not only has the characteristics of the classical bistable potential model but also has the ability to adjust the potential width, barrier height, and wall steepness independently. Simulated data are used to demonstrate the proposed new SR method. The results indicate that the weak fault characteristics can be effectively extracted from simulated signals with heavy noise. Experiments on the bearings and planetary gearboxes demonstrate that the proposed SR method can correctly diagnose the faults of rotating machinery and moreover has higher spectrum peak and better recognition degree compared with the CBSR method.

The vibration test is often used method in fault diagnosis of mechanical equipment, but the mechanical equipment is usually affected by heavy noise, multi vibration source excitation and response mutual coupling, human disturbance, and so on [

In Section

Most of the research studies on SR focus on the traditional bistable model. The CBSR can be explained as a particle driven by noises and the cycle force. As the particles are driven by noise, the cycle force will be enhanced. The governing equation that illustrates such a phenomenon under the assumption of overdamped condition can be introduced by the following equation:

The bistable potential function with different parameters.

To analyze the difference and similarity between the classical bistable potential and the bistable confining potential, the bistable potential curves with different potential parameters are described in Figure

The bistable confining potential function with different parameters: (a) the bistable confining potential change with parameter

Substitute Equation (

The

The essential of SR phenomenon is that the particles move in the potential well under the interaction of potential force, periodic force, and noise to consequently achieve the enhancement of system output. In three forces, the periodic force and the noise are fixed. Thus, the effect of SR depends on the change in potential force. In general, if the distance between two potential walls is too far, the particles could not arrive to the wall edge that it is must give the greater reverse elasticity. On the other hand, when the distance between two potential walls is too narrow, the particles cannot arrive to the potential wall edge, so it will go back prematurely. Similarly, if the potential wall is too steep, the particles may rebound rapidly due to an intense reverse elasticity. When the restoring speed is too fast, the periodic oscillation could not catch up with the restoring speed. If the wall is too gentle, it cannot give a large enough acceleration to cause the particles to periodic oscillation. Therefore, when the potential model has the optimal structure, the system output can have the best enhancement effect.

With these features, to enhance the periodic signal to an extreme, the potential model should be well adjusted. For the classical bistable potential (Equation (

To optimally extract the target signal from the strong noise, the optimal input signal can be obtained by tuning the potential parameters. Here, the performance of the weak signal detection is evaluated by employing the output SNR as a criterion. Subsequently, a strategy to extract the weak fault characteristics signal based on SR with bistable confining potential is synthetically presented as shown in Figure

Signal preprocessing: some common technologies include filtering part of noises, detecting the driving frequency by signal demodulation when the original signal is modulated, and the shifting-frequency rescaling transform which can be performed to satisfy the small parameter condition when the driving frequency is greater than 1.

Parameter initialization: initialize the computed ranges of the bistable confining potential parameters

Output calculation: compute the output waveform by using Equation (

Output assessment: look for the maximum SNR in ternary array that corresponds to the varying variables

Postprocessing of the processed signal: the processed signal is input to the SR system to calculate the final output and realize the fault diagnosis of the mechanical equipment.

Proposed strategy to detect weak signal based on SR with bistable confining potential.

To intuitively illustrate the generated interaction between the system output and the potential parameters, a simulated failure bearing signal is produced and processed by the BCPSR method as mentioned above. As the rolling bearing is often in the form of impact, the periodic unilateral attenuation pulse signal with AGWN is selected as the simulated signal waveform. The simulated waveform is generated by virtue of the below equation:

Simulated bearing fault signal: (a) periodic impulse signal without the AGWN, (b) periodic impulse signal with the AGWN, and (c) corresponding spectrum.

After that, the SR equation with the bistable confining potential for dealing with the simulated bearing failure signal is as follows:

Since the simulated bearing fault signal is modulated, the Hilbert transform (HT) can be employed to detect the distinct fault-induced impulse features. Meanwhile, the envelop signal of the simulated failure bearing obtained by using HT is depicted in Figure

(a) Time-domain waveform of envelope signal and (b) corresponding spectrum.

(a) Optimal output signal using the proposed BCPSR method and (b) corresponding power spectrum; (c) optimal output signal using the CBSR method and (d) corresponding power spectrum.

To prove the validity of the BCPSR method, the new method is carried over into the fault characteristic frequency extraction of a slight damage inner race of bearing. To further analyze the powerful effect of the new method, the experimental data of missing tooth and broken tooth of planetary gearbox are used to prove the validation of the new method.

Rolling bearing is the key component of rotating equipment, and the working state of the rolling bearing is bound up with the operation reliability of the rotating machinery. However, the rolling bearing is one of the most easily damaged parts in the machine. In this case, the extraction of the slight damage inner race of bearing signal is used to confirm the effectiveness of the BCPSR method. Mechanical equipment failure comprehensive test bench was used in the experiment, as shown in Figure

Mechanical equipment failure comprehensive test bench.

Vibration signal and the spectrum of slight damage inner race of bearing: (a) time-domain waveform, (b) power spectrum, and (c) envelope spectrum.

A slight damage inner race of bearing: (a) optimal output signal using the proposed BCPSR method and (b) corresponding power spectrum; (c) optimal output signal using the CBSR method and (d) corresponding power spectrum.

Comparing with ordinary fixed-axis gearbox, planetary gearbox structure is more complex and has unique operation mode and common complex dynamic load forces. Therefore, the fault diagnosis problem of planetary gearbox should be taken seriously. In this case, the fault extraction experiment on missing tooth and broken tooth of the planetary gearbox is used to prove the correctness of the BCPSR method. The test platform of planetary gearbox is used in the experiment. The three-dimensional model and physical model of the system bench are shown in Figures

The experimental platform of planetary gearbox: (a) three-dimensional model and (b) physical map.

Parameters of planetary gearbox.

Forms of gear | Sun gear | Planet gear | Gear ring | Number of planetary gear |
---|---|---|---|---|

Teeth number of stage 1 | 20 | 40 | 100 | 3 |

Teeth number of stage 2 | 28 | 36 | 100 | 4 |

The physical map of experimental gears: (a) missing tooth and (b) broken tooth.

Motor speed and the characteristic frequency of the first-stage planetary gearbox.

Motor speed | Rotating frequency of sun gear ( |
Rotating frequency of planet gear ( |
Rotating frequency of gear ring ( |
Meshing frequency ( |
---|---|---|---|---|

600r/min | 10 Hz | 1.667 Hz | 0 Hz | 166.667 Hz |

The time-domain, power spectrum, and envelope spectrum of the collected missing tooth and broken tooth signals are shown in Figure

Vibration signal and the spectrum of the missing tooth: (a) time-domain waveform, (b) power spectrum, and (c) envelope spectrum. Vibration signal and the spectrum of the broken tooth: (d) time-domain waveform, (e) power spectrum, and (f) envelope spectrum.

The missing tooth of planetary gearbox: (a) optimal output signal using the proposed BCPSR method and (b) corresponding power spectrum; (c) optimal output signal using the CBSR method and (d) corresponding power spectrum.

The broken tooth of planetary gearbox: (a) optimal output signal using the proposed BCPSR method and (b) corresponding power spectrum; (c) optimal output signal using the CBSR method and (d) corresponding power spectrum.

As the system parameters of the CBSR method are optimized, it is difficult to obtain the perfect potential model structure due to the simultaneous change of the barrier height and width, which hinders the enhancement of the weak fault characteristic frequency and reduces the output SNR. In order to overcome this shortcoming, a new method for the BCPSR is proposed. The following conclusions are drawn:

The BCPSR system model is analyzed and its potential function and structural characteristics are described. Compared with the CBSR system, the BCPSR system parameters can not only be independently adjusted for the local shape of the potential model but also have a higher theoretical output SNR, which implies that it can better detect and extract the weak signal in the strong background noise.

When the barrier height of the CBSR system potential model is adjusted, the potential width is changed and vice versa. When the potential width and barrier height are changed synchronously, the potential structure will not be perfect. The BCPSR system potential model can make the potential width, barrier height, and wall steepness adjusted independently and simultaneously, so the potential model has better matching parameters.

The proposed BCPSR method is applied to a slight damage inner race of bearing and the missing tooth and broken tooth of the planetary gearbox. The experiment indicates that the BCPSR method not only can extract the weak characteristic frequency but also has better recognition, higher fault characteristic frequency spectrum peak, and better SNR than the CBSR method.

The proposed BCPSR method is based on the overdamped state. The SNR output and the weak fault diagnosis method of the BCPSR on the underdamped state and the factors of affecting the saturation for the CBSR are the focus of further work in the future. In addition, the proposed BCPSR method is applied to the actual engineering verification of the planetary gearbox to further determine the application effect of the method in engineering.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This research was supported by the National Science Foundation of China (51805275).