Weak signal detection based on underdamped stochastic resonance with an exponential bistable potential

Stochastic resonance (SR) is a vital approach to detect weak signals submerged in strong background noise, which is useful for mechanical fault diagnosis. The underdamped bistable SR (UBSR) is a kind of the most used SR, however, their potential structures are deficient to match with the complicated and diverse mechanical vibration signals and their parameters are selected subjectively which probably resulting in poor performance of UBSR. To overcome these shortcomings, this paper proposes an underdamped SR with exponential potential (UESR) which is generalized by using a harmonic model and a Gaussian potential (GP) model. The dynamics in UESR system is evaluated by the signal-to-noise ratio (SNR) which represents the effectiveness of noise utilization. Then, the effects of system parameters on system performance are investigated by output SNR versus noise intensity D for different parameters. Finally, the proposed method is used to process bearing experimental data and further perform bearing fault diagnosis. The experimental results demonstrate that a larger output SNR and higher spectrum peaks at fault characteristic frequencies can be obtained by the proposed method compared with the UBSR method, which confirm the effectiveness of the proposed method.     

Stochastic resonance in a single-well system with exponential potential driven by Levy noise

Noise and potential function are vital to stochastic resonance (SR). This paper attempts to broaden the research of the SR and explore a better potential function. Based on the absolute and exponential potentials, a generalized exponential type single-well potential function is constructed. Then the characteristics of the corresponding exponential type single-well SR (ESR) system driven by Levy noise is analyzed numerically. Firstly, the effects of the characteristic index α, symmetric parameter β and noise intensity D of Levy noise on the input signal to noise ratio (SNRi) are investigated. Then, the effects of system parameters a, b, r and noise intensity D on the resonant output is explored. Finally, the ESR system is applied to the fault characteristic extraction of rolling element bearings. The simulation results show that the SR phenomenon is able to be excited by tuning the parameters a, b, r and D under different values of α and β. The larger b (or a) widens the parameter interval of a (or b) which can induce SR. The ESR system is able to solve the problem that the traditional systems fail to achieve SR due to the improper selection of parameters. In bearing fault detection, the detection effect of the ESR system is superior to the bistable SR system.     

Stochastic resonance in an asymmetric bistable system driven by multiplicative and additive Gaussian noise and its application in bearing fault detection

The phenomenon of stochastic resonance (SR) in a new asymmetric bistable model is investigated. Firstly, a new asymmetric bistable model with an asymmetric term is proposed based on traditional bistable model and the influence of system parameters on the asymmetric bistable potential function is studied. Secondly, the signal-to-noise ratio (SNR) as the index of evaluating the model are researched. Thirdly, Applying the two-state theory and the adiabatic approximation theory, the analytical expressions of SNR is derived for the asymmetric bistable system driven by a periodic signal, unrelated multiplicative and additive Gaussian noise. Finally, the asymmetric bistable stochastic resonance (ABSR) is applied to the bearing fault detection and compared with classical bistable stochastic resonance (CBSR) and classical tri-stable stochastic resonance (CTSR). The numerical computations results show that:(1) the curve of SNR as a function of the additive Gaussian noise and multiplicative Gaussian noise first increased and then decreased with the different influence of the parameters a, b, r and A; This demonstrates that the phenomenon of SR can be induced by system parameters; (2) by parameter compensation method, the ABSR performs better in bearing fault detection than the CBSR and CTSR with merits of higher output SNR, better anti-noise and frequency response capability.     

Asymmetric delay feedback stochastic resonance detection method based on prior knowledge particle swarm optimization

For the adjustable parameters stochastic resonance system, the selection of the structural parameters plays a decisive role in the performance of the detection method. The vibration signal of rotating machinery is non-linear and unstable, and its weak fault characteristics are easily concealed by noise. Under strong background noise interference, the detection of fault features is particularly challenging. Therefore, a type of weak fault feature extraction method, named knowledge-based particle swarm optimization algorithm for asymptotic delayed feedback stochastic resonance (abbreviated as KPSO-ADFSR) is proposed. Through deduction under adiabatic approximation, we observe that both the asymmetric parameters, the length of delay and the feedback strength, impact the potential function. After adjusting the asymmetric parameters of the system, the output signal-to-noise ratio (SNR) is used as the fitness function, and the setting of the relationship between the noise intensity and barrier height is used as the prior knowledge of the particle swarm algorithm. Through this algorithm, the delay length and the feedback strength are optimized. This method achieves global optimization of system parameters in a short time; it overcomes the shortcomings of the traditional stochastic resonance method, which has a long convergence time and tends to easily fall into local optimization. It can effectively improve the detection of weak fault features. In the bearing rolling body pitting corrosion failure experiment and steel field engineering experiment, the proposed method could extract the characteristics of a weak fault more effectively than the traditional stochastic resonance method based on the standard particle swarm algorithm.     

A new method for weak fault feature extraction based on piecewise mixed stochastic resonance

In a continuous bistable system, when the input signal is continuously increased, the output signal tends to be stable and no longer increases. At this time, the weak signal under strong background noise is difficult to be extracted, which means saturation occurs. Aiming at the saturation characteristics of stochastic resonance (SR), the proposed piecewise nonlinear bistable system (PNBSR) model has achieved certain results. However, the potential barrier in the middle of the PNBSR method still completely uses the potential function of classical bistable stochastic resonance (CBSR). There is no fundamental solution to the fourth-order limitation. This paper explores an improved piecewise mixed stochastic resonance (PMSR) potential model. The fourth-order potential function that restricts particle motion in CBSR is improved to a piecewise second-order potential function. This potential function subverts the shape of the traditional potential function and presents a symmetrical double-hook shape. Based on PMSR model, this paper uses particle swarm optimization (PSO) to select system parameters and elaborates the characteristics of the potential function curve in detail. Under the same conditions, the output signal-to-noise ratio (SNR) curve of the improved system is generally higher than that of the CBSR and PNBSR systems. Experiments on bearings and gears show that the proposed method can accurately extract weak fault features, and the effect is better than the PNBSR method.     

Adaptive bistable stochastic resonance and its application in mechanical fault feature extraction

Stochastic resonance (SR) is an important approach to detect weak vibration signals from heavy background noise. In order to increase the calculation speed and improve the weak feature detection performance, a new bistable model has been built. With this model, an adaptive and fast SR method based on dyadic wavelet transform and least square system parameters solving is proposed in this paper. By adding the second-order differential item into the traditional bistable model, noise utilization can be increased and the quality of SR output signal can be improved. The iteration algorithm for implementing the adaptive SR is given. Compared with the traditional adaptive SR method, this algorithm does not need to set up the searching range and searching step size of the system parameters, but only requires a few iterations. The proposed method, discrete wavelet transform and the traditional adaptive SR method are applied to analyzing simulated vibration signals and extracting the fault feature of a rotor system. The contrastive results verify the superiority of the proposed method, and it can be effectively applied to weak mechanical fault feature extraction.     

三稳系统的动态响应及随机共振

以平衡点参数p, q构造出一类对称三稳势函数, 进而提出微弱信号和噪声共同驱动的三稳系统模型. 深入研究并总结参数p, q对势垒高度ΔU₁, ΔU₂及两势垒高度差的影响. 从定常输入的角度提出了系统稳态解曲线的概念, 并进一步研究低频谐波信号输入时系统的输出动态响应. 引入噪声, 三稳系统在合适的参数条件下实现随机共振, 从稳态解曲线的角度分析了噪声诱导的三稳系统随机共振机理. 最后研究了阻尼比k和平衡点参数p, q对系统随机共振的影响.     

Research and application of coupled two-dimensional asymmetric bistable stochastic resonance system

Considering that the vibration signal of rolling bearing is very weak and difficult to be extracted under the working environment noise, a two-dimensional asymmetric bistable system (TDAB) is proposed. The system is coupled by two systems, and the weak signal is enhanced by adjusting the control system (monostable system) to adjust the controlled system (asymmetric bistable system). First, under the condition of adiabatic approximation system, the signal-to-noise ratio (SNR) is deduced, and analyzes the influence of different system parameters on the shape of potential function. Then the effects of each parameter on the SNR of the system are analyzed theoretically. Finally, combined with the adaptive intelligent algorithm, the parameters of the controlled system are optimized, and then the coupling coefficient and control system parameters are adjusted to obtain better system performance. The TDAB system obtained is applied to different bearing fault diagnosis and compared with different coupling systems. The experimental results show that the method can extract the characteristic frequency effectively and has good spectrum amplification performance and anti-noise capability. It is proved that the TDAB system can also have good detection effect in practical application.     

Intelligent Fault Diagnosis of Rolling-Element Bearings Using a Self-Adaptive Hierarchical Multiscale Fuzzy Entropy

The fuzzy-entropy-based complexity metric approach has achieved fruitful results in bearing fault diagnosis. However, traditional hierarchical fuzzy entropy (HFE) and multiscale fuzzy entropy (MFE) only excavate bearing fault information on different levels or scales, but do not consider bearing fault information on both multiple layers and multiple scales at the same time, thus easily resulting in incomplete fault information extraction and low-rise identification accuracy. Besides, the key parameters of most existing entropy-based complexity metric methods are selected based on specialist experience, which indicates that they lack self-adaptation. To address these problems, this paper proposes a new intelligent bearing fault diagnosis method based on self-adaptive hierarchical multiscale fuzzy entropy. On the one hand, by integrating the merits of HFE and MFE, a novel complexity metric method, named hierarchical multiscale fuzzy entropy (HMFE), is presented to extract a multidimensional feature matrix of the original bearing vibration signal, where the important parameters of HMFE are automatically determined by using the bird swarm algorithm (BSA). On the other hand, a nonlinear feature matrix classifier with strong robustness, known as support matrix machine (SMM), is introduced for learning the discriminant fault information directly from the extracted multidimensional feature matrix and automatically identifying different bearing health conditions. Two experimental results on bearing fault diagnosis show that the proposed method can obtain average identification accuracies of 99.92% and 99.83%, respectively, which are higher those of several representative entropies reported by this paper. Moreover, in the two experiments, the standard deviations of identification accuracy of the proposed method were, respectively, 0.1687 and 0.2705, which are also greater than those of the comparison methods mentioned in this paper. The effectiveness and superiority of the proposed method are verified by the experimental results.     

A novel piecewise tri-stable stochastic resonance system with time-delayed feedback and its application

It is of great significance to judge whether mechanical equipment has faults, so it is necessary to study the extraction of mechanical fault characteristic signals. Stochastic resonance (SR) has been applied diffusely in feature extraction because of its excellent output performance, but there are few studies on SR with time-delay feedback (TF) terms. In some cases, the output of the system will be improved when the TF term is added to the SR system, so it is meaningful to study the SR with TF term. Because piecewise tri-stable system has good characteristics of overcoming output saturation, on the basis of piecewise tri-stable SR (PTSR), the time-delay feedback PTSR (TFPTSR) is proposed, and for purpose of further studying the internal mechanism of this system, its generalized potential function and the law that the parameter causes its change are derived and studied. Then the probability density function (PDF) of the proposed model and its mean first-passage time (MFPT) are calculated and compared with the variation of the generalized potential function together with the Signal to noise ratio (SNR), through such research, the difficulty of the system to produce stochastic resonance and the degree of the output performance are directly related to the system parameters. Finally, the proposed TFPTSR method processes the same signal as the PTSR method, and it is found that the TFPTSR method can get better output SNR.     

Wayside acoustic defective bearing detection based on improved Dopplerlet transform and Doppler transient matching

The diagnosis of train bearing defects plays a significant role in maintaining the safety of railway transport. However, the phenomenon of Doppler Effect in the acoustic signal recorded by the wayside Acoustic Defective Bearing Detector (ADBD) system leads to the difficulty for fault diagnosis of train bearings with a high moving speed. This paper proposes a double-searching solution based on improved Dopplerlet transform and Doppler transient matching to overcome the difficulty in wayside acoustic bearing diagnosis. In the solution, the first searching procedure is to extract necessary parameters of Doppler Effect under the situation with very low signal-to-noise ratio (SNR) based on an improved Dopplerlet transform. Using the obtained parameters, the Doppler Effect can be embedded into the constructed periodic Laplace wavelet transient models. Subsequently, the second searching procedure is conducted to search fault impact period of the defective bearing through an operation, called Doppler transient matching, which is to calculate the correlation coefficient between the Doppler transient model and the filtered raw signal with the Doppler Effect. The proposed double-searching algorithm can adapt to the real Doppler Effect situation and extract the exact fault impact period from the Doppler distorted signal, and thus shows powerful capability to analyze wayside acoustic signals from train bearings. The proposed wayside acoustic diagnostic scheme is verified by means of a simulated Doppler distorted signal with a very low SNR (−20 dB) and the experiments conducted on train bearings. The results indicate that the proposed algorithm is effective and has obvious advantages for ADBD system.     

A multiscale noise tuning stochastic resonance for fault diagnosis in rolling element bearings

Condition monitoring of rotating machinery is important to extend the mechanical system's reliability and operational life. However, in many cases, useful information is often overwhelmed by strong background noise and the defect frequency is difficult to be extracted. Stochastic resonance (SR) is used as a noise-assisted tool to amplify weak signals in nonlinear systems, which can detect weak signals of interest submerged in the noise. The multiscale noise tuning SR (MSTSR), which is originally based on discrete wavelet transform (DWT), has been applied to identify the fault characteristics and has also increased the signal-to-noise ratio (SNR) improvement of SR. Therefore, a novel tri-stable SR method with multiscale noise tuning (MST) is proposed to extract fault signatures for fault diagnosis of rotating machinery. The wavelet packets transform (WPT) based MST can obtain better denoising effect and higher SNR of resonance output compared with the traditional SR method. Thus the proposed method is well-suited for enhancement of rotating machine fault identification, whose effectiveness has been verified by means of practical vibration signals carrying fault information from bearings. Finally, it can be concluded that the proposed method has practical value in engineering.     

Double Gaussian potential stochastic resonance method and its application in centrifugal fan blade crack detection

Stochastic resonance (SR) has been extensively utilized in the field of weak fault signal detection for its characteristic of enhancing weak signals by transferring the noise energy. Aiming at solving the output saturation problem of the classical bistable stochastic resonance (CBSR) system, a double Gaussian potential stochastic resonance (DGSR) system is proposed. Moreover, the output signal-to-noise ratio (SNR) of the DGSR method is derived based on the adiabatic approximation theory to analyze the effect of system parameters on the DGSR method. At the same time, for the purpose of overcoming the drawback that the traditional SNR index needs to know the fault characteristic frequency (FCF), the weighted local signal-to-noise ratio (WLSNR) index is constructed. The DGSR with WLSNR can obtain optimal parameters adaptively, thereby establishing the DGSR system. Ultimately, a DGSR method is proposed and applied in centrifugal fan blade crack detection. Through simulations and experiments, the effectiveness and superiority of the DGSR method are verified.     

A novel parameter-induced adaptive stochastic resonance system based on composite multi-stable potential model

Stochastic resonance (SR) is used widely as a weak signal detection method by using noise in many fields. In order to improve the weak signal processing capability of SR, a novel composite multi-stable model is proposed, which is constructed by the joint of the tristable model and the Gaussian Potential (GP) model. The SR system based on this model is constructed and the signal-to-noise ratio (SNR) is regarded as the index to measure the SR effect. The differential brain storm optimization (DBSO) algorithm is used to optimize the system parameters collaboratively to achieve parameter-induced adaptive SR. The influences of the system parameters V and R and the noise intensity D on the output response of SR system are analyzed under Gaussian white noise and α stable noise environments, and the advantages of the composite multi-stable SR system over the traditional tristable system are verified. For different levels of weak signals, the output performances of SR systems based on composite multi-stable model, traditional tristable model, composite tristable model are compared and analyzed. The results prove that the proposed model has better performance. Meanwhile, the adaptive detection of the multiple high-frequency weak signal is realized using the composite multi-stable SR system. The simulation results show that the proposed system has strong weak signal processing capability and good immunity to noise types, which widens the application range of SR in practical engineering.     

An improved multiscale noise tuning of stochastic resonance for identifying multiple transient faults in rolling element bearings

Stochastic resonance (SR), a noise-assisted tool, has been proved to be very powerful in weak signal detection. The multiscale noise tuning SR (MSTSR), which breaks the restriction of the requirement of small parameters and white noise in classical SR, has been applied to identify the characteristic frequency of a bearing. However, the multiscale noise tuning (MST), which is originally based on discrete wavelet transform (DWT), limits the signal-to-noise ratio (SNR) improvement of SR and the performance in identifying multiple bearing faults. In this paper, the wavelet packet transform (WPT) is developed and incorporated into the MSTSR method to overcome its shortcomings and to further enhance its capability in multiple faults detection of bearings. The WPT-based MST can achieve a finer tuning of multiscale noise and aims at detecting multiple target frequencies separately. By introducing WPT into the MST of SR, this paper proposes an improved SR method particularly suited for the identification of multiple transient faults in rolling element bearings. Simulated and practical bearing signals carrying multiple characteristic frequencies are employed to validate the performance improvement of the proposed method as compared to the original DWT-based MSTSR method. The results confirm the good capability of the proposed method in multi-fault diagnosis of rolling element bearings.     

A Bearing Fault Diagnosis Method Based on PAVME and MEDE

When rolling bearings have a local fault, the real bearing vibration signal related to the local fault is characterized by the properties of nonlinear and nonstationary. To extract the useful fault features from the collected nonlinear and nonstationary bearing vibration signals and improve diagnostic accuracy, this paper proposes a new bearing fault diagnosis method based on parameter adaptive variational mode extraction (PAVME) and multiscale envelope dispersion entropy (MEDE). Firstly, a new method hailed as parameter adaptive variational mode extraction (PAVME) is presented to process the collected original bearing vibration signal and obtain the frequency components related to bearing faults, where its two important parameters (i.e., the penalty factor and mode center-frequency) are automatically determined by whale optimization algorithm. Subsequently, based on the processed bearing vibration signal, an effective complexity evaluation approach named multiscale envelope dispersion entropy (MEDE) is calculated for conducting bearing fault feature extraction. Finally, the extracted fault features are fed into the k-nearest neighbor (KNN) to automatically identify different health conditions of rolling bearing. Case studies and contrastive analysis are performed to validate the effectiveness and superiority of the proposed method. Experimental results show that the proposed method can not only effectively extract bearing fault features, but also obtain a high identification accuracy for bearing fault patterns under single or variable speed.     

The analysis of stochastic resonance and bearing fault detection based on linear coupled bistable system under lévy noise

In this paper, the stochastic resonance (SR) phenomenon of the linear coupled bistable system induced by Lévy noise is analyzed. Meanwhile, the characteristics of Lévy noise is also analyzed according to its probability density functions (PDFs) of different stability index α, symmetry parameter β, scale parameter σ and location index μ. The mean of signal-noise ratio increase (MSNRI) is regarded as an index to measure the SR phenomenon. Then, the rules for MSNRI affected by noise intensity D are explored under different charastic indexes of Lévy noise, system parameters a, b, c and coupling coefficient r. The results are beneficial to the numerical simulation of single-frequency and multi-frequency weak signals detection based on single bistable system and linear coupled system respectively. It is found that the performance of the proposed system is better than single bistable system and results of bearing fault detection could also verify the conclusion.     

Analysis of weak signal detection based on tri-stable system under Levy noise

Stochastic resonance system is an effective method to extract weak signal.However,system output is directly influenced by system parameters.Aiming at this,the Levy noise is combined with a tri-stable stochastic resonance system.The average signal-to-noise ratio gain is regarded as an index to measure the stochastic resonance phenomenon.The characteristics of tri-stable stochastic resonance under Levy noise is analyzed in depth.First,the method of generating Levy noise,the effect of tri-stable system parameters on the potential function and corresponding potential force are presented in detail.Then,the effects of tri-stable system parameters w,a,b,and Levy noise intensity amplification factor D on the resonant output can be explored with different Levy noises.Finally,the tri-stable stochastic resonance system is applied to the bearing fault detection.Simulation results show that the stochastic resonance phenomenon can be induced by tuning the system parameters w,a,and b under different distributions of Levy noise,then the weak signal can be detected.The parameter intervals which can induce stochastic resonances are approximately equal.Moreover,by adjusting the intensity amplification factor D of Levy noise,the stochastic resonances can happen similarly.In bearing fault detection,the detection effect of the tri-stable stochastic resonance system is superior to the bistable stochastic resonance system.     

Bearing Fault Diagnosis Method Based on Convolutional Neural Network and Knowledge Graph

An effective fault diagnosis method of bearing is the key to predictive maintenance of modern industrial equipment. With the single use of equipment failure mechanism or operation of data, it is hard to resolve multiple complex variable working conditions, multiple types of fault and equipment malfunctions and failures related to knowledge and data. In order to solve these problems, a fault diagnosis method based on the fusion of deep learning with a knowledge graph is proposed in this paper. Firstly, the knowledge rules of bearing data is used for entity extraction. Next, the multiscale optimized convolutional neural network (MOCNN) proposed in this paper is used for fault classification to achieve relationship extraction. Finally, the fault diagnosis graph of the bearing is constructed for fault-assisted decision-making as well as the detailed display of fault information. According to experiment analysis, the fault diagnosis model based on MOCNN proposed in this paper, which integrates the end-to-end convolutional neural network and the attention mechanism, still achieves an accuracy of 97.86% under the data set of 160 types of faults. Compared with the deep learning models such as Resnet and Inception in the noise environment of multiple working conditions and variable working conditions, the model proposed in this paper not only shows a faster convergence speed and stable performance, but also a higher accuracy in evaluation indicators, which is beneficial to practical use.     

Reliable Fault Diagnosis of Bearings Using an Optimized Stacked Variational Denoising Auto-Encoder

Variational auto-encoders (VAE) have recently been successfully applied in the intelligent fault diagnosis of rolling bearings due to its self-learning ability and robustness. However, the hyper-parameters of VAEs depend, to a significant extent, on artificial settings, which is regarded as a common and key problem in existing deep learning models. Additionally, its anti-noise capability may face a decline when VAE is used to analyze bearing vibration data under loud environmental noise. Therefore, in order to improve the anti-noise performance of the VAE model and adaptively select its parameters, this paper proposes an optimized stacked variational denoising autoencoder (OSVDAE) for the reliable fault diagnosis of bearings. Within the proposed method, a robust network, named variational denoising auto-encoder (VDAE), is, first, designed by integrating VAE and a denoising auto-encoder (DAE). Subsequently, a stacked variational denoising auto-encoder (SVDAE) architecture is constructed to extract the robust and discriminative latent fault features via stacking VDAE networks layer on layer, wherein the important parameters of the SVDAE model are automatically determined by employing a novel meta-heuristic intelligent optimizer known as the seagull optimization algorithm (SOA). Finally, the extracted latent features are imported into a softmax classifier to obtain the results of fault recognition in rolling bearings. Experiments are conducted to validate the effectiveness of the proposed method. The results of analysis indicate that the proposed method not only can achieve a high identification accuracy for different bearing health conditions, but also outperforms some representative deep learning methods.