Bearing fault diagnosis method of multi-stable system random resonance with time-delay feedback control

Abstract

The application discloses a bearing fault diagnosis method of multistable system random resonance of time-delay feedback control, and belongs to the field of early fault diagnosis of rotary machinery. The method is realized as follows: an original rolling bearing vibration signal is acquired; color noise is added in the rolling bearing vibration signal; an improved signal-to-noise ratio ISNR of the bearing vibration signal is taken as an optimization index, a quantum particle swarm optimization algorithm is adopted to perform combined optimization on system parameters, damping coefficients, time-delay lengths, displacement feedback gain coefficients and speed feedback gain coefficients, and an optimal under-damped multistable system random resonance model of time-delay feedback control containing displacement and speed is obtained; the random resonance model is used to perform random resonance processing on the rolling bearing vibration signal, and the improved signal-to-noise ratio of the bearing vibration signal is improved; and spectrum analysis is performed on the bearing vibration signal output after the random resonance, so that accurate extraction of early weak fault features of the rolling bearing and accurate fault identification in a strong noise background are realized.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of early fault diagnosis of rotating machinery, and particularly relates to a bearing fault diagnosis method for a multi-stable system with random resonance of time-delay feedback control. BACKGROUND

[0002] Rolling bearings are key components for supporting rotation in mechanical equipment, and are prone to failure due to frequent operation in harsh environments with high speed, heavy load and poor sealing. Once a rolling bearing fails, it may reduce the reliability and availability of the equipment, and even threaten the safety of people's lives and property. At present, measuring bearing vibration signals and extracting fault features are common methods for early fault monitoring and diagnosis of bearings. However, rolling bearing vibration signals have prominent characteristics such as strong nonlinearity, non-stationarity and strong background noise, while the characteristic signals representing early bearing failure are very weak. In addition, the characteristic signals of early bearing failure are mixed with strong noise from other components of the mechanical equipment, so it is difficult to determine whether the rolling bearing is faulty and the type of fault by time or frequency domain analysis.

[0003] The early fault feature information extraction method of rolling bearings is based on noise suppression and signal decomposition, such as wavelet transform, empirical mode decomposition, variational mode decomposition and other methods. At present, these methods suppress noise while also weakening useful fault feature information to some extent, and inevitably damage useful signals while suppressing noise. Compared with wavelet transform, empirical mode decomposition, variational mode decomposition and other methods, the random resonance method is a new method for weak signal detection that has emerged in recent years with the rapid development of nonlinear dynamics and statistical physics theory. It is widely used in fault diagnosis of rotating machinery and has achieved good results. Unlike traditional weak signal detection methods based on noise suppression and signal decomposition, the random resonance method does not directly reduce noise, but through the best matching between signal, nonlinear system and noise, it converts part of the noise energy to useful signal to realize the detection of weak signal. However, in actual engineering, the traditional random resonance method has poor detection effect in some complex noise environments, and is prone to serious sideband interference problems. SUMMARY

[0004] To address the aforementioned technical problems and enable the detection of subtle bearing faults under strong noise conditions, the main objective of this invention is to provide a bearing fault diagnosis method based on stochastic resonance in a multisteady-state system with time-delay feedback control. Using the improved signal-to-noise ratio (ISNR) of the bearing vibration signal as the optimization metric, a quantum particle swarm optimization algorithm is employed to optimize the multisteady-state system parameters, damping coefficient, time delay length, displacement feedback gain coefficient, and velocity feedback gain coefficient, resulting in an optimal underdamped multisteady-state stochastic resonance model incorporating displacement and velocity time-delay feedback control. This stochastic resonance model is then used to perform stochastic resonance processing on the rolling bearing vibration signal, reducing the noise contained in the bearing vibration signal and improving its improved SNR. A short-time Fourier transform is performed on the high SNR rolling bearing vibration signal output after stochastic resonance to obtain its spectrum. Signal analysis is then performed on the spectrum to facilitate the capture of the evolution patterns of fault characteristic signals. This invention provides a stochastic resonance method for multi-steady-state systems with time-delay feedback control for bearing fault diagnosis. This method improves the signal-to-noise ratio of bearing vibration signals, eliminates severe sideband interference near the bearing fault characteristic frequency, and makes the peak value of the obtained target signal characteristic frequency more prominent, thus significantly improving the accuracy of detecting weak bearing fault signals under strong noise background.

[0005] The objective of this invention is achieved through the following technical solution.

[0006] The bearing fault diagnosis method for stochastic resonance in a multisteady system with time-delay feedback control disclosed in this invention includes the following steps:

[0007] Step 1: Obtain the original rolling bearing vibration signal.

[0008] Step Two: The actual noise is colored noise, which is more complex than Gaussian white noise. The presence of colored noise allows the stochastic resonance dynamics model of the multi-stable system to incorporate time memory, making the stochastic resonance process non-Markovian. Compared to Gaussian white noise excitation, the presence of colored noise causes the system to exhibit more diverse characteristics. Furthermore, compared to Gaussian white noise, the presence of Gaussian colored noise enhances the stochastic resonance phenomenon. Therefore, Gaussian colored noise is added to the original bearing vibration signal.

[0009] Step 3: Preprocess the bearing vibration signal containing colored noise to ensure that the bearing vibration signal containing colored noise meets the parameter constraints of random resonance.

[0010] The preprocessing method is as follows: perform Hilbert transform on the bearing vibration signal containing colored noise, extract the envelope for demodulation, and then use the double sampling method to scale the signal so that the bearing vibration signal containing colored noise meets the parameter constraints of random resonance.

[0011] Step four: according to the improved signal-to-noise ratio formula, the improved signal-to-noise ratio ISNR of the bearing vibration signal containing color noise after pretreatment in step three is calculated.

[0012] The improved signal-to-noise ratio formula is shown as formula (1):

[0013]

[0014] In the formula, f T is the actual frequency of the characteristic signal, S(f T ) is the actual power at the characteristic frequency, n represents the number of asymmetric data points near the characteristic signal frequency, H is the total power of the characteristic signal and noise in the local region from i to n near the characteristic signal frequency, and H-S(f T ) represents the power of noise in the local region.

[0015] Step five: taking the improved signal-to-noise ratio ISNR of the rolling bearing vibration signal as the comprehensive optimization objective function of the quantum particle swarm algorithm, searching for the maximum value of the comprehensive optimization objective function by the quantum particle swarm algorithm, selecting the multi-stable system potential function U(x), considering the mutual influence between the system parameters a, b, c, the damping coefficient γ, the time delay length τ, the displacement feedback gain coefficient β and the velocity feedback gain coefficient ν, and using the quantum particle swarm algorithm to find the optimal parameter combination of the under-damped multi-stable system random resonance model with displacement and velocity time delay feedback control. Set the range of optimal parameters and the maximum number of optimization. When the parameters corresponding to the maximum improved signal-to-noise ratio ISNR meet the optimal parameter range, save the combination of system parameters a, b, c, the damping coefficient γ, the time delay length τ, the displacement feedback gain coefficient β and the velocity feedback gain coefficient ν and execute step six; otherwise, continue to optimize until the condition is met or the maximum number of optimization is reached.

[0016] The method for finding the optimal parameter combination of the under-damped multi-stable system random resonance model with displacement and velocity time delay feedback control using the quantum particle swarm algorithm includes the following steps:

[0017] The optimization range of the parameters a, b, c, γ, β, ν, τ of the under-damped multi-stable system random resonance model with displacement and velocity time delay feedback control, the population size N, the maximum number of iterations T max and the search space dimension S are specified, the parameter optimization range is set at each dimension, and the initial position of the population individuals is initialized.

[0018] The pre-processed rolling bearing vibration signal containing Gaussian color noise is inputted into the under-damped multi-stable system random resonance model containing displacement and velocity time-delay feedback control, the improved signal-to-noise ratio ISNR of the rolling bearing vibration signal output by random resonance is taken as the fitness function of the quantum particle swarm algorithm, the fitness value of the initial position of each particle is calculated according to the calculation formula of the improved signal-to-noise ratio ISNR, and taken as the individual optimal position fitness value P best(i) of the first generation of particles. best(i) The maximum value in P g is taken as the global optimal position fitness value P max .

[0019] The average optimal position M of the particle swarm is calculated, the fitness value of the current position is calculated, the individual optimal position fitness value is updated, and compared with the fitness value of the global optimal position, if better than the global optimal position, then taken as the new global optimal.

[0020] The optimization process of the quantum particle swarm optimization algorithm on the multi-stable system random resonance model is repeated, and it is judged whether the iteration number T reaches the maximum iteration number T max pre-set, if the maximum iteration number is reached, step six is entered, if the iteration number does not reach T max , the operation of the above steps is continued.

[0021] The combination of the system parameters a, b, c, the damping coefficient γ, the time delay length τ, the displacement feedback gain coefficient β and the velocity feedback gain coefficient v at the time when the improved signal-to-noise ratio ISNR is maximum is saved.

[0022] Step six: according to the optimal combination of the system parameters, the damping coefficient, the time delay length, the displacement feedback gain coefficient and the velocity feedback gain coefficient obtained by optimization in step five, the under-damped multi-stable system random resonance model containing displacement and velocity time-delay feedback control is constructed, the random resonance model is used to process the rolling bearing vibration signal, the noise contained in the bearing vibration signal is weakened, the side frequency interference near the characteristic frequency is eliminated, the amplitude of the characteristic frequency and the improved signal-to-noise ratio of the bearing vibration signal are improved, that is, the high signal-to-noise ratio rolling bearing vibration signal is output after random resonance.

[0023] The under-damped multi-stable system random resonance model with displacement and velocity time-delay feedback control driven by Gaussian color noise and periodic excitation is represented by the following formula:

[0024]

[0025] Where U(x) = ax 2 / 2-bx 4 / 4+cx 6 / 6 is the potential function of the three-stable system, and a, b, c are all positive real numbers. x τ= x(t - τ), where τ is the time delay length, β is the displacement feedback gain coefficient, v is the velocity feedback gain coefficient, ε is the periodic excitation amplitude, ω is the periodic excitation frequency. η(t) is Gaussian colored noise and satisfies the following statistical properties:

[0026]

[0027] where D, τ1represent the noise intensity and correlation time of the colored noise, respectively.

[0028] The above multi-stable potential function U(x) has three stable equilibrium points (x mi ,0)i=(1,2,3) and two unstable equilibrium points (x ni ,0)i=(1,2):

[0029]

[0030] By introducing the generalized harmonic function, the displacement term and the velocity term in equation (2) are expressed as

[0031]

[0032] where x mi (H)(i=1,2,3) is the stable equilibrium point, A(H) and ω(H) represent the energy-dependent amplitude and frequency, respectively.

[0033] By harmonic transformation of equation (3), the displacement and velocity time delay feedback terms in equation (2) are approximately expressed as

[0034]

[0035] Substituting equation (4) into equation (2) gives:

[0036]

[0037] where

[0038]

[0039] According to equation (5), the potential function and the total energy are obtained:

[0040]

[0041]

[0042] In equation (6), ω(H) is obtained according to the following equation

[0043]

[0044] where x1and x2denote the minimum and maximum values of displacement determined by H = U(A) without time-delayed feedback control, the energy-dependent frequency needs to be calculated according to the shape of the potential function. According to three different periodic motions of the multi-stable potential function, the energy-dependent frequency is obtained by iterative calculation.

[0045] According to equation (5), two first-order differential equations of velocity and energy are obtained as follows:

[0046]

[0047]

[0048] The total energy H(t) is an approximate Markov process, and by the method of stochastic averaging of energy envelope, the total energy H(t) is written in the form of Ito equation as follows:

[0049] dH = m(H)dt + σ(H)dB(t) (12)

[0050] where B(t) is a standard Wiener process, m(H) and σ(H) represent the drift term and diffusion term, respectively, and have the following forms:

[0051]

[0052]

[0053] where <·> t denotes the time average:

[0054]

[0055] and,

[0056]

[0057] According to equation (12), the stationary probability density of energy is obtained as:

[0058]

[0059] where N is a normalization constant. Since

[0060]

[0061] The joint quasi-stationary probability density function of system displacement and velocity is derived as:

[0062]

[0063] According to equation (19), the generalized potential function is obtained as:

[0064]

[0065] Let Equation (5) is rewritten in the following form:

[0066]

[0067] When the noise, periodic signal excitation and time delay feedback are not considered, the eigenvalues of the linearized matrix of equation (21) at the stable equilibrium point are:

[0068]

[0069] The eigenvalues of the linearized matrix at the unstable equilibrium point are:

[0070]

[0071] The transition rates between different potential wells are obtained using the particle transition probability formula of two-dimensional dynamical systems:

[0072]

[0073]

[0074]

[0075]

[0076] Under the adiabatic approximation condition, equation (5) describes a discrete Markov process of the transition of probability p i (i = 1, 2, 3) and the corresponding master equation is obtained:

[0077]

[0078] where P = (p1 p2 p3) T , the transition matrix W has the following form:

[0079]

[0080] For small amplitude ε, the first-order approximation of P is:

[0081]

[0082] where Δp i is the first-order correction term of p i , and the transition matrix W is expanded as W = W0 + εΔWcosωt, which is substituted into equation (28) together with equation (30):

[0083]

[0084] where W 0 and transition matrix and probability, respectively, for ε = 0, corresponding element of matrix W 0 corresponding element of σ 0 wherein:

[0085]

[0086] The solution of equation (31) in the long-time limit has the form:

[0087] Δp i = q i sin(ωt) + w i cos(ωt) (33)

[0088] Substituting equation (33) into equation (31) gives:

[0089]

[0090]

[0091] where ε k and ζ k are the eigenvalues and eigenvectors of W 0 , respectively,

[0092] The steady-state response of the under-damped multi-stable system (1) in the long-time limit is obtained from equation (33):

[0093]

[0094] According to equation (36), the power spectrum amplification factor is defined as η = (V / ε) 2

[0095]

[0096] When the power spectrum amplification factor reaches a maximum value, optimal stochastic resonance occurs, the noise contained in the bearing vibration signal is weakened, the sideband interference near the characteristic frequency is eliminated, the amplitude of the characteristic frequency is improved, and the improved signal-to-noise ratio of the bearing vibration signal is improved, and a high signal-to-noise ratio rolling bearing vibration signal is output.

[0097] ​Step seven: the high signal-to-noise ratio rolling bearing vibration signal output after random resonance in step six is subjected to short-time Fourier transform, and a high signal-to-noise ratio rolling bearing vibration signal spectrum is obtained. The spectrum is subjected to signal analysis to facilitate the capture of the fault characteristic signal evolution law, and then the bearing fault diagnosis is performed based on the random resonance method of the multi-stable system with time delay feedback control, the signal-to-noise ratio of the bearing vibration signal is improved, the serious side frequency interference existing near the bearing fault characteristic frequency is eliminated, the target signal characteristic frequency peak value obtained is more prominent, and the accuracy of bearing weak signal fault detection under strong noise background is significantly improved.

[0098] Step eight: according to the characteristic frequency in the rolling bearing vibration signal spectrum obtained in step seven, the early rolling bearing fault in the working process can be effectively and accurately judged under the strong noise background, which is beneficial to assist the staff to maintain the mechanical device, and can effectively prevent the reduction of equipment precision and production quality caused by the rolling bearing fault.

[0099] Advantages:

[0100] Since the bearing works in a harsh environment, the early fault characteristic information of the rolling bearing is often submerged in strong noise and difficult to extract, mainly due to two reasons. One is that the early fault characteristic signal of the rolling bearing is weak and the fault characteristic signal evolution law is difficult to capture. The other reason is that the noise under the harsh external environment is too large. To solve the above technical problems, the bearing fault diagnosis method of the multi-stable system with time delay feedback control random resonance has the following advantages:

[0101] 1. For the problems of difficult extraction of early fault characteristic signals of rolling bearings and difficult identification of early fault types of rolling bearings, the bearing fault diagnosis method of the multi-stable system with time delay feedback control random resonance disclosed in the application takes the improved signal-to-noise ratio ISNR of the bearing vibration signal as an optimization index, adopts a quantum particle swarm optimization algorithm to perform combined optimization on the multi-stable system parameters, damping coefficients, time delay lengths, displacement feedback gain coefficients and speed feedback gain coefficients, and obtains an optimal under-damped multi-stable random resonance model with displacement and speed time delay feedback control. The random resonance model is used for random resonance processing of the rolling bearing vibration signal to weaken the noise contained in the bearing vibration signal and improve the improved signal-to-noise ratio of the bearing vibration signal. The high signal-to-noise ratio rolling bearing vibration signal output after random resonance is subjected to short-time Fourier transform, and a high signal-to-noise ratio rolling bearing vibration signal spectrum is obtained. The spectrum is subjected to signal analysis to facilitate the capture of the fault characteristic signal evolution law. The bearing fault diagnosis is performed by the random resonance method of the multi-stable system with time delay feedback control, the signal-to-noise ratio of the bearing vibration signal is improved, the serious side frequency interference existing near the bearing fault characteristic frequency is eliminated, the target signal characteristic frequency peak value obtained is more prominent, and the accuracy of bearing weak fault signal detection under strong noise background is significantly improved.

[0102] 2. The bearing fault diagnosis method based on the multistable system random resonance with time delay feedback control disclosed in the application considers that the actual noise is more complex than the Gaussian white noise, and the existence of the color noise makes the multistable system random resonance dynamic model contain the memory of time, so that the random resonance process has non-Markov property. Compared with the Gaussian white noise excitation, the existence of the color noise makes the system present more different characteristics, and the existence of the Gaussian color noise makes the random resonance phenomenon enhance, which is more conducive to the diagnosis of bearing weak faults.

[0103] 3. The bearing fault diagnosis method based on the multistable system random resonance with time delay feedback control disclosed in the application adds time delay feedback control in the multistable system random resonance model. In the process of bearing weak fault diagnosis of the multistable system with time delay feedback, it is found that the existence of time delay feedback enhances the detection performance. The quantum particle swarm optimization algorithm is combined with the under-damped multistable random resonance method with displacement and velocity time delay feedback control to improve the signal-to-noise ratio ISNR as the optimization objective function, the system parameters, damping coefficient, time delay length, displacement feedback gain coefficient and velocity feedback gain coefficient are combined and optimized, an adaptive under-damped multistable random resonance method with displacement and velocity time delay feedback control is proposed, and it is applied to bearing weak fault diagnosis, so that the problem that bearing weak fault characteristics are difficult to extract in strong noise background is solved.

[0104] 4. The bearing fault diagnosis method based on the multistable system random resonance with time delay feedback control disclosed in the application can effectively and accurately judge whether the early rolling bearing appears fault in work in the strong noise background on the basis of realizing the beneficial effects 1, 2 and 3, which is conducive to assisting workers to maintain the mechanical device, and can effectively prevent the reduction of equipment precision and production quality caused by rolling bearing fault. BRIEF DESCRIPTION OF DRAWINGS

[0105] Figure 1 is a bearing fault diagnosis method flow chart of the multistable system random resonance with time delay feedback control;

[0106] Figure 2 is a raw rolling bearing outer ring fault vibration signal;

[0107] Figure 3 is a raw rolling bearing inner ring fault vibration signal;

[0108] Figure 4 is a rolling bearing outer ring fault vibration signal containing color noise;

[0109] Figure 5 is a rolling bearing inner ring fault vibration signal containing color noise

[0110] Figure 6 is the time-delay feedback control multistable system stochastic resonance method applied to bearing outer ring fault signal output time-frequency chart;

[0111] Figure 7 is the time-delay feedback control multistable system stochastic resonance method applied to bearing inner ring fault signal output time-frequency chart;

[0112] Figure 8 is the time-delay feedback control multistable system stochastic resonance method applied to bearing outer ring fault signal output time-frequency chart;

[0113] Figure 9 is the time-delay feedback control multistable system stochastic resonance method applied to bearing inner ring fault signal output time-frequency chart; DETAILED DESCRIPTION

[0114] In order to better illustrate the purpose and advantages of the present application, the content of the application is further described below in combination with the drawings and examples.

[0115] The present application is suitable for early fault diagnosis of rotating machinery, especially early fault diagnosis of rolling bearings. In order to verify the feasibility of the present application, the experimental data uses bearing data from Case Western Reserve University in the United States, and the bearing model used is 6205-2RSJEMSKF deep groove ball bearing. The rolling bearing specifications are shown in Table 1. The bearing data with a rotational speed N r = 1797 r / min is used, and the sampling frequency and sampling time are 12000 Hz and 1 s, respectively.

[0116] Table 1 Rolling bearing specifications Size: (inches)

[0117]

[0118] The theoretical fault frequencies of the bearing outer and inner rings are determined by the following formulas, respectively:

[0119]

[0120]

[0121] According to formula (38) and formula (39), the theoretical fault frequency of the bearing outer ring is f outside = 108 Hz, and the theoretical fault frequency of the bearing inner ring is f inside = 162 Hz.

[0122] As Figure 1 shown, the bearing fault diagnosis method of time-delay feedback control multistable system stochastic resonance disclosed in the embodiment is specifically implemented as follows:

[0123] Step one: Obtain the original rolling bearing vibration signal, and use the acceleration data of the driving end of SKF type rolling bearing of Case Western Reserve University in the United States. The fault type data used is the outer ring and inner ring fault of the rolling bearing, and the fault characteristic frequency is 108 Hz and 162 Hz, respectively. As shown in FIG. 1, it is the original rolling bearing outer ring fault vibration signal. As shown in FIG. 2, it is the original rolling bearing inner ring fault vibration signal. Figure 2 Figure 3

[0124] Step two: The actual noise is a color noise more complex than the Gaussian white noise. The existence of the color noise makes the random resonance vibration mechanics model of the multi-stable system contain the memory of time, so that the random resonance process has non-Markov property. Compared with the Gaussian white noise excitation, the existence of the color noise makes the system show more different characteristics. Compared with the Gaussian white noise, the existence of the Gaussian color noise makes the random resonance phenomenon enhance. Therefore, the Gaussian color noise with noise intensity D = 0.1 is added to the original bearing vibration signal. As shown in FIG. 3, it is the rolling bearing outer ring fault vibration signal containing color noise; as shown in FIG. 4, it is the rolling bearing inner ring fault vibration signal containing color noise. Figure 4 Figure 5

[0125] Step three: The bearing vibration signal containing color noise is preprocessed to make the bearing vibration signal containing color noise meet the parameter limit condition of random resonance. The preprocessing method is to perform Hilbert transform on the bearing vibration signal containing color noise, extract the envelope line for demodulation, and then use the twice sampling method to perform scale transformation on the signal, so that the bearing vibration signal containing color noise meets the parameter limit condition of random resonance.

[0126] Step four: According to formula (1), the improved signal-to-noise ratio ISNR of the bearing vibration signal containing color noise preprocessed in step three is calculated.

[0127] Step five: Taking the improved signal-to-noise ratio ISNR of the rolling bearing vibration signal as the comprehensive optimization objective function of the quantum particle swarm algorithm, searching the maximum value of the comprehensive optimization objective function by the quantum particle swarm algorithm, selecting the multi-stable system potential function U(x), considering the mutual influence between the system parameters a, b, c, damping coefficient γ, time delay length τ, displacement feedback gain coefficient β and velocity feedback gain coefficient v, and using the quantum particle swarm algorithm to find the optimal parameter combination of the under-damped multi-stable system random resonance model containing displacement and velocity time delay feedback control. Set the range of the optimal parameters and the maximum optimization times. When the parameters corresponding to the maximum improved signal-to-noise ratio ISNR meet the range of the optimal parameters, save the combination of the system parameters a, b, c, damping coefficient γ, time delay length τ, displacement feedback gain coefficient β and velocity feedback gain coefficient v and execute step six; otherwise, continue to optimize until the condition is met or the maximum optimization times are reached. ​​​​

[0128] The steps of using quantum particle swarm algorithm to find the optimal parameter combination of the under-damped multi-stable system random resonance model with displacement and velocity time-delay feedback control are as follows:

[0129] The optimization range of the parameters a, b, c, γ, β, ν, τ of the under-damped multi-stable system random resonance model with displacement and velocity time-delay feedback control, the population number N, and the maximum iteration number T are specified. max The parameter optimization range is set at each dimension, and the initial position of the population individual is initialized.

[0130] The preprocessed rolling bearing vibration signal with Gaussian color noise is input into the under-damped multi-stable system random resonance model with displacement and velocity time-delay feedback control, the improved signal-to-noise ratio ISNR of the random resonance output rolling bearing vibration signal is taken as the fitness function of the quantum particle swarm algorithm, the fitness value of the initial position of each particle is calculated according to the calculation formula of the improved signal-to-noise ratio ISNR, and it is taken as the individual optimal position fitness value P of the first generation particle. best(i) Meanwhile, the maximum value in P best(i) is taken as the global optimal position fitness value P g .

[0131] The average optimal position M of the particle swarm is calculated, the fitness value of the current position is calculated, the individual optimal position fitness value is updated, and it is compared with the fitness value of the global optimal position. If it is better than the global optimal position, it is taken as the new global optimal position.

[0132] The optimization process of the quantum particle swarm optimization algorithm for the multi-stable system random resonance model is repeated, and it is judged whether the iteration number T reaches the maximum iteration number T max . If the maximum iteration number is reached, step six is entered. If the iteration number does not reach T max , the operation of the above steps is continued.

[0133] The combination of the system parameters a, b, c, the damping coefficient γ, the time-delay length τ, the displacement feedback gain coefficient β, and the velocity feedback gain coefficient ν at the time of the maximum improved signal-to-noise ratio ISNR is saved.

[0134] Step six: according to the optimal combination of the system parameters, the damping coefficient, the time-delay length, the displacement feedback gain coefficient, and the velocity feedback gain coefficient obtained by the optimization in step five, the under-damped multi-stable system random resonance model with displacement and velocity time-delay feedback control is constructed, the random resonance model is used to process the rolling bearing vibration signal, the noise contained in the bearing vibration signal is weakened, the sideband interference near the characteristic frequency is eliminated, the amplitude of the characteristic frequency and the improved signal-to-noise ratio of the bearing vibration signal are improved, that is, the high signal-to-noise ratio rolling bearing vibration signal is output after random resonance.

[0135] The stochastic resonance model of an underdamped multi-stable system with displacement and velocity time-delay feedback control driven by Gaussian colored noise and periodic excitation is given by

[0136]

[0137] where U(x) = ax 2 / 2-bx 4 / 4+cx 6 / 6 is the potential function of the three-stable system, and a, b, c are positive real numbers. τ = x(t-τ), τ is the time-delay length, β is the displacement feedback gain coefficient, v is the velocity feedback gain coefficient, ε is the amplitude of periodic excitation, and ω is the frequency of periodic excitation. η(t) is Gaussian colored noise and satisfies the following statistical properties:

[0138]

[0139] where D and τ1 are the noise intensity and correlation time of the colored noise, respectively.

[0140] The above multi-stable potential function U(x) has three stable equilibrium points (x mi ,0)i = (1, 2, 3) and two unstable equilibrium points (x ni ,0)i = (1, 2):

[0141]

[0142]

[0143] By introducing the generalized harmonic function, the displacement and velocity terms in equation (40) are expressed as

[0144]

[0145] where x mi (H)(i = 1, 2, 3) is the stable equilibrium point, and A(H) and ω(H) are the energy-dependent amplitude and frequency, respectively.

[0146] Through harmonic transformation (41), the displacement and velocity time-delay feedback terms in equation (40) are approximately expressed as

[0147]

[0148] Substituting equation (42) into equation (40) gives

[0149]

[0150] where

[0151]

[0152] The potential function and the total energy are obtained from equation (43):

[0153]

[0154]

[0155] The ω(H) in equation (44) is obtained from

[0156]

[0157] where x1and x2represent the minimum and maximum of the displacement determined by H = U(A) without time-delayed feedback control, the energy-dependent frequency needs to be calculated according to the shape of the potential function. According to three different periodic motions of the multi-stable potential function, the energy-dependent frequency is obtained from iterative calculation.

[0158] According to equation (43), the two first-order differential equations of velocity and energy are obtained as follows:

[0159]

[0160]

[0161] The total energy H(t) is an approximate Markov process, and by the method of stochastic averaging of energy envelope, the total energy H(t) is written in the form of Ito equation as follows:

[0162] dH = m(H)dt + σ(H)dB(t) (50)

[0163] where B(t) is a standard Wiener process, m(H) and σ(H) represent the drift term and the diffusion term, respectively, and have the following forms:

[0164]

[0165]

[0166] where <·> t represents the time average:

[0167]

[0168] and,

[0169]

[0170] The stationary probability density of energy is obtained from equation (50) as:

[0171]

[0172] where N is a normalization constant. Since

[0173]

[0174] The joint quasi-stationary probability density function of the system displacement and velocity is derived as

[0175]

[0176] The generalized potential function is obtained from equation (57) as

[0177]

[0178] Let Equation (43) is rewritten in the following form:

[0179]

[0180] When the noise, periodic signal excitation and time-delay feedback are not considered, the eigenvalues of the linearized matrix of equation (59) at the stable equilibrium point are

[0181]

[0182] The eigenvalues of the linearized matrix at the unstable equilibrium point are

[0183]

[0184] The transition rates between different potential wells are obtained using the particle transition probability formula of two-dimensional dynamical systems as

[0185]

[0186]

[0187]

[0188]

[0189] Under the adiabatic approximation, equation (43) is regarded as a discrete Markov process describing the transition of probabilities p i (i = 1, 2, 3) and the corresponding master equation is obtained as

[0190]

[0191] where P = (p1p2p3) T The transition matrix W has the following form:

[0192]

[0193] For small amplitude ε, the first order approximation of P is:

[0194]

[0195] where Δp i is the first order correction term of p i , and the transition matrix W is expanded as W = W0+ εΔWcosωt. Substituting equation (68) into equation (66) gives:

[0196]

[0197] where W 0 and are the transition matrix and probability at ε = 0, respectively, is the corresponding element of the matrix W 0 , and is the corresponding element of σ 0 , where:

[0198]

[0199] The solution of equation (69) in the long time limit has the following form:

[0200] Δp i = q i sin(ωt) + w i cos(ωt) (71)

[0201] Substituting equation (71) into equation (69) gives:

[0202]

[0203]

[0204] where ε k and ζ k are the eigenvalue and eigenvector of W 0 , respectively,

[0205] The steady state response of system (40) in the long time limit is obtained from equation (71):

[0206]

[0207] According to equation (74), the power spectral amplification factor is defined as η = (V / ε) 2

[0208]

[0209] When the power spectrum amplification factor appears the maximum value, the optimal stochastic resonance is generated, the noise contained in the bearing vibration signal is weakened, the side frequency interference near the characteristic frequency is eliminated, the amplitude of the characteristic frequency is improved, the improved signal-to-noise ratio of the bearing vibration signal is improved, and the high signal-to-noise ratio rolling bearing vibration signal is output.

[0210] Step seven: the high signal-to-noise ratio rolling bearing vibration signal output after the random resonance in step six is subjected to short-time Fourier transform, and the frequency spectrum diagram of the high signal-to-noise ratio rolling bearing vibration signal is obtained. The signal analysis is carried out on the frequency spectrum diagram, so as to capture the evolution law of the fault characteristic signal, and then the bearing fault diagnosis is carried out based on the random resonance method of the multi-stable system with time delay feedback control, the signal-to-noise ratio of the bearing vibration signal is improved, the serious side frequency interference existing near the bearing fault characteristic frequency is eliminated, the peak value of the target signal characteristic frequency obtained is more prominent, and the accuracy of the bearing weak fault detection under the strong noise background is significantly improved.

[0211] Step eight: according to the characteristic frequency in the rolling bearing vibration signal frequency spectrum diagram obtained in step seven, the early rolling bearing fault in the working process can be effectively and accurately judged under the strong noise background, which is beneficial to assist the staff to maintain the mechanical device, and can effectively prevent the reduction of the equipment precision and the production quality caused by the rolling bearing fault.

[0212] As shown in Figure 6 , it is the time-frequency diagram of the bearing outer ring fault signal output by the random resonance method of the multi-stable system with time delay feedback control. Figure 6 In the figure, the frequency corresponding to the highest spectral peak is 108Hz, which is the same as the theoretical fault characteristic frequency of the rolling bearing outer ring, and the early fault such as fatigue wear existing in the rolling bearing outer ring can be judged. Figure 7 As shown in Figure 7 , it is the time-frequency diagram of the bearing inner ring fault signal output by the random resonance method of the multi-stable system with time delay feedback control. In the figure, the frequency corresponding to the highest spectral peak is 162Hz, which is the same as the theoretical fault characteristic frequency of the rolling bearing inner ring, and the early fault such as fatigue wear existing in the rolling bearing inner ring can be judged.

[0213] At the same time, in order to further illustrate the superiority of the method of the application, as shown in Figure 8 , it is the time-frequency diagram of the bearing outer ring fault signal output by the random resonance method of the multi-stable system without time delay feedback control; as shown in Figure 9 , it is the time-frequency diagram of the bearing inner ring fault signal output by the random resonance method of the multi-stable system without time delay feedback control. Figure 6 and Figure 8 It can be seen that, Figure 6 the improved signal-to-noise ratio of the rolling bearing vibration signal is Figure 8The improved rolling bearing vibration signal has higher signal-to-noise ratio and more prominent peak frequency. Figure 7 Compared with Figure 9 It is obvious that, compared with the multi-stable system random resonance method without time delay feedback control, the multi-stable system random resonance method with time delay feedback control can effectively and accurately determine whether the early rolling bearing is in failure in the strong noise background, is conducive to assisting the staff to maintain the mechanical device, can effectively prevent the decrease of equipment precision and production quality caused by the failure of the rolling bearing, and can protect the production state and system operation to a certain extent, and protect the life and property safety of the on-site staff.

[0214] It should be noted that the above description is only to illustrate some basic principles of the present application, and it is easy for those skilled in the same technical field to make some modifications and changes on the basis. Therefore, the present specification is not intended to limit the present application to the specific structure and application range shown and described, and all possible corresponding modifications and equivalents belong to the patent scope of the present application.

Claims

1. A bearing fault diagnosis method for stochastic resonance in a multisteady system with time-delay feedback control, characterized in that: Includes the following steps, Step 1: Obtain the original rolling bearing vibration signal; Step 2: The actual noise is colored noise, which is more complex than Gaussian white noise. The presence of colored noise makes the stochastic resonance dynamics model of the multi-stable system include time memory, making the stochastic resonance process non-Markovian. Compared with Gaussian white noise excitation, the presence of colored noise makes the system exhibit more different characteristics. Compared with Gaussian white noise, the presence of Gaussian colored noise enhances the stochastic resonance phenomenon. Gaussian colored noise is added to the original bearing vibration signal. Step 3: Preprocess the bearing vibration signal containing colored noise to ensure that the bearing vibration signal containing colored noise meets the parameter constraints of random resonance; Step 4: Calculate the improved signal-to-noise ratio (ISNR) for the bearing vibration signal containing colored noise after the preprocessing in Step 3, according to the improved signal-to-noise ratio formula. Step 5: Using the improved signal-to-noise ratio (ISNR) of the rolling bearing vibration signal as the comprehensive optimization objective function of the quantum particle swarm optimization (QPSO) algorithm, the QPSO algorithm searches for the maximum value of the comprehensive optimization objective function. The potential function U(x) of the multi-stable system is selected. Considering the mutual influence between system parameters a, b, c, damping coefficient γ, time delay length τ, displacement feedback gain coefficient β, and velocity feedback gain coefficient ν, the QPSO algorithm is used to find the optimal parameter combination for the stochastic resonance model of the underdamped multi-stable system containing displacement and velocity time delay feedback control. The range of optimal parameters and the maximum number of optimization attempts are set. When the parameters corresponding to the maximum improved ISNR satisfy the optimal parameter range, the combination of system parameters a, b, c, damping coefficient γ, time delay length τ, displacement feedback gain coefficient β, and velocity feedback gain coefficient ν is saved, and Step 6 is executed. Otherwise, the optimization continues until the conditions are met or the maximum number of optimization attempts is reached. Step Six: Based on the optimal combination of system parameters, damping coefficient, time delay length, displacement feedback gain coefficient, and velocity feedback gain coefficient obtained in Step Five, construct a stochastic resonance model of an underdamped multi-steady-state system containing displacement and velocity time delay feedback control. Use this stochastic resonance model to perform stochastic resonance processing on the rolling bearing vibration signal to reduce the noise contained in the bearing vibration signal, eliminate sideband interference near the characteristic frequency, improve the characteristic frequency amplitude, and improve the signal-to-noise ratio of the bearing vibration signal. That is, after stochastic resonance, a high signal-to-noise ratio rolling bearing vibration signal is output. Step 7: Perform a short-time Fourier transform on the high signal-to-noise ratio (SNR) rolling bearing vibration signal output after random resonance in Step 6 to obtain the spectrum of the high SNR rolling bearing vibration signal; perform signal analysis on the spectrum to capture the evolution law of fault characteristic signals, and then perform bearing fault diagnosis based on the random resonance method of multi-steady-state systems with time-delay feedback control, improve the SNR of the bearing vibration signal, eliminate severe sideband interference near the bearing fault characteristic frequency, and make the peak value of the obtained target signal characteristic frequency more prominent.

2. The bearing fault diagnosis method for stochastic resonance in a multisteady system with time-delay feedback control as described in claim 1, characterized in that: It also includes step eight, Based on the characteristic frequencies in the spectrum of the rolling bearing vibration signal obtained in step seven, it is possible to accurately determine whether the rolling bearing has malfunctioned during operation in the early stages, even in the context of strong noise. This helps to assist workers in the maintenance of mechanical devices and prevents the reduction in equipment precision and production quality caused by rolling bearing failure.

3. The bearing fault diagnosis method for stochastic resonance in a multisteady system with time-delay feedback control as described in claim 1 or 2, characterized in that: In step three, The preprocessing method is as follows: perform Hilbert transform on the bearing vibration signal containing colored noise, extract the envelope for demodulation, and then use the double sampling method to scale the signal so that the bearing vibration signal containing colored noise meets the parameter constraints of random resonance.

4. The bearing fault diagnosis method for stochastic resonance in a multisteady system with time-delay feedback control as described in claim 3, characterized in that: In step four, The improved signal-to-noise ratio formula is shown in equation (1): In the formula, f T It is the actual frequency of the characteristic signal, S(f T ) is the actual power at the characteristic frequency, n represents the number of asymmetric data points near the characteristic signal frequency, H is the total power of the characteristic signal and noise in the local region from i to n near the characteristic signal frequency, HS(f T This indicates the power of the noise in this local area.

5. The bearing fault diagnosis method for stochastic resonance in a multisteady system with time-delay feedback control as described in claim 4, characterized in that: In step five, The specific steps of the method for finding the optimal parameter combination of a stochastic resonance model of an underdamped multi-steady-state system with displacement and velocity time-delay feedback control using the quantum particle swarm optimization algorithm are as follows: Specify the optimization range, population size N, and maximum number of iterations T for the parameters a, b, c, γ, β, ν, τ of the stochastic resonance model of an underdamped multisteady system with displacement and velocity time-delay feedback control. max Given the search space dimension S, set the parameter optimization range for each dimension and initialize the initial position of the population individuals; The preprocessed rolling bearing vibration signal containing Gaussian noise is input into a stochastic resonance model of an underdamped multi-steady-state system with displacement and velocity time-delay feedback control. The improved signal-to-noise ratio (ISNR) of the stochastic resonance output rolling bearing vibration signal is used as the fitness function of the quantum particle swarm optimization algorithm. The fitness value of the initial position of each particle is calculated according to the formula for calculating the improved ISNR, and this value is used as the individual optimal position fitness value P of the first generation of particles. best(i) At the same time, P best(i) The maximum value in the range is used as the global optimal fitness value P. g ; Calculate the average optimal position M of the particle swarm; calculate the fitness value of the current position, update the fitness value of the individual's optimal position, and compare it with the fitness value of the global optimal position. If it is better than the global optimal position, then take it as the new global optimal. Repeat the above quantum particle swarm optimization algorithm for the optimization process of the stochastic resonance model of the multistable system, and determine whether the number of iterations T has reached the preset maximum number of iterations T. max If the maximum number of iterations is reached, proceed to step six; otherwise, proceed to step T. max If so, continue with the above steps; Save the system parameters a, b, c, damping coefficient γ, time delay length τ, displacement feedback gain coefficient β, and velocity feedback gain coefficient ν when the improved signal-to-noise ratio (ISNR) is at its maximum.

6. The bearing fault diagnosis method for stochastic resonance in a multisteady system with time-delay feedback control as described in claim 5, characterized in that: Step six is ​​implemented as follows: Driven by Gaussian colored noise and periodic excitation, the stochastic resonance model of an underdamped multisteady-state system with displacement and velocity time-delay feedback control is expressed by the following equation: Where U(x) = ax 2 / 2-bx 4 / 4+cx 6 / 6 represents the potential function of the tristable system, where a, b, and c are all positive real numbers; x τ =x(t-τ), τ is the time delay length, β is the displacement feedback gain coefficient, ν is the velocity feedback gain coefficient, ε is the periodic excitation amplitude, ω is the periodic excitation frequency; η(t) is Gaussian colored noise and satisfies the following statistical properties: Where D and τ1 represent the noise intensity and correlation time of the colored noise, respectively; The aforementioned multistable state function U(x) has three stable equilibrium points (x0, x ...). mi ,0)i=(1,2,3) and two unstable equilibrium points (x ni ,0)i=(1,2): By introducing a generalized harmonic function, the displacement and velocity terms in equation (2) can be expressed as follows: Where x mi (H)(i=1,2,3) is the stable equilibrium point, and A(H) and ω(H) represent the energy-dependent amplitude and frequency, respectively; Through harmonic transformation equation (3), the displacement and velocity time delay feedback terms in equation (2) are approximately expressed as follows: Substituting equation (4) into equation (2), we get: in According to equation (5), the potential function and total energy are obtained: In equation (6), ω(H) is obtained from the following equation. Where x1 and x2 represent the minimum and maximum displacements determined by H=U(A) without time-delay feedback control, and the calculation of the energy-dependent frequency requires consideration of the shape of the potential function; Based on the three different periodic motions of the multistable state function, the energy-dependent frequency is obtained by iterative calculation; According to equation (5), the following two first-order differential equations for velocity and energy are obtained: The total energy H(t) is an approximate Markov process. Using the stochastic averaging method of the energy envelope, the total energy H(t) can be written in the form of the Iton equation as follows: dH=m(H)dt+σ(H)dB(t) (12) Where B(t) is a standard Wiener process, m(H) and σ(H) represent the drift term and diffusion term, respectively, and have the following forms: in· t Indicates time average: and, According to equation (12), the stationary probability density of energy is: Where N is the normalization constant; due to Derive the joint quasi-stationary probability density function of the system displacement and velocity: According to equation (19), the generalized potential function is: make Equation (5) can be rewritten in the following form: When noise, periodic signal excitation, and time-delay feedback are not considered, the linearized matrix eigenvalues ​​of the deterministic equation (21) at the stable equilibrium point are: The eigenvalues ​​of the linearized matrix at the unstable equilibrium point are: The transition rates between different potential wells can be obtained using the particle transition probability formula for a two-dimensional dynamical system: Under the adiabatic approximation, equation (5) describes the probability p. i The discrete Markov process transitioning from (i = 1, 2, 3) yields the corresponding master equation: Where P = (p1p2p3) T The transition matrix W has the following form: For small amplitudes ε, the first-order approximation of P is: Where, Δp i It is p i The first-order correction term expands the transition matrix W to W = W0 + εΔWcosωt, which, together with equation (30), is substituted into equation (28): Among them, W 0 and Let be the transition matrix and probability when ε = 0, respectively. For matrix W 0 The corresponding element, For σ 0 The corresponding element, where: The solution to equation (31) under the long-time limiting condition has the following form: Δp i =q i sin(ωt)+w i cos(ωt) (33) Substituting equation (33) into equation (31), we get: Where ε k and ζ k W respectively 0 eigenvalues ​​and eigenvectors, According to equation (33), the steady-state response of the underdamped multi-steady-state system (1) over a long time is obtained: According to equation (36), the power spectral amplification factor is defined as η = (V / ε). 2 When the power spectrum amplification factor reaches its maximum value, optimal random resonance is generated, which weakens the noise contained in the bearing vibration signal, eliminates sideband interference near the characteristic frequency, improves the characteristic frequency amplitude and the signal-to-noise ratio of the bearing vibration signal, and outputs a high signal-to-noise ratio rolling bearing vibration signal.

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