A bearing fault diagnosis method based on bivariate weighted mapping

Abstract

The application discloses a bearing fault diagnosis method based on a bivariate weighted mapping diagram, which is based on the autocorrelation principle to process the spectral coherence information of a signal, uses the autocorrelation trend to distribute appropriate weights for each spectral frequency of the spectral coherence, and proposes an improved spectral weighting method. The information point with the maximum amplitude in each cyclic frequency slice is extracted, a one-dimensional cyclic frequency diagram is reconstructed, and a cyclic frequency weighting function is proposed to enhance the energy of the fault characteristic frequency. Based on the improved spectral weighting method and the cyclic frequency weighting function, a two-dimensional weighted bivariate diagram is constructed and is mapped to a spectral coherence two-dimensional diagram to enhance the fault characteristics, thereby improving the accuracy of bearing fault diagnosis.

Description

Technical Field

[0001] This invention belongs to the field of fault prediction and health management technology, and relates to a bearing fault diagnosis method based on a bivariate weighted mapping diagram. Background Technology

[0002] Rolling bearings, as one of the fundamental components of rotating machinery, are widely used in various industrial applications. However, because they typically operate under harsh conditions such as heavy loads and high temperatures, they are prone to failure. Once a rolling bearing fails, it can lead to the failure of the entire system, resulting in economic losses and increased time costs. Therefore, the health monitoring and diagnosis of rolling bearings are crucial.

[0003] Bearing vibration signals exhibit typical second-order cyclostationary characteristics. Spectral correlation and its normalized representation (i.e., spectral coherence) can reveal this second-order cyclostationarity by mapping the vibration signal to a bivariate graph composed of spectral frequencies and cyclic frequencies. The spectral frequencies reflect the inherent characteristics of the carrier wave, while the cyclic frequencies reveal the hidden periodic modulation within the signal. Abboud et al. proved that the enhanced envelope spectrum obtained by integrating spectral coherence along the spectral frequencies outperforms the Hilbert envelope spectrum. However, when bearing fault characteristics are weak, the enhanced envelope spectrum struggles to reveal fault information due to noise interference. Therefore, researchers have introduced weighting techniques to construct weighted spectral coherence to enhance fault characteristics. For example, Zhang et al. proposed a new index to quantify the average signal-to-noise ratio (SNR) of the envelope spectrum and then combined this index with spectral coherence to construct a weighted envelope spectrum. Chen et al. proposed frequency domain SNR to measure fault information within a frequency band, thereby enhancing characteristics at the resonant frequency band.

[0004] While weighted spectral coherence can enhance fault characteristics, it still has limitations. Existing methods typically require prior knowledge, such as fault characteristic frequencies, when constructing fault information evaluation indicators. However, in practical applications, fault characteristic frequencies are usually unknown, making it difficult to directly apply existing methods to actual operating conditions. Therefore, there is an urgent need to propose a weighted method that does not require prior knowledge of fault characteristic frequencies to achieve fault diagnosis of bearings under actual operating conditions. Summary of the Invention

[0005] The purpose of this invention is to provide a bearing fault diagnosis method based on a bivariate weighted mapping graph, which solves the problems of fault diagnosis methods relying on prior knowledge and the difficulty in suppressing noise within the signal band.

[0006] The technical solution adopted in this invention is a bearing fault diagnosis method based on a bivariate weighted mapping. This method processes the spectral coherence information of the signal based on the autocorrelation principle, assigns appropriate weights to each spectral frequency of the spectral coherence using the autocorrelation trend, and proposes an improved spectral weighting method. It extracts the information point with the largest amplitude in each cyclic frequency slice, reconstructs a one-dimensional cyclic frequency map, and proposes a cyclic frequency weighting function to enhance the energy of fault characteristic frequencies. Based on the improved spectral weighting method and the cyclic frequency weighting function, a bivariate weighted mapping is constructed and mapped onto the two-dimensional spectral coherence map to enhance fault characteristics, thereby improving the accuracy of bearing fault diagnosis.

[0007] S1 spectrum coherence;

[0008] If the second-order statistical moments (i.e., covariance function) of a vibration signal are periodic, then the signal can be called a second-order cyclostationary signal. Spectral correlation and spectral coherence play crucial roles in second-order cyclostationary signals, converting the signal waveform into a bivariate map composed of spectral frequencies and cyclic frequencies. Assuming the vibration signal is x(t), the instantaneous autocorrelation function of x(t) is as follows:

[0009] C x (t,τ)=E{x(t)x(t-τ) *}=C x (t+T,τ)

[0010] Where E{·} is the averaging operator. Since the instantaneous autocorrelation function of x(t) changes periodically with time t, C x (t,τ) can be expressed in the form of a Fourier series:

[0011]

[0012] Where α is the cycle frequency, τ is the time delay, and the coefficient is... Let C be the cyclic autocorrelation function, and let C be the instantaneous autocorrelation function. x The magnitude of (t, τ) at the cyclic frequency α. The cyclic frequency α is a multiple of the reciprocal of the signal period and can be represented by the function α = i / T (i = 0, 1, 2, ...), where T is the signal period. Cyclic autocorrelation function The definition is as follows:

[0013]

[0014] Where W is the signal duration. Spectral correlation can be obtained by performing a double Fourier transform of the instantaneous autocorrelation function with respect to time t and time delay τ, as shown in the following formula:

[0015] S x (α,f)=∫∫C x (t,τ)e-j2παt e -j2πft dtdτ

[0016] As shown in the formula above, spectral correlation consists of the spectral frequency f and the cyclic frequency α. The cyclic frequency reflects the modulation information of the signal, while the spectral frequency reflects the inherent characteristics of the carrier wave. For a cyclostationary signal that satisfies the generalized cyclostationary property, the spectral correlation can be expressed as follows:

[0017]

[0018] in This is the cyclic spectrum. As shown in the formula above, the spectral correlation exhibits a continuously distributed spectral line at the cyclic frequency α. This spectral line is parallel to the frequency axis. In practical applications, normalized spectral correlation (i.e., spectral coherence) is typically used to analyze signals, with amplitudes between 0 and 1. The spectral coherence formula is shown below:

[0019]

[0020] Where S x (0,f) represents the power spectral density.

[0021] S2 weighted envelope spectrum;

[0022] S2.1 Circular Weighted Function

[0023] As can be seen from the spectral coherence characteristics, when the spectral frequency f is close to the resonant frequency, the spectral coherence has a maximum value at the cyclic frequency of zero. Furthermore, when the periodicity of the spectral coherence increases along the cyclic frequency direction, it has a local maximum at the fault characteristic frequency and its multiples. Otherwise, the function only has a peak at the cyclic frequency of zero.

[0024] Therefore, based on the above characteristics, the position of the fault characteristic frequency in the enhanced envelope spectrum can be determined using the value of α. The proposed cyclic weighting function is shown below:

[0025]

[0026] Where γ x (α,f max The maximum value in each cyclic frequency slice is represented by the value of . Theoretically, fault characteristic frequencies and their harmonics have higher amplitudes than noise spectrum lines. The proposed cyclic weighting function can increase the amplitude difference between fault-related components and other components.

[0027] S2.2 Improved Spectral Weighting Method

[0028] Measuring the periodicity along the cyclic frequency direction of the spectral coherence is a crucial step in constructing the weighting function. Since the autocorrelation function of a periodic function has the same period as its original function, but its periodicity is significantly enhanced, a second-order statistic is a suitable intermediate variable for designing the weighting function. Unlike the original signal autocorrelation defined based on the time delay τ, the autocorrelation in this invention is defined based on the cyclic frequency lag η of the spectral coherence. The autocorrelation function of the spectral coherence is shown below:

[0029]

[0030] Where η represents the cyclic frequency lag, and E{·} is the average operator.

[0031] The autocorrelation function of spectral coherence exhibits the same periodicity as spectral coherence itself. When the spectral frequency f approaches the resonance frequency, the autocorrelation function reaches its maximum value at zero hysteresis. Furthermore, when the periodicity of the spectral coherence increases along the direction of the cyclic frequency, a local maximum exists at the fault characteristic frequency and its multiples. Otherwise, the function has only a peak at zero hysteresis.

[0032] To enhance the resonance band, an improved spectral weighting method was further developed, the specific formula of which is shown below:

[0033]

[0034] The improved spectral weighting method assigns weights to each spectral frequency, with higher weights to frequency slices with stronger periodicity (i.e., resonant frequencies) and lower weights to frequency slices with weaker periodicity. This improved method expands the weight of the resonant band in the integration process, enhancing fault characteristics.

[0035] S2.3 Constructing a bivariate weighted mapping graph

[0036] The cyclic weighting function can increase the amplitude difference between the fault-related component and other components, and the improved spectral weighting method can enhance the resonance band information. Based on the cyclic weighting function and the improved spectral weighting method, a bivariate weighted mapping diagram is constructed, and its formula is shown below:

[0037]

[0038] The bivariate weighted mapping diagram mentioned above comprehensively considers all variables of spectral coherence and enhances fault characteristics from both the spectral frequency and cyclic frequency directions. Therefore, its performance is superior to the existing univariate weighted method.

[0039] S2.4 Calculate the weighted envelope spectrum.

[0040] Mapping the aforementioned bidirectional weighting function onto a two-dimensional spectral coherence graph yields the weighted spectral coherence. Integrating the weighted spectral coherence across the entire frequency band, we obtain the proposed bidirectional weighted enhanced envelope spectrum, as shown in the following formula:

[0041]

[0042] The bivariate weighted mapping W(α,f) enhances fault characteristics by comprehensively considering two variables of spectral coherence. Specifically, it highlights the resonant frequency band by weighting the spectral frequencies, and strengthens the fault characteristic frequencies and the amplitude of their harmonics by weighting the cyclic frequencies.

[0043] The steps of the bearing fault diagnosis method based on bivariate weighted mapping diagram in S3 are as follows:

[0044] The first step is to collect bearing vibration signals. An accelerometer is used to collect signals from a test bench with bearings exhibiting different faults, obtaining the vibration acceleration signal x(t).

[0045] The second step is to calculate the spectral coherence of the signal. Based on the spectral coherence formula, a double Fourier transform is performed on the vibration acceleration signal x(t) to convert the time-domain signal into a two-dimensional spectral coherence graph composed of the spectral frequency and the cyclic frequency.

[0046] The third step is to calculate the bivariate weighted mapping. The autocorrelation function of the spectral coherence is calculated, and an improved spectral weighting method is obtained using the autocorrelation trend. The information points with the largest amplitudes in each cyclic frequency slice are extracted, and a one-dimensional cyclic frequency map is reconstructed to obtain the cyclic frequency weighting function. Based on the improved spectral weighting method and the cyclic frequency weighting function, a bivariate weighted mapping is constructed. The obtained bivariate weighted mapping is then mapped onto the two-dimensional spectral coherence map to obtain the weighted spectral coherence.

[0047] The fourth step is to construct the proposed weighted envelope spectrum. The weighted spectrum is coherently integrated across the entire frequency band to obtain the proposed weighted envelope spectrum for fault diagnosis.

[0048] Compared with the prior art, the present invention has the following beneficial effects.

[0049] This invention proposes a novel method for bearing fault diagnosis based on a bivariate weighted mapping. While existing spectral coherence weighting methods can effectively enhance fault features, they require prior knowledge such as fault feature frequencies and only consider the frequency of a single component of the spectral coherence, thus amplifying in-band noise along with fault features. To overcome the dependence of existing weighting methods on prior knowledge, an improved spectral weighting method is designed, expanding its applicability. This method processes the spectral coherence information of the signal based on the autocorrelation principle, assigning appropriate weights to all spectral frequencies of the spectral coherence using the autocorrelation trend. To suppress in-band noise, the information point with the largest amplitude in each cyclic frequency slice is extracted, a one-dimensional cyclic frequency diagram is reconstructed, and a cyclic frequency weighting function is proposed. Based on the improved spectral weighting method and the cyclic frequency weighting function, a bivariate weighted mapping is constructed and mapped onto a two-dimensional spectral coherence diagram to enhance fault features, thereby improving the accuracy of bearing fault diagnosis. Attached Figure Description

[0050] Figure 1 Flowchart of bearing fault diagnosis method based on bivariate weighted mapping graph.

[0051] Figure 2 These are index values ​​obtained by processing simulation signals of outer ring fault bearings under different signal-to-noise ratios using various methods. (a) Gini index, (b) Negative entropy.

[0052] Figure 3 These are index values ​​obtained by processing simulation signals of inner race fault bearings under different signal-to-noise ratios using various methods. (a) Gini index, (b) Negative entropy.

[0053] Figure 4 The results are from an experiment conducted at Case Western Reserve University on a bearing with an outer race failure. Detailed Implementation

[0054] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0055] Figure 1 This is a flowchart of the bearing fault diagnosis method based on a bivariate weighted mapping diagram according to the present invention. The principle of the bearing fault diagnosis method based on the bivariate weighted mapping diagram will be explained in detail below with reference to the flowchart.

[0056] (1) Acquiring bearing vibration signals. Using an accelerometer, signals from a bearing test bench with different faults are acquired to obtain vibration acceleration signals x(t).

[0057] (2) Calculate the spectral coherence of the signal. According to the spectral coherence formula, perform a double Fourier transform on the vibration acceleration signal x(t) to convert the time-domain signal into a two-dimensional spectral coherence graph composed of the spectral frequency and the cyclic frequency.

[0058] (3) Calculate the bivariate weighted mapping. Calculate the autocorrelation function of the spectral coherence and use the autocorrelation trend to obtain an improved spectral weighting method; extract the information point with the largest amplitude in each cyclic frequency slice, reconstruct the one-dimensional cyclic frequency map, and obtain the cyclic frequency weighting function; construct the bivariate weighted mapping based on the improved spectral weighting method and the cyclic frequency weighting function. Map the obtained bivariate weighted mapping onto the two-dimensional spectral coherence map to obtain the weighted spectral coherence.

[0059] (4) Construct the proposed weighted envelope spectrum. Integrate the weighted spectrum coherence along the entire frequency band of the spectrum, and finally obtain the proposed weighted envelope spectrum to realize fault diagnosis.

[0060] Figure 2 These are index values ​​obtained by processing simulation signals of outer ring fault bearings under different signal-to-noise ratios using various methods. (a) Gini index, (b) Negative entropy.

[0061] Figure 3 These are index values ​​obtained by processing simulation signals of inner race fault bearings under different signal-to-noise ratios using various methods. (a) Gini index, (b) Negative entropy.

[0062] Figure 4 These are the results obtained from experimental signals of an outer ring faulty bearing at Case Western Reserve University. (a) EES, (b) WES, (c) WEES, (d) the proposed method.

Claims

1. A bearing fault diagnosis method based on a bivariate weighted mapping graph, characterized in that: The method includes the following steps: S1 spectrum coherence; If the second-order statistical moments, i.e., the covariance function, of the bearing vibration signal are periodic, then the bearing vibration signal is a second-order cyclic stationary signal. The bearing vibration signal waveform is converted into a bivariate mapping composed of the spectral frequency and the cyclic frequency. The bearing vibration signal is x(t), and the instantaneous autocorrelation function C of x(t) is... x (t,τ) is shown below: C x (t,τ)=E{x(t)x(t-τ) * }=C x (t+T,τ) Where E{·} is the averaging operator; the instantaneous autocorrelation function of x(t) changes periodically with time t, C x (t,τ) can be expressed in the form of a Fourier series: Where α is the cycle frequency; τ is the time delay; and the coefficient is... Let C be the cyclic autocorrelation function, and let C be the instantaneous autocorrelation function. x The magnitude of (t, τ) at the cyclic frequency α; the cyclic frequency α is a multiple of the reciprocal of the bearing vibration signal period, represented by the function α = i / T, i = 0, 1, 2, ..., where T is the period of the bearing vibration signal; cyclic autocorrelation function. The definition is as follows: Where W is the duration of the bearing vibration signal; The spectral correlation is obtained by performing a double Fourier transform of the instantaneous autocorrelation function with respect to time t and time delay τ, as shown in the following formula: S x (α,f)=∫∫C x (t,τ)e -j2παt e -j2πft dtdt The spectral correlation consists of the spectral frequency f and the cyclic frequency α; where the cyclic frequency reflects the modulation information of the bearing vibration signal, and the spectral frequency reflects the inherent characteristics of the carrier wave; for a cyclostationary bearing vibration signal that satisfies the generalized cyclostationary characteristics, the spectral correlation is expressed in the following form: in For a cyclic spectrum; α = 0, ±1, ±2, ..., according to the above formula, the spectral correlation has a continuously distributed spectral line at the cyclic frequency α. The spectral line is parallel to the frequency axis; normalized spectral correlation, i.e., spectral coherence, is used to analyze the bearing vibration signal. The spectral coherence formula is shown below: Where S x (0,f) represents the power spectral density; S2 weighted envelope spectrum; S2.1 cyclic weighted function; When the spectral frequency f is close to the resonance frequency, the spectral coherence has a maximum value at the cyclic frequency of zero, and when the periodicity of the spectral coherence increases along the cyclic frequency direction, there is a local maximum at the fault characteristic frequency and its multiples; otherwise, the cyclic weighting function has only one peak at the cyclic frequency of zero. The position of the fault characteristic frequency in the enhanced envelope spectrum is determined using the value of α. The cyclic weighting function is shown below: Where γ x (α,f max The maximum value in each cyclic frequency slice; the cyclic weighting function can increase the amplitude difference between the fault-related component and other components; S2.2 Improved Spectral Weighting Method Measuring the periodicity along the cyclic frequency direction of the spectral coherence is key to constructing the weighting function. The autocorrelation is defined based on the cyclic frequency lag η of the spectral coherence; the autocorrelation function of the spectral coherence is shown below: Where η represents the cyclic frequency lag, and E{·} is the average operator; The autocorrelation function of spectral coherence has the same periodicity as spectral coherence; the autocorrelation function has a maximum value at zero hysteresis when the spectral frequency f is close to the resonance frequency. Furthermore, when the periodicity of the spectral coherence increases along the direction of the cyclic frequency, there is a local maximum at the fault characteristic frequency and its multiples; otherwise, the function has only one peak at zero hysteresis. To enhance the resonance band, an improved spectral weighting method is constructed, the formula of which is shown below: The improved spectral weighting method assigns weights to each spectral frequency, with high weights to frequency slices with strong periodicity (i.e., resonant frequencies) and low weights to frequency slices with weak periodicity. The improved spectral weighting method also increases the weight of the resonant band in the integration process, thereby enhancing fault characteristics. S2.3 Construct a bivariate weighted mapping graph; The cyclic weighting function can increase the amplitude difference between the fault-related component and other components, and the improved spectral weighting method can enhance the resonance band information. Based on the cyclic weighting function and the improved spectral weighting method, a bivariate weighted mapping diagram is constructed, as shown in the following formula: S2.4 Calculate the proposed weighted envelope spectrum; Mapping the aforementioned bidirectional weighting function to a two-dimensional spectral coherence graph yields the weighted spectral coherence; integrating the weighted spectral coherence across the entire frequency band, we obtain the proposed bidirectional weighted enhanced envelope spectrum, as shown in the following formula: The bivariate weighted mapping W(α,f) enhances fault characteristics by comprehensively considering two variables of spectral coherence; it highlights the resonant frequency band by weighting the spectral frequencies, and strengthens the fault characteristic frequencies and the amplitude of their harmonics by weighting the cyclic frequencies. S3 uses a bivariate weighted mapping based on S2 for bearing fault diagnosis; the specific implementation steps are as follows: The first step is to collect bearing vibration signals; use an accelerometer to collect bearing vibration signals with different faults to obtain the bearing vibration signal x(t); The second step is to calculate the spectral coherence of the bearing vibration signal. Based on the spectral coherence formula, a double Fourier transform is performed on the bearing vibration signal x(t) to convert the time-domain signal into a two-dimensional spectral coherence graph composed of the spectral frequency and the cyclic frequency. The third step involves calculating the bivariate weighted mapping; calculating the autocorrelation function of the spectral coherence, and using the autocorrelation trend to obtain an improved spectral weighting method; extracting the information point with the largest amplitude in each cyclic frequency slice, reconstructing the one-dimensional cyclic frequency map, and obtaining the cyclic frequency weighting function; constructing the bivariate weighted mapping based on the improved spectral weighting method and the cyclic frequency weighting function; and mapping the obtained bivariate weighted mapping onto the two-dimensional spectral coherence map to obtain the weighted spectral coherence. The fourth step is to construct the proposed weighted envelope spectrum; integrate the weighted spectrum coherently along the entire frequency band of the spectrum, and finally obtain the proposed weighted envelope spectrum to realize bearing fault diagnosis.

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