Flexible optimal demodulation method under strong interference environment based on Ramanujan theory

Abstract

The invention discloses a flexible optimal demodulation method under a strong interference environment based on a Ramanujan theory, and aims to effectively extract fault feature information from a complex fault signal. Relates to the field of signal processing and feature recognition. Comprising the steps of collecting fault bearing vibration signals in a strong interference environment; searching a frequency spectrum division boundary by adopting a self-adaptive frequency band division technology; the method comprises the following steps of: constructing a generalized filter (GF) based on a boundary, and outputting an optimal filtering component of each frequency band by using a unique multi-stage filtering strategy of the GF; the signal is recovered through a Ramanujan theory, and a time domain component is obtained; and constructing a reweighted Ramanujan spectrum negentropy index, selecting an optimal demodulation frequency band, and extracting fault features by using an envelope spectrum. According to the method, the generalized filter and the reweighted Ramanujan spectrum negentropy index cooperate with each other, accurate extraction of fault feature information in a complex interference environment can be realized, high noise robustness is achieved, and all interference information can be filtered out to a certain extent through a flexible filtering strategy.

Description

Technical Field

[0001] This invention relates to the field of optimal demodulation frequency band and its fault feature extraction, specifically a flexible optimal demodulation method based on Ramanujan's theory under strong interference conditions. Background Technology

[0002] Bandwidth demodulation is a commonly used method for revealing the state characteristics of fault signals. It mainly includes frequency band allocation, filtering, and optimal frequency band selection. For many years, researchers have been dedicated to developing methods to accurately and effectively extract fault features from raw signals. They have continuously proposed more reasonable frequency band allocation methods, improved filtering techniques, and effective feature indices to evaluate fault feature information. This paper addresses the problem that traditional bandwidth demodulation methods struggle to effectively extract fault features in complex interference environments. The performance of filtering and noise reduction significantly affects the fault information content in the filtered components; evaluation indices are not sensitive enough to fault information and have poor noise robustness, directly impacting the accuracy of index evaluation. The use of a single filtering strategy and evaluation indices lacking robustness are also issues.

[0003] Problems with existing optimal frequency band demodulation methods:

[0004] Traditional filtering methods (such as finite impulse response filters and time-varying morphological filters) lack interference adaptability and exhibit poor filtering performance under multiple interference components. Traditional evaluation metrics (such as kurtosis, energy, and negative entropy) lack the ability to deeply mine fault information and have poor stability in single-scale quantization. Existing band demodulation methods have poor adaptability to complex interference components and lack flexibility. Evaluation metrics lack the ability to quantify detailed feature information, and the stability of single-scale evaluation perspectives is poor. Summary of the Invention

[0005] To address the shortcomings of existing frequency band demodulation methods in filtering and indexing, this invention provides a more flexible optimal demodulation method: the generalized reweighted Ramanujan gram. The method provided by this invention enables dynamic adjustment of the filtering strategy under complex interference environments, significantly improving the flexibility of the filtering process. Furthermore, the reweighted Ramanujan spectral negative entropy index possesses multi-scale fault feature quantification characteristics, resulting in higher evaluation accuracy and sensitivity to fault information.

[0006] The generalized reweighted Ramanujan gram method has the following steps:

[0007] Step 1: Collect the vibration signal x(t) of the faulty bearing under strong interference conditions.

[0008] Step 2: Set the initial number of partitioned layers to k = 1, the maximum number of partitioned layers to K = 9, and the number of frequency bands in the k-th layer to k + 2. Next, use adaptive frequency band partitioning technology to find the set of normalized frequency band partitioning boundaries f for the k-th layer.i (i = 0, 1, 2, ···, k + 2), where f0 = 0 and f k+2 = π.

[0009] Step 3: Construct a generalized filter based on the Ramanujan period transformation theory to obtain the generalized filtering components λik(t) in each frequency band.

[0010] (1) Construct a generalized filter and obtain the filtering components in each frequency band. Its expression is as follows:

[0011]

[0012] where the value range of i is [0, k + 2], G i (f) represents a raised cosine filter, η i (f) represents a zero-phase filter, H i (f) represents a FIR filter, and S(f) represents the Ramanujan Fourier transform of the signal.

[0013] 1) Cosine filter

[0014] The cosine filter is faster and the filter shape is more compact, making the energy of the fault signal more concentrated and the transition band smooth, so that there will be no loss or distortion of useful signals due to the sudden cut-off of the filter. The low-pass cosine filter is expressed as:

[0015]

[0016] where the influence factor α controls the width of the transition region, and α ∈ [0, 1].

[0017] When designing a filter bank with multiple different cut-off frequencies f1, f2, ···, f M , it is necessary to ensure that the transition bands of consecutive filters do not overlap, that is, αf i + αf i+1 < f i+1 - f i , 1 ≤ i < M. Therefore, α is expressed as:

[0018]

[0019] The expression of the high-pass cosine filter is:

[0020] h i (f) = 1 - g i (f) (9)

[0021] In summary, the complete cosine filter is shown in formula (6), where h0(f) = 0.

[0022] Gi =g i (f)h i-1 (f) (10)

[0023] 2) Zero-phase filter

[0024] Constructing a zero-phase filter bank η i (f), as shown in Formula 7.

[0025]

[0026] 3) FIR filter

[0027] An FIR filter is constructed using a windowing technique, and its frequency response is defined as:

[0028]

[0029] In the formula, f i-1 and f i These are the lower and upper cutoff frequencies of the passband, respectively. τ is the group delay coefficient.

[0030] (2) Then, the time-domain component λ(t) (i = 1, 2, ..., k+2) is recovered using the inverse Ramanujan Fourier transform, where λik is the i-th component of the k-th layer.

[0031]

[0032] A generalized filter capable of flexible filtering and noise reduction was constructed, which can adapt to complex and dynamic signal environments and adaptively adjust the filtering module to provide more accurate and efficient filtering results.

[0033] Step 4: Construct a reweighted Ramanujan spectral negative entropy index and select the optimal filtering component. The Ramanujan spectral negative entropy and squared envelope negative entropy calculation formulas are as follows.

[0034]

[0035] Where RS represents the Ramanujan spectrum of the signal, and SE represents the square envelope of the signal.

[0036] The specific steps for calculating the reweighted Ramanujan spectral negative entropy are as follows:

[0037] (1) Calculate the RS and SE of the reconstructed component x(n).

[0038] (2) Divide the signal into equal segments and set the initial average number of segments Q = 1.

[0039] (3) Divide the RS and SE of the signal into Q segments on average to obtain the segment set RS. q and SE q , where q=1,2,···,Q.

[0040] (4) Calculate the negative entropy NEq RS and the square envelope negative entropy NEq SE of each small segment of the Ramanujan spectrum according to formulas (5) and (6).

[0041] (5) Sort all elements in NEq RS in ascending order to obtain the ascending vector NERSsorta of RS. Similarly, sort all elements in NEq SE in ascending order to obtain the ascending vector NSEsorta of SE.

[0042] (6) Calculate the weight of each individual element in NEq RS relative to the sum of elements to obtain the weight vector WQRS of RS. Its calculation expression is shown in formula (13).

[0043]

[0044] Using the same calculation method, we obtain the weight vector NEq SE of SE.

[0045] (7) Sort all elements in WQRS in descending order to obtain the descending vector WRSsortd of RS. Similarly, sort all elements in WQSE in descending order to obtain the descending vector WSEsortd of SE.

[0046] (8) Calculate the Ramanujan spectral negative entropy RSNE of the reconstructed signal x(n) using the following formula.

[0047]

[0048] Where ⊙ represents the inner product operation.

[0049] Repeat the above process to obtain the squared envelope negative entropy SENE of the reconstructed signal.

[0050] (9) In summary, the reweighted Ramanujan spectral negative entropy RRSNE of x(n) is obtained. Q :

[0051] RRSNE Q =(RSNE+SENE) / 2 (15)

[0052] (10) Finally, determine RRSNE Q Is it optimal? If RRSNE Q ≤RRSNE Q+1 Then let Q = Q + 1, and repeat steps (2) to (9). If RRSNE Q>RRSNE Q+1 The output RRSNE k =RRSNE Q The reweighted Ramanujan spectral negative entropy is used to reconstruct the components.

[0053] Step 5: Determine if the decomposition is complete.

[0054] If k < K, then let k = k + 1, and repeat steps 2 to 4. If k = K, then let k = k + 1, the loop decomposition process ends, and step 6 is executed.

[0055] Step 6: Select RRSNE in all layers k The frequency band with the highest value is selected as the optimal demodulation frequency band. Then, envelope spectrum analysis is performed to extract fault characteristics.

[0056] A reweighted Ramanujan spectral negative entropy index with multi-scale fault feature quantification characteristics is proposed. This index dynamically evaluates fault information through reweighted statistical features, optimizes fault distribution characteristics, and makes its evaluation performance more accurate and stable.

[0057] Compared with existing known technologies, the technical solution provided by this invention has the following significant advantages:

[0058] This invention proposes a flexible optimal demodulation method under strong interference conditions based on Ramanujan's theory. This method features a multi-level filtering strategy, which can dynamically adjust filtering characteristics, improving the flexibility of the filtering process; furthermore, the reweighting of indicators endows it with multi-scale quantization characteristics, enhancing the sensitivity and robustness to fault information. Specifically,

[0059] 1) A generalized filter employing a multi-stage filtering strategy was constructed. This filter can not only flexibly adapt to complex and dynamic signal environments, but also, in conjunction with a reweighted Ramanujan spectral negative entropy index, adjust the filtering module, thereby providing more accurate and efficient filtering results.

[0060] 2) A reweighted Ramanujan spectral negative entropy index is defined, which enables multi-scale quantitative evaluation of fault characteristics. This index dynamically evaluates the fault information content and optimizes fault distribution characteristics through reweighted statistics. This method makes the evaluation performance more accurate and stable.

[0061] 3) Finally, Ramanujan's theory was introduced to improve the periodicity extraction capability of the method. At the same time, the method was applied to the field of mechanical fault diagnosis. Experimental results show that the model has superior fault feature diagnosis performance, flexible and excellent filtering performance, and stable and sensitive index evaluation. It is an efficient diagnostic tool.

[0062] This invention achieves the first effective extraction of mechanical fault feature information under complex interference environments through a triple innovation of Ramanujan Fourier transform, generalized filter, and reweighted Ramanujan spectral negative entropy, overcoming the shortcomings of existing technologies in terms of noise reduction, robustness, and feature extraction. Attached Figure Description

[0063] Figure 1 This is a flowchart illustrating a flexible optimal demodulation method (GRRgram) under strong interference conditions based on Ramanujan theory according to the present invention.

[0064] Figure 2 (a) Time-domain plot and (b) Envelope spectrum of the simulated signal;

[0065] Figure 3 The diagnostic results of the GRRgram method of the present invention are as follows: (a) GRRgram marker spectrogram; (b) time-domain plot; (c) envelope spectrum;

[0066] Figure 4 For the present invention Figure 3 (a) Original spectral plot of GRRgram;

[0067] Figure 5 Diagnostic results of the CERRgram method: (a) CERRgram labeled spectrogram; (b) time-domain plot; (c) envelope spectrum;

[0068] Figure 6 for Figure 5 (a) Original CERRgram spectrum;

[0069] Figure 7 Diagnostic results of the Ramanujan-gram method: (a) Ramanujan-gram labeled spectrogram; (b) time-domain plot; (c) envelope spectrum;

[0070] Figure 8 for Figure 7 (a) Original spectrum of Ramanujan-gram;

[0071] Figure 9 Diagnostic results for the Autogram method: (a) Autogram labeled spectrogram; (b) Time-domain plot; (c) Envelope spectrum;

[0072] Figure 10 for Figure 9 (a) Original Autogram spectrum;

[0073] Figure 11 This invention provides a fault simulation test bench for low-speed heavy-duty axle and wheel system.

[0074] Figure 12 Measured signal for outer ring fault: (a) Time domain plot; (b) Envelope spectrum;

[0075] Figure 13 The diagnostic results of the GRRgram method of the present invention are as follows: (a) GRRgram marker spectrogram; (b) time-domain plot; (c) envelope spectrum;

[0076] Figure 14 for Figure 13 (a) Original spectral plot of GRRgram;

[0077] Figure 15 Diagnostic results of the CERRgram method: (a) CERRgram labeled spectrogram; (b) time-domain plot; (c) envelope spectrum;

[0078] Figure 16 for Figure 15 (a) Original CERRgram spectrum;

[0079] Figure 17 Diagnostic results of the Ramanujan-gram method: (a) Ramanujan-gram labeled spectrogram; (b) time-domain plot; (c) envelope spectrum;

[0080] Figure 18 for Figure 17 (a) Original spectrum of Ramanujan-gram;

[0081] Figure 19 Diagnostic results for the Autogram method: (a) Autogram labeled spectrogram; (b) Time-domain plot; (c) Envelope spectrum;

[0082] Figure 20 for Figure 5 (a) Autogram marker spectrogram. Detailed Implementation

[0083] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0084] Example 1

[0085] The generalized reweighted Ramanujan gram technique in this embodiment includes the following steps:

[0086] Step 1: Construct the bearing fault simulation signal x(t) to simulate the fault diagnosis performance in the theoretical environment.

[0087] Step 2: Set the initial number of partition layers as k = 1, the maximum number of partition layers as K = 9, and the number of frequency bands in the k-th layer as k + 2. Then, adopt the adaptive frequency band partitioning technique to find the set of normalized frequency band partitioning boundaries f i (i = 0, 1, 2, ···, k + 2), where f0 = 0 and f k+2 = π.

[0088] Step 3: Construct a generalized filter based on the Ramanujan period transform theory to obtain the generalized filtered components λik(t) in each frequency band.

[0089] (1) Construct a generalized filter and obtain the filtered components in each frequency band. Its expression is as follows:

[0090]

[0091] where the value range of i is [0, k + 2], G i (f) represents a raised cosine filter, η i (f) represents a zero-phase filter, H i (f) represents a FIR filter, and S(f) represents the Ramanujan Fourier transform of the signal.

[0092] 1) Cosine filter

[0093] The cosine filter is faster, the filter shape is more compact, making the energy of the fault signal more concentrated, and the transition band is smooth, so there will be no loss or distortion of useful signals due to the sudden cut-off of the filter. The low-pass cosine filter is expressed as:

[0094]

[0095] where the influencing factor α controls the width of the transition region, α ∈ [0, 1].

[0096] When designing a filter bank with multiple different cut-off frequencies f1, f2, ···, f M , it is necessary to ensure that the transition bands of consecutive filters do not overlap, that is, αf i + αf i+1 < f i+1 - f i , 1 ≤ i < M. Therefore, α is expressed as:

[0097]

[0098] The expression of the high-pass raised cosine filter is:

[0099] h i (f)=1-g i (f) (9)

[0100] In summary, the complete raised cosine filter is shown in equation (10), where h0(f) = 0.

[0101] G i =g i (f)h i-1 (f) (10)

[0102] 2) Zero-phase filter

[0103] Constructing a zero-phase filter bank η i (f), as shown in Formula 7.

[0104]

[0105] 3) FIR filter

[0106] An FIR filter is constructed using a windowing technique, and its frequency response is defined as:

[0107]

[0108] In the formula, f i-1 and f i These are the lower and upper cutoff frequencies of the passband, respectively. τ is the group delay coefficient.

[0109] (2) Then, the time-domain component λ(t) (i = 1, 2, ..., k+2) is recovered using the inverse Ramanujan Fourier transform, where λik is the i-th component of the k-th layer.

[0110]

[0111] By using a generalized filter with flexible noise reduction performance according to formula (1) to process fault signals, the single filtering process is transformed into a composite filtering process guided by indicators; GF i (f) represents the result of multi-stage filtering by the generalized filter, which is a set of filtered components; S(f) represents the Ramanujan-Fourier transform of the signal, and the set of filtered components S is obtained through noise reduction processing by the generalized filter. i (f); Then, Equations (3) and (4) are the process of introducing the reweighted Ramanujan spectral negative entropy index into the generalized filter; quantization of the filter component set S i The fault information content of (f) guides the output of the optimal filtering result as the generalized filtering component; at this time, the complete generalized filtering process ends, and the final output generalized filtering component is the generalized filter result.

[0112] Step 4: Construct a reweighted Ramanujan spectral negative entropy index and select the optimal filter component for the k-th layer.

[0113] First, the formulas for calculating the Ramanujan spectral negative entropy and the squared envelope negative entropy are as follows.

[0114]

[0115] Where RS represents the Ramanujan spectrum of the signal, and SE represents the square envelope of the signal.

[0116] The specific steps for calculating the reweighted Ramanujan spectral negative entropy are as follows:

[0117] (1) Calculate the RS and SE of the reconstructed component x(n).

[0118] (2) Divide the signal into equal segments and set the initial average number of segments Q = 1.

[0119] (3) Divide the RS and SE of the signal into Q segments on average to obtain the segment set RS. q and SE q , where q=1,2,···,Q.

[0120] (4) Calculate the negative entropy NEq RS and the square envelope negative entropy NEq SE of each small segment of the Ramanujan spectrum according to formulas (16) and (17).

[0121] (5) Sort all elements in NEq RS in ascending order to obtain the ascending vector NERSsorta of RS. Similarly, sort all elements in NEq SE in ascending order to obtain the ascending vector NSEsorta of SE.

[0122] (6) Calculate the weight of each individual element in NEq RS relative to the sum of elements to obtain the weight vector WQRS of RS. Its calculation expression is shown in formula (13).

[0123]

[0124] Using the same calculation method, we obtain the weight vector NEq SE of SE.

[0125] (7) Sort all elements in WQRS in descending order to obtain the descending vector WRSsortd of RS. Similarly, sort all elements in WQSE in descending order to obtain the descending vector WSEsortd of SE.

[0126] (8) Calculate the Ramanujan spectral negative entropy RSNE of the reconstructed signal x(n) using the following formula.

[0127]

[0128] Where ⊙ represents the inner product operation.

[0129] Repeat the above process to obtain the squared envelope negative entropy SENE of the reconstructed signal.

[0130] (9) In summary, the reweighted Ramanujan spectral negative entropy RRSNE of x(n) is obtained. Q :

[0131] RRSNE Q =(RSNE+SENE) / 2 (15)

[0132] (10) Finally, determine RRSNE Q Is it optimal? If RRSNE Q ≤RRSNE Q+1 Then let Q = Q + 1, and repeat steps (2) to (9). If RRSNE Q >RRSNE Q+1 The output RRSNE k =RRSNE Q The reweighted Ramanujan spectral negative entropy is used to reconstruct the components.

[0133] The constructed reweighted Ramanujan spectrum negative entropy index comprehensively considers the fault feature information contained in the time and frequency domains, and calculates the Ramanujan spectrum and square envelope of the signal. Then, it is given multi-scale fault feature quantification characteristics through reweighting, optimizes the fault distribution characteristics, and improves noise robustness.

[0134] By combining formulas (5) and (6), the Ramanujan spectral negative entropy and the square envelope negative entropy are calculated, giving the reweighted Ramanujan spectral negative entropy accurate fault information quantification capability, effectively extracting fault features at various scales and visualizing them;

[0135] Dual-domain negative entropy contains more comprehensive fault information, and the reweighting strategy gives multi-scale fault feature quantification capability and stronger anti-interference capability. Finally, Ramanujan spectral negative entropy and square envelope negative entropy effectively extract fault feature information and visualize it.

[0136] In summary, the fault information quantification capability of the reweighted Ramanujan spectral negative entropy is more accurate and stable.

[0137] Step 5: Determine if the decomposition is complete.

[0138] If k < K, then let k = k + 1, and repeat steps 2 to 4. If k = K, then let k = k + 1, the loop decomposition process ends, and step 6 is executed.

[0139] Step 6: Select RRSNE in all layers k The frequency band with the highest value is selected as the optimal demodulation frequency band. Then, envelope spectrum analysis is performed to extract fault characteristics.

[0140] To verify the effectiveness of this invention in the field of fault diagnosis, this embodiment conducted a simulated signal diagnosis experiment, and used Ramanujan gram (CERRgram), Autogram, and Ramanujan-gram as comparative methods to illustrate the effectiveness and superiority of the method. The aforementioned three methods are all prior art.

[0141] The search links and method details for the three comparison methods are as follows:

[0142] 1) Eigen-energy ratio Ramanujan Gram: A novel optimal multi-band demodulation method; Search link:

[0143] https: / / ieeexplore.ieee.org / document / 10752557;

[0144] The feature energy ratio Ramanujan Gram method includes the following steps:

[0145] Step 1: Set the maximum number of division layers K = 9, and set the number of frequency bands divided by layer k = 1 to i = 3.

[0146] Step 2: Obtain the boundary set J using the improved frequency band division method. i (i = 0, 1, 2, ..., k+2).

[0147] (1) Calculate the power spectral density S of the signal x(t). xx (f).

[0148]

[0149] (2) The envelope curve of the power spectral density is optimized by sequential order statistical filter, simplifying the distribution of the envelope curve and highlighting the location of fault characteristics.

[0150] (3) Extract the positions of the first D = k + 2 large points in the optimized envelope curve, and use the smallest local minimum point between adjacent large points as the dividing boundary. Finally, normalize the data to obtain the boundary set J. i (i = 0, 1, 2, ..., k+2).

[0151] Step 3: Construct a zero-phase filter using the empirical Ramanujan decomposition method to obtain the filter components λik(t) for each frequency band.

[0152] (1) Constructing a zero-phase filter bank η i (a) Obtain the filtered signal S i (f).

[0153]

[0154] Where S(f) represents the Ramanujan Fourier transform of the signal x(t).

[0155] (2) The inverse Ramanujan Fourier transform is used to recover the time-domain filtered component set λ(t) (i=1,2,…,k+2).

[0156]

[0157] Where λik(t) represents the recovered i-th temporal component of the k-th layer, and λ(t) represents the set of temporal components of the current layer.

[0158] Step 4: Calculate the characteristic energy ratio of each component, and then construct the characteristic energy ratio Ramanujan gram.

[0159] (1) Calculate the characteristic energy ratio.

[0160]

[0161] P(f)=|IRS(f)| 2 +|IRS(2f)| 2 +…+|IRS(ωf)| 2 (twenty four)

[0162] Where P(f) represents the sum of harmonic energies, f represents the fault characteristic frequency, IRS(·) represents the amplitude of the fault frequency in the iterative Ramanujan spectrum, N represents the spectral length, and ω represents the order of the harmonics (the energy of the fault characteristic harmonics in the demodulated band envelope spectrum decreases with increasing order. The main fault characteristics are usually concentrated in the lower harmonics (such as the first, second, and third harmonics), which have higher energy and more obvious fault characteristics). Therefore, this paper selects ω = 3.

[0163] (2) Considering the certain deviation between the actual fault characteristic frequency and the theoretical value, the corrected harmonic sum is expressed as:

[0164]

[0165] Where Δ represents the error range, which is usually taken as 1.

[0166] Step 5: Determine if the decomposition is complete.

[0167] If k < K, then set k = k + 1 and repeat steps 2 - 4; if k = K, the decomposition process is complete and step 6 is executed.

[0168] Step 6: Select the frequency band with the largest characteristic energy ratio as the optimal demodulation frequency band, and perform iterative Ramanujan spectral analysis to extract composite fault features.

[0169] The above is the fault diagnosis process of the characteristic energy ratio Ramanujan gram method.

[0170] 2) Autogram: An effective method for selecting the optimal demodulation frequency band in rolling bearing diagnosis; retrieval link:

[0171] https: / / www.sciencedirect.com / science / article / pii / S0888327017306441;

[0172] Regarding the Autogram method, it includes the following steps:

[0173] Step 1: According to the binary tree structure, divide the time-domain data into frequency bands through wavelet transform (WT). Wavelets have very good local characteristics in both the time domain and the frequency domain, and WT can be used as an effective filter to divide the signal into different frequency bands and center frequencies. Remove the downsampling step in the discrete wavelet packet transform (DWPT) by using the maximum overlap discrete wavelet packet transform (MODWPT). Apply MODWPT as a filter to the studied time history, and thus a series of signals are generated at each decomposition level. The filtered signals, each corresponding to a frequency band and a center frequency (node), are the input for the following step 2.

[0174] Step 2: Calculate the unbiased autocorrelation (AC) of the squared envelope of the signal x(t).

[0175]

[0176] where X is the squared envelope of the filtered signal, q = 0, 1, …, N - 1, and the delay factor τ = q / f s .

[0177] Step 3: Find the most suitable demodulation frequency band. To quantify the impulsiveness of the AC of each node, three modified versions of the equation based on the kurtosis equation are proposed as follows:

[0178]

[0179] where N is the length of the original signal, |·| + and |·| -This indicates that only positive and negative values ​​are accepted, and all other data points are set to zero; in addition... This represents the threshold level, which is defined here as a moving average.

[0180]

[0181] Where k represents the length of the windowed signal during the averaging process.

[0182] Step 4: Finally, perform a Fourier transform on the squared envelope of the signal associated with the node selected in Step 3 to extract the fault characteristic frequency and perform bearing diagnosis.

[0183] 3) Ramanujan-gram: An autonomous weak-periodic tomographic extraction method under strong noise conditions; Search link:

[0184] https: / / sage.cnpereading.com / paragraph / article / ?doi=10.1177 / 14759217231197806;

[0185] The Ramanujan-gram method includes the following steps:

[0186] Step 1: Obtain the fault vibration signal x(t).

[0187] Step 2: Set the maximum number of layers in the Ramanujan-gram to N (N is set to 9 in this paper), and set the number of frequency bands in the k-th (k=1,…,N) layer to D=k+2.

[0188] Step 3: Adaptively segment the k-th layer Ramanujan-gram frequency band to obtain boundary B. i (i = 1, 2, ..., k+1).

[0189] The steps of the adaptive frequency band segmentation method are as follows:

[0190] (1) Perform a Fourier transform on x(t) to obtain its Fourier spectrum X(f).

[0191] (2) Use the sequential order statistical filtering method to envelop X(f) and obtain the spectral envelope component.

[0192] OSF(n) = max(Y n (f)) (28)

[0193] Where OSF(n) represents the result of the sequential order statistical filtering method, Y n (f) represents all the data of the spectrum X(f) in the nth window.

[0194] (3) The spectrum envelope component is optimized by smoothing to remove first-order non-differentiable points and avoid their influence on the segmentation results, thereby obtaining a smooth spectrum envelope component.

[0195]

[0196] Where y s (i) is the smoothed value at the i-th point, and M is the value of y. s (i) The number of adjacent data points on both sides, where 2M+1 is the span.

[0197] (4) Set the number of segments to D, and then find the extreme points of the smooth spectral envelope.

[0198] Take the first D maxima of all maxima points, and then locate the boundary of the minimum value closest to the midpoint between two adjacent maxima. This is the segmentation position of the smooth spectrum envelope component.

[0199] (5) Standardize the boundary to obtain boundary B. i (i = 1, 2, ..., k+1).

[0200] Step 4: Reconstruct all frequency band components in the k-th layer using Ramanujan feature extraction technology.

[0201] (1) B i Normalize to the range [0,π] and supplement the endpoint boundaries.

[0202] (2) Constructing the filter bank η i (a) and obtain the filtered signal R i (g)

[0203]

[0204] Where R(g) represents the Ramanujan Fourier transform of the signal.

[0205] (3) The time-domain filtered component set r is recovered by using the inverse Ramanujan Fourier transform. i (t).

[0206] r i (t)=R -1 [R i (g)] (32)

[0207]

[0208] Where ri(t) represents the recovered i-th temporal component of the k-th layer, and r(t) represents the set of temporal components of the current layer.

[0209] Step 5: Construct an adaptive squared envelope spectrum weighted kurtosis index to quantify the signal content in the time-domain components.

[0210] (1) Calculate the squared envelope spectrum of the filtered component r(t).

[0211] SE(t)=r(t) 2 +H[r(t)] 2 (34)

[0212] SES(f)=F(SE(t)) (35)

[0213] Where H[·] represents the Hilbert transform and F[·] represents the Fourier transform.

[0214] (2) Divide SES(f) into J segments with an initial value of 1.

[0215] (3) Calculate each SES segment j The kurtosis value K of (j=1,2,...,J) j (j = 1, 2, ..., J).

[0216]

[0217] (4) K j The ascending order of kurtosis vector K is obtained by sorting (j = 1, 2, ..., J) in ascending order. V .

[0218] (5) Calculate K j The weight of each value in (j = 1, 2, ..., J).

[0219]

[0220] (6) W j The weights are obtained by arranging the j = 1, 2, ..., J in descending order. V .

[0221] (7) Calculate the adaptive squared envelope spectrum weighted kurtosis index ASESSK J .

[0222] ASESWK J =K V ×(W V ) T (38)

[0223] (8) Repeat steps (1)-(7) until J = 8. Maximum ASESSWK J Defined as ASESWK, as shown below:

[0224] ASESWK = max(ASESWK)J (40)

[0225] Step 6: Select the reconstructed component corresponding to the maximum ASESSK value, plot the square envelope spectrum, and extract the fault characteristic frequencies.

[0226] The above is an introduction to three comparative methods. The inventors of the first and third comparative methods are both listed in journal articles. The second method is based on existing publicly known literature. The following section is the experimental analysis. The aforementioned scheme elucidates the core approach regarding the three methods for extracting fault feature frequencies.

[0227] Experiments verified that the constructed simulation signal expression is shown in formula (16).

[0228]

[0229] In the formula, x(t) is the fault simulation signal, composed of the fault pulse signal x1(t) and Gaussian white noise interference component n(t). x1(t) is the fault pulse part, which is the superposition of M exponentially decaying sine waves, each corresponding to one fault impact response; a represents the impact sequence number (a = 1, 2, ..., M), representing the a-th fault impact; A represents the amplitude modulation term, with amplitude A changing periodically with time, simulating the characteristic of the fault impact being modulated by the bearing rotation (the impact intensity is greatest when the fault location rotates past the sensor; the impact intensity weakens as it moves away). Furthermore, n(t) represents Gaussian white noise with a signal-to-noise ratio of -15dB; f r The frequency represents the rotation frequency; f is the fault characteristic frequency; M represents the total number of cyclic pulses at the fault characteristic frequency, M = 3f; f n ξ is the resonant frequency of the bearing fault pulse; ξ is the damping ratio, controlling the signal attenuation rate (the larger ξ is, the faster the attenuation); δ is the small fluctuation generated when the rolling element fault pulse occurs, and δ follows a uniform distribution. The various parameter settings for the simulation signal are shown in Table 1.

[0230] Table 1 Simulation Signal Parameters

[0231]

[0232] Figure 2 The time-domain plot and envelope spectrum results of the simulated signal are shown. The spectral distribution characteristics of the fault feature information in the envelope spectrum are as follows: the spectral lines have obvious peaks at the fault feature frequency f and its harmonics. Due to the addition of -15dB Gaussian white noise, the periodic pulse information in the signal is completely submerged, and there are no obvious peaks at the fault feature frequency f and its harmonics in the signal's envelope spectrum. Therefore, the fault feature information cannot be diagnosed.

[0233] Figure 3The above figures show the diagnostic results of the GRRgram method in the simulated signal experiment. The upper left of Figure (a) shows the frequency response spectrum of the GRNN neural network, and the lower left shows a schematic diagram of the optimal position of the marked frequency response spectrum of the GRNN neural network; the horizontal axis represents frequency (unit: Hz), and the vertical axis represents signal strength (level). Different gray levels in the spectrum correspond to the signal energy distribution at different frequencies: high-brightness areas (such as high-frequency bands) represent stronger energy for that frequency component, reflecting the GRNN network's response preference to specific frequency signals; low-brightness areas correspond to weaker energy frequency components. High brightness represents the optimal solution. Figure 3 (a) It can be seen that the optimal demodulation frequency band position diagnosed is located in the first band of the sixth layer. Figure 3 (b) is the time domain diagram of this frequency band. Figure 3 (c) shows the optimal demodulated frequency band envelope spectrum. The fault characteristic frequency f and its harmonics exhibit prominent spectral peaks, allowing for the effective extraction of fault characteristic frequency f and its harmonics 2f, 3f, and 4f. Therefore, the GRRgram method can effectively diagnose fault characteristic information under strong interference conditions. Figure 4 for Figure 3 (a) The original spectrogram, without the red box marking.

[0234] Figure 5 , 7 The diagnostic results for Ramanujan gram, Autogram, and Ramanujan-gram are characterized by energy ratios of 9. Figure 5 (c) Figure 7 (c) and Figure 9 (c) shows many prominent interference frequency peaks, while the peak at the fault characteristic frequency f is not prominent, and the harmonic information is also buried by the noise spectrum. The diagnostic results of each method fail to effectively extract the fault characteristic information. Therefore, the comparative method's diagnosis fails. Figure 3-11 Both 13-20 are color views, used to represent the true results obtained from the experiment, and Figure 5 , 7 The positions of 9(a) and 9(a) are highlighted in red, and the positions highlighted in red represent the optimal positions. The corresponding (b) and (c) are the frequency band time domain diagram and frequency modulation band envelope spectrum of the corresponding optimal solutions. Figure 6 , 8 and 10 Figure 5 , 7 The original spectrum of 9(a) is not highlighted in red. Color graphs are used to highlight the authenticity of the experimental results and to emphasize the results.

[0235] In summary, the GRRgram method has good bearing fault feature diagnosis capability. Comparative tests have verified that this method has strong frequency band demodulation capability and better frequency band identification capability.

[0236] Example 2

[0237] To verify the effectiveness of this invention in the field of fault diagnosis, this embodiment conducted a measured signal diagnosis experiment and used Ramanujan gram, Autogram, and Ramanujan-gram as comparative methods to demonstrate the effectiveness and superiority of the method.

[0238] Figure 11 This is a low-speed heavy-duty axle and wheel system fault simulation test bench for Anhui University of Technology. It uses color-coded drawings to represent the actual system, with red labels indicating directions, and marks the various components of the simulation test bench. The time-domain plot and envelope spectrum of the acquired bearing outer ring fault signal are shown below. Figure 12 As shown, Figure 12 (b) The spectral peak at the fault characteristic frequency f is submerged by the surrounding interference components, and the noise effect is severe.

[0239] Figure 13 The results are diagnostic results of the GRRgram method based on the measured signal. Figure 13 (a) It can be seen that the optimal demodulation frequency band diagnosed is located in the fifth band of the fourth layer, and its time domain diagram is as follows. Figure 13 As shown in (b). Figure 13 (c) shows the envelope spectrum results, from which it can be observed that the spectral peak at the fault characteristic frequency f is optimally prominent, effectively extracting the dominant frequency f information. Furthermore, the harmonics of the fault characteristic frequency 2f, 3f, 4f, 5f, and 6f are also very prominent within their respective frequency ranges, unaffected by interference components. Therefore, the GRRgram method can successfully diagnose the outer ring fault characteristic information.

[0240] Figure 15 , 17 19 represents the diagnostic results of three comparison methods. Figure 15 (c) and Figure 19 (c) There is a more prominent interference frequency spectrum peak to the right of the fault characteristic frequency f, which masks the spectral characteristics of the fault information. Figure 17 (c) A spectral peak exists at the fault characteristic frequency f, but a prominent interference peak is observed around 100Hz in the entire envelope spectrum. Furthermore, the amplitude difference between this peak and f is small, making them difficult to distinguish effectively in the spectrum. Therefore, the envelope spectrum cannot definitively determine whether this method has extracted fault feature information. In summary, the comparative method cannot effectively extract fault-related feature information from the results; therefore, the outer ring fault signal feature extraction by the comparative method failed.

[0241] This invention employs a generalized fusion approach to construct a generalized filter with a multi-level filtering strategy. It is more flexible and stable in performance than traditional filters. Secondly, a multi-scale fault feature information quantization and reweighting Ramanujan spectral negative entropy index without prior conditions is defined. The reweighting strategy makes the index's fault feature information evaluation performance more stable and superior. Finally, simulation and experimental signals both verify that GRRgram is an effective fault feature extraction method. Comparative experimental results further demonstrate that GRRgram has excellent filtering and noise reduction performance as well as outstanding fault information quantization and evaluation capabilities.

[0242] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any modifications or equivalent changes made based on the technical essence of the present invention shall still fall within the scope of protection claimed by the present invention.

Claims

1. A flexible optimal demodulation method under strong interference environment based on Ramanujan theory, characterized in that: Comprising the following steps: Step 1: Signal acquisition: collect the fault bearing vibration signal x(t) under strong interference conditions; Step 2: Adaptive frequency band division: set the initial division layer number as k, the maximum division layer number as K, and the frequency band number of the kth layer as k+2; then, adopt the adaptive frequency band division technology to find the normalized frequency band division boundary set f i (i = 0, 1, 2, ···, k+2); Step 3: Construct a generalized filter based on the Ramanujan periodic transform theory, so as to obtain the generalized filter components λik(t) of each frequency band; (1) Construct a generalized filter and obtain the filter components of each frequency band; The expression is as follows: Filtering and denoising: where i is in the range [0, k+2]; G i (f) denotes a raised cosine filter; η i (f) denotes a zero-phase filter; H i (f) denotes a FIR filter; S(f) denotes the Ramanujan Fourier transform of the signal; (2) Signal reconstruction: restore the time domain components λ(t) (i = 1, 2,..., k+2) by using inverse Ramanujan Fourier transform, wherein λik is the i-th component of the k-th layer; Filter component: Step 4: Construct a re-weighted Ramanujan spectral negative entropy index, and select the optimal filter component; Calculate RRSNE: Firstly, the Ramanujan spectral negative entropy and the square envelope negative entropy calculation formula are as follows: Wherein, RS represents the Ramanujan spectrum of the signal, and SE represents the square envelope of the signal; Step 5: Determine whether the decomposition is ended; If k < K, let k = k + 1, and then repeat the steps 2-4; if k = K, let k = k + 1, then the decomposition process is ended, and the subsequent step 6 is executed; Step 6: Select RRSNE in all layers k The frequency band with the largest value is the optimal demodulation frequency band; then the envelope spectrum analysis is drawn to extract the fault features.

2. The flexible optimal demodulation method under strong interference environment based on Ramanujan theory of claim 1, characterized in that: In step 3, the lifting cosine filter has faster speed and more compact filter shape, so that the energy of the fault signal is more concentrated, and the transition band is smooth, so that the loss or distortion of useful signals caused by the sudden cutoff of the filter will not occur; a. The low-pass cosine filter is expressed as: Wherein, the influence factor α controls the width of the transition zone, and α ∈ [0, 1]; When designing a filter bank with multiple different cut-off frequencies f1, f2, ···, f M , it is necessary to ensure that the transition bands of consecutive filters do not overlap, that is, αf i + αf i+1 < f i+1 - f i , 1 ≤ i < M; Therefore, α is expressed as: b. The expression of the high-pass cosine filter is: h i (f) = 1 - g i (f) (9) G i = g i (f) h i-1 (f) (10) In summary, the complete cosine filter is shown in formula (10), wherein h0(f) = 0.

3. The flexible optimal demodulation method under strong interference environment based on Ramanujan theory of claim 1, characterized in that: In step 3, the zero-phase filter is: Constructing zero-phase filter bank η i (f), as shown in equation (11).

4. The flexible optimal demodulation method under strong interference environment based on Ramanujan theory of claim 1, characterized in that: In step 3, the FIR filter is: The FIR filter is constructed by windowing technology, and its frequency response is defined as: where f i-1 and f i are the lower and upper cutoff frequencies of the passband, respectively, and τ is the group delay factor.

5. The flexible optimal demodulation method under strong interference environment based on Ramanujan theory according to claim 1, characterized in that: In step 2, the initial number of division layers is set to k = 1, and the maximum number of division layers is set to K = 9; Normalized set of band division boundaries f i , i = 0, 1, 2, ···, k+2, where f0= 0 and f k+2 = π.

6. The flexible optimal demodulation method under strong interference environment based on Ramanujan theory of claim 1, wherein, The calculation steps of the re-weighted Ramanujan spectral negative entropy are as follows: (1) Calculate the RS and SE of the reconstructed component x(n); (2) Perform equal division processing on the signal, and set the initial average division segment number Q = 1; (3) dividing the RS and SE of the signal into Q segments averagely to obtain a segmented set RS q and SE q wherein q = 1, 2, ···, Q; (4) Calculate the Ramanujan spectral negative entropy NEq RS and the square envelope negative entropy NEqSE according to formula (5) and (6); (5) Arrange all elements in NEq RS in ascending order to obtain the ascending vector NERS sorta of RS, and arrange all elements in NEqSE in ascending order to obtain the ascending vector NESE sorta of SE; (6) Calculate the weight of all single elements in NEq RS relative to the total sum of elements to obtain the weight vector WQRS of RS, and the calculation expression is shown in formula (13); In the same way, the weight vector NEq SE of SE is obtained; (7) Arrange all elements in WQRS in descending order to obtain the descending vector WRSsortd of RS, and arrange all elements in WQRS in descending order to obtain the descending vector WSEsortd of SE; All elements in WQ SE are arranged in descending order to obtain the descending vector WSE sortd of SE; (8) The Ramanujan spectral negentropy RSNE of the reconstructed signal x(n) is calculated, and the calculation formula is as follows: Where ⊙ represents the inner product operation; The above process is repeated to obtain the square envelope negentropy SENE of the reconstructed signal; (9) In summary, the re-weighted Ramanujan spectral negentropy RRSNE of x(n) is obtained Q : RRSNE Q = (RSNE + SENE) / 2 (15) (10) Finally, determine if the RRS NE Q is optimal; if RRSNE Q ≤ RRSNE Q+1 then let Q = Q + 1, and re-execute (2) - (9) of Step 4. if RRSNE Q >RRSNE Q+1 , then output RRSNE k = RRSNE Q is the reweighted Ramanujan spectral negentropy of the reconstructed component.

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