Quadratic Manhattan entropy calculation method and nonlinear vibration system complexity measurement method

Abstract

The invention discloses a quadratic Manhattan entropy calculation method and a nonlinear vibration system complexity measurement method, and belongs to the technical field of rotating machinery fault detection and diagnosis. According to the method, phase-space reconstruction is carried out on an original time sequence, the distance between subsequences is measured by using a quadratic Manhattan distance, the difference between elements is represented by measuring the distance, the similarity of obtained distance values is further represented by introducing a perception function, and finally the quadratic Manhattan entropy is defined through an information entropy theorem. The quadratic Manhattan entropy has good consistency, complexity estimation capability and noise immunity for nonlinear vibration system signals of the rotating machine, and can accurately represent characteristic parameters reflecting fault essence in the vibration signals.

Description

Technical Field

[0001] The present invention relates to the technical field of rotating machinery fault detection and diagnosis, and in particular to a quadratic Manhattan entropy calculation method and a method for measuring the complexity of a nonlinear vibration system. Background Art

[0002] Vibration signals are sensitive to subtle faults in rotating machinery and can comprehensively characterize the operating status of rolling bearings. Monitoring and evaluating vibration signals can detect potential faults in advance, making them a crucial area of ​​fault diagnosis research. However, in practice, the collected vibration signals exhibit significant complexity, nonlinearity, and nonstationarity. Entropy, a quantitative indicator of the degree of disorder in a system, was first proposed by German scientists to describe the chaotic state of molecules. With the continuous improvement and expansion of research, entropy theory has been widely applied in various fields. As a sensitive indicator, entropy methods can quantify the irregularities and complexity of nonlinear systems and have been widely used in rotating machinery fault diagnosis.

[0003] The application of entropy in rolling bearing fault diagnosis can be broadly categorized into three types: 1) Basic entropy features, which are calculated directly from the raw vibration signal. These include metrics such as sample entropy (SE), permutation entropy (PE), fuzzy entropy (FE), dispersion entropy (DE), and attention entropy (AE). However, these basic entropy features fail to fully reflect fault information. 2) Entropy features are calculated after vibration signal preprocessing, including wavelet entropy, energy entropy after time-frequency decomposition, morphological spectrum entropy, and singular spectrum entropy. While these methods can mitigate the effects of noise on entropy features, their calculations rely on complex signal processing techniques. 3) Improved entropy features, including consideration of multiscale entropy, composite multiscale entropy, and hierarchical entropy.

[0004] Uniformity of information complexity is considered a key property of entropy methods, meaning that entropy values ​​must be consistent with dynamic complexity. High vibration signal complexity corresponds to high entropy values, while low signal complexity corresponds to low entropy values. While currently used entropy methods can estimate the complexity of arbitrary time series and extract fault signatures from vibration signals, they still suffer from drawbacks, such as the inconsistency between the entropy values ​​of certain deterministic systems and their dynamic complexity. This is inconsistent with the original definition of information entropy, and the resulting entropy feature refinement methods will also suffer from inconsistencies. Summary of the Invention

[0005] To solve the above problems, the present invention proposes a method for calculating quadratic Manhattan entropy and a method for measuring the complexity of a nonlinear vibration system.

[0006] To achieve the above object, the present invention provides the following technical solutions:

[0007] A method for calculating the quadratic Manhattan entropy and a method for measuring the complexity of nonlinear vibration systems. Time series , the calculation steps of the quadratic Manhattan entropy are as follows:

[0008] Step 1: Through phase space reconstruction, the reconstructed trajectory can better reveal the dynamic characteristics of the system. Get the reconstruction matrix as follows:

[0009]

[0010] in, represents the embedding dimension, , ,……, represents the reconstructed subsequence, , , .

[0011] Step 2: Use the quadratic Manhattan distance to measure the distance between any two subsequences and The sum of the distances between corresponding elements is defined as follows:

[0012] (1)

[0013] in, , yes The elements in yes Elements in .

[0014] Step 3: Construct a perception function to further characterize the similarity of the obtained distance values, which is defined as follows:

[0015] (2)

[0016] in, represents the perception function, Represents the perception factor.

[0017] Step 4: Define the function for

[0018] (3)

[0019] Step 5: Calculate the embedding dimension as hour, The probability of the quadratic Manhattan distance of the subsequences is as follows:

[0020] (4)

[0021] Step 6: Similarly, let the embedding dimension be , repeat steps (1)-(5) to obtain ;

[0022] Step 7. Finally, according to the definition of information entropy, the definition of Quadratic Manhattan Entropy (QME) is obtained as

[0023] (5)

[0024] when When is a finite number, the above formula is expressed as

[0025] (6)

[0026] As a further technical solution of the present invention: the embedding dimension and perception factors The threshold is adjustable, usually The value ranges from 2 to 7. The value range is 2~6.

[0027] A method for measuring the complexity of a nonlinear vibration system is proposed, using the aforementioned quadratic Manhattan entropy method for calculation. The method tests the ability of the quadratic Manhattan entropy to measure and distinguish different signal states of a nonlinear vibration system. The quadratic Manhattan entropy of vibration signals in different states is calculated and used as input to a pattern recognition classifier.

[0028] Compared with the prior art, the present invention offers the following advantages: a method for calculating quadratic Manhattan entropy and measuring the complexity of nonlinear vibration systems reconstructs the original time series in phase space, uses quadratic Manhattan distance to measure the distance between subsequences, and measures the differences between elements. The similarity of the obtained distance values ​​is further characterized by the introduction of a perceptual function, and the quadratic Manhattan entropy is defined using the information entropy theorem. Quadratic Manhattan entropy exhibits excellent consistency, complexity estimation capabilities, and noise immunity for signals from nonlinear vibration systems of rotating machinery. This method enriches and develops entropy theory and can accurately characterize characteristic parameters in vibration signals that reflect the nature of faults. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 This is a flow chart for calculating the quadratic Manhattan entropy of the present invention;

[0030] Figure 2 The time domain waveform diagram of the normal and fault signals of the bearing according to the present invention;

[0031] Figure 3 This is a diagram showing the effect of the parameter size on QME according to the present invention;

[0032] Figure 4 This is the logic mapping diagram of μ∈3.5-4.0 described in the present invention;

[0033] Figure 5 This is a graph showing the consistency analysis results of logical mapping using different entropy methods described in the present invention;

[0034] Figure 6 This is the time domain waveform diagram of the AM / FM signal of the present invention;

[0035] Figure 7 This is a complexity estimation diagram for AM / FM signals using different entropy methods described in the present invention;

[0036] Figure 8 The time domain waveform diagram of the time-varying noise simulation signal of the present invention;

[0037] Figure 9 This is a graph showing the complexity measurement entropy value of a time-varying noise signal using different entropy methods described in the present invention;

[0038] Figure 10 is the growth rate curve of different entropy methods described in the present invention;

[0039] Figure 11 The time domain waveform diagram of the bearing signal in different states according to the present invention;

[0040] Figure 12 This is a graph showing the complexity measurement entropy value of bearing signals in different states using different entropy methods described in the present invention; DETAILED DESCRIPTION

[0041] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0042] Example 1: Figure 1 As shown in the figure, a method for calculating quadratic Manhattan entropy and measuring the complexity of nonlinear vibration systems includes: first, reconstructing the phase space of the original time series; using quadratic Manhattan distance to measure the distance between subsequences, and measuring the distance to represent the difference between elements; introducing a perceptual function to further characterize the similarity of the obtained distance values; and finally defining quadratic Manhattan entropy using the information entropy theorem. The specific steps are as follows:

[0043] Step 1: Through phase space reconstruction, the reconstructed trajectory can better reveal the dynamic characteristics of the system. Get the reconstruction matrix as follows:

[0044]

[0045] in, represents the embedding dimension, , ,……, represents the reconstructed subsequence, , , .

[0046] Step 2: Use the quadratic Manhattan distance to measure the distance between any two subsequences and The sum of the distances between corresponding elements is defined as follows:

[0047] (1)

[0048] in, , yes The elements in yes Elements in .

[0049] Step 3: Construct a perception function to further characterize the similarity of the obtained distance values, which is defined as follows:

[0050] (2)

[0051] in, represents the perception function, Represents the perception factor.

[0052] Step 4: Define the function for

[0053] (3)

[0054] Step 5: Calculate the embedding dimension as hour, The probability of the quadratic Manhattan distance of the subsequences is as follows:

[0055] (4)

[0056] Step 6: Similarly, let the embedding dimension be , repeat steps (1)-(5) to obtain ;

[0057] Step 7. Finally, according to the definition of information entropy, the definition of Quadratic Manhattan Entropy (QME) is obtained as

[0058] (5)

[0059] when When is a finite number, the above formula is expressed as

[0060] (6)

[0061] Example 2: Based on Example 1, to further illustrate the parameter selection of the quadratic Manhattan entropy, two sets of real rolling bearing signals are taken: This study uses the rolling bearing health data and fault data of Case Western Reserve University in the United States for testing. The rolling bearing model in the rolling bearing fault database of Case Western Reserve University (CWRU) is SKF6205. The experimental sampling frequency is selected as 12kHz and the speed is 1797r / min. Figure 2 The time domain waveforms of the vibration signals of the bearing in different health states are obtained. The bearing signals are divided into 100 groups of non-overlapping windows and the quadratic Manhattan entropy value of each group is calculated. The window length is 3000. The first 50 groups are normal signals and the last 50 groups are fault signals.

[0062] The entropy calculation results are as follows Figure 3 As shown in Figure 2, the QME values ​​of the 1st to 50th groups of normal signals are generally lower than the QME values ​​of the 50th to 100th groups of fault signals, indicating that the QME of the fault signals is generally greater than the QME value of the normal signals. Figure 3 It can be seen that the larger the embedding dimension m, the smaller the entropy of the signal. It is worth noting that if m is too small, the intrinsic dynamic characteristics of the system may not be captured. If m is too large, the computational complexity will increase, and it may also lead to overfitting and data processing difficulties. Therefore, considering the performance and computation time, m = 4 is selected. Figure 3 It can be seen that the larger the perception factor s is, the larger the entropy value of the signal is. The change trend of the entropy value curve of the signal is basically consistent. The s value can increase the difference between signals. Therefore, the present invention selects s ​​= 6.

[0063] Based on Examples 1 and 2, the proposed method for calculating quadratic Manhattan entropy and the method for measuring the complexity of nonlinear vibration systems were experimentally verified using simulated signals and real bearing vibration signals:

[0064] 1. Analysis and verification of consistency between entropy value and system complexity:

[0065] This paper defines consistency as a positive correlation between the entropy estimate and the system's dynamic complexity, meaning that entropy should increase with increasing system complexity. To evaluate the consistency of different entropy methods, this paper conducted a logistic mapping simulation experiment. The logistic chaotic map is the simplest one-dimensional chaotic map. The classic logistic function is shown below:

[0066] (7)

[0067] Where μ is the adjustment parameter of the dynamic behavior of the logistic map. As μ changes, the time series of the logistic map exhibits a variety of dynamic phenomena from periodic to chaotic. In this invention, 3.5≤μ<4 is selected, μ is set to 0.001, the initial value μ = 0.5, the number of iterations is 1000, and the bifurcation diagram of the logistic map is as follows: Figure 4 As shown. Within the range of μ∈ (3.4, 4), the key nodes where the system shows periodic phenomena are: μ=3.544 (4 cycles - 8 cycles); μ=3.627 (chaos - 6 cycles); μ=3.738 (chaos - 5 cycles); μ=3.828 (chaos - 3 cycles). At this time, the complexity of the system will change significantly, and the overall complexity of the logical data will gradually increase. At the same time, the complexity of the periodic system should be less than that of the non-periodic system, and the entropy value corresponding to the system should be consistent with the system complexity. The present invention uses logical mapping to verify the consistency of the proposed quadratic Manhattan entropy. In addition, QME is compared with the existing AE (attention entropy), SE, PE, DE, and FE. The remaining entropy methods are set to their respective optimal parameters. The calculation results are shown in the figure. Figure 5 shown.

[0068] See Figure 5 QME can accurately reflect the complexity change process of the logical mapping. The QME value of the periodic state is lower than that of the chaotic state. At the same time, the QME values ​​of the logical mapping of different periodic states are also different. The QME value corresponding to the 8-period system is higher than that of the 4-period system. The AE, SE, PE, DE, and FE methods show inconsistencies at different stages in the complexity estimation of the logical mapping. However, the QME proposed in this paper has good consistency in estimating system complexity.

[0069] 2. Complexity estimation based on simulated signals:

[0070] AM / FM signals are typical components of signals generated by nonlinear vibration systems. In order to verify the ability of the proposed QME to measure the complexity of nonlinear systems, AM / FM simulation signals are used for verification. The signal is represented as:

[0071] (8)

[0072] The time domain waveform of the signal is Figure 6 As shown, QME and the existing AE (attention entropy), SE, PE, DE, and FE are also used to calculate the time series generated by the logistic mapping. The parameter settings of different methods are consistent with the previous article. The calculation results are as follows Figure 7 shown.

[0073] See Figure 6 It can be seen that the amplitude and frequency of the AM / FM signal change continuously with periodicity, and its complexity is determined by the AM signal and the FM signal. Its complexity first decreases and then increases, and then decreases again and then gradually increases. Figure 7 The results show that the QME entropy curve is consistent with the complexity variation characteristics of AM / FM signals. The test results show that QME is more sensitive and accurate than other methods and has good performance in signal complexity estimation.

[0074] 3. Noise immunity test:

[0075] To test the performance of QME in noisy signals, we created additive white Gaussian noise with different powers using a modulated sinusoidal signal as a benchmark. The first 0.2 seconds of the sequence were free of noise, and then white Gaussian noise was added to the signal. The noise power increased every 0.1 seconds. The signal is represented as follows:

[0076] (9)

[0077] The time domain curve of the noise signal is obtained as Figure 8 As shown in Figure 2, as the noise increases, the complexity of the signal gradually increases, and the corresponding entropy value should also increase. The smoother the curve changes, the higher the noise resistance of the entropy method. The entropy curve obtained is as follows Figure 9 As shown in the figure, by comparison, the changing trend of QME conforms to the law of gradually increasing signal complexity.

[0078] In addition, in this experiment, the growth rate is defined as To evaluate the noise resistance of different entropy methods. The growth rate is calculated as follows:

[0079] (10)

[0080] Where E is the current entropy value and E0 represents the initial entropy value. In the present invention, the initial entropy value E0 is the entropy value when no noise is added. The growth rate can quantitatively evaluate the noise resistance of each method under different initial values. Calculate the growth rate of different entropy methods and draw a curve as shown in Figure 10 As shown in the figure, the comparison shows that the growth rate of QME is relatively stable, which means that QME can better evaluate the complexity of signals under different noise levels and has better noise resistance.

[0081] 4. Real rolling bearing signal:

[0082] Different types of rolling bearing faults in the CWRU bearing dataset are selected to verify the effectiveness of the QME algorithm. Table 1 shows the relevant data information. The number of samples under each bearing state is 100, of which 50% are randomly selected as the training set and 50% as the test set, totaling 100*7=700 samples, and the sample length is 3000. Figure 11 The time domain waveforms of faults of different degrees are given. In this study, -15dB white noise is injected into the data of each healthy state to simulate the fault signal information under strong background noise.

[0083] Table 1 CWRU bearing health status dataset

[0084]

[0085] Figure 12 The entropy value curves of vibration signal complexity measurement of QME and comparison methods for training set data under strong background noise are given. The results show that QME has good noise resistance and can extract features representing fault information in strong background noise, while the traditional entropy method has poor feature extraction ability in noisy environment, indicating that QME has superior complexity measurement ability for vibration signals in different health states.

[0086] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the invention can be embodied in other specific forms without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered in all respects as illustrative and non-restrictive, and the scope of the invention is defined by the appended claims, not the foregoing description, and all variations within the meaning and range of equivalents of the claims are intended to be included therein. Any reference sign in a claim should not be construed as limiting the claim to which it relates.

Claims

1. A method for calculating quadratic Manhattan entropy and a method for measuring the complexity of a nonlinear vibration system, characterized in that: For length Time series , the calculation steps of the quadratic Manhattan entropy are as follows: Step 1: Through phase space reconstruction, the reconstructed trajectory can better reveal the dynamic characteristics of the system and obtain the reconstruction matrix as follows: ； in, represents the embedding dimension, , ,……, represents the reconstructed subsequence, , , ; Step 2: Use the quadratic Manhattan distance to measure the distance between any two subsequences and The sum of the distances between corresponding elements is defined as follows: (1) in, , yes The elements in yes Elements in Step 3: Construct a perception function to further characterize the similarity of the obtained distance values, which is defined as follows: (2) in, represents the perception function, represents the perception factor; Step 4: Define the function for (3) Step 5: Calculate the embedding dimension as hour, The probability of the quadratic Manhattan distance of the subsequences is as follows: (4) Step 6: Similarly, let the embedding dimension be , repeat steps (1)-(5) to obtain ; Step 7. Finally, according to the definition of information entropy, the definition of Quadratic Manhattan Entropy (QME) is obtained as (5) when When is a finite number, the above formula is expressed as (6) The embedding dimension and perception factors The threshold is adjustable, usually The value ranges from 2 to 7. The value range is 2~6; Finally, the signal entropy value is calculated using the quadratic Manhattan entropy as a measurement parameter of the complexity of the nonlinear vibration system; the measurement and discrimination capabilities of the quadratic Manhattan entropy for different signal states of the nonlinear vibration system are tested; and the quadratic Manhattan entropy of vibration signals in different states is calculated, which can be used as the input of a pattern recognition classifier.

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