Beam structure wavelet energy sample entropy crack positioning method based on bilinear characteristics

Abstract

According to the method, beam structure crack positioning is carried out by using bilinear characteristics caused by a breathing effect of fatigue cracks, vibration response is decomposed into a closed component and an open component by using technologies such as zero point search and signal splitting and overturning, irregularity in split components is amplified by using wavelet transform, and the accuracy of the beam structure crack positioning is improved. A new crack damage index is constructed on the basis, a model reference of a healthy beam structure is not needed, vibration information in a healthy state does not need to be tested, interference of common steps, grooves and other factors on the beam structure can be avoided, and the method has the advantages of being good in noise robustness, rapid and reliable.

Description

Technical Field

[0001] The invention belongs to the field of crack diagnosis of beam structures, and in particular relates to a wavelet energy sample entropy fatigue crack location method of beam structures based on bilinear characteristics. Technical Background

[0002] Beam structures are prone to fatigue crack damage under long-term alternating loads or random loads. If not identified and maintained in time, it may lead to serious engineering accidents and economic losses. Therefore, in order to ensure the safe and reliable use of beam structures, early detection of cracks and providing information on crack location and severity are of great significance.

[0003] The damage location method based on modal and derived parameter method can comprehensively detect various types of damage in the structure and is suitable for damage detection of most beam structures. However, the damage location method based on modal and derived parameter method is greatly affected by linear factors, and the location method is often not accurate enough. When crack damage occurs in the beam structure, the stiffness and damping of the crack damage location area will change, especially when the crack is a breathing crack. When the beam structure reciprocates under the action of external force, the breathing effect of the crack will enhance the nonlinear vibration response of the beam structure. The open crack will lead to a decrease in stiffness and discontinuity at the crack location, while the closed crack will produce axial pressure due to structural bending, which is similar to the intact state. These are the two sources of bilinear disturbance or irregularity of the vibration response caused by cracks. If the nonlinear behavior of cracks can be further utilized, it will be more conducive to nonlinear damage identification. The researchers proposed a crack frequency domain location method based on bilinearity. "Jiang Yang, Lu Zhiwen, Feng Kewei. A method for locating breathing cracks in beam structures based on bilinear frequency separation [J]. Modern Manufacturing Engineering, 2024(003):000." The crack location was successfully achieved. However, the proposed frequency domain method uses the frequency difference of the separated signal as the damage indicator. The traditional Fourier transform has limited ability to process nonlinear signals, which will lead to inaccurate spectrum analysis results and even nonlinear distortion, making it difficult to accurately reflect the characteristic behavior of the original signal. For this reason, the present invention proposes a damage indicator based on wavelet energy sample entropy, which uses the changes in complexity caused by cracks at different positions to achieve crack location. The method does not need to extract the frequency components of the separated signal, so it can avoid the problem of spectrum distortion, and wavelet decomposition can more effectively capture the transient characteristics of the signal. Summary of the invention

[0004] Based on the above background, the present invention proposes a wavelet energy sample entropy fatigue crack location method for beam structures based on bilinear characteristics. The method makes full use of multi-sensor information and uses zero-point search, signal flipping and signal segmentation techniques to decompose the signal of each measuring point into two split signals. The two split signals carry different dynamic information, corresponding to the opening and closing states of the crack. In the time domain, the irregularity in the signal is increased by using wavelet transform, and the damage index is constructed using energy sample entropy to locate the crack. The irregularity of the vibration response can provide information about the crack location. The wavelet coefficients obtained by wavelet transform are energy-normalized to obtain the ratio of specific coefficient energy to total energy. The energy sample entropy is used to compare the entropy of the cantilever beam at different positions of the open component and the closed component in the split component to achieve the purpose of precise positioning. The proposed crack location method is not limited to cantilever beams, but can be applied to situations where the relationship between the opening frequency, closing frequency and breathing frequency of the structure satisfies the bilinear theory. The proposed method is baseline-free and utilizes the bilinearity of the model. Through mathematical segmentation technology, the positive response component obtained represents the crack opening state, and the negative response component represents the crack closing state. This avoids the problem of being unable to obtain healthy baseline data in actual engineering.

[0005] The present invention provides the following technical solutions to achieve the above technical functions:

[0006] A beam structure wavelet energy sample entropy crack location method based on bilinear characteristics, the method comprising the following steps:

[0007] S1: Evenly distribute s sensor points on the beam structure and extract the vibration response x of different sensors j , j=1,2,…,s.

[0008] S2: Analyze the s sensors separately and track the zero crossing points of the vibration response of the jth sensor in sequence. Each zero crossing point has an upper zero point and a lower zero point, which are represented by x(i) j ≥0&x(i+1) j <0 and x(i) j <0&x(i+1) j ≥0 means, where i is the i-th point in the measurement point response.

[0009] S3: Use the upper and lower zero points to split the original signal of each sensor into two half-cycle signals, one with a positive response value and one with a negative response value. The upper and lower zero points themselves are included in the half-cycle signal with a positive response value and the half-cycle signal with a negative response value.

[0010] S4: Continue processing the split signal obtained in S3, eliminate the time interval after the split in the half-period signal, and define the eliminated signal as a time-continuous half-period signal. During the elimination process, the last value after any half-period response is followed by the first value of the next half-period response.

[0011] S5: Flip the odd-order or even-order half-period responses of all time-continuous half-period signals to form a signal with a complete period, which is defined as an opening component and a closing component, collectively referred to as a splitting component.

[0012] S6: After obtaining the split components based on the time domain method, wavelet transform is performed on the open component and the closed component, and the wavelet transform energy ratio is calculated.

[0013] S7: Perform sample entropy analysis on the signal obtained in S6, and calculate a relative entropy damage index.

[0014] S8: Obtain damage indicators of the time domain positioning method.

[0015] S9: Obtain the crack position, which corresponds to the peak position of the damage index.

[0016] Furthermore, S1 is specifically:

[0017] The dynamic model of a cantilever beam with a breathing crack is established using ANSYS APDL. One end of the beam is fixed, and a concentrated harmonic force is applied to the free end of the beam to uniformly extract the lateral acceleration response of several nodes at the root. The exciting force formula is F = fsin (2πωt). The amplitude of the exciting force on the cantilever beam is 100N and the frequency is 10Hz. The sampling frequency is 1000Hz and the sampling time is 5s.

[0018] Furthermore, S5 is specifically:

[0019] The vibration response of the cracked beam contains M cycles, and each response cycle contains two frequency components, namely, the upper half axis (upper part of the time axis) of the X-axis (time axis) represents the opening component f when breathing is opened u The lower half of the X-axis represents the closed component of the breathing closure f d , the time interval of each component in a cycle is defined as T a and T b ,have:

[0020]

[0021] Assume that the duration of crack opening and crack closing in the whole response process is T N1 and T N2 .but:

[0022] T N1 =Ta +T a +…+T a =MT a (3)

[0023] Where T a is the crack opening time in one cycle.

[0024] T N2 =T b +T b +…+T b =MT b (4)

[0025] Where T b is the crack closure time in one cycle.

[0026] Total response time T N for:

[0027]

[0028] Where M is the number of vibration response cycles of the cracked beam.

[0029] The positive response component and the negative response component represent the vibration characteristics of the system when the crack is open and closed.

[0030] Furthermore, S6 is specifically:

[0031] WT maps the time series signal f(t) to a two-dimensional domain. The result of WT is W f (a,b) is expressed as:

[0032]

[0033] Where φ() is the wavelet function, is a set of basis functions of the wavelet function φ() using scale parameter a and translation parameter b.

[0034] "Symlet2" is used as the wavelet function. Generally, b is used as the default conversion parameter. Only the scale parameter a needs to be adjusted. In this chapter, the scale parameter is set to 100.

[0035] For the wavelet coefficients Cij of the CWT, the total energy is calculated as:

[0036]

[0037] In the formula, index i represents the coefficient of each scale, and index j represents the jth scale.

[0038] Then the energy ratio of each wavelet coefficient can be expressed as:

[0039]

[0040] p ij The value corresponds to a specific coefficient C ij The ratio of energy to total energy. ij The value is like a probability distribution of energy. Therefore, p ij The values ​​sum to 1.

[0041] Furthermore, S7 is specifically:

[0042] Sample entropy (SampEn) is a tool to quantify the complexity of a time series by measuring the probability of generating new patterns in a signal. For a time series consisting of N samples, that is, x(n) = x(1), x(2), ..., x(N), the calculation process of SampEn is as follows:

[0043] The template vector of length m (m is also called the embedding dimension) is defined as:

[0044]

[0045] The Chebyshev distance between X(i) and X(j) is defined as follows, where 1≤i≤Nm, 1≤j≤Nm, i≠j.

[0046] d[X(i),X(j)]=max k＝0,…,m-1 (|x(i+k)-x(j+k)|) (10)

[0047] The number of j is between X(i) and X(j) (1≤i≤Nm,1≤j≤Nm,i≠j) and is less than the predetermined threshold r, which is recorded as Right now:

[0048]

[0049] Where r is a predetermined tolerance value, r=k×SD(X), where k is a constant (generally 0.1≤k≤0.25), and SD is the standard deviation.

[0050] all Average B m (r) can be calculated as:

[0051]

[0052] Increasing the dimension to m+1, similar to formula (4-13), formula Calculated as:

[0053]

[0054] In the formula, 1≤i≤Nm, 1≤j≤Nm, i≠j.

[0055] Similar to formula (4-14), all The average value A m+1 (r) can be calculated as:

[0056]

[0057] The SampEn value of the input time series signal can be calculated as.

[0058]

[0059] Furthermore, S8 is specifically:

[0060] The relative entropy comparison value of the two split components of positive response and negative response is used as the damage index, and the entropy value of the positive component Ent d and the entropy value Ent of the negative component h The percentage of the entropy value increment. Using the same WT scale parameters to obtain the DI (damage index) of the beam at different test positions, assuming that the number of measuring points is s, the damage index DI1 of each measuring point is:

[0061]

[0062] Where d and h represent the entropy of the cantilever beam under positive response (damaged state) and negative response (healthy state), respectively. d -Ent h Represents relative entropy.

[0063] The present invention adopts the above technical solution to bring the following beneficial effects:

[0064] The present invention applies bilinear characteristics to fatigue crack location of beam structures for the first time, analyzes the bilinearity generated by fatigue crack nonlinearity, uses zero-point search and signal splitting and flipping techniques to decompose the vibration response into two components, closed and open, uses wavelet transform to amplify the irregularities in the split components, and constructs a new crack damage index on this basis. The method does not require a model reference of a healthy cantilever beam, nor does it require testing vibration information under a healthy state, and can avoid interference from common steps, grooves and other factors on the cantilever beam, and has the advantages of good noise robustness, fast and reliable. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 To locate the flow chart;

[0066] Figure 2 This is a schematic diagram of the cantilever beam with breathing crack and the arrangement of measuring points;

[0067] Figure 3 Schematic diagram of bilinear characteristics;

[0068] Figure 4 Time domain comparison chart of different nodes;

[0069] Figure 5 Frequency domain comparison chart of different nodes;

[0070] Figure 6 Positioning result diagram with different parameters;

[0071] Figure 7 Schematic diagram of a stepped beam with a breathing crack;

[0072] Figure 8 Step beam positioning result diagram;

[0073] Fig. 9 Add noise localization result map. DETAILED DESCRIPTION

[0074] In order to make the technical solution of the present invention more clear, the invention will be further described below in conjunction with the accompanying drawings and embodiments thereof. The specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0075] The fatigue crack location process of the beam structure of the present invention is as follows: Figure 1 .

[0076] The present application provides a beam structure wavelet energy sample entropy fatigue crack location method based on bilinear characteristics, the method comprising the following steps:

[0077] S1: Evenly distribute s sensor measurement points on the cantilever beam of the breathing crack and extract the vibration response x of different sensors j , j = 1, 2, ..., s. For details, see Figure 2 As shown;

[0078] Built using ANSYS APDL Figure 2 The dynamic model of the cantilever beam with breathing cracks shown in the figure has one end of the beam as a fixed end. A concentrated harmonic force is applied to the free end of the beam, and the lateral acceleration response of the 24 nodes at the root is uniformly extracted. The exciting force formula is F = fsin (2πωt). The amplitude of the exciting force on the cantilever beam is 100N and the frequency is 10Hz. The sampling frequency is 1000Hz and the sampling time is 5s.

[0079] S2: Analyze the s sensors separately and track the zero crossing points of the vibration response of the i-th sensor in sequence. Each zero crossing point has an upper zero point and a lower zero point, which are represented by x(i) j ≥0&x(i+1) j <0 and x(i) j <0&x(i+1) j ≥0 means, where i is the i-th point in the measurement point response.

[0080] S3: Use the upper and lower zero points to split the original signal of each sensor into two half-cycle signals, one with a positive response value and one with a negative response value. The upper and lower zero points themselves are included in the half-cycle signal with a positive response value and the half-cycle signal with a negative response value.

[0081] S4: Continue processing the split signal obtained in S3, eliminate the time interval after the split in the half-period signal, and define the eliminated signal as a time-continuous half-period signal. During the elimination process, the last value after any half-period response is followed by the first value of the next half-period response.

[0082] S5: Flip the odd-order or even-order half-period responses of all time-continuous half-period signals to form a signal with a complete period, which is defined as an opening component and a closing component, collectively referred to as a splitting component.

[0083] When a beam structure with breathing cracks is subjected to simple harmonic excitation, its response can be regarded as a process of continuous alternation of crack opening and closing. The bilinear function can simulate this behavior well. For a beam with breathing cracks, its vibration response contains two components, the positive half component corresponds to the crack opening component, and the negative half component corresponds to the crack closing component. It has obvious bilinear behavior. The bilinear characteristics are as follows: Figure 3 shown.

[0084] set up Figure 3 The corresponding cracked beam vibration response contains M cycles, which is given by Figure 3 As shown in the figure, each response cycle contains two frequency components, namely, the upper half axis (upper part of the time axis) of the X-axis (time axis) represents the opening component f when breathing is opened u The lower half of the X-axis represents the closed component of the breathing closure f d , the time interval of each component in a cycle is defined as T a and T b ,have:

[0085]

[0086] Assume that the duration of crack opening and crack closing in the whole response process is T N1 and T N2 .but:

[0087] T N1 =T a +T a +…+T a =MT a (3)

[0088] Where T a is the crack opening time in one cycle.

[0089] T N2 =T b +T b +…+T b =MT b (4)

[0090] Where T b is the crack closure time in one cycle.

[0091] Total response time T N for:

[0092]

[0093] Where M is the number of vibration response cycles of the cracked beam.

[0094] The positive response component and the negative response component represent the vibration characteristics of the system when the crack is open and closed.

[0095] Figure 4 Shown are the numerical results of the original and split signals for nodes 8 and 4. Figure 4 (a) The corresponding node is far away from the fixed end and has a large amplitude. Figure 4 (b) Near the crack, the nonlinearity is stronger. Using the signal segmentation method described in S2-S5, the processed signal is as follows Figure 4 As shown in (c) and (d). Figure 4 (c) and (d) are the time domain signals after splitting. It can be clearly seen from here that Figure 4 In (d), the corresponding opening and closing components change irregularly within a period. The irregularity here means that the opening and closing periods are not synchronized, and the closing frequency component period is shorter than the opening frequency period. Figure 4 (c) It is difficult to observe the period difference between the opening and closing components because it is far away from the crack location and the bilinearity is weak, but it is observed that the amplitude of the opening frequency is different from the amplitude of the closing frequency. Figure 5 (a) and (b) are the logarithmic spectra of the split components of node 8 and node 4, respectively. There are superharmonic components in both nodes, which indicates that there are cracks in the structure. Figure 5 (a) The mid-base frequency (10 Hz) does not show split signal separation, and a weak split frequency separation can be seen at the double frequency. Figure 5 The fundamental frequency and superharmonics in (b) show obvious signal separation. This is due to Figure 5 (b) Closer to the crack. Figure 5 It can be seen that the opening and closing components carry different information and are sensitive to the crack location.

[0096] S6: After obtaining the split components based on the time domain method, wavelet transform is performed on the open component and the closed component, and the wavelet transform energy ratio is calculated.

[0097] The specific theory of wavelet transform is as follows:

[0098] WT maps the time series signal f(t) to a two-dimensional domain. The result of WT is W f (a,b) is expressed as:

[0099]

[0100] Where φ() is the wavelet function, is a set of basis functions of the wavelet function φ() using scale parameter a and translation parameter b.

[0101] "Symlet2" is used as the wavelet function. Generally, b is used as the default conversion parameter. Only the scale parameter a needs to be adjusted. In this chapter, the scale parameter is set to 100.

[0102] For the wavelet coefficients Cij of the CWT, the total energy is calculated as:

[0103]

[0104] In the formula, index i represents the coefficient of each scale, and index j represents the jth scale.

[0105] Then the energy ratio of each wavelet coefficient can be expressed as:

[0106]

[0107] p ij The value corresponds to a specific coefficient C ij The ratio of energy to total energy. ij The value is like a probability distribution of energy. Therefore, p ij The values ​​sum to 1.

[0108] S7: Perform sample entropy analysis on the signal obtained in S6, and calculate the relative entropy damage index.

[0109] Sample entropy (SampEn) is a tool to quantify the complexity of a time series by measuring the probability of generating new patterns in a signal. For a time series consisting of N samples, that is, x(n) = x(1), x(2), ..., x(N), the calculation process of SampEn is as follows:

[0110] The template vector of length m (m is also called the embedding dimension) is defined as:

[0111]

[0112] The Chebyshev distance between X(i) and X(j) is defined as follows, where 1≤i≤Nm, 1≤j≤Nm, i≠j.

[0113] d[X(i),X(j)]=max k＝0,…,m-1 (|x(i+k)-x(j+k)|) (10)

[0114] The number of j is between X(i) and X(j) (1≤i≤Nm,1≤j≤Nm,i≠j) and is less than the predetermined threshold r, which is recorded as Right now:

[0115]

[0116] Where r is a predetermined tolerance value, r=k×SD(X), where k is a constant (generally 0.1≤k≤0.25), and SD is the standard deviation.

[0117] all Average B m (r) can be calculated as:

[0118]

[0119] Increasing the dimension to m+1, similar to formula (4-13), formula Calculated as:

[0120]

[0121] In the formula, 1≤i≤Nm, 1≤j≤Nm, i≠j.

[0122] Similar to formula (4-14), all The average value A m+1 (r) can be calculated as:

[0123]

[0124] The SampEn value of the input time series signal can be calculated as.

[0125]

[0126] S8: Obtain damage indicators of the time domain positioning method.

[0127] The irregularity of the vibration response of the relative entropy damage indicator can provide information about the crack location. The wavelet coefficients obtained by wavelet transform are energy normalized to obtain the ratio of specific coefficient energy to total energy. The energy sample entropy is used to compare the entropies of the cantilever beam at different positions of the open component and the closed component in the split component to achieve the purpose of precise positioning.

[0128] The relative entropy comparison value of the two split components of positive response and negative response is used as the damage index, and the entropy value of the positive component Ent d and the entropy value Ent of the negative component h The percentage of the entropy value increment. Using the same WT scale parameters to obtain the DI (damage index) of the beam at different test positions, assuming that the number of measuring points is s, the damage index DI1 of each measuring point is:

[0129]

[0130] Where d and h represent the entropy of the cantilever beam under positive response (damaged state) and negative response (healthy state), respectively. d -Ent h Represents relative entropy.

[0131] S9: Get the crack position, which corresponds to the peak position of the damage index, such as Figure 6 shown.

[0132] Depend on Figure 6 It can be seen that the damage factors at different positions, depths, excitation frequencies, and excitation amplitudes show obvious peaks. Therefore, the stability and reliability of the method can be judged. Figure 7 The figure shows a schematic diagram of a stepped beam. The results are shown in Figure 8 As shown in the figure, the damage factor has an obvious peak, which is consistent with the location of the real crack, and eliminates the interference caused by the decrease in stiffness of the stepped shaft, which can illustrate the accuracy of crack positioning of this method. Fig. 9 In order to add noise to the localization result, the method has a certain robustness to noise.

[0133] The above is only a preferred embodiment of the present invention. It should be noted that, for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered as the protection scope of the present invention. All components not specified in this embodiment can be implemented by existing technologies.

Claims

1. A wavelet energy sample entropy crack location method for beam structures based on bilinear characteristics, characterized in that: The method comprises the following steps: S1: Evenly distribute s sensor points on the beam structure and extract the vibration response x of different sensors j , j = 1, 2, ..., s; S2: Analyze the s sensors separately and track the zero crossing points of the vibration response of the jth sensor in sequence. Each zero crossing point has an upper zero point and a lower zero point, which are represented by x(i) j ≥0&x(i+1) j <0 and x(i) j <0&x(i+1) j ≥0 means, where i is the i-th point in the measurement point response; S3: Split the original signal of each sensor into two half-cycle signals using the upper and lower zero points, one with a positive response value and the other with a negative response value. The upper and lower zero points themselves are included in the half-cycle signal with a positive response value and the half-cycle signal with a negative response value. S4: Continue processing the split signal obtained in S3, eliminate the time interval after the split in the half-period signal, and define the eliminated signal as a time-continuous half-period signal. During the elimination process, the last value after any half-period response is followed by the first value of the next half-period response; S5: flip the odd-order or even-order half-period responses of all time-continuous half-period signals to form a signal with a complete period, which is defined as an opening component and a closing component, collectively referred to as a splitting component; S6: After obtaining the split components based on the time domain method, wavelet transform is performed on the open component and the closed component, and the wavelet transform energy ratio is calculated; S7: Perform sample entropy analysis on the signal obtained in S6, and calculate the relative entropy damage index; S8: Obtain damage indicators of the time domain positioning method; S9: Obtain the crack position, which corresponds to the peak position of the damage index.

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