A Structural Fatigue Crack Identification Method Based on the Collaboration of VMD and HHT

Abstract

The present invention provides a method for identifying structural fatigue cracks based on the collaboration of VMD-HHT, comprising the following steps: applying a harmonic excitation at the end position of the structure to be identified, collecting the acceleration response data at different positions of the structure, and using wavelet packet transform to perform decomposition, noise suppression and reconstruction processing in sequence to obtain a reconstructed acceleration signal with noise reduction; performing variational mode decomposition on the reconstructed acceleration signal to obtain intrinsic mode functions; performing Hilbert-Huang transform on each intrinsic mode function respectively to obtain a Hilbert spectrum set; when multi-order secondary high-frequency signals appear in the Hilbert spectrum set, a fatigue crack occurs at the response data acquisition position of the structure, and the greater the instantaneous frequency amplitude of the multi-order secondary high-frequency signals, the greater the fatigue crack. This method combines the advantages of VMD and HHT, overcomes the problem of false intrinsic mode function information, improves the ability to decouple different frequency components in narrow-band signals, and can effectively distinguish frequency differences.

Description

Technical Field

[0001] The present invention relates to the technical field of structural health monitoring and safety warning, and particularly relates to a method for identifying structural fatigue cracks based on the cooperation of VMD-HHT. Background Art

[0002] While the construction of infrastructure, civil, commercial, industrial, etc. has developed rapidly, building structures are developing in the direction of multi-function, long life, large scale, and complexity. These structures are often in complex service environments, facing various sudden external factors, facing the change of structural function state and the increasingly serious problem of crack accumulation. To ensure the normal, safe, and stable operation of the structure, the safety problem of the construction structure has become the focus of people's attention. The demand for structural health monitoring has shown rapid growth. With the structural health monitoring system collecting a large amount of data, how to analyze the state of the structure based on the data has become the core content of structural health monitoring. At present, with the development of science and technology, especially the rapid development of information data analysis and processing technology, analysis methods with data-driven as the core have rapidly emerged, triggering new thinking in structural health monitoring.

[0003] During the long-term service of the structure, a large number of tiny fatigue cracks are generated due to the complex environment. The cracks vibrate with the structure, showing a "breathing" effect of contact-separation. After the cracks accumulate, they continuously expand into an always-open state, changing the structural stiffness and energy distribution, resulting in fracture failure. At present, the commonly used fatigue crack identification methods are mainly based on known systems and their dynamic output signals. Using the response data collected by monitoring sensors, methods such as Fourier transform and EMD-HHT (empirical mode decomposition) are used for crack identification. Fourier transform can realize the identification of cracks, but it lacks the ability to process non-stationary signals and is easily affected by environmental noise; EMD-HHT can reflect the change of crack structure frequency, but the unknown ripples in the results obtained by EMD-HHT destroy the time-frequency characteristics, leading to difficult interpretation of the results, and several unsatisfactory intrinsic mode functions are generated in the low-frequency range, which is deceptive in data interpretation. Summary of the Invention

[0004] To solve the above problems, the present invention proposes a method for identifying structural fatigue cracks based on the cooperation of VMD-HHT. This method combines the advantages of variational mode decomposition (VMD) and Hilbert-Huang transform (HHT), overcomes the existence of false intrinsic mode function information, improves the ability to decouple different frequency components in narrowband signals, and can effectively distinguish frequency differences.

[0005] To achieve the above object, the present invention provides the following technical solutions.

[0006] A method for identifying structural fatigue cracks based on the cooperation of VMD-HHT includes the following steps:

[0007] Apply a harmonic excitation at the end position of the structure to be identified, and collect the acceleration response data at different positions of the structure;

[0008] Perform wavelet packet transform on the acceleration response data at each position of the structure, and successively perform decomposition, noise suppression, and reconstruction processing to obtain a reconstructed acceleration signal with noise reduction;

[0009] Perform variational mode decomposition on the reconstructed acceleration signal to obtain intrinsic mode functions; perform Hilbert-Huang transform on each intrinsic mode function respectively to obtain a Hilbert spectrum set arranged from high frequency to low frequency;

[0010] When multiple orders of secondary high-frequency signals appear in the Hilbert spectrum set, a fatigue crack occurs at the position where the response data of the structure is collected, and the greater the instantaneous frequency amplitude of the multiple orders of secondary high-frequency signals, the greater the fatigue crack.

[0011] Preferably, the collecting the acceleration response data at different positions of the structure includes the following steps:

[0012] Arrange multiple acceleration sensors on the structure to be identified, and electrically connect the multiple acceleration sensors to a processor respectively;

[0013] The processor is electrically connected to a control switch and a power supply. The multiple acceleration sensors form a detection group. Apply a harmonic excitation at the end position of the structure to be tested. The processor collects the response signals on each acceleration sensor and establishes an acceleration response database at different positions by using the response signals.

[0014] Preferably, the performing wavelet packet transform on the acceleration response data at each position of the structure, and successively performing decomposition, noise suppression, and reconstruction processing to obtain a reconstructed acceleration signal with noise reduction includes the following steps:

[0015] Recursively construct a wavelet function:

[0016]

[0017]

[0018] where h(k) and g(k) are the orthogonal reflection filters of an arbitrarily selected mother wavelet, i is the modulation parameter, j is the scale parameter, k is the translation parameter, and t is the time;

[0019] Construct a wavelet packet function according to the wavelet function:

[0020]

[0021] where It includes three wavelet packet functions, where i is the modulation parameter, j is the scale parameter, and k is the translation parameter. is the wavelet function;

[0022] Calculate the wavelet packet coefficients:

[0023]

[0024] In the formula, is the wavelet packet coefficient, x(t) is the acceleration signal, is the wavelet packet function, i represents the modulation parameter, j represents the scale parameter, and k represents the translation parameter;

[0025] Reconstruct the acceleration signal:

[0026]

[0027]

[0028] In the formula, X(t) is the reconstructed signal, is the wavelet packet reconstructed signal, is the wavelet packet coefficient, is the wavelet packet function, i represents the modulation parameter, j represents the scale parameter, and k represents the translation parameter.

[0029] Preferably, the step of performing variational mode decomposition on the reconstructed acceleration signal to obtain the intrinsic mode function includes the following steps:

[0030] Construct a variational constraint model:

[0031] u k (t) = A k (t)cos(Φ k (t))

[0032]

[0033] In the formula, u k (t) is the signal after variational mode decomposition into each modal frequency modulation - amplitude modulation signal, A k (t) is the instantaneous amplitude of u k (t), Φ k (t) is used as the phase; ω t (ω k (t) = Φ k (t)) is the instantaneous frequency, which is slowly varying with respect to the phase compared to the instantaneous amplitude; k is each modal component, k = 1, 2, 3,...; k, f is the reconstructed signal, and δ(t) is the Dirac distribution;

[0034] Introduce the quadratic penalty factor α and the Lagrange multiplier λ:

[0035]

[0036] Wherein, f(t) is the reconstructed signal, and ω k is the k-th order frequency;

[0037] The operation is performed by the method of alternating mathematical multiplication operators to obtain the output of K intrinsic mode function IMF components.

[0038] Preferably, performing the Hilbert-Huang transform on each intrinsic mode function respectively includes the following steps:

[0039] Performing the Hilbert-Huang transform on each intrinsic mode function respectively:

[0040]

[0041] z(t) = x j (t) + iy(t) = a(t)e iθ(t)

[0042]

[0043] θ(t) = arctan(y(t) / x j (t))

[0044] ω(t) = dθ / dt

[0045] Wherein, x j (t) is the reconstructed IMF real signal, y(t) is the imaginary signal obtained by the convolution of x j (t) and 1 / t of time, z(t) is the amplitude of the coupled signal of x j (t) and y(t), a(t) is the instantaneous energy amplitude, θ(t) is the phase function, ω(t) is the instantaneous frequency, and t is the time;

[0046] The Hilbert spectrum set is obtained according to the above formula.

[0047] Advantages of the present invention:

[0048] Compared with the traditional method, the method of the present invention cannot process non-stationary signals, overcomes the existence of false intrinsic mode function information, improves the ability to decouple different frequency components in narrow-band signals, and can effectively distinguish frequency differences.

[0049] The present invention reconstructs non-stationary signals through wavelet packet transform, decomposes the reconstructed acceleration signals by using VMD to generate a set of intrinsic mode functions (IMFs), then extracts the instantaneous frequencies of each group of intrinsic mode functions by using Hilbert-Huang to obtain a Hilbert spectrum set arranged from high frequency to low frequency; the high-frequency subset of the Hilbert spectrum set is used as an effective information source reflecting the fatigue cracks of the structure, where the amplitude of the instantaneous frequency in the Hilbert spectrum reflects the degree of fatigue cracks; based on the fact that fatigue cracks will modulate the signal and trigger multi-order secondary high-frequency signals, the greater the change in the amplitude of the instantaneous frequency, the deeper the degree of fatigue cracks, so as to visually and accurately identify the location and degree of fatigue breathing cracks, thereby revealing the fatigue crack condition. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 is a flowchart of the method of the embodiment of the present invention;

[0051] Figure 2 (a) is the Hilbert spectrum set of the numerically simulated structure without damage in the embodiment of the present invention;

[0052] Figure 2 (b) is the Hilbert spectrum set of the numerically simulated beam with a crack depth of 20% of the beam height in the embodiment of the present invention;

[0053] Figure 2 (c) is the Hilbert spectrum set of the numerically simulated beam with a crack depth of 41% of the beam height in the embodiment of the present invention;

[0054] Figure 3 (a) is the Hilbert spectrum set of the numerically simulated structure without damage in the embodiment of the present invention;

[0055] Figure 3 (b) is the Hilbert spectrum set of the numerically simulated crack located in the middle of the cantilever beam in the embodiment of the present invention;

[0056] Figure 3 (c) is the Hilbert spectrum set of the numerically simulated crack located at the end of the cantilever beam in the embodiment of the present invention;

[0057] Figure 4 (a) is the Hilbert spectrum set of the structure without damage in the experiment of the embodiment of the present invention;

[0058] Figure 4 (b) is the Hilbert spectrum set of the crack with a depth of 20% of the beam height in the experiment of the embodiment of the present invention;

[0059] Figure 4 (c) is the Hilbert spectrum set of the crack with a depth of 41% of the beam height in the experiment of the embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0060] To make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0061] Embodiment 1

[0062] A method for identifying structural fatigue cracks based on the cooperation of VMD-HHT of the present invention specifically includes the following steps as Figure 1 shown:

[0063] S1: Apply a harmonic excitation at the end position of the structure to be identified, and collect the acceleration response data at different positions of the structure.

[0064] Arrange a plurality of acceleration sensors on the structure to be identified, and electrically connect the plurality of acceleration sensors to a processor respectively; the processor is electrically connected to a power supply through a control switch, and the plurality of acceleration sensors form a detection group. Apply a harmonic excitation at the end position of the tested structure, and the processor collects the response signals on each acceleration sensor, and uses the response signals to establish an acceleration response database at different positions.

[0065] S2: Perform wavelet packet transform on the acceleration response data at each position of the structure, and perform decomposition, noise suppression and reconstruction processing in sequence to obtain a reconstructed acceleration signal with noise reduction. Specifically:

[0066] (1) Recursively construct a wavelet function:

[0067]

[0068]

[0069] In the formula, h(k) and g(k) are the orthogonal reflection filters of any selected mother wavelet (such as Harr wavelet, Daubechies), i is the modulation parameter, j is the scale parameter, k is the translation parameter, and t is the time.

[0070] (2) Construct a wavelet packet function according to the wavelet function:

[0071]

[0072] In the formula, contains three wavelet packet functions, i is the modulation parameter, j is the scale parameter, k is the translation parameter, is the wavelet function.

[0073] (3) Calculate the wavelet packet coefficients:

[0074]

[0075] In the formula, is the wavelet packet coefficient, x(t) is the acceleration signal, is the wavelet packet function, i represents the modulation parameter, j represents the scale parameter, and k represents the translation parameter.

[0076] (4) Reconstruct the acceleration signal:

[0077]

[0078]

[0079] In the formula, X(t) is the reconstructed signal, is the wavelet packet reconstructed signal, is the wavelet packet coefficient, is the wavelet packet function, i represents the modulation parameter, j represents the scale parameter, and k represents the translation parameter.

[0080] S3: Perform variational mode decomposition on the reconstructed acceleration signal to obtain the intrinsic mode functions; perform Hilbert-Huang transform on each intrinsic mode function respectively to obtain the Hilbert spectrum set arranged from high frequency to low frequency.

[0081] (1) VMD signal decomposition

[0082] u k (t) = A k (t)cos(Φ k (t))

[0083]

[0084] This is the variational constraint model. In the formula, u k (t) is the signal after variational mode decomposition into each modal frequency modulation - amplitude modulation signal, A k (t) is the instantaneous amplitude of u k (t), Φ k (t) is used as the phase; ω t (ω k (t) = Φ k (t)) is the instantaneous frequency, which is slowly varying with respect to the phase compared to the instantaneous amplitude; k is each modal component, k = 1, 2, 3,...; k, f is the reconstructed signal, and δ(t) is the Dirac distribution.

[0085] In order to obtain the optimal solution of the variational constraint model, introduce the quadratic penalty factor α and the Lagrange multiplier λ:

[0086]

[0087] In the formula, f(t) is the reconstructed signal, ω k is the k-th order frequency;

[0088] The operation is carried out by the method of alternating mathematical multiplication operators to obtain the output of K intrinsic mode function IMF components.

[0089] (2) HHT Hilbert-Huang transform

[0090]

[0091] z(t) = x j (t) + iy(t) = a(t)e iθ(t)

[0092]

[0093] θ(t) = arctan(y(t) / x j (t)) θ(t) = arctan(y(t) / x j (t))

[0094] ω(t) = dθ / dt ω(t) = dθ / dt

[0095] In the formula, x j (t) is the reconstructed IMF real signal, y(t) is the imaginary signal obtained by the convolution of x j (t) and 1 / t of time, z(t) is the amplitude of the coupled signal of x j (t) and y(t), a(t) is the instantaneous energy amplitude, θ(t) is the phase function, ω(t) is the instantaneous frequency, and t is the time. Thus, the Hilbert spectrum set is obtained.

[0096] S4: Use the high-frequency subset of the obtained Hilbert spectrum set as the effective information source reflecting the fatigue crack of the structure. Among them, the amplitude of the instantaneous frequency in the Hilbert spectrum reflects the degree of fatigue crack. Combine the sensor position corresponding to the acceleration signal to determine the fatigue crack position. If there is no fatigue crack, no high-frequency signal will be generated. If there is a fatigue crack, it will trigger multi-order secondary high-frequency signals, and the greater the crack degree, the greater the amplitude of the instantaneous frequency of the multi-order secondary high-frequency signals generated by modulation.

[0097] Example 1

[0098] In this example, the harmonic excitation applied to the beam end is Gaussian distribution excitation, which is implemented as fatigue cracks under different depth and position conditions that may occur in the beam structure. Numerically simulate and experimentally verify the nonlinear dynamic response characteristics of the beam structure, and use the acceleration data decomposition and conversion analysis method to obtain the instantaneous frequency and energy amplitude. The detailed calculation process is as follows:

[0099] In terms of numerical simulation:

[0100] 1. Use ABAQUS to construct a cantilever beam. The geometric dimensions of the beam are: length (L = 300 mm), width (B = 25 mm), and thickness (H = 10 mm).

[0101] 2. Set the crack as a fatigue crack.

[0102] The opening and closing of the fatigue crack are considered as a local contact problem. By regarding one crack surface as the master surface and the other as the slave surface, the interaction between the fatigue crack surfaces is modeled. During the vibration of the beam, three contact states occur for the fatigue crack:

[0103] (i) The crack is fully open, which means there is no contact between the master surface and the slave surface.

[0104] (ii) All nodes on the slave crack surface and the master crack surface are in contact, and the crack is fully closed.

[0105] (iii) The slave crack surface and the master crack surface are partially in contact.

[0106] 3. Set the crack position along the length direction of the beam. The distance of the crack position from the end is x c , and the depth of the crack is a. To facilitate the definition of the crack position and crack depth, the crack position and crack depth are defined as:

[0107] q = x c / L

[0108] p = a / H

[0109] where x c represents the position of the crack from the end, L is the length of the beam, q represents the relative position of the crack from the end, a represents the depth of the crack, H represents the thickness of the beam, and p represents the relative depth of the crack.

[0110] Three crack degrees p are set. For the crack degree p, p = 7%, p = 20%, and p = 41% crack degrees; three crack relative positions q, q = 0.040, q = 0.450, q = 0.773, and the crack-free structure is set as a control.

[0111] 4. Set sensors along the length direction of the beam and select 10 points as the sensor positions.

[0112] 5. Apply a harmonic excitation to the free end of the cantilever beam.

[0113] 6. Collect the acceleration response through 10 points.

[0114] 7. The wavelet packet transform and the signal reconstruction method are as follows:

[0115] (1) Recursively construct the wavelet function

[0116]

[0117]

[0118] In the formula, h(k) and g(k) are orthogonal reflection filters for selecting any mother wavelet (such as Harr wavelet, Daubechies), i represents the modulation parameter, j represents the scale parameter, k represents the translation parameter, and t represents time.

[0119] (2) Construct wavelet packet function

[0120]

[0121] In the formula, It contains three wavelet packet functions. i represents the modulation parameter, j represents the scale parameter, and k represents the translation parameter. is the wavelet function.

[0122] (3) Calculate wavelet packet coefficients

[0123]

[0124] In the formula, is the wavelet packet coefficient, x(t) is the acceleration signal, is the wavelet packet function. i represents the modulation parameter, j represents the scale parameter, and k represents the translation parameter.

[0125] (4) Reconstruct the signal

[0126]

[0127]

[0128] In the formula, X(t) is the reconstructed signal, is the reconstructed signal of wavelet packet (j, i), is the wavelet packet coefficient, is the wavelet packet function. i represents the modulation parameter, j represents the scale parameter, and k represents the translation parameter.

[0129] 8. Perform VMD (Variational Mode Decomposition) on the reconstructed signal. The obtained Intrinsic Mode Function IMF after decomposition is subjected to Hilbert-Huang transform for each Intrinsic Mode Function to obtain the instantaneous frequency and energy amplitude at different times, forming a Hilbert spectrum set. The method is as follows:

[0130] (1) VMD signal decomposition

[0131] u k (t) = A k (t)cos(Φ k(t))

[0132]

[0133] This is a variational constraint model. u k (t) is the signal decomposed by VMD into each modal frequency modulation - amplitude modulation signal. A k (t) is u k (t)'s instantaneous amplitude, Φ k (t) is used as the phase; the instantaneous frequency ω t (ω k (t) = Φ k (t)) is slow - varying with respect to the phase relative to the instantaneous amplitude. k is each modal component (k = 1, 2, 3,..., k), f is the reconstructed signal in step 2, and δ(t) is the Dirac distribution.

[0134] To obtain the optimal solution of the variational constraint model, a quadratic penalty factor α and a Lagrange multiplier λ are introduced, and we get:

[0135]

[0136] u k (t) is the signal decomposed by VMD into each modal frequency modulation - amplitude modulation signal, α is the quadratic penalty factor, λ is the Lagrange multiplier, f(t) is the reconstructed signal, δ(t) is the Dirac distribution, and ω k is the k - th order frequency. To obtain the above - mentioned model, the method of alternating mathematical multiplication operators is used for calculation to obtain the output of K intrinsic mode function IMF components.

[0137] (2) HHT Hilbert - Huang transform

[0138]

[0139] z(t) = x j (t)+iy(t) = a(t)e iθ(t)

[0140]

[0141] θ(t) = arctan(y(t) / x j (t))

[0142] ω(t) = dθ / dt ω(t) = dθ / dt

[0143] Among them, y(t) is the imaginary signal obtained by the convolution of the reconstructed IMF real signal x j (t) and 1 / t in time, and z(t) is x j(t) is the amplitude of the coupled signal with y(t), a(t) represents the instantaneous energy amplitude, θ(t) represents the phase function, ω(t) represents the instantaneous frequency, and t represents time. Thus, the Hilbert spectrum set is obtained.

[0144] 9. Figure 2 (a), (b), and (c) are the Hilbert spectrum sets of the numerically simulated beams with crack depths of non-damaged, 20%, and 41% of the beam height, respectively. Figure 3 (b) and (c) are the Hilbert spectrum sets of the numerically simulated cracks located in the middle and end of the cantilever beam, respectively. Figure 3 (a) is the Hilbert spectrum set in the non-damaged state.

[0145] From Figure 2 , Figure 3 It is shown that the frequency amplitude in the obtained Hilbert spectrum set increases compared with the instantaneous frequency amplitude in the Hilbert spectrum set in the normal state. As the crack degree increases, the instantaneous frequency also increases. This method is based on the fact that fatigue cracks will modulate the signal, triggering multi-order secondary high-frequency signals. The greater the change in the instantaneous frequency amplitude, the deeper the fatigue crack degree, achieving an intuitive and accurate identification of the location and degree of fatigue breathing cracks, thereby revealing the fatigue crack condition.

[0146] Example 2

[0147] In terms of experiments:

[0148] 1. Use high-performance structural adhesive to bond three steel blocks together to form a cantilever beam with different crack parameters. The dimensions of the beam are taken as L (300 mm) × B (25 mm) × H (10 mm).

[0149] 2. Set the crack position along the length direction of the beam. The distance of the crack position from the end is x c , and the depth of the crack is a. To facilitate the definition of the crack position and crack depth, the crack position and crack depth are defined as:

[0150] q = x c / L

[0151] p = a / H

[0152] Among them, x c represents the position of the crack from the end, L is the length of the beam, q represents the relative position of the crack from the end, a represents the depth of the crack, H represents the thickness of the beam, and p represents the relative depth of the crack.

[0153] Three crack degrees p are set. For the crack degree p, p = 7%, p = 20%, and p = 41% crack degrees; three crack relative positions q, q = 0.040, q = 0.450, q = 0.773, and the crack-free structure is set as a control.

[0154] 3. Select 10 points along the length of the beam using a laser vibration measuring instrument. The positions of the 10 points are consistent with the numerical simulation, and reflective powder is applied at the positions of these 10 points.

[0155] 4. Apply a harmonic excitation at the free end of the cantilever beam using an exciter.

[0156] 5. Automatically analyze through the laser vibration measuring instrument processor to obtain the acceleration response signals of 10 points.

[0157] 6. The method for wavelet packet transform and signal reconstruction is as follows:

[0158] (1) Recursively construct wavelet functions

[0159]

[0160]

[0161] In the formula, h(k) and g(k) are orthogonal reflection filters for selecting any mother wavelet (such as Harr wavelet, Daubechies), i represents the modulation parameter, j represents the scale parameter, k represents the translation parameter, and t represents time.

[0162] (2) Construct wavelet packet functions

[0163]

[0164] In the formula, It contains three wavelet packet functions. i represents the modulation parameter, j represents the scale parameter, k represents the translation parameter, is the wavelet function.

[0165] (3) Calculate wavelet packet coefficients

[0166]

[0167] In the formula, is the wavelet packet coefficient, x(t) is the acceleration signal, is the wavelet packet function, i represents the modulation parameter, j represents the scale parameter, and k represents the translation parameter.

[0168] (4) Reconstruct the signal

[0169]

[0170]

[0171] In the formula, X(t) is the reconstructed signal, is the reconstructed signal of wavelet packet (j, i), is the wavelet packet coefficient, is a wavelet packet function, where i represents the modulation parameter, j represents the scale parameter, and k represents the translation parameter.

[0172] 7. Perform VMD (Variational Mode Decomposition) on the reconstructed signal. After decomposition, the obtained Intrinsic Mode Functions (IMFs) are subjected to Hilbert-Huang Transform (HHT) respectively to obtain the instantaneous frequencies and energy amplitudes at different times, forming a Hilbert spectrum set. The method is as follows:

[0173] (1) VMD signal decomposition

[0174] u k (t) = A k (t)cos(Φ k (t))

[0175]

[0176] This is the variational constraint model. u k (t) is the signal decomposed into each modal frequency-modulated and amplitude-modulated signal by VMD. A k (t) is the instantaneous amplitude of u k (t), and Φ k (t) is used as the phase; the instantaneous frequency ω t (ω k (t) = Φ k (t)) is slowly varying with respect to the instantaneous amplitude and the phase. k is each modal component (k = 1, 2, 3,..., k), f is the reconstructed signal, and δ(t) is the Dirac distribution.

[0177] To obtain the optimal solution of the variational constraint model, a quadratic penalty factor α and a Lagrange multiplier λ are introduced, and the following is obtained:

[0178]

[0179] where u k (t) is the signal decomposed into each modal frequency-modulated and amplitude-modulated signal by VMD, α is the quadratic penalty factor, λ is the Lagrange multiplier, f(t) is the reconstructed signal in step 2, δ(t) is the Dirac distribution, and ω k is the k-th order frequency. To obtain the above model, the method of alternating mathematical multiplication operators is used for calculation to obtain the output of K Intrinsic Mode Function (IMF) components.

[0180] (2) HHT Hilbert-Huang Transform

[0181]

[0182] z(t) = x j (t) + iy(t) = a(t)e iθ(t)

[0183]

[0184] θ(t) = arctan(y(t) / x j (t))

[0185] ω(t) = dθ / dt

[0186] where y(t) is the imaginary signal obtained by convolving the reconstructed IMF real signal x j (t) with 1 / t of time, z(t) is the amplitude of the coupled signal of x j (t) and y(t), a(t) represents the instantaneous energy amplitude, θ(t) represents the phase function, ω(t) represents the instantaneous frequency, and t represents time. Thus, the Hilbert spectrum set is obtained.

[0187] 8. Figure 4 (a), (b), and (c) are respectively the Hilbert spectrum sets of the test specimens with crack depths of non-destructive, 20%, and 41% of the beam height.

[0188] From Figure 4 the frequency amplitudes in the obtained Hilbert spectrum set are larger than the instantaneous frequency amplitudes in the Hilbert spectrum set under normal conditions. As the crack degree increases, the instantaneous frequency also increases. This method is based on the fact that fatigue cracks will modulate the signal, triggering multi-order secondary high-frequency signals. The greater the change in the instantaneous frequency amplitude, the deeper the fatigue crack degree, achieving an intuitive and accurate identification of the position and degree of fatigue breathing cracks, thereby revealing the fatigue crack condition and being suitable for extension to practical applications.

[0189] In summary, from Figure 2 , Figure 3 , Figure 4 it can be obtained that:

[0190] Under harmonic excitation of the cracked beam, it is proved from the numerical simulation and experimental perspectives that the "breathing" effect of fatigue cracks is obvious. Fatigue cracks will modulate the signal, triggering multi-order secondary high-frequency signals. The greater the change in the instantaneous frequency amplitude, the deeper the fatigue crack degree, achieving an intuitive and accurate identification of the position and degree of fatigue breathing cracks, thereby revealing the fatigue crack condition and being suitable for extension to practical applications.

[0191] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A structural fatigue crack identification method based on the cooperation of VMD-HHT, characterized in that, Including the following steps: Apply a harmonic excitation at the end position of the structure to be identified, and collect the acceleration response data at different positions of the structure; Perform wavelet packet transform on the acceleration response data at each position of the structure, and successively perform decomposition, noise suppression, and reconstruction processing to obtain a reconstructed acceleration signal with noise reduction; Perform variational mode decomposition on the reconstructed acceleration signal to obtain intrinsic mode functions; perform Hilbert-Huang transform on each intrinsic mode function respectively to obtain a Hilbert spectrum set arranged from high frequency to low frequency; When multiple-order secondary high-frequency signals appear in the Hilbert spectrum set, a fatigue crack occurs at the position where the response data of the structure is collected, and the larger the instantaneous frequency amplitude of the multiple-order secondary high-frequency signals, the larger the fatigue crack.

2. The structural fatigue crack identification method based on VMD-HHT collaboration according to claim 1, wherein The step of collecting the acceleration response data at different positions of the structure includes the following steps: Arrange multiple acceleration sensors on the structure to be identified, and electrically connect the multiple acceleration sensors to a processor respectively; The processor is electrically connected to a control switch and a power supply. The multiple acceleration sensors form a detection group. Apply a harmonic excitation at the end position of the structure under test. The processor collects the response signals on each acceleration sensor, and uses the response signals to establish an acceleration response database at different positions.

3. A method for identifying structural fatigue cracks based on the cooperation of VMD-HHT according to claim 1, characterized in that The step of performing wavelet packet transform on the acceleration response data at each position of the structure, and successively performing decomposition, noise suppression, and reconstruction processing to obtain a reconstructed acceleration signal with noise reduction includes the following steps: Recursively construct a wavelet function: where h(k) and g(k) are the orthogonal reflection filters of an arbitrarily selected mother wavelet respectively, i is a modulation parameter, j is a scale parameter, k is a translation parameter, and t is time; Construct a wavelet packet function according to the wavelet function: In the formula, includes three wavelet packet functions, where i is the modulation parameter, j is the scale parameter, and k is the translation parameter, is the wavelet function; Calculate the wavelet packet coefficients: wherein, is the wavelet packet coefficient, x(t) is the acceleration signal, is the wavelet packet function, i represents the modulation parameter, j represents the scale parameter, and k represents the translation parameter; Reconstruct the acceleration signal: where \(X(t)\) is the reconstructed signal, is the wavelet packet reconstructed signal, is the wavelet packet coefficient, is the wavelet packet function, \(i\) represents the modulation parameter, \(j\) represents the scale parameter, and \(k\) represents the translation parameter.

4. A method for identifying structural fatigue cracks based on VMD-HHT collaboration according to claim 1, characterized in that, The step of performing variational mode decomposition on the reconstructed acceleration signal to obtain intrinsic mode functions includes the following steps: Construct a variational constraint model: u k (t) = A k (t) cos(Φ k (t)) Where, u k (t) is the signal after variational mode decomposition into each modal frequency modulation - amplitude modulation signal, A k (t) is the instantaneous amplitude of u k (t), and Φ k (t) is used as the phase; ω t (ω k (t) = Φ k (t)) is the instantaneous frequency, which is slowly varying with respect to the phase compared to the instantaneous amplitude; k is each modal component, k = 1, 2, 3, …; k, f is the reconstructed signal, and δ(t) is the Dirac distribution; Introduce a quadratic penalty factor α and a Lagrange multiplier λ: where \(f(t)\) is the reconstructed signal, and \(\omega_{ k}\) is the \(k\)-th order frequency; k ​ Use the method of alternating mathematical multiplication operators for operation to obtain the output of K intrinsic mode function IMF components.

5. A method for identifying structural fatigue cracks based on VMD-HHT collaboration according to claim 4, characterized in that, The step of performing Hilbert-Huang transform on each intrinsic mode function respectively includes the following steps: Perform Hilbert-Huang transform on each intrinsic mode function respectively: z(t) = x j (t) + iy(t) = a(t)e iθ(t) θ(t) = arctan(y(t) / x j (t)) ω(t) = dθ / dt where x j (t) is the reconstructed IMF real signal, y(t) is the imaginary signal obtained by convolving x j (t) with 1 / t in time, z(t) is the amplitude of the coupled signal of x j (t) and y(t), a(t) is the instantaneous energy amplitude, θ(t) is the phase function, ω(t) is the instantaneous frequency, and t is time; Obtain the Hilbert spectrum set according to the above formula.

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