A time-varying structure weak modal identification method based on adaptive frequency modulation modal decomposition

Abstract

The application belongs to the technical field of civil engineering structure health monitoring data analysis, and relates to a time-varying structure weak modal identification method based on adaptive frequency modulation modal decomposition. First, the collected time-varying acceleration response is segmented, the power of the response data of each time period is obtained by using the Welch power spectrum, and the time-frequency distribution of the time-varying signal is obtained. Then, the ridge line is extracted by searching for the time-frequency energy extreme value, the initial center frequency of the maximum energy mode of the energy is located, and then the adaptive frequency modulation modal decomposition is used to extract the high-energy mode of the order, and the extraction result is subtracted from the original signal, and the obtained residual signal is used as the original signal for frequency positioning and modal extraction. Finally, the modal components are segmented and principal component analysis is used to realize the identification of the time-varying mode shape. The application uses adaptive frequency modulation modal decomposition to preliminarily extract strong modes, and can complete the identification of time-varying result modal parameters even in the case that there are weak modes in the signal.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of civil engineering structure health monitoring data analysis, and relates to a modal identification method for a time-varying structure with weak modes, in particular to a time-varying structure weak modal identification method based on adaptive chirp mode decomposition. BACKGROUND

[0002] Structural modal parameters including modal frequencies, mode shapes and damping ratios can reflect the dynamic characteristics of a structure. Under the joint action of environment and time, civil engineering structures will be damaged or even destroyed, and the structural modal parameters will also change accordingly. Therefore, correctly identifying the structural modal parameters is the key to structural health monitoring. Traditional modal parameter identification methods include time domain methods and frequency domain methods. However, the dynamic characteristics of civil engineering structures in operation will often change over time, and thus the modal parameters exhibit time-varying characteristics. Therefore, time-frequency analysis methods are more suitable for identifying the modal parameters of time-varying structures.

[0003] The variational mode decomposition algorithm is a time-frequency analysis method for solving the identification of time-varying modal parameters. This algorithm separates the modes by solving the optimal solution of the variational constraint problem, but cannot directly obtain the parameters of the time-varying structure. The variational nonlinear chirp mode decomposition method can complete the decomposition of the nonlinear chirp signal at one time, and can directly obtain the instantaneous frequency and amplitude of the time-varying mode, but the extraction result depends on the preset number of modes. The adaptive chirp mode decomposition (ACMD) method does not need to define the number of modes in advance, and extracts a single-order mode according to the specified initial frequency.

[0004] In actual modal identification, the energy of each order mode often differs greatly. When noise exists in the measured signal, the weak features with small energy may be submerged by noise or high-energy modes, resulting in the inability to extract weak modes. To solve this problem, it is of great engineering significance to study a time-varying structure weak modal parameter identification method based on adaptive chirp mode decomposition. SUMMARY

[0005] In view of the above technical problems, a time-varying structure weak modal identification method based on adaptive chirp mode decomposition is provided. The technical means adopted by the application are as follows:

[0006] A time-varying structure weak modal identification method based on adaptive chirp mode decomposition comprises the following steps:

[0007] Step 1: Collect the acceleration response data of the structure, divide the vibration response data into partially overlapping data segments, and calculate the power spectrum estimate P of each small data segment using the Welch method. xx(e jω );

[0008] Step 2, arrange the Welch power spectrum of each segment according to the time period to form the time-frequency distribution of the response data, find the position with the maximum energy in the time-frequency distribution (t m ,f m ) as the mth point on the ridge line, and take the frequency of this point as the frequency estimation value f L and f R of the ridge points on the left and right sides of the mth point; then for the time t L on the left side of the mth point, find the frequency corresponding to the point with the maximum energy within the maximum allowed frequency variation range as the frequency f L of this ridge point, and assign it to the left side point at the same time. In this way, the adjacent ridge points are continuously found, the extraction of the ridge line with the highest energy is completed by finding the position with the maximum energy, and the frequency of the ridge line is taken as the initial center frequency f

[0009] Step 3, for the mth modal to be extracted, establish the model of the frequency-modulated signal s m (t);

[0010] Step 4, construct the constraint problem corresponding to the mth modal component and discretize it;

[0011] Step 5, given the penalty factor and the limit value of the collected signal, iteratively update the mth modal response, instantaneous frequency, and instantaneous amplitude. When the stop criterion parameter ξ k reaches the limit value, output the modal response, instantaneous frequency, and instantaneous amplitude of the mth modal;

[0012] Step 6, after the extraction of the strong modal is completed, subtract this modal from the original signal to obtain the residual signal , and use it as the original signal for the extraction of the high-energy modal component, denoted as s r (t)→s(t). Repeat this step until the extraction of all strong modal components is completed.

[0013] Step 7, after the extraction of all modal components is completed, obtain the instantaneous frequencies and instantaneous amplitudes of each order. Divide each modal component into equal-length time segments and perform centering processing to obtain the mth normalized data in the i th segment . Calculate the modal shape in each time interval by principal component analysis, connect all the time segment shapes obtained, and obtain the time-varying modal shape of the structure, completing the identification of the modal parameters of each order.

[0014] Further, in step 1, the power spectrum estimate P xx (e jω ) is calculated as follows:

[0015]

[0016] where M is the length of each data segment, U is a normalization factor, L is the total number of data segments, x(p) is the signal, w(p) is a window function, e denotes the exponential function, j denotes the unit complex number, ω denotes the angular frequency variable, and p denotes the time variable;

[0017] In step 3, a model of the frequency-modulated signal is established by the following way:

[0018]

[0019] where a m (t) and b m (t) are the corresponding demodulated signals, and their expressions are as follows: A m (t) is the amplitude function, f m (t) is the frequency function, is the frequency estimation value, θ m is the phase function, and are two coordination operators, τ is the integral operator, and t represents time;

[0020] In step 4, a discretized constraint problem is constructed by the following way:

[0021]

[0022] where, denotes the minimum value of the objective function with respect to the variables u m ,f m , is a second-order difference matrix, a m = [a m (t0),..., a m (t N-1 )] T , b m = [b m (t0),..., b m (t N-1 )] T , x = [x(t0),..., x(t N-1 )] T , denotes the original signal, [·] T denotes the transpose operation, and α is a parameter for adjusting the bandwidth, which is determined according to the initial frequency The matrix diag[·] denotes the operation of obtaining diagonal elements;

[0023] In step 5, the stopping criterion parameters are calculated as follows:

[0024]

[0025] in, This is the result of the k-th iteration of the m-th modal response.

[0026] In step 6, the structural mode shape is calculated as follows:

[0027] Constructing decentralized and normalized modal responses Covariance matrix:

[0028]

[0029] Where E() represents taking the expectation and performing eigenvalue decomposition on the covariance matrix:

[0030]

[0031] Where λ is a diagonal matrix composed of eigenvalues, V = [v1, v2, K, v l ] is derived from the feature vector v i The orthogonal matrix composed of i = 1, ..., l, where l is the number of principal components, and the first principal direction vector v1 is the estimate of the mode shape vector.

[0032] The beneficial effects of this invention are:

[0033] This invention segments the acquired time-varying acceleration response signal and uses Welch power spectrum to obtain the power of the response data in each time period, thus obtaining a higher resolution time-frequency distribution result. By finding the time-frequency energy extrema for ridge extraction, a more accurate initial center frequency is obtained. The center frequency of the highest energy mode is located each time, and ACMD is used to extract the high-energy mode. The extracted result is subtracted from the original signal to obtain the residual signal, which is then used for further mode extraction. After repeated extraction of strong modes, weak modes are exposed, thus realizing the extraction of weak modes. Instantaneous frequency and instantaneous amplitude can be directly obtained through ACMD, and principal component analysis is used to extract the structural vibration mode, thereby realizing the identification of time-varying weak mode parameters. Attached Figure Description

[0034] Figure 1 The time-frequency distribution of the response in Embodiment 1 of the present invention

[0035] Figure 2 Arrangement of acceleration sensors for steel truss bridge in Embodiment 2 of the present invention Detailed Implementation

[0036] The following will be described by specific embodiments in combination with technical solutions to illustrate the implementation of the patent.

[0037] A time-varying structure weak modal identification method based on adaptive frequency modulation modal decomposition, comprising the following steps:

[0038] Step 1, collect the acceleration response data of the structure, divide the vibration response data s(t) with a total length of N into X segments, each segment has a length of P, the overlapping part has a length of Q, then divide and window process each small segment by Welch data, divide each small segment data into L sections, each section has a length of M, draw the Welch power spectrum of each segment response data, and the power spectrum expression of each section is P xx (e jω ) is the power spectrum estimation of each section signal x(p), U is a normalization factor, and w(p) is a window function.

[0039] Step 2, draw the time-frequency distribution |TF(t,f)| of the response data according to the Welch power spectrum of each segment obtained in the first step, find the position (t m ,f m ) with the maximum energy in the time-frequency distribution as the mth point on the ridge line, and take the frequency of this point as the frequency estimation value f L and f R of the ridge points on the left and right sides of the mth point; then for the left time t L of the mth point, find the frequency corresponding to the point with the maximum energy in the range [f m -Δf,f m +Δf] as the ridge point frequency f L , and at the same time, assign it to the left point, wherein Δf represents the maximum allowed frequency change between two consecutive points. In this way, adjacent ridge points are continuously found. The extraction of the highest-order ridge line is completed by finding the position of the maximum energy, and the ridge line frequency is taken as the initial center frequency ω

[0040] Step 3, the original response signal s(t) can be represented as the sum of K modal components s i (t): Where A i (t) and f i (t) represent the instantaneous amplitude and instantaneous frequency of the i-th modal, respectively, and θ i is the initial phase of the i-th modal. For the mth modal to be extracted, Where, and are two coordination operators, f m (t) is the frequency function, is the frequency estimation value, and am (t) and b m (t) is the corresponding demodulated signal;

[0041] Step 4, solve the constraint problem corresponding to the mth modal component, and assume that the response signal is discretized in time; the discretized minimization constraint problem can be expressed as: wherein, is a second-order difference matrix, a m = [a m (t0),..., a m (t N-1 )] T b m = [b m (t0),..., b m (t N-1 )] T s = [s(t0),..., s(t N-1 )] T [·] T denotes the transpose operation, and a is a parameter adjusting the bandwidth, determined according to the initial frequency in the second step Construct the matrix wherein, diag[·] denotes an operation of obtaining diagonal elements;

[0042] Step 5, for the collected signal s(t), given the parameters a and the limit value e, after j iterations, the vector u i is updated to After filtering, the mth modal estimation is The instantaneous frequency The frequency increment estimation value and the instantaneous amplitude When the stop criterion parameter ξ satisfies , the instantaneous frequency The instantaneous amplitude and the modal component are outputted.

[0043] Step 6, after the extraction of the strong modal is completed, the mth modal is subtracted from the original signal to obtain the residual signal which is used as the original signal for the extraction of the high-energy modal component, denoted as s r (t)→s(t), and the step is repeated until the extraction of all strong modals is completed. In this way, the central frequency positioning and the extraction of the modal component are performed in the order of energy from large to small, which can avoid missing modals, and finally the weak modals can be exposed to realize the extraction of the weak modals;

[0044] Step 7, extraction of all modal components is achieved by ACMD, and the instantaneous frequency of each order is obtained and the instantaneous amplitude Then the modal components of each order are divided into equal-length time segments, and the decentering processing is performed to obtain the normalized data of the mth order in the ith segment The structural mode shape in each time interval is calculated by principal component analysis to obtain the characteristic matrix V and the eigenvalue matrix λ as follows: wherein, is the covariance matrix, λ is a diagonal matrix composed of eigenvalues, V = [v1, v2, K, v l ] is an orthogonal matrix composed of eigenvectors, and l is the number of principal components. The principal direction vector v1 is the estimation of the mode shape vector. The time-varying mode shape of the structure is obtained by connecting the mode shapes of all time segments, and the identification of the modal parameters of each order is completed.

[0045] Example 1

[0046] A time-varying four-degree-of-freedom shear building model is taken, and its mass, damping, and stiffness matrices are respectively: wherein, m1 = m2 = m3 = m4 = 4 kg, c1 = c2 = c3 = c4 = 0.5 N / (m -1 s), k1 and k2 are time-varying stiffness parameters, k1 = (1-0.04t)×10 4 N / m, k2 = (1.2-0.03t)×10 4 N / m, k3 = k4 = 1.2×10 4 N / m, a random excitation is applied to the structure, and noise with a signal-to-noise ratio of 30 is added. The initial velocity and acceleration of the system are both 0: wherein [·] T T represents the transpose operation on a vector. The time history data of each floor is collected by an acceleration sensor, and the sampling frequency is 1000 Hz.

[0047] The acceleration response data of the structure is collected, and the total length of the vibration response data s(t) is divided into 161 segments, each with a length of 2000, and the overlapping part has a length of 1950. Each small segment is then divided into L sections by Welch data division and windowing processing, and each small segment data is divided into L sections, each with a length of 5000. The Welch power spectrum of each segment of response data is plotted, and the power spectrum expression of each section is wherein P xx (e jω ) is the power spectrum estimate of each section of signal x(p), U is a normalization factor, and w(p) is a Hamming window.

[0048] The time-frequency distribution |TF(t,f)| of the response data is plotted according to the Welch power spectrum of each segment obtained in the first step, as shown in Figure 1 The position (t m ,f m ) with the maximum energy in the time-frequency distribution is found as the mth point on the ridge line, and the frequency of this point is taken as the frequency estimate f L and f R of the ridge points to the left and right of the mth point; subsequently, for the time t L to the left of the mth point, the frequency corresponding to the point with the maximum energy in the range [f m -Δf,f m +Δf] is taken as the frequency f L of the ridge point, and is assigned to the left point at the same time, where Δf represents the maximum allowed frequency variation between two consecutive points, and is set to 1 Hz. The adjacent ridge points are found in this way. The extraction of the ridge line with the highest energy is completed by finding the position of the point with the maximum energy, and the frequency of the ridge line is taken as the initial center frequency f

[0049] Four intrinsic mode functions will be obtained through the signal decomposition by the time-frequency distribution, and thus the original response signal s(t) can be expressed as the sum of the 4th order modal components s i (t): where A i (t) and f i (t) represent the instantaneous amplitude and the instantaneous frequency of the ith mode respectively, and θ i is the initial phase of the ith mode. For the 1st order mode to be extracted, where, and are two coordination operators, f1(t) is the frequency function thereof, is the frequency estimate, and a1(t) and b1(t) are the corresponding demodulation signals;

[0050] The constraint problem corresponding to the 1st order modal component is solved, and the response signal is assumed to be discretized in time; the discretized minimization constraint problem can be expressed as: where, is a second-order difference matrix, a1=[a1(t0),...,a1(t N-1 )] T ,b1=[b1(t0),...,b1(t N-1 )] T ,s=[s(t0),...,s(t N-1 )] T , [·] Tdenotes the transpose operation, and a is a parameter that regulates the bandwidth, determined from the initial frequency of the second step Constructing matrix where,

[0051] For the collected signal s(t), given parameter a = 1 x 10 -8 and limit value e = 1 x 10 -11 After j iterations, the vector u1 is updated as After filtering, the first-order modal estimate is Obtain the instantaneous frequency The frequency increment estimate value and the instantaneous amplitude When the stop criterion parameter ξ satisfies , then output the instantaneous frequency the instantaneous amplitude and the modal component The extraction of the maximum energy order modal is completed;

[0052] After the extraction of the strong modal is completed, the order modal is subtracted from the original signal to obtain the residual signal and it is used as the original signal to continue to extract the high-energy modal component, denoted as s r (t)→s(t), repeat the step until the extraction of all strong modes is completed. In this way, the center frequency is located and the modal component is extracted in the order of energy from large to small, which can avoid missing modes, and finally the fourth-order weak modal is exposed, realizing the extraction of the weak modal;

[0053] Through ACMD, the extraction of all modal components is realized, and the instantaneous frequency of each order and the instantaneous amplitude Subsequently, each order modal component is divided into equal-length time intervals, and the decentering process is performed to obtain the mth-order i-th segment of normalized data Through principal component analysis, the structural mode shape in each time interval can be obtained, and the following characteristic matrix V and eigenvalue matrix λ are obtained: where, is the covariance matrix, λ is the diagonal matrix composed of eigenvalues, and V is the orthogonal matrix composed of eigenvectors, and each principal direction vector is the estimate of the mode shape. Connecting the obtained mode shapes of all time intervals, the time-varying mode shape of the structure can be obtained After the identification of each order modal parameter is completed, it can be used for further evaluation of the health status of civil engineering structures.

[0054] Example 2

[0055] To further verify the effectiveness of the method, this section selects the test results of a practical simply supported steel truss bridge to achieve comparative verification. The simply supported steel truss bridge has a main span of 59.2 m, a net height of 8.2 m at the midspan, a bridge deck width of 3.6 m, and a single lane design. Eight single-axis acceleration sensors are arranged on both sides of the bridge to collect response signals generated during the test. The specific arrangement of the sensors and the general situation of the bridge are shown in Figure 2 , wherein 1-8 are the sensor arrangement positions. Comparative analysis is performed using the response data obtained during the ambient excitation test. The outdoor temperature on the test day was basically maintained at about 15°C, and the influence of temperature changes on the test data is ignored. The sampling frequency of the sensors during the test was 200 Hz, and the sampling duration was 50 s. Therefore, the response signal collected by each sensor consists of 10,000 data points.

[0056] To simulate corrosion fracture or overload fracture that may occur during actual use of the bridge, artificial cutting and welding repair are performed on the steel truss. Five different working conditions are created and tested. The first working condition is the bridge in perfect condition without damage. The second working condition is cutting a vertical truss member at position 9 in Figure 2 to half its cross-sectional depth. The third working condition is cutting the same vertical truss member at position 9 in Figure 2 to full cross-section. The fourth working condition is repairing the cut vertical truss member at position 9 in Figure 2 . The fifth working condition is cutting a vertical truss member at position 10 in Figure 2 to full cross-section. The five working conditions are performed sequentially to simulate the time-varying characteristics of the structure. The 50 s acceleration response data under the five working conditions are connected in sequence for analysis. The sampling frequency of the sensors is 200 Hz, and the sample size of the time series is 50,000 data points.

[0057] The acceleration response data of the structure are collected. The total length of the vibration response data s(t) is 50,000, which is divided into 161 segments, each with a length of 10,000, and the overlapping part has a length of 9,750. Each small segment is then divided into L sections, each with a length of 25,000. The Welch power spectrum of each response data segment is plotted. The power spectrum expression of each section is , wherein P xx (e jω ) is the power spectrum estimate of each section of signal x(p), U is the normalization factor, and w(p) is the Hamming window.

[0058] The time-frequency distribution of the response data is plotted based on the Welch power spectrum of each segment obtained in the first step. The position (t m ,f m ) with the maximum energy in the time-frequency distribution is taken as the mth point on the ridge line, and the frequency of this point is taken as the frequency estimate f of the ridge points to the left and right of the mth point.L and f R ; then for the left side of m points at time t L , find the frequency corresponding to the energy maximum point in the range of [f m -Δf, f m +Δf] as the ridge point frequency f L , and at the same time assign it to the left side point, where Δf represents the maximum allowed frequency change between two consecutive points, set to 1 Hz, continue to find the adjacent ridge point by this method, complete the extraction of the highest order ridge line by finding the position of the energy maximum value, and take the ridge line frequency as the initial center frequency of ACMD extraction

[0059] Through time-frequency distribution, it can be concluded that signal decomposition will obtain 7 intrinsic modal functions, so the original response signal s(t) can be expressed as the sum of 7 modal components s i (t): Where A i (t) and f i (t) represent the instantaneous amplitude and instantaneous frequency of the i-th modal, respectively, and θ i is the initial phase of the i-th modal. For the first modal to be extracted, Where, and are two coordination operators, f1(t) is its frequency function, is the frequency estimation value, and a1(t) and b1(t) are the corresponding demodulation signals;

[0060] Solve the constraint problem corresponding to the first modal component, and assume that the response signal is discretized in time; the discretized minimization constraint problem can be expressed as: Where, is a second-order difference matrix, a1 = [a1(t0),..., a1(t N-1 )] T , b1 = [b1(t0),..., b1(t N-1 )] T , s = [s(t0),..., s(t N-1 )] T , [·] T represents the transpose operation, and α is a parameter for adjusting the bandwidth, determined according to the initial frequency Construct the matrix Where,

[0061] For the collected signal s(t), given parameter α = 1 × 10 -8and the threshold value ε = 1 x 10 -11 After j iterations, the vector u1 is updated as After filtering, the first order modal estimation is The instantaneous frequency is obtained The frequency increment estimation value And the instantaneous amplitude When the stop criterion parameter ξ satisfies Then the instantaneous frequency is output The instantaneous amplitude And the modal component The extraction of the maximum energy order modal is completed;

[0062] After the extraction of the strong modal is completed, the order modal is subtracted from the original signal to obtain the residual signal And it is used as the original signal to continue to extract the high energy modal component, denoted as s r (t)→s(t), repeat the step until the extraction of all strong modes is completed. In this way, the center frequency is located and the modal component is extracted in the order of energy from large to small, which can avoid missing modes, and finally the 7th weak modal is exposed, realizing the extraction of weak modes;

[0063] Through ACMD, the extraction of all modal components is realized, and the instantaneous frequency of each order And the instantaneous amplitude Then each order modal component is divided into equal length time intervals, and the mth order i segment normalized data is obtained after the centering processing Through principal component analysis, the structural mode shape in each time interval can be obtained, and the following characteristic matrix V and eigenvalue matrix λ can be obtained: In the formula, Covariance matrix, λ is a diagonal matrix composed of eigenvalues, and V is an orthogonal matrix composed of eigenvectors, and each principal direction vector is the estimation of the mode shape. By connecting the mode shapes of all time intervals, the time-varying mode shape of the structure can be obtained After the identification of each order modal parameter is completed, it can be used for further evaluation of the health state of civil engineering structures.

Claims

1. A time-varying structural weak mode identification method based on adaptive frequency modulation mode decomposition, characterized in that, Includes the following steps: Step 1: Collect acceleration response data of the structure, divide the vibration response data into partially overlapping data segments, and use the Welch method to calculate the power spectrum estimate P for each small segment. xx (e jω ); Step 2: Arrange the Welch power spectrum segments according to time periods to form a time-frequency distribution of the response data, and find the position with the highest energy in the time-frequency distribution (t). m ,f m ) is the m-th point on the ridge line, and the frequency of this point is used as the frequency estimate f of the ridge points to the left and right of point m. L and f R Subsequently, for time t to the left of point m... L Within the maximum permissible frequency variation range, the frequency corresponding to the point of maximum energy is found and taken as the ridge frequency f. L This value is then assigned to the left-hand point, and the process of finding adjacent ridge points continues. The extraction of the highest-order energy ridge is completed by determining the location of the maximum energy value, and the ridge frequency is used as the initial center frequency for ACMD extraction. Step 3: For the m-th mode to be extracted, establish the frequency modulation signal model s. m (t); Step 4: Construct the constraint problem corresponding to the m-th modal component and discretize it; Step 5: Given the penalty factor and limit value of the acquired signal, iteratively update the m-th modal response, instantaneous frequency, and instantaneous amplitude. When the stopping standard parameter ξ... k When the limit value is reached, the modal response, instantaneous frequency, and instantaneous amplitude of the m-th mode are output. Step 6: After extracting the strong mode, subtract this mode from the original signal to obtain the residual signal. This signal is then used as the original signal for the extraction of high-energy mode components, denoted as s. r (t)→s(t), repeat this step until all strong modes have been extracted. in, Modal components; Step 7: After extracting all modal components, the instantaneous frequencies and amplitudes of each order can be obtained. The modal components of each order are divided into equal-length time segments, and then decentralized to obtain the normalized data of the m-th order i-th segment. By calculating the structural mode shapes within each time interval through principal component analysis, and connecting all the obtained mode shapes over all time intervals, the time-varying mode shapes of the structure can be obtained, thus completing the identification of the modal parameters of each order.

2. The time-varying structural weak mode identification method based on adaptive frequency modulation mode decomposition according to claim 1, characterized in that, In step 1, the power spectrum estimate P is calculated as follows: xx (e jω ): Where M is the length of each data segment, U is the normalization factor, L is the total number of data segments, x(p) is the signal, w(p) is the window function, e represents the exponential function, j represents the unit complex number, ω represents the angular frequency variable, and p represents the time variable; In step 3, the model of the frequency modulation signal is established as follows: Among them, a m (t) and b m (t) is the corresponding demodulated signal, and its expression is: A m (t) is the magnitude function, f m (t) is the frequency function. For the frequency estimate, θ m For phase function, and There are two coordination operators, τ is the integration operator, and t represents time; In step 4, the discretized constraint problem is constructed in the following manner: in, Indicates the relationship between variable u m ,f m Find the minimum value of the objective function. It is a second-order difference matrix. a m =[a m (t0),...,a m (t N-1 )] T b m =[b m (t0),...,b m (t N-1 )] T , x=[x(t0),...,x(t N-1 )] T , representing the original signal, [·] T This represents the transpose operation, where α is a parameter for adjusting the bandwidth, based on the initial frequency determined in step two. Constructing a matrix diag[·] represents the operation of retrieving diagonal elements; In step 5, the stopping criterion parameters are calculated in the following manner: in, This is the result of the k-th iteration of the m-th modal response; In step 6, the structural mode shape is calculated in the following manner: Constructing decentralized and normalized modal responses Covariance matrix: Where E() represents taking the expectation and performing eigenvalue decomposition on the covariance matrix: Where λ is a diagonal matrix composed of eigenvalues, V = [v1, v2, K, v l ] is derived from the feature vector v i The orthogonal matrix composed of i = 1, ..., l, where l is the number of principal components, and the first principal direction vector v1 is the estimate of the mode shape vector.

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