Weak earthquake modal identification method for concrete dam

Abstract

This invention relates to the field of dam modal analysis and identification technology, and discloses a method for identifying the modal responses of concrete dams under weak seismic excitation. The method includes: acquiring the vibration response signal of a concrete dam under weak seismic excitation, converting it into an analytical signal, and obtaining a global whitening matrix. The analytical signal is adaptively segmented into several sub-segments. The global whitening matrix is ​​used to perform a whitening transformation on the analytical signal corresponding to each sub-segment, obtaining each whitened signal sub-segment. Based on the Riemann stationarity variance of each sub-segment, corresponding weights are applied to each time delay covariance matrix, and joint approximate diagonalization is performed to obtain the modal matrix and modal response signal of the concrete dam. Using the modal response signal as input, its time delay covariance matrix is ​​calculated, and the modal parameters of the concrete dam are identified using a covariance-driven random subspace method. This invention effectively estimates the modal responses and modal parameters of concrete dams using weak seismic data, providing reliable technical support for the health diagnosis and seismic safety assessment of concrete dams.

Description

Technical Field

[0001] This invention relates to the field of dam modal analysis technology, and in particular to a method for identifying weak seismic modes of concrete dams. Background Technology

[0002] Modal parameters of concrete dams are key dynamic indicators for evaluating their structural health. Identifying modal parameters such as natural frequency, damping ratio, and mode shape using weak earthquake records requires no artificial excitation and does not rely on rare strong earthquakes. This method can economically and continuously obtain the true dynamic characteristics of dams, which is of great significance for daily safety monitoring.

[0003] Related modal identification methods based on weak earthquakes largely draw on output-only identification techniques under environmental excitation. Their general process includes: First, preprocessing the dam's weak earthquake acceleration response signal by detrending and resampling. Second, applying modal identification algorithms such as the random subspace method or frequency domain decomposition method to directly estimate the system's state-space model or power spectral density function from the preprocessed dam vibration response data. Finally, extracting modal parameters such as natural frequencies, damping ratios, and mode shapes through stability plots or singular spectra.

[0004] While existing methods can achieve good modal identification results under environmental excitation, their core premise is that the input excitation is wide-spectrum stationary white noise. This assumption is incompatible with the non-stationary narrowband characteristics of seismic signals, weakening the identification performance of these methods under weak seismic conditions. Specifically, the stability graph of the random subspace method has a large number of spurious poles, making it difficult to form a clear sequence of stable poles; and the modal spectral peaks in the singular spectrum of the frequency domain decomposition method are easily masked by spurious clutter. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention provides a method for identifying weak seismic modes in concrete dams. This method aims to solve the incompatibility between the wide-spectrum stationary white noise excitation assumption of the output mode identification method and the non-stationary narrow-band characteristics of the vibration response signal.

[0006] This invention provides a method for identifying weak seismic modes of concrete dams, comprising the following steps: The vibration response signal of the concrete dam under weak seismic excitation is obtained, converted into an analytical signal, and the global whitening matrix is ​​obtained based on the analytical signal. The analytic signal is adaptively segmented into several sub-segments based on its non-stationary characteristics; the segmentation logic of the sub-segments is to minimize the stationarity variance of the signal covariance matrix on the Riemann manifold within the candidate sub-segments. The global whitening matrix is ​​used to perform whitening transformation on the analytical signal corresponding to each segment to obtain each whitened signal segment. The time delay covariance matrix of each whitened signal segment under different time delays is calculated. According to the Riemann stationarity variance of each segment, the corresponding weights are applied to each time delay covariance matrix, and joint approximate diagonalization is performed to obtain the modal matrix and modal response signal of the concrete dam. Using the modal response signal as input, its time delay covariance matrix is ​​calculated, and the modal parameters of the concrete dam are identified by the covariance-driven random subspace method.

[0007] Optionally, the analytic signal can be adaptively segmented into several sub-segments based on its non-stationary characteristics, including the following steps: Each segment is determined sequentially starting from the signal's origin, with the start time of each segment being the midpoint of the previous segment. For the current sub-segment, select multiple candidate window lengths within the preset minimum and maximum window length range and select them by a preset step size. Generate each candidate data segment starting from the start time of the current sub-segment. Calculate the Riemann stationarity variance for each candidate data segment; The candidate data segment with the smallest Riemann stationarity variance is selected as the final segmentation scheme for the current sub-segment; Repeat this process until the segments are complete.

[0008] Optionally, the Riemann stationarity variance of each candidate data segment is calculated, including the following steps: Windowing is applied to the candidate data segments; The windowed candidate data segment is divided into multiple sub-segments, and the covariance matrix of the signal in each sub-segment is calculated. Calculate the Riemannian centroid of the covariance matrix of each segment; The Riemann stationarity variance of the candidate data segment is determined based on the affine invariant Riemann distance from the covariance matrix of each sub-segment to the Riemann barycenter. Riemann stationarity variance Calculate using the following formula: , in, This represents the number of sub-segments in the data segment. Let be the covariance matrix of each sub-segment. For Riemann's center of gravity, the symbol ( ) is the trace of the matrix.

[0009] Optionally, the window treatment can use Hanning windows or Hanming windows.

[0010] Optionally, the Riemann centroid is estimated iteratively using the Riemann gradient descent algorithm, including the following steps: The initial value of the Riemannian centroid is set according to the following arithmetic mean. : , For the In the next iteration, the Riemann centroid is updated according to the following iteration update rules: , In the formula, This is the iteration step size; The iteration ends and the Riemannian centroid is finally obtained when the following termination condition is met. ; , in, This is the threshold for iterative convergence.

[0011] Optionally, weights are applied to each time delay covariance matrix based on the Riemann stationarity variance of each segment, including the following steps: The weight of each sub-segment is the normalized value of the inverse of its Riemann stationarity variance; Multiply the time delay covariance matrix of each segment under each time delay by the weight of the corresponding segment to obtain the weighted time delay covariance matrix of each segment under each time delay.

[0012] Optionally, joint approximate diagonalization is performed to obtain the modal matrix and modal response signal of the concrete dam, including the following steps: Multiply all weighted time delay covariance matrices sequentially by a series of complex Givens matrices to obtain a joint diagonalized matrix; The whitened signal is processed using a joint diagonalization matrix to obtain the modal matrix and modal response signal of the concrete dam.

[0013] Optionally, the modal parameters of the concrete dam can be identified using a covariance-driven stochastic subspace method, including the following steps: The time delay covariance matrix is ​​calculated from the modal response signal, and a block Topplitz matrix is ​​constructed. Singular value decomposition is performed on the block Toplitz matrix to estimate the extended observable matrix and the system matrix; The system matrix is ​​decomposed into eigenvalues, and the natural frequencies and damping ratios are calculated from the eigenvalues. The mode shapes are then calculated from the extended energy matrix.

[0014] Optionally, the vibration response signal can be converted into an analytical signal, including the following steps: The vibration response signal is detrended and resampled to obtain a preprocessed signal; The preprocessed signal is converted into an analytic signal using the Hilbert transform.

[0015] Optionally, the global whitening matrix can be obtained from the analytical signal, including the following steps: Calculate the covariance matrix of the analytic signal; The covariance matrix is ​​decomposed into eigenvector matrix and eigenvalue diagonal matrix. A global whitening matrix is ​​then constructed based on the eigenvector matrix and eigenvalue diagonal matrix.

[0016] The technical solution provided by the embodiments of the present invention has the following advantages compared with the prior art: This invention provides a method for identifying weak seismic modes in concrete dams. Addressing the challenges of low excitation energy and non-stationary narrow-band characteristics in weak seismic excitation, which are incompatible with the stationary white noise assumptions of existing methods, the method first collects vibration response signals under weak seismic excitation and converts them into analytical signals, calculating a global whitening matrix to consider more general complex modal damping scenarios. Second, the analytical signal is logically adaptively segmented based on minimizing the stationar variance of the signal covariance matrix within candidate segments on the Riemannian manifold. This makes the statistical characteristics of the signal within each segment as close to stationary as possible, thereby decomposing the globally non-stationary signal into several locally stationary segments and eliminating the interference of non-stationarity on mode identification. Then, the global whitening matrix is ​​used to perform whitening transformation on each segment, and the covariance matrix of each whitened signal segment under different time delays is calculated. Based on the Riemann stationarity variance of each segment, corresponding weights are applied to the time delay covariance matrix; segments with better stationarity have larger weights, and vice versa. This weakens the adverse effects of non-stationary segments on joint approximate diagonalization, and then the modal matrix and modal response signal of the concrete dam are accurately estimated through weighted joint approximate diagonalization. Finally, using the extracted modal response signal as input, its time delay covariance matrix is ​​calculated, and modal parameters are identified using the covariance-driven random subspace method. This invention's method, through the synergistic mechanism of adaptive segmentation and weighted diagonalization, transforms non-stationary seismic responses into a processable form. Thus, without artificial excitation or strong earthquake events, and within a framework of output-only identification, it stably extracts modal responses and modal parameters from weak earthquake data with higher occurrence frequency, achieving high-frequency tracking of the structural state of concrete dams and providing reliable technical support for health diagnosis and seismic safety assessment. Attached Figure Description

[0017] To more clearly illustrate the embodiments and design schemes of the present invention, the accompanying drawings required for this embodiment will be briefly described below. The drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 A flowchart of a method for identifying weak seismic modes of a concrete dam, provided in an embodiment of the present invention; Figure 2 This is a layout diagram of the seismic monitoring system for a concrete gravity arch dam according to an embodiment of the present invention; Figure 3This is a time history diagram of seismic acceleration at a free field point of a concrete gravity arch dam in an embodiment of the present invention; Figure 4 This is a time history diagram of seismic acceleration recorded at measuring point 9-1 on the crest of the concrete gravity arch dam in this embodiment of the invention; Figure 5 This is a diagram of the Riemann adaptive segmentation process in an embodiment of the present invention; Figure 6 It is the autocorrelation function of the modal response of the concrete gravity arch dam in the embodiments of the present invention; Figure 7 This is the power spectral density function of the modal response of the concrete gravity arch dam in this embodiment of the invention; Figure 8 These are the first 6 mode shapes identified in the concrete gravity arch dam embodiment of the present invention.

[0019] Figure 9 These are the stability graphs and singular spectra of the concrete gravity arch dam obtained by the random subspace method and the frequency domain decomposition method in the embodiments of the present invention. Detailed Implementation

[0020] To enable those skilled in the art to better understand and implement the technical solutions of the present invention, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. The following embodiments are only used to more clearly illustrate the technical solutions of the present invention and should not be construed as limiting the scope of protection of the present invention.

[0021] The technical solutions provided by the various embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0022] First, this invention provides a method for identifying weak seismic modes of concrete dams, specifically as follows: Figure 1 As shown, it includes the following steps: The vibration response signal of a concrete dam under seismic excitation is acquired, converted into an analytical signal, and a global whitening matrix is ​​obtained based on the analytical signal. Vibration response signals under weak seismic excitation are acquired, converted into analytical signals, and the global whitening matrix is ​​calculated to preserve signal phase information and provide a unified transformation benchmark for subsequent whitening processing.

[0023] The analytical signal is adaptively segmented into several sub-segments based on its non-stationary characteristics. The segmentation logic of the sub-segments is to minimize the stationarity variance of the signal covariance matrix in the candidate sub-segment on the Riemann manifold, so that the statistical characteristics of the signal in each sub-segment are as close to stationar as possible. This decomposes the global non-stationary signal into several locally stationary segments, thus eliminating the interference of non-stationarity on mode recognition.

[0024] The global whitening matrix is ​​used to perform a whitening transformation on the analytical signals corresponding to each segment, resulting in whitened signal segments. The time delay covariance matrix of each whitened signal segment under different time delays is calculated. Based on the Riemann stationarity variance of each segment, corresponding weights are applied to the time delay covariance matrix, and joint approximate diagonalization is performed to obtain the modal matrix and modal response signal of the concrete dam. The global whitening matrix is ​​used to perform a whitening transformation on each segment, and the covariance matrix of each whitened signal segment under different time delays is calculated. Based on the Riemann stationarity variance of each segment, corresponding weights are applied to the time delay covariance matrix; segments with better stationarity have larger weights, and vice versa. This reduces the adverse effects of non-stationary segments on the joint approximate diagonalization, and then the modal matrix and modal response signal of the concrete dam are accurately estimated through weighted joint approximate diagonalization.

[0025] Using the modal response signal as input, its time delay covariance matrix is ​​calculated, and the modal parameters of the concrete dam are identified using the covariance-driven stochastic subspace method. Using the extracted modal response signal as input, its time delay covariance matrix is ​​calculated, and the modal parameters are identified using the covariance-driven stochastic subspace method.

[0026] This invention provides a method for identifying weak seismic modes in concrete dams. Through a collaborative mechanism of adaptive segmentation and weighted diagonalization, the non-stationary seismic response is transformed into a form that can be effectively processed. Thus, without the need for artificial excitation or strong earthquake events, the method stably extracts modal responses and modal parameters of each order using weak seismic data that occur more frequently. This enables high-frequency tracking of the structural state of concrete dams and provides reliable technical support for health diagnosis and seismic safety assessment.

[0027] Optionally, the analytic signal can be adaptively segmented into several sub-segments based on its non-stationary characteristics, including the following steps: Each segment is determined sequentially starting from the signal's origin, with the start time of each segment being the midpoint of the previous segment. For the current sub-segment, within the preset minimum window length and maximum window length Within the range, according to the preset step size Multiple candidate window lengths are selected, and candidate data segments are generated starting from the start time of the current sub-segment. , and All are determined based on the calculation results; Calculate the Riemann stationarity variance for each candidate data segment; The candidate data segment with the smallest Riemann stationarity variance is selected as the final segmentation scheme for the current sub-segment; Repeat this process until the segments are complete.

[0028] This invention achieves optimal local stationary segmentation of non-stationary seismic signals with low computational cost through a sliding starting point plus local optimal window search strategy. Specifically, starting from the signal's starting point, the start time of each segment is set to the midpoint of the previous segment. This overlapping recursive mechanism allows the segment boundaries to move smoothly with changes in the signal's statistical characteristics, avoiding the loss or abrupt changes in local information caused by fixed segmentation or hard boundary cutting. Simultaneously, candidate data segments are enumerated in steps within a preset window length, and the window with the smallest Riemannian stationarity variance is selected as the current segment, ensuring that each segment has optimal local stationarity in the Riemannian manifold sense. Compared to directly dividing the entire signal into uniform blocks or relying on empirically set windows, this method requires no prior knowledge of the signal and can adaptively track the evolution of time-varying statistical characteristics during seismic excitation. This makes the covariance matrix within each segment more concentrated around the Riemannian centroid, thus providing a more accurate basis for subsequent weighted joint approximate diagonalization. Segments with smaller stationarity variance have larger weights, and their corresponding time delay covariance matrices contribute more reliably to the joint diagonalization, directly improving the estimation accuracy of the modal matrix and modal response signal. Furthermore, this recursive strategy only needs to search for the optimal window within the current segment, without global optimization, resulting in high computational efficiency and meeting the real-time processing requirements of engineering projects. Ultimately, this refined adaptive segmentation enables the concrete dam weak seismic mode identification method provided in this embodiment to obtain the modal parameters of the concrete dam with higher accuracy and stability when facing low-energy, time-varying weak seismic signals, providing a more reliable basis for subsequent health diagnosis.

[0029] Optionally, the Riemann stationarity variance of each candidate data segment is calculated, including the following steps: Windowing is applied to the candidate data segments; Divide the data segment into Given three equal-length sub-segments, calculate the covariance matrix of each sub-segment. ; Calculate the Riemann centroid using the Riemann gradient descent algorithm. ; Calculate the Riemann stationarity variance using the following formula. : , Among them, the symbol ( ) is the trace of the matrix.

[0030] This invention first applies a windowing process to the candidate data segment to suppress spectral leakage caused by signal truncation, ensuring the numerical stability of the covariance matrix estimation. Then, the windowed data segment is divided into several sub-segments, and the covariance matrix of each sub-segment is calculated. These covariance matrices are distributed on the Riemannian manifold as Emyer positive definite matrices. Based on this, the Riemannian centroids of these covariance matrices are calculated, and the stationarity variance is defined by the affine invariant Riemannian distance from the covariance matrix of each sub-segment to the centroid. This achieves a precise characterization of the time-varying nature of the statistical characteristics within the data segment. When the statistical characteristics of each sub-segment tend to be consistent, their covariance matrices are tightly clustered at the centroid on the Riemannian manifold, resulting in a smaller stationarity variance; conversely, they are more dispersed and have a larger variance. The affine invariant Riemannian distance can fully utilize the geometric structure of the Emyer positive definite matrix, avoiding measurement distortion caused by matrix scale differences or non-positive definite disturbances, and exhibits stronger robustness to signal amplitude variations and noise interference. This refined metric method enables candidate window search to make optimal decisions based on real manifold geometry, ensuring that the selected segments have the best local stationarity in a statistical sense. This provides a more reliable basis for segment partitioning and weight allocation for weighted joint approximate diagonalization, and ultimately further improves the accuracy of modal parameter identification under weak seismic conditions.

[0031] Optionally, the windowing process can use either the Hanning window or the Hamming window. By selecting an appropriate window function type, the numerical quality of the covariance matrix estimation is significantly improved, thereby ensuring the accuracy and reliability of the entire Riemann stationarity variance measure.

[0032] Optionally, the Riemann centroid is estimated iteratively using the Riemann gradient descent algorithm, including the following steps: The initial value of the Riemann centroid is set according to the following arithmetic mean: .

[0033] For the In the next iteration, the Riemann centroid is updated according to the following iteration update rules: , In the formula, The iteration step size is determined based on the calculation results.

[0034] The iteration ends and the Riemannian centroid is finally obtained when the following termination condition is met. .

[0035] , in, The iterative convergence threshold is determined based on the calculation results.

[0036] The data segment with the smallest Riemann stationarity variance among the candidate data segments is selected as the data sub-segment.

[0037] The Riemann barycenter estimated iteratively using the Riemann gradient descent algorithm described above provides a numerical method for accurately and efficiently estimating the Riemann barycenter on manifolds with positive Emilt definite matrices. Specifically, Riemann manifolds composed of positive Emilt definite matrices have a nonlinear curvature structure, and their Riemann barycenter does not have a closed analytical solution. The arithmetic mean in traditional Euclidean space can lead to deviations from the true barycenter and underestimation of the manifold's dispersion. This invention addresses this by iteratively solving the problem using the Riemann gradient descent algorithm. First, an initial value is set as the current Riemann barycenter. Then, the steepest descent direction on the manifold is calculated, and the barycenter is updated along this direction. The optimal value is output when the Riemann distance between the barycenters of two adjacent iterations is less than a preset threshold. This method not only ensures that the iterative convergence process strictly follows the geometric structure of the Riemann manifold, avoiding accumulated errors caused by linear approximation, but also flexibly balances computational accuracy and iterative efficiency by introducing a convergence threshold, exhibiting high controllability and adaptability in engineering applications. This precise Riemann centroid estimation makes the Riemann distance from the covariance matrix of each segment to the centroid more realistically reflect the dispersion of the covariance matrix on the manifold. This makes the Riemann stationarity variance measurement of the candidate data segments more accurate, further ensuring that adaptive segmentation can accurately locate the local signal segments with the most stable statistical characteristics within a preset window. Even under conditions of low energy excitation from weak seismic events and strong non-stationarity, it can still provide a high-quality segmentation basis for weighted joint approximate diagonalization, improving the accuracy and stability of modal parameter identification for concrete dams.

[0038] Optionally, weights are applied to each time delay covariance matrix based on the Riemann stationarity variance of each segment, including the following steps: The weight of each sub-segment is the normalized value of the inverse of its Riemann stationarity variance; Multiply the time delay covariance matrix of each segment under each time delay by the weight of the corresponding segment to obtain the weighted time delay covariance matrix of each segment under each time delay.

[0039] This invention constructs an adaptive weighting mechanism based on Riemannian geometric metrics, which precisely links the contribution of each sub-segment in the joint approximate diagonalization to its local stationarity quality. This addresses the fundamental contradiction at the algorithmic level between the stationary white noise assumption in related technologies and the non-stationary characteristics of seismic signals. Specifically, this invention defines the weight of each sub-segment as the normalized inverse of its Riemannian stationarity variance. This means that sub-segments with better stationarity (higher concentration of covariance matrix on the Riemannian manifold and smaller variance) receive larger weights, while sub-segments with poorer stationarity have their weights adaptively reduced. Subsequently, the time delay covariance matrix of each sub-segment at each time delay is multiplied by its corresponding weight to obtain a weighted time delay covariance matrix used for subsequent joint approximate diagonalization. The non-stationarity of weak seismic signals manifests in the variation of their statistical characteristics over time. The quality of system modal information carried by the time delay covariance matrix at different time intervals varies considerably. If all segments are treated with equal weight, the time delay covariance matrix of non-stationary segments will introduce a large number of modality-independent time-varying components into the joint diagonalization process, causing the diagonalization result to deviate from the true modal direction. This invention, through adaptive weighting driven by Riemann stationarity variance, mathematically achieves information-theoretic filtering of signal quality, ensuring that the time delay covariance matrix of high-weight segments dominates the joint diagonalization objective function, while effectively suppressing the adverse effects of low-weight segments. Compared to simply eliminating non-stationary segments or using fixed threshold truncation, this normalized weighting strategy retains all available data information, avoiding insufficient data volume due to excessive elimination and thus reducing the statistical significance of modal identification. Ultimately, this weighting mechanism enables the present invention to stably extract modal responses and modal parameters from response data with varying quality under non-stationary narrow-band conditions of weak seismic excitation, significantly improving the accuracy of modal identification results and providing more reliable dynamic characteristic data for the health diagnosis of concrete dams.

[0040] Optionally, joint approximate diagonalization is performed to obtain the modal matrix and modal response signal of the concrete dam, including the following steps: Multiply all weighted time delay covariance matrices sequentially by a series of complex Givens matrices to obtain a joint diagonalized matrix; The whitened signal is processed using a joint diagonalization matrix to obtain the modal matrix and modal response signal of the concrete dam.

[0041] Specifically, the analytical signal of each segment is whitened to obtain the whitened signal segment. The modal matrix and modal response signal of the concrete dam are obtained through weighted joint approximate diagonalization, including the following steps: Step S51: From the global whitening matrix Obtain the whitened signal segment ; For the sub-segment, whitened signal sub-segment Calculate using the following formula: , in, To analyze the signal sub-segments.

[0042] Step S52, from the first subsegment signal The calculation time delay is The time delay covariance matrix : , in, The length of the sub-segment.

[0043] Step S53: Calculate the Riemann stationarity variance corresponding to each whitened signal segment. ; Step S54: Weight all time delay covariance matrices based on the Riemann stationarity variance of each segment; Corresponding weights Calculate using the following formula: .

[0044] Step S55: For all weighted time delay covariance matrices Multiplying by a series of complex Givens matrices on the right in sequence yields the joint diagonalization matrix. ; The transformation unit of the complex Givens matrix is ​​of the following form: , in, and The rotation angle and phase angle are obtained by minimizing the sum of the squares of the absolute values ​​of the off-diagonal elements after the transformation.

[0045] Step S56: Estimate the modal matrix of the concrete dam according to the following formula. and modal response signal : , .

[0046] Optionally, the modal parameters of the concrete dam can be identified using a covariance-driven stochastic subspace method, including the following steps: The time delay covariance matrix is ​​calculated from the modal response signal, and a block Topplitz matrix is ​​constructed. Singular value decomposition is performed on the block Toplitz matrix to estimate the extended observable matrix and the system matrix; The system matrix is ​​decomposed into eigenvalues, and the natural frequencies and damping ratios are calculated from the eigenvalues. The mode shapes are then calculated from the extended energy matrix.

[0047] Specifically, the calculation of the time delay covariance matrix of the modal response signal, and the calculation of the modal parameters of the concrete dam using the covariance-driven stochastic subspace method, includes the following steps: Step S61, from modal response Calculate the time delay covariance matrix using the following formula. : , in, This is a time delay.

[0048] Step S62: Based on the time delay covariance matrix Construct a block Topletz matrix As shown in the following formula: , in, This represents the number of blocks in the block Toplitz matrix.

[0049] Step S63: For the block Topplitz matrix Perform singular value decomposition using the following formula: , in, and They are orthogonal matrices, It is a diagonal matrix composed of singular values.

[0050] Step S64: Estimate the extended observability matrix and system matrix As shown in the following formula: , , in, express After deletion The matrix obtained after row, express Before deletion The matrix obtained after row, for The number of columns, the symbol ( ) is the pseudo-inverse of the matrix.

[0051] Step S65: Perform system matrix analysis. Eigenvalue decomposition is performed according to the following formula: , in, The unitary matrix formed by the eigenvectors. for eigenvalues , is a diagonal matrix with diagonal elements.

[0052] Step S66: From eigenvalues Calculate the natural frequency using the following formula. Damping ratio : , , in, This represents the sampling time interval.

[0053] Step S67: Calculate the mode shape matrix using the following formula. : .

[0054] Using the high-purity modal response signal extracted through pre-weighted joint approximate diagonalization as input, and through systematic state-space modeling and eigenvalue decomposition, the final output is key modal parameters such as natural frequency, damping ratio, and mode shape that can be directly used for engineering evaluation, thus completing a complete closed loop from the original weak seismic response to the modal parameters usable in engineering.

[0055] Optionally, the vibration response signal can be converted into an analytical signal, including the following steps: The vibration response signal is detrended and resampled to obtain a preprocessed signal; The preprocessed signal is converted into an analytic signal using the Hilbert transform.

[0056] Optionally, the global whitening matrix can be obtained from the analytical signal, including the following steps: Calculate the covariance matrix of the analytic signal; The covariance matrix is ​​decomposed into eigenvector matrix and eigenvalue diagonal matrix. A global whitening matrix is ​​then constructed based on the eigenvector matrix and eigenvalue diagonal matrix.

[0057] Specifically, converting the vibration response signal of a concrete dam into an analytic signal using Hilbert transform includes the following steps: Vibration response signal Convert to an analytical signal according to the following formula : , in, For discrete time sequence numbers. for Hilbert transform, ; Specifically, the covariance matrix of the analytic signal is calculated to obtain the global whitening matrix, including the following steps: Analyzing signals The whitening signal is obtained by performing whitening treatment according to the following formula. : , in, This is the whitening matrix; Whitening matrix Calculate using the following formula: , in, To analyze the signal The covariance matrix, This indicates the conjugate transpose.

[0058] The following example, a prototype monitoring and analysis project of a concrete dam, illustrates how to use the technical methods of this invention to identify weak seismic modes in a concrete dam.

[0059] The concrete dam is a 178m high gravity arch dam with a water-retaining front of 396m. The top width and maximum bottom width are 15m and 80m respectively. The foundation and top elevations are 2432m and 2610m respectively. The normal high water level in front of the dam is 2600m.

[0060] like Figure 2 As shown, a seismic monitoring system is installed on the concrete dam body, including 12 monitoring points inside the main dam and 1 free-field monitoring point in the valley. The 12 monitoring points inside the dam (numbered 4-1, 4-3, 7-1, 9-1, 9-2, 9-3, 9-5, 11-1, 13-1, 13-3, left-1, and right-1) are located in four main dam galleries at elevations of 2600m, 2560m, 2530m, and 2463m, respectively. The free-field point is located in the powerhouse access tunnel at an elevation of 2510m. Each monitoring point is configured in three directions (BAE, BAN, BAZ), with radial, tangential, and vertical configurations, totaling 39 monitoring directions. Three-dimensional force-balanced accelerometers are used. Signal transmission lines from each monitoring point are led to the auxiliary powerhouse protection room, where acceleration signals are acquired at 200Hz using signal acquisition equipment.

[0061] The concrete dam seismic monitoring system recorded a reservoir-induced earthquake of magnitude 1.6. The epicenter was 1 km from the dam site, with a focal depth of 10 km. At the time of the earthquake, the upstream reservoir water level was 2570.3 m. Figure 3 and Figure 4 The time histories of triaxial acceleration under seismic action are shown at the free field point and at the 9-1 measuring point on the dam crest.

[0062] Modal identification of the concrete dam was carried out using 10-second acceleration time history data from 30 channels collected at 10 monitoring points (numbered 4-1, 4-3, 7-1, 9-1, 9-2, 9-3, 9-5, 11-1, 13-1, and 13-3) on the main dam under seismic loading.

[0063] First, detrending is performed, followed by resampling. The Nyquist frequency after resampling is 25 Hz. The vibration response signal of the concrete dam is converted into an analytic signal using a Hilbert transform. The covariance matrix of the analytic signal is calculated to obtain the global whitening matrix. A minimum window length is then set. and minimum window length 1s and 5s respectively, window step size The time is 0.2s, and the number of equal-length sub-segments is... The iteration step size of the Riemann gradient descent algorithm is 5. The iterative convergence threshold is 0.5. 10 -6 The Hanning window was chosen as the window function. Based on the minimum Riemann stationarity variance, the whitened signal was adaptively divided into 5 sub-segments using the Riemann adaptive segmentation method. The Riemann adaptive segmentation process and results are as follows: Figure 5 As shown in Table 1.

[0064] Table 1. Results of Riemann Adaptive Segmentation Segmentation order Start time End time Window length Riemann stationarity variance Weight 1 0 4 4 45.05 0.180 2 2 4.8 2.8 31.63 0.215 3 3.4 6.6 3.2 33.76 0.208 4 5 8.4 3.4 36.84 0.199 5 6.7 9.5 2.8 37.88 0.197 Five sub-segments of the analytical signal were whitened to obtain whitened signal sub-segments, and time delay covariance matrices were calculated. The number of time delays in each sub-segment was set to 50, resulting in a total of 250 time delay covariance matrices. All time delay covariance matrices were weighted according to the Riemann stationarity variance of each sub-segment, as shown in Table 1. Weighted joint approximate diagonalization of all matrices yielded the modal matrix and modal response signal of the concrete dam. The autocorrelation function and autopower spectral density function of the modal response signal are shown in Table 1. Figure 6 and Figure 7 As shown in the figure, the autocorrelation function exhibits decaying oscillation characteristics, and the autopower spectral density function shows concentrated spectral peaks, indicating that the present invention has good modal response estimation capabilities. The covariance matrix is ​​calculated for the modal response signal, and the modal parameters of the concrete dam are calculated using the covariance-driven random subspace method. The number of blocks in the block Topplitz matrix is... The value is set to 15. The modal parameters of the first six modes of the concrete dam were identified. The natural frequencies of the first six modes are 3.23 Hz, 4.41 Hz, 5.89 Hz, 6.35 Hz, 7.95 Hz, and 9.62 Hz, respectively, and the damping ratios of the first six modes are 3.1%, 1.7%, 1.6%, 3.1%, 2.8%, and 1.9%, respectively. The mode shapes of the first six modes are as follows: Figure 8As shown. Among them, the 1st, 3rd, 4th, and 6th mode shapes are symmetric mode shapes, while the 2nd and 5th mode shapes are antisymmetric mode shapes. To illustrate the superiority of this invention, the random subspace method and frequency domain decomposition method are used for comparison. The stability graph poles and singular spectra are shown below. Figure 9 The midpoint and solid line are shown. It can be seen that the stability diagram of the random subspace method has a large number of spurious poles, making it difficult to form a clear sequence of stable poles. In the singular spectrum of the frequency domain decomposition method, it is difficult to distinguish between modal spectral peaks and spurious clutter peaks. Under weak seismic conditions, existing techniques are unable to identify effective modal parameter results.

[0065] In summary, the weak seismic mode identification method for concrete dams proposed in this invention can effectively solve the problem of incompatibility between the basic assumptions of existing mode identification methods and the non-stationary narrow-band characteristics of vibration response signals. It can effectively estimate the modal responses and modal parameters of concrete dams using weak seismic data, providing reliable technical support for the health diagnosis and seismic safety assessment of concrete dams.

[0066] The above-described embodiments are merely specific examples of the present invention. However, the embodiments of the present invention are not limited thereto, and any variations that can be conceived by those skilled in the art should fall within the protection scope of the present invention.

Claims

1. A method for identifying weak mode of a concrete dam under earthquake, characterized in that, Includes the following steps: The vibration response signal of a concrete dam under weak seismic excitation is obtained, converted into an analytical signal, and a global whitening matrix is ​​obtained based on the analytical signal. The analytical signal is adaptively segmented into several sub-segments based on its non-stationary characteristics; wherein the segmentation logic of the sub-segments is that the stationarity variance of the signal covariance matrix in the candidate sub-segment is minimized on the Riemann manifold. The global whitening matrix is ​​used to perform whitening transformation on the analytical signal corresponding to each segment to obtain each whitened signal segment. The time delay covariance matrix of each whitened signal segment under different time delays is calculated. According to the Riemann stationarity variance of each segment, the corresponding weights are applied to each time delay covariance matrix, and joint approximate diagonalization is performed to obtain the modal matrix and modal response signal of the concrete dam. Using the modal response signal as input, its time delay covariance matrix is ​​calculated, and the modal parameters of the concrete dam are identified by the covariance-driven random subspace method.

2. The concrete dam weak seismic modal identification method of claim 1, wherein, The analytical signal is adaptively segmented into several sub-segments based on its non-stationary characteristics, including the following steps: Each sub-segment is determined sequentially starting from the signal start point, and the start time of each sub-segment is the midpoint of the previous sub-segment; For the current sub-segment, select multiple candidate window lengths within the preset minimum and maximum window length range and select them by a preset step size. Generate each candidate data segment starting from the start time of the current sub-segment. Calculate the Riemann stationarity variance for each candidate data segment; The candidate data segment with the smallest Riemann stationarity variance is selected as the final segmentation scheme for the current sub-segment; Repeat this process until the segments are complete.

3. The concrete dam weak seismic modal identification method of claim 2, wherein, The calculation of the Riemann stationarity variance for each candidate data segment includes the following steps: Windowing is applied to the candidate data segments; The windowed candidate data segment is divided into multiple sub-segments, and the covariance matrix of the signal in each sub-segment is calculated. Calculate the Riemannian centroid of the covariance matrix of each segment; determining a Riemann stationarity variance of the candidate data segment from an affine-invariant Riemann distance of the covariance matrices of the subsegments to the Riemann center of mass the Riemann stationarity variance is computed according to the following formula: ， wherein is the number of subsegments of the data segment, is the covariance matrix of each subsegment, is the Riemann center of mass, and is the trace of the matrix.

4. The concrete dam weak seismic modal identification method of claim 3, wherein, The windowing process uses Hanning windows or Hanming windows.

5. The concrete dam weak seismic modal identification method of claim 3, wherein, The Riemann centroid is estimated iteratively using the Riemann gradient descent algorithm, including the following steps: The Riemann center of mass initial value is set according to the following arithmetic mean : ， For the first iteration, the Riemann center of mass is updated according to the following iterative update rule: For the first iteration, the Riemann center of mass is updated according to the following iterative update rule: ， In the formula, is the iteration step size; The iteration ends and the Riemann center of gravity is finally obtained when the following termination condition is satisfied ; ， wherein, is an iteration convergence threshold.

6. The method for identifying weak seismic modes of concrete dams as described in claim 1, characterized in that, The step of applying corresponding weights to each time delay covariance matrix based on the Riemann stationarity variance of each segment includes the following steps: The weight of each sub-segment is the normalized value of the inverse of its Riemann stationarity variance; Multiply the time delay covariance matrix of each segment under each time delay by the weight of the corresponding segment to obtain the weighted time delay covariance matrix of each segment under each time delay.

7. The method for weak seismic mode identification of concrete dams based on Riemann blind subspace identification as described in claim 1, characterized in that, The joint approximate diagonalization process to obtain the modal matrix and modal response signal of the concrete dam includes the following steps: Multiply all weighted time delay covariance matrices sequentially by a series of complex Givens matrices to obtain a joint diagonalized matrix; The whitened signal is processed using the joint diagonalization matrix to obtain the modal matrix and modal response signal of the concrete dam.

8. The method for identifying weak seismic modes of concrete dams as described in claim 1, characterized in that, The method of identifying modal parameters of concrete dams using the covariance-driven stochastic subspace method includes the following steps: The time delay covariance matrix is ​​calculated from the modal response signal, and a block Topplitz matrix is ​​constructed. Singular value decomposition is performed on the block Toplitz matrix to estimate the extended observable matrix and the system matrix; The system matrix is ​​decomposed into eigenvalues, and the natural frequency and damping ratio are calculated from the eigenvalues. The mode shape is then calculated from the extended energy matrix.

9. The method for identifying weak seismic modes of concrete dams as described in claim 1, characterized in that, Converting the vibration response signal into an analytical signal includes the following steps: The vibration response signal is subjected to detrending and resampling processing to obtain a preprocessed signal; The preprocessed signal is converted into an analytic signal using a Hilbert transform.

10. The method for identifying weak seismic modes of concrete dams as described in claim 1, characterized in that, The global whitening matrix is ​​obtained based on the analyzed signal, including the following steps: Calculate the covariance matrix of the analytic signal; The covariance matrix is ​​subjected to eigenvalue decomposition to obtain the eigenvector matrix and the eigenvalue diagonal matrix. The global whitening matrix is ​​then constructed based on the eigenvector matrix and the eigenvalue diagonal matrix.

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