A method for identifying modal parameters of high arch dam under non-stationary working conditions

Abstract

The application discloses a high-arch dam modal parameter identification method under non-stationary working conditions. The method first arranges sensors on the high-arch dam to collect multi-channel vibration data, and removes non-stationary components in the signals based on a simple empirical Fourier decomposition method. Then, a parameter self-adaptive optimization multivariate modal decomposition method is used to process the data to obtain multi-channel intrinsic modal components. Finally, the random decrement method and Hilbert transform are combined for modal parameter identification, and the working modal parameters are output. The method effectively separates the non-stationary components by using the frequency cutoff strategy and the zero-phase filter, and realizes efficient and accurate decomposition of the multi-channel signals by using the correlation variance contribution rate fusion and the adaptive optimization algorithm. The method overcomes the shortcomings of the traditional method, such as insufficient interference processing of multi-channel data and lack of consideration of signal non-stationarity, and can more comprehensively and accurately identify the modal parameters of the high-arch dam, thereby providing reliable data support for the safety monitoring and stable operation of the high-arch dam.

Description

Technical Field

[0001] This invention relates to the field of modal parameter identification technology for high arch dams, specifically a method for identifying modal parameters of high arch dams under non-stationary working conditions. Background Technology

[0002] For high arch dams with a height exceeding 200 meters, flow-induced vibrations are inevitable during flood discharge. Severe vibrations can even lead to structural damage, threatening the safe operation and maintenance of the high arch dam. Therefore, studying the actual behavior of high arch dam structures during the discharge process is of great engineering value and practical significance for ensuring the safe operation of high arch dams.

[0003] In recent years, with the rapid development of prototype monitoring technology, the prototype monitoring data of high arch dam structures has shifted from traditional static monitoring to real-time dynamic monitoring, enabling a more comprehensive reflection of the dynamic characteristics and actual performance of the structure during service. However, the analysis technology for the dynamic response data of high arch dams under real-time dynamic monitoring mainly processes vibration data from a single channel, making it difficult to simultaneously identify the modal parameters of high arch dams under multi-channel data. This increases the identification error of high arch dam modal parameters due to interference components from different channels, and also increases the computational cost of high arch dam modal parameter identification. Furthermore, for vibration response signals with low amplitude, strong noise, and non-stationary characteristics, existing research focuses primarily on how to handle the identification of high arch dam working modal parameters under strong background noise conditions, neglecting the non-stationarity of the vibration response signal. This means there is an assumption of stationarity in the vibration response signal during prototype acquisition, which can easily lead to deviations in the identification of high arch dam working modal parameters. Currently, there is a lack of research on the identification of high arch dam working modal parameters under non-stationary conditions; therefore, it is urgent to develop an effective method to achieve reliable identification of high arch dam modal parameters under non-stationary conditions. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies, this invention provides a method for identifying modal parameters of high arch dams under non-stationary conditions. This method involves introducing a frequency cutoff strategy to determine the spectral segmentation boundary, filtering the signal using a zero-phase filter, and employing inverse Fourier transform to obtain the non-stationary components and reconstructed signal of the high arch dam vibration response signal. Subsequently, the reconstructed signal is processed, and the correlation variance contribution rate is used to fuse the multi-channel reconstructed signals. By establishing constraints on the deviation between the center frequency and the pickup frequency of the intrinsic modal components under different structural mode numbers, the normalized mutual information coefficient of the intrinsic modal components under different quadratic penalty factors is calculated to minimize the quadratic penalty factor, thereby achieving adaptive decomposition of the multi-channel high arch dam vibration response signal. Finally, based on the decomposed multi-channel intrinsic modal components of the high arch dam, a reliable identification of the high arch dam modal parameters is achieved by combining the random decrement method and Hilbert transform.

[0005] To achieve the above objectives, the present invention adopts the following technical solution.

[0006] A method for identifying modal parameters of high arch dams under non-stationary conditions includes the following steps:

[0007] Step S1: Deploy multiple dynamic displacement sensors or acceleration sensors on the high arch dam to collect vibration response signals at different locations during the discharge process of the high arch dam, and obtain multi-channel vibration data of the high arch dam.

[0008] Step S2: Process the multi-channel vibration data of the high arch dam based on the simplified empirical Fourier decomposition method to remove the non-stationary components of the vibration response signal of the high arch dam.

[0009] The simplified empirical Fourier decomposition method is based on a simplified spectrum segmentation boundary strategy. It performs simplified segmentation of the power spectrum of the autoregressive AR model of the original vibration response signal, and then performs inverse Fourier transform on each segmentation interval, thereby separating the non-stationary vibration response signal component from the original vibration response signal.

[0010] Step S3: The PAO-MVMD multivariate variational mode decomposition method based on parameter adaptive optimization is used to process the multi-channel vibration data of the high arch dam after removing non-stationary components, and the multi-channel intrinsic mode components of the high arch dam structure are obtained.

[0011] The PAO-MVMD method based on parameter adaptive optimization modally modulates the multi-channel vibration response signal into multi-channel intrinsic mode components with the same center frequency. It combines the correlation variance contribution rate fusion algorithm and the peak method to determine the number of center frequencies and their peak values. The normalized mutual information coefficient is used as the optimization objective to achieve adaptive updating of the number of modes and the secondary penalty factor of the high arch dam structure, thereby decomposing the multi-channel intrinsic mode components of the high arch dam structure.

[0012] Step S4: Combine the random decrement method and Hilbert transform to identify the modal parameters of the multi-channel intrinsic modal components of the decomposed high arch dam structure, and output the working modal parameters of the high arch dam structure under non-stationary conditions.

[0013] Specifically, step S2 involves separating the non-stationary vibration response signal component from the original vibration response signal, including the following:

[0014] Step S21: Power spectrum estimation based on autoregressive AR model for vibration response signal of high arch dam. Frequency domain processing was performed to obtain the AR model power spectrum of the vibration response signal of the high arch dam. :

[0015] (1);

[0016] In the above formula, For frequencies in the AR model; is the variance of white noise; P is the order in the AR model parameters. Let k' be the order of the AR model. Let be the k'th autoregressive coefficient; e be the natural base; j be the imaginary unit, denoted as ;

[0017] Step S22: Introduce a frequency cutoff strategy to segment the power spectrum of the AR model;

[0018] First, the power spectrum of the AR model is divided into three intervals, including: the low-frequency interval of water flow noise and non-stationary components [0, ... ], the natural frequency range of high arch dam structures [ , ], and the high-frequency range of white noise [ Then, determine the low-frequency range [0, +∞] respectively. ] and high frequency range [ Does the frequency domain range [, +∞] contain peak values? If there are no peak values ​​in the low-frequency or high-frequency range, then the frequency domain range boundary... and Take respectively and If a peak exists in the low-frequency or high-frequency range, then the minimum point after the peak is found to determine the boundary of the frequency domain range. and :

[0019] (2);

[0020] (3);

[0021] In the above formula, For frequency AR model power spectral density values;

[0022] Step S23: Use a zero-phase filter to filter the segmented signal and separate the frequency domain interval boundaries. and Used as the cutoff frequency to construct a zero-phase filter bank :

[0023] (4);

[0024] In the above formula, For the nth frequency; This refers to the Nth frequency; n is the number of frequencies; N is the total number of frequencies.

[0025] By constructing a zero-phase filter bank, the Fourier spectrum of the vibration response signal is converted into the spectrum within the target frequency domain. :

[0026] (5);

[0027] In the above formula, The Fourier spectrum value is obtained by performing a discrete Fourier transform on the vibration response signal.

[0028] Step S24: Perform an inverse Fourier transform on the filtered vibration response signal to output the non-stationary component and the reconstructed signal.

[0029] Perform an inverse Fourier transform on the vibration response signal after it has been filtered by the zero-phase filter to obtain the filtered signal. :

[0030] (6);

[0031] Further obtain the reconstructed signal :

[0032] (7);

[0033] In the above formula, K is the number of signal components of the high arch dam structure obtained by the simplified spectrum segmentation boundary strategy.

[0034] Specifically, the process of obtaining the multi-channel intrinsic mode components of the high arch dam structure in step S3 is as follows:

[0035] Step S31: An adaptive method for determining the number of modes of a high arch dam structure based on correlation variance contribution rate data-level fusion, using multi-channel vibration response signals as input for dynamic data fusion;

[0036] for Vibration response signal of each data channel , Let represent the vibration response signal of the g-th data channel, and t represent the time sequence number of the vibration response signal. The normalized signal is obtained through energy normalization. :

[0037] (8);

[0038] In the above formula, The vibration response signal acquired by the i-th sensor; The total energy of the vibration response signal collected by the i-th sensor on the high arch dam structure is calculated by correlation.

[0039] Will Discretized , Let be the total number of samples in the signal, and define the variance contribution rate of the q-th data point in the normalized signal. Represented as:

[0040] (9);

[0041] In the above formula, and These represent the mean and variance of the normalized signal, respectively. The normalized signal of the q-th data;

[0042] Based on the variance contribution rate of signals from different measurement points over a certain time period, the fusion coefficient of the q-th data collected by the i-th sensor is... Represented as:

[0043] (10);

[0044] Further, the q-th data point after dynamic data fusion is obtained. Represented as:

[0045] (11);

[0046] Thus, the fused signal is obtained. ;

[0047] Step S32: Calculate the fused signal The power spectral density of the AR model is obtained, and the frequency domain result of the target domain is obtained. The peak value method is used to pick the peak value of the target domain and obtain the corresponding number of peak values ​​and peak frequency.

[0048] Step S33: Decompose the multi-channel vibration response signal using a quadratic penalty factor, calculate the center frequencies of different intrinsic mode components, filter the results using the peak-to-peak method, and calculate the signal decomposition energy difference. :

[0049] (12);

[0050] In the above formula, Indicates the structure of the high arch dam. Energy of one IMF component; This represents the total energy of the vibration response signal collected by the i-th sensor on the high arch dam structure; This represents the true modal numbers of the high arch dam structure;

[0051] The magnitude of the value is positively correlated with the degree of signal decomposition; if the signal decomposition energy difference value... A larger value indicates that the signal has been over-decomposed; if the signal decomposition energy difference is large... When the value approaches 0, it indicates that the signal decomposition is reasonable, and the optimal number of modes for the high arch dam structure can be determined accordingly.

[0052] Step S34: Calculate the normalized mutual information coefficients of the structural components of the high arch dam under the optimal structural mode number. The optimal quadratic penalty factor is determined by minimizing the normalized mutual information coefficient.

[0053] (13);

[0054] In the above formula, For the first c The mutual information coefficients of the IMF components i' and j' of each channel; Indicates the number of data channels; Indicates the number of IMF components in the high arch dam structure;

[0055] Step S35: Based on the determined number of structural modes and the second-order penalty factor, obtain the multi-channel intrinsic modal components of the high arch dam structure by solving the constrained variational model of the multi-channel vibration response signal decomposition. :

[0056] (14);

[0057] In the above formula, This represents the k-th intrinsic mode component of the g-th channel; This represents the center frequency of the k-th component; Represents the Lagrange multiplier for the g-th channel; Indicates a secondary penalty factor; This indicates the partial derivative, used to calculate the gradient of the variational constraint model; express The unilateral spectrum; express Norm; This represents the vibration response signal of the g-th data channel, where t represents the number of time-series vibration response signals. This represents the Lagrange multiplier operator for the g-th channel corresponding to the time sequence number t of the vibration response signal; This indicates that the vibration signal contains k-order structural feature information. .

[0058] Compared with the prior art, the present invention has the following beneficial effects:

[0059] 1. This invention introduces a simplified empirical Fourier decomposition method with a frequency cutoff strategy, which can accurately identify and separate low-frequency trend terms (non-stationary components) and high-frequency white noise in vibration signals. Unlike traditional methods that assume signal stationarity, this invention removes the main interference that causes the deviation in modal parameter identification from the source, providing a clean and stable signal basis for subsequent analysis and significantly improving the applicability and robustness of the entire method under non-stationary conditions.

[0060] 2. The method of this invention uses a correlation variance contribution rate fusion algorithm and peak value method to determine the number of center frequencies and their peak values. The dynamic weighted fusion of vibration signals from multiple sensor channels effectively suppresses the unique random noise of each channel and enhances the common signal components that reflect the overall dynamic characteristics of the structure, thereby improving the signal-to-noise ratio in the data input stage and overcoming the limitation of traditional single-channel processing methods that ignore the correlation between channels.

[0061] 3. The parameter adaptive optimization multivariate variational mode decomposition method PAO-MVMD of the present invention differs from previous multivariate variational mode decomposition methods. It modulates and oscillates multi-channel signals into multi-channel intrinsic mode components with the same center frequency. It combines the correlation variance contribution rate fusion algorithm and the peak value method to determine the number of center frequencies and their peak values. The normalized mutual information coefficient is used as the optimization objective to achieve adaptive updating of the number of structural modes and the secondary penalty factor. Thus, the multi-channel intrinsic mode components of the high arch dam structure are decomposed. This ensures that the signal can be fully decomposed to extract all major modes, while avoiding over-decomposition and the generation of false components. It greatly improves the accuracy and reliability of the decomposition results and provides reliable data support for the safety monitoring and stable operation of high arch dams. Attached Figure Description

[0062] Figure 1 This is a flowchart of a method for identifying modal parameters of high arch dams under non-stationary working conditions according to the present invention;

[0063] Figure 2 This is a flowchart of the simplified empirical Fourier decomposition algorithm of the present invention;

[0064] Figure 3 This is a flowchart of the multivariate variational mode decomposition algorithm with adaptive parameter optimization according to the present invention.

[0065] Figure 4 This is a diagram showing the prototype project and vibration test layout of the Ertan arch dam in this embodiment of the invention.

[0066] Figure 5 This is a schematic diagram of the dynamic displacement time history curve and normalized power spectral density of measuring point B1 in this embodiment of the invention;

[0067] Figure 6This is a schematic diagram of the dynamic displacement time history curve and normalized power spectral density of measuring point B2 in this embodiment of the invention;

[0068] Figure 7 This is a schematic diagram of the dynamic displacement time history curve and normalized power spectral density of measuring point B7 in this embodiment of the invention;

[0069] Figure 8 This is a schematic diagram of the dynamic displacement time history curves and normalized power spectral density of the stability term component and trend term component of measurement point B1 in an embodiment of the present invention.

[0070] Figure 9 This is a schematic diagram of the dynamic displacement time history curves and normalized power spectral density of the stability and trend components of measuring point B7 in this embodiment of the invention.

[0071] Figure 10 This is a schematic diagram of the multi-channel signal processing results of the arch dam in an embodiment of the present invention;

[0072] Figure 11 This is a schematic diagram of the dynamic displacement time history curve and normalized power spectral density of the intrinsic mode components at measurement point B1 in this embodiment of the invention.

[0073] Figure 12 This is a schematic diagram of the dynamic displacement time history curve and normalized power spectral density of the intrinsic mode components at measurement point B7 in this embodiment of the invention.

[0074] Figure 13 This is a graph showing the free decay response and amplitude logarithmic fitting curve of IMF3 in an embodiment of the present invention. Detailed Implementation

[0075] To facilitate understanding and implementation of the present invention by those skilled in the art, the various steps of the method proposed in this invention are described in detail below. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. Furthermore, it should be understood that after reading the teachings of this invention, those skilled in the art can make various modifications or alterations to the invention, and these equivalent forms also fall within the scope defined by the appended claims.

[0076] Example 1

[0077] like Figure 1 As shown, this invention discloses a method for identifying modal parameters of high arch dams under non-stationary conditions, comprising the following steps:

[0078] Step S1: Deploy multiple dynamic displacement sensors or acceleration sensors on the high arch dam to collect vibration response signals at different locations during the discharge process of the high arch dam, and obtain multi-channel vibration data of the high arch dam.

[0079] Step S2: Process the multi-channel vibration data of the high arch dam based on the simplified empirical Fourier decomposition method to remove the non-stationary components of the vibration response signal of the high arch dam.

[0080] The simplified empirical Fourier decomposition method is based on a simplified spectrum segmentation boundary strategy. It performs simplified segmentation of the power spectrum of the autoregressive AR model of the original vibration response signal, and then performs inverse Fourier transform on each segmentation interval, thereby separating the non-stationary vibration response signal component from the original vibration response signal.

[0081] Step S3: The PAO-MVMD multivariate variational mode decomposition method based on parameter adaptive optimization is used to process the multi-channel vibration data of the high arch dam after removing non-stationary components, and the multi-channel intrinsic mode components of the high arch dam structure are obtained.

[0082] The PAO-MVMD method based on parameter adaptive optimization modally modulates the multi-channel vibration response signal into multi-channel intrinsic mode components with the same center frequency. It combines the correlation variance contribution rate fusion algorithm and the peak method to determine the number of center frequencies and their peak values. The normalized mutual information coefficient is used as the optimization objective to achieve adaptive updating of the number of modes and the secondary penalty factor of the high arch dam structure, thereby decomposing the multi-channel intrinsic mode components of the high arch dam structure.

[0083] Step S4: Combine the random decrement method and Hilbert transform to identify the modal parameters of the multi-channel intrinsic modal components of the decomposed high arch dam structure, and output the working modal parameters of the high arch dam structure under non-stationary conditions.

[0084] like Figure 2 As shown, step S2 involves separating the non-stationary vibration response signal component from the original vibration response signal, including the following:

[0085] Step S21: Power spectrum estimation based on autoregressive AR model for vibration response signal of high arch dam. Frequency domain processing was performed to obtain the AR model power spectrum of the vibration response signal of the high arch dam. :

[0086] (1);

[0087] In the above formula, For frequencies in the AR model; is the variance of white noise; P is the order in the AR model parameters. Let k' be the order of the AR model. Let be the k'th autoregressive coefficient; e is the natural base; j is the imaginary unit, which is ;

[0088] Step S22: Introduce a frequency cutoff strategy to segment the power spectrum of the AR model;

[0089] First, the power spectrum of the AR model is divided into three intervals, including: the low-frequency interval of water flow noise and non-stationary components [0, ... ], the natural frequency range of high arch dam structures [ , ], and the high-frequency range of white noise [ Then, determine the low-frequency range [0, +∞] respectively. ] and high frequency range [ Does the frequency domain range [, +∞] contain peak values? If there are no peak values ​​in the low-frequency or high-frequency range, then the frequency domain range boundary... and Take respectively and If a peak exists in the low-frequency or high-frequency range, then the minimum point after the peak is found to determine the boundary of the frequency domain range. and :

[0090] (2);

[0091] (3);

[0092] In the above formula, For frequency AR model power spectral density values;

[0093] Step S23: Use a zero-phase filter to filter the segmented signal and remove the frequency domain interval boundary. and Used as the cutoff frequency to construct a zero-phase filter bank :

[0094] (4);

[0095] In the above formula, For the nth frequency; This refers to the Nth frequency; n is the number of frequencies; N is the total number of frequencies.

[0096] By constructing a zero-phase filter bank, the Fourier spectrum of the vibration response signal is converted into the spectrum within the target frequency domain. :

[0097] (5);

[0098] In the above formula, The Fourier spectrum value is obtained by performing a discrete Fourier transform on the vibration response signal.

[0099] Step S24: Perform an inverse Fourier transform on the filtered vibration response signal to output the non-stationary component and the reconstructed signal.

[0100] Perform an inverse Fourier transform on the vibration response signal after it has been filtered by the zero-phase filter to obtain the filtered signal. :

[0101] (6);

[0102] Further obtain the reconstructed signal :

[0103] (7);

[0104] In the above formula, K is the number of signal components of the high arch dam structure obtained by the simplified spectrum segmentation boundary strategy.

[0105] like Figure 3 As shown, the process of obtaining the multi-channel intrinsic mode components of the high arch dam structure in step S3 is as follows:

[0106] Step S31: An adaptive method for determining the number of modes of a high arch dam structure based on correlation variance contribution rate data-level fusion, using multi-channel vibration response signals as input for dynamic data fusion;

[0107] for Vibration response signal of each data channel , Let represent the vibration response signal of the g-th data channel, and t represent the time sequence number of the vibration response signal. The normalized signal is obtained through energy normalization. :

[0108] (8);

[0109] In the above formula, The vibration response signal acquired by the i-th sensor; The total energy of the vibration response signal collected by the i-th sensor on the high arch dam structure is calculated by correlation.

[0110] Will Discretized , Let be the total number of samples in the signal, and define the variance contribution rate of the q-th data point in the normalized signal. Represented as:

[0111] (9);

[0112] In the above formula, and These represent the mean and variance of the normalized signal, respectively. The normalized signal of the q-th data;

[0113] Based on the variance contribution rate of signals from different measurement points over a certain time period, the fusion coefficient of the q-th data collected by the i-th sensor is... Represented as:

[0114] (10);

[0115] Further, the q-th data point after dynamic data fusion is obtained. Represented as:

[0116] (11);

[0117] Thus, the fused signal is obtained. ;

[0118] Step S32: Calculate the fused signal The power spectral density of the AR model is obtained, and the frequency domain result of the target domain is obtained. The peak value method is used to pick the peak value of the target domain and obtain the corresponding number of peak values ​​and peak frequency.

[0119] Step S33: Decompose the multi-channel vibration response signal using a quadratic penalty factor, calculate the center frequencies of different intrinsic mode components, filter the results using the peak-to-peak method, and calculate the signal decomposition energy difference. :

[0120] (12);

[0121] In the above formula, Indicates the structure of the high arch dam. Energy of one IMF component; This represents the total energy of the vibration response signal collected by the i-th sensor on the high arch dam structure; This represents the true modal numbers of the high arch dam structure;

[0122] The magnitude of the value is positively correlated with the degree of signal decomposition; if the signal decomposition energy difference value... A larger value indicates that the signal has been over-decomposed; if the signal decomposition energy difference is large... When the value approaches 0, it indicates that the signal decomposition is reasonable, and the optimal number of modes for the high arch dam structure can be determined accordingly.

[0123] Step S34: Calculate the normalized mutual information coefficients of the structural components of the high arch dam under the optimal structural mode number. The optimal quadratic penalty factor is determined by minimizing the normalized mutual information coefficient.

[0124] (13);

[0125] In the above formula, For the first c The mutual information coefficients of the IMF components i' and j' of each channel; Indicates the number of data channels; Indicates the number of IMF components in the high arch dam structure;

[0126] Step S35: Based on the determined number of structural modes and the second-order penalty factor, obtain the multi-channel intrinsic modal components of the high arch dam structure by solving the constrained variational model of the multi-channel vibration response signal decomposition. :

[0127] (14);

[0128] In the above formula, This represents the k-th intrinsic mode component of the g-th channel; This represents the center frequency of the k-th component; Represents the Lagrange multiplier for the g-th channel; Indicates a secondary penalty factor; This indicates the partial derivative, used to calculate the gradient of the variational constraint model; express The unilateral spectrum; express Norm; This represents the vibration response signal of the g-th data channel, where t represents the number of time-series vibration response signals. This represents the Lagrange multiplier operator for the g-th channel corresponding to the time sequence number t of the vibration response signal; This indicates that the vibration signal contains k-order structural feature information. .

[0129] The technical effectiveness of the method of the present invention will be verified by conducting a vibration test on a prototype high arch dam.

[0130] like Figure 4 As shown in (a), the arch dam is a 240-meter-high concrete hyperbolic arch dam with foundation and crest elevations of 965m and 1205m, respectively. The maximum arc length of the crest is 774.65m. The thickness of the arch crown beam gradually increases from 11m at the top to 55.74m at the bottom. The arc-to-height ratio and thickness-to-height ratio are 3.23 and 0.232, respectively. The normal upstream water level is 1200m.

[0131] Prototype vibration tests were conducted on a high arch dam under flood discharge excitation, and a radial dynamic displacement signal acquisition system was deployed, such as... Figure 4 Image (b) shows the sensor measurement point arrangement of the arch ring of a high arch dam, as follows: Figure 4Image (c) shows the sensor measurement point arrangement of the arch crown beam of the high arch dam. Seven dynamic displacement sensors, numbered B1 to B7, are arranged at the top of the arch ring, and four dynamic displacement sensors, numbered B8 to B11, are arranged at the arch crown beam, for a total of 11 dynamic displacement sensors. The sampling frequency of the sensors is set to 200Hz, and the model is DP type low-frequency dynamic displacement sensor. The water level conditions for the vibration test are: upstream water level 1196.0m, downstream water level 1014.5m.

[0132] like Figures 5-7 As shown, the dynamic displacement time history curves and normalized power spectral densities of measuring points B1, B2, and B7 on the arch ring are presented respectively. The dynamic displacement measurement results show that the vibration amplitude of the measuring points near the dam abutment is significantly lower than that of measuring point B2, indicating that the vibration amplitude gradually decreases from the center of the arch ring towards the dam abutment. Furthermore, measuring points B1 and B7, closer to the dam abutment, contain more burrs, meaning that the vibration information collected at these points is subject to more severe noise interference. The normalized power spectral density measurement results show that measuring points B1 and B7 contain significant strong low-frequency noise and trend components, with their power spectral peaks mainly concentrated around 0.09 Hz and 0.90 Hz. Measuring point B2, on the other hand, has four relatively clear power spectral peaks: 1.51 Hz, 2.19 Hz, 2.81 Hz, and 3.58 Hz, and does not exhibit significant low-frequency water flow noise or high-frequency white noise interference.

[0133] Furthermore, the simplified empirical Fourier decomposition algorithm proposed in this invention was used to process the vibration test results of the 11 measuring points, and the results were then analyzed. and To filter out non-stationary components and high-frequency noise, it should be noted that water flow noise interference was not considered in the low-frequency range. This is mainly because the 0.5Hz filtering for non-stationary components does not cover water flow noise components exceeding 0.5Hz. The power spectrum of the AR model is segmented based on a frequency cutoff strategy, such as... Figures 5-7 As shown, the dividing line of the frequency cutoff interval is given. The upper and lower limits of the integral in formula (6) are determined by the dividing line of the frequency cutoff interval, thereby revealing the non-stationary and stationary components in the signal. Figures 8-9 As shown, the time history curves and normalized power spectral density results for the stable and trend components at measurement points B1 and B7 are presented respectively. Figure 8 and Figure 9Comparison reveals that measurement points B1 and B7 both contain significant trend components, with their power spectral peaks mainly concentrated around 0.09 Hz, indicating significant non-stationary characteristics in the original signal. Comparison of the stable components at measurement points B1 and B7 shows that the time history curves do not exhibit baseline deviation, and the normalized power spectral density results indicate three significant peaks: 0.97 Hz, 1.45 Hz, and 2.19 Hz. This demonstrates that by processing the dynamic displacement time history curves containing significant non-stationary characteristics using the simplified empirical Fourier decomposition algorithm proposed in this invention, the stable components reflecting the structural dynamic characteristics can be accurately obtained.

[0134] Furthermore, the multivariate variational mode decomposition algorithm with adaptive parameter optimization proposed in this invention is used to process the multichannel vibration data of high arch dams after removing non-stationary components, such as... Figure 10 As shown, Figure 10 Figure (a) presents the dynamic fusion coefficients of multi-channel vibration signals calculated based on the correlation variance contribution rate. Figure 10 (b) and Figure 10 Figure (c) presents the dynamic displacement time history curves and normalized power spectral density of the arch dam after data fusion, respectively. Figure 10 As shown in (c), the signal power spectral function contains 6 peaks in the target domain, with peak frequencies of 0.88 Hz, 1.45 Hz, 2.19 Hz, 2.81 Hz, 3.58 Hz, and 4.39 Hz, respectively. Considering that low-frequency water flow noise was not filtered out when removing non-stationary components, i.e., the 0.88 Hz low-frequency water flow noise component in the fused signal, after removing it, the number of structural frequency peaks in the target domain is 5. Therefore, it can be determined that the signal contains at least 5 structural dominant frequency components, i.e., the true number of modes of the high arch dam structure in the multivariate variational mode decomposition. The value should be calculated starting from 5, and increased by increasing the number of structural modes. The values ​​are used to calculate the peak frequency requirement and signal decomposition energy difference for high arch dam structures. The conditional secondary penalty factor. For example... Figure 10 As shown in Figure (d), the modal number of a high arch dam structure is optimized by combining the peak method and the signal decomposition energy difference method. The normalized information coefficients of different quadratic penalty factors under the optimal modal number of the high arch dam structure are calculated, and the optimization solution curve of the quadratic penalty factor is obtained. Figure 10 As shown in (d), the minimum value of the normalized information coefficient corresponds to a quadratic penalty factor of 13100, and the number of modes of the high arch dam structure is 7 at this time. Based on parameter adaptive optimization, the number of modes and the quadratic penalty factor of the high arch dam structure are determined. By solving the constrained variational model of multi-channel signal decomposition, the intrinsic mode components of the multi-channel vibration signal of the high arch dam are obtained.

[0135] like Figures 11-12As shown, the time history curves of the dynamic displacement of the intrinsic mode components and the normalized power spectral density of measuring points B1 and B7 are presented; taking IMF3 as an example, as... Figure 13 As shown, the free decay response of IMF3 is extracted by random decrement method (RDT) and Hilbert transform, and the logarithmic fitting curve of amplitude is obtained. By analyzing the relationship between amplitude and phase with time, the natural frequency and damping ratio can be obtained, as shown in Table 1 below.

[0136] Table 1. Identification results of natural frequencies and damping ratios of arch dams

[0137] ;

[0138] The identification results of the present invention are compared with other existing methods, including the improved random subspace method (ISSI) and the improved frequency domain decomposition method (IFDD), and the results are shown in Table 2 below.

[0139] Table 2. Comparison of natural frequencies of arch dams obtained by different methods (unit: Hz)

[0140] ;

[0141] As shown in Table 2, when the modal order is 2, both the method of this invention and the ISSI method can identify the dense frequency components of the structure around 1.51 Hz, significantly outperforming the IFDD method. When the modal order is 6, the method of this invention and the IFDD method exhibit good performance in identifying high-order modal information, while the ISSI method cannot capture the high-frequency modal components of the structure. Therefore, compared to the ISSI and IFDD methods, the method of this invention can capture the modal parameter information of the arch dam more comprehensively and accurately, indicating that the method of this invention can effectively achieve accurate identification of the modal parameters of high arch dams under non-stationary conditions.

[0142] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A method for identifying modal parameters of high arch dams under non-stationary conditions, characterized in that, Includes the following steps: Step S1: Deploy multiple dynamic displacement sensors or acceleration sensors on the high arch dam to collect vibration response signals at different locations during the discharge process of the high arch dam, and obtain multi-channel vibration data of the high arch dam. Step S2: Process the multi-channel vibration data of the high arch dam based on the simplified empirical Fourier decomposition method to remove the non-stationary components of the vibration response signal of the high arch dam. The simplified empirical Fourier decomposition method is based on a simplified spectrum segmentation boundary strategy. It performs simplified segmentation of the power spectrum of the autoregressive AR model of the original vibration response signal, and then performs inverse Fourier transform on each segmentation interval, thereby separating the non-stationary vibration response signal component from the original vibration response signal. Step S3: The PAO-MVMD multivariate variational mode decomposition method based on parameter adaptive optimization is used to process the multi-channel vibration data of the high arch dam after removing non-stationary components, and the multi-channel intrinsic mode components of the high arch dam structure are obtained. The process by which the multivariate variational mode decomposition method PAO-MVMD, based on parameter adaptive optimization, obtains the multi-channel intrinsic mode components of a high arch dam structure is as follows: Step S31: An adaptive method for determining the number of modes of a high arch dam structure based on correlation variance contribution rate data-level fusion, using multi-channel vibration response signals as input for dynamic data fusion; Step S32: Calculate the fused signal The power spectral density of the AR model is obtained, and the frequency domain result of the target domain is obtained. The peak value method is used to pick the peak value of the target domain and obtain the corresponding number of peak values ​​and peak frequency. Step S33: Decompose the multi-channel vibration response signal using a quadratic penalty factor, calculate the center frequencies of different intrinsic mode components, filter the results using the peak-to-peak method, and calculate the signal decomposition energy difference. : ； In the above formula, Indicates the structure of the high arch dam. Energy of one IMF component; This represents the total energy of the vibration response signal collected by the i-th sensor on the high arch dam structure; This represents the true modal numbers of the high arch dam structure; The magnitude of the value is positively correlated with the degree of signal decomposition; if the signal decomposition energy difference value... A larger value indicates that the signal has been over-decomposed; if the signal decomposition energy difference is large... When the value approaches 0, it indicates that the signal decomposition is reasonable, and the optimal number of modes for the high arch dam structure can be determined accordingly. Step S34: Calculate the normalized mutual information coefficients of the structural components of the high arch dam under the optimal structural mode number. The optimal quadratic penalty factor is determined by minimizing the normalized mutual information coefficient. ； In the above formula, For the first c The mutual information coefficients of the IMF components i' and j' of each channel; Indicates the number of data channels; Indicates the number of IMF components in the high arch dam structure; Step S35: Based on the determined number of structural modes and the second-order penalty factor, obtain the multi-channel intrinsic modal components of the high arch dam structure by solving the constrained variational model of the multi-channel vibration response signal decomposition. : ； In the above formula, This represents the k-th intrinsic mode component of the g-th channel; This represents the center frequency of the k-th component; Represents the Lagrange multiplier for the g-th channel; Indicates a secondary penalty factor; This indicates the partial derivative, used to calculate the gradient of the variational constraint model; express The unilateral spectrum; express Norm; This represents the vibration response signal of the g-th data channel, where t represents the number of time-series vibration response signals. This represents the Lagrange multiplier operator for the g-th channel corresponding to the time sequence number t of the vibration response signal; This indicates that the vibration signal contains k-order structural feature information. ; Step S4: Combine the random decrement method and Hilbert transform to identify the modal parameters of the multi-channel intrinsic modal components of the decomposed high arch dam structure, and output the working modal parameters of the high arch dam structure under non-stationary conditions.

2. The method for identifying modal parameters of high arch dams under non-stationary conditions according to claim 1, characterized in that, Step S2 involves separating the non-stationary vibration response signal components from the original vibration response signal, including the following: Step S21: Power spectrum estimation based on autoregressive AR model for vibration response signal of high arch dam. Frequency domain processing was performed to obtain the AR model power spectrum of the vibration response signal of the high arch dam. : ； In the above formula, For frequencies in the AR model; is the variance of white noise; P is the order in the AR model parameters. Let k' be the order of the AR model. Let be the k'th autoregressive coefficient; e be the natural base; j be the imaginary unit, denoted as ; Step S22: Introduce a frequency cutoff strategy to segment the power spectrum of the AR model; Step S23: Use a zero-phase filter to filter the segmented signal and remove the frequency domain interval boundary. and Used as the cutoff frequency to construct a zero-phase filter bank : ； In the above formula, For the nth frequency; This refers to the Nth frequency; n is the number of frequencies; N is the total number of frequencies. By constructing a zero-phase filter bank, the Fourier spectrum of the vibration response signal is converted into the spectrum within the target frequency domain. : ； In the above formula, The Fourier spectrum value is obtained by performing a discrete Fourier transform on the vibration response signal. Step S24: Perform an inverse Fourier transform on the filtered vibration response signal to output the non-stationary component and the reconstructed signal. Perform an inverse Fourier transform on the vibration response signal after it has been filtered by the zero-phase filter to obtain the filtered signal. : ； Further obtain the reconstructed signal : ； In the above formula, K is the number of signal components of the high arch dam structure obtained by the simplified spectrum segmentation boundary strategy.

3. The method for identifying modal parameters of high arch dams under non-stationary conditions according to claim 2, characterized in that, The frequency cutoff strategy introduced in step S22 to segment the power spectrum of the AR model is as follows: First, the power spectrum of the AR model is divided into three intervals, including: the low-frequency interval of water flow noise and non-stationary components [0, ... ], the natural frequency range of high arch dam structures [ , ], and the high-frequency range of white noise [ Then, determine the low-frequency range [0, +∞] respectively. ] and high frequency range [ Does the frequency domain range [, +∞] contain peak values? If there are no peak values ​​in the low-frequency or high-frequency range, then the frequency domain range boundary... and Take respectively and If a peak exists in the low-frequency or high-frequency range, then the minimum point after the peak is found to determine the boundary of the frequency domain range. and : ； ； In the above formula, For frequency The power spectrum value of the AR model.

4. The method for identifying modal parameters of high arch dams under non-stationary conditions according to claim 3, characterized in that, The data dynamic fusion process described in step S31 is as follows: for Vibration response signal of each data channel , Let represent the vibration response signal of the g-th data channel, and t represent the time sequence number of the vibration response signal. The normalized signal is obtained through energy normalization. : ； In the above formula, The vibration response signal acquired by the i-th sensor; The total energy of the vibration response signal collected by the i-th sensor on the high arch dam structure is calculated by correlation. Will Discretized , Let be the total number of samples in the signal, and define the variance contribution rate of the q-th data point in the normalized signal. Represented as: ； In the above formula, and These represent the mean and variance of the normalized signal, respectively. The normalized signal of the q-th data; Based on the variance contribution rate of signals from different measurement points over a certain time period, the fusion coefficient of the q-th data collected by the i-th sensor is... Represented as: ； Further, the q-th data point after dynamic data fusion is obtained. Represented as: ； Thus, the fused signal is obtained. .

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