Automatic modal parameter identification method based on stochastic subspace method and DBSCAN clustering

Abstract

The application discloses an automatic modal parameter identification method based on a random subspace method and DBSCAN clustering, calculates average regularized power spectral density based on structural dynamic response signals to obtain frequency domain features of structural responses, performs modal calculation by using a covariance random subspace identification method, and jointly evaluates model orders by using a toplyiz matrix information entropy increment and a modified Akaike information criterion, determines a maximum model order of random subspace identification, and generates a stability diagram, extracts candidate modal poles, and constructs an improved normalized modal distance measurement which comprehensively considers modal frequencies, damping ratios and modal shapes, determines a density clustering parameter range by analyzing near-neighbor modal distance distribution between candidate modal samples, performs clustering analysis under different parameter combinations to automatically determine an optimal clustering parameter combination, filters candidate modes based on modal energy and clustering cluster dimensions, and obtains structural representative modal parameters by combining modal confidence criteria and modal overlap coefficients to judge modal consistency.

Description

Technical Field

[0001] This invention relates to the field of structural engineering and structural health monitoring technology, and particularly to an automatic modal parameter identification method based on the random subspace method and DBSCAN clustering. Specifically, this invention provides a technical method for automated operational modal analysis of structures, applicable to the identification of structural dynamic characteristic parameters and the assessment of structural operating status under environmental excitation conditions. Background Technology

[0002] Operational Modal Analysis (OMA) is a technique for identifying the dynamic characteristics of a structure by analyzing its vibration response signals under environmental excitations. It can be used to obtain key dynamic parameters such as the structure's natural frequencies, damping ratios, and mode shapes. Because it does not require external artificial excitation, this method is widely used in health monitoring and operational status assessment of engineering structures such as bridges, long-span spatial structures, and building structures. Among these methods, Stochastic Subspace Identification (SSI) has become one of the commonly used modal identification techniques in OMA due to its good numerical stability and identification accuracy.

[0003] However, in practical applications, the stable graphs obtained from random subspace identification often contain a large number of spurious modes caused by noise or computational errors, and the selection of physical modes still largely relies on human experience. Meanwhile, automatic identification methods such as density clustering still suffer from problems in practical applications, including reliance on experience for model order selection, difficulty in adaptively determining clustering parameters, and challenges in identifying mode splitting under complex modal conditions, thus affecting the stability and reliability of automatic modality identification. Therefore, it is necessary to propose a modality identification computational framework that can adaptively (fully automatically) determine key aspects such as model order determination, key clustering parameters, and true / false mode identification, thereby achieving fully automatic modality parameter identification. Summary of the Invention

[0004] The purpose of this invention is to provide an automatic modal parameter identification method and system based on the random subspace method and DBSCAN clustering, to solve the problems of relying on manual experience for modal selection, significant noise interference, and difficulty in reasonably determining the model order in existing structural operational modal analysis. In structural operational modal analysis, the covariance random subspace identification method can effectively identify the natural frequencies, damping ratios, and mode shapes of a structure. However, in practical applications, random subspace identification algorithms often require manual determination of the model order and reliance on manual selection of poles in the stability plot. This not only increases the analysis cost but may also lead to instability in the modal identification results.

[0005] To address the aforementioned problems, this invention proposes an automatic modal parameter identification method based on the random subspace method and DBSCAN clustering. This method first calculates the average regularized power spectral density based on the structural dynamic response signal to obtain the frequency domain characteristics of the structural response. Then, it uses the covariance random subspace identification method to perform modal calculations and jointly evaluates the model order using the Toplitz matrix information entropy increment and the modified Akaike information criterion, thereby automatically determining the maximum model order for random subspace identification and generating a stability graph. Based on the stability graph, candidate modal poles are extracted, and an improved normalized modal distance metric that comprehensively considers modal frequency, damping ratio, and mode shape information is constructed. The parameter range of the density clustering algorithm is determined by analyzing the nearest neighbor modal distance distribution among candidate modal samples, and clustering analysis is performed under different parameter combinations. The optimal clustering parameter combination is automatically determined through clustering quality evaluation indicators. After completing the candidate mode clustering, this invention further uses mode energy and cluster dimension to filter the clustering results, thereby automatically identifying candidate physical modes; at the same time, it evaluates mode consistency through mode confidence criteria and mode overlap coefficient, thereby suppressing mode splitting phenomenon and finally determining the representative mode of the structure.

[0006] A first aspect of the present invention is to provide an automatic modal parameter identification method based on the random subspace method and DBSCAN clustering, comprising:

[0007] S1, acquire the structural dynamic response signal of the structure under test, and calculate the average regularized power spectral density of the signal in each test channel;

[0008] S2, based on the structural dynamic response signal, the covariance random subspace identification method is used to perform modal calculation, and the model order is initially estimated by the information entropy increment of the Toplitz matrix.

[0009] S3 evaluates the model order based on the modified Akaike information criterion and automatically determines the maximum model order for random subspace identification by combining the information entropy increment, thereby obtaining a stable graph for modal pole formation;

[0010] S4, extract candidate modal poles from the stable graph, and determine the minimum neighborhood sample number parameter range of the density clustering algorithm based on the candidate modal feature dimension and the total number of samples;

[0011] S5. Taking into account modal frequencies, damping ratios, and mode shape information, an improved normalized modal distance metric is constructed between candidate modes. The improved normalized modal distance is composed of eigenvalue difference terms and mode shape difference terms. Based on the fact that the thresholds of both the eigenvalue distance component and the mode shape distance component are [0,1], the defined modal distance is strictly bounded to the interval [0,1]. When the poles of any two candidate modes are more similar, their modal distance is closer to 0. When the difference between the two is greater, their modal distance is closer to 1. The bounded improved normalized modal distance can uniformly characterize the difference between eigenvalue and mode shape information, and at the same time maintain the consistency of numerical range with the parameter identification process of subsequent density clustering analysis, thereby improving the stability and physical interpretability of clustering analysis.

[0012] S6, calculate the nearest neighbor distance distribution between candidate modes based on the modal distance, and determine the candidate range of the neighborhood radius parameter of the density clustering algorithm;

[0013] S7. Perform density clustering analysis by traversing different parameter combinations within the parameter range, and determine the optimal clustering parameter combination based on the clustering quality evaluation index.

[0014] S8. Using the optimal clustering parameter combination, cluster analysis is performed on the candidate modal poles to obtain a set of stable modes after removing noise interference.

[0015] S9 filters the stable mode set based on modal energy characteristics and cluster dimension, and judges modal consistency through modal confidence criteria and modal overlap coefficient, thereby obtaining representative modal parameters of the median damping ratio index.

[0016] Preferably, S1 includes:

[0017] Calculate the average regularized power spectral density The calculation is shown in equation (1):

[0018] (1);

[0019] (2);

[0020] In the formula, Discrete frequency; Number of test channels; For the first Power spectral density of each test channel; For the first Normalized power spectral density of each test channel; For the first The maximum power spectral density of each test channel.

[0021] Preferably, S2 includes:

[0022] S21, parameter settings, including:

[0023] Based on the fundamental frequency of the structure and sampling frequency Determined based on rules of thumb The experience value (defaulting to the lower limit) is:

[0024] (3);

[0025] S22, for a given parameter Calculate the constructed Toplitz matrix Information entropy increment :

[0026] (4);

[0027] (5);

[0028] (6);

[0029] In the formula, The first singular value decomposition of the Topletz matrix is ​​obtained by... One singular value; For the first The normalized weights corresponding to the singular values; For the front The information entropy corresponding to each singular value; Index for model order. Maximum model order. It can be represented as:

[0030] (7);

[0031] In the formula, The derivative of the information entropy increment. This is the asymptotic threshold at which the derivative of the information entropy increment approaches zero; the default value is 1e-6.

[0032] Preferably, S3 includes:

[0033] S31, according to the Tollitz matrix The singular value decomposition results are approximated by the mean of the squared residuals, and the order of computation is [missing information]. Model residual variance :

[0034] (8);

[0035] In the formula, The order is The residual variance of the model under the following conditions For matrix The One eigenvalue;

[0036] S32, Calculation of Akaike Information Criteria :

[0037] (9);

[0038] In the formula, The approximate number of parameters for the state-space model ( ), This represents the total number of test data samples.

[0039] S33, Calculate the corrected Akaike information criterion :

[0040] (10);

[0041] Calculate the minimum Corresponding model order :

[0042] (11);

[0043] S34, Automatically determine the maximum model order of the covariance random subspace algorithm:

[0044] The model order is automatically determined based on two methods: information entropy increment and modified Akaike information criterion. The even number corresponding to the larger of the two values ​​is taken as the maximum model order of the covariance random subspace method.

[0045] (12);

[0046] In the formula, Take even numbers.

[0047] Preferably, S4 includes:

[0048] Let the set of all pole samples in the frequency stability plot be:

[0049] (13);

[0050] In the formula, The total number of modal pole samples to be clustered; The feature dimension of the sample is defined as follows:

[0051] (14);

[0052] The range of values ​​for the minimum neighborhood sample number Mp in density clustering is:

[0053] (15).

[0054] Preferably, S5 includes:

[0055] Calculate the modal distance between any two poles in the stability graph. The calculation formula is as follows:

[0056] (16);

[0057] In this invention, the modal distance is composed of both eigenvalue difference terms and mode shape difference terms. Because the eigenvalue distance component ( ) and mode distance component The threshold values ​​are all [0,1], therefore the defined modal distance The modal distance is strictly bounded to the interval [0,1]. The more similar the poles of any two candidate modes, the closer their modal distance is to 0; the greater the difference between them, the closer their modal distance is to 1. This bounded modal distance can not only uniformly characterize the difference between eigenvalues ​​and mode shapes, but also maintain numerical consistency with the subsequent k-dist-based density clustering parameter identification process, thus improving the stability and physical interpretability of the clustering analysis.

[0058] In equation (16), and These refer to the j-th and k-th mode shape vectors, respectively. and These refer to the eigenvalues ​​of the j-th and k-th continuous state-space equations, respectively, and their calculation expressions are as follows:

[0059] (17);

[0060] In equation (16), This refers to extending the Modal Mode Confidence Criterion (MAC) from the traditional real modal space to the complex modal space, and its calculation formula is as follows:

[0061] (18);

[0062] In the formula, yes conjugate, yes transpose, yes The conjugate transpose of .

[0063] Preferably, S6 includes:

[0064] S61, for For each value in the graph, the k-th modal distance between each data point and its k-th nearest neighbor is calculated based on the modal distance metric between any two poles in the stable graph (Formula (16)), and the data points are arranged in descending order to form a k-dist graph.

[0065] S62, For the data in the k-dist graph, perform clustering using binary K-means; wherein the basic calculation formula for binary K-means clustering is shown in the following formula (19):

[0066] (19);

[0067] In the formula, and They represent potential physical modes and spurious modes, respectively. This represents the modal distance of the k-th nearest neighbor dataset in the one-dimensional modal distance dataset calculated according to equation (16). The initial ideal cluster centers that can be objectively determined are 0 and 1, respectively, representing the k-th mode distance between the physical mode and the spurious mode.

[0068] S63, for any Mp, using equation (19) for bisection k-means clustering, automatically identify the knee point of the k-dist graph (the intersection point of the two clusters in the bisection clustering result), i.e., the corresponding... ;Calculate the value of Mp by traversing As a result, The range of values ​​for the parameter combination.

[0069] Preferably, S7 includes:

[0070] S71, for each group Density clustering results under parameter combinations, and calculation of clustering quality evaluation indicators. :

[0071] (20);

[0072] In the formula, Refers to sample points The average distance to other points in the same cluster. Sample points The average distance to all points in the nearest neighbor cluster, where:

[0073] (twenty one);

[0074] In the formula, It is the extreme point The cluster it belongs to, ( =[1,2]) is the distance from the pole. Recent clustering, It is the two poles and The Euclidean distance between them. The clustering quality evaluation metric is calculated by analyzing all extreme point samples. The value is obtained by averaging, and its range is [-1, 1]. The closer the value is to 1, the better the clustering quality.

[0075] S72, traverse all parameters within the preset parameter space Density clustering under combination automatically selects the parameter combination that maximizes the clustering quality evaluation index as the optimal solution;

[0076] S73, based on optimal parameter combination Density clustering is performed using normalized modal distance metrics to obtain a stable map with noise removed.

[0077] Preferably, S8 includes:

[0078] S81, derived from normalized modal energy and cluster dimension The initial cluster centers of the two-dimensional input sample dataset are objectively determined, namely, the initial cluster centers of the two-dimensional samples representing the physical mode and the spurious mode are vectors {1,1} and {0,0}, respectively.

[0079] The calculation formula is shown in equation (22):

[0080] (twenty two);

[0081] In the formula, ; For matrix A ( ) One right eigenvector; For matrix The OK; The system matrix in continuous state space The 1 eigenvalue, ; For conjugate operations; matrices C and G are the output matrix and the output covariance matrix of the next state, respectively, where matrix C is the expression... The former Line, matrix G can be extended by a controllable matrix. After Column determined;

[0082] S82, the iterative optimization process based on the binary K-means clustering retains candidate polar axes with high cluster dimensions and modal energy, thereby automatically filtering the clusters in which candidate physical modes reside.

[0083] Preferably, S9 includes:

[0084] S91, determine that the modal confidence criterion (MAC) and modal overlap coefficient (MOF) are used as indicators to quantify the degree of overlap between any two modes, wherein the modal confidence criterion (MAC) is quantified from the perspective of mode shape coefficients, and the modal overlap coefficient (MOF) is quantified from the perspective of frequency and damping ratio;

[0085] The modal confidence criterion (MAC) and the modal overlap coefficient (MOF) are calculated as shown in equations (23) and (24), respectively:

[0086] (twenty three);

[0087] In the formula, H represents the Hermitian transpose. and Representing the first and the First-order mode shape;

[0088] (twenty four);

[0089] In the formula, and These represent the frequencies of a pair of modes to be quantized. Refers to the first Damping ratio of the first mode;

[0090] S92, to avoid modal splitting, this invention uses the Modal Confidence Criterion (MAC) and Modal Overlap Factor (MOF) to evaluate the consistency of candidate modes. When the following conditions are met... and In this case, the candidate modality cluster with the smaller cluster dimension will be eliminated.

[0091] S93, Extract representative modes indexed by the median damping ratio and perform modal parameter identification based on the representative modes. The modal parameter identification includes frequency, damping ratio, and linearly normalized modal energy. and cluster dimension .

[0092] A second aspect of the present invention provides an automatic modal parameter identification system based on the random subspace method and DBSCAN clustering, for implementing the method of the first aspect, comprising:

[0093] The structural dynamic response acquisition and frequency domain feature calculation module (101) is used to perform spectrum calculation based on the acquired structural dynamic response of each channel;

[0094] The random subspace modal calculation module (102) is used to initially identify modes and perform modal calculations on the structural dynamic response based on the time-domain covariance random subspace method to obtain the original stability diagram;

[0095] The automatic model order determination module (103) based on information entropy increment is used to automatically determine the key parameters of the covariance random subspace method and the maximum model order.

[0096] The minimum neighborhood sample number range determination module (104) is used to take all the poles in the stable graph as input samples for the DBSCAN density clustering, and determine the minimum neighborhood sample number range for density clustering based on the total number of samples and the dimension of the modality features to be clustered.

[0097] An improved normalized modal distance determination module (105) is used to comprehensively consider frequency, damping ratio and mode shape to calculate the normalized modal distance between any two poles in the stability diagram;

[0098] The module (106) for determining the range of combinations of minimum neighborhood sample number and neighborhood radius parameters is used to calculate the neighborhood radius under different minimum neighborhood sample number conditions, thereby forming the range of combinations of minimum neighborhood sample number and neighborhood radius parameters.

[0099] The density clustering parameter optimization module (107) is used to substitute the optimal parameter combination and the normalized modal distance into the density clustering algorithm to obtain a stable map with noise removed.

[0100] Candidate physical mode screening module (108) is used for mode energy based on linear normalization. and cluster dimension Multiple sets of two-dimensional input samples are formed, and the multiple sets of two-dimensional input samples are substituted into binary K-means clustering to automatically filter the cluster in which the candidate physical mode belongs.

[0101] The modal consistency discrimination module (109) is used to effectively evaluate whether the modes exhibiting modal splitting belong to the same order based on the synergistic effect of the modal confidence criterion (MAC) and the modal overlap coefficient (MOF). The MAC is evaluated from the perspective of mode shape coefficients, and the MOF is evaluated from the perspective of frequency and damping ratio. Finally, representative modes indexed by the median damping ratio are extracted.

[0102] A third aspect of the present invention provides an electronic device including a processor and a memory, the memory storing a plurality of instructions, the processor being configured to read the instructions and execute the method as described in the first aspect.

[0103] A fourth aspect of the present invention provides a computer-readable storage medium storing a plurality of instructions which can be read by a processor and executed as described in the first aspect.

[0104] The beneficial effects of the method and system of the present invention are as follows:

[0105] (1) By combining the information entropy increment with the modified Akaike information criterion, the order of the random subspace identification model is automatically determined;

[0106] (2) Construct an improved normalized modal distance metric that comprehensively considers modal frequency, damping ratio and mode shape information to improve the rationality of modal similarity judgment;

[0107] (3) Automatic clustering analysis of candidate modes is achieved through an adaptive selection mechanism for density clustering parameters;

[0108] (4) Candidate modes are screened in multiple layers by modal energy, cluster dimension and modal consistency index to improve the stability and accuracy of modal recognition.

[0109] Therefore, this invention can automatically identify structural modal parameters under complex environmental excitation conditions, thereby improving the automation level of structural modal analysis. Attached Figure Description

[0110] To more clearly illustrate the technical solutions in the specific embodiments or related technologies of the present invention, the drawings used in the description of the specific embodiments or related technologies will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0111] Figure 1 This is a 4-DOF mass-spring-damping system according to Embodiment 1 of the present invention;

[0112] Figure 2 Embodiment 1 of the present invention automatically determines the order of the covariance random subspace model based on the information entropy increment;

[0113] Figure 3 In Embodiment 1 of this invention, the order of the covariance random subspace model is automatically determined based on the modified Akaike information criterion.

[0114] Figure 4 This is the k-dist graph that is traversed and calculated for different k values ​​in Embodiment 1 of the present invention;

[0115] Figure 5 This is different from Embodiment 1 of the present invention. Clustering quality evaluation index diagram under parameter combinations;

[0116] Figure 6 This is the frequency-damping ratio diagram obtained by density clustering with optimal clustering parameters in Embodiment 1 of the present invention;

[0117] Figure 7 This is an image showing the automatic screening results of candidate modality clustering in Embodiment 1 of the present invention;

[0118] Figure 8 This is the MAC diagram of all candidate modes in Embodiment 1 of the present invention;

[0119] Figure 9 This is the MOF diagram of all candidate modes in Embodiment 1 of the present invention;

[0120] Figure 10 This is the final stable graph of the fully automatic modal recognition in Embodiment 1 of the present invention;

[0121] Figure 11 This is a relative error diagram of all 4th-order modal parameters identified in Embodiment 1 of the present invention;

[0122] Figure 12 This refers to the steel truss pedestrian bridge in Embodiment 2 of the present invention;

[0123] Figure 13 This is a sensor arrangement diagram of the steel truss pedestrian bridge in Embodiment 2 of the present invention;

[0124] Figure 14 This is the k-dist graph with different k values ​​that is traversed and calculated in Embodiment 2 of the present invention;

[0125] Figure 15 This is different in Embodiment 2 of the present invention. Clustering quality evaluation index diagram under parameter combinations;

[0126] Figure 16 This is the final stable graph of the fully automatic modal recognition in Embodiment 2 of the present invention;

[0127] Figure 17 This is a system architecture diagram of an automatic modal parameter identification system based on improved density provided in an embodiment of the present invention;

[0128] Figure 18 This is a structural diagram of an electronic device provided according to an embodiment of the present invention. Detailed Implementation

[0129] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0130] In the description of this invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0131] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0132] A first aspect of the present invention is to provide an automatic modal parameter identification method based on the random subspace method and DBSCAN clustering, comprising:

[0133] S1, acquire the structural dynamic response signal of the structure under test, and calculate the average regularized power spectral density of the signal in each test channel;

[0134] S2, based on the structural dynamic response signal, the covariance random subspace identification method is used to perform modal calculation, and the model order is initially estimated by the information entropy increment of the Toplitz matrix.

[0135] S3 evaluates the model order based on the modified Akaike information criterion and automatically determines the maximum model order for random subspace identification by combining the information entropy increment, thereby obtaining a stable graph for modal pole formation;

[0136] S4, extract candidate modal poles from the stable graph, and determine the minimum neighborhood sample number parameter range of the density clustering algorithm based on the candidate modal feature dimension and the total number of samples;

[0137] S5. Taking into account modal frequencies, damping ratios, and mode shape information, an improved normalized modal distance metric is constructed between candidate modes. The improved normalized modal distance is composed of eigenvalue difference terms and mode shape difference terms. Based on the fact that the thresholds of both the eigenvalue distance component and the mode shape distance component are [0,1], the defined modal distance is strictly bounded to the interval [0,1]. When the poles of any two candidate modes are more similar, their modal distance is closer to 0. When the difference between the two is greater, their modal distance is closer to 1. The bounded improved normalized modal distance can uniformly characterize the difference between eigenvalue and mode shape information, and at the same time maintain the consistency of numerical range with the parameter identification process of subsequent density clustering analysis, thereby improving the stability and physical interpretability of clustering analysis.

[0138] S6, calculate the nearest neighbor distance distribution between candidate modes based on the modal distance, and determine the candidate range of the neighborhood radius parameter of the density clustering algorithm;

[0139] S7. Perform density clustering analysis by traversing different parameter combinations within the parameter range, and determine the optimal clustering parameter combination based on the clustering quality evaluation index.

[0140] S8. Using the optimal clustering parameter combination, cluster analysis is performed on the candidate modal poles to obtain a set of stable modes after removing noise interference.

[0141] S9 filters the stable mode set based on modal energy characteristics and cluster dimension, and judges modal consistency through modal confidence criteria and modal overlap coefficient, thereby obtaining representative modal parameters of the median damping ratio index.

[0142] In a preferred embodiment, S1 includes:

[0143] Calculate the average regularized power spectral density The calculation is shown in equation (1):

[0144] (1);

[0145] (2);

[0146] In the formula, Discrete frequency; Number of test channels; For the first Power spectral density of each test channel; For the first Normalized power spectral density of each test channel; For the first The maximum power spectral density of each test channel.

[0147] In a preferred embodiment, S2 includes:

[0148] S21, parameter settings, including:

[0149] Based on the fundamental frequency of the structure and sampling frequency Determined based on rules of thumb The experience value (defaulting to the lower limit) is:

[0150] (3);

[0151] S22, for a given parameter Calculate the constructed Toplitz matrix Information entropy increment :

[0152] (4);

[0153] (5);

[0154] (6);

[0155] In the formula, The first singular value decomposition of the Topletz matrix is ​​obtained by... One singular value; For the first The normalized weights corresponding to the singular values; For the front The information entropy corresponding to each singular value; Index for model order. Maximum model order. It can be represented as:

[0156] (7);

[0157] In the formula, The derivative of the information entropy increment. This is the asymptotic threshold at which the derivative of the information entropy increment approaches zero; the default value is 1e-6.

[0158] In a preferred embodiment, S3 includes:

[0159] S31, according to the Tollitz matrix The singular value decomposition results are approximated by the mean of the squared residuals, and the order of computation is [missing information]. Model residual variance :

[0160] (8);

[0161] In the formula, The order is The residual variance of the model under the following conditions For matrix The One eigenvalue;

[0162] S32, Calculation of Akaike Information Criteria :

[0163] (9);

[0164] In the formula, The approximate number of parameters for the state-space model ( ), This represents the total number of test data samples.

[0165] S33, Calculate the corrected Akaike information criterion :

[0166] (10);

[0167] Calculate the minimum Corresponding model order :

[0168] (11);

[0169] S34, Automatically determine the maximum model order of the covariance random subspace algorithm:

[0170] The model order is automatically determined based on two methods: information entropy increment and modified Akaike information criterion. The even number corresponding to the larger of the two values ​​is taken as the maximum model order of the covariance random subspace method.

[0171] (12);

[0172] In the formula, Take even numbers.

[0173] In a preferred embodiment, S4 includes:

[0174] Let the set of all pole samples in the frequency stability plot be:

[0175] (13);

[0176] In the formula, The total number of modal pole samples to be clustered; The feature dimension of the sample is defined as follows:

[0177] (14);

[0178] The range of values ​​for the minimum neighborhood sample number Mp in density clustering is:

[0179] (15).

[0180] In a preferred embodiment, S5 includes:

[0181] Calculate the modal distance between any two poles in the stability graph. The calculation formula is as follows:

[0182] (16);

[0183] In this invention, the modal distance is composed of both eigenvalue difference terms and mode shape difference terms. Because the eigenvalue distance component ( ) and mode distance component The threshold values ​​are all [0,1], therefore the defined modal distance The modal distance is strictly bounded to the interval [0,1]. The more similar the poles of any two candidate modes, the closer their modal distance is to 0; the greater the difference between them, the closer their modal distance is to 1. This bounded modal distance can not only uniformly characterize the difference between eigenvalues ​​and mode shapes, but also maintain numerical consistency with the subsequent k-dist-based density clustering parameter identification process, thus improving the stability and physical interpretability of the clustering analysis.

[0184] In equation (16), and These refer to the j-th and k-th mode shape vectors, respectively. and These refer to the eigenvalues ​​of the j-th and k-th continuous state-space equations, respectively, and their calculation expressions are as follows:

[0185] (17);

[0186] In equation (16), This refers to extending the Modal Mode Confidence Criterion (MAC) from the traditional real modal space to the complex modal space, and its calculation formula is as follows:

[0187] (18);

[0188] In the formula, yes conjugate, yes transpose, yes The conjugate transpose of .

[0189] In a preferred embodiment, S6 includes:

[0190] S61, for For each value in the graph, the k-th modal distance between each data point and its k-th nearest neighbor is calculated based on the modal distance metric between any two poles in the stable graph (Formula (16)), and the data points are arranged in descending order to form a k-dist graph.

[0191] S62, For the data in the k-dist graph, perform clustering using binary K-means; wherein the basic calculation formula for binary K-means clustering is shown in the following formula (19):

[0192] (19);

[0193] In the formula, and They represent potential physical modes and spurious modes, respectively. This represents the modal distance of the k-th nearest neighbor dataset in the one-dimensional modal distance dataset calculated according to equation (16). The initial ideal cluster centers that can be objectively determined are 0 and 1, respectively, representing the k-th mode distance between the physical mode and the spurious mode.

[0194] S63, for any Mp, using equation (19) for bisection k-means clustering, automatically identify the knee point of the k-dist graph (the intersection point of the two clusters in the bisection clustering result), i.e., the corresponding... ;Calculate the value of Mp by traversing As a result, The range of values ​​for the parameter combination.

[0195] In a preferred embodiment, S7 includes:

[0196] S71, for each group Density clustering results under parameter combinations, and calculation of clustering quality evaluation indicators. :

[0197] (20);

[0198] In the formula, Refers to sample points The average distance to other points in the same cluster. Sample points The average distance to all points in the nearest neighbor cluster, where:

[0199] (twenty one);

[0200] In the formula, It is the extreme point The cluster it belongs to, ( =[1,2]) is the distance from the pole. Recent clustering, It is the two poles and The Euclidean distance between them. The clustering quality evaluation metric is calculated by analyzing all extreme point samples. The value is obtained by averaging, and its range is [-1, 1]. The closer the value is to 1, the better the clustering quality.

[0201] S72, traverse all parameters within the preset parameter space Density clustering under combination automatically selects the parameter combination that maximizes the clustering quality evaluation index as the optimal solution;

[0202] S73, based on optimal parameter combination Density clustering is performed using normalized modal distance metrics to obtain a stable map with noise removed.

[0203] In a preferred embodiment, S8 includes:

[0204] S81, derived from normalized modal energy and cluster dimension The initial cluster centers of the two-dimensional input sample dataset are objectively determined, namely, the initial cluster centers of the two-dimensional samples representing the physical mode and the spurious mode are vectors {1,1} and {0,0}, respectively.

[0205] The calculation formula is shown in equation (22):

[0206] (twenty two);

[0207] In the formula, ; For matrix A ( ) One right eigenvector; For matrix The OK; The system matrix in continuous state space The 1 eigenvalue, ; For conjugate operations; matrices C and G are the output matrix and the output covariance matrix of the next state, respectively, where matrix C is the expression... The former Line, matrix G can be extended by a controllable matrix. After Column determined;

[0208] S82, the iterative optimization process based on the binary K-means clustering retains candidate polar axes with high cluster dimensions and modal energy, thereby automatically filtering the clusters in which candidate physical modes reside.

[0209] In a preferred embodiment, S9 includes:

[0210] S91, determine that the modal confidence criterion (MAC) and modal overlap coefficient (MOF) are used as indicators to quantify the degree of overlap between any two modes, wherein the modal confidence criterion (MAC) is quantified from the perspective of mode shape coefficients, and the modal overlap coefficient (MOF) is quantified from the perspective of frequency and damping ratio;

[0211] The modal confidence criterion (MAC) and the modal overlap coefficient (MOF) are calculated as shown in equations (23) and (24), respectively:

[0212] (twenty three);

[0213] In the formula, H represents the Hermitian transpose. and Representing the first and the First-order mode shape;

[0214] (twenty four);

[0215] In the formula, and These represent the frequencies of a pair of modes to be quantized. Refers to the first Damping ratio of the first mode;

[0216] S92, to avoid modal splitting, this invention uses the Modal Confidence Criterion (MAC) and Modal Overlap Factor (MOF) to evaluate the consistency of candidate modes. When the following conditions are met... and In this case, the candidate modality cluster with the smaller cluster dimension will be eliminated.

[0217] S93, Extract representative modes indexed by the median damping ratio and perform modal parameter identification based on the representative modes. The modal parameter identification includes frequency, damping ratio, and linearly normalized modal energy. and cluster dimension .

[0218] Application Examples

[0219] Automatic modal parameter identification methods based on random subspace method and DBSCAN clustering include:

[0220] S1, based on the structural dynamic response obtained from the structure under test, calculate the frequency domain average regularized power spectral density of the signal in each test channel;

[0221] S2, based on the time-domain covariance random subspace method, performs modal calculations on the structural dynamic response, and preliminarily automatically determines the maximum model order of the covariance random subspace method by calculating the asymptotic value of the information entropy increment of the Toplitz matrix.

[0222] S3. Based on the modified Akaike Information Criterion (AICC) and the information entropy increment, the maximum model order of the covariance random subspace method is determined, and the frequency stability diagram is obtained by substituting it into the covariance random subspace method.

[0223] S4, take all the poles in the frequency stability graph as input samples for the density clustering, and define the range of the minimum neighborhood sample number Mp for density clustering according to the candidate modality feature dimension and the total number of samples.

[0224] S5, taking into account frequency, damping ratio, and mode shape, calculate the normalized modal distance between any two poles in the stability graph after the initial cleaning; the improved normalized modal distance is composed of eigenvalue difference term and mode shape difference term. Based on the fact that the thresholds of the eigenvalue distance component and the mode shape distance component are both [0,1], the defined modal distance is strictly bounded to the interval [0,1]. When any two candidate modal poles are more similar, their modal distance is closer to 0; when the difference between the two is greater, their modal distance is closer to 1. The bounded improved normalized modal distance can uniformly characterize the difference between the two types of modal information, namely eigenvalue and mode shape, and at the same time, it can maintain the consistency of numerical range with the parameter identification process of subsequent density clustering analysis, thereby improving the stability and physical interpretability of clustering analysis.

[0225] S6. By calculating the modal distance between each pole and its k-th nearest neighbor, and based on the distribution of the modal distances, binary K-means clustering is used to determine the neighborhood radius value under the corresponding k-dist graph. Thus forming The range of values ​​for parameter combinations;

[0226] S7, calculates different values ​​by traversing... Density clustering with parameter combinations automatically selects the parameter combination that yields the highest clustering quality evaluation index as the optimal density clustering parameters. And by substituting the normalized modal distance metric into the density clustering algorithm, a stable graph with noise removed is obtained;

[0227] S8, Modal Energy Based on Linear Normalization and cluster dimension Multiple sets of two-dimensional input samples are formed, and the multiple sets of two-dimensional input samples are substituted into binary K-means clustering to automatically filter the cluster in which the candidate physical mode belongs.

[0228] S9. Based on the synergistic effect of the Modal Confidence Criterion (MAC) and the Modal Overlap Factor (MOF), it is effectively assessed whether modes exhibiting modal splitting belong to the same order. The MAC is evaluated from the perspective of mode shape coefficients, and the MOF is evaluated from the perspectives of frequency and damping ratio. Finally, the modal parameter features of the median damping ratio index are extracted as representative modes for modal identification. These modal parameter identification features include frequency, damping ratio, and linearly normalized modal energy. and cluster dimension .

[0229] To verify the effectiveness of the proposed automatic modal parameter identification method, a four-degree-of-freedom linear vibration system was constructed as a numerical example, such as... Figure 1 As shown. The system consists of four layers of mass-spring-damping units, with each floor having a mass of 10 kg, an inter-floor stiffness of 10000 N / m, and a damping coefficient of 10 N / (ms). In this simulation model, random excitation with Gaussian white noise is applied to each floor to simulate the dynamic response of the structure under environmental excitation. The output response signals of each degree of freedom of the system are denoted as... Data acquisition employed a 50 Hz sampling frequency and a sampling duration of 300 s to obtain output response time history data for operational modal analysis. The specific implementation steps for structural modal parameter identification are described below:

[0230] Based on the output response signals of each layer The structural dynamic response under environmental excitation is obtained.

[0231] The dynamic response signal is substituted into the covariance-driven stochastic subspace algorithm for modal calculation. The number of rows and blocks in the Toplitz matrix is ​​determined based on an empirical lower bound. =58, substituting this into the information entropy increment calculation, the maximum model order automatically determined is 50. Figure 2 Substituting the modified Akaike Information Criterion (AICC) into the calculation, the maximum model order automatically determined is only 18. Figure 3 Therefore, N increases by 2 from 2 to 50.

[0232] The stability graph is initially cleaned using hard index criteria, and the average regularized power spectral density of each degree of freedom is calculated based on the response signal and superimposed on the stability graph.

[0233] Based on the 202 candidate poles in the initial cleaned stability plot, k = Mp = [6, 18].

[0234] Calculate the improved normalized modal distance between any two poles in the stability graph.

[0235] The modal distance between each pole and its k-th nearest neighbor is calculated through iteration. Based on the distribution of modal distances, k-dist graphs are generated for different k values. Binary K-means clustering is then used to determine the neighborhood radius value for the corresponding k-dist graph. Thus forming Range of parameter combinations ( Figure 4 ).

[0236] Different traversal calculations Density clustering under parameter combinations: calculate the clustering quality evaluation index for the corresponding parameter combinations, and automatically select the parameter combination with the largest clustering quality evaluation index as the optimal density clustering parameters to obtain the clustering result. Figure 5 Among them, the four candidate modality clusters (legend "M") and their corresponding representative modalities (legend "RM") are distinguished by different colors, and noise points identified as false modalities are represented by black asterisks. Figure 6 ).

[0237] Calculate the normalized modal energy ( ) and cluster dimension ( The four sets of two-dimensional data, consisting of the two components, were substituted into the binary K-Means clustering algorithm. The initial ideal cluster centers for the physical modality (PM) and the spurious modality (SM) were {1,1} and {0,0}, respectively. All four candidate modality clusters were automatically identified as the physical modality. Figure 7 ).

[0238] The synergistic effect of the Modal Confidence Criterion (MAC) and the Modal Overlap Factor (MOF) is further used to control modal splitting, which did not occur in this example.

[0239] in Figure 8 The MAC diagram of all candidate modes in Embodiment 1 of the present invention is shown; Figure 9 The MOF diagrams of all candidate modes in Embodiment 1 of the present invention are shown; Figure 10 The final stable graph of fully automatic modal recognition in Embodiment 1 of the present invention is shown.

[0240] Table 1 lists the representative modal results for all candidate modes based on the median damping ratio, including frequency, damping ratio, MEL, and other parameters. By comparing the frequencies, damping ratios, and mode shapes of the four representative modes with their corresponding theoretical values, the errors in frequencies and mode shapes were extremely small (relative errors close to 0), and the average error in damping ratios was 9.63%. Figure 11 ).

[0241] Table 1

[0242]

[0243] Example 2

[0244] To further verify the applicability of the automatic modal parameter identification method proposed in this invention to actual engineering structures, the DH steel truss pedestrian bridge in the United States is used as a case study. Figure 12 As shown in the image, this bridge structure is one of the widely used vibration monitoring benchmark platforms in the field of structural health monitoring. The bridge is approximately 44m long and 3.7m wide. Eight vertical piezoelectric accelerometers are arranged along the bottom of the bridge structure, as shown in the image. Figure 13 As shown. This structural health monitoring system has been operating continuously since 2010, with a data sampling frequency of 128Hz. The monitoring system collects data using a timed triggering method, automatically recording structural vibration response data for the five minutes preceding each hour. This embodiment selects a sampling data point from the first week as the analysis sample to verify the automatic modal parameter identification method proposed in this invention.

[0245] The first six modal parameters of the structure were automatically identified, and the results are shown in Table 2, which lists the representative modal results. The identification results are largely consistent with the benchmark identification results for this benchmark. Specifically, the modal distance between each pole and its k-th nearest neighbor was automatically determined through traversal calculation. The range of parameter combinations is as follows: Figure 14 As shown, the parameter combination with the highest clustering quality evaluation index is automatically selected as the optimal density clustering parameter. Figure 15 As shown, the final frequency stability plot results are as follows: Figure 16 As shown.

[0246] Table 2

[0247]

[0248] like Figure 17 As shown, this embodiment provides an automatic modal parameter identification system based on the random subspace method and DBSCAN clustering, for implementing the corresponding method, including:

[0249] The structural dynamic response acquisition and frequency domain feature calculation module (101) is used to perform spectrum calculation based on the acquired structural dynamic response of each channel;

[0250] The random subspace modal calculation module (102) is used to initially identify modes and perform modal calculations on the structural dynamic response based on the time-domain covariance random subspace method to obtain the original stability diagram;

[0251] The automatic model order determination module (103) based on information entropy increment is used to automatically determine the key parameters of the covariance random subspace method and the maximum model order.

[0252] The minimum neighborhood sample number range determination module (104) is used to take all the poles in the stable graph as input samples for the DBSCAN density clustering, and determine the minimum neighborhood sample number range for density clustering based on the total number of samples and the dimension of the modality features to be clustered.

[0253] An improved normalized modal distance determination module (105) is used to comprehensively consider frequency, damping ratio and mode shape to calculate the normalized modal distance between any two poles in the stability diagram;

[0254] The module (106) for determining the range of combinations of minimum neighborhood sample number and neighborhood radius parameters is used to calculate the neighborhood radius under different minimum neighborhood sample number conditions, thereby forming the range of combinations of minimum neighborhood sample number and neighborhood radius parameters.

[0255] The density clustering parameter optimization module (107) is used to substitute the optimal parameter combination and the normalized modal distance into the density clustering algorithm to obtain a stable map with noise removed.

[0256] Candidate physical mode screening module (108) is used for mode energy based on linear normalization. and cluster dimension Multiple sets of two-dimensional input samples are formed, and the multiple sets of two-dimensional input samples are substituted into binary K-means clustering to automatically filter the cluster in which the candidate physical mode belongs.

[0257] The modal consistency discrimination module (109) is used to effectively evaluate whether the modes exhibiting modal splitting belong to the same order based on the synergistic effect of the modal confidence criterion (MAC) and the modal overlap coefficient (MOF). The MAC is evaluated from the perspective of mode shape coefficients, and the MOF is evaluated from the perspective of frequency and damping ratio. Finally, representative modes indexed by the median damping ratio are extracted.

[0258] The present invention also provides a memory that stores multiple instructions for implementing the method as described in Embodiment 1.

[0259] like Figure 18 As shown, the present invention also provides an electronic device, including a processor 301 and a memory 302 connected to the processor 301. The memory 302 stores a plurality of instructions, which can be loaded and executed by the processor to enable the processor to perform the methods as described in the application embodiment.

[0260] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. An automatic modal parameter identification method based on random subspace method and DBSCAN clustering, characterized in that, include: S1, acquire the structural dynamic response signal of the structure under test, and calculate the average regularized power spectral density of the signal in each test channel; S2, based on the structural dynamic response signal, the covariance random subspace identification method is used to perform modal calculation, and the model order is initially estimated by the information entropy increment of the Toplitz matrix. S3 evaluates the model order based on the modified Akaike information criterion and automatically determines the maximum model order for random subspace identification by combining the information entropy increment, thereby obtaining a stable graph for modal pole formation; S4, extract candidate modal poles from the stable graph, and determine the minimum neighborhood sample number parameter range of the density clustering algorithm based on the candidate modal feature dimension and the total number of samples; S5. Taking into account modal frequencies, damping ratios, and mode shape information, an improved normalized modal distance metric is constructed between candidate modes. The improved normalized modal distance is composed of eigenvalue difference terms and mode shape difference terms. Based on the fact that the thresholds of both the eigenvalue distance component and the mode shape distance component are [0,1], the defined modal distance is strictly bounded to the interval [0,1]. When the poles of any two candidate modes are more similar, their modal distance is closer to 0. When the difference between the two is greater, their modal distance is closer to 1. The bounded improved normalized modal distance can uniformly characterize the difference between eigenvalue and mode shape information, and at the same time maintain the consistency of numerical range with the parameter identification process of subsequent density clustering analysis, thereby improving the stability and physical interpretability of clustering analysis. S6, calculate the nearest neighbor distance distribution between candidate modes based on the modal distance, and determine the candidate range of the neighborhood radius parameter of the density clustering algorithm; S7. Perform density clustering analysis by traversing different parameter combinations within the parameter range, and determine the optimal clustering parameter combination based on the clustering quality evaluation index. S8. Using the optimal clustering parameter combination, cluster analysis is performed on the candidate modal poles to obtain a set of stable modes after removing noise interference. S9 filters the stable mode set based on modal energy characteristics and cluster dimension, and judges modal consistency through modal confidence criteria and modal overlap coefficient, thereby obtaining representative modal parameters of the median damping ratio index.

2. The automatic modal parameter identification method based on random subspace method and DBSCAN clustering according to claim 1, characterized in that, S1 includes: Calculate the average regularized power spectral density The calculation is shown in equation (1): （1）； （2）； In the formula, Discrete frequency; Number of test channels; For the first Power spectral density of each test channel; For the first Normalized power spectral density of each test channel; For the first The maximum power spectral density of each test channel.

3. The automatic modal parameter identification method based on random subspace method and DBSCAN clustering according to claim 2, characterized in that, S2 includes: S21, parameter settings, including: Based on the fundamental frequency of the structure and sampling frequency Determined based on rules of thumb The experience value (defaulting to the lower limit) is: （3）； S22, for a given parameter Calculate the constructed Toplitz matrix Information entropy increment : （4）； （5）； （6）； In the formula, The first singular value decomposition of the Topletz matrix is ​​obtained by... One singular value; For the first The normalized weights corresponding to the singular values; For the front The information entropy corresponding to each singular value; For model order index, the maximum model order. It can be represented as: （7）； In the formula, The derivative of the information entropy increment. This is the asymptotic threshold at which the derivative of the information entropy increment approaches zero; the default value is 1e-6.

4. The automatic modal parameter identification method based on random subspace method and DBSCAN clustering according to claim 3, characterized in that, S3 includes: S31, according to the Tollitz matrix The singular value decomposition results are approximated by the mean of the squared residuals, and the order of computation is [missing information]. Model residual variance : （8）； In the formula, The order is The residual variance of the model under the following conditions For matrix The One eigenvalue; S32, Calculation of Akaike Information Criteria : （9）； In the formula, The approximate number of parameters for the state-space model ( ), This represents the total number of test data samples. S33, Calculate the corrected Akaike information criterion : （10）； Calculate the minimum Corresponding model order : （11）； S34, Automatically determine the maximum model order of the covariance random subspace algorithm: The model order is automatically determined based on two methods: information entropy increment and modified Akaike information criterion. The even number corresponding to the larger of the two values ​​is taken as the maximum model order of the covariance random subspace method. （12）； In the formula, Take even numbers.

5. The automatic modal parameter identification method based on random subspace method and DBSCAN clustering according to claim 4, characterized in that, S4 includes: Let the set of all pole samples in the frequency stability plot be: （13）； In the formula, The total number of modal pole samples to be clustered; The feature dimension of the sample is defined as follows: （14）； The range of values ​​for the minimum neighborhood sample number Mp in density clustering is: （15）。 6. The automatic modal parameter identification method based on random subspace method and DBSCAN clustering according to claim 5, characterized in that, S5 includes: Calculate the improved normalized modal distance between any two poles in the stability graph. The calculation formula is as follows: （16）； In equation (16), and These refer to the j-th and k-th mode shape vectors, respectively. and These refer to the eigenvalues ​​of the j-th and k-th continuous state-space equations, respectively, and their calculation expressions are as follows: （17）； In equation (16), This refers to extending the Modal Mode Confidence Criterion (MAC) from the traditional real modal space to the complex modal space, and its calculation formula is as follows: （18）； In the formula, yes conjugate, yes transpose, yes The conjugate transpose of .

7. The automatic modal parameter identification method based on random subspace method and DBSCAN clustering according to claim 6, characterized in that, S6 includes: S61, for For each value in the graph, the k-th modal distance between each data point and its k-th nearest neighbor is calculated based on the modal distance metric between any two poles in the stable graph (Formula (16)), and the data points are arranged in descending order to form a k-dist graph. S62, For the data in the k-dist graph, perform clustering using binary K-means; wherein the basic calculation formula for binary K-means clustering is shown in the following formula (19): （19）； In the formula, and They represent potential physical modes and spurious modes, respectively. This represents the modal distance of the k-th nearest neighbor dataset in the one-dimensional modal distance dataset calculated according to equation (16). The initial ideal cluster centers that can be objectively determined are 0 and 1, respectively, representing the k-th mode distance between the physical mode and the spurious mode. S63, for any Mp, using equation (19) for bisection k-means clustering, automatically identify the knee point of the k-dist graph (the intersection point of the two clusters in the bisection clustering result), i.e., the corresponding... ;Calculate the value of Mp by traversing As a result, The range of values ​​for the parameter combination.

8. The automatic modal parameter identification method based on random subspace method and DBSCAN clustering according to claim 7, characterized in that, S7 includes: S71, for each group Density clustering results under parameter combinations, and calculation of clustering quality evaluation indicators. : （20）； In the formula, Refers to sample points The average distance to other points in the same cluster. Sample points The average distance to all points in the nearest neighbor cluster, where: （21）； In the formula, It is the extreme point The cluster it belongs to, ( =[1,2]) is the distance from the pole. Recent clustering, It is the two poles and The Euclidean distance between them, the clustering quality evaluation index is obtained by analyzing all extreme point samples. The value is obtained by averaging, and its range is [-1, 1]. The closer the value is to 1, the better the clustering quality. S72, traverse all parameters within the preset parameter space Density clustering under combination automatically selects the parameter combination that maximizes the clustering quality evaluation index as the optimal solution; S73, based on optimal parameter combination Density clustering is performed using normalized modal distance metrics to obtain a stable map with noise removed.

9. The automatic modal parameter identification method based on random subspace method and DBSCAN clustering according to claim 8, characterized in that, S8 includes: S81, derived from normalized modal energy and cluster dimension The initial cluster centers of the two-dimensional input sample dataset are objectively determined, namely, the initial cluster centers of the two-dimensional samples representing the physical mode and the spurious mode are vectors {1,1} and {0,0}, respectively. The calculation formula is shown in equation (22): （22）； In the formula, ; For matrix A ( ) One right eigenvector; For matrix The OK; The system matrix in continuous state space The 1 eigenvalue, ; For conjugate operations; matrices C and G are the output matrix and the output covariance matrix of the next state, respectively, where matrix C is the expression... The former Line, matrix G can be extended by a controllable matrix. After Column determined; S82, the iterative optimization process based on the binary K-means clustering retains candidate polar axes with high cluster dimensions and modal energy, thereby automatically filtering the clusters in which candidate physical modes reside.

10. The automatic modal parameter identification method based on random subspace method and DBSCAN clustering according to claim 9, characterized in that, S9 includes: S91, determine that the modal confidence criterion (MAC) and modal overlap coefficient (MOF) are used as indicators to quantify the degree of overlap between any two modes, wherein the modal confidence criterion (MAC) is quantified from the perspective of mode shape coefficients, and the modal overlap coefficient (MOF) is quantified from the perspective of frequency and damping ratio; The modal confidence criterion (MAC) and the modal overlap coefficient (MOF) are calculated as shown in equations (23) and (24), respectively: （23）； In the formula, H represents the Hermitian transpose. and Representing the first and the First-order mode shape; （24）； In the formula, and These represent the frequencies of a pair of modes to be quantized. Refers to the first Damping ratio of the first mode; S92, to avoid modal splitting, this invention uses the Modal Confidence Criterion (MAC) and Modal Overlap Factor (MOF) to evaluate the consistency of candidate modes. When the following conditions are met... and In this case, the candidate modality cluster with the smaller cluster dimension will be eliminated; S93, Extract representative modes indexed by the median damping ratio and perform modal parameter identification based on the representative modes. The modal parameter identification includes frequency, damping ratio, and linearly normalized modal energy. and cluster dimension .

Images