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

Abstract

The invention discloses an automatic modal parameter identification method based on a random subspace method and DBSCAN clustering. The method comprises the following steps: arranging an acceleration sensor on a to-be-measured structure to obtain a structure dynamic response; performing modal calculation based on a covariance random subspace method of a time domain to obtain an original stability graph; obtaining a preliminarily cleaned stable graph based on a hard index criterion, and superposing and drawing the average regularization power spectrum density of a frequency domain; all the poles are used as input samples of DBSCAN clustering, and the minimum neighborhood sample number is determined; calculating a normalized modal distance between any two poles; calculating a modal distance and distribution to determine a neighborhood radius value; substituting into DBSCAN clustering to obtain a noise-rejected stability graph; on the basis of linear normalized modal energy and clustering cluster dimensions, performing bisection K-means clustering to screen clusters of candidate physical modals; evaluating modal splitting according to the modal confidence factor MAC and the modal overlap factor MOF; a representative modal of the damping ratio median index is extracted and modal parameters are identified. The invention further discloses a system, electronic equipment and a computer readable storage medium.

Description

Technical Field

[0001] The present invention relates to the technical field of structural engineering and health monitoring, and particularly to an automatic modal parameter identification method based on the stochastic subspace method and DBSCAN clustering, specifically to a technical method for automated operational modal analysis of structures, which is applicable to modal parameter identification under ambient excitation and structural condition assessment. Background Art

[0002] Operational modal analysis (OMA) is an important means of structural health monitoring. By using the output dynamic signals under ambient excitation, the natural frequencies, damping ratios, and modal shapes of structures can be identified. As the research basis for vibration response analysis, health monitoring, damage detection, and structural dynamics optimization design, OMA has played an important role in the fields of civil engineering, aerospace, and machinery. Compared with traditional input-output modal analysis, OMA avoids the complexity and high cost of external excitation and has been widely applied in large-scale buildings such as bridges and buildings. However, in the existing OMA process, the distinction between physical modes and pseudo-modes, the stability screening of modal parameters, and the final confirmation of representative modes often rely on manual judgment, which not only increases the analysis cost but also may affect the analysis accuracy.

[0003] In recent years, with the development of automation technology, automated operational modal analysis (AOMA) has gradually become a research hotspot, and the stochastic subspace identification (SSI) algorithm has been widely used. However, the degree of automation of AOMA is still limited, and there are still many challenges in the automatic elimination of pseudo-modes, the identification of stable pole columns, and parameter screening under complex excitation conditions. Therefore, there is an urgent need for a more efficient, stable, and intelligent automated analysis method to further improve the efficiency and reliability of modal analysis of complex structures. Summary of the Invention

[0004] The purpose of the present invention is to provide an automatic modal parameter identification method based on the stochastic subspace identification (SSI) algorithm and the density-based spatial clustering of applications with noise (DBSCAN) algorithm. The constructed automated algorithm framework can effectively solve key problems such as modal separation, parameter selection, and pseudo-mode screening, and is applicable to the health monitoring and modal analysis of complex structures. Starting from the basic principles of the stability diagram, binary K-means clustering, and DBSCAN clustering, the automated algorithm framework defines an improved modal distance based on MACXP to control the aliasing problem between spatially dense modes; then constructs a new k-dist graph, and finally automatically determines the key parameters of DBSCAN through binary K-means clustering; performs binary K-means clustering on candidate modes according to the normalized clustering dimension and modal energy to achieve automatic selection of complex modes or weakly excited modes; and uses the synergistic effect of MOF and MAC to intelligently control the modal splitting phenomenon.

[0005] The first aspect of the present invention is to provide an automatic modal parameter identification method based on the random subspace method and DBSCAN clustering, including:

[0006] S1, arranging acceleration sensors on the structure to be measured, and the acceleration sensors are used to obtain the structural dynamic response;

[0007] S2, performing modal calculation on the structural dynamic response based on the covariance random subspace method in the time domain to obtain the original stability diagram;

[0008] S3, cleaning the original stability diagram based on the hard threshold criterion to obtain a preliminarily cleaned stability diagram, calculating the average normalized power spectral density in the frequency domain of each degree of freedom according to the structural dynamic response signal, and plotting the average normalized power spectral density as a reference on the preliminarily cleaned stability diagram;

[0009] S4, taking all the poles in the preliminarily cleaned stability diagram as the input samples of the DBSCAN clustering, and determining the minimum number of neighborhood samples of the DBSCAN clustering according to the total number of samples;

[0010] S5, comprehensively considering frequency, damping ratio and modal shape, calculating the normalized modal distance between any two poles in the preliminarily cleaned stability diagram;

[0011] S6, by calculating the modal distance between each pole and its k-th nearest neighbor, determining the value of the neighborhood radius Eps according to the distribution of the modal distance;

[0012] S7, substituting the obtained minimum number of neighborhood samples, neighborhood radius and normalized modal distance into the DBSCAN clustering algorithm to obtain a stability diagram with noise removed;

[0013] S8, based on the linearly normalized modal energy and the dimension of the clustering cluster , forming multiple groups of two-dimensional input samples, substituting the multiple groups of two-dimensional input samples into the binary K-means clustering to automatically screen the clustering where the candidate physical modes are located;

[0014] S9, effectively evaluating whether the modes with modal splitting phenomenon belong to the same order according to the synergistic effect of the modal confidence factor MAC and the modal overlap factor MOF, where the modal confidence factor MAC is evaluated from the perspective of the modal coefficient, and the modal overlap factor MOF is evaluated from the perspectives of frequency and damping ratio;

[0015] S10, extracting the representative modes indexed by the median value of the damping ratio from all the candidate modes and performing modal parameter identification based on the representative modes, and the modal parameter identification includes frequency, damping ratio, linearly normalized modal energy and the dimension of the clustering cluster 。

[0016] Preferably, the S2 includes:

[0017] S21, constructing a Toeplitz matrix, including:

[0018] Assuming that the output data is ergodic, the output covariance matrix and the Toeplitz matrix constructed according to the covariance matrix are respectively expressed as: respectively expressed as:

[0019] (2);

[0020] In the formula, represents the observation vector of the N-degree-of-freedom (DOF) system at the discrete time k, that is, the measured signal time history can be acceleration, velocity or displacement, where is the number of test channels; j corresponds to the total length of the measurement data; l is the number of test channels;

[0021] (3);

[0022] In the formula, the parameter represents the number of row blocks of the Toeplitz matrix, and the dimension of the matrix is , which is smaller than the dimension of the original Hankel matrix , thus reducing the computational amount and memory requirements; similarly, is expressed as:

[0023] (4);

[0024] S22, performing singular value decomposition on the Toeplitz matrix, including:

[0025] The model order N is another parameter that needs to be defined by the user. By ignoring small singular values, the small singular values are equal to the number of singular values obtained from the singular value decomposition of the Toeplitz matrix (Equation (5));

[0026] (5);

[0027] In the formula, and are orthogonal matrices, is a diagonal matrix; is a non-singular matrix, usually taken as the identity matrix, i.e., T = I, and we can obtain:

[0028] (6);

[0029] (7);

[0030] Where, and are the extended observable matrix and the extended controllable matrix respectively;

[0031] S23, identification system matrix, including:

[0032] The expression of the system matrix A is:

[0033] (8);

[0034] The modal parameters of the discrete-time system are obtained from the system matrix A and the output matrix C, where the output matrix C is equal to the value in equation (6): The first l lines of

[0035] S24, determine the modal parameters, including:

[0036] Decompose the system matrix A through eigenvalues to obtain:

[0037] (9);

[0038] In formula (9), The system poles of the discrete-time system are The diagonal matrix composed of corresponds to the right eigenvector of the matrix A; while the continuous-time eigenvalue and the eigenvalues in the discrete state The following relations are satisfied:

[0039] (10);

[0040] Where, represents the sampling time interval; finally, the natural frequency of the system , damping ratio and vibration mode It can be determined by formula (11):

[0041] (11);

[0042] S25, parameter setting, including:

[0043] In the matrix When performing singular value decomposition, the matrix The number of row blocks i and the model order N must satisfy the following relationship in the algorithm:

[0044] (12);

[0045] According to the fundamental frequency of the structure and the sampling frequency , the lower limit value determined based on the rule of thumb is:

[0046] (13).

[0047] Preferably, the S3 includes:

[0048] S31, the candidate modes that do not simultaneously satisfy the hard index judgment criterion in Equation (14) are excluded as mathematical false modes:

[0049] (14);

[0050] wherein, is the eigenvalue obtained by eigenvalue decomposition of the system matrix A in the discrete state, is conjugate;

[0051] S32, calculate the average regularized power spectral density , and the calculation is as shown in Equation (15) below:

[0052] (15);

[0053] wherein, is the number of degrees of freedom of the test; is the PSD of the signal of the i-th test degree of freedom; is the k-th discrete frequency; is the total number of discrete frequencies.

[0054] Preferably, the S5 includes:

[0055] S51, calculate the modal distance between any two poles in the stability diagram; the calculation formula is as follows:

[0056] (16);

[0057] wherein, refers to the j-th mode vector, refers to the j-th eigenvalue of the continuous state space equation, and its calculation expression is Equation (17):

[0058] (17);

[0059] Extend the modal assurance criterion MAC in the traditional real modal space to the complex modal space, which is called , and the calculation formula is as shown in Equation (18) below:​

[0060] (18);

[0061] In formula (18), is the conjugate of, is the transpose of, is the conjugate transpose of.

[0062] Preferably, the said S6 includes:

[0063] S61, calculating the distance between each data point and its k-th nearest neighbor, i.e., the k-th modal distance, based on the modal distance between any two poles in the stability diagram;

[0064] S62, constructing a k-dist function graph by using binary K-means clustering, where the k-dist function graph is used for the adaptive determination of parameters; the basic calculation formula of the binary K-means clustering is shown in the following formula (19):

[0065] (19);

[0066] In formula (19), and respectively represent the potential physical mode and the spurious mode, where represents the modal distance of the k-th nearest neighbor of the one-dimensional modal distance dataset calculated according to formula (16), represents the objectively determinable initial ideal clustering center, that is, the initial clustering centers of the k-th modal distances representing the physical mode and the spurious mode are 0 and 1 respectively.

[0067] Preferably, the said S8 includes:

[0068] S81, forming a two-dimensional input sample dataset from the normalized modal energy and the clustering cluster dimension and objectively determining the initial clustering center of the two-dimensional input sample dataset, that is, the initial clustering centers of the two-dimensional samples representing the physical mode and the spurious mode are the vectors {1, 1} and {0, 0} respectively;

[0069] The calculation formula of

[0070] is shown in the following formula (20):

[0071] In formula (20), ; is the matrix A ( ​The i-th right eigenvector of ()); is the matrix of the i-th row; refers to the i-th eigenvalue of the system matrix in the continuous state space, ; is the conjugate operation; the matrices C and G are the output matrix and the output covariance matrix of the next state respectively, where the matrix C is the first l rows of Equation and the matrix G can be determined by the last l columns of the extended controllable matrix ;

[0072] S82, based on the iterative optimization process of the binary K-means clustering, retain the candidate polar axes with higher clustering cluster dimensions and modal energies, so as to automatically screen the clusters where the candidate physical modes are located.

[0073] Preferably, the S9 includes:

[0074] S91, determine that the modal assurance criterion MAC and the modal overlap factor MOF are used as indicators to quantify the overlap degree of any two modes, where the modal assurance criterion MAC is quantified from the perspective of the mode shape coefficient, and the modal overlap factor MOF is quantified from the perspectives of frequency and damping ratio;

[0075] where the calculation formulas of the modal assurance criterion MAC and the modal overlap factor MOF are shown in the following formulas (21) and (22) respectively:

[0076] (21);

[0077] In formula (21), H represents the Hermitian transpose, and represent the i-th and j-th order mode shapes respectively;

[0078] (22);

[0079] In formula (22), and represent the frequencies of a pair of modes to be quantified respectively, refers to the damping ratio of the i-th order mode;

[0080] S92, if the modal overlap factor MOF of any pair of modes exceeds 0.1 and the modal assurance criterion MAC exceeds 0.9, then automatically identify that the two belong to the same order mode, and at this time, the candidate mode cluster with a smaller clustering cluster dimension among the two will be eliminated.

[0081] The second aspect of the present invention is to provide an automatic modal parameter identification system based on the random subspace method and DBSCAN clustering for implementing the method of the first aspect, including:

[0082] A structural dynamic response acquisition module (101) for arranging acceleration sensors on the structure to be measured, and the acceleration sensors are used to acquire the structural dynamic response;

[0083] A modal estimation module (102) for performing modal calculations on the structural dynamic response based on the covariance random subspace method in the time domain to obtain an original stability diagram;

[0084] A stability diagram acquisition module (103) for cleaning the original stability diagram based on a hard index criterion to obtain a preliminarily cleaned stability diagram, calculating the average normalized power spectral density in the frequency domain of each degree of freedom according to the structural dynamic response signal, and plotting the average normalized power spectral density as a reference on the preliminarily cleaned stability diagram;

[0085] A minimum neighborhood sample number determination module (104) for taking all the poles in the preliminarily cleaned stability diagram as the input samples of the DBSCAN clustering, and determining the minimum neighborhood sample number of the DBSCAN clustering according to the total number of samples;

[0086] A normalized modal distance determination module (105) for comprehensively considering frequency, damping ratio, and modal shape to calculate the normalized modal distance between any two poles in the preliminarily cleaned stability diagram;

[0087] A neighborhood radius determination module (106) for determining the neighborhood radius value by calculating the modal distance between each pole and its k-th nearest neighbor and according to the distribution of the modal distances;

[0088] A stability diagram noise removal module (107) for substituting the obtained minimum neighborhood sample number, neighborhood radius, and normalized modal distance into the DBSCAN clustering algorithm to obtain a stability diagram with noise removed;

[0089] A clustering module (108) for forming multiple groups of two-dimensional input samples based on linearly normalized modal energy and the dimension of the clustering cluster , substituting the multiple groups of two-dimensional input samples into the binary K-means clustering to automatically screen the clusters where the candidate physical modes are located;

[0090] A modal same - order determination module (109) is configured to effectively evaluate whether the modes with modal splitting phenomenon belong to the same order according to the combined action of the modal assurance criterion (MAC) and the modal overlap factor (MOF), where the modal assurance criterion (MAC) is evaluated from the perspective of mode - shape coefficients, and the modal overlap factor (MOF) is evaluated from the perspectives of frequency and damping ratio;

[0091] A modal parameter identification module (110) is configured to extract representative modes indexed by the median damping ratio from all the candidate modes and perform modal parameter identification based on the representative modes, where the modal parameter identification includes frequency, damping ratio, linearly - normalized modal energy and the dimension of the clustering cluster .

[0092] A third aspect of the present invention provides an electronic device, including a processor and a memory, where the memory stores multiple instructions, and the processor is configured to read the instructions and execute the method as described in the first aspect.

[0093] A fourth aspect of the present invention provides a computer - readable storage medium, where the computer - readable storage medium stores multiple instructions, and the multiple instructions can be read and executed by a processor to perform the method as described in the first aspect.

[0094] Advantages of the method and system of the present invention:

[0095] (1) According to the normalized clustering dimension and modal energy loss (MEL), the present invention performs binary K - means clustering on candidate modes, which can effectively handle the situation of "only physical modes" in candidate - mode clustering, thus avoiding the need to artificially define a certain number of empty sets to achieve automatic screening of modes in this case.

[0096] (2) The present invention can effectively distinguish the influence of spatially - dense modes; through the combined action of MOF and MAC, the modal splitting phenomenon can be accurately identified.

[0097] (3) The present invention realizes the accurate identification of physical modes and spurious modes, provides a more efficient, stable and intelligent automated analysis method, has significant advantages in improving the efficiency, accuracy and applicability of structural operational modal analysis, and provides strong technical support for the field of structural health monitoring. BRIEF DESCRIPTION OF THE DRAWINGS

[0098] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the related art, the following will briefly introduce the drawings required for use in the description of the specific embodiments or the related art. Obviously, the following - described drawings are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0099] Figure 1 Flow chart of an automatic modal parameter identification method based on the random subspace method and DBSCAN according to an embodiment of the present invention.

[0100] Figure 2 7-degree-of-freedom mass-spring-damper system of Embodiment 1 of the present invention;

[0101] Figure 3 a and Figure 3 b are respectively the stability diagrams before and after cleaning by the hard index criterion in Embodiment 1 of the present invention;

[0102] Figure 4 k-dist function diagram in Embodiment 1 of the present invention;

[0103] Figure 5 Frequency-damping ratio diagram obtained by adaptive DBSCAN clustering in Embodiment 1 of the present invention;

[0104] Figure 6 Automatic screening result diagram of candidate mode clustering in Embodiment 1 of the present invention;

[0105] Figure 7 MAC diagram of all candidate modes in Embodiment 1 of the present invention;

[0106] Figure 8 MOF diagram of all candidate modes in Embodiment 1 of the present invention;

[0107] Figure 9 Final stability diagram of full-automatic modal identification in Embodiment 1 of the present invention;

[0108] Figure 10 Relative error diagram of all 7th-order modal parameters identified in Embodiment 1 of the present invention;

[0109] Figure 11 Dowling Hall steel truss pedestrian bridge in Embodiment 2 of the present invention;

[0110] Figure 12 Sensor layout diagram of the Dowling Hall steel truss pedestrian bridge in Embodiment 2 of the present invention;

[0111] Figure 13 Frequency-damping ratio diagram obtained by adaptive DBSCAN clustering in Embodiment 2 of the present invention;

[0112] Figure 14 Final stability diagram of full-automatic modal identification in Embodiment 2 of the present invention;

[0113] Figure 15The architecture diagram of the automatic modal parameter identification system based on the random subspace method and DBSCAN according to an embodiment of the present invention;

[0114] Figure 16 The structural diagram of the electronic device according to an embodiment of the present invention. Detailed implementation manners

[0115] The technical solutions of the present invention will be described clearly and completely below with reference to the accompanying drawings. Apparently, the described embodiments are some but not all of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0116] In the description of the present invention, it should be noted that the orientation or positional relationship indicated by the terms "center", "upper", "lower", "left", "right", "vertical", "horizontal", "inner", "outer", etc. is based on the orientation or positional relationship shown in the drawings, and is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation to the present invention. In addition, the terms "first", "second", and "third" are only used for descriptive purposes and cannot be understood as indicating or implying relative importance.

[0117] In the description of the present invention, it should be noted that unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be directly connected, or indirectly connected through an intermediate medium, and it can be the communication inside two elements. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific situations.

[0118] This embodiment provides an automatic modal parameter identification method based on the random subspace method and DBSCAN clustering, including:

[0119] S1, arranging acceleration sensors on the structure to be measured, and the acceleration sensors are used to obtain the structural dynamic response;

[0120] S2, performing modal calculation on the structural dynamic response based on the covariance random subspace method in the time domain (SSI-Cov) to obtain the original stability diagram;

[0121] S3. Clean the original stability diagram based on the hard index criterion to obtain a preliminarily cleaned stability diagram. Calculate the average normalized power spectral density ANPSD in the frequency domain of each degree of freedom according to the structural dynamic response signal, and plot the average normalized power spectral density ANPSD as a reference on the preliminarily cleaned stability diagram;

[0122] S4. Take all the poles in the preliminarily cleaned stability diagram as the input samples for the DBSCAN clustering. Determine the key parameter of the DBSCAN clustering - the minimum number of neighborhood samples MinPts according to the total number of samples, as shown in the following formula (1):

[0123] MinPts = ln(P) (1);

[0124] In formula (1), P is the number of samples of all candidate modes in the preliminarily cleaned stability diagram obtained after the hard index cleaning;

[0125] S5. Comprehensively consider the frequency, damping ratio and mode shape, and calculate the normalized modal distance between any two poles in the preliminarily cleaned stability diagram;

[0126] S6. By calculating the modal distance between each pole and its k-th nearest neighbor, determine the value of the neighborhood radius Eps according to the distribution of the modal distance;

[0127] S7. Substitute the obtained minimum number of neighborhood samples MinPts, neighborhood radius Eps and normalized improved modal distance into the DBSCAN clustering algorithm to obtain a stability diagram with noise removed;

[0128] S8. Based on the linearly normalized modal energy and the clustering cluster dimension , form multiple groups of two-dimensional input samples, and substitute the multiple groups of two-dimensional input samples into the binary K-means clustering to automatically screen the clusters where the candidate physical modes are located;

[0129] S9. Effectively evaluate whether the modes with modal splitting phenomena belong to the same order according to the synergistic effect of the modal confidence factor MAC and the modal overlap factor MOF, where the modal confidence factor MAC is evaluated from the perspective of the mode shape coefficient, and the modal overlap factor MOF is evaluated from the perspectives of frequency and damping ratio;

[0130] S10. Extract the representative modes indexed by the median damping ratio from all the candidate modes and perform modal parameter identification based on the representative modes. The modal parameter identification includes frequency, damping ratio, linearly normalized modal energy and the clustering cluster dimension .

[0131] As a preferred embodiment, S2 includes:

[0132] S21, constructing a Toeplitz matrix, including:

[0133] Assuming that the output data is ergodic, the output covariance matrix and the Toeplitz matrix constructed according to the covariance matrix are respectively expressed as:

[0134] (2);

[0135] In the formula, represents the observation vector of the N-degree-of-freedom (DOF) system at the discrete time k, that is, the time history of the measurement signal can be acceleration, velocity or displacement, where is the number of test channels; j corresponds to the total length of the measurement data; l is the number of test channels;

[0136] (3);

[0137] In the formula, the parameter represents the number of row blocks of the Toeplitz matrix, and the dimension of the matrix is , which is smaller than the dimension of the original Hankel matrix , thereby reducing the computational amount and memory requirements; similarly, is expressed as:

[0138] (4);

[0139] S22, performing singular value decomposition on the Toeplitz matrix, including:

[0140] The model order N is another parameter that needs to be defined by the user. By ignoring small singular values, the small singular values are equal to the number of singular values obtained from the singular value decomposition of the Toeplitz matrix (Equation (5));

[0141] (5);

[0142] In the formula, and are orthogonal matrices, is a diagonal matrix; is a non-singular matrix, usually taken as the identity matrix, that is, T = I, and we can get:

[0143] (6);

[0144] ​(7);

[0145] Wherein, and are respectively the extended observable matrix and the extended controllable matrix;

[0146] S23, identifying the system matrix, including:

[0147] The expression of the system matrix A is:

[0148] (8);

[0149] The modal parameters of the discrete-time system are obtained from the system matrix A and the output matrix C, where the output matrix C is equal to the first l rows in Equation (6) of;

[0150] S24, determining the modal parameters, including:

[0151] Decompose the system matrix A through eigenvalues to obtain:

[0152] (9);

[0153] In Equation (9), is a diagonal matrix composed of the system poles of the discrete-time system; corresponds to the right eigenvector of matrix A; and the continuous-time eigenvalue and the eigenvalue in the discrete state satisfy the following relationship:

[0154] (10);

[0155] Wherein, represents the sampling time interval; finally, the natural frequency of the system, the damping ratio and the vibration mode can be determined by Equation (11) as:

[0156] (11);

[0157] S25, parameter setting, including:

[0158] When performing singular value decomposition on the matrix , the number of row blocks i of the matrix and the model order N need to satisfy the following relationship algorithmically:

[0159] (12);

[0160] According to the fundamental frequency and sampling frequency , determined based on the rule of thumb The lower limit value of is:

[0161] (13).

[0162] As a preferred embodiment, the S3 includes:

[0163] S31, as shown in the following formula (14), the candidate modes that do not simultaneously meet the hard index judgment criteria in formula (14) are excluded as mathematical false modes:

[0164] (14);

[0165] In the formula, is the eigenvalue obtained by eigenvalue decomposition of the system matrix A in the discrete state, is conjugate;

[0166] S32, calculate the average regularized power spectral density , the calculation is as shown in the following formula (15):

[0167] (15);

[0168] In the formula, is the number of degrees of freedom of the test; is the PSD of the signal of the i-th test degree of freedom; is the k-th discrete frequency; is the total number of discrete frequencies.

[0169] As a preferred embodiment, the S5 includes:

[0170] S51, calculate the modal distance between any two poles in the stability diagram ; The calculation formula is as follows:

[0171] (16);

[0172] In the formula, refers to the j-th mode vector, refers to the j-th eigenvalue of the continuous state space equation, and its calculation expression is formula (17):

[0173] (17);

[0174] It should be noted that when the number of measurement points or the sensor position is limited, the modal shapes are often unable to be effectively distinguished. Therefore, this algorithm extends the modal assurance criterion MAC in the traditional real modal space to the complex modal space, called , the calculation formula is as shown in the following formula (18):

[0175] (18);

[0176] In formula (18), is the conjugate of is the transpose of is the conjugate transpose; The criterion integrates the pole information related to the modal shape by considering pole weighting to achieve better modal discrimination; the basic idea of the criterion is to fully consider the free decay response associated with each pole, rather than making a modal similarity judgment only based on the modal shape coefficients; as a normalized modal distance metric, its range is from 0 to 1.

[0177] As a preferred embodiment, the S6 includes:

[0178] S61, calculating the distance between each data point and its k-th nearest neighbor, the k-th modal distance, based on the modal distance between any two poles in the stability diagram ;

[0179] S62, constructing a k-dist function graph using binary K-means clustering, and the k-dist function graph is used for the adaptive determination of parameters; the basic calculation formula of the binary K-means clustering is as shown in the following formula (19):

[0180] (19);

[0181] In formula (19), and respectively represent the potential physical mode and the spurious mode, where represents the modal distance of the k-th nearest neighbor of the one-dimensional modal distance data set calculated according to formula (16), represents the objectively determinable initial ideal clustering center, that is, the initial clustering centers of the k-th modal distances representing the physical mode and the spurious mode are 0 and 1 respectively.

[0182] As a preferred embodiment, the S8 includes:

[0183] S81, forming a two-dimensional input sample data set from the normalized modal energy and the clustering cluster dimension and objectively determining the initial clustering center of the two-dimensional input sample data set, that is, the initial clustering centers of the two-dimensional samples representing the physical mode and the spurious mode are the vectors {1, 1} and {0, 0} respectively;

[0184] In this embodiment, The calculation formula of

[0185] is as follows: (20);

[0186] In formula (20), ; is the i-th right eigenvector of matrix A ( ); is the i-th row of matrix ; refers to the i-th eigenvalue of the system matrix ( ) in the continuous state space; is the conjugate operation; matrices C and G are the output matrix and the output covariance matrix of the next state respectively, where matrix C is the first l rows of formula , and matrix G can be determined by the last l columns of the extended controllable matrix ; considering that according to the above formula, the value range of MEL is not from 0 to 1, so linear normalization is uniformly adopted for it in this algorithm.

[0187] S82. Based on the iterative optimization process of the binary K-means clustering, retain the candidate polar axes with higher clustering cluster dimensions and modal energies, so as to automatically screen the clusters where the candidate physical modes are located.

[0188] As a preferred implementation manner, the said S9 includes:

[0189] S91. Determine that the modal confidence factor MAC and the modal overlap factor MOF are used as indicators to quantify the overlap degree of any two modes, where the modal confidence factor MAC is quantified from the perspective of the mode shape coefficient, and the modal overlap factor MOF is quantified from the perspectives of frequency and damping ratio;

[0190] The calculation formulas of the modal confidence factor MAC and the modal overlap factor MOF are shown in formulas (21) and (22) below respectively:

[0191] (21);

[0192] In formula (21), H represents the Hermitian transpose, and represent the i-th and j-th order modal shapes respectively;

[0193] (22);

[0194] In formula (22), and represent the frequencies of a pair of modes to be quantified respectively, It refers to the damping ratio of the i-th order mode.

[0195] S92. If the modal overlap factor MOF of any pair of modes exceeds 0.1 and the modal assurance criterion MAC exceeds 0.9, it is automatically recognized that they belong to the same order of mode. At this time, the candidate mode clustering with a smaller cluster dimension among the two will be eliminated.

[0196] The implementation process of the automatic modal parameter identification technology proposed by the present invention is as Figure 1 shown. Taking a 7-degree-of-freedom linear damped coupled vibration frame system model ( Figure 2 ) as the research object: In this simulation system model, the mass and stiffness of each floor are respectively: m n = 1 unit (e.g., kg), k n = 10 units (e.g., N / m) (n = 1,…,7); Considering Rayleigh damping, the damping coefficient matrix is C n = 0.01M n + 0.003K n , where M n and K n are the mass matrix and stiffness matrix of the model respectively. By applying load inputs to each floor, Gaussian white noise excitation f i (t) with a mean of 0 and a variance of 1 can be input based on the lsim function of the MATLAB language to simulate the influence of environmental excitation. The output responses of each floor are y i (t), thus completing the construction of the operational modal analysis in the environment of only output signals. The specific implementation steps for describing the structural modal parameter identification are as follows:

[0197] Based on the output response signals y i (t) of each floor, the structural dynamic response under environmental excitation is obtained.

[0198] Substitute the dynamic response signal into the covariance-driven stochastic subspace algorithm for modal calculation to obtain the original stability diagram. Among them, according to the empirical rule, the number of row blocks i of the Toeplitz matrix is determined to be 50, and the model order N ranges from 2 to 100 with an increment of 2.

[0199] Use the hard threshold criterion to initially clean the stability diagram. Calculate the average normalized power spectral density ANPSD of each degree of freedom according to the response signal and superimpose it on the stability diagram, as Figure 3 a and Figure 3 b shown.

[0200] According to the 971 candidate poles in the initially cleaned stability diagram, determine k = MinPts = ceil {ln (971)} = 7.

[0201] Calculate the normalized modal distance between any two poles in the stability diagram.

[0202] Calculate the modal distance between each pole and its k-th (k = 7) nearest neighbor, and form a k-dist function graph based on the distribution of the modal distances; use binary K-means clustering, where the initial clustering centers representing the spurious modes and physical modes for the k-th modal distance are 1 and 0 respectively, and the updated clustering centers are 0.305 and 0.038 respectively, and calculate Eps = 0.171 ( Figure 4 ).

[0203] Substitute the obtained minimum neighborhood sample number (MinPts), neighborhood radius (Eps), and normalized improved modal distance into the DBSCAN algorithm to obtain the frequency-damping ratio graph with noise removed ( Figure 5 ). Among them, 9 candidate modal clusters (legend "M") and their corresponding representative modes (legend "RM") are distinguished by different colors, and the noise points representing spurious modes are indicated by black asterisks.

[0204] Calculate the normalized modal energy ( ), and the clustering cluster dimension ( ), and substitute the 9 groups of two-dimensional data composed of the two into binary K-Means clustering, where the initial ideal clustering centers for physical modes (PM) and spurious modes (SM) are {1, 1} and {0, 0} respectively, and after updating, they are {0.624, 0.815} and {0.329, 0.172} respectively. Among them, 2 modes are identified as spurious modes ( Figure 6 ).

[0205] The combined effect of the modal confidence factor (MAC) and the modal overlap factor (MOF) further confirms the modal splitting phenomenon: the modal overlap coefficients MOF (inf means infinity) between modes 6 and 7 and between modes 8 and 9 reach 0.388 and 1.027 respectively, while the modal confidence factor (MAC) is close to 1. Therefore, modes 6 and 8 with smaller clustering dimensions are automatically removed.

[0206] Among them Figure 7 shows the MAC graph of all candidate modes in Embodiment 1 of the present invention; Figure 8 shows the MOF graph of all candidate modes in Embodiment 1 of the present invention; Figure 9 shows the final stability diagram of the full-automatic modal identification in Embodiment 1 of the present invention.

[0207] Table 1 lists the representative modal results of all 9 candidate modes indexed by the median damping ratio, including frequency, damping ratio, MEL, . By comparing the frequencies, damping ratios, and modal vibration modes of the finally determined 7 representative modes with their corresponding theoretical values, the relative errors are asFigure 10 As shown. The results show that the errors of both the frequency and the modal shape are extremely small (the relative error is close to 0), and the average error of the damping ratio is 13.83%.

[0208] Table 1 Representative modal results of all candidate modes indexed by the median damping ratio

[0209]

[0210] Note: (* represents a spurious mode)

[0211] Example 2

[0212] Taking the Dowling Hall steel truss pedestrian bridge in the United States ( Figure 11 ) as an example, the automatic modal parameter identification method proposed by the present invention is further demonstrated. This bridge provides a Benchmark platform for vibration-based structural health monitoring research. The bridge is 44 meters long and 3.7 meters wide. Eight vertical piezoelectric 393B04 accelerometers ( Figure 12 ) are installed at the bottom of the bridge. The monitoring system has been running continuously since 2010, sampling at a sampling frequency of 2048 Hz and resampling to 128 Hz. Data timing sampling is performed once every 5 minutes of each whole hour. This Benchmark dataset includes 17 weeks of ambient excitation monitoring data. In this example, the proposed algorithm is verified based on the dataset of the first week (number a100106_10000).

[0213] The automatic identification process of structural modal parameters is as Figure 1 shown. The first six-order modal parameters of the structure are automatically identified, and the statistics are shown in Table 2. The identification results are basically consistent with the benchmark identification results of this Benchmark. Among them, the frequency-damping ratio results identified by adaptive DBSCAN clustering are as Figure 13 shown, and the final frequency stability diagram results are as Figure 14 shown.

[0214] Table 2 Representative modal results of automatic identification

[0215]

[0216] As Figure 15 shown, this example aims to provide an automatic modal parameter identification system based on the stochastic subspace method and DBSCAN clustering for implementing the corresponding method, including:

[0217] A structural dynamic response acquisition module 101, configured to arrange accelerometers on the structure to be measured, and the accelerometers are used to acquire the structural dynamic response;

[0218] The modal estimation module 102 is used to perform modal calculations on the structural dynamic response based on the covariance stochastic subspace method in the time domain (SSI-Cov) to obtain the original stability diagram;

[0219] The stability diagram acquisition module 103 is used to clean the original stability diagram based on the hard index criterion to obtain the preliminarily cleaned stability diagram, calculate the average normalized power spectral density ANPSD in the frequency domain of each degree of freedom according to the structural dynamic response signal, and plot the average normalized power spectral density ANPSD as a reference on the preliminarily cleaned stability diagram;

[0220] The minimum neighborhood sample number determination module 104 is used to take all the poles in the preliminarily cleaned stability diagram as the input samples of the DBSCAN clustering, and determine the key parameter of the DBSCAN clustering - the minimum neighborhood sample number MinPts according to the total number of samples, as shown in the following formula (1):

[0221] MinPts = ln(𝑃) (1);

[0222] In formula (1), P is the number of samples of all candidate modes in the preliminarily cleaned stability diagram obtained after cleaning by the hard index;

[0223] The normalized modal distance determination module 105 is used to comprehensively consider frequency, damping ratio and modal shape to calculate the normalized modal distance between any two poles in the preliminarily cleaned stability diagram;

[0224] The neighborhood radius determination module 106 is used to calculate the modal distance between each pole and its k-th nearest neighbor, and determine the value of the neighborhood radius Eps according to the distribution of the modal distance;

[0225] The stability diagram noise removal module 107 is used to substitute the obtained minimum neighborhood sample number MinPts, neighborhood radius Eps and normalized improved modal distance into the DBSCAN clustering algorithm to obtain the stability diagram with noise removed;

[0226] The clustering module 108 is used to form multiple groups of two-dimensional input samples based on the linearly normalized modal energy and the dimension of the clustering cluster , substitute the multiple groups of two-dimensional input samples into the binary K-means clustering, and automatically screen the clusters where the candidate physical modes are located;

[0227] The modal same-order determination module 109 is used to effectively evaluate whether the modes showing the modal splitting phenomenon belong to the same order according to the synergistic effect of the modal confidence factor MAC and the modal overlap factor MOF, where the modal confidence factor MAC is evaluated from the perspective of the modal shape coefficient, and the modal overlap factor MOF is evaluated from the perspectives of frequency and damping ratio;

[0228] A modal parameter identification module 110 is configured to extract a representative mode indexed by the median damping ratio from all the candidate modes and perform modal parameter identification based on the representative mode, where the modal parameter identification includes frequency, damping ratio, linearly normalized modal energy and the dimension of the clustering cluster .

[0229] The present invention also provides a memory storing multiple instructions for implementing the method according to Embodiment 1.

[0230] As Figure 16 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 multiple instructions that can be loaded and executed by the processor, so that the processor can execute the method according to Embodiment 1.

[0231] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements 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 the random subspace method and DBSCAN clustering, characterized in that, Including: S1. Arrange an acceleration sensor on the structure to be measured, and the acceleration sensor is used to obtain the structural dynamic response. S2. Perform modal calculation on the structural dynamic response based on the covariance-based stochastic subspace identification method in the time domain to obtain the original stability diagram. S3. Clean the original stability diagram based on the hard index criterion to obtain the preliminarily cleaned stability diagram. Calculate the average normalized power spectral density in the frequency domain of each degree of freedom according to the structural dynamic response signal, and plot the average normalized power spectral density as a reference on the preliminarily cleaned stability diagram. S4. Take all the poles in the preliminarily cleaned stability diagram as the input samples of the DBSCAN clustering, and determine the minimum number of neighborhood samples of the DBSCAN clustering according to the total number of samples. S5. Considering frequency, damping ratio, and modal shape comprehensively, calculate the normalized modal distance between any two poles in the preliminarily cleaned stability diagram. S6. By calculating the modal distance between each pole and its k-th nearest neighbor, determine the neighborhood radius value according to the distribution of the modal distance. S7. Substitute the obtained minimum number of neighborhood samples, neighborhood radius, and normalized modal distance into the DBSCAN clustering algorithm to obtain the stability diagram with noise removed. S8, Modal energy based on linear normalization and the dimension of the clustering cluster , forming multiple groups of two-dimensional input samples, substituting the multiple groups of two-dimensional input samples into the binary K-means clustering, and automatically screening the clusters where the candidate physical modes are located; S9. Effectively evaluate whether the modes with modal splitting phenomenon belong to the same order according to the synergistic effect of the modal assurance criterion MAC and the modal overlap factor MOF, where the modal assurance criterion MAC is evaluated from the perspective of modal shape coefficients, and the modal overlap factor MOF is evaluated from the perspectives of frequency and damping ratio. S10. Extract a representative mode indexed by the median damping ratio from all the candidate modes and perform modal parameter identification based on the representative mode. The modal parameter identification includes frequency, damping ratio, linearly normalized modal energy and the dimension of the clustering cluster .

2. The automatic modal parameter identification method based on the random subspace method and DBSCAN clustering according to claim 1, characterized in that The S2 includes: S21. Construct a Toeplitz matrix, including: Assume that the output data is ergodic, and the output covariance matrix and the Toeplitz matrix constructed according to the covariance matrix are respectively expressed as: are respectively expressed as: （2）； In the formula, represents the observation vector of the N - degree - of - freedom (DOF) system at the discrete time instant k. That is, the time history of the measured signal can be acceleration, velocity, or displacement, where is the number of test channels; j corresponds to the total length of the measurement data; l is the number of test channels; （3）； In the formula, the parameter represents the number of row blocks of the Toeplitz matrix. The matrix has a dimension of , which is smaller than the dimension of the original Hankel matrix , thus reducing the computational amount and memory requirements. Similarly, is expressed as: （4）； S22. Perform singular value decomposition on the Toeplitz matrix, including: The model order N is another parameter that needs to be defined by the user. By ignoring small singular values, the small singular values are equal to the number of singular values obtained from the singular value decomposition of the Toeplitz matrix (Equation (5)); （5）； In the formula, and are orthogonal matrices, is a diagonal matrix; is a non-singular matrix, usually taken as the identity matrix, i.e., T = I, and we can obtain: （6）； （7）； wherein, and are the extended observable matrix and the extended controllable matrix, respectively; S23. Identify the system matrix, including: The expression of the system matrix A is: （8）； The modal parameters of the discrete-time system are obtained from the system matrix A and the output matrix C, where the output matrix C is equal to the first l rows in Equation (6). ; S24. Determine the modal parameters, including: Decompose the system matrix A through eigenvalue decomposition to obtain: （9）； In Equation (9), is a diagonal matrix composed of the system poles of the discrete-time system ; corresponds to the right eigenvector of matrix A; and the continuous-time eigenvalues and the eigenvalues in the discrete state satisfy the following relationship: （10）； wherein, represents the sampling time interval; finally, the natural frequency of the system , damping ratio and mode shape can be determined by Equation (11) as follows: （11）； S25. Parameter setting, including: When performing singular value decomposition on the matrix , the number of row blocks i of the matrix and the model order N need to satisfy the following relationship algorithmically: （12）； Based on the fundamental frequency of the structure and the sampling frequency , determined based on the rule of thumb The lower limit value is: （13）。 3. An automatic modal parameter identification method based on the random subspace method and DBSCAN clustering according to claim 2, characterized in that The S3 includes: S31. The candidate modes that do not simultaneously meet the hard index judgment criterion in Equation (14) are excluded as mathematical spurious modes. （14）； In the formula, is the eigenvalue obtained by the eigenvalue decomposition of the system matrix A in the discrete state, is 's conjugate; S32, calculate the average normalized power spectral density , which is calculated as shown in the following formula (15): （15）； In the formula, is the number of degrees of freedom for testing; is the PSD of the signal of the i-th degree of freedom for testing; is the k-th discrete frequency; is the total number of discrete frequencies.

4. An automatic modal parameter identification method based on the random subspace method and DBSCAN clustering according to claim 3, characterized in that, The S5 includes: S51, Calculate the modal distance between any two poles in the stability diagram ; The calculation formula is as follows: （16）； In the formula, represents the j-th mode vector, represents the eigenvalue of the j-th continuous state space equation, and its calculation expression is Equation (17): （17）； The modal assurance criterion (MAC) in the traditional real modal space is extended to the complex modal space, which is called , and the calculation formula is shown in the following formula (18): （18）； In formula (18), is the conjugate of, is the transpose of, is the conjugate transpose of.

5. The automatic modal parameter identification method based on the random subspace method and DBSCAN clustering according to claim 4, characterized in that The S6 includes: S61, based on the modal distance between any two poles in the stability diagram Calculate the distance between each data point and its k-th nearest neighbor, the k-th modal distance; S62. Use the binary K-means clustering to construct a k-dist function graph, and the k-dist function graph is used for the adaptive determination of parameters; the basic calculation formula of the binary K-means clustering is shown in Equation (19) below. （19）； In Equation (19), and represent the potential physical mode and the spurious mode, respectively, where represents the modal distance of the k-th nearest neighbor of the one-dimensional modal distance dataset calculated according to Equation (16), represents the initially ideal clustering center that can be objectively determined, that is, the initial clustering centers representing the k-th modal distances of the physical mode and the spurious mode are 0 and 1, respectively.

6. An automatic modal parameter identification method based on the random subspace method and DBSCAN clustering according to claim 5, characterized in that, The S8 includes: S81, consisting of normalized modal energy and the dimension of the clustering cluster to form a two-dimensional input sample data set and objectively determine the initial clustering centers of the two-dimensional input sample data set. That is, the initial clustering centers of the two-dimensional samples representing physical modes and false modes are the vectors {1, 1} and {0, 0} respectively; The calculation formula is as shown in the following formula (20): （20）； In formula (20), ; is the i-th right eigenvector of matrix A ( ); is the i-th row of matrix ; denotes the i-th eigenvalue of the system matrix in the continuous state space, ; is the conjugate operation; matrices C and G are the output matrix and the output covariance matrix of the next state respectively, where matrix C is the first l rows of formula , and matrix G can be determined by the last l columns of the extended controllable matrix ; S82. Based on the iterative optimization process of the binary K-means clustering, retain the candidate polar axes with higher clustering cluster dimensions and modal energies, so as to automatically screen the clusters where the candidate physical modes are located.

7. An automatic modal parameter identification method based on the random subspace method and DBSCAN clustering according to claim 6, characterized in that The S9 includes: S91. Determine that the modal assurance criterion MAC and the modal overlap factor MOF are used as indicators to quantify the overlap degree of any two modes, where the modal assurance criterion MAC is quantified from the perspective of modal shape coefficients, and the modal overlap factor MOF is quantified from the perspectives of frequency and damping ratio. The calculation formulas of the modal assurance criterion MAC and the modal overlap factor MOF are shown in Equations (21) and (22) below respectively. （21）； In Equation (21), H represents the Hermitian transpose, and represent the i-th and j-th order modal vibration modes, respectively; （22）； In Equation (22), and respectively represent the frequencies of a pair of modes to be quantified, denotes the damping ratio of the i-th order mode; S92. If the modal overlap factor MOF of any pair of modes exceeds 0.1 and the modal assurance criterion MAC exceeds 0.9, it is automatically recognized that the two belong to the same order of mode. At this time, the candidate mode clustering with a smaller cluster dimension among the two will be eliminated.

8. An automatic modal parameter identification system based on the random subspace method and DBSCAN clustering for implementing the method according to any one of claims 1-7, comprising: A structural dynamic response acquisition module (101) for arranging acceleration sensors on the structure to be measured, and the acceleration sensors are used to acquire the structural dynamic response; A modal estimation module (102) for performing modal calculation on the structural dynamic response based on the covariance random subspace method in the time domain to obtain an original stability diagram; A stability diagram acquisition module (103) for cleaning the original stability diagram based on a hard threshold criterion to obtain a preliminarily cleaned stability diagram, calculating the average normalized power spectral density in the frequency domain of each degree of freedom according to the structural dynamic response signal, and plotting the average normalized power spectral density as a reference on the preliminarily cleaned stability diagram; A minimum neighborhood sample number determination module (104) for using all the poles in the preliminarily cleaned stability diagram as input samples for the DBSCAN clustering, and determining the minimum neighborhood sample number of the DBSCAN clustering according to the total number of samples; A normalized modal distance determination module (105) for comprehensively considering frequency, damping ratio and modal vibration mode, and calculating the normalized modal distance between any two poles in the preliminarily cleaned stability diagram; A neighborhood radius determination module (106) for determining the neighborhood radius value by calculating the modal distance between each pole and its k-th nearest neighbor and according to the distribution of the modal distance; A stability diagram noise removal module (107) for substituting the obtained minimum neighborhood sample number, neighborhood radius and normalized modal distance into the DBSCAN clustering algorithm to obtain a stability diagram with noise removed; A clustering module (108) for forming multiple groups of two-dimensional input samples based on linearly normalized modal energy and the dimension of the clustering cluster and substituting the multiple groups of two-dimensional input samples into binary K-means clustering to automatically screen the clusters where the candidate physical modes are located; A modal same-order determination module (109) for effectively evaluating whether the modes with modal splitting phenomenon belong to the same order according to the synergistic effect of the modal assurance criterion MAC and the modal overlap factor MOF, wherein the modal assurance criterion MAC is evaluated from the perspective of modal shape coefficients, and the modal overlap factor MOF is evaluated from the perspectives of frequency and damping ratio; A modal parameter identification module (110) is configured to extract a representative mode indexed by a median damping ratio from all the candidate modes and perform modal parameter identification based on the representative mode. The modal parameter identification includes frequency, damping ratio, linearly normalized modal energy and the dimension of the clustering cluster .

9. An electronic device, characterized in that, Comprising a processor and a memory, the memory stores multiple instructions, and the processor is configured to read the instructions and execute the method according to any one of claims 1-7.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores multiple instructions, and the multiple instructions can be read and executed by the processor to execute the method according to any one of claims 1-7.

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