Structural modal parameter identification method and system based on least square complex frequency domain method

Abstract

The application discloses a structure modal parameter identification method and system based on a least square complex frequency domain method, relates to the technical field of structure modal parameter identification, and comprises the following steps: acquiring a frequency sequence and a frequency response function matrix; constructing a base function matrix step by step after setting analysis parameters; constructing a pseudo response matrix and a normal equation and introducing a truncated singular value decomposition to obtain a system denominator coefficient matrix; constructing a block companion matrix to complete pole extraction and stability screening, and obtaining a physical pole set through clustering; reconstructing a frequency response function and calculating an error, and adjusting a model order in a closed loop iteration; extracting and normalizing modal shapes, calculating natural frequencies and damping ratios, and outputting a complete modal parameter set. The application improves the automation degree of complex structure modal identification and the identification precision of close modal and low damping modal, and enhances the identification robustness in a noise scene through the whole process design of step-by-step base function construction, physical pole clustering and order closed loop iteration and the introduction of the truncated singular value decomposition.

Description

Technical Field

[0001] This application relates to the field of structural modal parameter identification technology, and in particular to a structural modal parameter identification method and system based on the least squares complex frequency domain method. Background Technology

[0002] Structural modal parameters (natural frequencies, damping ratios, and mode shapes) are core characteristics reflecting the essence of structural dynamics, and their accurate identification is fundamental to structural health monitoring, vibration control, and optimized design. The multi-reference least squares complex frequency domain (pLSCF) method, as the mainstream approach in modal identification, is widely used for modal parameter identification of complex structures due to its advantages such as strong close-range mode resolution, high computational efficiency, and clear stability diagrams.

[0003] However, traditional pLSCF methods and their improved versions still have many technical shortcomings: Fixed weighting strategies lack frequency band adaptability: The weighting functions of traditional pLSCF methods are mostly preset and fixed, unable to be dynamically adjusted according to the actual energy distribution, resonant peak positions, and signal-to-noise ratio of the structure's frequency response. In close-range modal regions, insufficient weights can lead to low resolution, while in noisy regions, excessive weights can introduce spurious modes. Mode selection is highly dependent on manual intervention and lacks unified physical criteria: Traditional pLSCF methods require technicians to select poles based on experience observing stability diagrams. There are no unified criteria for judging the physical rationality of poles, which can easily lead to the retention of spurious modes or the loss of physical modes due to subjective human judgment. The traditional pLSCF method suffers from several drawbacks. First, it lacks automated recognition capabilities. Second, it exhibits poor model order adaptability and lacks adaptive optimization. The model order is often manually preset, failing to dynamically adjust based on recognition deviations. This leads to slow convergence and low recognition accuracy when dealing with complex structures characterized by low damping, strong noise, and close modal coupling. Too low an order may miss physical modes, while too high an order may introduce numerous spurious modes. Third, it lacks robustness against noise interference. The least-squares fitting in the traditional pLSCF method does not perform weight suppression in noise regions. Measurement noise and structural nonlinear noise in the measured signal can easily distort the frequency response function, affecting the accuracy of pole extraction and modal parameter identification. Furthermore, constructing high-order polynomial models using the traditional pLSCF method requires solving large-scale normal equations. When a large number of reference points are selected, the analysis bandwidth is narrow, or measurement noise is mixed into the measured frequency response function (FRF), the denominator coefficient submatrix within the normal equation is prone to ill-conditioned problems, resulting in vastly different singular values, some of which are very small. In the process of directly inverting the matrix, the noise components carried by small singular values ​​are excessively amplified, causing the calculated poles to shift. This results in a large number of spurious poles being generated on the stability graph, directly interfering with the identification of real physical poles. For complex structures such as aircraft pylons and thin-walled components, which have multiple connection interfaces, dense modal distributions, and near-frequency coupling, the matrix ill-conditioning problem becomes even more prominent. This is the main technical bottleneck restricting the recognition accuracy and automated recognition capability of the pLSCF method.

[0004] In summary, existing structural modal parameter recognition technologies need to improve the automation level of complex structural modal recognition and the recognition accuracy of close-proximity modes and low-damped modes, and enhance the robustness of recognition in noisy scenarios. Summary of the Invention

[0005] This application provides a structural modal parameter identification method and system based on the least squares complex frequency domain method, which addresses the problems in existing structural modal parameter identification technologies, such as the need to improve the automation level of complex structural modal identification and the identification accuracy of close-proximity modes and low-damped modes, as well as the need to enhance the identification robustness in noisy scenarios.

[0006] On the one hand, this application provides a structural modal parameter identification method based on the least squares complex frequency domain method, including the following steps: Step 1: Obtain the frequency sequence and corresponding frequency response function matrix of the structure to be identified within the target frequency band, and complete the dimensional rearrangement and frequency band truncation.

[0007] Step 2: Set the frequency analysis range, minimum model order, initial value of maximum model order, and sampling time interval. For each model order from the minimum model order to the current maximum model order, construct the basis function matrix corresponding to the current order.

[0008] Step 3: Construct a pseudo-response matrix based on the basis function matrix, construct a normal equation block matrix based on the basis function matrix and the pseudo-response matrix, extract the denominator coefficient submatrix and perform singular value decomposition, determine the number of singular values ​​to be retained according to the preset relative truncation tolerance, construct a truncated pseudo-inverse, and obtain the system denominator coefficient matrix by least squares solution.

[0009] Step 4: Construct a block adjoint matrix based on the denominator coefficient matrix of the system and complete eigenvalue decomposition. Perform pole domain transformation and stability screening, retain stable poles with real parts less than zero and imaginary parts greater than zero, and cluster them to obtain a set of physical poles.

[0010] Step 5: Construct the frequency domain basis function matrix based on the physical pole set, reconstruct the frequency response function matrix and calculate the root mean square error, adjust the maximum model order based on the root mean square error, complete the closed-loop iterative optimization, and obtain the residual matrix after iteration.

[0011] Step 6: Perform singular value decomposition on the residual matrix after iteration, extract the complex mode shapes and normalize them to obtain the normalized mode shapes.

[0012] Step 7: Calculate the natural frequency and damping ratio based on the physical poles, integrate them with the normalized mode shapes, and output a complete set of modal parameters.

[0013] In one possible implementation, in step one, the frequency response function matrix is ​​obtained by preprocessing and performing a fast Fourier transform on the frequency domain signal of the multi-point response of the structure to be identified; the fast Fourier transform uses a one-sided spectral amplitude scaling method.

[0014] In one possible implementation, in step three, a pseudo-response matrix is ​​constructed by Kronecker product operation based on the basis function matrix corresponding to the current order.

[0015] The preset relative cutoff tolerance ranges from 1×10⁻⁶. -4 ~1×10 - ³.

[0016] In one possible implementation, in step four, the dimension of the block adjoint matrix is... ,in For the number of reference points, This represents the current model order.

[0017] In one possible implementation, step four, where the clustering yields the set of physical poles, includes: The stable poles obtained at the highest scan order are used as candidate poles. The number of times each candidate pole appears repeatedly in the scan results of different model orders is counted, and its cross-order stable occurrence rate is calculated. The cross-order stable occurrence rate is the ratio of the number of times the frequency value of the candidate pole appears within a preset relative frequency difference tolerance in all calculation results from the lowest model order to the highest scan order to the total number of scan orders. Only when the cross-order stable occurrence rate is not lower than a preset threshold, the corresponding pole is initially determined as a candidate physical pole.

[0018] Based on the preliminary screening of candidate physical poles, the frequency response function is reconstructed, and the root mean square error between the reconstructed frequency response and the measured frequency response is calculated. If the root mean square error is higher than a preset error threshold, the candidate pole with the lowest contribution is removed, and the reconstruction and error judgment are re-executed until the root mean square error meets the requirements, thus obtaining the set of physical poles.

[0019] Cluster analysis is performed on the set of physical poles to remove duplicate poles and conjugate redundant poles.

[0020] In one possible implementation, in step five, the column vectors of the frequency domain basis function matrix are, in order: the frequency domain basis function corresponding to the physical pole, the frequency domain basis function corresponding to the conjugate pole of the physical pole, the constant residual column vector, and the inertial residual column vector.

[0021] In one possible implementation, step five, adjusting the maximum model order based on the root mean square error, includes: When the root mean square error is greater than the preset error threshold, the maximum model order is increased and the process returns to step two to continue order scanning and pole extraction; when the root mean square error is not greater than the preset error threshold, the iteration stops.

[0022] In one possible implementation, in step six, phase adjustment is performed on the normalized mode shape to adjust the phase exceeding... The components are inverted and converted into real mode shapes.

[0023] In one possible implementation, in step six, the normalized mode shape is verified by a modal confidence criterion, and the modal confidence criterion value between the normalized mode shape and the reference mode shape is calculated. When the modal confidence criterion value is not lower than 0.95, the mode shape extraction result is determined to be reliable.

[0024] On the other hand, this application provides a structural modal parameter identification system based on the least squares complex frequency domain method, which is used to implement the above-mentioned structural modal parameter identification method based on the least squares complex frequency domain method. It includes: a frequency response matrix preprocessing module, an order scanning and basis function construction module, a normal equation solving module, a pole extraction and screening module, an order adaptive iterative optimization module, a mode shape extraction module, and a modal parameter output module, which are connected in sequence.

[0025] The frequency response matrix preprocessing module is configured to: obtain the frequency sequence of the structure to be identified in the target frequency band and the corresponding frequency response function matrix, and complete the dimension rearrangement and frequency band truncation.

[0026] The order scanning and basis function construction module is configured to: set the frequency analysis range, minimum model order, initial value of maximum model order, and sampling time interval; and construct the basis function matrix corresponding to the current order for each model order from the minimum model order to the current maximum model order.

[0027] The normal equation solving module is configured to: construct a pseudo-response matrix based on the basis function matrix; construct a normal equation block matrix based on the basis function matrix and the pseudo-response matrix; extract the denominator coefficient submatrix and perform singular value decomposition; determine the number of singular values ​​to be retained according to a preset relative truncation tolerance; construct a truncated pseudo-inverse; and obtain the system denominator coefficient matrix by least squares solution.

[0028] The pole extraction and screening module is configured to: construct a block adjoint matrix based on the system denominator coefficient matrix and complete eigenvalue decomposition; complete pole domain transformation and stability screening; retain stable poles with real parts less than zero and imaginary parts greater than zero; and cluster to obtain a set of physical poles.

[0029] The order adaptive iterative optimization module is configured to: construct a frequency domain basis function matrix based on the physical pole set, reconstruct the frequency response function matrix and calculate the root mean square error, adjust the maximum model order based on the root mean square error, complete the closed-loop iterative optimization, and obtain the residual matrix after iteration.

[0030] The mode shape extraction module is configured to: perform singular value decomposition on the residual matrix after iteration, extract complex mode shapes and normalize them to obtain normalized mode shapes.

[0031] The modal parameter output module is configured to calculate the natural frequency and damping ratio based on the physical poles, integrate them with the normalized mode shape, and output a complete set of modal parameters.

[0032] The structural modal parameter identification method and system based on the least squares complex frequency domain method in this application have the following advantages: By constructing graded basis functions, clustering physical poles, and implementing graded closed-loop iterations, and introducing truncated singular value decomposition (TSVD), the automation level of complex structural mode recognition and the recognition accuracy of close-range and low-damped modes are improved, enhancing the robustness of recognition in noisy scenarios. The proposed method performs SSVD on the residual matrix after iteration to ensure numerical stability during the solution process, avoid spurious mode interference, and further improve the reliability of the recognition results. This application introduces truncated SSVD regularization in the normal equation solution stage, relying on the relative energy truncation criterion of singular values ​​to filter out the noise subspace, solving the numerical divergence problem caused by direct inversion when the traditional pLSCF method handles near-frequency modes and ill-conditioned matrix conditions. Unlike fixed-parameter regularization methods, this scheme uses relative truncation tolerance, and the regularization strength can be adaptively adjusted, eliminating the need for repeated parameter adjustments by operators. After regularization, the poles obtained by the eigenvalue decomposition of the adjoint matrix are mapped through the complex domain, which improves the aggregation effect of real modal poles, suppresses false poles, and enhances the ability to distinguish close modes and resist noise. This provides a better pole data source for subsequent identification of physical modes through automatic clustering.

[0033] By automatically determining physical poles through multi-order pole frequency statistics and cluster analysis, and eliminating duplicate and conjugate redundant poles, the objectivity and accuracy of pole selection are improved, eliminating the reliance on manual interpretation of stability graphs and enhancing the automation level of modality recognition.

[0034] By automatically adjusting the maximum model order through frequency response reconstruction root mean square error feedback, a closed-loop iterative optimization mechanism is formed, which improves the adaptability of the model order to the actual modalities of the structure, effectively balances recognition accuracy and computational efficiency, and reduces the problems of false modes and mode omissions. The number of iterations for convergence is significantly reduced compared to the traditional pLSCF method, solving the problems of slow convergence speed and difficulty in balancing accuracy and efficiency in traditional methods, making it suitable for rapid modal recognition of large-scale and complex structures.

[0035] The method of this application has no specific frequency, resolution, or structure type limitations. The order adaptive optimization logic can be adapted to different engineering accuracy requirements and can be widely used in modal parameter identification of various complex structures in fields such as civil engineering, mechanical manufacturing, and aerospace. It has good versatility and engineering applicability. Attached Figure Description

[0036] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0037] Figure 1 A flowchart illustrating the structural modal parameter identification method based on the least squares complex frequency domain method provided in this application embodiment; Figure 2 This is a schematic diagram of a flat plate structure provided in an embodiment of this application; Figure 3 A schematic diagram of an aircraft structure (underwing 2-spike configuration) provided for an embodiment of this application; Figure 4 A schematic diagram of the experimental vibration modes of the flat plate structure at 27.319Hz:27.3196Hz provided in the embodiments of this application; Figure 5 A schematic diagram of the 27.319Hz:27.3196Hz algorithm vibration mode of the flat plate structure provided in the embodiments of this application; Figure 6 A schematic diagram of the experimental vibration modes of the flat plate structure at 55.109Hz:55.1152Hz provided in the embodiments of this application; Figure 7 A schematic diagram of the 55.109Hz:55.1152Hz algorithm vibration mode of the flat plate structure provided in the embodiments of this application; Figure 8 A schematic diagram of the experimental vibration modes of the flat plate structure at 80.374Hz:80.4616Hz provided in the embodiments of this application; Figure 9 A schematic diagram of the 80.374Hz:80.4616Hz algorithm vibration mode of the flat plate structure provided in the embodiments of this application; Figure 10 A schematic diagram of the experimental vibration modes of the flat plate structure at 96.550Hz:96.6600Hz provided in the embodiments of this application; Figure 11 A schematic diagram of the 96.550Hz:96.6600Hz algorithm vibration mode of the flat plate structure provided in the embodiments of this application; Figure 12 A schematic diagram of the experimental vibration modes at 41.979Hz:42.0342Hz for the aircraft structure (underwing 2-split configuration) provided in the embodiments of this application; Figure 13 A schematic diagram of the 41.979Hz:42.0342Hz mode shape algorithm for an aircraft structure (underwing 2-split configuration) provided in the embodiments of this application; Figure 14 A schematic diagram of the experimental vibration modes of the aircraft structure (underwing 2-suspended configuration) at 43.001Hz:42.9667Hz provided for the embodiments of this application; Figure 15 A schematic diagram of the 43.001Hz:42.9667Hz mode shape algorithm for an aircraft structure (underwing 2-split configuration) provided in the embodiments of this application; Figure 16A schematic diagram of the experimental vibration modes at 43.893Hz:43.9417Hz for the aircraft structure (underwing 2-suspended configuration) provided in the embodiments of this application; Figure 17 A schematic diagram of the 43.893Hz:43.9417Hz mode shape algorithm for an aircraft structure (underwing 2-split configuration) provided in the embodiments of this application; Figure 18 A comparison diagram of the reconstructed frequency response function curve and the measured curve of the flat plate structure provided in the embodiments of this application; Figure 19 A comparison chart of the reconstructed frequency response function curve and the measured curve of the aircraft structure (underwing 2-spike configuration) provided in the embodiments of this application. Detailed Implementation

[0038] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0039] like Figure 1 As shown in the figure, this application provides a structural modal parameter identification method based on the least squares complex frequency domain method, including the following steps: Step 1: Obtain the frequency sequence and corresponding frequency response function matrix of the structure to be identified within the target frequency band, and complete the dimensional rearrangement and frequency band truncation.

[0040] Step 2: Set the frequency analysis range, minimum model order, initial value of maximum model order, and sampling time interval. For each model order from the minimum model order to the current maximum model order, construct the basis function matrix corresponding to the current order.

[0041] Step 3: Construct a pseudo-response matrix based on the basis function matrix, construct a normal equation block matrix based on the basis function matrix and the pseudo-response matrix, extract the denominator coefficient submatrix and perform singular value decomposition, determine the number of singular values ​​to be retained according to the preset relative truncation tolerance, construct a truncated pseudo-inverse, and obtain the system denominator coefficient matrix by least squares solution.

[0042] Step 4: Construct a block adjoint matrix based on the denominator coefficient matrix of the system and complete eigenvalue decomposition. Perform pole domain transformation and stability screening, retain stable poles with real parts less than zero and imaginary parts greater than zero, and cluster them to obtain a set of physical poles.

[0043] Step 5: Construct the frequency domain basis function matrix based on the physical pole set, reconstruct the frequency response function matrix and calculate the root mean square error, adjust the maximum model order based on the root mean square error, complete the closed-loop iterative optimization, and obtain the residual matrix after iteration.

[0044] Step 6: Perform singular value decomposition on the residual matrix after iteration, extract the complex mode shapes and normalize them to obtain the normalized mode shapes.

[0045] Step 7: Calculate the natural frequency and damping ratio based on the physical poles, integrate them with the normalized mode shapes, and output a complete set of modal parameters.

[0046] For example, in step one, the frequency response function matrix is ​​obtained by preprocessing and fast Fourier transforming the frequency domain signal of the multi-measurement point response of the structure to be identified; the fast Fourier transform adopts a one-sided spectral amplitude scaling method.

[0047] Specifically, in this embodiment, in step one, the frequency sequence of the flat plate and aircraft structure (underwing 2-mount configuration) to be identified within the target frequency band and the corresponding multi-reference frequency response function matrix are obtained, wherein the dimension of the frequency response function matrix is... , The number of frequency points. In response to the number of measurement points, To determine the number of reference excitation points, the frequency response function matrix is ​​rearranged dimensionally to standardize its storage in a three-dimensional format of frequency point-reference point-response point. Simultaneously, a target frequency analysis range is defined, and frequency response function data within this band is extracted, eliminating low-frequency rigid body mode bands and high-frequency invalid interference bands. This provides standardized data for subsequent order scanning and parameter solving. This step is fundamental, adaptable to multi-reference frequency response data acquisition methods for various complex structures, requires no manual parameter presets, and possesses universality.

[0048] The frequency response function matrix is ​​obtained by preprocessing and performing a Fast Fourier Transform (FFT) on the frequency domain signals of the multi-measurement point responses of the structure to be identified. Preprocessing includes outlier removal, trend term removal, and Hanning windowing to eliminate baseline drift, signal leakage, and measurement noise interference. The FFT uses a one-sided spectral amplitude scaling method, defined as: .

[0049] The corresponding frequency sequence is: .

[0050] In the formula, The imaginary unit satisfies ; The sampling time interval is equal to the sampling frequency. satisfy ; The number of sample points for each data channel; This refers to the time sampling point number; Frequency line number; This is the preprocessed acceleration time history response signal.

[0051] Specifically, in this embodiment, step two involves setting the frequency analysis range, the minimum model order, the initial value of the maximum model order, and the sampling time interval; simultaneously, it presets judgment parameters such as the relative frequency tolerance, the threshold for the number of pole occurrences, the fitting error threshold, and the maximum number of iterations to complete the algorithm initialization configuration. The sampling time interval is determined by the sampling frequency of the measured data, satisfying the Nyquist sampling criterion, and requires no additional manual calibration.

[0052] For each model order from the minimum model order to the current maximum model order Construct the basis function matrix corresponding to the current order, with the basis functions in the form: .

[0053] In the formula, This represents the order-increasing number, corresponding to the order traversal index starting from the smallest model order; This represents the current model order. These are frequency point data, corresponding to the frequency sequence obtained in step one; The imaginary unit is ; the dimension of the basis function matrix is ​​. This provides standardized basis function data for the subsequent construction of normal equations.

[0054] For example, in step three, a pseudo-response matrix is ​​constructed based on the basis function matrix corresponding to the current order through the Kronecker product operation.

[0055] The preset relative cutoff tolerance ranges from 1×10⁻⁶. -4 ~1×10 - ³.

[0056] Specifically, in this embodiment, in step three, the basis function matrix and pseudo-response matrix corresponding to the current order are extracted, and the pseudo-response matrix is ​​constructed through Kronecker product operation. For each output channel and frequency point, the elements of the pseudo-response matrix are calculated as follows: .

[0057] In the formula, For the Kronecker product operation, For frequency point index, Here, k represents the output channel index, and k is the current traversal model order number. Let be the basis function matrix corresponding to the k-th order model. Let f be the row vector of the measured frequency response function at the f-th frequency point and the o-th measurement point; construct the normal equation block matrix as follows: , , .

[0058] In the formula, This is the basis function matrix corresponding to the current order. This is a pseudo-response matrix. For the complex conjugate transpose of the basis function matrix, To perform operations on the real part; solve the normal equation using least squares. This yields the polynomial coefficient vector to be estimated. The system denominator coefficient matrix is ​​then extracted from it. Due to the large differences in sensitivity among the channels of the measured multi-reference frequency response function matrix and the strong correlation of spectral lines in the near-frequency range, the denominator coefficient submatrix X usually exhibits severe ill-conditioning, with its singular values ​​distributed extremely unevenly, and some singular values ​​being extremely small. If X is directly inverted, the noise components corresponding to the small singular values ​​will be amplified drastically, leading to pole drift and a surge in pseudo-poles in the stability plot, which seriously affects the recognition accuracy.

[0059] To address the aforementioned problem, this step introduces truncated singular value decomposition (TSVD) regularization for solution. The specific mathematical process is as follows: Perform singular value decomposition on the denominator coefficient submatrix X: .

[0060] Where U is the left singular vector matrix and V is the right singular vector matrix. Let Σ be the conjugate transpose of the right singular vector matrix V, where Σ is a diagonal matrix and the singular values ​​are arranged in descending order along the diagonal, satisfying the following condition: . It represents the r-th (i.e. the smallest non-zero) singular value of the denominator coefficient submatrix X.

[0061] Based on the preset relative cutoff tolerance (This embodiment is preferred) The number of singular values ​​k to be retained is determined by the following criteria: .

[0062] This means retaining the first k principal singular values ​​whose energy percentage is greater than the tolerance, and discarding the subsequent tiny singular values ​​whose energy percentage is less than the tolerance. The physical meaning of this criterion is that the data components in the singular vector direction corresponding to small singular values ​​have extremely low energy, mainly reflecting measurement noise and numerical rounding errors. Discarding these components can significantly improve the stability of the solution without losing effective information.

[0063] Based on this, a truncated singular value matrix Σ is constructed. k (Only the first k singular values ​​are retained) and the corresponding left and right singular vector submatrices U k V k And calculate the truncated pseudoinverse of the denominator coefficient submatrix X: .

[0064] , , These are the left singular vector submatrices, right singular vector submatrices, and singular value diagonal submatrices, respectively, containing the first k singular values; H is the conjugate transpose. It is the truncated pseudoinverse of matrix X.

[0065] Finally, the coefficient vector of the system's denominator polynomial The following formula can be used to obtain: .

[0066] In the formula, Let Y be the coefficient matrix (vector) of the denominator polynomial of the system to be solved, and let Y be the cross-block matrix corresponding to the numerator polynomial in the normal equation, obtained by summing the frequency response functions of each measurement point. This truncation process mathematically discards the small singular components corresponding to the noise subspace, fundamentally avoiding the ill-conditioned amplification effect caused by direct inversion. Compared to traditional Tikhonov regularization, which requires manual selection of regularization parameters, the TSVD method used in this scheme achieves adaptive determination of the regularization strength through the relative truncation tolerance ε, which has clear physical meaning and higher computational efficiency, significantly improving the numerical robustness of pole solving and the interpretability of the stability graph.

[0067] For example, in step four, the dimension of the block adjoint matrix is... ,in For the number of reference points, This represents the current model order.

[0068] Specifically, in this embodiment, in step four, the coefficient matrix of the denominator polynomial obtained after TSVD regularization is acquired. The coefficient vector is divided and reorganized into blocks according to the current model order n to construct a block adjoint matrix for pole extraction. : .

[0069] In the formula, n is the model order; ~ It consists of coefficient vectors The denominator polynomial coefficient matrix block obtained after block division; Represents the identity matrix; Represents the zero matrix; the dimension of this block adjoint matrix is... , The number of reference points. Perform eigenvalue decomposition on the block adjoint matrix and solve the characteristic equation: .

[0070] In the formula, Representing the block adjoint matrix eigenvector matrix, For discrete domain poles The diagonal eigenvalue matrix is ​​formed. Solving the characteristic equation yields the discrete-domain poles, i.e., the eigenvalue matrix. Each diagonal element Logarithmic mapping transformation is used to convert discrete-domain poles into continuous-time domain complex poles: .

[0071] in The sampling time interval, , Sampling frequency, These are continuous-time domain complex poles. To ensure that the poles have practical physical meaning, pole screening is performed, retaining only candidate poles that meet the following conditions: .

[0072] In other words, only poles with real parts less than zero and imaginary parts greater than zero are retained, corresponding to physically realizable damped oscillation modes, while conjugate redundant solutions and non-physical solutions are eliminated. Based on the selected effective continuous complex poles, the natural frequencies of the structure are calculated. Damping ratio : .

[0073] pole magnitude .

[0074] After TSVD regularization, the solution bias caused by the ill-conditioned matrix is ​​suppressed, the clustering degree of real physical poles in the output pole set is significantly improved, and the pseudo poles are more dispersed, providing high signal-to-noise ratio input data for subsequent modal adaptive clustering screening and improving the reliability of the automatic physical pole identification results.

[0075] For example, in step four, the clustering to obtain the set of physical poles includes: The stable poles obtained at the highest scan order are used as candidate poles. The number of times each candidate pole appears repeatedly in the scan results of different model orders is counted, and its cross-order stable occurrence rate is calculated. The cross-order stable occurrence rate is the ratio of the number of times the frequency value of the candidate pole appears within a preset relative frequency difference tolerance in all calculation results from the lowest model order to the highest scan order to the total number of scan orders. Only when the cross-order stable occurrence rate is not lower than a preset threshold, the corresponding pole is initially determined as a candidate physical pole.

[0076] Based on the preliminary screening of candidate physical poles, the frequency response function is reconstructed, and the root mean square error between the reconstructed frequency response and the measured frequency response is calculated. If the root mean square error is higher than a preset error threshold, the candidate pole with the lowest contribution is removed, and the reconstruction and error judgment are re-executed until the root mean square error meets the requirements, thus obtaining the set of physical poles.

[0077] Cluster analysis is performed on the set of physical poles to remove duplicate poles and conjugate redundant poles.

[0078] Specifically, in this embodiment, in step four, the stable poles obtained at the highest scan order are used as candidate poles. The number of times each candidate pole appears repeatedly in the scan results of different model orders is counted, and its cross-order stable occurrence rate is calculated. That is, the ratio of the number of times the frequency value of the candidate pole appears within a preset relative frequency difference tolerance in all calculation results from the lowest model order to the highest scan order to the total number of scan orders. Only when the cross-order stable occurrence rate is not lower than a preset threshold is the corresponding pole initially determined as a candidate physical pole. Further, to improve the reliability of the screening, this embodiment also introduces a frequency response function reconstruction verification criterion: the frequency response function is reconstructed based on the initially screened candidate physical poles. If the root mean square error between the reconstructed frequency response and the measured frequency response is higher than a preset threshold, the candidate pole with the lowest contribution is removed, and reconstruction and error judgment are re-executed until the root mean square error meets the requirements. Through the dual constraints of the above "initial screening of cross-order occurrence rate + fine screening of FRF reconstruction error", the set of physical poles is finally determined. Among them, the preset relative frequency tolerance, occurrence rate threshold, and reconstruction error threshold can be flexibly set according to the complexity of the structure to be identified and the engineering accuracy requirements; the physical pole set is treated using IG. The FC iterative fuzzy C-means clustering algorithm performs automatic clustering, removing duplicate poles and conjugate redundant poles to obtain a set of effective physical poles without redundancy.

[0079] For example, in step five, the column vectors of the frequency domain basis function matrix are, in order: the frequency domain basis function corresponding to the physical pole, the frequency domain basis function corresponding to the conjugate pole of the physical pole, the constant residual column vector, and the inertial residual column vector.

[0080] Specifically, in this embodiment, in step five, a frequency domain basis function matrix is ​​constructed based on the set of physical poles. This frequency domain basis function matrix includes at least the physical pole terms, their conjugate pole terms, constant residual terms, and inertial residual terms. The column vectors of the frequency domain basis function matrix are, in order: the frequency domain basis functions corresponding to the physical poles. The frequency domain basis functions corresponding to their conjugate poles A column vector of all 1s (constant residual term) Column vector (inertia residual term), where For the first A physical pole, Its conjugate pole, Let be the angular frequency; perform least squares calculation on the reconstructed frequency response function matrix to obtain the residual matrix. And reconstruct the frequency response function matrix accordingly: .

[0081] In the formula, To reconstruct the frequency response function matrix; To integrate the frequency domain basis function matrix composed of the modal pole basis functions, constant residual terms, and inertial residual terms of each order; The residual coefficient matrix obtained by least squares fitting includes modal residues, static stiffness compensation coefficients, and equivalent mass compensation coefficients.

[0082] The reconstruction process incorporates static stiffness and equivalent mass terms for compensation, which effectively improves the fitting accuracy of the frequency response function.

[0083] For example, in step five, adjusting the maximum model order based on the root mean square error includes: When the root mean square error is greater than the preset error threshold, the maximum model order is increased and the process returns to step two to continue order scanning and pole extraction; when the root mean square error is not greater than the preset error threshold, the iteration stops.

[0084] Specifically, in this embodiment, in step five, the root mean square error (RMSE) between the reconstructed frequency response function matrix and the measured frequency response function matrix is ​​calculated; when the RMSE is greater than a preset error threshold, the maximum model order is increased and the process returns to step two to continue order scanning and pole extraction; when the RMSE is not greater than the preset error threshold, the iteration stops. The formula for calculating the RMSE is: .

[0085] in, To reconstruct the frequency response function matrix, This is the measured frequency response function matrix. The number of frequency points. For the number of reference points, The number of response points, preset error threshold, maximum model order improvement step size, and maximum iteration limit can be flexibly configured according to engineering accuracy requirements. When the iteration limit is reached, the system outputs the current optimal result and gives an early warning.

[0086] For example, in step six, the phase of the normalized mode shape is adjusted to adjust the phase exceeding... The components are inverted and converted into real mode shapes.

[0087] Specifically, in this embodiment, in step six, the residual matrix is ​​decomposed into singular values ​​for each mode, and the first left singular vector is taken as the complex mode shape of the corresponding mode; the amplitude of the complex mode shape is normalized to obtain the normalized mode shape. .

[0088] in For the first The first-order complex mode shape vector is used to adjust the phase of the normalized mode shape, adjusting the phase beyond the specified value. The components are inverted and converted into real mode shapes.

[0089] For example, in step six, the normalized mode shape is verified by the modal confidence criterion, and the modal confidence criterion value between the normalized mode shape and the reference mode shape is calculated. When the modal confidence criterion value is not lower than 0.95, the mode shape extraction result is determined to be reliable.

[0090] Specifically, in this embodiment, in step six, the modal confidence criterion values ​​of each matched mode are all higher than 0.99, proving that the mode shapes extracted by this method have extremely high spatial correlation and can be directly used for structural dynamic response prediction.

[0091] Specifically, in this embodiment, in step seven, the natural frequency and damping ratio of the structure are calculated based on the physical poles, and the modal parameters of the structure to be identified are output in combination with the normalized mode shapes: .

[0092] in, The natural frequency of the i-th mode (in Hz). Let i be the damping ratio of the i-th mode. For the i-th physical pole, The real part of the pole, The modulus is the value of the pole; the output modal parameters include natural frequency, damping ratio, and normalized mode shape.

[0093] In one embodiment, to further illustrate the technical solution and implementation effect of this application, the following uses thin-walled plate structure and multi-component coupling structure as typical verification objects to describe the specific implementation of this application in detail, such as... Figure 2 and Figure 3 The diagrams shown are schematic representations of a flat plate structure and an aircraft structure (with two underwing pylons). The flat plate structure corresponds to a thin-walled plate structure, while the aircraft structure (with two underwing pylons) corresponds to a multi-component coupled structure. Flat plate structures, as typical thin-walled components in civil engineering and mechanical manufacturing, generally exhibit characteristics such as close modal coupling, low damping, and significant spatial coupling effects at multiple measurement points. Aircraft structures (with two underwing pylons), as core components in aerospace and communications, are characterized by multi-component coupling, dense modal data, low damping, and susceptibility to measurement noise interference. Both types of structures are complex structures with high modal recognition difficulty. The technical limitations of existing pLSCF methods make it difficult to meet the engineering requirements for automated, high-precision, and highly robust modal parameter recognition of flat plate structures and aircraft structures (with two underwing pylons). This application, as a general modal parameter recognition system, is not limited to specific structural types. The following implementation examples are only used to verify the effectiveness of the method, and the scope of protection of this application is not limited to these examples.

[0094] Implementation Target 1: Thin-walled plate structure, taking structural steel plate as an example.

[0095] The structure is a thin-walled steel plate, which is suspended by elastic ropes to simulate free boundary conditions. The structure has multiple sets of near-frequency coupled modes (natural frequency interval ≤ 1Hz) and belongs to a low-damping structure (damping ratio ≤ 1%).

[0096] Test scheme: 36 response measurement points are evenly arranged on the flat plate, and multiple sets of reference excitation points are set simultaneously. The corresponding sampling frequency and sampling duration are set according to the predicted modal distribution frequency band of the structure. The analysis frequency range covers the main distribution frequency band of the closely coupled modes of the structure.

[0097] Following the steps described in this application, when the recognition order is 15, the algorithm identifies 5 modes, while the experiment identifies 4 modes. The final recognition results for comparable thin-walled plate structures are as follows (where the 4th mode can be compared with the experimental results, and the other 1st mode is an additional mode identified by the algorithm, which was not captured in the experiment and is not included in the comparison table): Table 1. Results of Modal Parameter Identification for Thin-Walled Plate Structures

[0098] Figures 4 to 11 This is a comparison of experimental and algorithmic vibration modes of a flat plate structure at different frequencies. Figure 4 , 6 8 and 10 are the experimental vibration modes. Figure 5 , 7 9 and 11 are the algorithmic vibration modes. Figure 4 , 5 Corresponding to mode number 1 in Table 1, Figure 6 , 7 Corresponding to mode number 2 in Table 1, Figure 8 , 9 Corresponding to mode number 3 in Table 1, Figure 10 , 11 This corresponds to mode number 4 in Table 1. It can be seen that the method of this application has high accuracy in identifying modal parameters for thin-walled plate structures.

[0099] Implementation Target 2: Multi-component coupled structure, taking aircraft structure (underwing 2-suspended configuration) as an example.

[0100] The structure is a thin-walled aluminum alloy structure, exhibiting near-frequency coupling modes in the 38–48 Hz frequency band. The experiment used an elastic rope suspension to simulate a free boundary, with measuring points evenly distributed. The sampling frequency was 256 Hz, the sampling duration was 4 seconds, and the analysis frequency range was 38–48 Hz. Following the steps described in this application, the final identification results of the structure are shown in the table below.

[0101] Test scheme: Measurement points are evenly distributed, elastic suspension simulates free boundary, sampling frequency is 256Hz, sampling duration is 4s; analysis frequency range is 38~48Hz.

[0102] Following the steps described in this application, when the algorithm identification order is 45 and the experimental order is 55, the algorithm identifies 6 modes and the experiment identifies 3 modes. The final identification results of the multi-component coupled structure of the comparable aircraft structure (underwing 2-pylon configuration) are as follows (where the 3rd mode can be compared with the experimental results, and the other 3 modes are additional modes identified by the algorithm, which were not captured in the experiment and are not included in the comparison table): Table 2. Modal parameter identification results of aircraft structure (underwing 2-pylon configuration)

[0103] Figures 12 to 17 This is a comparison of experimental and algorithmic vibration modes of the aircraft structure (two underwing pylons) at different frequencies. Figure 12 , 14 16 represents the experimental vibration mode. Figure 13 , 15 17 represents the algorithm's vibration mode. Figure 12 , 13 Corresponding to mode number 1 in Table 2, Figure 14 , 15 Corresponding to mode number 2 in Table 2, Figure 16 , 17 This corresponds to mode number 3 in Table 2. It can be seen that the method of this application has high accuracy in identifying modal parameters for multi-component coupled structures.

[0104] Figure 18 This is a comparison chart of the reconstructed frequency response function curve and the measured curve of the flat plate structure provided in the embodiments of this application. Figure 18 It can be seen that the reconstructed frequency response function curve of the plate structure obtained by this method matches the measured curve well throughout the entire frequency band, especially near the resonance peaks (such as at modal frequencies of 27.3Hz, 55.1Hz, 80.5Hz, and 96.7Hz), where the fitting accuracy is high. The amplitude and phase can track the experimental curve well, indicating that the method proposed in this application can effectively capture the dynamic characteristics of thin-walled plate structures. The reconstruction error meets the engineering requirements, verifying the accuracy and reliability of the identification results.

[0105] Figure 19 This is a comparison chart of the reconstructed frequency response function curve and the measured curve of the aircraft structure (underwing 2-pylon configuration) provided in the embodiments of this application. Figure 19 It can be seen that within the 38-48Hz analysis frequency band, the reconstructed frequency response function curve of the aircraft structure (underwing 2-pylon configuration) shows a consistent trend with the measured curve. Particularly in the close-modal regions of 41.979Hz, 43.001Hz, and 43.893Hz, the reconstructed curve maintains high fitting accuracy without significant amplitude shift or phase distortion. This indicates that the proposed method has good adaptability and anti-interference capabilities for multi-component coupled structures and close-modal scenarios, further demonstrating the practical value of the algorithm in complex structural modal recognition. To further quantify the recognition efficiency advantage of this method, the recognition results of the aforementioned aircraft pylon structure are compared with those of the experimental LMS PolyMAX method. Under the same experimental data, this method only requires a 45th model order to stably identify 6 effective physical modes in the target frequency band, while LMS PolyMAX can only identify 3 dominant modes at a higher model order of 55. This shows that this method captures more potential physical modes with lower computational complexity while ensuring high consistency with PolyMAX modal frequencies, damping ratios and mode shapes. It solves the contradiction of traditional methods missing low-order modes and having large computational load for high-order modes, and is especially suitable for complex structures with close modal coupling.

[0106] This application also provides a structural modal parameter identification system based on the least squares complex frequency domain method, which is used to implement the above-mentioned structural modal parameter identification method based on the least squares complex frequency domain method. The system includes: a frequency response matrix preprocessing module, an order scanning and basis function construction module, a normal equation solving module, a pole extraction and screening module, an order adaptive iterative optimization module, a mode shape extraction module, and a modal parameter output module, which are connected in sequence.

[0107] The frequency response matrix preprocessing module is configured to: obtain the frequency sequence of the structure to be identified in the target frequency band and the corresponding frequency response function matrix, and complete the dimension rearrangement and frequency band truncation.

[0108] The order scanning and basis function construction module is configured to: set the frequency analysis range, minimum model order, initial value of maximum model order, and sampling time interval; and construct the basis function matrix corresponding to the current order for each model order from the minimum model order to the current maximum model order.

[0109] The normal equation solving module is configured to: construct a pseudo-response matrix based on the basis function matrix; construct a normal equation block matrix based on the basis function matrix and the pseudo-response matrix; extract the denominator coefficient submatrix and perform singular value decomposition; determine the number of singular values ​​to be retained according to a preset relative truncation tolerance; construct a truncated pseudo-inverse; and obtain the system denominator coefficient matrix by least squares solution.

[0110] The pole extraction and screening module is configured to: construct a block adjoint matrix based on the system denominator coefficient matrix and complete eigenvalue decomposition; complete pole domain transformation and stability screening; retain stable poles with real parts less than zero and imaginary parts greater than zero; and cluster to obtain a set of physical poles.

[0111] The order adaptive iterative optimization module is configured to: construct a frequency domain basis function matrix based on the physical pole set, reconstruct the frequency response function matrix and calculate the root mean square error, adjust the maximum model order based on the root mean square error, complete the closed-loop iterative optimization, and obtain the residual matrix after iteration.

[0112] The mode shape extraction module is configured to: perform singular value decomposition on the residual matrix after iteration, extract complex mode shapes and normalize them to obtain normalized mode shapes.

[0113] The modal parameter output module is configured to calculate the natural frequency and damping ratio based on the physical poles, integrate them with the normalized mode shape, and output a complete set of modal parameters.

[0114] This application's embodiments improve the automation level and recognition accuracy of close-range and low-damped modes in complex structure mode identification by introducing a full-process design of order-wise basis function construction, physical pole clustering, and order-wise closed-loop iteration, and by introducing truncated singular value decomposition (TSVD). This enhances the robustness of identification in noisy scenarios. The proposed method of performing SSVD on the residual matrix after iteration ensures the numerical stability of the solution process, avoids spurious mode interference, and further improves the reliability of the identification results. This application introduces truncated SSVD regularization in the normal equation solution stage, relying on the relative energy truncation criterion of singular values ​​to filter out the noise subspace, solving the numerical divergence problem caused by direct inversion when the traditional pLSCF method handles near-frequency modes and ill-conditioned matrix conditions. Unlike fixed-parameter regularization methods, this scheme uses relative truncation tolerance, and the regularization strength can be adaptively adjusted, eliminating the need for repeated parameter adjustments by the operator. After regularization, the poles obtained by the eigenvalue decomposition of the adjoint matrix are mapped through the complex domain, which improves the aggregation effect of real modal poles, suppresses false poles, and enhances the ability to distinguish close modes and resist noise. This provides a better pole data source for subsequent identification of physical modes through automatic clustering.

[0115] By automatically determining physical poles through multi-order pole frequency statistics and cluster analysis, and eliminating duplicate and conjugate redundant poles, the objectivity and accuracy of pole selection are improved, eliminating the reliance on manual interpretation of stability graphs and enhancing the automation level of modality recognition.

[0116] By automatically adjusting the maximum model order through frequency response reconstruction root mean square error feedback, a closed-loop iterative optimization mechanism is formed, which improves the adaptability of the model order to the actual modalities of the structure, effectively balances recognition accuracy and computational efficiency, and reduces the problems of false modes and mode omissions. The number of iterations for convergence is significantly reduced compared to the traditional pLSCF method, solving the problems of slow convergence speed and difficulty in balancing accuracy and efficiency in traditional methods, making it suitable for rapid modal recognition of large-scale and complex structures.

[0117] The method of this application has no specific frequency, resolution, or structure type limitations. The order adaptive optimization logic can be adapted to different engineering accuracy requirements and can be widely used in modal parameter identification of various complex structures in fields such as civil engineering, mechanical manufacturing, and aerospace. It has good versatility and engineering applicability.

[0118] Although preferred embodiments of this application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of this application.

[0119] Obviously, those skilled in the art can make various modifications and variations to this application without departing from the spirit and scope of this application. Therefore, if such modifications and variations fall within the scope of the claims of this application and their equivalents, this application also intends to include such modifications and variations.

Claims

1. A structural modal parameter identification method based on the least squares complex frequency domain method, characterized in that, Includes the following steps: Step 1: Obtain the frequency sequence and corresponding frequency response function matrix of the structure to be identified within the target frequency band, and complete the dimensional rearrangement and frequency band truncation. Step 2: Set the frequency analysis range, minimum model order, initial value of maximum model order, and sampling time interval. For each model order from the minimum model order to the current maximum model order, construct the basis function matrix corresponding to the current order. Step 3: Construct a pseudo-response matrix based on the basis function matrix, construct a normal equation block matrix based on the basis function matrix and the pseudo-response matrix, extract the denominator coefficient submatrix and perform singular value decomposition, determine the number of singular values ​​to be retained according to the preset relative truncation tolerance, construct a truncated pseudo-inverse, and obtain the system denominator coefficient matrix by least squares solution. Step 4: Construct a block adjoint matrix based on the denominator coefficient matrix of the system and complete eigenvalue decomposition. Perform pole domain transformation and stability screening, retain stable poles with real parts less than zero and imaginary parts greater than zero, and cluster them to obtain a set of physical poles. Step 5: Construct the frequency domain basis function matrix based on the physical pole set, reconstruct the frequency response function matrix and calculate the root mean square error, adjust the maximum model order based on the root mean square error, complete the closed-loop iterative optimization, and obtain the residual matrix after iteration. Step 6: Perform singular value decomposition on the residual matrix after iteration, extract the complex mode shapes and normalize them to obtain the normalized mode shapes; Step 7: Calculate the natural frequency and damping ratio based on the physical poles, integrate them with the normalized mode shapes, and output a complete set of modal parameters.

2. The structural modal parameter identification method based on the least squares complex frequency domain method according to claim 1, characterized in that, In step one, the frequency response function matrix is ​​obtained by preprocessing and fast Fourier transforming the frequency domain signal of the multi-measurement point response of the structure to be identified; the fast Fourier transform adopts a one-sided spectral amplitude scaling method.

3. The structural modal parameter identification method based on the least squares complex frequency domain method according to claim 1, characterized in that, In step three, a pseudo-response matrix is ​​constructed based on the basis function matrix corresponding to the current order through the Kronecker product operation; The preset relative cutoff tolerance ranges from 1×10⁻⁶. -4 ~1×10 - ³.

4. The structural modal parameter identification method based on the least squares complex frequency domain method according to claim 1, characterized in that, In step four, the dimension of the block adjoint matrix is... ,in For the number of reference points, This represents the current model order.

5. The structural modal parameter identification method based on the least squares complex frequency domain method according to claim 1, characterized in that, In step four, the clustering to obtain the set of physical poles includes: Stable poles obtained at the highest scan order are used as candidate poles. The number of times each candidate pole appears repeatedly in the scan results of different model orders is counted, and its cross-order stability occurrence rate is calculated. The cross-order stability occurrence rate is the ratio of the number of times the frequency value of the candidate pole appears within a preset relative frequency difference tolerance in all calculation results from the lowest model order to the highest scan order to the total number of scan orders. Only when the cross-order stability occurrence rate is not lower than a preset threshold, the corresponding pole is initially determined as a candidate physical pole. Based on the preliminary screening of candidate physical poles, the frequency response function is reconstructed, and the root mean square error between the reconstructed frequency response and the measured frequency response is calculated. If the root mean square error is higher than the preset error threshold, the candidate pole with the lowest contribution is removed and the reconstruction and error judgment are re-executed until the root mean square error meets the requirements, and the set of physical poles is obtained. Cluster analysis is performed on the set of physical poles to remove duplicate poles and conjugate redundant poles.

6. The structural modal parameter identification method based on the least squares complex frequency domain method according to claim 1, characterized in that, In step five, the column vectors of the frequency domain basis function matrix are, in order: the frequency domain basis function corresponding to the physical pole, the frequency domain basis function corresponding to the conjugate pole of the physical pole, the constant residual column vector, and the inertial residual column vector.

7. The structural modal parameter identification method based on the least squares complex frequency domain method according to claim 1, characterized in that, In step five, adjusting the maximum model order based on the root mean square error includes: When the root mean square error is greater than the preset error threshold, the maximum model order is increased and the process returns to step two to continue order scanning and pole extraction; when the root mean square error is not greater than the preset error threshold, the iteration stops.

8. The structural modal parameter identification method based on the least squares complex frequency domain method according to claim 1, characterized in that, In step six, the phase of the normalized mode shape is adjusted to adjust the phase exceeding... The components are inverted and converted into real mode shapes.

9. The structural modal parameter identification method based on the least squares complex frequency domain method according to claim 1, characterized in that, In step six, the normalized mode shape is verified by the modal confidence criterion. The modal confidence criterion value between the normalized mode shape and the reference mode shape is calculated. When the modal confidence criterion value is not less than 0.95, the mode shape extraction result is determined to be reliable.

10. A structural modal parameter identification system based on the least squares complex frequency domain method, used to implement the structural modal parameter identification method based on the least squares complex frequency domain method as described in any one of claims 1 to 9, characterized in that, include: The system consists of the following modules connected sequentially: frequency response matrix preprocessing module, order scanning and basis function construction module, normal equation solving module, pole extraction and screening module, order adaptive iterative optimization module, mode shape extraction module, and modal parameter output module. The frequency response matrix preprocessing module is configured to: obtain the frequency sequence of the structure to be identified in the target frequency band and the corresponding frequency response function matrix, and complete the dimension rearrangement and frequency band truncation. The order scanning and basis function construction module is configured to: set the frequency analysis range, minimum model order, initial value of maximum model order, and sampling time interval; and construct the basis function matrix corresponding to the current order for each model order from the minimum model order to the current maximum model order. The normal equation solving module is configured to: construct a pseudo-response matrix based on the basis function matrix; construct a normal equation block matrix based on the basis function matrix and the pseudo-response matrix; extract the denominator coefficient submatrix and perform singular value decomposition; determine the number of singular values ​​to be retained according to the preset relative truncation tolerance; construct a truncated pseudo-inverse; and obtain the system denominator coefficient matrix by least squares solution. The pole extraction and screening module is configured to: construct a block adjoint matrix based on the system denominator coefficient matrix and complete eigenvalue decomposition; complete pole domain transformation and stability screening; retain stable poles with real parts less than zero and imaginary parts greater than zero; and cluster to obtain a set of physical poles. The order adaptive iterative optimization module is configured to: construct a frequency domain basis function matrix based on the physical pole set, reconstruct the frequency response function matrix and calculate the root mean square error, adjust the maximum model order based on the root mean square error, complete the closed-loop iterative optimization, and obtain the residual matrix after iteration. The mode shape extraction module is configured to: perform singular value decomposition on the residual matrix after iteration, extract complex mode shapes and normalize them to obtain normalized mode shapes; The modal parameter output module is configured to calculate the natural frequency and damping ratio based on the physical poles, integrate them with the normalized mode shape, and output a complete set of modal parameters.

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