Underdetermined modal identification method based on block term decomposition and adaptive kurtosis harmonic separation

Abstract

This invention provides an underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation, belonging to the field of structural health monitoring and operating mode identification technology. The method acquires multi-channel vibration responses, calculates the covariance matrices under different time delays, and stacks them into a third-order tensor. Block term tensor decomposition is used to extract the time-domain autocovariance function sequence of each potential component. The natural frequencies and damping ratios of each component are initially estimated using local peak finding and logarithmic decay methods. The minimum truncation length is adaptively calculated based on the coupling relationship between period and damping ratio. Kurtosis values ​​are calculated for the truncated autocovariance sequence, and harmonic interference is eliminated based on a kurtosis threshold while retaining the true structural modes. Finally, mode shapes are extracted from the spatial block matrix of the retained components, outputting high-precision natural frequencies, damping ratios, and mode shapes. This invention can achieve accurate identification of dense modes and high-damped modes under complex operating conditions with severe underdeterminacy and harmonic interference, improving the robustness of operating mode parameter identification.

Description

Technical Field

[0001] This invention relates to the field of structural health monitoring and operational mode identification technology, specifically to an underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation. Background Technology

[0002] In the health monitoring and dynamic testing of large-scale engineering structures (such as bridges, high-rise buildings, and aerospace vehicles), due to practical engineering limitations such as high sensor deployment costs, limited installation space, or uneven deployment, the dimension of the actually observable vibration response is often much smaller than the true active modal order of the structure, thus forming a typical problem of underdetermined operating modal parameter identification. To address this challenge, existing research has mainly developed three mainstream methods: sparsity methods, compressed sensing methods, and tensor decomposition methods.

[0003] Sparse Component Analysis (SCA), by assuming the sparsity of the source signal in the time-frequency domain, estimates independent sources from underdetermined aliasing signals using sparse optimization and geometric clustering. It has achieved some success in processing field test data and modally dense environments. However, when faced with conditions common in real-world structures, such as extremely dense modes, highly overlapping frequency bands, or high damping, the sparsity of each modal component rapidly decays, making this method highly susceptible to modal aliasing or the generation of spurious modes. Similarly, compressed sensing methods that model underdetermined identification as sparse signal reconstruction (such as L1-regularized modal damping identification optimization algorithms) also face the bottleneck of decreased reconstruction accuracy in complex and non-rigidly sparse real-world dynamic environments.

[0004] In contrast, tensor decomposition methods utilize the spatiotemporal physical structure characteristics of multi-channel response data to construct high-dimensional tensors for joint decomposition. In particular, the identification framework based on Block Term Decomposition (BTD) proposed in recent years utilizes the low-order block representation characteristics of vibration signals, which not only better maps the physical characteristics of structural complex conjugate pole pairs, but also ensures the uniqueness of tensor decomposition under the blind source separation framework, thereby improving the identifiability of source signal separation under underdetermined conditions.

[0005] However, this method still has limitations in practical engineering applications: on the one hand, the selection of model parameters (such as the number of extracted components and the order of the factor matrix) in block term decomposition depends on experience, which has a huge impact on the identification results; on the other hand, in large-scale structures, environmental excitations often include not only random white noise, but also a large number of strong periodic harmonic interferences caused by mechanical operation. Although existing block term tensor decomposition can separate potential sources, it is extremely difficult to automatically distinguish which are real structural physical modes and which are false harmonic interferences. In traditional engineering, kurtosis is often used as a discrimination index, but when dealing with dense or highly damped modes, if the separated signal is truncated with a fixed length, the excessive attenuation of the tail end of the highly damped signal will introduce a large amount of computational noise, causing serious distortion of the kurtosis statistical regularity of the real mode, which will then be misjudged as harmonics and eliminated by the algorithm.

[0006] In view of the above, this application is hereby submitted. Summary of the Invention

[0007] This invention provides an underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation, which can at least partially improve the above-mentioned problems.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] An underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation, comprising: The multi-channel vibration response signal matrix collected by the preset sensor components is obtained, the spatial covariance matrix corresponding to the multi-channel vibration response signal matrix under different time delays is calculated, and the spatial covariance matrix is ​​stacked sequentially along the time delay dimension to obtain the third-order covariance tensor. The covariance third-order tensor is decoupled and optimized using a pre-defined block term tensor decomposition model to obtain a sequence of time-domain autocovariance functions. The time-domain autocovariance function sequence is preliminarily analyzed using the time-domain logarithmic decay method or the frequency-domain single-mode fitting method to obtain the corresponding damping ratio, actual natural frequency and natural period. Based on the damping ratio and natural period, the adaptive minimum cutoff time length is calculated, and the adaptive minimum cutoff time length is converted into the number of discrete data sampling points; The time-domain autocovariance function sequence is truncated according to the number of discrete data sampling points to obtain an adaptively truncated effective signal vector, and the adaptive kurtosis eigenvalue is calculated based on the effective signal vector. Based on adaptive kurtosis eigenvalues, each potential source is classified and identified to obtain the retained structural physical mode components. Then, by combining the retained structural physical mode components, the corresponding damping ratio, the actual natural frequency, and the spatial hybrid block matrix, the complete structural underdetermined operating mode parameters are output.

[0010] In summary, this invention provides an underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation, which can achieve effective separation and high-precision parameter identification of dense modes and external harmonics under underdetermined conditions. This method is used to achieve accurate separation and high-precision parameter identification of the actual physical modes of a structure from external harmonics under complex operating conditions with a limited number of sensors and strong periodic harmonic interference. First, multi-channel vibration response signals of the structure are acquired, and their spatial covariance matrices at different time delays are calculated. Then, a third-order covariance tensor containing the spatiotemporal evolution characteristics of the system is constructed by stacking along the time delay dimension. Subsequently, within the framework of underdetermined blind source separation, a rank-wise... The Block Term Tensor Decomposition (BTD) model decouples the covariance tensor to extract independent spatial mixed block matrices of the mapping structure's conjugate complex pole pairs and their corresponding latent component time-domain autocovariance function sequences. In the initial feature extraction stage, local extremum peak finding and logarithmic decay methods are combined to estimate the time-frequency features of each covariance sequence, obtaining the natural frequency and damping ratio of each latent independent component. The frequency features characterize the periodic oscillation of the component, while the damping ratio reveals the rate of energy decay and dissipation.

[0011] To eliminate the severe interference of excessive attenuation tail noise of high-damped signals on the statistical distribution, based on the aforementioned mathematical coupling relationship between period and damping ratio, the minimum truncation time length is adaptively calculated for each potential component, and strict dynamic data truncation is implemented on the autocovariance sequence. The truncated high-fidelity effective data segments are input into an adaptive kurtosis evaluation framework as the core statistical constraint for identifying physical modes and spurious harmonics. During harmonic removal and parameter inversion, adaptive kurtosis features are obtained by calculating the ratio of the fourth and second central moments of each truncated sequence signal. Decision threshold boundaries are used to accurately capture and remove external periodic harmonic interference exhibiting uniform distribution characteristics. Finally, for the retained true physical modes exhibiting Gaussian tail distribution, dominant feature vectors are extracted from their corresponding block matrices to reconstruct the structural spatial mode shapes. Combined with the high-precision frequency and damping ratio features output, a pure and complete set of underdetermined structural operating mode parameters is finally output. This achieves dynamic filtering of harmonic interference and restoration of physical consistency of modal characteristics under blind source aliasing conditions, providing highly reliable data support for structural health monitoring in complex environments. Attached Figure Description

[0012] Figure 1 This is a flowchart illustrating the underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation provided in an embodiment of the present invention.

[0013] Figure 2 This is a flowchart of the underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation provided in the embodiments of the present invention.

[0014] Figure 3 This is a schematic diagram of a three-degree-of-freedom spring oscillator structure model under dense modes provided in an embodiment of the present invention.

[0015] Figure 4 This is a comparison diagram of the mode shapes for identifying a three-degree-of-freedom spring oscillator structure under dense modes, provided in an embodiment of the present invention.

[0016] Figure 5 This is the MAC matrix diagram for identifying the mode shape of a three-degree-of-freedom spring oscillator structure under dense modes, provided in an embodiment of the present invention. Detailed Implementation

[0017] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0018] refer to Figure 1 , Figure 2 As shown, the first embodiment of the present invention discloses an underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation, which can be executed by an underdetermined mode identification device based on block term decomposition and adaptive kurtosis harmonic separation (hereinafter referred to as identification device), specifically, by one or more processors within the identification device, to implement the following method: S1. Obtain the multi-channel vibration response signal matrix collected by the preset sensor components, calculate the spatial covariance matrix corresponding to the multi-channel vibration response signal matrix under different time delays, and stack the spatial covariance matrix sequentially along the time delay dimension to obtain the third-order covariance tensor. Specifically, step S1 further includes: acquiring the multi-channel discrete-time vibration response signals of the engineering structure synchronously collected by sensor components deployed on the engineering structure under unknown environmental excitation and harmonic interference, and splicing them into a multi-channel vibration response signal matrix at time t. , Let M be the set of real numbers, and M be the number of sensor channels (spatial dimension). The total number of discrete-time sampling points (time dimension); Calculate the multi-channel vibration response signal matrix separately. The formula for the spatial covariance matrix under different time delays is: , , k is the index of the time delay, and K is the total number of time delay points. For the k-th time delay The spatial covariance matrix is ​​given below, with each time delay having a corresponding spatial covariance matrix. The sampling time interval, For transpose; The resulting multiple (K) spatial covariance matrices are stacked sequentially along the time delay dimension (the third dimension) to construct a third-order covariance tensor containing spatial mixing features and temporal decay features. This provides a high-dimensional data foundation for subsequent block tensor blind source separation.

[0019] In this embodiment, multiple acceleration or displacement sensors are arranged on the engineering structure. Under the combined effects of random excitation from an unknown environment and unknown periodic harmonic interference, a limited number of sensors deployed on the surface of the structure are used to synchronously acquire multi-channel discrete-time vibration response signals of the engineering structure. The signals from all channels are then stitched together in chronological order to form a multi-channel vibration response signal matrix.

[0020] To construct a high-dimensional data format suitable for blind source separation, the spatial covariance matrices of the multi-channel vibration response signal matrices under different time delays were calculated. Subsequently, all the calculated spatial covariance matrices were sequentially stacked along the time delay dimension (i.e., the third dimension) to construct a third-order covariance tensor containing both spatial mixing and temporal decay characteristics. This tensor encompasses both the instantaneous spatial coupling relationship between different measurement points and records the evolution of covariance with time delay, providing information-rich input for subsequent underdetermined source separation. This step, by elevating the one-dimensional time delay information to an independent dimension of the tensor, significantly enhances the amount of identifiable statistical information, enabling the extraction of sufficient independent source features from limited observations even under underdetermined conditions where the number of sensors is less than the number of physical modes.

[0021] S2, use the preset block term tensor decomposition model to decouple and optimize the third-order covariance tensor to obtain a sequence of time-domain autocovariance functions; Specifically, step S2 further includes: processing the third-order covariance tensor using a preset block term tensor decomposition model, and minimizing the reconstruction error objective function using an alternating least squares algorithm. , , The third-order covariance tensor is decoupled and optimized. After convergence, R independent rank-(2,2,1) block structures are extracted and decomposed; where R is the total number of potential active components (including structural modes and harmonic sources), and r is the index of the potential active component. Let r be the spatial mixing block matrix corresponding to the r-th potential active component. For tensor outer product, Let r be the sequence of time-domain autocovariance functions corresponding to the r-th potential active component. The square of the Frobenius norm of the tensor; Wherein, the rank of the block term tensor decomposition model is , , The rank of the block matrix is ​​set to accurately characterize the complex conjugate pole properties of the structural physical modes. Specifically, this is because the modes of a single-degree-of-freedom system require two linearly independent basis functions (cosine and sine components) in the real domain to fully describe its complex pole pairs containing frequency and damping. Therefore, the corresponding covariance subspace rank is 2, and the matrix rank needs to be set in the block tensor decomposition. The physical characteristics of the conjugate complex pole pairs in the single-degree-of-freedom response of the mapped structure.

[0022] In this embodiment, block term tensor decomposition is used to decouple the potential independent components in the tensor to overcome the underdetermined blind source aliasing problem where the number of sensors is less than the number of sources, as well as the difficulty in separating densely structured modes. Block term decomposition is performed on the third-order covariance tensor. Through this decomposition process, the third-order covariance tensor is decoupled into the sum of multiple independent blocks. The first two dimensions of the spatial mixing matrix of each potential component are obtained, along with the third-dimensional sequence of independent component vectors. This sequence of independent component vectors represents the separated potential sources from the covariance function.

[0023] Specifically, firstly, for the third-order covariance tensor, a decomposition model based on Block Term Tensor Decomposition (BTD) is established. Considering that each physical mode of the engineering structure requires two linearly independent basis functions (e.g., cosine and sine components) in the real domain to fully characterize its complex conjugate pole pairs, this invention sets the rank of the block matrix to 2. Simultaneously, the total number R of potential active components of the system is pre-estimated, including the actual structural physical modes and possible periodic harmonic interference sources.

[0024] Subsequently, the alternating least squares algorithm is used to minimize the reconstruction error objective function, thereby performing decoupled optimization on the third-order covariance tensor. This objective function is in the form of the square of the Frobenius norm of the tensor representing the difference between the original third-order covariance tensor and the reconstruction results of each block after decomposition. The decomposition result is represented as the sum of the tensor outer products of the spatial mixing block matrix corresponding to all potential active components and the sequence of temporal autocovariance functions. During the optimization process, the alternating least squares algorithm iteratively updates the temporal autocovariance function sequence while fixing the spatial mixing block matrix, and then updates the spatial mixing block matrix while fixing the temporal autocovariance function sequence, repeating this iterative process until the reconstruction error converges.

[0025] After optimization and convergence, for each potential active component, this method extracts its corresponding spatial mixing block matrix and the third-dimensional temporal autocovariance function sequence. The spatial mixing block matrix has a size equal to the number of sensors multiplied by the number of sensors. Since the rank of the block matrix is ​​2, it possesses a rank-2 subspace structure, which precisely maps the inherent complex conjugate characteristics of a single-degree-of-freedom mode. The temporal autocovariance function sequence is a vector of length K, representing the autocorrelation decay law of the potential component under different time delays.

[0026] Through the aforementioned block term tensor decomposition, this invention can effectively decouple aliased vibration response signals into a series of independent components under underdetermined conditions where the number of sensors is less than the total number of potential active components. Compared with traditional sparse component analysis or parallel factor decomposition methods, the block term decomposition model with a rank of -(2,2,1) introduced in this step fully respects the physical essence of the structural vibration response—that is, the second-order subspace characteristics of each mode in the real domain. Therefore, it can maintain stable separation performance even under dense mode or high-damping conditions, avoiding the generation of mode aliasing or spurious modes. In addition, the spatial mixed block matrix obtained by decomposition retains complete directional information for subsequent mode shape reconstruction, while the time-domain autocovariance function sequence provides a high-fidelity one-dimensional signal for frequency, damping ratio estimation, and harmonic identification. Through this decoupling optimization, subsequent steps no longer need to deal with complex multidimensional aliased signals; only the analysis of each independent time-domain autocovariance function sequence is required to efficiently and accurately complete the preliminary estimation of modal parameters and harmonic discrimination.

[0027] S3. Use the time-domain logarithmic decay method or the frequency-domain single-mode fitting method to perform preliminary feature analysis on the time-domain autocovariance function sequence to obtain the corresponding damping ratio, actual natural frequency and natural period. Specifically, step S3 further includes: processing the temporal autocovariance function sequence corresponding to the r-th potential active component. The free decay response, considered as a single-degree-of-freedom system, is mathematically characterized as follows: , Let be the amplitude corresponding to the r-th potential active component, and e be the base of the natural logarithm. Let r be the damping ratio corresponding to the r-th potential active component. Let r be the undamped natural frequency corresponding to the r-th potential active component. Let r be the damped natural frequency corresponding to the r-th potential active component. This represents the phase corresponding to the r-th potential active component; The local extremum peak-finding algorithm is used to extract envelope feature points from the free decay response of a single-degree-of-freedom system. Specifically, the time-domain autocovariance function sequence... Search and record all peaks (i.e., local maxima) in the middle, and obtain the total number of valid peaks. , Extract the time sequence corresponding to the peak. and amplitude envelope sequence ; Based on the time node sequence Calculate the mean time difference between adjacent peaks to preliminarily estimate the damped natural period corresponding to the r-th potential active component. and damped natural frequency , The time point of the first peak, For the first The time point of each peak; Based on the amplitude envelope sequence Calculate the logarithmic decay rate corresponding to the r-th potential active component. The initial damping ratio corresponding to the r-th potential active component is obtained by converting the logarithmic decay rate. , For natural logarithm operations, This represents the amplitude value corresponding to the first wave peak. For the first The amplitude values ​​corresponding to each wave peak; among them, in order to suppress the noise interference of a single wave peak, the span calculation method of the first and last wave peaks is adopted.

[0028] Based on the damped natural frequency corresponding to the r-th potential active component The damping ratio corresponding to the r-th potential active component By using the structural dynamics conversion formula, the actual natural frequency corresponding to the r-th potential active component, which is ultimately used for determining the cutoff length, is calculated. and natural cycles .

[0029] In this embodiment, the modal parameters of each potential independent component are initially estimated for each potential source, based on the covariance function sequence. Preliminary feature analysis is performed using the time-domain logarithmic decay method or the frequency-domain single-mode fitting method to estimate the initial natural frequency and initial damping ratio corresponding to the r-th potential source (i.e., component), and the natural period corresponding to the potential source is calculated from the initial natural frequency.

[0030] Specifically, firstly, the time-domain autocovariance function sequence corresponding to the r-th potential active component is considered as the free decay response of a single-degree-of-freedom system. The mathematical analytical form of this response is described by the amplitude, an exponential decay term consisting of the base e of the natural logarithm, the damping ratio, the undamped natural frequency, the damped natural frequency, and the initial phase. This model accurately characterizes the physical law of the structure's free vibration amplitude decaying exponentially with time and oscillating at the damped natural frequency, providing a theoretical basis for subsequent parameter extraction.

[0031] Subsequently, a local extremum peak-finding algorithm is used to extract envelope feature points from the aforementioned free decay response. Specifically, all local maxima, i.e., peak positions, are automatically searched and recorded in the time-domain autocovariance function sequence. To ensure the stability of parameter estimation, the total number of extracted valid peaks must be no less than two. After the search is completed, the time node sequences corresponding to all peaks and the amplitude envelope sequence at each peak are obtained. This peak-finding process requires no manual intervention, adaptively captures the main oscillation characteristics of the signal, and can reliably identify the true peak positions even under noise interference.

[0032] Next, based on the obtained time node sequence, the damped natural period of the r-th component is initially estimated by calculating the average time difference between adjacent peaks, thus obtaining its damped natural frequency. Specifically, the time span between the first and last peaks is taken and divided by the number of peak intervals (i.e., the total number of peaks minus 1) to obtain the average period. This first-and-last span calculation method effectively suppresses time deviations caused by local noise in individual peaks, improving the robustness of damped period estimation. Simultaneously, the logarithmic decay rate of the r-th component is calculated based on the extracted amplitude envelope sequence. To further suppress noise interference on single amplitude values, this invention employs the first-and-last peak span calculation method, i.e., taking the natural logarithm of the ratio of the amplitude of the first peak to the amplitude of the last peak, and then dividing it by the number of peak intervals to obtain the average logarithmic decay rate. Based on this logarithmic decay rate, the damping ratio of the r-th component can be calculated using a conversion formula commonly used in structural dynamics. This damping ratio reflects the rate of energy dissipation of structural vibration over time and is one of the core parameters for subsequent adaptive truncation length calculation.

[0033] Finally, based on the obtained damped natural frequencies and damping ratios, the actual undamped natural frequency and its corresponding natural period of the r-th component are calculated using the standard conversion formula in structural dynamics. The actual natural frequency is equal to the damped natural frequency divided by the square root of (1 minus the square of the damping ratio), and the natural period is its reciprocal. This conversion eliminates the influence of damping on the frequency observations, restoring the intrinsic dynamic parameters of the structure, thus providing a physically meaningful input constraint for the adaptive truncation length calculation based on the coupling relationship between the period and the damping ratio in subsequent steps.

[0034] S4. Calculate the adaptive minimum cutoff time length based on the damping ratio and natural period, and convert the adaptive minimum cutoff time length into the number of discrete data sampling points. Specifically, step S4 further includes: based on the damping ratio corresponding to the r-th potential active component The natural cycle corresponding to the r-th potential active component The conditional determination is made for the interval in which the damping ratio value is located; When judged At that time, the adaptive minimum truncation time length required for the r-th potential active component in the time dimension is calculated according to the mathematical coupling criterion. To ensure that the truncation length covers a sufficiently damped oscillation envelope; When judged When low-damped harmonics or calculation errors exist, (to prevent the truncation length from tending to infinity) a mandatory constraint is set to calculate the adaptive minimum truncation time length required for the r-th potential active component in the time dimension. ; Adaptive minimum truncation time length Convert to discrete data sampling points , This is a round-down operation. This represents the sampling frequency of the sensor.

[0035] In this embodiment, in order to eliminate the serious interference of the excessive attenuation tail of the high-damped signal on the kurtosis statistical distribution characteristics, the adaptive minimum truncation time length of the required data to be truncated is calculated separately for the r-th potential source based on the natural period and the damping ratio.

[0036] Specifically, the damping ratio and natural period are first obtained. Since the damping ratio directly determines the decay rate of the time-domain autocovariance function sequence, while the natural period reflects the time scale of the oscillation, this invention determines the required truncation time length independently for each component based on the mathematical coupling relationship between the two. This avoids the problems of over-truncation of high-damped signals or insufficient truncation of low-damped signals caused by using a uniform fixed length in traditional methods. Next, the interval in which the damping ratio value falls is conditionally determined. When it is determined that the damping ratio of the component is greater than or equal to 0.1%, it indicates that the component has a certain energy dissipation capability, and its autocovariance sequence will gradually decay with the increase of time. At this time, this invention calculates the adaptive minimum truncation time length required for the component in the time dimension according to the mathematical coupling determination criterion. Specifically, the ratio of the natural period to the damping ratio is multiplied by a preset coefficient (e.g., 30 times) to ensure that the truncation length can cover a sufficient amount of decaying oscillation envelope, so that the data segment on which the subsequent kurtosis calculation is based contains both the main oscillation information and avoids the invalid region with only noise after excessive decay of the signal tail.

[0037] When the damping ratio of a component is determined to be less than 0.1%, it indicates that the component exhibits extremely low damping characteristics (possibly due to external harmonic interference or calculation errors). If the above criteria are still applied, the truncation time will tend to infinity, rendering the actual truncation meaningless. To prevent this, this invention sets a mandatory constraint: a preset reference damping ratio (e.g., 0.1%) is used to replace the actual damping ratio when calculating the truncation time. This ensures that the algorithm obtains a finite and reasonable truncation length under any circumstances, while also preventing the introduction of a large amount of terminal noise due to excessively long truncation of extremely low damping components, thus protecting the robustness of kurtosis statistics. After calculating the adaptive minimum truncation time (in seconds), it needs to be converted into the number of sampling points suitable for discrete signal processing. Specifically, this time length is multiplied by the sensor's sampling frequency, and a floor operation is performed to obtain the corresponding number of discrete data sampling points. Floor operation ensures that the number of truncation points will not exceed the actual length of the original sequence, avoiding index out-of-bounds errors.

[0038] S5, the time-domain autocovariance function sequence is truncated according to the number of discrete data sampling points to obtain the effective signal vector after adaptive truncation, and the adaptive kurtosis eigenvalue is calculated based on the effective signal vector. Specifically, step S5 further includes: based on the number of discrete data sampling points For the temporal autocovariance function sequence corresponding to the r-th potential active component Apply dynamic data truncation, retaining only the first few lines. By sampling points, the effective signal vector after adaptive truncation is obtained. ; Since the self-covariance sequence is a zero-mean process, based on the effective signal vector The adaptive kurtosis eigenvalue corresponding to the r-th potential active component is calculated using the statistical higher-order central moment formula. , Let be the j-th effective signal vector. Here, the numerator represents the empirical estimate of the fourth central moment of the truncated signal vector, and the denominator represents the square of the empirical estimate of the second central moment of the truncated signal vector.

[0039] In this embodiment, dynamic data truncation and adaptive kurtosis are performed. The autocovariance function sequence of the r-th potential source is strictly truncated to the adaptive minimum truncation time length, eliminating end noise data that has no physical meaning and lowers the kurtosis value. The kurtosis value of the potential source is calculated using the effective autocovariance function data segment retained after truncation. The kurtosis value is the ratio of the fourth central moment of the truncated signal to the square of the second central moment.

[0040] Specifically, the first step is to obtain the number of discrete data sampling points. This number represents the number of valid sampling points that should be retained from the original time-domain autocovariance function sequence. This invention applies dynamic data truncation to the complete time-domain autocovariance function sequence of this component, i.e., only retaining the first... The first sampling point is selected, and all subsequent sampling points are discarded. Through this operation, an effective signal vector after adaptive truncation is obtained. This effective signal vector eliminates the invalid part at the end of the original sequence, which is reduced to noise due to excessive signal attenuation. The remaining data segment precisely covers the main decaying oscillation process of the component. Compared with the traditional method of using a uniform fixed length for all components, this invention performs personalized truncation based on the dynamic parameters (damping ratio and natural period) of each component. This can effectively avoid the high-damped signal from being lowered in kurtosis statistics due to tail noise interference, and also prevent the low-damped signal from losing its proper distribution characteristics due to excessively short truncation, thus providing high-fidelity input data for subsequent kurtosis calculation.

[0041] Subsequently, based on the aforementioned effective signal vector, the adaptive kurtosis eigenvalue corresponding to the r-th potential active component is calculated using the higher-order central moment formula in statistics. Since the time-domain autocovariance function sequence has already achieved zero-mean processing after truncation, the ratio of the fourth-order central moment to the square of the second-order central moment of this vector can be directly calculated. Specifically, the numerator is the sum of the fourth powers of each element in the effective signal vector divided by the number of sampling points, and the denominator is the sum of the squares of each element in the effective signal vector divided by the number of sampling points and then squared. This ratio is the kurtosis eigenvalue. This eigenvalue can sensitively reflect the tail characteristics of the signal in the probability density distribution: for periodic harmonic signals that follow a uniform or approximately uniform distribution, the kurtosis value tends to be close to 1.5; while for real structural modal response signals exhibiting Gaussian tail characteristics, the kurtosis value is usually greater than or equal to 3.0.

[0042] S6 classifies and identifies each potential source based on adaptive kurtosis eigenvalues, obtains the retained structural physical mode components, and outputs complete structural underdetermined operating mode parameters by combining the retained structural physical mode components, the corresponding damping ratio, the actual natural frequency and the spatial hybrid block matrix.

[0043] Specifically, step S6 further includes: (i.e., for the entire acquired system) (For each potential independent component, a threshold decision boundary is set) to obtain preset harmonic feature reference values ​​and structural physical mode feature reference values, where the harmonic feature reference value is 1.5 and the structural physical mode feature reference value is 3.0. When judged When (i.e., the kurtosis value approaches the preset harmonic threshold / approaches a uniform distribution characteristic of 1.5) The r-th potential active component is determined to be an externally driven periodic harmonic interference signal, and the frequency, damping and block matrix data corresponding to the potential active component are removed and masked. When judged When (i.e., the kurtosis value is greater than or equal to the preset structural mode threshold / approaching or greater than 3.0 Gaussian distribution tail feature) The r-th potential active component is determined to be the real physical modal component of the engineering structure and is retained in the preset set of effective structural modes.

[0044] For each component in the effective structural mode set, the actual natural frequency is... Damping ratio As the final frequency and damped output; Using SVD or principal component extraction algorithms, from the spatial mixing block matrix A one-dimensional dominant feature vector is extracted and used as the high-fidelity mode shape vector corresponding to the final mode of the structure. It outputs complete underdetermined structural modal parameters. Specifically, for the retained structural physical modal components, it extracts high-precision structural mode shapes from their corresponding first two-dimensional block subspace, and combines them with their autocovariance function to output the final high-precision natural frequencies and damping ratios, thereby completing the identification of working modal parameters under underdetermined conditions.

[0045] In this embodiment, harmonic adaptive separation and final modal parameter extraction based on kurtosis thresholds are performed, and each potential source is classified and identified according to its kurtosis value. First, two preset feature reference values ​​are obtained: the harmonic feature reference value is set to 1.5, and the structural physical modal feature reference value is set to 3.0. These two thresholds are empirical boundaries derived from the statistical characteristics of a large number of signals—the probability density distribution of periodic harmonic signals tends to be uniform, and its kurtosis value is close to 1.5; while the actual structural modal response, due to the presence of an exponentially decaying envelope, exhibits a Gaussian tail characteristic in its distribution, and its kurtosis value is usually not lower than 3.0. This invention uses these as the discrimination criteria to classify each potential active component.

[0046] When the adaptive kurtosis eigenvalue of the r-th potential active component is less than 2.5 (i.e., a uniform distribution characteristic approaching 1.5), the component is determined to be an externally driven periodic harmonic interference signal. For such components, this invention completely removes and masks their corresponding frequency, damping ratio, and spatial mixing block matrix data to ensure that they are not mixed into the final modal parameter identification results. This removal operation requires no manual intervention and is automatically completed based entirely on data-driven statistical features, thereby effectively eliminating the contamination of structural modal identification by external periodic excitations such as mechanical operation and electromagnetic interference.

[0047] When the adaptive kurtosis eigenvalue of the r-th potential active component is determined to be greater than or equal to 2.5 (i.e., approaching or greater than the Gaussian distribution tail feature of 3.0), the component is determined to be a true physical mode component of the engineering structure and is retained in the preset set of effective structural modes. Since steps S4 and S5 have eliminated the influence of high-damped signal tail noise on kurtosis statistics through adaptive truncation, the kurtosis value of the true mode can be accurately calculated, avoiding the risk of the true mode being mistakenly identified as a harmonic and incorrectly removed due to improper truncation.

[0048] After classifying and screening all potential active components, for each retained component in the effective structural mode set, this invention directly uses the actual natural frequency and damping ratio estimated in step S3 as the final high-precision frequency and damping output. These parameters have already undergone robust estimation using local extremum peak finding and logarithmic decay methods in the preliminary analysis stage and have clear physical meaning. Simultaneously, to obtain complete modal parameters, it is also necessary to output the spatial mode shape information for each retained component. This invention utilizes singular value decomposition (SVD) or principal component extraction algorithms to extract a one-dimensional dominant eigenvector from the spatial hybrid block matrix corresponding to the component. Since the rank of this block matrix is ​​preset to 2, it contains information reflecting the dominant direction of the structural spatial distribution. Through the above decomposition, the mode shape vector characterizing the relative vibration amplitude and phase relationship between each measuring point can be obtained. This vector is used as the final high-fidelity mode shape output corresponding to the structural mode. Thus, this invention completely outputs the natural frequency, damping ratio, and spatial mode shape vector for each mode for all retained structural physical modal components, thereby achieving high-precision identification of working modal parameters under underdetermined conditions.

[0049] Please see Figures 3 to 5 Specifically, in this embodiment, to verify the correctness and effectiveness of the method, a dense-mode three-degree-of-freedom spring oscillator model and a three-degree-of-freedom linear time-invariant spring oscillator mechanical model are established in Matlab / Simulink, and white noise excitation is applied to block 3. Newmark is used. The method involves numerical integration of the system to solve for the displacement response at each node on the beam. In this experiment, sensor data from blocks 2 and 3 were used to simulate n. d The condition is underdetermined with M=3 and M=2. Parameter settings are shown in Table 1. Table 1 Simulation parameters for the three-DOF dense modal dataset:

[0050] Evaluation metrics include Modal Confidence Criterion (MAC) and Relative Error of Natural Frequencies. The Modal Confidence Criterion (MAC) formula is shown below: The formula for the relative error of the natural frequency is: .

[0051] The experimental results are shown in Table 2.

[0052] Table 2:

[0053] In summary, this method is used to achieve high-precision extraction of the true physical modes of a structure and restoration of its physical parameter consistency under conditions of limited sensor quantity and strong external periodic harmonic interference. First, a limited number of sensors are used to collect multi-channel vibration response signals of the structure under the combined effects of unknown environmental excitation and harmonic interference. The input multi-sensor vibration signals are preprocessed to calculate their spatial covariance matrices at different time delays, and these matrices are stacked along the time delay dimension to generate a third-order covariance tensor containing the spatiotemporal evolution characteristics of the system, providing a data foundation for subsequent high-dimensional separation. To address the underdetermined blind source aliasing problem caused by insufficient sensor quantity, a rank-wise... The Block Term Tensor Decomposition (BTD) model performs preliminary decoupling of the covariance tensor to extract the block subspace of the spatial mixing matrix and the temporal autocovariance function sequence of each potential component, enabling the model to fully learn and map the physical coupling characteristics of the structured dense modes and complex conjugate poles.

[0054] In the signal feature modeling and analysis stage, an adaptive truncation and statistical kurtosis joint constraint is constructed to uniformly fuse and encode time decay features, oscillation period features, and higher-order statistical distribution features. The extracted autocovariance function sequence is preliminarily analyzed using time-domain logarithmic decay or frequency-domain fitting to estimate the natural period and damping ratio of each potential component. Based on the mathematical coupling relationship between period and damping ratio, the adaptive minimum truncation time length is calculated for each independent component. Specifically, the oscillation period feature characterizes the temporal evolution of the potential components; the time decay feature (damping ratio) reflects the temporal dissipation rate of signal energy; and the higher-order statistical distribution feature (kurtosis) reveals the Gaussian tail or uniform distribution property of the signal in probability density. These features are jointly input into the adaptive separation framework as physical discrimination constraints, guiding the accurate separation of true modes and spurious harmonics in the structure.

[0055] During parameter separation and identification (i.e., in the parameter solution and discrimination stage), the attenuation distribution of each component is initially analyzed through local peak finding and logarithmic attenuation algorithms. To address the computational noise problem caused by excessive attenuation at the tail of high-damped signals, the model, based on the mathematical coupling relationship between period and damping ratio, rigorously calculates and applies an adaptive minimum cutoff time length for each potential component. In the decision-making process, a dual-threshold classification criterion is used for optimization. The harmonic threshold feature is used to accurately capture and eliminate pseudo-modal interference in the environment, while the structural threshold feature ensures that the true physical response is completely locked, thereby learning a high-fidelity decoupled mapping from the aliased tensor to the pure structural components. In simple terms, adaptive kurtosis assessment is used as the core framework for harmonic identification. A strict data truncation of corresponding length is applied to the extracted autocovariance function, and multi-dimensional feature constraints are introduced in the elimination and inversion process, including subspace mapping features, dynamic truncation features, and kurtosis threshold features. Among them, the subspace mapping features are used to characterize the complex conjugate poles of the dense modes of the structure and the high-precision mode shape association. The dynamic truncation features capture the true statistical distribution pattern by eliminating the noise in the excessive attenuation tail of the high-damped signal. The kurtosis threshold features, combined with a numerical classifier, realize the accurate boundary perception between physical modes and periodic harmonics.

[0056] In the final modal identification stage, the autocovariance sequence to be extracted, guided by adaptively truncated data, undergoes high-order central moment statistical evaluation and attribute determination. By dynamically filtering out harmonic components and extracting dominant subspace features, the physical reconstruction of missing pure signals is achieved. The final output is a high-precision set of underdetermined working modal parameters, ensuring that the identification results maintain a high degree of consistency with the actual physical state of the structure in terms of spatial mode shapes, attenuation laws, and frequency characteristics. This provides a highly reliable data foundation for the dynamic analysis, damage diagnosis, and health assessment of complex structures, significantly improving the accuracy and robustness of modal identification for large-scale engineering structures under severe interference conditions. In short, through the joint constraints of the aforementioned multi-dimensional tensors and adaptive statistical features, the separation algorithm is guided to eliminate spurious harmonics and generate high-precision structurally pure modal parameters during the identification process. The method of this invention can effectively decouple the vibration signals of multi-sensor structures under complex working conditions with severe underdeterminacy and strong external periodic harmonic interference, improve the accuracy of dense mode and high-damped mode parameter identification, and provide highly reliable mode identification algorithm support for engineering scenarios where complex structures or environmental excitation limitations lead to the failure of traditional blind source separation.

[0057] The above description represents the preferred embodiments of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications are also considered to be within the scope of protection of the present invention.

Claims

1. A block term decomposition and adaptive kurtosis harmonic separation based underdetermined modal identification method, characterized in that, include: The multi-channel vibration response signal matrix collected by the preset sensor components is obtained, the spatial covariance matrix corresponding to the multi-channel vibration response signal matrix under different time delays is calculated, and the spatial covariance matrix is ​​stacked sequentially along the time delay dimension to obtain the third-order covariance tensor. The covariance third-order tensor is decoupled and optimized using a pre-defined block term tensor decomposition model to obtain a sequence of time-domain autocovariance functions. The time-domain autocovariance function sequence is preliminarily analyzed using the time-domain logarithmic decay method or the frequency-domain single-mode fitting method to obtain the corresponding damping ratio, actual natural frequency and natural period. Based on the damping ratio and natural period, the adaptive minimum cutoff time length is calculated, and the adaptive minimum cutoff time length is converted into the number of discrete data sampling points; The time-domain autocovariance function sequence is truncated according to the number of discrete data sampling points to obtain an adaptively truncated effective signal vector, and the adaptive kurtosis eigenvalue is calculated based on the effective signal vector. Based on adaptive kurtosis eigenvalues, each potential source is classified and identified to obtain the retained structural physical mode components. Then, by combining the retained structural physical mode components, the corresponding damping ratio, the actual natural frequency, and the spatial hybrid block matrix, the complete structural underdetermined operating mode parameters are output.

2. The underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation according to claim 1, characterized in that, Obtain the multi-channel vibration response signal matrix collected by the preset sensor components, calculate the spatial covariance matrix corresponding to the multi-channel vibration response signal matrix under different time delays, and sequentially stack the spatial covariance matrices along the time delay dimension to obtain the third-order covariance tensor, specifically: The sensor array deployed on the engineering structure synchronously acquires multi-channel discrete-time vibration response signals of the structure under unknown environmental excitation and harmonic interference, and then splices these signals into a multi-channel vibration response signal matrix at time t. , Let M be the set of real numbers, and M be the number of sensor channels. This represents the total number of discrete-time sampling points. Calculate the multi-channel vibration response signal matrix separately. The formula for the spatial covariance matrix under different time delays is: , , k is the index of the time delay, and K is the total number of time delay points. For the k-th time delay The spatial covariance matrix is ​​given below, with each time delay having a corresponding spatial covariance matrix. The sampling time interval, For transpose; The obtained spatial covariance matrices are stacked sequentially along the time delay dimension to construct a third-order covariance tensor containing spatial mixing features and temporal decay features. .

3. The underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation according to claim 2, characterized in that, The third-order covariance tensor is decoupled and optimized using a pre-defined block term tensor decomposition model to obtain a sequence of time-domain autocovariance functions, specifically: The covariance third-order tensor is processed using a pre-defined block term tensor decomposition model, and the reconstruction error objective function is minimized using an alternating least squares algorithm. , , The third-order covariance tensor is decoupled and optimized, where R is the total number of potential active components and r is the index of the potential active components. Let r be the spatial mixing block matrix corresponding to the r-th potential active component. For tensor outer product, Let r be the sequence of time-domain autocovariance functions corresponding to the r-th potential active component. The square of the Frobenius norm of the tensor; Wherein, the rank of the block term tensor decomposition model is , , Let be the rank of the block matrix.

4. The underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation according to claim 3, characterized in that, Preliminary feature analysis of the time-domain autocovariance function sequence was performed using the time-domain logarithmic decay method or the frequency-domain single-mode fitting method to obtain the corresponding damping ratio, actual natural frequency, and natural period, specifically: The temporal autocovariance function sequence corresponding to the r-th potential active component The free decay response, considered as a single-degree-of-freedom system, is mathematically characterized as follows: , Let be the amplitude corresponding to the r-th potential active component, and e be the base of the natural logarithm. Let r be the damping ratio corresponding to the r-th potential active component. Let r be the undamped natural frequency corresponding to the r-th potential active component. Let r be the damped natural frequency corresponding to the r-th potential active component. This represents the phase corresponding to the r-th potential active component; The local extremum peak-finding algorithm is used to extract envelope feature points from the free decay response of a single-degree-of-freedom system. Specifically, the time-domain autocovariance function sequence... Search and record all peaks in the middle, obtaining the total number of valid peaks. , Extract the time sequence corresponding to the peak. and amplitude envelope sequence ; Based on the time node sequence Calculate the mean time difference between adjacent peaks to preliminarily estimate the damped natural period corresponding to the r-th potential active component. and damped natural frequency , The time point of the first peak, For the first The time point of each peak; Based on the amplitude envelope sequence Calculate the logarithmic decay rate corresponding to the r-th potential active component. The damping ratio corresponding to the r-th potential active component is obtained by converting the logarithmic decay rate. , For natural logarithm operations, This represents the amplitude value corresponding to the first wave peak. For the first The amplitude value corresponding to each wave peak; Based on the damped natural frequency corresponding to the r-th potential active component The damping ratio corresponding to the r-th potential active component By using the structural dynamics conversion formula, the actual natural frequency corresponding to the r-th potential active component, which is ultimately used for determining the cutoff length, is calculated. and natural cycles .

5. The underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation according to claim 4, characterized in that, Based on the damping ratio and natural period, the adaptive minimum cutoff time length is calculated, and then converted into the number of discrete data sampling points, specifically: Based on the damping ratio corresponding to the r-th potential active component The natural cycle corresponding to the r-th potential active component The conditional determination is made for the interval in which the damping ratio value is located; When judged At that time, the adaptive minimum truncation time length required for the r-th potential active component in the time dimension is calculated according to the mathematical coupling criterion. ; When judged When setting mandatory constraints, calculate the adaptive minimum cutoff time length required for the r-th potential active component in the time dimension. ; Adaptive minimum truncation time length Convert to discrete data sampling points , This is a round-down operation. This represents the sampling frequency of the sensor.

6. The underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation according to claim 5, characterized in that, The time-domain autocovariance function sequence is truncated based on the number of discrete data sampling points to obtain an adaptively truncated effective signal vector. Adaptive kurtosis eigenvalues ​​are then calculated based on this effective signal vector. Specifically: Based on the number of discrete data sampling points For the temporal autocovariance function sequence corresponding to the r-th potential active component Apply dynamic data truncation, retaining only the first few lines. By sampling points, the effective signal vector after adaptive truncation is obtained. ; Based on effective signal vector The adaptive kurtosis eigenvalue corresponding to the r-th potential active component is calculated using the statistical higher-order central moment formula. , Let be the j-th valid signal vector.

7. The underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation according to claim 6, characterized in that, Based on adaptive kurtosis eigenvalues, each potential source is classified and identified to obtain the retained structural physical mode components. Combining the retained structural physical mode components, corresponding damping ratios, actual natural frequencies, and spatial hybrid block matrices, the complete structural underdetermined operating mode parameters are output, specifically: Obtain preset harmonic characteristic reference values ​​and structural physical mode characteristic reference values, wherein the harmonic characteristic reference value is 1.5 and the structural physical mode characteristic reference value is 3.0; When judged When the r-th potential active component is determined to be an externally driven periodic harmonic interference signal, the frequency, damping and block matrix data corresponding to the potential active component are removed and masked. When judged When the r-th potential active component is determined to be a real physical modal component of the engineering structure, it is retained in the preset set of effective structural modes.

8. The underdetermined mode identification method based on block term decomposition and adaptive kurtosis harmonic separation according to claim 7, characterized in that, Also includes: For each component in the effective structural mode set, the actual natural frequency is... Damping ratio As the final frequency and damped output; Using SVD or principal component extraction algorithms, from the spatial mixing block matrix A one-dimensional dominant feature vector is extracted and used as the high-fidelity mode shape vector corresponding to the final structural mode. Output complete structural underdetermined operating mode parameters.

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