Blind Modal Identification Research Articles

Utilizing data-driven algorithms for blind modal parameter identification of structures from output-only video measurements

In structural dynamics and vibration analysis, modal identification plays a pivotal role in understanding and characterizing the dynamic behavior of complex systems. Traditional modal analysis techniques heavily rely on accurate sensor placement and knowledge of the excitation forces, which may not always be feasible or practical in real-world scenarios. Recently, non-contact video-based modal analysis methods for structures with arbitrary complexity using computer vision techniques have gained much importance. But these methods need to know ahead of time where the structure's natural frequency ranges are, and they use steerable pyramids, which are complicated multi-scale image decomposition filters. To address these issues, a blind modal identification technique is suggested to extract the modal parameters (modal frequency and mode shapes) from the recorded structural vibration video signal. The proposed algorithm for unsupervised machine learning is the Non-Negative Matrix Factorization (NNMF) method, which is combined with the Generalized Complexity Pursuit (GCP) method for blind source separation. The NNMF algorithm is directly applied to the raw pixel-time series made from the video data to get the temporal components, which are then separated using GCP to find the individual modal frequencies and mode shapes. The above algorithm is first validated on a numerical model and then implemented on laboratory scale models (cantilever, single-degree, and multi-degree frame models). Finally, the proposed methodology is implemented on real-world recorded structural vibration videos to determine its noise sensitivity, such as the Tacoma Narrows bridge and the London Millennium footbridge. The modal parameters extracted are compared with those from the available literature for validation. The estimation errors obtained from all the validations are well below 1%, which makes the technique quite suitable and reliable for structural vibration monitoring by identifying and reproducing complex vibration modes.

An unsupervised learning algorithm for computer vision-based blind modal parameters identification of output-only structures from video measurements

Operational modal analysis (OMA) is required to maintain large-scale and necessary civil infrastructures. The non-contact method of OMA using digital image correlation and point tracking algorithms requires a speckle pattern placed on the structure. Alternatively, advanced computer vision methods like optical flow and phase-based video motion magnification (PBVMM) techniques are used to measure modal parameters. Despite the importance of PBVMM, the users should know the range of frequencies in which the natural structure frequency lies. A methodology based on an unsupervised machine learning technique is developed to extract the modal parameters blindly from its recorded digital video. The proposed methodology uses complex steerable pyramids and an unsupervised machine learning technique, also known as principal component analysis, and analytical mode decomposition with a random decrement technique to blindly extract the modal parameters of a structure. This study validated the proposed methodology using a multi-degree of freedom (DOF) numerical model. The results are compared with theoretical and estimated values and are in good agreement. Furthermore, it is implemented on a laboratory benchmark SDOF, MDOF, and real-time videos of the London Millennium and Tacoma Narrows bridges for blindly extracting the modal frequencies and damping ratios.

Blind modal identification for decentralized sensor network by modified sparse component analysis in frequency-domain subspace

It is practical to develop a modal identification method using a decentralized sensor network for field tests. Sparse component analysis (SCA) is a potential method in the framework of blind source separation to overcome the difficulty of using limited sensors. Nevertheless, it is still challenging for SCA to identify a larger number of modes than the number of measurements using the decentralized sensor network. This is because the partial mode shapes may be similar. It is convenient to perform SCA in frequency-domain subspaces to improve SCA further for decentralized sensor implementation; this is owing to the distinguishability of different partial mode shapes. In this study, a modal identification method applied to a largely underdetermined case caused by a decentralized sensor network is proposed by modifying SCA by automatic segmentation of frequency-domain subspaces. First, a method for determining the segmenting point is developed by identifying the valleys of the energy fluctuation spectrum using the scale-space peak detection method. Second, single-mode-points in the time–frequency plane are detected to facilitate mode shape estimation in the frequency-domain subspaces. Third, density peak clustering with the clustering number determination method is developed to estimate mode shapes, particularly for closely spaced modes. Finally, the modal frequencies and damping ratios are identified through the inverse operator in the frequency-domain subspaces and the curve fitting of the expected instant spectrum. The proposed method is verified by a 10 degree-of-freedom system, a structure with close modes, and an actual cable-stayed bridge. The investigations verified the effectiveness of the proposed method for decentralized sensor networks and practical applications.

Input motion extraction and closely-spaced modes recognition through partial system identification: An orthogonality-aided blind source separation approach

This research proposes an output-only procedure to extract seismic input motion and identify systems with severe closely-spaced modes incorporating a partial system identification approach. The proposed framework employs new methods to detect free vibration portion of the response histories and clustering the candidate points to find the first mode shape in a blind modal identification (BMI) procedure. The first mode frequency and damping along with mass matrix distribution of the system are calculated then using an orthogonality-based optimization process. The mentioned identified features are utilized next for the input motion extraction without any need to know the higher mode characteristics of the system. Next, higher mode characteristics are detected in succeeding steps considering the removal of each identified mode share from the total response at the end of each step. The perfect performance of the framework was proved against two simulated synthetic structures with mild and severe closely-spaced modes.

Compressed sensing based on dictionary learning for underdetermined modal identification

With the engineering structures becoming more complicated, it is difficult to obtain complete measurement responses with limited sensors. Thus, carrying out the underdetermined modal identification will have practical engineering application values. In this paper, a new approach for underdetermined blind modal identification based on dictionary learning in the framework of compressed sensing (CS) is proposed. The principal idea is to estimate modal shapes using a clustering technique, and recover modal responses combing the estimated mode shapes matrix and the learned dictionary. The experiment results on a typical cantilever beam structure illustrate that the proposed method can perform accurate dynamic parameters identification whether in underdetermined case or determined case.

Sparse component analysis with optimized clustering for underdetermined blind modal identification

Modal identification performs one of the most important roles in structural dynamics analysis and structural health monitoring, especially when the input excitations are not measurable. Most of the traditional blind source separation approaches can only handle determined or overdetermined blind modal identification, where the number of observed sensors is equal to or greater than the number of active modes. When the number of observed sensors is less than the number of active modes, new methods to perform underdetermined blind modal identification should be considered. To tackle this issue, a novel operational modal identification method based on an enhanced sparse component analysis with optimized clustering is proposed. Firstly, a robust K-means clustering with differential evolution algorithm is put forward to estimate the mode shape matrix utilizing the sparse property of observed mixtures. Secondly, the modal responses are recovered by the least squares method from the incomplete knowledge of the mode shape matrix and the system outputs. Subsequently, the modal responses are transformed into a time domain through time-frequency transformation, where the modal parameters are extracted. Finally, numerical simulation and experimental verification demonstrate that in both the determined and underdetermined case, the proposed method can perform accurate and robust parameter identification of structural systems.

Blind Modal Identification Using Limited Sensors through Modified Sparse Component Analysis by Time‐Frequency Method

AbstractOutput‐only modal identification with limited sensors available may be encountered in practical applications. Sparse component analysis (SCA), which is one of the most popular methods among blind source separation (BSS) techniques, is known as its capability for handling underdetermined BSS problems and nonstationary signals. Single‐source‐points (SSPs) where only one mode makes contribution are adopted to estimate the modal matrix. However, the accuracy of SSP detection is influenced by the number of sensors, because the detecting condition for SSPs may be satisfactory for multiple‐source‐points (MSPs). Besides, the estimation of frequencies and damping ratios is based on the sparse reconstruction of time‐frequency domain modal responses, where the number of sensors will have an impact on the accuracy. To improve the performance of blind modal identification using limited sensors, a method which is called SCA‐TF by modifying SCA through time‐frequency method is proposed in this paper. The TF representation in SCA is employed to modify the SSP detection process and identify frequencies and damping ratios instead of recovering the modal responses. In order to improve the accuracy of modal matrix estimation, the result of SSP detection is used as a preliminary for further refinement. Then the false SSPs are removed from the preliminary result through the power spectra which are constructed by short‐time Fourier transform coefficients. Subsequently, the estimation of modal matrix is accomplished by clustering the refined SSPs. Frequencies and damping ratios are extracted directly from the generalized spectra matrix which is formed by the short‐time Fourier transform coefficients. Because the identification of modal parameters is implemented directly in time‐frequency domain, the reconstruction of modal responses is omitted. The results of the numerical studies show that SCA‐TF can identify the modal parameters accurately. To investigate the performance of SCA‐TF in practical structures, the acceleration measurements from Tianjin Yonghe Bridge are analyzed to identify the modal parameters. The results of the Yonghe Bridge demonstrate the effectiveness of the proposed method to perform blind modal identification in practical structures.

Blind Modal Identification in Frequency Domain Using Independent Component Analysis for High Damping Structures with Classical Damping

AbstractOutput‐only modal identification methods are practical for large‐scale engineering. Recently, independent component analysis (ICA) which is one of the most popular techniques of blind source separation (BSS) has been used for output‐only modal identification to directly separate the modal responses and mode shapes from vibration responses. However, this method is only accurate for undamped or lightly damped structures. To improve the performance of ICA for high damping structures, this article presents an extended ICA‐based method called ICA‐F, which establishes a BSS model in frequency domain. First, the basic idea of BSS and ICA applied in modal identification is introduced in detail. The free vibration responses and the correlation functions of ambient responses can be cast into the frequency‐domain BSS framework just by mapping the time history responses to frequency domain through fast Fourier transform (FFT). Then, an ICA‐based method in frequency domain called ICA‐F is proposed to accurately extract mode shapes and modal responses for both light and high damping structures. A simulated 3 degree of freedom mass‐spring system and a 4‐story simulated benchmark model developed by the IASC‐ASCE Task Group in Health Monitoring are employed to verify the effectiveness of the proposed method. The results show that the proposed method can perform accurate modal identification for both light and high damping structures. Finally, the IASC‐ASCE experimental benchmark structure is also utilized to illustrate the proposed method applied to practical structure.

An Investigation of Soil-Structure Interaction Effects Observed at the MIT Green Building

The soil-foundation impedance function of the MIT Green Building is identified from its response signals recorded during an earthquake. Estimation of foundation impedance functions from seismic response signals is a challenging task, because: (1) the foundation input motions (FIMs) are not directly measurable, (2) the as-built properties of the super-structure are only approximately known, and (3) the soil-foundation impedance functions are inherently frequency-dependent. In the present study, aforementioned difficulties are circumvented by using, in succession, a blind modal identification (BMID) method, a simplified Timoshenko beam model (TBM), and a parametric updating of transfer functions (TFs). First, the flexible-base modal properties of the building are identified from response signals using the BMID method. Then, a flexible-base TBM is updated using the identified modal data. Finally, the frequency-dependent soil-foundation impedance function is estimated by minimizing the discrepancy between TFs (of pairs instrumented floors) that are (1) obtained experimentally from earthquake data and (2) analytically from the updated TBM. Using the fully identified flexible-base TBM, the FIMs as well as building responses at locations without instruments can be predicted, as demonstrated in the present study.

Blind modal identification of non-classically damped structures under non-stationary excitations

In the present paper, we extend a previously developed blind modal identification method to systems with non-classical damping, which have complex-valued mode shapes. Unlike conventional output-only identification methods, blind modal identification method can provide modal property estimates when the input excitations are unknown and non-stationary (e.g., for systems equipped with added dampers or soil–structure problems). In earlier work, we have developed a technique that can be used for non-classically damped systems, but it is only applicable to stationary (i.e., ambient) excitations. Herein, we present an extension to non-classically damped systems under non-stationary (e.g., seismic) excitations. The proposed method yields mode shapes, natural frequencies and damping ratios, sequentially. A critical ingredient in this new method is a combination of generalized eigen-decomposition and rough–fuzzy c-means clustering techniques, which are employed to identify the complex-valued mode shapes. The accuracy of the proposed method is verified through a simulated but realistic example. Copyright © 2016 John Wiley & Sons, Ltd.

Underdetermined blind modal identification of structures by earthquake and ambient vibration measurements via sparse component analysis

Sparse component analysis (SCA) approach was adopted to handle underdetermined blind modal identification of structures, where the number of sensors is less than the number of active modes. To exploit the sparsity of structural responses in time–frequency domain, Short Time Fourier Transform (STFT) was used in this study. The proposed SCA-based approach has two main stages: modal matrix estimation and modal displacement estimation. In the first stage, hierarchical clustering algorithm was used to estimate the modal matrix. The clustering algorithm was preceded by a preprocessing step to select the points in time–frequency domain that only one mode makes contribution in the responses. These points were fed to the clustering algorithm as an input. Performing this analysis enhanced the modal matrix estimation accuracy and reduced the computational cost while conducting clustering analysis. Having estimated mixing matrix, the complex-valued modal responses in the transformed domain were recovered via Smoothed zero-norm (SL-0) algorithm. In a broad sense, using the SL-0 algorithm permits researchers to use any kind of transform in seeking sparsity, regardless of obtaining real-valued or complex-valued signals in transformed domain. Natural frequencies and damping ratios were extracted from the recovered modal responses. Performance of the proposed method was investigated using a synthetic example and a benchmark structure with earthquake and ambient excitation, respectively.

Extended Blind Modal Identification Technique for Nonstationary Excitations and Its Verification and Validation

AbstractRecently the authors developed an output-only (or blind) identification technique for the modal identification of structures under ambient/free vibrations. In the present study, this method is extended to incorporate nonstationary unknown inputs—a feature that existing output-only methods are unable to replicate. The method employs a so-called PARAllel FACtor technique to decompose the third-order tensor that is built using the spatial time–frequency matrices of the response signals in order to find the mode shapes and modal coordinates’ auto time–frequency distribution (TFD). The proposed method is first verified using a synthetic data set and then used to extract the modal parameters from vibration data recorded during shaking table tests on the International Institute of Earthquake Engineering and Seismology (IIEES) test structure. The accuracy of the blind identification results is assessed through comparisons with a well-established input–output identification method, namely, a well-known com...

Blind modal identification of output-only non-proportionally-damped structures by time-frequency complex independent component analysis

Recently, a new output-only modal identification method based on time-frequency independent component analysis (ICA) has been developed by the authors and shown to be useful for even highly-damped structures. In many cases, it is of interest to identify the complex modes of structures with non-proportional damping. This study extends the time-frequency ICA based method to a complex ICA formulation for output-only modal identification of non-proportionally-damped structures. The connection is established between complex ICA model and the complex-valued modal expansion with sparse time-frequency representation, thereby blindly separating the measured structural responses into the complex mode matrix and complex-valued modal responses. Numerical simulation on a non-proportionally-damped system, laboratory experiment on a highly-damped three-story frame, and a real-world highly-damped base-isolated structure identification example demonstrate the capability of the time-frequency complex ICA method for identification of structures with complex modes in a straightforward and efficient manner.

Blind Modal Identification of Non-Classically Damped Systems from Free or Ambient Vibration Records

A significant segment of system identification literature on civil structures is devoted to response-only identification, simply because lack of measurements of input excitations for civil structures is a fairly common scenario. In recent years, several researchers have successfully adapted a second-order blind identification (SOBI) technique—a method originally developed for “blind source separation” of audio signals—to response-only identification of mechanical and civil structures. However, this development had been confined to fully instrumented classically damped systems. While several approaches have been proposed recently for extending SOBI to non-classically damped systems, they all require additional data such as velocity or analytic signals. Herein, we present a version of SOBI that requires only acceleration signals recorded during free or ambient vibration tests, and yields the system's complex mode shapes, natural frequencies, and damping ratios. Performance of the proposed technique is demonstrated through two synthetic examples: a ten-story structure possessing a passive control system, and a soil-structure system with seven degrees of freedom (seven-DOF).

The identification of a dam's modal parameters under random support excitation based on the Hankel matrix joint approximate diagonalization technique

Modal analysis is an effective method for monitoring a dam's health. Modal parameters can be identified from the measured vibration response of a dam under ambient excitation, such as that from an earthquake. In this paper, we first use the state space model to analyze the vibration of a dam under ambient support excitation and conclude that the nature excitation technique (NExT) can be used to the measured absolute acceleration response of the dam under band-limited stochastic support excitation to obtain its impulse response. To overcome some of the limitations of the traditional modal identification method for a structure under ambient excitation, we propose a modal parameter identification method based on the Hankel matrix joint approximate diagonalization (HJAD) technique. In this method, the Hankel matrix is defined as the covariance matrix of a vector, which is composed of measured acceleration responses, and their time-lagged data. This modal parameter identification method can be regarded as an improvement to the traditional time domain method because it introduces the joint approximate diagonalization (JAD) technique into the original method. On the other hand, the method can be regarded as an improvement to the SOBI-based modal identification method because it uses the Hankel matrix instead of the covariance matrix of response to perform the JAD. Therefore, this method combines the advantages of the two existing modal identification methods and overcomes some of their limitations. Compared with the SOBI-based modal parameter identification method, the implementation of the method presented in this paper is very convenient because we need only to add time-lagged response data to the analysis. The numerical analysis results show that the proposed modal parameter identification method based on HJAD technology not only has the advantage of a traditional blind modal parameter identification algorithm, but can also overcome the limitation of not being able to estimate more active modes than the number of available sensors. According to the satisfactory performance of this method in the analysis of strong-motion earthquake observation data for a gravity dam, the modal parameter identification method proposed in this paper has the potential for application in water conservancy engineering.

Modal Identification Using Blind Source Separation Based on Moving Average Method

The paper discusses the basic principles of blind source separation and modal identification of structures, analyses the feasibility that using blind source separation techniques for modal parameter identification. According to the noisy features of the measured data in experiments, a second-order blind identification algorithm based on moving average method is proposed. By moving average method the noises are efficiently eliminated. It greatly improves the separation performance of this algorithm. The cantilever experiments verify the stability and the applicability of the algorithm.

Blind modal identification of structures from spatially sparse seismic response signals

SUMMARY Response-only identification of civil structures has attracted much attention during recent years, as input excitations are rarely measurable for ambient vibrations. Although various techniques have been developed by which identification can be carried out using ambient responses, these techniques are generally not applicable to non-stationary excitations that structures experience during moderate-to-severe earthquakes. Recently, the authors proposed a new response-only modal identification method that is applicable to strong shaking data. This new method is highly attractive for cases in which the true input motions are unavailable. For example, when soil–structure interaction effects are non-negligible, neither the free-field motions nor the recorded foundation responses may be assumed as input. Even in the absence of soil–structure interaction, in many instances, the foundation responses are not recorded (or are recorded with low signal-to-noise ratios). Thus far, the said method has been only applicable to fully instrumented systems wherein the number of sensors is equal to or greater than the number of active modes. In this study, we offer various improvements, including an extension that enables the treatment of sparsely instrumented systems. Specifically, a cluster-based underdetermined time–frequency method is employed at judiciously selected auto-source points to determine the mode shapes. The mode shape matrix identified in this manner is not square, which precludes the use of simple matrix inversion to extract the modal coordinates. As such, natural frequencies and damping ratios are identified from the recovered modal coordinates' time–frequency distributions using a subspace method. Simulated data are used for verifying the proposed identification method. Copyright © 2013 John Wiley & Sons, Ltd.

Blind modal identification of output‐only structures in time‐domain based on complexity pursuit

SUMMARYOutput‐only modal identification is needed when only structural responses are available. As a powerful unsupervised learning algorithm, blind source separation (BSS) technique is able to recover the hidden sources and the unknown mixing process using only the observed mixtures. This paper proposes a new time‐domain output‐only modal identification method based on a novel BSS learning algorithm, complexity pursuit (CP). The proposed concept—independent ‘physical systems’ living on the modal coordinates—connects the targeted constituent sources (and their mixing process) targeted by the CP learning rule and the modal responses (and the mode matrix), which can then be directly extracted by the CP algorithm from the measured free or ambient system responses. Numerical simulation results show that the CP method realizes accurate and robust modal identification even in the closely spaced mode and the highly damped mode cases subject to non‐stationary ambient excitation and provides excellent approximation to the non‐diagonalizable highly damped (complex) modes. Experimental and real‐world seismic‐excited structure examples are also presented to demonstrate its capability of blindly extracting modal information from system responses. The proposed CP is shown to yield clear physical interpretation in modal identification; it is computational efficient, user‐friendly, and automatic, requiring little expertise interactions for implementations. Copyright © 2013 John Wiley & Sons, Ltd.

Time-Frequency Blind Source Separation Using Independent Component Analysis for Output-Only Modal Identification of Highly Damped Structures

Output-only algorithms are needed for modal identification when only structural responses are available. The recent years have witnessed the fast development of blind source separation (BSS) as a promising signal processing technique, pursuing to recover the sources using only the measured mixtures. As the most popular tool solving the BSS problem, independent component analysis (ICA) is able to directly extract the time-domain modal responses, which are viewed as virtual sources, from the observed system responses; however, it has been shown that ICA loses accuracy in the presence of higher-level damping. In this study, the modal identification issue, which is incorporated into the BSS formulation, is transformed into a time-frequency framework. The sparse time-frequency representations of the monotone modal responses are proposed as the targeted independent sources hidden in those of the system responses which have been short-time Fourier-transformed (STFT); they can then be efficiently extracted by ICA, whereby the time-domain modal responses are recovered such that the modal parameters are readily obtained. The simulation results of a multidegree-of-freedom system illustrate that the proposed output-only STFT-ICA method is capable of accurately identifying modal information of lightly and highly damped structures, even in the presence of heavy noise and nonstationary excitation. The laboratory experiment on a highly damped three-story frame and the analysis of the real measured seismic responses of the University of Southern California hospital building demonstrate the capability of the method to perform blind modal identification in practical applications.

An analytic formulation for blind modal identification

In this paper a complex-valued formulation of the modal superposition equation is provided and shown to be equivalent to the original, real-valued Blind Modal IDentification (BMID) problem. The complex-valued variant involves the analytic form of the physical and modal responses. The formulation is shown to be more concise and straightforward than the original. It is noted that complex-valued mode shapes can be obtained using a complex version of the two-step Joint Approximate Diagonalization (JAD) algorithm. Using this approach the modal response pairing step of the original BMID method is eliminated. Since the development of the original BMID method, several new, one-step JAD algorithms have been devised. Many of the algorithms can be extended to identify complex mixing matrices. A complex version of the one-step JAD method known as the Weighted Exhaustive Diagonalization with Gauss itErations algorithm is utilized to solve for the complex mode shapes and modal responses. By using this simplified formulation, the whitening step is eliminated, as well as the modal response pairing step, which is necessary in the original BMID algorithm. Performance of the new Complex BMID (CBMID) algorithm is evaluated by application to synthesized data from a three-degrees-of-freedom system with complex modes, application to measured laboratory data on a structural frame and application to measured output-only data from the Heritage Court Tower building. It is seen that the CBMID method results in essentially the same estimates of modal responses, complex mode shapes, natural frequencies and modal damping compared to results from BMID. Furthermore, it is shown that modal parameters from BMID and CBMID are very consistent with those obtained from state-of-the-art methods, such as the Eigensystem Realization Algorithm and the covariance-driven Stochastic Subspace Identification method.