Stochastic Subspace Identification Research Articles

Vibration-based damage detection of beams using Hybrid Genetic Algorithm with combined l0 and l1 regularization

Vibration-based damage detection is an effective way of identifying beam damages before they become severe and potentially catastrophic. This paper introduces a novel and robust method for detecting multiple cracks in beams using a Hybrid Genetic Algorithm (HGA). The objective function of the proposed method, based on dynamic properties, can be employed with partial modal information and is enhanced by including the initial residuals between numerical and experimental models along with combined l0 and l1 regularization. This enhancement mitigates the impact of limited modal data, experimental noise, and modeling inaccuracies. Additionally, a method of selecting an appropriate regularization parameter by analyzing residual and solution norm curves is presented. The performance of the proposed method is evaluated and verified using both numerical simulations and experimental studies on a steel cantilever beam under four damage scenarios. In the experimental study, natural frequencies and mode shapes were determined under ambient vibration conditions using Enhanced Frequency Domain Decomposition (EFDD) and the Stochastic Subspace Identification (SSI) method. Finite element models of the beam were developed in ABAQUS software for numerical analysis, and a comparative study of numerical and experimental data was conducted. The proposed approach successfully identifies, locates and quantifies single and multiple damages in the beam, demonstrating its reliability and effectiveness, particularly in scenarios with limited modal data and significant experimental noise.

An enhanced stochastic subspace identification incorporating time variables and its application to large-span sea-crossing suspension bridges

Structural health monitoring and assessment are essential to large-span sea-crossing bridges, and modal identification is the most effective method for achieving them, with the stochastic subspace identification (SSI) method dominating. However, when analyzed in open traffic scenarios with SSI for a large-span sea-crossing bridge, the contour of the vibration mode is distorted. Related affecting factors such as the measuring conditions, user-defined parameters, and traffic are explored, and traffic is verified as the essential component. To address this issue, an enhanced SSI algorithm incorporating time variables is created to automatically identify and rectify vibration mode shapes. Feature heterotopic signals are first matched with a common reference signal to perform modal identification. Advanced particle swarm optimization is applied to estimate critical parameters. The SSI algorithm with the density-based spatial clustering of applications with noise (DBSCAN) algorithm is then utilized to extract the modal characteristics. Both simulated misalignment and real records are applied to verify the suggested approach's performance. Comparative analysis with state-of-the-art techniques, such as the error norm, cross correlation, phase transform-β, mean phase deviation, and genetic algorithm, shows that the recommended approach is dependable and successful at identifying the large-span sea-crossing bridge's vibration mode shapes and computationally more accurate and efficient.

Automated identification and long-term tracking of modal parameters for a super high-rise building

Structural modal parameter identification plays an important role in wind resistance design, damage identification, and health monitoring. At present, the modal identification process still suffers from low automation and low computational efficiency, which are not conducive to the continuous processing of monitoring data. To address this, an automated modal parameter identification approach is proposed by introducing the fast density and grid based (DGB) clustering, and applied to the Shanghai Tower, which is the tallest building in China. Moreover, in order to determine the optimal model order of the stochastic subspace identification (SSI), the trend of the logarithmic reduction rate for singular values is extracted by empirical mode decomposition (EMD). The numerical simulation of a cantilever beam and the field measurement of Shanghai Tower under the influence of Typhoon In-Fa are conducted to validate the effectiveness of the proposed approach. Results demonstrate that the approach can obtain modal parameters without human intervention and has the advantages of simple process, fast computation and high precision. Furthermore, based on the data collected from the structural health monitoring system of the Shanghai Tower over a period of 5.5 years, the approach is applied to analyze the correlation between natural frequencies and ambient temperature, as well as the long-term evolution law of modes, discovering that multi-order frequencies show a decreasing trend over time. The finite element analysis suggests that this fact might be attributed to the increase in structural mass caused by the rising usage rate and the degradation in material properties.

Image-assisted 2D partition denoising method for distributed optical fiber vibration measurements

The traditional one-dimensional denoising method, commonly applied in vibration signal processing, would encounter issues as the calculation complexity, difficulty in maintaining spatial consistency and inability to address noise distributed spatially when measurements are from multiple points. Distributed optical fiber vibration sensing technology produces measurements densely distributed on both time and spatial axes, accompanied by inevitable noise from demodulation error, uneven packaging of sensing components and environmental interference, in which condition proposing new two-dimensional denoising method becomes particularly important. Focusing on the above spatiotemporal denoising target, this paper proposes a new image-assisted 2D partition denoising method by introducing the morphological methods, improving the traditional empirical mode decomposition, and raising the partition denoising strategy. The experimental and simulated distributed vibration measurements of a cantilever beam were carried out to verify the proposed method from the perspective of modal analysis implemented by the DATA-driven stochastic subspace identification. The analysis results indicate that the proposed method could handle the spatiotemporal noise simultaneously and is in favor of improving the continuity of mode identification results especially.

Full life-cycle vibration-based monitoring of a full-scale masonry arch bridge with increasing levels of damage

Structural health monitoring (SHM) is of great significance to maintain the safe operation of infrastructure, particularly for historic structures such as masonry arch bridges, as their material properties are deteriorating under effects such as climate change, but the demands of modern transport systems increase significantly. Vibration-based SHM has become a prevalent method for assessing the vulnerability of masonry arch bridges. However, one of the most critical limitations lies in the inability of correlate bridge’s damage accumulation with the variation trends in its modal parameters. To address this issue, in the present study, full life-cycle vibration-based monitoring was conducted on a full-scale masonry arch bridge, progressing from an undamaged state to failure under laboratory conditions. Accumulated damage was induced by applying point loads with increasing magnitudes until the bridge failed. Vibration data, collected after each loading test, were analysed utilizing the stochastic subspace identification (SSI) method, and frequency and damping ratio for the first five modes were identified. The results demonstrated that the first-order frequency had the most pronounced correlation with the stiffness degradation and the damage accumulation in the masonry arch bridge. The evolution of early-stage damage, the formation of the first hinge, and the activation of a four-hinge mechanism were successfully captured through the frequency degradation. The findings of the study may provide valuable insights into the damage assessment of historic masonry arch bridge infrastructures.

A subspace parameter identification method for nonlinear structures under oversampling conditions

In engineering practice, sampling frequencies much higher than the Nyquist frequency are often used to obtain accurate measurements of the dynamic characteristics of nonlinear systems. However, it may lead to oversampling when using the nonlinear subspace identification method (NSIM). This study proposes a subspace parameter identification method to address the identification of nonlinear structures under oversampling conditions. The proposed method first employs the prediction error method (PEM) to estimate the linear part of the state space model. In noisy environments, the PEM can achieve more accurate results than the NSIM by minimizing an error criterion function. The initial values fed into the PEM are obtained by the stochastic subspace identification method. The nonlinear part of the state space model is estimated by solving a linear regression equation. The proposed method effectively reduces the impact of high sampling frequencies on identification accuracy by combining data with different sampling frequencies to construct the linear regression equation. Among them, the low sampling frequency data is obtained by downsampling the high sampling frequency measurement data. The proposed method effectively reduces the effect of high sampling frequencies and environmental noise on identification accuracy by accurately estimating both the linear and nonlinear parts of the state space model. The identification accuracy of the proposed method is numerically verified by a chain-type structure with five degrees of freedom and a cantilever beam structure. Additionally, experimental verification is carried out on a two degrees of freedom floor structure with local nonlinearities. The results show that the proposed method achieves high identification accuracy in oversampling conditions and noisy environments.

Wear parameters identification of hydrodynamic bearing based on operational modal analysis and on-rotor sensing technology

Abrasive wear is a predominant fault in hydrodynamic bearings that can deteriorate bearing performance and damage the rotating machine eventually. The present work proposes a novel wear parameters identification method for hydrodynamic bearings based on operational modal analysis (OMA) and on-rotor sensing (ORS) technology. With this framework, the objective function is defined as the sum of squared errors between the theoretical modal frequencies and the identified modal frequencies. For the experiment verification, the ORS technology is used to collect rotor vibration directly. Additionally, the cepstrum editing procedure (CEP) is applied to enhance modal information. The different wear parameters are successfully identified using combined stochastic subspace identification (SSI) and particle swarm optimization (PSO) algorithms in both simulation and experimental signals.

A new fully automated operational modal analysis algorithm intended for large civil structures

Abstract A new fully automated operational modal analysis (AOMA) algorithm based on the time-domain covariance-driven stochastic subspace identification (cov-SSI) system identification method is proposed. The new algorithm is intended for large civil structures, such as long-span suspension bridge for which an application example is included. It is shown that the new algorithm is capable of consistently detecting all expected structural modes of the Hardanger bridge, without requiring any prior tuning or parameter selection. The dominating three-step approach to AOMA algorithms in published literature on this topic is highlighted and used as a basis for the new algorithm; it first removes spurious poles using a Gaussian mixture model by analysing their distance to their nearest neighbour. Secondly, hierarchical clustering is used to regroup similar poles, and finally, the groups representing physical modes are selected through a similarity analysis of the size of hierarchical clusters.

Finite element model updating and response prediction of a frame structure based on optimal sensor placement

This paper proposes an approach for finite element (FE) model updating and response prediction of frame structures based on optimal sensor placement (OSP), which integrates sensor placement optimization, mode expansion, and model updating techniques. Firstly, sensor optimization layout and modal testing analysis are conducted on a bolted laboratory frame structure. The covariance-driven stochastic subspace identification (SSI-COV) method identifies the real-world structure’s natural frequencies, modal damping, and mode shapes. Secondly, the complete mode shapes are expanded using the measured incomplete modal data from the limited number of sensors. Thirdly, a multi-objective function based on frequency and mode shapes is established to adjust the parameters of the FE model. This ensures that the updated model accurately represents the dynamic properties of the actual structure within a specific frequency range. Finally, the Rayleigh damping of the frame structure is estimated, and the damping matrix is assembled to enhance the accuracy of dynamic response prediction in the updated model. By comparing the response prediction results of the updated FE model with and without considering the updated damping effects to the measurement data of the real-world structure, it is demonstrated that the proposed method considering updated damping effects can more effectively predict the structural response.

A mechanics-informed neural network method for structural modal identification

Modal identification is one of the core topics within the realm of structural health monitoring (SHM). In this study, we summarize four modal mechanical properties and propose a mechanics-informed neural network (MINN) method for structural modal identification. The proposed MINN method incorporates the sparsity of the data in the time–frequency domain and cross-correlation minimization in the time domain into the neural network to obtain modal parameters, which uses sparsity constraint and cross-correlation minimization constraint to obtain the accurate modal responses and mode shapes. Subsequently, modal frequencies and damping ratios can be derived from the modal responses. The proposed MINN method is verified by numerical simulations and two actual suspension bridges. Compared with traditional methods, the proposed MINN method has two major advantages. Firstly, the proposed MINN method presents explicit mathematical equations to distinguish the modes and the spurious modes, which obviates the necessity for priori information such as model order or time-consuming manual intervention to distinguish the modes and the spurious modes. Therefore, it can be implemented adaptively to determine the modal order and obtain the modal parameters. Secondly, the proposed MINN method can obtain a greater number of accurate modal parameters than traditional methods and achieves an increase of 102.6%, 43.4%, and 31.5% in the number of accurate results when compared to covariance-driven stochastic subspace identification (SSI-COV), data-driven stochastic subspace identification (SSI-DATA) and the natural excitation technique and the eigensystem realization algorithm (NExT-ERA), respectively. Therefore, the proposed MINN method provides an adaptively modal identification method that has clear modal mechanical properties to distinguish the modes and the spurious modes and can obtain a greater number of accurate results.

Experimental validation of a short-term damping estimation method for wind turbines in nonstationary operating conditions

Abstract. Modal properties and especially damping of operational wind turbines can vary over short time periods as a consequence of environmental and operational variability. This study seeks to experimentally test and validate a recently proposed method for short-term damping and natural frequency estimation of structures under the influence of varying environmental and operational conditions from measured vibration responses. The method is based on Gaussian process time-dependent auto-regressive moving average (GP-TARMA) modelling and is tested via two applications: a laboratory three-storey shear frame structure with controllable, time-varying damping and a flutter test of a full-scale 7 MW wind turbine prototype, in which two edgewise modes become unstable. Damping estimates for the shear frame compare well with estimates obtained with stochastic subspace identification (SSI) and standard impact hammer tests. The efficacy of the GP-TARMA approach for short-term damping estimation is illustrated through comparison to short-term SSI estimates. For the full-scale flutter test, GP-TARMA model residuals imply that the model cannot be expected to be entirely accurate. However, the damping estimates are physically meaningful and compare well with a previous study. The study shows that the GP-TARMA approach is an effective method for short-term damping estimation from vibration response measurements, given that there are enough training data and that there is a representative model structure.

The Determination of Effective Modes of Wind-Bridge Coupling System Based on Stabilization Clustering Method

Modal identification is the core content of the full-bridge aeroelastic model wind tunnel test. Modal consistency is the most direct and effective basis to ensure that the model and the real bridge meet the similarity criteria. Accurate, simple and fast identification of modal parameters has very important practical significance for the evaluation and correction of aeroelastic models. In this paper, the eigensystem realization algorithm (ERA) and stochastic subspace identification (SSI) is improved, and the automatic identification theory of effective modes of bridge aeroelastic model based on stable clustering method is proposed. The stable diagram eliminates the unstable modes by comparing the results of different system orders, and the pedigree clustering further polarizes the stable modes to obtain multi-order and high-precision system modes. Numerical simulation examples and wind tunnel engineering examples show that the method has high accuracy and good adaptability, and can solve the problem of unstable damping identification results to a certain extent. In the wind tunnel test, the stable clustering method is one of the effective methods to identify the modal of the bridge aeroelastic model excited by the wind environment.

Fully automated operational modal identification based on scale-space peak picking algorithm and power spectral density estimation

Modal analysis is a fundamental and essential research direction in the field of structural engineering. The ultimate goal is to determine the modal parameters of the structures. However, the existing modal analysis algorithms often require a large number of parameter adjustments and manual intervention during operation, which cannot be fully automated. In order to realize the automatic identification of modal parameters, the automatic operational modal identification method (AOMI) is proposed based on the interpolated power spectral density estimation (IPSE). To achieve more precise spectrum analysis in the low-frequency band, IPSE is employed to perform Fourier transform on the original frequency domain segment with optimized frequency resolution. This enhances the sharpness of the obtained spectrum in the low-frequency range, making peak frequencies more discernible. Subsequently, the scale-space peak picking algorithm is used to automatically obtain the peak of the power spectral density (PSD), thus enabling the automatic identification of the natural frequency. Finally, the frequency domain decomposition method (FDD) is used to identify modal parameters based on the natural frequencies. The effectiveness of AOMI is verified through the modal identification of the old steel truss bridge and the three layer framework. Under the environmental excitation, the frequencies identified by the IPSE method is close to that of FDD, Bayesian fast fourier transform (FFT) and covariance driven stochastic subspace identification (SSI-COV). Furthermore, the PSD obtained through IPSE has sharper peak than that of FDD and the Welch’s method. The damping ratio identification accuracy and modal assurance criterion (MAC) are satisfactory in AOMI, which can improve the automatic performance.

An improved stochastic subspace identification approach for automated operational modal analysis of high-rise buildings

The Stochastic Subspace Identification (SSI) method has attracted significant attention in the field of Operational Modal Analysis (OMA) for large civil engineering structures such as high-rise buildings. State-space-based identification methods often encounter the challenge of spurious modes, leading to extensive efforts to mitigate this issue, commonly through stabilization diagram techniques. Nonetheless, the accuracy of the stabilization diagrams can be compromised by various interferences and the presence of spurious poles (i.e., outliers) in the high-order region. Consequently, this diminishes the precision of the modal identification. This study introduces an improved automated OMA technique based on SSI to enhance the reliability and accuracy of modal estimation. The proposed approach encompasses modal identification using SSI coupled with automated interpretation of SSI outputs, employing rigorous validation criteria and clustering techniques. An essential component of this approach involves signal segmentation during the data preprocessing stage, enabling the effective removal of spurious poles and thus a more robust interpretation of modal identification results via the stabilization diagram in an automated fashion. A numerical example was employed to demonstrate the effectiveness and applicability of the method. Subsequently, the proposed methodology is validated using field measurements conducted on two high-rise buildings.

Mode Shape-Based Damage Detection of Twin Skewed Bridges Based on OMA and FE Model Updating

A comparative study of the dynamic behavior of twin single-span bridges was carried out based on operational modal analysis and a linear finite element model updating with the aim of finding a potential damage indicator. The bridges are skewed, 30 m long and with prestressed concrete girders as the main support structure, but one of them were collided and suffered damage in two of its I-shape girders. Ambient vibration tests were conducted by deploying 14 uniaxial accelerometers along the sidewalks, experimental modal parameters were identified using the COVariance-driven Stochastic Subspace Identification method (SSI-COV). Six vertical modes were clearly identified for each structure. Additionally, some extra modes with non-proportional damping were identified. It was found that the modal frequencies are sensitive to the support conditions and the overall stiffness of the bridge but not to the localized girder damage, which, on the contrary, is reflected in the discrepancy in the fundamental bending mode shape amplitude between the lateral sides of the bridge, and a so-called Continuous Symmetry Measure adopted as a symmetry index seems to be a good damage indicator for this type of straight-alignment simply supported bridges, given the symmetric nature of the bending mode shapes and the asymmetric nature of the damage.

Tracking Modal Parameters of Structures Online Using Recursive Stochastic Subspace Identification under Ambient Excitations

Continuous and autonomous system identification is an alternative to regular inspection during operations, which is essential for structural integrity management (SIM) as well as structural health monitoring (SHM). In this regard, online (or real-time) system identification techniques that have recently received considerable attention can be used to assess the current condition and performance during operations and, in the meantime, can be utilized to detect any damage or deterioration. For example, stochastic subspace identification (SSI), based on recursive formulation, has proven its capability in tracking modal parameters as well as time-variant dynamic behaviors. This study proposes the implementation of recursive SSI (RSSI) using the matrix inversion lemma to track slow time-varying parameter changes under ambient excitations. Subsequently, some investigations for practical implementation are examined and discussed. For verifying the reliability of SHM applications based on the proposed methods, two datasets measured from different experiments are exploited to identify the modal parameters reclusively. The results from both numerical simulations and experimental investigations demonstrated the effectiveness of tracking the modal parameters exhibiting time-varying dynamic characteristics under white noise excitations (or ambient excitations).

On the Effect of Nonlinear Damping Sources in Output-Only Identification Methods Applied to Floating Wind Turbines

Output-only methods for modal identification are only strictly valid if a set of requirements are fulfilled regarding both structural and environmental conditions. A particularly challenging effect in wind turbine dynamics is the significant presence of nonlinear damping sources coming from aerodynamic forces and, in offshore applications, hydrodynamic forces on the substructure. In this work, the impact of these terms is firstly discussed in analytical terms, and then the corresponding effect on the performance of the covariance-driven stochastic subspace identification is evaluated on a single-degree-of-freedom model. The analysis is then extended to a full hydro-aeroelastic simulation of a 5 MW floating wind turbine using the open source software OpenFAST, mimicking the structural response in free decay tests and in parked conditions with turbulent wind fields. The results show that output-only identification methods are applicable in these challenging scenarios, but the results obtained must be carefully interpreted, since their dependence on the environmental conditions and motion amplitude imply that they are not directly translated into the structure properties, although still closely related to them.

Automated operational modal analysis for supertall buildings based on a three-stage strategy and modified hierarchical clustering

Several automated operational modal analysis (AOMA) methods have been developed to evaluate the dynamic characteristics of large-scale structures in real-time under environmental and operational variations. Typically, these methods need to manually set some thresholds to achieve automatic identification. However, there is a lack of uniform standards for setting these thresholds, which may result in the adoption of inappropriate settings in AOMA, thereby introducing errors in modal identification. To address this issue, this paper proposes a novel AOMA framework to automatically identify modal parameters, which mainly involves three steps. Firstly, the covariance-driven stochastic subspace identification (Cov-SSI) algorithm is employed to perform modal estimation based on measured ambient vibration records. Then, a three-stage strategy, including hard validation criteria, Gaussian mixture model clustering-driven soft validation criteria, and the local outlier factor algorithm, is presented to remove spurious modes and determine real physical modes. Finally, a modified hierarchical clustering based on the k-nearest neighbor information is applied to group the determined physical modes to achieve automatic modal identification. The effectiveness of the proposed AOMA framework in automatically identifying modal parameters is validated through a numerical simulation study for a five-floor frame structure. Furthermore, the proposed framework is applied to evaluate the modal properties of a 600 m high supertall building based on long-term monitoring records. The results show that the AOMA framework can effectively remove spurious modes and group physical modes, thus identifying the modal parameters with high accuracy. In short, the proposed AOMA framework is an effective tool for modal identification applied in field measurements or structural health monitoring.

Subspace Identification of Bridge Frequencies Based on the Dimensionless Response of a Two-Axle Vehicle.

As an essential reference to bridge dynamic characteristics, the identification of bridge frequencies has far-reaching consequences for the health monitoring and damage evaluation of bridges. This study proposes a uniform scheme to identify bridge frequencies with two different subspace-based methodologies, i.e., an improved Short-Time Stochastic Subspace Identification (ST-SSI) method and an improved Multivariable Output Error State Space (MOESP) method, by simply adjusting the signal inputs. One of the key features of the proposed scheme is the dimensionless description of the vehicle-bridge interaction system and the employment of the dimensionless response of a two-axle vehicle as the state input, which enhances the robustness of the vehicle properties and speed. Additionally, it establishes the equation of the vehicle biaxial response difference considering the time shift between the front and the rear wheels, theoretically eliminating the road roughness information in the state equation and output signal effectively. The numerical examples discuss the effects of vehicle speeds, road roughness conditions, and ongoing traffic on the bridge identification. According to the dimensionless speed parameter Sv1 of the vehicle, the ST-SSI (Sv1 < 0.1) or MOESP (Sv1 ≥ 0.1) algorithm is applied to extract the frequencies of a simply supported bridge from the dimensionless response of a two-axle vehicle on a single passage. In addition, the proposed methodology is applied to two types of long-span complex bridges. The results show that the proposed approaches exhibit good performance in identifying multi-order frequencies of the bridges, even considering high vehicle speeds, high levels of road surface roughness, and random traffic flows.

A robust probabilistic approach to stochastic subspace identification

Modal parameter estimation of operational structures is often a challenging task when confronted with unwanted distortions (outliers) in field measurements. Atypical observations present a problem to operational modal analysis (OMA) algorithms, such as stochastic subspace identification (SSI), severely biasing parameter estimates and resulting in misidentification of the system. Despite this predicament, no simple mechanism currently exists capable of dealing with such anomalies in SSI. Addressing this problem, this paper first introduces a novel probabilistic formulation of stochastic subspace identification (Prob-SSI), realised using probabilistic projections. Mathematically, the equivalence between this model and the classic algorithm is demonstrated. This fresh perspective, viewing SSI as a problem in probabilistic inference, lays the necessary mathematical foundation to enable a collection of new OMA approaches. To this end, a statistically robust SSI algorithm (robust Prob-SSI) is developed, capable of providing a principled and automatic way of handling outlying or anomalous data in the measured timeseries, such as may occur in field recordings, e.g. intermittent sensor dropout. Robust Prob-SSI is shown to outperform conventional SSI when confronted with ‘corrupted’ data, exhibiting improved identification performance and higher levels of confidence in the found poles when viewing stabilisation diagrams. Similar benefits are also demonstrated on the Z24 Bridge benchmark dataset, highlighting enhanced performance on measured systems.