Stochastic Subspace Identification Method Research Articles

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.

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.

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.

Evaluation of vibration properties of an 18-story mass timber–concrete hybrid building by on-site vibration tests

Timber–concrete hybrid structural systems are a practical option to provide tall mass timber buildings with a lateral load-resisting system. This paper discusses the dynamic behavior of an 18-story timber–concrete hybrid building based on the vibration properties evaluated by on-site vibration tests. First, microtremor measurements and human-powered excitation tests were carried out and the obtained vibration data were analyzed using a stochastic subspace identification method to derive natural frequencies, damping ratios, and mode shapes. Then, a finite-element (FE) model was developed based on detailed structural design information, and its eigenvalues and eigenvectors were compared with the test results. The vibration test results showed various mode shapes, including in-plane deformation of the floor diaphragm composed of cross-laminated timber (CLT) panels. The damping ratios in all the modes were scattered between 1 and 3%, and no frequency dependency was observed. The modal properties of the FE model agreed well with the test results by considering the additional stiffness of non-structural components. In order to simulate the in-plane deformation of the CLT floor diaphragm, detailed modeling of the connection between each CLT floor panel and the connection between CLT floor panels and concrete cores is recommended. The findings provide practitioners with an insight into dynamic properties and FE modeling methods of tall timber–concrete hybrid buildings.

The added mass of a biaxial tensioned membrane in still air

It is well known that added mass significantly impacts the vibration of membrane structures in still air. Numerous theoretical, numerical, and experimental studies have been conducted on the added mass of axial tensioned membranes. However, the added mass of biaxial tensioned membranes subjected to different tensile forces in two directions in still air remains unclear. Therefore, this study focuses on investigating the added mass of biaxial tensioned membranes in still air. Firstly, this paper proposes a simplified theoretical model for the added mass of biaxial tensioned membranes in still air based on the energy conservation law. To validate the accuracy of the proposed model, a modal testing system for a rectangular membrane is designed to identify the frequencies of biaxial tensioned membranes in still air. In the experimental system, the utilization of a laser displacement sensor array is to obtain the time-displacement information of observation points. Then, a combination of Fast Fourier Transform (FFT) and Stochastic Subspace Identification (SSI) methods is employed for modal identification. Comparative analysis between the experimental and theoretical model results demonstrates an excellent agreement in frequencies. Additionally, the proposed added mass model is closer to the experimental added mass than other models, with a maximum deviation of only 4.77 %. This research provides a reference for the study of the impact of air on the ground vibration of membranes.

Numerical and experimental study on the dynamic equivalent methodology of a membrane antenna structure and a grid membrane structure

The dynamic equivalent methodology utilizing a grid membrane substitute for a planar membrane for vibration test verification on the ground has been demonstrated in small sizes with simple constraints. The method lacks accuracy and validity for a real complex membrane antenna. This paper thus focuses on the applicability of the equivalence approach to a membrane antenna and the factors affecting its dynamic equivalence to verify whether the grid membrane structure can replace the membrane antenna structure for modal identification on the ground. To this end, an experimental system is developed to conduct vibration tests on the grid membrane structure in air, using the grid membrane model with unidirectional tension as a substitute for the membrane array. The connecting strips attach the grid membrane to thin-walled lenticular booms (TWLBs) to allow for unidirectional tension. A laser displacement sensor array is employed to obtain the vibration data. A combination of the Fast Fourier Transfer (FFT) and Stochastic Subspace Identification (SSI) methods is applied to identify the structural frequencies and mode shapes. Vibration tests on the grid membranes with different added masses and tensile forces are conducted. Meanwhile, the numerical results for the rectangular membrane are chosen as a reference for comparison with the experimental results for the grid membrane. In conclusion, the numerical results for the membrane antenna structure and the experimental results for the grid membrane structure show good agreement in terms of frequency and mode shape. Additionally, both the added masses in the intersection areas of the grid membrane and the tensile force have a significant effect on the dynamic equivalence.

Correlation Analysis of Large-Span Cable-Stayed Bridge Structural Frequencies with Environmental Factors Based on Support Vector Regression

The dynamic characteristics of bridge structures are influenced by various environmental factors, and exploring the impact of environmental temperature and humidity on structural modal parameters is of great significance for structural health assessment. This paper utilized the Covariance-Driven Stochastic Subspace Identification method (SSI-COV) and clustering algorithms to identify modal frequencies from four months of acceleration data collected from the health monitoring system of the Jintang Hantan Twin-Island Bridge. Furthermore, a correlation analysis is conducted to examine the relationship between higher-order frequency and environmental factors, including temperature and humidity. Subsequently, a Support Vector Machine Regression (SVR) model is employed to analyze the effects of environmental temperature on structural modal frequencies. This study has obtained the following conclusions: 1. Correlation analysis revealed that temperature is the primary influencing factor in frequency variations. Frequency exhibited a strong linear correlation with temperature and little correlation with humidity. 2. SVR regression analysis was performed on frequency and temperature, and an evaluation of the fitting residuals was conducted. The model effectively fit the sample data and provided reliable predictive results. 3. The original structural frequencies underwent smoothing, eliminating the influence of temperature-induced frequency data generated by the SVR model. After eliminating the temperature effects, the fluctuations in frequency within a 24 h period significantly decreased. The data presented in this paper can serve as a reference for further health assessments of similar bridge structures.

Enhanced semi-automated Bayesian modal identification approach for Ultra-high voltage transmission towers

The accuracy and efficiency of the modal identification for Ultra-high voltage (UHV) transmission towers under harsh operational environments is seriously affected by environmental noises. In general, the initial natural frequency drowned in the noises is evaluated by the experience of engineers before performing a Bayesian modal identification approach. It will lead to potential errors and limitations to the application of the Bayesian modal identification approach. In this study, a coherent clustering enhanced semi-automated Bayesian modal identification method is proposed and then implemented in the operational modal analysis of an instrumented UHV transmission tower under ambient excitation. The coherent clustering is adopted to obtain the quantitative value of the initial natural frequency, in place of the empirical evaluation. Then, the Fast Bayesian Fast Fourier Transform (FBFFT) method could accurately identify the most probable values (MPV) and the corresponding posterior uncertainty of modal parameters. Moreover, the first ten modes of the UHV transmission tower are identified, and the results of modal parameters are compared to the stochastic subspace identification (SSI) method. The results show that the coherent clustering enhanced semi-automated Bayesian method could accurately identify modal parameters with posterior uncertainty, even for modes weakly excited. The proposed method could provide a reliable basis for the modal updating, damage detection, and dynamic reliability assessment of UHV transmission towers.

An edge-computing framework for operational modal analysis of offshore wind-turbine tower

With massive construction of offshore wind turbines (OWT), structural health monitoring (SHM) techniques have been applied to estimate the working conditions and provide early warnings of structural damage. This study proposes an edge-computing-based SHM system for operational modal analysis (OMA) of an OWT. ThssCe system comprised an open-source data acquisition (DAQ) device, a Raspberry Pi single-board computer, and associated drivers, protocols, and algorithms. The DAQ device can measure the acceleration response of an operating OWT subjected to environmental excitation and transmit raw signals to the Raspberry Pi according to the UDP protocol. The data are further processed to obtain the first-order frequency via an OMA algorithm developed by combining Kalman filtering, the random decrement technique (RDT), and the stochastic subspace identification (SSI) method. The proposed SHM system was implemented in a scaled OWT model to verify its feasibility for field use. The experimental results indicate that (1) the edge-computing-based SHM system can accurately measure the response of the OWT; (2) the proposed algorithm can estimate the modal information of the OWT during operation; (3) the variation in the first-order frequency induced by lateral loads can be identified by the SHM system. Unlike a centralized SHM system, the proposed edge-computing-based SHM system does not require massive data transmission, making it promising for offshore environments.

Development of a software platform for bridge modal and damage identification based on ambient excitation

Modal and damage identification based on ambient excitation can greatly improve the efficiency of high-speed railway bridge vibration detection. This paper first describes the basic principles of stochastic subspace identification, peak-picking, and frequency domain decomposition method in modal analysis based on ambient excitation, and the effectiveness of these three methods is verified through finite element calculation and numerical simulation. Then the damage element is added to the finite element model to simulate the crack, and the curvature mode difference and the curvature mode area difference square ratio are calculated by using the stochastic subspace identification results to verify their ability of damage identification and location. Finally, the above modal and damage identification techniques are integrated to develop a bridge modal and damage identification software platform. The final results show that all three modal identification methods can accurately identify the vibration frequency and mode shape, both damage identification methods can accurately identify and locate the damage, and the developed software platform is simple and efficient.

Amplitude dependence of natural frequency and damping ratio for 5 supertall buildings with moderate-to-strong typhoon-induced vibrations

Modal parameters play a significant role in the wind-resistant design and vibration control of high-rise buildings. Results from previous studies demonstrate that the natural frequency and damping ratio of high-rise buildings usually exhibit nonlinear variation characteristics with the increase of structural amplitude. However, due to lacking relevant field measurements, the features of natural frequency and damping ratio for super-tall buildings under large wind-induced amplitudes have been less explored. This paper presents a field study on typhoon-induced structural responses of five supertall buildings whose structural heights all exceed 300 m. The peak acceleration responses at the measurement height of most buildings exceeded 10 cm/s2 (gal), with the maximum response reaching 20 gal. Two frequently used methods, i.e., the stochastic subspace identification (SSI) and random decrement technique (RDT) methods are utilized to compare and analyze the amplitude-dependent characteristics of target modal parameters. It is found that with the increase of wind-induced structural vibration amplitude, the fundamental natural frequency generally first decreased and then leveled off, while the damping ratio for the first-order mode of the buildings except the one equipped with a tuned mass damper (TMD) system first increased while then tended to decrease. More interestingly, the critical normalized structural vibration amplitude associated with the maximum damping ratio is found to be in the range of 10−4∼10−5, which agrees well with the theory of “critical tip drift ratio”. Potential explanations for the above phenomena are presented. Furthermore, the obtained results are further compared with existing statistical models, and associated discrepancies suggest the complexities of the dynamic features of supertall buildings.

Modal analysis of an operational offshore wind turbine using enhanced Kalman filter‐based subspace identification

AbstractOperational modal analysis (OMA) is an essential tool for understanding the structural dynamics of offshore wind turbines (OWTs). However, the classical OMA algorithms require the excitation of the structure to be stationary white noise, which is often not the case for operational OWTs due to the presence of periodic excitation caused by rotor rotation. To address this issue, several solutions have been proposed in the literature, including the Kalman filter‐based stochastic subspace identification (KF‐SSI) method which eliminates harmonics through estimation and orthogonal projection. In this paper, an enhanced version of the KF‐SSI method is presented that involves a concatenation step, allowing multiple datasets with similar environmental conditions to be used in the identification process, resulting in higher precision. This enhanced framework is applied to an operational OWT and compared to other OMA methods, such as the modified least‐squares complex exponential and PolyMAX. Using field data from a multi‐megawatt operational OWT, it is shown that the enhanced framework is able to accurately distinguish the first three bending modes with more stable estimates and lower variance compared to the original KF‐SSI algorithm and follows a similar trend compared to other approaches.

A Real-Time Monitoring System for Cable Tension with Vibration Signals Based on an Automated Algorithm to Sieve Out Reliable Modal Frequencies

Cables or suspenders are the critical force-transmitting components of cable-supported bridges, and their timely tension monitoring is consequently the most important issue in the corresponding structural health monitoring. However, very few works regarding the full automation of vibration-based tension estimation have been reported in the literature. To develop a monitoring system of cable tension based on real-time vibration signals, this research first employs an efficient stochastic subspace identification (SSI) method with tailored parameter selection to continuously identify the three frequencies of adjacent modes for the cables of Mao-Luo-Hsi Bridge. More importantly, an automated sieving algorithm is delicately established to obtain the stable modal frequencies by making the best of the specific modal frequency distribution for cables. The ratios between the frequency values identified from SSI analysis are exhaustively checked to systematically extract the qualified cable frequencies and decide their corresponding mode orders. The tension is finally computed with one available cable frequency according to the priority order predetermined by the statistics of identification rate. Demonstrated by analyzing the vibration signals measured from the stay cable of Mao-Luo-Hsi Bridge in real time for two full years, the effectiveness and robustness of this real-time monitoring system have been extensively testified. The long-term success rates for the immediate determination of dependable tension are found to be perfect for 15 of the 18 investigated cables. As for the other three cables, their corresponding success rates are still higher than 99.99% with very few cases of absent or false tension values.

A novel dynamic model parameter updating methodology of a sluice based on vibration response

Accurate dynamic model of the sluice plays an important role in the structural health monitoring and performance evaluation. It is difficult for the established model to truly and comprehensively reflect the overall mechanical characteristics of the sluice because of parameter uncertainty. Structural dynamic model updating based on vibration response is a feasible approach, but it relies on the modal parameters identification and reliable surrogate model. Therefore, a novel dynamic model parameter updating methodology of a sluice is proposed. Firstly, an improved stochastic subspace identification method based on adaptive variational mode decomposition is proposed to accurately obtain sluice modal parameters. Secondly, Kriging surrogate model based on orthogonal test method and particle swarm optimization is introduced. The mathematical surrogate model reflecting the strong non-linear relationship between modal parameters and material parameters with high sensitivity is constructed. Finally, taking the minimum relative error between the calculated and identified values of the sluice modal parameters as the objective function, whale optimization algorithm is used to optimize and the dynamic model parameter of the sluice is updated. A case of sluice physical model shows that the proposed method is feasible and reliable, which lays a great foundation for the structural health diagnosis of sluices.

Influence of distance between the first sensor and support in determining tension of fixed-end cable using multiple synchronized vibration measurement

Non-destructive evaluation has been developed to identify cable tension for any end-rotational stiffness condition. One of the existing methods uses multiple vibration measurements. This paper aims to study the effect of shifting the closest sensor to the support in multiple vibration measurements through numerical simulation. A cable fixed at both ends vibrates under artificial random excitation is simulated. Three cases of five synchronized signal recording arrangements are applied to the cable, where the distance from the support to the first recorder (d1) is different for each case. The distance d1 of Case 1, Case 2, and Case 3 are d, 2d, and 3d, respectively, where d is the distance between the recorder, and d is taken as 7% of the total length and the same for all cases. Cable dynamical properties are analyzed using a stochastic subspace identification method, and effective lengths are figured out by curve fitting. Under the same frequency range, seven effective lengths are obtained in Case 1, while eight are detected in Cases 2 and 3. Besides, it can be concluded that Case 1 generates larger errors in predicting tension compared to other cases.

Modal identification of structures with a dynamic behaviour characterised by global and local modes at close frequencies

Identification of modal parameters of a space frame structure is a complex assignment due to a large number of degrees of freedom, close natural frequencies, and different vibrating mechanisms. Research has been carried out on the modal identification of rather simple truss structures. So far, less attention has been given to complex three-dimensional truss structures. This work develops a vibration-based methodology for determining modal information of three-dimensional space truss structures. The method uses a relatively complex space truss structure for its verification. Numerical modelling of the system gives modal information about the expected vibration behaviour. The identification process involves closely spaced modes that are characterised by local and global vibration mechanisms. To distinguish between local and global vibrations of the system, modal strain energies are used as an indicator. The experimental validation, which incorporated a modal analysis employing the stochastic subspace identification method, has confirmed that considering relatively high model orders is required to identify specific mode shapes. Especially in the case of the determination of local deformation modes of space truss members, higher model orders have to be taken into account than in the modal identification of most other types of structures.

Noise effects analysis on subspace-based damage detection with neural networks

Structural health monitoring (SHM) is nowadays a significant and topical issue in the civil engineering field, whose first-level task is the damage detection job. Recent advances in artificial intelligence (AI) combined with non-destructive assessments based on output-only vibration signals represent the frontier of SHM. The present research study has handled an AI-based multi-class damage classification task on a numerical benchmark beam problem with three different procedures. The main purpose was to investigate the noise effects of the analyzed damage detection procedures simulating real-world accelerometer sensors. The first damage identification method leverages simple statistical features of vibration signals, whereas the second method additionally considers a subspace-based damage indicator. The third method relies on a dataset composed of a specific set of damage indicators. Two deep learning (DL) techniques have been adopted to compare and verify the results, i.e. a multi-layer perceptron (MLP) and a one-dimensional convolutional network (1D-CNN). Three groups of different reasonable signal-to-noise ratio (SNR) levels were set up to simulate micro electro-mechanical system (MEMS) accelerometer sensors in real-world conditions to evaluate noise effects. In a nutshell, the results demonstrated that the DL models based on stochastic subspace identification (SSI) methods have good noise immunity performance for damage detection purposes.

Fast online implementation of covariance-driven stochastic subspace identification

Fast online system identification is critical for the timeliness of downstream applications such anomaly detection, condition assessment and decision making. Although some very efficient algorithms, such as the fast Fourier transform (FFT), exist, they are unable to identify damping ratios or mode shapes. The stochastic subspace identification (SSI) method is currently a widely used system identification algorithm. However, its time-consuming operation prevents downstream applications from running online in (near) real time. This paper aims to develop a fast online implementation of covariance-driven SSI (COV-SSI). Time complexity is utilized to analyze the computational cost of COV-SSI. Two steps of COV-SSI are determined to be time-consuming and are accelerated. They are (i) the computation of the covariance matrix (accelerated through incremental aggregation in the dataflow model) and (ii) the singular value decomposition (SVD) of the Toeplitz matrix (accelerated through a fast truncated SVD approach). These strategies do not require any additional parameters; the error between the proposed method and the original COV-SSI method is negligible, but the method is significantly accelerated. Finally, some numerical examples are used to validate the speed and accuracy of the fast online strategies.

Dynamic material parameter inversion of high arch dam under discharge excitation based on the modal parameters and Bayesian optimised deep learning

Vibration-based dam safety monitoring methods have increasingly become a research hotspot for assessing dam safety. The accurate material parameters of a high arch dam and its foundation are critical for monitoring vibration safety. Accordingly, a dynamic material parameter inversion framework for an arch dam is developed based on modal parameters and deep learning. Firstly, an determined-order stochastic subspace identification method with adaptive variational modal decomposition is proposed based on the prototype vibration signal obtained under the discharge excitation, which can effectively identify the modal parameters. Secondly, the sensitivity of the dynamic elastic modulus (DEM) in different regions to the modal parameters of the arch dam was analysed using the orthogonal test method and variance analysis method, and it was utilised to ascertain the DEM to be inverted. Finally, a Bayesian optimised multi-output long short-term memory neural network is used to establish a nonlinear mapping relationship between the DEM and the modal parameters as an alternative to finite element calculations, and the identified modal parameters are employed as network inputs to inverse the actual DEM of each zoning. An engineering example shows that the proposed DEM inversion method for high arch dams is effective and accurate, providing a good basis for the vibration safety analysis of arch dams. This study overcomes the limitations of difficult to effectively extract modal parameters of arch dams under discharge excitation, and advances the application of deep learning technology in hydraulic engineering by combining modal parameters.

Modal Parameters Prediction and Damage Detection of Space Grid Structure under Environmental Effects Using Stacked Ensemble Learning

A stacked ensemble learning model is developed to predict the modal parameters of space grid steel structures under environmental effects. Potential damage is detected via statistical analysis of the prediction residuals. For this purpose, five standalone heterogeneous machine learning models were trained for predicting natural frequencies; each model used the principal components of the environmental data as input parameters. Next, a stacked ensemble learner was built using the outputs of the five standalone models as its inputs. Finally, a damage indicator combining the predicted residuals of multiple orders of natural frequencies is proposed and statistically analyzed for accurate damage detection. To verify the effectiveness of the proposed method, a space grid model was created in the field environment and measured for a period. Dynamic and environmental data were collected, such as ambient temperature, humidity, wind speed and direction, and structural surface temperature. An automated procedure of the covariance-driven stochastic subspace identification method was conducted to identify bulk mode. The environmental dependence of the natural frequencies, damping ratios, and vibration modes was analyzed. Then, the method was validated based on short-term monitoring data from the baseline health state and unknown future states. The results show that the natural frequencies and damping ratios of space grid structures fluctuate significantly on a daily basis due to environmental influences. Stacked ensemble learning utilizes predictions from multiple heterogeneous models to produce a better predictive model. The statistical analysis of the prediction residuals by ensemble learning effectively removes the environmental influences, allowing for timely structural damage detection.