Stochastic Subspace Identification Method Research Articles

Dynamic response analysis of a real-world operating offshore wind turbine under earthquake excitations

This paper describes the dynamic monitoring campaign conducted on a monopile offshore wind turbine (OWT). An unexpected harvest is that the dynamic responses of the OWT under earthquake excitations were captured. Therefore, particular emphasis is placed on any abnormal behavior of the OWT. Operational modal analysis with the stochastic subspace identification method is used to reveal any variations in the modal parameters before, during and after the earthquake. The influence of the time-varying yaw angle on modal identification is derived. The results indicate that the earthquake had a colossal impact on the OWT as compared to the wind loading. The earthquake allows the identification of some of the high-frequency bending and twisting modes. Another critical finding is that the first frequency of the OWT increased after the earthquake, possibly implying some quick consolidation of the seabed adjacent to the OWT. Although earthquakes are often considered harmful to the safety of OWTs, it is possible for some special cases to improve the bearing capacity and stability of the monopile foundations. This paper could be the first study on the dynamic response of a real-world OWT under earthquake excitations, in which some beneficial effects of an earthquake are observed.

Propeller's operational modal parameters identification based on Fiber Bragg Grating sensing

The propeller will generate a lot of harmful vibration and noise underwater. In order to effectively solve the problem of vibration and noise, it is of great significance to accurately obtain the modal parameters of propeller for the design and optimization of the propeller structure. Since it is difficult to measure the load of propeller when it is running underwater, only operational modal analysis (OMA) method can be used to identify the operational modal parameters of propeller. This paper is aimed at solving this problem by taking the metal propeller as the research object, an open-water test platform was built, and the Fiber Bragg Grating (FBG) sensors were used to obtain the underwater strain information of the propeller. Finally, the operational modal parameters of propeller are identified by the Stochastic subspace identification (SSI) method. In addition, the modal simulation analysis of propeller was carried out by using finite element simulation software, and the experimental results were compared with the simulation results. The results show that the stochastic subspace identification method can accurately identify the operational modal parameters of propeller.

MEMD-Based Hybrid Modal Identification for High-Rise Structures with Multi-Sensor Vibration Measurements

Although widely used in various fields due to its powerful capability of signal processing, empirical mode decomposition has to decompose signals separately, which limits its application for multivariate data such as the structural monitoring data recorded by multiple sensors. In order to avoid this shortcoming, a multivariate extension of empirical mode decomposition is proposed to deal with the multidimensional signals synchronously by employing a real-valued projection on hyperspheres. This study presents a hybrid modal identification method combining the multivariate empirical mode decomposition with stochastic subspace identification and fast Bayesian FFT methods to more conveniently and accurately identify structural dynamic parameters from multi-sensor vibration measurements. Deployed as a preprocessing tool, the multivariate signals are decomposed into several aligned intrinsic mode functions, which contain only a dominant component in the frequency domain. Then, the modal parameters can be identified by advanced fast Bayesian FFT and stochastic subspace identification directly. The combined method is first validated by a numerical illustration of a frame structure and then is applied in a shaking table test and a full-scale measurement under nonstationary earthquake excitation. Compared with the finite element method, the peak–pick, the half-power bandwidth methods, and Hilbert–Huang transform method, the results show that this hybrid method is more robust and reliable in the modal parameters identification. The main contribution of this paper is to develop a more effective integrated approach for accurate modal identification with the output-only multi-dimensional nonstationary signal.

Determining the Dynamic Characteristics of a Multi-Story RC Building Located in Chile: A Comparison of the Results between the Nonparametric Spectral Analysis Method and the Parametric Stochastic Subspace Identification Method

Estimating the dynamic characteristics of instrumented built structures from seismic vibration data collected from built civil structures is essential input information for structural model updating and assessing the health of structures. This study focuses on the earthquake acceleration time histories obtained from several events recorded during its construction phase by accelerometers placed throughout an office building located in Viña del Mar (Chile) to determine its modal features. To this end, the data obtained were analyzed to compare the building’s dynamic properties obtained with two different modal identification techniques. MATLAB programs were developed to implement both methods. The stochastic subspace identification technique for linear systems developed by van Overschee and de Moor was used to study the dynamic properties of the building. In contrast, the nonparametric method employed herein uses correlations and spectral analysis based on the Welch transform in the frequency domain. The investigation demonstrated that both methods identify similar frequencies and that the obtained translational mode shapes exhibit good agreement. Furthermore, the identified frequencies are congruent with the design frequencies.

Damping ratio identification using attenuation responses extracted by time series semantic segmentation

Damping ratio is an essential dynamic characteristic of bridge structures, and its properties have still not been well understood due to the difficulties in accurate calculation with structural health monitoring (SHM) data. It is widely accepted that the exponential attenuation (EA) method using free attenuation responses is a reliable way to calculate damping ratio. To identify the damping ratio from SHM data with EA method, a time series semantic segmentation method is proposed in this paper to extract free attenuation segments from the massive data of monitored acceleration responses. A labeled semantic dataset is firstly constructed based on the long-term monitoring data of the Z24 bridge which is a continuous bridge widely used as a benchmark model for SHM, and the labeling standard is introduced to illustrate the preference of selecting desirable attenuation segments. Then an adapted U-net model is developed and trained with the constructed dataset, and key hyper-parameters including the configuration of convolutional kernel and decision boundary are investigated and determined to meet the requirement of processing time series data of acceleration responses. A large number of attenuation segments have then been extracted by the well-trained model, based on which the damping ratio of the 1st vibration mode in 10 months is obtained with the EA method. The identified damping ratios are found reliable as compared with those identified by the stochastic subspace identification method, thus enabling further investigation of the properties of damping ratio. Finally, the influence of the temperature variation is explored, revealing an obvious correlation relationship between the temperature and the damping ratio. The results of this study not only provide a novel method to calculate damping ratio with SHM data but also could deepen the understanding of damping ratio properties and serve as a reference for more reasonable utilization of the damping ratio in structural dynamic analysis or condition assessment.

Modal measurement and identification of hexagonal air cushions

Air cushions have been widely utilised as roofs and facades for large-span buildings owing to their advantages such as ultra-light weight and aesthetic appearance. The dynamic performance of air cushions, which is a typical fluid-solid interaction problem, is complex, because the top layer, bottom layer, and inner air are highly coupled. However, there is a paucity of effective dynamic experimental studies to test the modal characteristics of entire air cushions. Therefore, in this study, a modal measurement and identification method was developed to simultaneously calculate the dynamic properties of both the top and bottom layers of air cushions. A laser displacement sensor array was employed to record the time-history displacement data at the observation points, and the combined application of the fast Fourier transform (FFT) and stochastic subspace identification (SSI) methods was used for modal identification. A hexagonal air cushion was adopted for testing, and the first eight frequencies and modes were successfully identified. The test results indicated that the first mode shape was significantly different from that of a single-layer foil. The one-wave vibration mode shape appeared after the multi-wave mode shapes, which was a unique phenomenon for air cushions. The frequencies and mode shapes of the top and bottom layers were almost identical. The mode deformation of the two layers can be in the same direction or opposite directions, and the frequencies corresponding to the mode shapes of layers vibrating in the same direction were higher than those of layers vibrating in opposite directions.

Comparison of methods for estimating short-term damping of idling wind turbine modes from measurements

Three methods for estimating time-varying (Hilbert-Huang transform, continuous wavelet transformation and partial Floquet analysis) and one for time-invariant (stochastic subspace identification) natural frequencies and viscous damping ratios based on free responses are tested and compared. The methods’ performance and sensitivities to several system characteristics are tested with a series of simulated responses. It is shown that, of the group of three methods, partial Floquet is the most suitable and Hilbert-Huang transform is unsuitable for the responses tested. Of the four methods tested, the stochastic subspace identification method is shown to be the method most robust to measurement noise, but its estimates contain notable bias error due to the short time series used. Natural frequencies and damping ratios of four edgewise blade modes are estimated based on measured responses from a single output location. The median of the estimated damping for these four modes have been calculated to be almost the same.

Determination to the period of a school building before and after its retrofitting using ambient vibration records

Determining the dynamic parameters of the structures allows us to obtain a substantial amount of information about the state of the building. Recently, the determination of these parameters were performed through Structural Health Monitoring Systems (SHMSs), which are non-destructive methods. In scope of this study, the period values prior to and following the retrofitting of a school building, which are among the dynamic parameters, were examined. The Operational Modal Analysis (OMA) method, which is based on the principle of measuring the responses of ambient effects in the structure in the experimental study, has been utilized. For this purpose, response records taken from the building under the effect of ambient vibration were used. Before and after retrofitting, the records of the building under the effect of ambient vibration were taken from the same locations. 3 sensors were used to record ambient vibrations. The data from these sensors were transferred to a computer by using a data logger. In order to measure and evaluate the responses Enhanced Frequency Domain Decomposition Method (EFDDM) in the frequency domain and Stochastic Subspace Identification Method (SSIM) in the time domain were used. Following the analysis, a comparison of how the period values had changed was conducted.

Optimal sensor placement and model updating applied to the operational modal analysis of a nonuniform wind turbine tower

Abstract. Test planning is a crucial step in the operational modal analysis (OMA) of wind turbines (WT), and it is an essential part of choosing the best positions for installing the sensors of the structures. On the other hand, updating the finite element model (FEM) with the OMA results implies a better prediction of the real structure's dynamic and vibrational behavior. This paper aims to show how the OMA of a nonuniform and two-section wind turbine tower can be performed more effectively, using the required test planning and optimal sensor placement. Then, accordingly, the OMA is used in operating and parked conditions to find the objective bending mode characteristics. Moreover, the updating of the applicable FEM of the multi-sectional wind turbine tower will be described. The tailor-made genetic algorithm (GA) is used to find the MEMS (micro electro-mechanical system) sensors' optimal positions of the WT under study. The OMA was performed and the acquired data analyzed using the stochastic subspace identification (SSI) method. Based on the OMA results, the FEM is updated by applying the sensitivity method. The results show that a tailor-made GA is a practical and quick approach to finding the optimal position of the sensors to obtain the best results for the objective modes of the WT. The OMA results, under operating and parked conditions, prove some modal characteristics of WTs. Based on the sensitivity analysis and engineering judgment, the modulus of elasticity was selected as a parameter for updating. Finally, we found that the updated FEM had less than 1 % error compared to the obtained frequencies from the test.

A detailed investigation of uplift and damping of a railway catenary span in traffic using a vision-based line-tracking system

For electrified railways, the catenary dynamic behavior is critical to ensure a robust and steady current collection quality for electric trains. The current collection is achieved as the catenary directly interacts with the pantograph, installed on the car-body roof to provide an electrical current to the engine. Damping plays an essential role in numerical simulations of pantograph-catenary interaction, especially for multiple pantographs. However, damping estimation of existing catenary sections is recognised as a challenge, and only a few studies have been published with single values of damping estimations. This study aimed to estimate the spatially distributed damping of an existing catenary span through full-span uplift measurements using a vision-based line-tracking system (VIBLITE). A detailed study was performed at critical locations along the catenary span. Sixty-nine single/double-pantograph train passages were acquired during scheduled train operation. Time series of uplift and acceleration were obtained through a line-tracking image-processing technique. The uplift amplitude was statistically analysed, where the damping ratios were identified using the covariance-driven stochastic subspace identification (Cov-SSI) method. Finally, the spatially distributed Rayleigh damping coefficients were successfully identified to quantify the important spatial variation in energy dissipation within a span. Arithmetic averages of damping coefficients over all measuring locations were obtained and recommended for future numerical simulations.

Experimental Modal Analysis of Angle Signals Based on the Stochastic Subspace Identification Method

This paper aims to verify the extraction of modal parameters from angle signals using the stochastic subspace identification (SSI) method. The use of angle signal-based mode shapes can reduce the loss of node information and enhance the robustness in curva-ture-based damage detection. In this regard, the system identification of angle signals should be first conducted prior to the damage detection. For large structures, an out-put-only system identification method should be considered for the modal analysis of an-gle signals, because artificial shaking excitation or impact excitation is practically impos-sible. In order to achieve this, the SSI method is used; it is one of the most powerful tools among the output-only system identification methods because it does not cover nonlinear problems. In order to demonstrate the system identification process of angle signals using the SSI method, the transformation matrix is assumed to represent the relationship be-tween the angular displacement and the normal displacement. Next, the modified block Hankel matrix that consists of angle signals, which can be expressed as a multiplication between the transformation matrix and displacement series vector, is constructed. The observability matrix can be estimated using the singular value decomposition for the pro-jection of the future part onto the past part of the modified block Hankel matrix. Finally, the natural frequencies and angle signal-based mode shapes are calculated using the state and observation matrices. In order to verify the results of the analytical studies, the modal properties estimated from the numerical simulation and the SSI method using angu-lar-velocity measurements are compared.

Dynamic analysis of Sitar: A comparative study of operational and experimental modal analysis

Dynamic analysis of Sitar, an Indian string instrument, is important for better understanding of the instrument behavior during performance. Sitar has complex geometry, and most of its components have anisotropic material properties, which generate a lot of challenges in performing numerical modal analysis. Considering this, an experimental approach of operational modal analysis (OMA) is performed on Sitar to extract its natural frequencies using the Stochastic Subspace Identification method. Hammer or shaker excitation required for conventional experimental modal analysis (EMA) has huge limitations of using harder hammer tips and high magnitude force as the instrument is delicate. However, to validate OMA results, EMA is performed with extreme care using an instrumented hammer with soft tip and with a very low excitation force. PolyMAX algorithm is used in EMA. It is observed that most of the correlated OMA and EMA modes lie in the audible frequency range. The maximum absolute percent error observed for these frequencies is 2.14%. All the modes obtained in OMA are significant as the string excitation simulate close to the real-life performing situation. Most of these modes map to musical note frequencies. Considering the detrimental effect of excitation required for EMA, OMA is a recommended method for extracting modal characteristics of Sitar.

Latent State Space Modeling of High-Dimensional Time Series With a Canonical Correlation Objective

High-dimensional time series are commonly encountered in modern control systems, especially in autonomous systems. In this work, a novel parsimonious latent state space (LaSS) model is proposed to characterize the latent dynamics, achieving general latent dynamic modeling with dimension reduction. The LaSS model is optimized by alternating estimations of the dimension reduction projection and the latent state space model. Precisely, the latent state dynamics are estimated by stochastic subspace identification methods. Furthermore, the canonical correlation analysis (CCA) objective is employed to acquire the optimal predictability for the extracted latent variables. The proposed LaSS-CCA algorithm is tested on a real industrial case for its effectiveness.

Fundamental mode shape estimation and element stiffness evaluation of girder bridges by using passing tractor-trailers

A comprehensive indirect method is proposed in this study to estimate the dynamic characteristics of the fundamental mode and evaluate the element stiffness of a girder bridge using a passing tractor with two identical trailers. The novelty of this study is the use of the dynamic interaction between the two identical trailers and the bridge rather than that between the truck and the bridge. One of the advantages of this is that the dynamic characteristics of trailers are easier to model and analyse. Under the proposed method, the residual acceleration of the two trailers is first used to eliminate the adverse effects of road surface roughness. The fundamental natural frequency of the bridge is obtained from the Fourier transform of the resulting signal. Next, the component corresponding to this fundamental frequency was extracted using bandpass filtering. The corresponding damping ratio and mode shape can be reconstructed through a short-time Fourier transform (STFT) under the assumption that the fundamental mode shape amplitude peak is located at the midspan. Finally, the element stiffness of the bridge was evaluated using the reconstructed fundamental mode shape. In this study, the proposed method is investigated via numerical examples and performs well in different scenarios, for example, in the presence of high-level measurement noise and low-level road surface roughness. Furthermore, the method was validated through field testing on the Li-Zi-Wan bridge in Chongqing. The extracted fundamental mode shape agrees well with that obtained using the stochastic subspace identification (SSI) method. The evaluated element stiffness matches very well with both the original design and the static deflection test results.

Modal Parameter Identification of Structures Using Reconstructed Displacements and Stochastic Subspace Identification

A method of modal parameter identification of structures using reconstructed displacements was proposed in the present research. The proposed method was developed based on the stochastic subspace identification (SSI) approach and used reconstructed displacements of measured accelerations as inputs. These reconstructed displacements suppressed the high-frequency component of measured acceleration data. Therefore, in comparison to the acceleration-based modal analysis, the operational modal analysis obtained more reliable and stable identification parameters from displacements regardless of the model order. However, due to the difficulty of displacement measurement, different types of noise interferences occurred when an acceleration sensor was used, causing a trend term drift error in the integral displacement. A moving average low-frequency attenuation frequency-domain integral was used to reconstruct displacements, and the moving time window was used in combination with the SSI method to identify the structural modal parameters. First, measured accelerations were used to estimate displacements. Due to the interference of noise and the influence of initial conditions, the integral displacement inevitably had a drift term. The moving average method was then used in combination with a filter to effectively eliminate the random fluctuation interference in measurement data and reduce the influence of random errors. Real displacement results of a structure were obtained through multiple smoothing, filtering, and integration. Finally, using reconstructed displacements as inputs, the improved SSI method was employed to identify the modal parameters of the structure.

Performance Assessment of a High-Speed Railway Bridge through Operational Modal Analysis

Modal parameters are widely recognized as valuable indicators for evaluating the performance of railway bridges in structural health monitoring. A major challenge in the mode-based performance assessment is to obtain modal parameters reliably because operational factors may cause significant identification errors or reduce modal identifiability. To reduce the operational effects on mode-based assessment, vibration responses are divided according to excitation types, and then two subspace identification techniques are developed for identifying modal parameters of the railway bridge. If the bridge is only acted on by the ambient excitation, the stochastic subspace identification (SSI) is taken in this paper. If the bridge is mainly acted on by the train excitation, modal parameters are difficult to identify due to the regularly spaced and highly energetic axle loads. In this case, a deterministic stochastic subspace identification (DSSI) method is developed for improving the modal identifiability of the railway bridge under train action. The monitoring responses of a high-speed railway bridge are analyzed in this paper to track long-term modal parameters. Besides, variations of modal parameters that are related to environmental factors, operational conditions, modal orders, vibration directions, and member types are extensively compared. The results show that modal parameters can be well-identified even though the nonwhite noise excitation exists, and the performance assessment can be well achieved through determining the optimal modal parameters.

Investigation of time-varying structural dynamic properties of high-rise buildings under typhoon conditions

This paper investigates the time-varying structural dynamic properties of high-rise buildings under typhoon conditions via the covariance-driven stochastic subspace identification (SSI–COV) method. Firstly, a numerical simulation study on a three-degree-of-freedom model is conducted, and the results prove that the SSI–COV method can provide accurate identification of the time-varying structural dynamic properties. Then, based on field measurements on a 420-m-tall high-rise building during Super Typhoon Mangkhut (No. 1822), the time-varying structural dynamic properties of the skyscraper are presented and discussed. The results indicate that the structural natural frequencies exhibited a trend of first decreasing and then increasing, while the damping ratios and modal directions basically remained unchanged during the typhoon process. Moreover, the variation of structural dynamic properties of the fundamental sway modes of the monitored building during 16 typhoons is investigated. The results show that the degree of variation in structural natural frequencies generally increased with the peak acceleration response of the building or the maximum 10-min-mean wind speed atop the building during a typhoon. While the damping ratios of the fundamental sway modes did not vary significantly during these 16 typhoons, which were around 0.64%. This paper aims to further the understanding of structural dynamic properties of high-rise buildings during typhoons.

Automatic modal parameters identification and uncertainty quantification based on block-bootstrap and multi-stage clustering under ambient excitation

This study proposes an algorithm for autonomous modal estimation to automatically eliminate false modes and quantify the uncertainty caused by the clustering algorithm and ambient factors. This algorithm belongs to the stochastic subspace identification (SSI) techniques and is based on the Block-Bootstrap and multi-stage clustering analysis. First, the Block-Bootstrap is introduced to decompose the response signal of the structure into M blocks of data. The covariance-driven stochastic subspace identification (SSI-Cov) method is used to process a random sample of data and obtain the corresponding M stabilization diagrams. In addition, the hierarchical clustering method is adopted to carry out the secondary clustering of the picked stable axis according to the defined distance threshold. Then, false modes are eliminated according to the proposed true and false modal discrimination index ( MDI). Finally, the above steps are repeated B times, and MDI is used to modify the initial modal parameters of group B. The mean value of elements in the cluster is taken as the recognition result of modal parameters, and the standard deviation is used to measure the accuracy of the recognition result. The numerical simulation results and the modal parameter identification of the Jing-yuan Yellow River Bridge show that, for identifying true and false modals, the proposed modal discrimination index is more effective than the threshold value of the traditional index. Also, it was found that the proposed method can eliminate the uncertainty introduced in the clustering process. In addition, this method can remove the influence of ambient noises, and it can improve the identification accuracy. It will be shown that this method has better anti-noise performance.

Dynamic assessment, FE modelling and parametric updating of a butterfly-arch stress-ribbon pedestrian bridge

The article focuses on the dynamic identification and finite element (FE) modelling of a butterfly-arch stress-ribbon pedestrian bridge in Fuzhou, Fujian, China. The Stochastic Subspace Identification method yields an estimate of the operational modal parameters. A highly synchronous tri-axial wireless sensor network was deployed on the bridge deck to record the structure’s ambient vibration. Eight stable modes in the frequency range 3.59–14.92 Hz were found, associated with prevalent bending and torsional deformations. Four distinct FE models of the bridge with progressive complexity and accuracy were developed to investigate the sensitivity of the modal features to the modelling choices. The FE model characterised by the fittest agreement with the experimental modal parameters was used for the automatic parametric optimisation based on a sensitivity-based algorithm. The stiffness of the springs simulating the soil-structure interaction, the elastic modulus of the concrete deck and the elastic modulus of the tendons were chosen as updating parameters for a total of eight parameters. The effect of non-structural elements (handrails) and prestress on the modal features are also investigated. The final advanced FE model developed can serve as baseline for a long-term monitoring of the bridge during its life-cycle, and also provides some recommendations to practitioners and scholars all over the world for the modelling and analysis of this particular kind of footbridges.

Optimal sensors placement for structural health monitoring based on system identification and interpolation methods

ABSTRACT Structural health monitoring (SHM) is required before and after major disasters to assess the safety of structures. In the stochastic subspace identification (SSI) method, the accuracy of the calculated frequency and damping ratio depends on the amount of collected data. However, setting sensors on each floor is time-consuming and difficult for high-rise structures. This study proposes an optimal sensor placement (OSP) method that can be used when a structure requires repeated monitoring, in order to reduce the number of sensors required, and to find the higher modal frequencies for the structure. Numerical results based on the dynamic responses of a ten-story shear frame were used to verify the proposed method. In addition, in-situ experiments in the Civil Engineering Research Building of the National Taiwan University were used to demonstrate the feasibility of this method.