Stochastic Subspace Identification Method Research Articles

Structural identification of cable-stayed bridge under back-to-back typhoons by wireless vibration monitoring

In this paper, structural identification of a cable-stayed bridge under back-to-back typhoons is performed using vibration responses measured by a wireless sensor system. Firstly, a cable-stayed bridge with the wireless monitoring system is described. Wireless vibration sensor nodes are utilized to measure accelerations from the deck, pylon and stay cables of the bridge. Secondly, dynamic responses of the cable-stayed bridge under the attack of two consecutive typhoons, Bolaven and Tembin, are analyzed under various wind speeds. Stochastic subspace identification method and short-time Fourier transform analysis are selected to examine wind-induced variations of the bridge’s responses based on the field measurements under the two typhoons. Finally, the structural identification of the bridge is performed under various wind velocities to examine the typhoon-induced variations of the bridge’s structural properties. The variations of dynamic characteristics due to the typhoons are turned into the changes of the bridge’s stiffness (or flexibility) via the fine-tuning process.

Continuous Modal Parameter Identification of a Cable-Stayed Bridge Based on Robustious Decomposition and Covariance-Driven Stochastic Subspace Identification

The objective of this paper was to develop an effective and efficient continuous health monitoring system. For this purpose, a novel improved ensemble empirical mode decomposition (EEMD) method has been introduced to decompose the nonlinear and non-stationary dynamic data. The improved EEMD method can commendably remove the noise from the original signals, and also it can alleviate the phenomenon of mode mixing. Then the reconstructed signal obtained after applying the improved EEMD technique was used to identify the continuous modal parameters such as frequency and damping ratio using the covariance-driven stochastic subspace identification method. The above techniques were then employed to the data obtained from a real-life cable-stayed bridge to identify its continuous modal parameters. The results validate that the proposed method is very effective in dealing with dynamic nonlinear and non-stationary data and can be used very effectually in real-life continuous health monitoring of cable-stayed bridges.

Identification of system damping in railway catenary wire systems from full-scale measurements

Damping is an important property in predicting the response of any civil or mechanical engineering structure, including railway catenary systems. Numerical models describing the pantograph–catenary interaction are dependent on a proper description of the structural damping for both systems to obtain accurate results. A proper description of the damping in different pantographs can easily be found in the literature. However, few studies have considered the damping properties in railway catenary systems even though such systems are considered to be lightly damped. The aim of this study was to identify the system damping of catenary sections by thoroughly analyse several recorded acceleration time series. These time series were sampled in several points on three different existing railway catenary systems, rendering a good description of the system damping in the frequency range of 0–20Hz. The covariance-driven stochastic subspace identification (Cov-SSI) method was used to analyse the time series, and Rayleigh coefficients were successfully identified for all three catenary sections.

Stochastic Subspace Method for Experimental Modal Analysis

The formula of stochastic subspace identification method is deduced in details and the program is written out. The two methods are verified by a vibration test on a 5-floor rigid frame model. In this test the gauss white noise generated from a shaker table to simulate the ambient vibration on the model, and the response signals are measured. Next, the response data of experiment are processed by auto-cross spectrum density method and stochastic subspace identification method respectively, the two methods are verified by comparing with the theory result. and bearing out the superiority of stochastic subspace identification method compared to auto-cross spectrum density method.

Experimental validation of the Kalman-type filters for online and real-time state and input estimation

In this study, a novel dual implementation of the Kalman filter proposed by Eftekhar Azam et al. (2014, 2015) is experimentally validated for simultaneous estimation of the states and input of structural systems. By means of numerical simulations, it has been shown that the proposed method outperforms existing techniques in terms of robustness and accuracy for the estimated displacement and velocity time histories. Herein, dynamic response measurements, in the form of displacement and acceleration time histories from a small-scale laboratory building structure excited at the base by a shake table, are considered for evaluating the performance of the proposed Dual Kalman filter and in order to compare this with available alternatives, such as the augmented Kalman filter (Lourens et al., 2012b) and the Gillijn De Moore filter (GDF) (2007b). The suggested Bayesian approach requires the availability of a physical model of the system in addition to output-only measurements from limited degrees of freedom. Two categories of such physical models are herein studied to evaluate the effect of model error on the filter performances; the first, is a model that comprises identified modal parameters, i.e., natural frequencies, mode shapes, modal damping ratios and modal participation factors; the second, is a model that is extracted from a recently developed subspace identification procedure, namely the transformed stochastic subspace identification method. The results are encouraging for the further use of the dual Kalman filter and its available alternatives for addressing the important problems of full response reconstruction and fatigue estimation in the entire body of linear structures, using a limited number of output-only vibration measurements.

Vibration-based identification of rotating blades using Rodrigues' rotation formula from a 3-D measurement

In this study, the geometrical setup of a turbine blade is tracked. A research-scale rotating turbine blade system is setup with a single 3-axes accelerometer mounted on one of the blades. The turbine system is rotated by a controlled motor. The tilt and rolling angles of the rotating blade under operating conditions are determined from the response measurement of the single accelerometer. Data acquisition is achieved using a prototype wireless sensing system. First, the Rodrigues\' rotation formula and an optimization algorithm are used to track the blade rolling angle and pitching angles of the turbine blade system. In addition, the blade flapwise natural frequency is identified by removing the rotation-related response induced by gravity and centrifuge force. To verify the result of calculations, a covariance-driven stochastic subspace identification method (SSI-COV) is applied to the vibration measurements of the blades to determine the system natural frequencies. It is thus proven that by using a single sensor and through a series of coordinate transformations and the Rodrigues\' rotation formula, the geometrical setup of the blade can be tracked and the blade flapwise vibration frequency can be determined successfully.

Modal parameter identification for a roof overflow powerhouse under ambient excitation

Modal parameter identification is a core issue in health monitoring and damage detection for hydraulic structures. For a roof overflow hydropower station with a bulb tubular unit under ambient excitation, a complex unit-powerhouse-dam coupling vibration system increases the difficulties of modal parameter identification. In this study, in view of the difficulties of modal order determination and the noise jamming caused by ambient excitation, along with false mode identification and elimination problems, the ensemble empirical mode decomposition (EEMD) method was used to decrease noise, the singular entropy increment spectrum was used to determine system order, and multiple criteria were used to eliminate false modes. The eigensystem realization algorithm (ERA) and stochastic subspace identification (SSI) method were then used to identify modal parameters. The results show that the relative errors of frequencies in the first four modes were within 10% for the ERA method, while those of SSI were over 10% in the second and third modes. Therefore, the ERA method is more appropriate for identifying the structural modal parameters for this particular powerhouse layout.

Dynamic and fatigue performances of a large-scale space frame assembled using pultruded GFRP composites

This paper describes the dynamic and fatigue performances of a large-scale space frame assembled using pultruded glass fiber reinforced polymer (GFRP) composites, with reference to pedestrian bridge application. The experimental structure was assembled by circular hollow section (CHS) GFRP members with the assistance of a novel steel connection system. The results from free vibration tests were analyzed using peak-picking (PP) and stochastic subspace identification (SSI) methods to extract modal parameters, i.e. natural frequencies, damping ratios, and mode shapes. From both experimental and validated FE analysis results, the proposed space frame structure satisfied the standard requirements for pedestrian bridge application in terms of natural frequency. The torsion mode as the first order mode shape can be avoided when the contribution of a bridge deck is considered. Furthermore, the structure was examined with 2.1 million fatigue load cycles and then statically loaded up to failure. The failure load showed no decrease when compared with that of a space frame without fatigue. The structural stiffness and strain of critical compressive members measured at 0.3 million fatigue loading intervals showed no significant variations, indicating that the applied fatigue did not degrade structural components and connections.

Operational Modal Analysis of a wing excited by transonic flow

Operational Modal Analysis (OMA) is a promising candidate for flutter testing and Structural Health Monitoring (SHM) of aircraft wings that are passively excited by wind loads. However, no studies have been published where OMA is tested in transonic flows, which is the dominant condition for large civil aircraft and is characterized by complex and unique aerodynamic phenomena. We use data from the HIRENASD large-scale wind tunnel experiment to automatically extract modal parameters from an ambiently excited wing operated in the transonic regime using two OMA methods: Stochastic Subspace Identification (SSI) and Frequency Domain Decomposition (FDD). The system response is evaluated based on accelerometer measurements. The excitation is investigated from surface pressure measurements. The forcing function is shown to be non-white, non-stationary and contaminated by narrow-banded transonic disturbances. All these properties violate fundamental OMA assumptions about the forcing function. Despite this, all physical modes in the investigated frequency range were successfully identified, and in addition transonic pressure waves were identified as physical modes as well. The SSI method showed superior identification capabilities for the investigated case. The investigation shows that complex transonic flows can interfere with OMA. This can make existing approaches for modal tracking unsuitable for their application to aircraft wings operated in the transonic flight regime. Approaches to separate the true physical modes from the transonic disturbances are discussed.

Stochastic subspace identification‐based approach for tracking inter‐area oscillatory modes in bulk power system utilising synchrophasor measurements

Stochastic subspace identification (SSI) methods have been widely employed for oscillatory mode identification on probing and ambient data and are reported to have good performances. This work proposes a novel SSI-based approach for identifying dominant oscillatory mode from measurement data and extends the application of SSI to ringdown condition. The proposed approach first constructs an initial cluster of eigenvalues from SSI with repetitive calculations and then utilises a novel hierarchical clustering method to extract the dominant modes from the initial cluster. The repetitive calculations within the SSI are performed through varying the model order over a range defined by a novel initial order determination process. By doing so the challenge of model order determination for SSI-based methods is resolved. Moreover, benefiting from the repetitive calculations and the clustering process, the proposed approach is highly immune to prevalent noises in the measurements. Finally, the proposed approach is applied and validated on the field-measurement data from the phasor measurement units of China Southern Power Grid (CSG) through comparisons with Prony, ARMAX, and Monte Carlo methods. Test results demonstrate that the proposed approach performs with high accuracy, robustness, and efficiency in CSG.

Full-scale tests of wind effects on a long span roof structure

Full-scale measurements are regarded as the most reliable method to evaluate wind effects on large buildings and structures. Some selected results are presented in this paper from the full-scale measurement of wind effects on a long-span steel roof structure during the passage of Typhoon Fanapi. Some field data, including wind speed and direction, acceleration responses, etc., were continuously and simultaneously recorded during the passage of the typhoon. Comprehensive analysis of the measured data is conducted to evaluate the typhoon-generated wind characteristics and its effects on a long-span steel roof. The first four natural frequencies and their vibration mode shapes of the Guangzhou International Sports Arena (GISA) roof are evaluated by the stochastic subspace identification (SSI) method and comparisons with those from finite element (FE) analysis are made. Meanwhile, damping ratios of the roof are also identified by the SSI method and compared with those identified by the random decrement method; the amplitude-dependent damping behaviors are also discussed. The fullscale measurement results are further compared with the corresponding wind tunnel test results to evaluate its reliability. The results obtained from this study are valuable for academic and professional engineers involved in the design of large-span roof structures.

Uncertainty quantification in operational modal analysis with stochastic subspace identification: Validation and applications

Identified modal characteristics are often used as a basis for the calibration and validation of dynamic structural models, for structural control, for structural health monitoring, etc. It is therefore important to know their accuracy. In this article, a method for estimating the (co)variance of modal characteristics that are identified with the stochastic subspace identification method is validated for two civil engineering structures. The first structure is a damaged prestressed concrete bridge for which acceleration and dynamic strain data were measured in 36 different setups. The second structure is a mid-rise building for which acceleration data were measured in 10 different setups. There is a good quantitative agreement between the predicted levels of uncertainty and the observed variability of the eigenfrequencies and damping ratios between the different setups. The method can therefore be used with confidence for quantifying the uncertainty of the identified modal characteristics, also when some or all of them are estimated from a single batch of vibration data. Furthermore, the method is seen to yield valuable insight in the variability of the estimation accuracy from mode to mode and from setup to setup: the more informative a setup is regarding an estimated modal characteristic, the smaller is the estimated variance.

Modal parameters identification of heavy-haul railway RC bridges - experience acquired

Traditionally, it is not easy to carry out tests to identify modal parameters from existing railway bridges because of the testing conditions and complicated nature of civil structures. A six year (2007-2012) research program was conducted to monitor a group of 25 railway bridges. One of the tasks was to devise guidelines for identifying their modal parameters. This paper presents the experience acquired from such identification. The modal analysis of four representative bridges of this group is reported, which include B5, B15, B20 and B58A, crossing the Caraj<TEX>$\acute{a}$</TEX>s railway in northern Brazil using three different excitations sources: drop weight, free vibration after train passage, and ambient conditions. To extract the dynamic parameters from the recorded data, Stochastic Subspace Identification and Frequency Domain Decomposition methods were used. Finite-element models were constructed to facilitate the dynamic measurements. The results show good agreement between the measured and computed natural frequencies and mode shapes. The findings provide some guidelines on methods of excitation, record length of time, methods of modal analysis including the use of projected channel and harmonic detection, helping researchers and maintenance teams obtain good dynamic characteristics from measurement data.

Effects of foundation damage and water-level change on vibration modal parameters of gravity-type caisson structure

In this paper, the effects of foundation damage and water-level change on vibration characteristics of gravity-type caisson structure are examined by analyzing modal parameters extracted from output-only information. To achieve the objective, the following approaches are implemented. Firstly, vibration response analysis methods are selected to estimate power spectral density and modal parameters such as natural frequency, damping ratio and mode shape of a lab-scale caisson structural system. Secondly, vibration tests on the lab-scale caisson system are performed under a series of test scenarios which include three water-level changes and three damage levels. Thirdly, experimental modal parameters corresponding to the damaging cases as well as the water level cases are extracted by frequency domain decomposition method and stochastic subspace identification method. Finally, the effects of the water-level variation and foundation damage on the extracted modal parameters are examined to assess the feasibility of the vibration-based damage detection in gravity-type caisson structures under water-level uncertainty.

Characterizing the Dynamic Response of a Chassis Frame in a Heavy-Duty Dump Vehicle Based on an Improved Stochastic System Identification

This paper presents an online method for the assessment of the dynamic performance of the chassis frame in a heavy-duty dump truck based on a novel stochastic subspace identification (SSI) method. It introduces the use of an average correlation signal as the input data to conventional SSI methods in order to reduce the noisy and nonstationary contents in the vibration signals from the frame, allowing accurate modal properties to be attained for realistically assessing the dynamic behaviour of the frame when the vehicle travels on both bumped and unpaved roads under different operating conditions. The modal results show that the modal properties obtained online are significantly different from the offline ones in that the identifiable modes are less because of the integration of different vehicle systems onto the frame. Moreover, the modal shapes between 7 Hz and 40 Hz clearly indicate the weak section of the structure where earlier fatigues and unsafe operations may occur due to the high relative changes in the modal shapes. In addition, the loaded operations show more modes which cause high deformation on the weak section. These results have verified the performance of the proposed SSI method and provide reliable references for optimizing the construction of the frame.

On the advances of automatic modal identification for SHM

Structural health monitoring of civil infrastructures has great practical importance for engineers, owners and stakeholders. Numerous researches have been carried out using long-term monitoring, for instance the Rio-Niterói Bridge in Brazil, the former Z24 Bridge in Switzerland, the Millau Bridge in France, among others. In fact, some structures are monitored 24/7 in order to supply dynamic measurements that can be used for the identification of structural problems such as the presence of cracks, excessive vibration, damage or even to perform a quite extensive structural evaluation concerning its reliability and life cycle. The outputs of such an analysis, commonly entitled modal identification, are the so-called modal parameters, i.e. natural frequencies, damping ratios and mode shapes. Therefore, the development and validation of tools for the automatic identification of modal parameters based on the structural responses during normal operation is fundamental, as the success of subsequent damage detection algorithms depends on the accuracy of the modal parameters estimates. The proposed methodology uses the data driven stochastic subspace identification method (SSI-DATA), which is then complemented by a novel procedure developed for the automatic analysis of the stabilization diagrams provided by the SSI-DATA method. The efficiency of the proposed approach is attested via experimental investigations on a simply supported beam tested in laboratory and on a motorway bridge.

Output-only cyclo-stationary linear-parameter time-varying stochastic subspace identification method for rotating machinery and spinning structures

Economical maintenance and operation are critical issues for rotating machinery and spinning structures containing blade elements, especially large slender dynamic beams (e.g., wind turbines). Structural health monitoring systems represent promising instruments to assure reliability and good performance from the dynamics of the mechanical systems. However, such devices have not been completely perfected for spinning structures. These sensing technologies are typically informed by both mechanistic models coupled with data-driven identification techniques in the time and/or frequency domain. Frequency response functions are popular but are difficult to realize autonomously for structures of higher order, especially when overlapping frequency content is present. Instead, time-domain techniques have shown to possess powerful advantages from a practical point of view (i.e. low-order computational effort suitable for real-time or embedded algorithms) and also are more suitable to differentiate closely-related modes. Customarily, time-varying effects are often neglected or dismissed to simplify this analysis, but such cannot be the case for sinusoidally loaded structures containing spinning multi-bodies. A more complex scenario is constituted when dealing with both periodic mechanisms responsible for the vibration shaft of the rotor-blade system and the interaction of the supporting substructure. Transformations of the cyclic effects on the vibrational data can be applied to isolate inertial quantities that are different from rotation-generated forces that are typically non-stationary in nature. After applying these transformations, structural identification can be carried out by stationary techniques via data-correlated eigensystem realizations. In this paper, an exploration of a periodic stationary or cyclo-stationary subspace identification technique is presented here for spinning multi-blade systems by means of a modified Eigensystem Realization Algorithm (ERA) via stochastic subspace identification (SSI) and linear parameter time-varying (LPTV) techniques. Structural response is assumed to be stationary ambient excitation produced by a Gaussian (white) noise within the operative range bandwidth of the machinery or structure in study. ERA-OKID analysis is driven by correlation-function matrices from the stationary ambient response aiming to reduce noise effects. Singular value decomposition (SVD) and eigenvalue analysis are computed in a last stage to identify frequencies and complex-valued mode shapes. Proposed assumptions are carefully weighted to account for the uncertainty of the environment. A numerical example is carried out based a spinning finite element (SFE) model, and verified using ANSYS® Ver. 12. Finally, comments and observations are provided on how this subspace realization technique can be extended to the problem of modal-parameter identification using only ambient vibration data.

Structural health monitoring of offshore wind turbines using automated operational modal analysis

This article will present and discuss the approach and the first results of a long-term dynamic monitoring campaign on an offshore wind turbine in the Belgian North Sea. It focuses on the vibration levels and modal parameters of the fundamental modes of the support structure. These parameters are crucial to minimize the operation and maintenance costs and to extend the lifetime of offshore wind turbine structure and mechanical systems. In order to perform a proper continuous monitoring during operation, a fast and reliable solution, applicable on an industrial scale, has been developed. It will be shown that the use of appropriate vibration measurement equipment together with state-of-the art operational modal analysis techniques can provide accurate estimates of natural frequencies, damping ratios, and mode shapes of offshore wind turbines. The identification methods have been automated and their reliability has been improved, so that the system can track small changes in the dynamic behavior of offshore wind turbines. The advanced modal analysis tools used in this application include the poly-reference least squares complex frequency-domain estimator, commercially known as PolyMAX, and the covariance-driven stochastic subspace identification method. The implemented processing strategy will be demonstrated on data continuously collected during 2 weeks, while the wind turbine was idling or parked.

Structural Identification of a Masonry Tower Based on Operational Modal Analysis

A wide program of structural assessment has been carried out by Politecnico di Milano on the historic bell tower of the church Chiesa Collegiata in Arcisate (Varese, northern Italy). Within this context, the first part of the article summarizes the results obtained from the application of the stochastic subspace identification method to ambient response data collected in two dynamic tests, carried out in June 2007 and June 2008. Next presented is the vibration-based methodology developed for the calibration of a three-dimensional finite element model of the tower, consisting in the successive application of systematic manual tuning, sensitivity analysis, and simple system identification algorithm.

Structural safety assessment of bowstring type RC arch bridges using ambient vibration testing and finite element model calibration

The structural safety evaluation of a bowstring type reinforced concrete (RC) arch bridge was investigated by ambient vibration testing and finite element model calibration in this study. The bridge named as Ali Çetinkaya Bridge in Samsun, Turkey, was selected for this investigation. The bridge constructed in 1937 consists of the seven discrete arches and its total length is approximately 250m. The bridge damaged by environmental effects and it was closed to the traffic loads. The presented study consists of four main parts: ambient vibration test, initial finite element modeling, finite element model calibration and structural analysis of the calibrated finite element model. The ambient vibration tests were performed using Operational Modal Analysis Method under the wind, human and water flow loads to identify the actual natural dynamic parameters, such as the natural frequencies and mode shapes, modal damping ratios. Enhanced Frequency Domain Decomposition and Stochastic Subspace Identification methods were used to extract the experimental dynamic characteristics. The initial finite element model was developed by including the soil effects in SAP2000 program to obtain the analytical dynamic characteristics. In the initial finite element model, the beam, plane and solid elements were used. After that, the initial finite element model was calibrated according to the ambient vibration test results considering the material properties and boundary sections as variable parameters. The calibrated model assumed to reflect the current state of the bowstring bridge was analyzed under dead, moving, traffic, water and earthquake loads. The maximum displacements, tensile stress, compressive stress and shear stress were attained each case and compared with each other.