Eigensystem Realization Algorithm Research Articles

A dynamic vehicle-bridge model based on the modal identification results of an existing EN57 train and bridge spans with non-ballasted tracks

This paper addresses the methodology of the bridge-vehicle dynamic model definition based on the free response measurements of an existing train and existing bridge spans. In the case of the railway vehicle, the methodology uses the impulse excitations of a single car by means of the wedge method. In the case of the bridge spans, free responses are collected after the passages of trains. The global modal parameters (frequencies and modal damping) of both structures are identified using two modal identification techniques, i.e., the eigensystem realization algorithm (ERA) and the peak picking method (PP). A simplified single-suspension model of the vehicle with 11 global dof’s is proposed that can be useful in the bridge-vehicle interaction analysis. Two steel spans having non-ballasted tracks are the bridge structures. Field tests are conducted to identify the modal parameters and validate the numerical models. Finally, numerical simulations of the train passing over the bridges are compared with the in situ measurements performed under operating conditions.

Modal parameter identification of a multiple-span post-tensioned concrete bridge using hybrid vibration testing data

The paper describes and evaluates application of output-only system identification to an eleven-span post-tensioned concrete bridge using hybrid excitation. A linear chirp sweeping force, induced by two light-weight electro-dynamic shakers, augmented environmental sources to excite the bridge during the hybrid testing exercise. To obtain the modal characteristics of the structure, two output-only time domain system identification methods were employed, namely auto-regressive (AR) time series model and eigensystem realization algorithm with observer/Kalman identification (ERA-OKID), with the traditional data-driven stochastic subspace identification method (SSI-data) providing a comparative benchmark. The accuracy and efficiency of both system identification algorithms when used on hybrid testing data are investigated and compared to the results from purely ambient vibration testing data. The study demonstrates that using both output-only identification algorithms the collected vibration responses induced by the proposed hybrid vibration testing methodology can be used for extracting modal parameters with enhanced accuracy and reliability (i.e. more identified modes) for the large-scale post-tensioned concrete bridge due to the increase in the excitation strength and better coverage of the relevant frequency bands. Compared to the classical SSI-data, the AR and ERA-OKID techniques were able to identify more modes at reduced computational cost when applied to voluminous data from multi-channel measurements.

Identification of first and second order models for the superstructure of base-isolated buildings using free vibration tests: A case study

Using a set of full-scale free vibration measurements, two separate state of the art identification strategies are implemented for the time domain identification of the modal properties of the superstructure of a three-story base-isolated building. In a first approach, the superstructure is modeled as a fixed-base linear 3-DOF system subjected to the base acceleration measured during the tests. Identification is performed using the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), a stochastic algorithm for non-linear, black-box optimization. In the second approach, a first order state-space representation of the superstructure is considered, and identification is carried out using the Eigensystem Realization Algorithm with Data Correlation, complemented by the Observer/Kalman filter Identification algorithm (ERA-DC/OKID). A single-input and multiple-output system is considered, the input being the recorded base acceleration and the output given by the measured floor accelerations. The results of the two procedures are compared leading to considerations about modeling assumptions that are typically adopted in the analysis and design of base-isolated buildings.

Data-driven selection of actuators for optimal control of airfoil separation

We present a systematic approach for determining the optimal actuator location for separation control from input-output response data, gathered from numerical simulations or physical experiments. The Eigensystem Realization Algorithm is used to extract state-space descriptions from the response data associated with a candidate set of actuator locations. These system realizations are then used to determine the actuator location among the set that can drive the system output to an arbitrary value with minimal control effort. The solution of the corresponding minimum energy optimal control problem is evaluated by computing the generalized output controllability Gramian. We use the method to analyze high-fidelity numerical simulation data of the lift and separation-angle responses to a pulse of localized body-force actuation from six distinct locations on the upper surface of a NACA 65(1)-412 airfoil. We find that the optimal location for controlling lift is different from the optimal location for controlling separation angle. In order to explain the physical mechanisms underlying these differences, we conduct controllability analyses of the flowfield by leveraging the dynamic mode decomposition with control algorithm. These modal analyses of flowfield response data reveal that excitation of coherent structures in the wake benefit lift control; whereas, excitation of coherent structures in the shear layer benefit separation-angle control.

In situ evaluation of dynamic characteristics of prefabricated ballastless track slab using EMA and OMA techniques

Abstract This study focused on the prefabricated ballastless slab tracks first applied in urban rail transit of a city in China. By combining the experimental modal analysis (EMA) and operational modal analysis (OMA) techniques, we evaluated the dynamic characteristics of the tracks using the eigensystem realization algorithm and stochastic subspace identification algorithm to identify modal parameters. This study attempted to determine the characteristics and correlation of the two modal analysis methods to comprehensively reflect the dynamic characteristics of the track slabs. We established a detailed finite element model to analyze the influence of the construction method and structural characteristics of the track slabs on the modal parameters. The results show that the in-situ testing of the track slab modes were subject to the influence of many uncertain factors, resulting in differences between the numerical calculation-based ideal mode shapes and the actual EMA- and OMA-based counterparts.

Modal Identification of Aerodynamic Models for Spanwise Lift Distribution Control

This work presents modal identification results for the unsteady three-dimensional flow around a wing featuring multiple trailing edge flaps. The unsteady spanwise lift distribution is modeled using the unsteady vortex lattice method, which yields a linear, time-invariant, high-order state-space model. The eigensystem realization algorithm is applied for model order reduction and modal identification, providing aerodynamic mode shapes and associated eigenvalues. The wing is considered rigid, so that the dynamic mode shapes associated with the spanwise lift distribution are purely aerodynamic. It is observed that the spatial-temporal discretization of the numerical model and flap dimensions inherently limit the number of modes that can be properly identified. Open-loop simulation results demonstrate the modal behavior of the aerodynamic system. Finally, aerodynamic mode shapes for different wing planforms are also presented for comparison. By expanding the system output (lift distribution) as a truncated superposition of aerodynamic mode shapes, a low-order multiple-input and multiple-output modal representation can be obtained, suitable for controller synthesis. Ultimately, this novel approach will be applied to gust alleviation, in order to keep a desired lift distribution profile using shape control techniques.

Identification of nonlinear aerodynamic systems with application to transonic aeroelasticity of aircraft structures

In this study, identification of nonlinear aerodynamic systems based on an improved nonlinear state-space modeling approach is presented to predict transonic aeroelastic behaviors of aircraft structures. It starts with identifying a linear state-space aerodynamic model by using the eigensystem realization algorithm and observer/Kalman filter identification methods. Subsequently, the remaining parameters of the nonlinear state-space aerodynamic model are determined through an output error-minimization procedure, based on a new training data generation methodology. To illustrate the approach, two- and three-dimensional transonic aeroelastic configurations are studied. The transonic aeroelastic behaviors predicted via the proposed approach agree well with those obtained via computational fluid dynamics technique in the interested range of dynamic pressure.

An improved nonlinear reduced-order modeling for transonic aeroelastic systems

In this study, an improved nonlinear reduced-order model composed of a linear part and a nonlinear part is explored for transonic aeroelastic systems. The linear part is identified via the eigensystem realization algorithm and the nonlinear part is obtained via the Levenberg–Marquardt algorithm. The impulsive signal is chosen as the training signal for the linear part and the sinusoidal signal is used to determine the order of the linear part. The training signal for the nonlinear part is selected as the filtered white Gaussian noise with the maximal amplitude and frequency range to be designed via the aeroelastic responses. An NACA64A010 airfoil and an NACA0012 airfoil are taken as illustrative examples to demonstrate the performance of the presented reduced-order model in modeling transonic aerodynamic forces. The aeroelastic behaviors of the two airfoils are obtained via computational fluid dynamics to solve the Euler equation and the Navier–Stokes equation, respectively. The numerical results demonstrate that the presented reduced-order model can successfully predict the nonlinear aerodynamic forces with and without viscous flows. Moreover, the presented reduced-order model is capable of capturing the flutter velocity and modeling complex aeroelastic behaviors, including limit-cycle oscillations, beat phenomena and nodal-shaped oscillations at the transonic Mach numbers with high accuracy.

Study on applicability of two modal identification techniques in irrelevant cases

Study on applicability of two modal identification techniques in irrelevant cases is made in this paper. The following techniques are considered: peak picking based on correlation analysis (PP-CA), dedicated for ambient vibrations and eigensystem realization algorithm (ERA), formulated for free-decay vibrations investigation. Irrelevant cases are found when analyzed signals are different than recommended to a given technique. The study is conducted on examples of two real structures: masonry tower and steel railway bridge. Both cases are diverse in age, material, excitation and vibrations energy. The signals measured on the tower are suitable for the PP-CA technique (ambient vibrations), while the signals measured on the bridge are suitable for the ERA (free-decay vibrations). However, both methods have been applied to both systems. Natural frequencies, mode shapes and damping ratios are identified and the effectiveness of the irrelevant technique is assessed in relation to the results obtained by the relevant method in each case.

Stability analysis of passive suppression for vortex-induced vibration

In this paper, we present a stability analysis of passive suppression devices for the vortex-induced vibration (VIV) in the laminar flow condition. A data-driven model reduction approach based on the eigensystem realization algorithm is used to construct a reduced-order model in a state-space format. From the stability analysis of the coupled system, two modes are found to be dominant in the phenomenon of self-sustained VIV: namely, the wake mode, with frequency close to that of the wake flow behind a stationary cylinder; and the structure mode, with frequency close to the natural frequency of the elastically mounted cylinder. The present study illustrates that VIV can be suppressed by altering the structure mode via shifting of the eigenvalues from the unstable to the stable region. This finding is realized through the simulations of passive control devices, such as fairings and connected-C devices, wherein the presence of appendages breaks the self-sustenance of the wake–body interaction cycle. A detailed proper orthogonal decomposition analysis is employed to quantify the effect of a fairing on the complex interaction between the wake features. From the assessment of the stability characteristics of appendages, the behaviour of a connected-C device is found to be similar to that of a fairing, and the trajectories of the eigenspectrum are nearly identical, while the eigenspectrum of the cylinder–splitter arrangement indicates a galloping behaviour at higher reduced velocities. Finally, we introduce a stability function to characterize the influence of geometric parameters on VIV suppression.

Damping estimates from reconstructed displacement for low-frequency dominant structures

This paper proposes a new concept for the estimation of modal damping ratios. This new concept is based on an operational modal analysis (OMA) technique that uses dynamic displacement reconstructed from measured acceleration. The reconstructed displacement physically suppresses the high-mode components in measured acceleration data and leads to modal participant factors that are sequentially arranged from low- to high-frequency modes. Thus, compared with the conventional approach that utilizes acceleration data, the proposed procedure provides a more reliable and robust damping estimate regardless of the model order selection. Furthermore, the use of reconstructed displacement guarantees the equilibrium of the system all the time. The OMA adopts a natural excitation technique (NExT) combined with an eigensystem realization algorithm (ERA). A finite element method (FEM)-based finite impulse response filter is adopted to reconstruct the dynamic displacement. The effectiveness and accuracy of the proposed approach is demonstrated for low-frequency dominant structures by using the numerical simulation study of a 9-story shear building and an actual field test of a cable-stayed bridge under operation in Korea. These examples allow a comparison of modal damping ratio estimates using reconstructed displacement and accelerations.

A realizable data-driven approach to delay bypass transition with control theory

The current work presents a realizable method to control streaky disturbances in boundary layer flows and delay transition to turbulence via active flow control. Numerical simulations of the nonlinear transitional regime in a Blasius boundary layer are performed where streaks are excited in the boundary layer via a high level of free-stream turbulence (FST). The occurring disturbances are measured via localized wall sensors and damped via near-wall actuators resembling plasma actuators. The time-varying amplitude of the signal of each actuator is computed by processing signals from the sensors. This processing is the result of two control laws: the Linear Quadratic Gaussian regulator (LQG) and the Inverse Feed-Forward Control technique (IFFC). The use of the first, LQG, requires a state-space representation of the system dynamics, so the flow is described via an operator that captures only the most relevant information of the dynamics and results in a reduced order model (ROM). The ROM is computed via the eigensystem realization algorithm, based on the impulse responses of the real system. Collecting such impulse responses may be unfeasible when considering FST because of the high dimensionality of the input forcing needed for a precise description of such a phenomenon. Here, a new method to identify the relevant system dynamics and generate the needed impulse responses is proposed, in a data-driven approach that would be realizable in experiments. Finally, the effectiveness of the technique in delaying bypass transition is shown. The work (i) presents a systematic way to deal with high dimensional disturbances to build ROMs for a control technique, and (ii) shows that even when considering constraints such as the type and size of actuators and sensors, it is possible to achieve at least as large delay of bypass transition as that obtained in more idealized cases found in literature.

A step excitation optimization method for intensive modal identification of cable net structures

Compared with the direct force measurement of cables, dynamic modal testing is more economic and applicable in engineering for monitoring the stiffness change of cable net structures caused by pretension deviation. Normally, it is sufficient to detect the variation of structural key stiffness by testing a few modes (target modes) that are, however, likely located in the region of dense modes. To effectively evaluate the stiffness of an existing cable net structure, the identification accuracy of these target modes should be improved when using conventional time-domain methods, e.g., the eigensystem realization algorithm (ERA). Structural free vibration can be induced by a step excitation that is generated by simultaneously releasing loads hung on the cable net. In this paper, the modal energy in the structural free vibration response is expounded to be determined by the work done by the step excitation loads in the direction of this mode shape. Therefore, the identification accuracy of a target mode can be improved by optimizing the layout and magnitude of the suspended loads to increase its energy proportion in the structural free vibration response and suppress those of its adjacent modes. For a certain set of degrees of freedom (DOFs) that loads are allowed to be suspended, a method is put forward to construct a step excitation load that can guarantee that the suppressed modes do not have an energy contribution to its induced free vibration. Referring to the effective independence (EI) method, an iterative algorithm is developed to select a given number of DOFs for the suspension of the step excitation loads that can ensure the target mode has a relatively large energy proportion. With the solution of generalized Rayleigh entropy, the magnitude of the loads suspended from the selected DOFs is further optimized to achieve the maximum energy proportion of the target mode. An illustrative saddle-shaped cable net roof is employed to validate the accuracy and validity of the proposed method by comparing the modal identification results under the optimized step excitation, conventional impact excitation and random excitation, respectively.

In-situ Ambient Vibration Study of a 900-kw Wind Turbine

ABSTRACT Dynamic measurements are undertaken on a utility scale wind turbine (900 kW rated power, 55 m hub height, and a 53.6 m rotor diameter) located at Oak Creek Energy Systems near Mojave, California, USA. The acceleration response to wind and other ambient sources is recorded by as many as 81 channels at locations along the height of the tower, on the foundation, and on the surface of the surrounding soil. This extensive dataset is accessible for further studies at https://www.designsafe-ci.org/data/browser/public/nees.public//NEES-2008-0661.groups. In this paper, modal parameters (natural frequencies, damping ratios, and mode shapes) are estimated with the Multiple-Reference Natural Excitation technique combined with the Eigensystem Realization Algorithm based on 19 h of data while the turbine is parked and 1.5 h when it is operating under normal conditions. Structural characteristics are identified from the parked and operational state recordings. These dynamic test series provides a unique opportunity to observe variability in modal parameters of the turbine and is the first to be published for a megawatt scale turbine. The experimentally observed properties are compared to results from a beam-column model, the most common approach for simulating global turbine dynamics. Agreement between the experimentally identified and numerically predicted modal parameters confirms that this modeling approach is representative of modern wind turbine dynamics and provides a basis for seismic response evaluations.

Integrated system identification and modal state estimation for control of flexible space structures

A novel approach of integrated system identification and modal state estimation is proposed for control of linear dynamical systems including flexible space structures. There are four steps involved in this approach. First, the relation between a stochastic state space model of a dynamical system and the coefficients of its autoregressive model with exogenous input is derived. Second, an adaptive least-squares transversal predictor is used to estimate the coefficients of the model. Third, a state space model and a steady state Kalman filter gain of the dynamical system are then identified from the coefficients of the model by using the eigensystem realization algorithm. Fourth, a modal state estimator is constructed using the modal parameters of the identified model. On-line implementation of this algorithm can continually improve the modal parameters and the filter gain. It can also gradually update the system model when the system characteristics are slowly changing. A numerical example is used to illustrate the feasibility of the new approach.

An eigensystem realization algorithm based data-driven approach for extracting electromechanical oscillation dynamic patterns from synchrophasor measurements in bulk power grids

In power systems, electromechanical oscillatory dynamics, dominant modes, mode shapes, participation factors and coherent groups of generators are all important modal parameters. Most of the existing measurement-based methods focus on estimating one or two aspects among dominant modes, mode shapes, participation factors, and coherent groups. Nevertheless, none of existing methods explore all the four aspects from synchrophasor measurements at the same time. In this work, an eigensystem realization algorithm (ERA) based data-driven approach is developed to estimate dominant modes, mode shapes, participation factors and coherent groups from synchrophasor measurements in a holistic framework. In the proposed approach, the reduced power system dynamic model is first estimated by ERA from multichannel synchrophasor measurements. Then, based on the estimated power system dynamic model, electromechanical oscillation modes, corresponding mode shapes and the left eigenvalue vectors are solved. Next, using the solved mode shapes and left eigenvalue vectors, the participation factors of the generators associated with the electromechanical oscillation modes are calculated. Further, the direction cosines among the generators representing coherent strength of generators are calculated by using the achieved mode shapes. Finally, in accordance with the calculated direction cosines, the coherent generators are then identified. The proposed approach is demonstrated with the simulation data of the 16-machine, 68-bus test system and the field Phasor Measurement Units (PMUs) data collected in Liaoning Electric Power Grid. The results demonstrate that the proposed data-driven approach captures dominant modes, mode shapes, participation factors, and coherent groups of generators from synchrophasor measurements with high accuracy and efficiency.

Investigation of dynamic characteristics of a monopile wind turbine based on sea test

Wind energy plays an important role in the field of renewable energy due to its lower cost and the maturity of its manufacturing technology. During the last few years, there has been an increase in offshore wind turbine (OWT) projects especially in China. Considering the fact that ocean environment where these offshore wind turbines operate usually has a negative impact on their structural health, the demand for investigating the dynamic characteristics of offshore wind turbines based on sea test is increasing. However, current studies are mainly focused on method development, theoretical analysis and numerical simulation, and few attention has been given to the study of the dynamic characteristics of an actual OWT in different operational states based on sea test. To improve the current studies, in September 2016, November 2016 and April 2017 a series of monitoring campaigns of an actual in service OWT and its monopile were conducted at an offshore wind farm in the sea area near Jiangsu Province, China. For the purpose of acquiring high-quality vibration signal, one type of artificial excitation was adopted to excite the OWT vibration. In these monitoring campaigns, with the help of a group of wireless health monitoring sensor nodes developed by Ocean University of China (OUC), we successfully acquired the vibration signal of the wind turbine in parked state and normal operational state and acquired the vibration signal of the monopile in construction, which means no tower was installed. Afterwards, two modal analysis methods, the eigensystem realization algorithm (ERA) and stochastic subspace identification (SSI), were used to obtain the modal parameters of the tested structure using the vibration signal under artificial excitation and random wave excitation, respectively. The results show that the ERA method can successfully identify the first two order modal parameters of the monopile and the first three order modal parameters of the wind turbine in parked or operational state. However, the SSI method failed to identify the third-order modal parameters of the wind turbine, regardless of whether it is in parked state or operational state (only the first two order modal parameters can be identified); meanwhile, SSI failed to identify the modal parameters of the monopile. Concerning the high-level environmental noise in an offshore site, we applied weak-mode identification (WMI) for the first time to obtain the modal parameters of OWT using vibration signal under both artificial excitation and random wave excitation. The results indicated that WMI has achieved the same good performance as the ERA when the vibration signal under artificial excitation is used. When the vibration signal under random wave is used, WMI successfully identified the first order modal parameters of the monopile and the first three order modal parameters of the wind turbine in parked or operational state, which indicated a better ability than that of the SSI method. We can conclude that the modal parameters of the wind turbine can be obtained by the WMI method and ERA method, and the ERA method is the best way to obtain the modal parameters of monopile under artificial excitation. These monitoring campaign of an actual OWT not only provide modal information support for structural maintenance and operation safety assessment, but also provide a real data reference for wind turbine design.

Mode identification by eigensystem realization algorithm through virtual frequency response function

Mode identification in civil engineering uses ambient excitation response, where the excitation is assumed to be stationary white noise. However, practical engineering conditions cannot satisfy this assumption of ideal excitation. This paper proposes an innovative method based on eigensystem realization algorithm and a proposed virtual frequency response function. The formulation of the virtual frequency response function is derived from the ratio of the cross-power and autopower spectral density functions of the measurement signals. The impulse responses are calculated by inverse Fourier transform of the product of the virtual frequency response function and the frequency response function of the reference point. After obtaining impulse responses, eigensystem realization algorithm is then performed to identify structural modes. The proposed virtual frequency response function and eigensystem realization algorithm are validated by a numerical example. The results show that the virtual frequency response function and eigensystem realization algorithm can give much a better impulse response than natural excitation technique and identify more precise mode parameters. Finally, the proposed method is applied to a practical engineering bridge, where the results by the virtual frequency response function and eigensystem realization algorithm can also give better results than those given by natural excitation technique and eigensystem realization algorithm.

Modal identification for superstructure using virtual impulse response

In civil engineering, structural modes are identified with the assumption of stationary white noise, which cannot be satisfied in practical engineering. This article proposes a new method, which contains the virtual impulse response and eigensystem realization algorithm. The formulation of virtual impulse response is derived from the inverse Fourier transform of the ratio of the cross-power to auto-power spectral density functions of the measurement responses, which is based on the concept of frequency response function. During the formulation derivation, a single point excitation is only considered. Frequency response function would not change with different excitations and responses, which means that the excitation cannot influence frequency response function. The impulse response is pointed out to only represent the behavior of superstructure. After obtaining impulse responses, eigensystem realization algorithm is then performed to identify the modes of superstructure. The proposed method is validated by a numerical example. The results show that virtual impulse response can have much better free decayed behavior than natural excitation technique and identify very precise modal parameters for superstructure.

A comparison among modal parameter extraction methods

An effective comparison among several methods for extraction of the modal parameters from the frequency response functions measurements is presented in this research. In particular, several curve fitting methods, which are the peak amplitude method, the circle fit method, the least square complex exponential method, the eigensystem realization algorithm method and the rational fraction polynomial method were implemented in Matlab environment and compared in terms of natural frequencies, modal damping and mode shapes. Measurements were performed on a carcass of the gearbox in free–free condition. A hammer has been used with a periodic impulsive excitation signal. The natural frequencies values obtained by all methods were very similar and the differences between the results were insignificant. The peak amplitude and the circle fit gave good results for the damping ratios. The rational fraction polynomial method did the best job in detecting the damping and frequency values. The results obtained by the least square complex exponential method and the eigensystem realization algorithm method were reasonable for both frequency and damping.