Improved bridge modal identification from vibration measurements using a hybrid empirical Fourier decomposition

In this paper, a new modal parameters identification method is presented based on NExT (natural excitation technique) and the response data of the structure under non-stationary ambient excitation the white noise excitation assumption for NExT is extended to a MA (q) model assumption to establish an extended NExT method. Then the assumption that the non-stationary excitation is composed of a (d-1)-th order polynomial and a MA(q) model is proposed. Taking d-th order difference of the non-stationary excitation yields a MA(q+d) model, which satisfies the assumption of the extended NExT method mentioned above. Thus, the extended NExT method can be used in the condition of non-stationary excitation. In order to reduce the error of difference calculation during identification implementation, the difference computation is performed in frequency-domain instead of time-domain. The identification results from the proposed method are compared with the standard NExT for a three DOFs (degrees of freedom) vibration system and the finite element (FE) model of Longtanhe Bridge. The results show that the method proposed in this paper can obtain better modal parameters of the system under non-stationary ambient excitation than the standard NExT.

The instrumented Canton Tower is a 610 m high-rise structure, which has been considered as a benchmark problem for structural health monitoring (SHM) research. In this paper, an improved automatic modal identification method is presented based on a natural excitation technique in conjunction with the eigensystem realization algorithm (NExT/ERA). In the proposed modal identification method, damping ratio, consistent mode indicator from observability matrices (CMI_O) and modal amplitude coherence (MAC) are used as criteria to distinguish the physically true modes from spurious modes. Enhanced frequency domain decomposition (EFDD), the data-driven stochastic subspace identification method (SSI-DATA) and the proposed method are respectively applied to extract the modal parameters of the Canton Tower under different environmental conditions. Results of modal parameter identification based on output-only measurements are presented and discussed. User-selected parameters used in those methods are suggested and discussed. Furthermore, the effect of environmental conditions on the dynamic characteristics of Canton tower is investigated.

<div class="htmlview paragraph">A new approach of estimating the modal parameters of operating piston engines is presented. The developed approach represents a combination of concepts from currently existing analyses such as the natural excitation technique (NExT), conditioned input analysis (CIA), and conditioned source analysis (CSA), and is hence termed “conditioned NExT analysis (CNA)”. NExT analysis can be employed to estimate modal parameters of structures in their naturally excited states. However, the existence of strong combustion induced periodic forcing makes the application of NExT analysis to operating engines difficult, if not impossible. CIA and CSA, built on concepts of partial and virtual coherence respectively, can effectively condition operating engine vibration data so as to remove any periodic energy associated with the process of combustion. Such conditioned operating engine vibration measurements, when combined with NExT analysis, offer a potential technique for estimating modal parameters of operating engines.</div> <div class="htmlview paragraph">The conditioned NExT methodology is initially demonstrated by computing the modal parameters of an analytically simulated frame structure subjected to a combination of random and known periodic forcing. Subsequently, the process is extended to a controlled experiment on a steel plate, designed to simulate an operating engine environment. Guided by the results of the controlled experiment, the developed technique is extended to estimate modal parameters of an operating piston engine. The results of analysis conducted on an engine under non-operating and operating conditions are compared. Results from this research indicate the potential of the developed approach to estimate operating engine modal damping ratios. Based on the findings of this research, suitable recommendations are advanced.</div>

Traditional modal parameter identification methods have many disadvantages, especially when used for processing nonlinear and non-stationary signals. In addition, they are usually not able to accurately identify the damping ratio and damage. In this study, methods based on the Hilbert-Huang transform (HHT) are investigated for structural modal parameter identification and damage diagnosis. First, mirror extension and prediction via a radial basis function (RBF) neural network are used to restrain the troublesome end-effect issue in empirical mode decomposition (EMD), which is a crucial part of HHT. Then, the approaches based on HHT combined with other techniques, such as the random decrement technique (RDT), natural excitation technique (NExT) and stochastic subspace identification (SSI), are proposed to identify modal parameters of structures. Furthermore, a damage diagnosis method based on the HHT is also proposed. Time-varying instantaneous frequency and instantaneous energy are used to identify the damage evolution of the structure. The relative amplitude of the Hilbert marginal spectrum is used to identify the damage location of the structure. Finally, acceleration records at gauge points from shaking table testing of a 12-story reinforced concrete frame model are taken to validate the proposed approaches. The results show that the proposed approaches based on HHT for modal parameter identification and damage diagnosis are reliable and practical.

In order to identify the modal parameters of time invariant three-dimensional engineering structures with damping and small nonlinearity, a novel isometric feature mapping (Isomap)-based three-dimensional operational modal analysis (OMA) method is proposed to extract nonlinear features in this paper. Using this Isomap-based OMA method, a low-dimensional embedding matrix is multiplied by a transformation matrix to obtain the original matrix. We find correspondence relationships between the low-dimensional embedding matrix and the modal coordinate response and between the transformation matrix and the modal shapes. From the low-dimensional embedding matrix, the natural frequencies can be determined using a Fourier transform and the damping ratios can be identified by the random decrement technique or natural excitation technique. The modal shapes can be estimated from the Moore–Penrose matrix inverse of the low-dimensional embedding matrix. We also discuss the effects of different parameters (i.e., number of neighbors and matrix assembly) on the results of modal parameter identification. The modal identification results from numerical simulations of the vibration response signals of a cylindrical shell under white noise excitation demonstrate that the proposed method can identify the modal shapes, natural frequencies, and ratios of three-dimensional structures in operational conditions only from the vibration response signals.

Performing forced vibration tests on full-scale structures is the most reliable way of determining the relevant modal parameters in structural dynamics, such as modal frequencies, mode shapes, modal damping, and modal masses. This study describes the modal identification of a double-level curved cable-stayed bridge with separate deck systems for pedestrians and vehicles via forced vibration tests. The steady-state structural responses to sinusoidal excitations produced by an electrodynamic shaker are recorded under varying excitation frequencies, and the frequency response functions are established. The measured frequency response functions are curve fitted to estimate the modal parameters. The numerical simulation of frequency response function–based modal parameter identification of an elastically multi-supported continuous beam structure is carried out, and the emphasis has been placed on the evaluation of the effect of an additional shaker mass, excitation frequency step and range, multi-mode vibration, and noise on identification results. Finally, the modal parameters for the first lateral mode of a double-level curved cable-stayed bridge are identified by forced vibration experiments, and the results are compared with those from ambient vibration tests and free vibration tests. The effect of the unmeasured wind excitation on identification is discussed. It is shown that the effect of ambient vibration is minor for wind velocity of 3–5 m/s. The damping ratios identified by forced and free vibration tests are comparable, while those from ambient vibration are subject to large variations. The modal mass obtained from forced vibration tests is in good agreement with finite element prediction, which provides design basis for mass-type dampers.

Modal Identification from response data only is studied for structural systems under nonstationary ambient vibration. The topic of this paper is the estimation of modal parameters from nonstationary ambient vibration data by applying the random decrement algorithm with time-varying threshold level. In the conventional random decrement algorithm, the threshold level for evaluating randomdec signatures is defined as the standard deviation value of response data of the reference channel. The distortion of randomdec signatures may be, however, induced by the error involved in noise from the original response data in practice. To improve the accuracy of identification, a modification of the sampling procedure in random decrement algorithm is proposed for modal-parameter identification from the nonstationary ambient response data. The time-varying threshold level is presented for the acquisition of available sample time history to perform averaging analysis, and defined as the temporal root-mean-square function of structural response, which can appropriately describe a wide variety of nonstationary behaviors in reality, such as the time-varying amplitude (variance) of a nonstationary process in a seismic record. Numerical simulations confirm the validity and robustness of the proposed modal-identification method from nonstationary ambient response data under noisy conditions.

Determination of the most meaningful structural modes and gaining insight into how these modes evolve are important issues for long-term structural health monitoring of the long-span bridges. To address this issue, modal parameters identified throughout the life of the bridge need to be compared and linked with each other, which is the process of mode tracking. The modal frequencies for a long-span bridge are typically closely-spaced, sensitive to the environment (e.g., temperature, wind, traffic, etc.), which makes the automated tracking of modal parameters a difficult process, often requiring human intervention. Machine learning methods are well-suited for uncovering complex underlying relationships between processes and thus have the potential to realize accurate and automated modal tracking. In this study, Gaussian mixture model (GMM), a popular unsupervised machine learning method, is employed to automatically determine and update baseline modal properties from the identified unlabeled modal parameters. On this foundation, a new mode tracking method is proposed for automated mode tracking for long-span bridges. Firstly, a numerical example for a three-degree-of-freedom system is employed to validate the feasibility of using GMM to automatically determine the baseline modal properties. Subsequently, the field monitoring data of a long-span bridge are utilized to illustrate the practical usage of GMM for automated determination of the baseline list. Finally, the continuously monitoring bridge acceleration data during strong typhoon events are employed to validate the reliability of proposed method in tracking the changing modal parameters. Results show that the proposed method can automatically track the modal parameters in disastrous scenarios and provide valuable references for condition assessment of the bridge structure.

Modal analysis has developed into a major technology for the study of structural dynamics in the past several decades. Through it, complex structural dynamics phenomena can be represented in terms of structural invariants, i.e., the modal parameters: natural frequencies, damping ratios and mode shapes. Operational Modal Analysis (OMA) deals with the estimation of modal parameters on vibration data measured for operational conditions, when the excitation on the structure is not measured. In this work, OMA is performed on a wind turbine blade undergoing wind tunnel testing. These tests included different wind speeds, pitch angles and also different health conditions of the blade, where masses of different magnitude were fixed to the blade on different locations to emulate damage conditions. In order to monitor the modal parameters across multiple days of varied tests in the wind tunnel, the Polymax modal parameter estimator was implemented, coupled with an Automated Modal Analysis methodology. This methodology included an automatic modal parameter selection technique, using a Machine Learning (ML) clustering algorithm, coupled with a modal tracking procedure which applied statistical thresholds on the modal parameters’ values. The tracking procedure searches for modes similar to the ones calculated for healthy conditions. The results show how the modal parameters of the wind turbine blade vary with the different measured conditions in the wind tunnel. Moreover, a damage detection methodology is implemented to differentiate between the healthy and damaged conditions on the blade, by leveraging an anomaly detection algorithm using the Multivariate Gaussian Distribution (MGD). This algorithm takes as input the modal parameters calculated by the previous Automated Modal Analysis methodology and detects statistical deviations among them which could indicate the presence of damage. All steps of this work contribute to developing an automatic framework able to detect damages on a wind turbine blade, and therefore perform Structural Health Monitoring (SHM) for different operational conditions.

The accuracy of modal parameter estimates from estimation methods built in the framework of experimental modal analysis (EMA) can be assessed by several methods provided the information of input loadings and output responses. However, a deeper investigation is necessary for estimation methods built in the operational modal analysis (OMA) framework to establish confidence boundaries for the modal parameter estimates. This paper aims to estimate uncertainties of the modal parameter estimates by testing the sensitivity of the damped natural frequencies, damping ratios, and mode shapes to the choice of the estimation method and data acquisition settings used in this method. The uncertainty estimates of the modal parameter estimates are tested for changes in the boundary conditions. The tests provide insights into tendencies for the accuracy of the modal parameter estimates when the boundary conditions provide a different set of “true” modal parameter values. A simple cantilever beam experimental setup is used to obtain benchmark modal parameter estimates from EMA that are used to estimate the accuracy of the modal parameter estimates from OMA. The method’s robustness in estimating uncertainties and accuracies for modal parameter estimates is tested for several independent experiments. The boundary conditions for the cantilever beam are changed to test the influence of altering modal parameter estimates on the estimated uncertainties and accuracies. The boundary conditions are changed using materials with different elastic properties and the connection pressure between the clamping device and beam. A method presented in Paulsen et al. (Proceedings of the 9th International Operational Modal Analysis Conference, IOMAC, 2022) calculates a representative modal parameter estimate from a set of modal parameter estimates based on weightings obtained from the modal assurance criterion. The method’s ability to reduce uncertainties and increase accuracies of the representative estimate is demonstrated by repeating a test setup ten times. The results present simultaneously a reduction of uncertainties and an increase of accuracies for the representative modal parameter when the weighting method is used. An evaluation of dependencies between the estimated uncertainties and accuracies finds a proportional relation for the damped natural frequency when the uncertainties are described as a relative standard deviation. The analysis presents higher accuracy for the damped natural frequency and mode shape estimates compared to the damping factor, and the accuracy of the damping factor is dependent on the actual value of the damping factor. These phenomena for accuracy are well known and indicate that the uncertainties of modal parameter estimates can be assessed through a sensitivity test of estimation methods and data acquisition parameters.

Operational modal analysis (OMA) has been widely used for large structures. However, measured signals are inevitably contaminated with noise and may not be clean enough for identifying the modal parameters with proper accuracy. The traditional methods to estimate modal parameters in noisy situation are usually absorbing the “noise modes” first, and then using the stability diagrams to distinguish the true modes from the “noise modes.” However, it is still difficult to sort out true modes because the “noise modes” will also tend to be stable as the model order increases. This study develops a noise reduction procedure for polyreference complex exponential (PRCE) modal analysis based on ambient vibration responses. In the procedure, natural excitation technique (NExT) is first applied to get free decay responses from measured (noisy) ambient vibration data, and then the noise reduction method based on solving the partially described inverse singular value problem (PDISVP) is implemented to reconstruct a filtered data matrix from the measured data matrix. In our case, the measured data matrix is block Hankel structured, which is constructed based on the free decay responses. The filtered data matrix should maintain the block Hankel structure and be lowered in rank. When the filtered data matrix is obtained, the PRCE method is applied to estimate the modal parameters. The proposed NExT-PDISVP-PRCE scheme is applied to field test of a jacket type offshore platform. Results indicate that the proposed method can improve the accuracy of OMA.

Modal tracking plays a vital role in structural health monitoring since changes in modal parameters help us understand a structure’s dynamic characteristics and may reflect the potential deterioration of structural performance. Although numerous modal parameter estimation (MPE) methods exist, it is not guaranteed that an MPE process will exclude all spurious modes and not lose any physical modes every time over a long-term monitoring period. Relatively large damping of a structure, poor data quality, and significant changes in structural modal parameters may make the estimated modal parameters spurious, missing, or misclassified. It makes long-term modal tracking semiautomated or manual, which constrains timely downstream applications such as anomaly detection, condition assessment, and decision making. This research aims to propose a long-term continuous automatic modal tracking algorithm based on Bayesian inference even when the modal parameters, damping, and data quality change significantly. Bayesian inference is used to determine the physical modes from the results of existing MPE methods. Both the modes identified from the most recent response set and the modal probability model from multiple previous response sets are considered in the Bayesian model to better determine the physical modes from the results of MPE. Moreover, the proposed algorithm requires only three extra hyperparameters compared to general modal tracking algorithms, and they can be quickly determined by a grid search method. The performance of the proposed algorithm is verified by a numerical example and a real-world civil structure Z24 Bridge benchmark.

The integrity of offshore structures is prone to the threat of drifting sea ice. By exciting the natural frequencies of the ice–structure interaction system, severe vibrations may occur. For a better understanding of how dynamic ice–structure interaction affects the system properties, a stochastic subspace system identification technique, SSI-cov is introduced to identify modal parameters during dynamic ice–structure interactions. Due to the uncertain input variables of SSI-cov, the identified modal parameters suffer from the quantitative judgment of the analyst. To address this problem, this study proposes an uncertainty analysis (UA) framework to obtain estimates of modal parameters. This framework constitutes both a sensitivity analysis (SA) and a UA. First, SA is applied to pick up the input variables that contribute the most to the identified modal parameters. Next, the important variables are left free to vary over their range of existence to obtain the modal parameters’ uncertainties. The framework is applied to full-scale tests from Norströmsgrund lighthouse and model tests carried out in Hamburg Ship Model Basin (HSVA). The comparison with traditional automated modal parameter identification shows its superiority in terms of the accuracy of modal parameters.

The ancient masonry city hall in L'Aquila, Italy (Margherita Palace) that was severely damaged during the L'Aquila earthquake in April 2009 has been subjected to ambient vibration tests to determine its dynamic characteristics such as the natural frequencies, mode shapes, and damping ratios. The processing of ambient vibration responses is performed with three output-only identification approaches: (1) the frequency-domain decomposition, (2) random decrement (RD) technique combined with the eigensystem realization algorithm (ERA), and (3) the natural excitation technique (NExT) combined with ERA. Measurements were made in the palace and in the civic tower using 15 sensors in three different configurations, and the ambient vibration tests involved the simultaneous measurements of longitudinal and lateral vibrations. Lateral vibration modes for the civic tower were identified clearly; a total of five modal shapes and frequencies were identified in the frequency range 0 to 7 Hz. The modal identification in the palace was more difficult because of the severe damage; however, the two bending modes in the perpendicular directions were identified. The comparison of the three experimental techniques shows good agreement, confirming the reliability of the three identification methods. The results of the modal identification have been used to update the finite-element model of the palace. DOI: 10.1061/(ASCE)ST.1943-541X.0000494. © 2012 American Society of Civil Engineers. CE Database subject headings: Historic buildings; Masonry; Finite element method; Vibration; Structural analysis; Earthquakes; Italy. Author keywords: Output-only modal identification; Frequency domain decomposition (FDD); Eigensystem realization algorithm (ERA); Random decrement (RD); Masonry structure; Finite-element model; Ambient vibration; Identification; EMA; Operational modal analysis.

Modal parameter identification via ambient vibration is popular but faces challenges from uncertainties due to unknown inputs and low signal‐to‐noise ratio. Bayesian methods are gaining increasing attention for operational modal identification due to their ability to quantify uncertainties. However, improvements in computational efficiency are needed, particularly when addressing numerous modes and degrees of freedom. To address this challenge, this study proposes an innovative approach, termed the “Bayesian spectral decomposition” method (BSD), employing the decompose‐and‐conquer strategy. This novel method, operating within the frequency domain, identifies each mode individually by exploiting their inherent separated modal characteristics. For each mode, the response spectrum matrix undergoes an eigenvalue decomposition, yielding crucial eigenvalues (incorporating frequency and damping information) and eigenvectors (containing mode shape information). Subsequently, statistical properties of the eigenvalues and eigenvectors are utilized to establish likelihood functions for Bayesian parameter identification. By combining prior information, the posterior probability distribution functions of modal parameters are derived. The optimal solution is then obtained by resolving the maximum posterior probability distribution function problem. To further quantify the uncertainty of modal parameters, Gaussian distributions are employed to approximate the posterior probability distribution functions. The adoption of the decomposition approach circumvents the joint identification of all modal parameters, substantially reducing the parameter dimensions for optimization. Consequently, this strategy leads to decreased computational complexity and significantly improved computational stability. The effectiveness of the BSD is confirmed through simulated data generated from an 8‐story shear building as well as measured data collected from both an experimental shear frame and the Canton Tower. The results demonstrate that the proposed method achieves high accuracy in identifying modal parameters, greatly improves computational efficiency, and effectively quantifies the uncertainties in modal parameters.