Operational Modal Analysis in Absence of a Random Distributed Load

Modified frequency and spatial domain decomposition method based on maximum likelihood estimation

Safety assessment of structures for near-field blast-induced ground excitations using operational modal analysis

Operational Modal Analysis (OMA) is a methodology to obtain the modal properties of a structure using the dynamical responses of the system only. That methodology is very useful in aeroelastic in-fligh testing where there are difficulties in measuring directly the aerodynamic loads on aircraft. For risk reduction in the future flight operations of the EOLO at ITA, wind-tunnel tests were performed. EOLO is an Unmanned Aerial Vehicle (UAV) with high aspect ratio and structural flexibility, designed to study aeroelastic phenomena and to evaluate the interaction of flexible effects with the aircraft flight dynamics. In this work The Frequency Domain OMA techniques: Decomposition (FDD), Enhanced Frequency Domain Decomposition (EFDD), Spatial and Frequency Domain Decomposition (SFDD) are applied to the vibrational data from wing tunnel test campaign in different operation conditions, using acceleration and strain measurements. The results between the OMA techniques are compared with previous modal characterization from Ground Vibration Test (GVT) and numerical aeroelastic analysis. The different generations of the frequency domain decomposition methods proved their suitability for use in aircraft aeroelastic characterization using accelerations and strain measurements.

<div class="htmlview paragraph">Operational Modal Analysis also known as Output Only Modal Analysis has in the recent years been used for extracting modal parameters of civil engineering structures and is now becoming popular for mechanical structures. The advantage of the method is that no artificial excitation needs to be applied to the structure or force signals to be measured. All the parameter estimation is based upon the response signals, thereby minimising the work of preparation for the test. This test case is a controlled lab set-up enabling different parameter estimation methods techniques to be used and compared to the Operational Modal Analysis. For Operational Modal Analysis two different estimation techniques are used: a non-parametric technique based on Frequency Domain Decomposition (FDD), and a parametric technique working on the raw data in time domain, a data driven Stochastic Subspace Identification (SSI) algorithm. These are compared to other methods such as traditional Modal Analysis. In the paper the method is illustrated by practical examples.</div>

The paper presents how to identify natural frequencies and mode shapes of structures by Operational Modal Analysis (OMA) technique, in which the Frequency Domain Decomposition (FDD) method is used. This method is an experimental method only base on the data of measuring the dynamic response of the structures under the excitation due to ambient forces and operational loads to determine the vibration characteristics. Measure vibration (acceleration) and determine spectral density matrix, using the singular values decomposition method of spectral density matrix to determine the natural frequencies and mode shapes of structures. The calculation results show that the natural frequencies, the mode shapes form determined by the OMA technique is consistent with the calculation results according to the theory and show the reliability of the method.KeywordsNatural frequencyMode shapeIdentifyEMAOMAFDD

The purpose of this study is to identify modal parameters of an elevator car by operational modal analysis, and to estimate the stiffness parameters of the elevator car based on the obtained modal parameters. It is assumed that the inertia properties such as mass, the center of gravity and the moment of inertia are well predicted based on the design condition. However, it is difficult to predict the stiffness parameters of an elevator car especially in the operating condition. Therefore, the stiffness coefficients are estimated based on the operational modal analysis in this study. The frequency range of interest is up to 4 Hz, where the assumption that the car behaves rigidly is satisfied. In order to check the assumption the Rigidity Indicator Function (RIF) is introduced as a function of frequency. Frequency Domain Decomposition (FDD) method is used for the operational modal analysis. The stiffness parameters are estimated by minimizing the sum of the errors of natural frequencies and the errors of mode shape vectors between the experimentally identified modal model and the analytical model.

The design and assessment of structural dynamics behaviour often rely on the ‘modal properties’ (primarily modal frequencies, damping ratios and mode shapes) of a structure. Modal properties are frequently identified in vibration related applications as their information can benefit design theory validation and in-situ property updating and monitoring. Given the prospect of flexibility and high implementation economy, operational modal analysis (OMA) aims to identify the modal properties using only ambient vibration response whose amplitude can be small. When instrument noise becomes a non-negligible part of measured data, the identified result can have significant uncertainty, calling for proper uncertainty quantification. Amidst the complex composition of ambient vibration data of civil structures, conventional noise assumption may not always apply. Motivated by issues about uncertainty modelling and computation, this thesis contributes to the theory and computational algorithm of OMA from a Bayesian perspective in the frequency domain. Focusing on the resonance band of vibration modes, a Bayesian probabilistic approach provides a fundamental mathematical framework for OMA. The ‘posterior’ (i.e., given data) most probable value (MPV) of modal properties maximising the likelihood of data informs about the best estimate. The uncertainty associated with the estimate is quantified by the posterior covariance matrix, which can be calculated as the inverse of the Hessian of negative log-likelihood function (NLLF) at the MPV. Existing formulations assume independent and identically distributed (i.i.d.) prediction errors among different channels, which account for both instrument noise and modelling error. While the assumption significantly simplifies theory and computation, and may not matter when the signal-to-noise (s/n) ratio is high, it is prone to be violated in OMA. For example, large disparity among noise intensities may exist, and significant correlation can occur because of the presence of unmodelled dynamics, which may affect identification results. To understand and manage the consequence of these potential violations, a comprehensive investigation on the effect of prediction error models with different statistical assumptions is presented in this work. Beyond the i.i.d. assumption, two generalised prediction error models have been considered to account for the disparity and correlation that may arise in applications. Such investigation was previously hindered by computational issues because the general prediction error models feature more parameters carrying somewhat different mathematical roles from the existing ones, making the computation of posterior MPV and covariance matrix more challenging in the first place. As a contribution to computation and enabling scientific investigation, new efficient algorithms applicable to general prediction error models are developed in this work for their computation. As a contribution to theory, a mathematical analogy for general noise is established with i.i.d. model for OMA problems. Not only does this provide a simple approximate way for handling general noise with existing tools based on i.i.d. assumption, it also allows the understanding and experience accumulated with i.i.d. model to be generalised. As a spin-off of this opportunity, asymptotic formulae for identification uncertainty are developed for general prediction error models in this work, which yield a generalised definition for s/n ratio that provides a fundamental quantification of data quality in OMA. The theory and algorithms are developed and tested against synthetic, laboratory and field data. As a further contribution to computation, motivated by existing overheads and hurdles in computer code development, this work proposes an alternative semi-analytical means for calculating the Hessian of NLLF in OMA. The proposed method explores a mathematical identity in the theory of expectation-maximisation (EM) algorithm to allow the Hessian matrix to be calculated via Monte Carlo simulation, which offers an alternative means for counter-checking/benchmarking during computer code development. Theoretical implications of the identity are also explored, and numerical examples are presented to illustrate implementation aspects.

Broadband responses under random excitations can reflect the modal characteristics of a structure. This paper investigates the applicability of operational modal analysis (OMA) to extract modal parameters of transformer windings under random excitations. Firstly, random vibration signals are extracted from measured vibration signals by estimating an accurate power frequency and eliminating vibration harmonics. Then, modal characteristics are obtained by applying OMA to the extracted random vibration signals. The numeric study on a winding axial vibration model demonstrates the effectiveness of the proposed method in obtaining its modal characteristics. Laboratory experiments show that the natural frequencies calculated by OMA are a subset of the actual natural frequencies of windings and the tank. The field test results of a 110 kV power transformer show that the number of natural frequencies increase when winding deformation occurs. OMA of transformer windings may pave a way for online winding condition monitoring.

This research investigates the dynamic behavior of a railway bridge using both experimental and numerical methods. Field tests were conducted to capture the bridge response to live loading conditions with acceleration data collected via uniaxial accelerometers placed at critical locations along the structure. The dynamic characteristics, including the natural frequencies and mode shapes, were determined using two analytical techniques: Fast Fourier Transform (FFT) and Operational Modal Analysis (OMA). While FFT provides a frequency domain analysis, OMA enables the estimation of modal parameters, such as natural frequencies, mode shapes, and damping ratios, using the bridge's response to operational forces. The results revealed that the fundamental frequency obtained from the OMA (2.163 Hz) was higher than that obtained from the FFT (1.95 Hz) and the Finite Element Analysis (FEA) model (1.65 Hz). Additionally, the OMA produced mode shapes that were closely aligned with those predicted by the FEA, validating the accuracy of the numerical model. This study highlights the advantages of OMA over FFT, particularly the ability to capture mode shapes, and underscores the importance of integrating OMA with FEA for a comprehensive dynamic assessment of bridge structures. These findings contribute to the growing body of knowledge on structural monitoring and provide practical insights into improving bridge safety and performance.

Operational modal analysis based dynamic parameters identification in milling of thin-walled workpiece

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.

This paper reports the step-by-step procedures of a full-scale ambient vibration test and the corresponding modal identification and Bayesian structural model updating of a coupled building. The building is characterized as a combination of a main part and a complementary part connected together by corridors in between. Compared with the main part, the volume of the complementary part is much smaller. Therefore, the influence on the dynamic properties of the complementary part from its counterpart is expected. To capture the dynamic properties of the coupled building, a 21-setup ambient vibration test was designed to cover all the degrees of freedom (DOFs) of interest. The modal parameters of each setup were identified following the frequency domain decomposition (FDD) method and the partial mode shapes from different setups were assembled following a least-squares method. To determine the stiffness of the linkage between the two parts, the coupled building was simulated with two linked shear buildings and updated utilizing the Markov chain Monte Carlo (MCMC)-based Bayesian model updating method. The identified modal parameters revealed interesting features about the coupled effects between the main part and complementary part and were discussed in detail. The good match between the model-predicted and identified modal parameters verified the validity of proposed shear building model. This study provides valuable experience in the area of structural model updating and structural health monitoring.

The work investigates the dynamic validation and seismic assessment of a bell tower located in Caserta Vecchia, emphasizing its relevance in structural engineering. Given the tower historical significance and possible vulnerability, understanding its structural behaviour under seismic loading is vital for preservation efforts. To capture the bell tower structural intricacies, advanced photogrammetric surveying techniques were utilized. This produced a detailed 3D model of the bell tower, which was subsequently integrated into the Abaqus software for finite element analysis. This approach facilitated a comprehensive examination of the tower structural dynamics and vibrational characteristics. The robustness of the study has been further enhanced through a validation process. Frequency analysis outcomes from Abaqus were cross-referenced with real-world seismic data collected using structural health monitoring (SHM) systems. This seismic data was analyzed using the Frequency Domain Decomposition (FDD) technique. This method ensured that the analytical model results were consistent with observed seismic responses, thereby validating the model’s accuracy and reliability. Further dynamic analyses were conducted to evaluate the bell tower potential vulnerabilities during high-intensity seismic events. These analyses are essential for predicting the structural response and potential damage patterns during earthquakes, facilitating better preparedness and mitigation strategies. The applicability of the methodologies used in similar contexts offers opportunities for advancing heritage preservation and optimizing the seismic safety of such complex structures.

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.

This paper describes some notable aspects regarding the dynamic monitoring planned for the Chiaravalle viaduct, linking SS76 and airport of Ancona in Falconara, in Italy. The viaduct is 875 m long and consists of four r.c. kinematic chains separated by joints. The deck is constituted by three beams supported by column bent piers constituted by two circular r.c. columns founded on a continuous footing on six concrete piles. The first purpose of the dynamic monitoring is to determine the experimental natural frequencies and mode shapes of the viaduct before a scheduled seismic rehabilitation. Then, comparison of the first results with those obtained after the retrofit of foundation and piers and finally after the substitution of the deck bearing, will contribute to evaluate efficiency of the executed works. Two different inputs are used for the dynamic tests, ambient and traffic vibrations. An Operational Modal Analysis is carried out starting from the collected data, using the Enhanced Frequency Domain Decomposition technique. Different sensor configurations are used to evaluate the overall behaviour of the viaduct. The Pre Global Estimation Re-scaling method is adopted in order to merge the different parts of mode shapes relevant to each configuration. Results of dynamic tests are interpreted through a 3D Finite Element model of the viaduct. Firstly, a conventional fixed base model is considered, then a model including the Soil-Structure Interaction phenomena is developed in order to improve consistency between numerical and experimental results. The modal parameters obtained by the model including Soil-Structure Interaction are in good agreement with those estimated experimentally.