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Structural Health Monitoring relies on accurate modal identification and effective damage detection to assess structural performance and safety. However, traditional sensor placement methods struggle to balance modal identification uncertainty, which arises from limited sensor coverage and measurement noise and damage detection sensitivity, which requires sensors to be optimally positioned to capture structural stiffness variations. To address this challenge, this study proposes a multi-objective sensor placement optimization method based on the Non-Dominated Sorting Genetic Algorithm. The method introduces two key objective functions: minimizing modal identification uncertainty by leveraging Bayesian modal identification theory and information entropy and maximizing damage detection sensitivity by incorporating an entropy-based measure to quantify the uncertainty in stiffness variation estimation. By formulating the problem as Pareto-based multi-objective optimization, the method efficiently explores a trade-off between the two competing objectives and provides a diverse set of optimal sensor placement solutions. The proposed approach is validated through numerical experiments on a simply supported beam and a benchmark bridge structure, demonstrating that different optimization objectives lead to distinct sensor placement patterns. The results show that solutions prioritizing modal identification distribute sensors across the structure to improve global response estimation, while solutions favoring damage detection concentrate sensors in critical areas to enhance sensitivity. The proposed method significantly improves sensor placement strategies by offering a systematic and flexible framework for SHM applications, enabling engineers to tailor monitoring strategies based on specific structural assessment needs.

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.

Summary Structural model updating aims at calculating the in‐situ structural properties (e.g., stiffness and mass) based on measured responses. One common approach is to first identify the modal parameters (i.e., natural frequencies and mode shapes) and then use them to update the structural parameters. In reality, the degrees of freedom that can be measured are usually limited by number of available sensors and accessibility of targeted measurement locations. Then, multiple setups are designed to cover all the degrees of freedom of interest and performed sequentially. Conventional methods do not account for identification uncertainty, which becomes critical when excitation information is not available. This is the situation in model updating utilizing ambient vibration data, in which the excitations, such as wind, traffic, and human activities, are random in nature and difficult to be measured. This paper develops a Bayesian model updating method incorporating modal identification information in multiple setups. Based on a recent fundamental two‐stage Bayesian formulation, the posterior uncertainty of modal parameters is incorporated into the updating process without heuristics that are commonly applied in formulating the likelihood function. Synthetic and experimental data are used to illustrate the proposed method.

In this paper the actual dynamic behavior of the civic Clock tower of Rotella, a little village in central Italy heavily damaged by the recent 2016 seismic sequence, is thoroughly investigated by means of a detailed numerical model built and calibrated using the experimental modal properties obtained through Ambient Vibration Tests. The goal is to update the uncertain parameters of two behavioral material models applied to the Finite Element Model (elastic moduli, mass densities, constraints, and boundary conditions) to minimize the discrepancy between experimental and numerical dynamic features. A sensitivity analysis was performed with the definition of a metamodel to reduce the computational strain and try to define the necessary parameters to use for the calibration process. Due to the high nonlinear dependency of the objective function of this optimization problem on the parameters, and the likely possibility to get trapped in local minima, a machine learning approach was meant. A fully automated Finite Element Model updating procedure based on genetic algorithms and global optimization is used, leading to tower uncertain parameters identification. The results allowed to create a reference numerical replica of the structure in its actual health state and to assess its dynamic performances allowing better control over their future evolution.

Modal identification consists of determining the natural frequencies, damping ratios and mode shapes of a build structure using measured dynamic data. The Bayesian approach is very appealing when the goal is not only to estimate the modal parameters but also their uncertainty, calculated from their joint probability distribution. This paper presents an example of Bayesian modal identification in time domain using the state space model and the software Stan.

An automated operational modal analysis (AOMA) method that considers the uncertainty in modal parameters is presented and data acquired from actual bridges are used to validate it. The proposed method processes stepwise, from SSI to pre-cleaning, clustering and the removal of outliers. The stochastic subspace identification (SSI) step also calculates the uncertainty of the modal parameters. In this step, the MAC (modal assurance criterion) index and its variability are additionally calculated by exploiting the alteration of the mode shapes. The pre-cleaning stage sorts out the spurious modes by means of the frequency, the coefficient of variation related to the frequency and the damping ratio, as well as the MAC index and its standard deviation. Under the assumption of normal distributions for the frequency and the MAC index, the clustering stage constructs clusters of identical modes with reference to the uncertainty of each mode. The outliers that may be contained in each of these clusters are then removed based upon the frequency, the MAC index and the damping ratio. Values for the parameters that make the proposed method applicable are suggested and are applied unilaterally to three instrumented bridges of different types. The results show that the proposed AOMA method provides accurate mode identification regardless of the bridge type.

Summary Considerable efforts have been made towards the development of robust and fully automated vibration‐based monitoring systems with the goal of extracting relevant information regarding the dynamic behaviour and health condition of the monitored structure. Satisfactory results have been obtained in this scientific domain by combining accurate measurement systems and fully automated output‐only modal identification techniques. In this context, the main objective of this contribution is to demonstrate in a full‐scale case study that with the application of sophisticated algorithms for the automatic tracking of modal parameters, it is possible to detect very small structural changes. Apart from describing the main features and capabilities of the autonomous monitoring system implemented to assess the structural condition of a peculiar football stadium suspension roof, this paper also outlines the main results obtained over the course of 4 years of monitoring carried out to assess the dynamic behaviour and the health condition of the roof structure. The routines developed for the online processing of the continuously collected acceleration time series include state‐of‐the‐art processing techniques, such as automated modal identification based on cluster analysis and principal components analyses combined with control charts for removal of environmental or operational effects and detection of structural changes, together with some innovative features in the context of continuous dynamic monitoring, such as the quantification of the uncertainties associated with each modal estimate and the estimation of the contribution of each mode to the measured structural response.

Operational modal analysis (OMA) has been widely used in many fields of study because it allows identifying the modal parameters of a flexible structure in its operating condition. The system is under unknown working loads assumed to be random with broadband spectral characteristics. These hypotheses are not always easy to fulfill, generating uncertainty about identified modal parameters. This study evaluates and compares the effectiveness of two OMA techniques, enhanced frequency-domain decomposition (EFDD) and Ibrahim time domain (ITD), in the accuracy of modal parameter estimation of an unmanned aerial vehicle (UAV) structure with output-only data obtained by flight testing. To evaluate the influence of the number of sensors used in the identification of the modes, different measurements setups were considered to carry out in-flight modal identification analyses. Some works have addressed uncertainty by focusing on retesting or subdivision of a single measurement record. This work innovates in presenting an uncertainty study considering the variables that intervene in the estimation of PSD. The uncertainty in the identified modal parameters is obtained using the variability of the values of the parameters found. The modal frequencies values observed employing EFDD and ITD do not present substantial variations associated with the PSD matrix estimates. The EFDD damping ratio values show significant variability because they are mainly affected by spectral leakage, while the ITD damping ratio values are less sensitive to Welch’s method parameters variation. The root mean square deviations (RMSDs) of the frequencies values for both techniques are compared with those resulting from ground vibration testing.

A Bayesian framework is presented for finite element model-updating using experimental modal data. A novel likelihood formulation is proposed regarding the inclusion of the mode shapes, based on a probabilistic treatment of the MAC value between the model predicted and experimental mode shapes. The framework is demonstrated by performing model-updating for the Metsovo bridge using a reduced high-fidelity finite element model. Experimental modal identification methods are used in order to extract the modal characteristics of the bridge from ambient acceleration time histories obtained from field measurements exploiting a network of reference and roving sensors. The Transitional Markov Chain Monte Carlo algorithm is used to perform the model updating by drawing samples from the posterior distribution of the model parameters. The proposed framework yields reasonable uncertainty bounds for the model parameters, insensitive to the redundant information contained in the measured data due to closely spaced sensors. In contrast, conventional Bayesian formulations which use probabilistic models to characterize the components of the discrepancy vector between the measured and model-predicted mode shapes result in unrealistically thin uncertainty bounds for the model parameters for a large number of sensors.

Summary A Bayesian statistical framework was previously developed for modal identification of well‐separated modes incorporating ambient vibration data, that is, operational modal analysis, from multiple setups. An efficient strategy was developed for evaluating the most probable value of the modal parameters using an iterative procedure. As a sequel to the development, this paper investigates the posterior uncertainty of the modal parameters in terms of their covariance matrix, which is mathematically equal to the inverse of the Hessian of the negative log‐likelihood function evaluated at the most probable value. Computational issues arising from the norm constraint of the global mode shape are addressed. Analytical expressions are derived for the Hessian so that it can be evaluated accurately and efficiently without resorting to finite difference. The proposed method is verified using synthetic and laboratory data. It is also applied to field test data, which reveals some challenges in operational modal analysis incorporating multiple setups. Copyright © 2014 John Wiley & Sons, Ltd.