Long-term continuous automatic modal tracking algorithm based on Bayesian inference

Modal parameter estimation plays an important role in vibration-based damage detection and is worth more attention and investigation, as changes in modal parameters are usually being used as damage indicators. This paper focuses on the problem of output-only modal parameter recursive estimation of time-varying structures based upon parameterized representations of the time-dependent autoregressive moving average (TARMA). A kernel ridge regression functional series TARMA (FS-TARMA) recursive identification scheme is proposed and subsequently employed for the modal parameter estimation of a numerical three-degree-of-freedom time-varying structural system and a laboratory time-varying structure consisting of a simply supported beam and a moving mass sliding on it. The proposed method is comparatively assessed against an existing recursive pseudolinear regression FS-TARMA approach via Monte Carlo experiments and shown to be capable of accurately tracking the time-varying dynamics in a recursive manner.

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 aim of the present paper is to evaluate structural health monitoring (SHM) techniques based on modal analysis for crack detection in small wind turbine blades. A finite element (FE) model calibrated to measured modal parameters will be introduced to cracks with different sizes along one edge of the blade. Changes in modal parameters from the FE model are compared with data obtained from experimental tests. These comparisons will be used to validate the FE model and subsequently discuss the usability of SHM techniques based on modal parameters for condition monitoring of wind turbine blades.

ABSTRACTThe structural stiffness of a blade changes when it is defective, which affects its modal parameters. Consequently, modal parameters can be crucial to investigating defects. Therefore, developing a model that can accurately analyze the modal parameters of healthy and defective blades is essential. This study proposes a high‐fidelity finite element (FE) model development method based on multiobjective optimization. Three 10‐kW‐class wind turbine blades were manufactured, and the modal parameter changes due to debonding damage were measured. The initial FE model was constructed using structural drawings and considering specific material properties. Subsequently, the modal analysis results were compared with the experimental results. The analysis errors of normal and abnormal blades were expressed as cost functions and multiobjective optimization was performed. Machine learning–based predictive models facilitated time‐ and cost‐effective optimization, enabling modal updating that simultaneously reflects modal parameter changes for various blade models. Because accurately predicting the dynamic characteristics of blades by various defect locations and sizes is crucial for blade damage detection, the proposed method is significant for damage detection research.

This article is an experimental study of the effects of multi-site damage on the vibration response of a composite beam damaged by low-energy impact. The variation of the modal parameters with different levels of impact energy and density of impact is studied. Specimens are impacted symmetrically in order to induce a global rate of damage. A damage detection tool Damage Index is introduced in order to verify the estimation of damping ratios. Design of Experiments is used to establish the sensitivity of both energy of impact and density of damage. The DOE analysis results (using natural frequency only) indicate that impact energy for 2nd, 3rd, and 4th bending modes is the most significant factor contributing to the changes in the modal parameters for this kind of symmetrical dynamic test.

Vibration-based damage detection approaches have received considerable attention in the field of structural health monitoring. Modal parameters are often adopted to define damage-sensitive features, since their strong physical meaning can help interpreting the structural condition. On the other hand, they are also sensitive to changes to environmental and operational conditions. This aspect is critical for an automatic damage detection, because changes in modal parameters due to varying external factors (e.g. temperature) can be greater than those caused by damage.In this context, this paper proposes an application to real data coming from long-term structural health monitoring of axially-loaded beam-like structures under realistic environment. These very common structural elements are usually subject to an axial load that changes under environmental and operating conditions. Since the axial load is not directly known in most real applications, assessing damage using modal-based damage features is a complicated task.In this paper, a data driven approach that does not require a knowledge of the axial load is proposed to filter out the environmental effects on modal-based damage features. The strategy has been successfully tested on data acquired in an uncontrolled environment, and resulted in being a promising solution for real structural health monitoring applications.KeywordsLong-term monitoringDamage detectionEnvironmental and operational variationsVibration-based featureStatistical pattern recognitionBeam-like structureTie-rod

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 presents an experimental study on the detection and localization of damage in a steel plate using vibration-based methods. Dynamic impulse tests were conducted on both undamaged and damaged plate to determine changes in modal parameters, such as eigenfrequencies and mode shapes. Several damage detection methods, including the Modal Assurance Criterion (MAC), Coordinate Modal Assurance Criterion (COMAC), Mode Shape Curvature (MSC), Mode Shape Curvature Square (MSCS), and Damage Index (DI), were applied. The results showed that while MAC and COMAC effectively detected damage, they did not provide information about the location. MSC and MSCS indices demonstrated improved damage localization, with the MSCS being particularly effective. The DI index provided the clearest and most accurate representation of the damage location. These findings highlight the potential of these methods for non-destructive testing and structural health monitoring.

Automated modal parameter estimation using correlation analysis and bootstrap sampling

Modal-based structural health monitoring (SHM) detects damage and degradation phenomena from the variations of the modal parameters over time. However, the modal parameter estimates are also influenced by environmental and operational variables (EOVs) whose effects have to be compensated. Modeling the influence of EOVs on modal parameters is very challenging, so black-box models, such as regression models, are often adopted as an alternative. However, in many applications, the set of measured EOVs is incomplete or the factors influencing the estimates cannot be identified or measured. In these conditions, output-only techniques for compensation of environmental and operational effects are an attractive alternative. Different linear as well as nonlinear methods for the compensation of the environmental and operational influence on modal parameters in the context of modal-based SHM are reviewed in the present paper. Real datasets collected from vibration-based monitoring systems are analyzed, and the results are presented and discussed to illustrate the applicative perspectives and possible drawbacks of the selected methods.KeywordsModal-based SHMEnvironmental influenceRegression modelPrincipal component analysisKernel PCASecond-order blind identification

Recent work with autonomous modal parameter estimation has shown great promise in the quality of the modal parameter estimation results when compared to results from traditional methods by experienced users. While autonomous modal parameter estimation means slightly different things to different researchers and practitioners, for the purpose of this discussion, autonomous will require an automated procedure which sorts and processes a large number of possible modal parameter solutions to yield one consistent estimate with no user interaction after initial thresholds are chosen. In the work discussed, this final, consistent set of modal parameters is identifiable due to the combination of temporal and spatial information in azdomain state vector of relatively high order (5–10). Since thiszdomain state vector has both complex modal frequency and modal vector information as embedded content, sorting consistent estimates from the multitude of possible solutions is relatively trivial. Because thiszdomain state vector can be developed from the results of any modal parameter estimation method, possible solutions from different traditional methods can be utilized in the autonomous procedure to yield one consistent set of modal parameters.

This paper reports the structural health monitoring benchmark study results for the Canton Tower using Bayesian methods. In this study, output-only modal identification and finite element model updating are considered using a given set of structural acceleration measurements and the corresponding ambient conditions of 24 hours. In the first stage, the Bayesian spectral density approach is used for output-only modal identification with the acceleration time histories as the excitation to the tower is unknown. The modal parameters and the associated uncertainty can be estimated through Bayesian inference. Uncertainty quantification is important for determination of statistically significant change of the modal parameters and for weighting assignment in the subsequent stage of model updating. In the second stage, a Bayesian model updating approach is utilized to update the finite element model of the tower. The uncertain stiffness parameters can be obtained by minimizing an objective function that is a weighted sum of the square of the differences (residuals) between the identified modal parameters and the corresponding values of the model. The weightings distinguish the contribution of different residuals with different uncertain levels. They are obtained using the Bayesian spectral density approach in the first stage. Again, uncertainty of the stiffness parameters can be quantified with Bayesian inference. Finally, this Bayesian framework is applied to the 24-hour field measurements to investigate the variation of the modal and stiffness parameters under changing ambient conditions. Results show that the Bayesian framework successfully achieves the goal of the first task of this benchmark study.

Modal tracking has evolved into a crucial research domain over the past decades. It synergizes with automatic modal parameter estimation (AMPE) to achieve automatic operational modal analysis (AOMA). Modal tracking aims to establish the dynamic correspondence between the currently identified mode and historical reference modes, thereby following the evolution of modal parameters over time. Established modal tracking methods involve two stages: reference mode determination and current mode linking. This paper reviews the developments of modal tracking from both methodological and application perspectives, emphasizing the pivotal challenges in existing results and suggestions for future research. Additionally, a detailed summary of the distinct strategies employed across two stages and the application of modal tracking in various types of structures is provided. Future research should focus on the adaptive updating of reference modes and linking thresholds while enhancing the capability to eliminate spurious modes and track rapidly changing physical modes. This review seeks to assist researchers in efficiently grasping the advancements in modal tracking and in developing more suitable and robust methods for civil engineering structures in the future.

Modal parameters estimation plays an important role for structural health monitoring. Accurately estimating the modal parameters of structures is more challenging as the measured vibration response signals are contaminated with noise. This study develops a mathematical algorithm of solving the partially described inverse singular value problem (PDISVP) combined with the complex exponential (CE) method to estimate the modal parameters. The PDISVP solving method is to reconstruct an L2-norm optimized (filtered) data matrix from the measured (noisy) data matrix, when the prescribed data constraints are one or several sets of singular triplets of the matrix. The measured data matrix is Hankel structured, which is constructed based on the measured impulse response function (IRF). The reconstructed matrix must maintain the Hankel structure, and be lowered in rank as well. Once the filtered IRF is obtained, the CE method can be applied to extract the modal parameters. Two physical experiments, including a steel cantilever beam with 10 accelerometers mounted, and a steel plate with 30 accelerometers mounted, excited by an impulsive load, respectively, are investigated to test the applicability of the proposed scheme. In addition, the consistency diagram is proposed to exam the agreement among the modal parameters estimated from those different accelerometers. Results indicate that the PDISVP-CE method can significantly remove noise from measured signals and accurately estimate the modal frequencies and damping ratios.

A structure's health or level of damage can be monitored by identifying changes in structural or modal parameters. This research directly identifies changes in structural stiffness due to modelling error or damage for a post-tensioned pre-cast reinforced concrete frame building with rocking beam column connections and added damping and stiffness (ADAS) elements. A structural health monitoring (SHM) method based on adaptive least mean squares (LMS) filtering theory is presented that identifies changes from a simple baseline model of the structure. This method is able to track changes in the stiffness matrix, identifying when the building is (1) rocking, (2) moving in a hybrid rocking–elastic regime, or (3) responding linearly. Results are compared for two different LMS-based SHM methods using an L2 error norm metric. In addition, two baseline models of the structure, one using tangential stiffness and the second a more accurate bi-linear stiffness model, are employed. The impact of baseline model complexity is then delineated. The LMS-based methods are able to track the non-linearity of the system to within 15% using this metric, with the error due primarily to filter convergence rates as the structural response changes regimes while undergoing the El Centro ground motion record. The use of a bi-linear baseline model for the SHM problem is shown to result in error metrics that are at least 50% lower than those for the tangential baseline model. Errors of 5–15% with this L2 error norm are fairly stringent compared to the greater than 2 × changes in stiffness undergone by the structure, however, in practice the usefulness of the results is dependent on the resolution required by the user. The impact of sampling rate is shown to be negligible over the range of 200–1000Hz, along with the choice of LMS-based SHM method. The choice of baseline model and its level of knowledge about the actual structure is seen to be the dominant factor in achieving good results. The methods presented require 2.8–14.0 Mcycles of computation and therefore could easily be implemented in real time. Copyright © 2005 John Wiley & Sons, Ltd.