Application of the least squares times domain algorithm for efficient modal parameter estimation

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 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.

A multivariate maximum likelihood procedure for the estimation of modal parameters is presented. The vibrating system is excited by a random force and sensors output only are used to estimate the natural frequencies and damping coefficients of the system. The method works in time domain and a vector autoregressive moving average (VARMA) process is used. The information about modal parameters is contained in the multivariate AR part, which is estimated using an iterative maximum likelihood algorithm. This algorithm uses a score technique and output data only. The order of the AR part is obtained via the Minimum Description Length associated with an instrumental variable procedure. Experimental results show the effectiveness of the method for model order selection and modal parameters estimation.

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.

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.

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.

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

In this paper, the ML-MM estimator, a multivariable frequency-domain maximum likelihood estimator based on a modal model formulation, will be represented and improved in terms of the computational speed and the memory requirements. Basically, the design requirements to be met in the ML-MM estimator were to have accurate estimate for both of the modal parameters and their confidence limits and, meanwhile, having a clear stabilization chart which enables the user to easily select the physical modes within the selected frequency band. The ML-MM method estimates the modal parameters by directly identifying the modal model instead of identifying a rational fraction polynomial model. In the ML-MM estimator, the confidence bounds on the estimated modal parameters (i.e., frequency, damping ratios, mode shapes, etc.) are derived directly by inverting the so-called Fisher information matrix and without using many linearization formulas that are normally used when identifying rational fraction polynomial-based models. Another advantage of the ML-MM estimator lies in its potential to overcome the difficulties that the classical modal parameter estimation methods face when fitting an FRF matrix that consists of many (i.e., 4 or more) columns, i.e., in cases where many input excitation locations have to be used in the modal testing. For instance, the high damping level in acoustic modal analysis requires many excitation locations to get sufficient excitation of the modes. In this contribution, the improved ML-MM estimator will be validated and compared with some other classical modal parameter estimation methods using simulated datasets and real industrial applications.

Two complete sets of records obtained from a strong-motion accclerograph array in the Gisborne Post Office building during earthquakes of 1982 and 1993 were analyzed by using systems identification techniques to estimate the modal parameters. Nonlinear responses of the soil-structure system in both horizontal and vertical directions were indicated by significant decreases in the estimated first mode frequencies for moderate to strong ground motion in 1993 compared to those for the small amplitude ground motion in 1982. The analysis of second floor and remote site records from a 1985 earthquake indicated that nonlinear responses also occurred then. Soil-structure interaction was found to be significant for both vertical and horizontal responses. Using a numerical model of the foundation-soil system, the foundation rocking frequency and first fixed-base modal frequency were estimated. It was found that frequencies of the building's first fixed-base mode, foundation translation and rocking all were reduced in the 1993 earthquake compared with those of the 1982 earthquake, indicating that nonlinear responses occurred in both the building and the soil in the 1993 earthquake. Possible evidence for a cutoff frequency at the soil site was also identified. Because of soil-structure interaction, the fixed-base horizontal response in the N44E direction was reduced by possibly 40-50% in the 1982 earthquake, and about 30% in the 1993 event, compared with the recorded first modal response of the soil-structure system.

This paper presents the use of piezoceramic transducers, Lead Zirconia Titanate (PZT) actuators and Polyvinylidene Fluoride (PVDF) sensors, for experimental modal testing of a simply supported plate. A series of rectangular-shaped PVDF films are evenly distributed over the plate and act as the sensing devices instead of traditional acceleration sensors. Pairs of rectangular PZT patches are bounded to opposite sides of the plate to form actuators and when they are excited by 180° out-of-phase stimulation, they will cause pure bending. The theoretical formulation of frequency response functions (FRFs) of the piezoceramic transducers is developed. The mode shape functions of the PZT actuator and the PVDF sensor are identified, respectively. Experiments are performed to obtain a column of FRF matrix. The structural modal parameters, including natural frequencies, modal damping ratios and mode shapes, can then be extracted by a modal parameter extraction method. Results show the modal parameters can be properly obtained and physically interpreted. This paper presents the concepts of smart structural testing (SST) with the use of piezoceramic transducers and leads to the applications of smart structures to health monitoring of structural systems.

A reduced-order method is proposed for estimating the modal parameters of structures. This method employs a matrix-free formulation of the rational Krylov projection. It constructs a reduced-order model by employing the transfer functions of the structure's near-field acoustic radiation at selected interpolation frequencies. The natural frequencies and modal damping ratios of the structure are derived from the eigenpairs of the reduced model, the computational cost is greatly reduced, and the valid range of the reduced model is discussed. The sound pressure transfer functions can be obtained from actual measurements, eliminating the need for the mass, stiffness, and damping matrices of the original system, which effectively mitigates errors introduced by simplifications and assumptions during the modelling process. Through tests of frequencies and modal damping ratios, non-physical modes introduced during the model simplification process are filtered out. The effectiveness of the method is validated through numerical examples.

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.

A two-stage, time-domain structural dynamics identification method, that circumvents some of the main drawbacks of current techniques, and combines good noise rejection properties with a low computational load and accurate modal parameter extraction, is proposed. In the first stage of the method a discrete model is obtained through a linear estimation scheme and a systematic and effective modeling strategy. In the second stage the modal parameters are evaluated according to a new procedure that is capable of detecting and correcting large errors that may be introduced into the modal parameter estimates through the discrete-to-continuous transformation. The proposed method yields consistent modal parameter estimates at a low computational load, and is well-suited for microcomputer implementation. Its effectiveness is finally demonstrated with both simulated and actual experimental data.

In this study, a frequency domain framework for estimating simultaneously the modal parameter and input load from the measured (modal) response of multi degree of freedom(MDOF) system is proposed. It is a technique to identify the modal and load model parameters, by theoretically developing a mathematical input model, and comparing it with the identified load from the assumed modal parameters. The proposed technique has inherent difference to the conventional modal identification in that it can effectively treat the shape of the load spectrum. In order to verify the estimation method proposed in this study, it was applied to a SDOF analytic model. From the simulation results, it was found that this technique has high reliability of modal parameter estimation and immunity to noise included in the response. It can be seen that it has robustness to the load model order and can stably estimate the model parameters. Furthermore, it is needed that this technique can be applied in various load conditions such as vortex induced load and buffeting load of full scale structure, later.

Damage detection and localization in civil or mechanical structures is a subject of active development and research. A few vibration‐based methods have been developed so far, requiring, for example, modal parameter estimates in the reference and damaged states of the investigated structure, and for localization in addition a finite element model. For structures in operation, temperature has been shown to be a major nuisance to the efficiency of such methods because the modal parameters are varying not only with damage but also due to temperature variations. For detection, a few rejection approaches have been developed. Besides the increased complexity, environmental variation is hardly taken into account in localization approaches. In this paper, we propose a sensitivity‐based correction of the identified modal parameters in the damaged state with respect to the temperature field in the reference state, on the basis of a sensitivity analysis with respect to temperature dependent Parameters of the finite element model in the reference state. The approach is then applied to the stochastic dynamic damage locating vector method, where its improved performance under nonuniform temperature variations is shown in a numerical application on a beam.