Automated modal parameter estimation using correlation analysis and bootstrap sampling

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.

Modal analysis techniques are widely used to characterize vibrations in structures, vehicles, machinery, etc. Modal parameters obtained through modal analysis are useful for structural health monitoring and condition monitoring. The common goal is to ensure the integrity of a mechanical system through monitoring of different operational conditions. Operational conditions described by spatial measurements can for instance be used to observe the vibration characteristics of the system and its parts. Therefore, the main originality of this work is to experimentally investigate if the presence of a turbulent gas flow in seal geometries can facilitate operational modal analysis for a coupled rotor–foundation system. A consistent and automatic way of obtaining the vibration characteristics is necessary to be included in the monitoring schemes. Therefore, this work also proposes a method to automatically obtain modal parameters for a bed plate of a rotor–foundation system when the rotor is affected by active magnetic bearings and gas seals. The automatic modal parameter algorithm is built on the stabilization diagram of modal parameters obtained from the stochastic subspace identification. Intermediate results from the automatic modal parameter algorithm show that some of the distance measures i.e., dωn,d, dζ, dMACX, dλ, dMACXωn,d, dMACXζ, and dMACXλ, for modal parameters in a stabilization diagram are better suited than others when distinguishing between possible physical modes and certainly spurious poles. The frequency band of interest for the system is found to influence the number of possible physical modes identified by the automatic algorithm. Measurements from 13 roving accelerometers are combined with the automatic modal parameter algorithm to obtain the vibration characteristics of the rotor and bed plate. For different operational conditions in the seal geometry, seven modes are consistently identified as physical modes by the automatic modal parameter algorithm. The algorithm uses (i) a combination of a white noise disturbance pattern intentionally introduced into the system through the active magnetic bearings and (ii) the operational disturbance coming from the seals, non-intentionally introduced by the turbulent gas flow. The seven physical modes are also identified when (iii) the rotor–foundation system is only disturbed from the turbulent gas flow in the seals. The seven physical modes identified by the algorithm under conditions (i), (ii), and (iii) compare well to the vibration characteristics found in experiments via experimental modal analysis and impact hammer testing as well as via a mathematical model of the rotor–foundation system.

The present paper deals with the novel approach of filtering technique using hybrid of empirical mode decomposition technique with stabilization diagram, that autonomously implemented within Matlab. Noise or unwanted signal is always present in the data and a bad signal-to-noise can cause a severe error in modal parameter extraction. With the recent developments of automated procedures without user interaction for the operational modal analysis (OMA), the corrupted input signals turn out to be a big issue in obtaining reliable results of automated modal parameter identification. The appearance of noise or unwanted modes due to environmental effects could affect the actual structural modes selection. There is a significant issue regarding “noise” (or spurious) modes and eliminating them from the raw signal remains to be solved and requires a lot of interaction with an expert user. In the parametric modal analysis, oversizing of a modal model is usually performed to minimize the bias on modal estimates by getting all physical modes in the frequency range of interest and help to obtain a good model fit to the data. However, this will introduce noise modes. Thus, authors take advantage of tools, such as the stabilization diagram, to illustrate, and decide, if a mode is physical or not. This selection is not a trivial task, but it may be difficult and time consuming depending on the quality of data, the performance of the estimator and the experience of the user. Since the extensive interaction between tools and user is inappropriate for monitoring purposes, image clustering tool is introduced to separate the physical poles from the others with short response time and low computational efforts compared to the available clustering algorithm. Meanwhile, Empirical mode decomposition (EMD) is then introduced to break down a signal into various components without leaving the time domain and purposely used for filtering. These are a great combination as well as an effective procedure in producing a good input signal that free from unwanted modes which can cause disruptive decision making for the actual modes selection.

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 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.

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-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

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.

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.

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.

High-speed digital image correlation is a camera-based displacement measurement technique that has experienced increasing popularity for modal analysis. It is due to a high spatial density of measurement that provides an outstanding interpretation of mode shapes or operational deflection shapes. However, its sensitivity is typically lower than other techniques and transducers such as accelerometers or laser vibrometry. With these measurements, the estimation of any transfer function in the frequency domain is noisier in valley regions of the response, including anti-resonances. Therefore, an accurate analytical model of the response should be used for modal identification. Modal analysis algorithms perform an identification of the modal parameters and a reconstruction of the response based on the transfer function between force excitation and displacement response. This is called the frequency response function. However, force measurements are not available in many experiments, where the motion excitation is recorded instead. This is the case when the vibration is applied to the specimen through a motion in its base. This transfer function, also known as the transmissibility function, relates the motion of excitation and response, whose shape differs from the frequency response function. This difference in the response function is relevant in a relatively noisy measurement like using digital image correlation. Hence, in this work, a transformation of the transmissibility function into equivalent frequency response function, based on theoretical models, is performed prior to modal identification in base motion tests. More suitable experimental response functions are obtained, increasing the accurateness of the modal parameter estimation.KeywordsModal analysisTransmissibility functionsFull-field measurement