Multi-channel sub-Nyquist cross-spectral estimation for modal analysis of vibrating structures

This article assesses numerically the potential of two different spectral estimation approaches supporting non-uniform-in-time data sampling at sub-Nyquist average rates (i.e. below the Nyquist frequency) to reduce data transmission payloads in wireless sensor networks for operational modal analysis of civil engineering structures. This consideration relaxes transmission bandwidth constraints in wireless sensor networks and prolongs sensor battery life since wireless transmission is the most energy-hungry on-sensor operation. Both the approaches assume acquisition of sub-Nyquist structural response acceleration measurements and transmission to a base station without on-sensor processing. The response acceleration power spectral density matrix is estimated directly from the sub-Nyquist measurements, and structural mode shapes are extracted using the frequency-domain decomposition algorithm. The first approach relies on the compressive sensing theory to treat sub-Nyquist randomly sampled data assuming that the acceleration signals are sparse/compressible in the frequency domain (i.e. have a small number of Fourier coefficients with significant magnitude). The second approach is based on a power spectrum blind sampling technique considering periodic deterministic sub-Nyquist “multi-coset” sampling and treating the acceleration signals as wide-sense stationary stochastic processes without posing any sparsity conditions. The modal assurance criterion is adopted to quantify the quality of mode shapes derived by the two approaches at different sub-Nyquist compression rates using computer-generated signals of different sparsity and field-recorded stationary data pertaining to an overpass in Zurich, Switzerland. It is shown that for a given compression rate, the performance of the compressive sensing–based approach is detrimentally affected by signal sparsity, while the power spectrum blind sampling–based approach achieves modal assurance criterion >0.96 independently of signal sparsity for compression ratios as low as 22% the Nyquist rate. It is concluded that the power spectrum blind sampling–based approach reduces effectively data transmission requirements in wireless sensor networks for operational modal analysis, without being limited by signal sparsity and without requiring a priori assumptions or knowledge of signal sparsity.

Motivated by a need to reduce energy consumption in wireless sensors for vibration-based structural health monitoring (SHM) associated with data acquisition and transmission, this paper puts forth a novel approach for undertaking operational modal analysis (OMA) and damage localization relying on compressed vibrations measurements sampled at rates well below the Nyquist rate. Specifically, non-uniform deterministic sub-Nyquist multi-coset sampling of response acceleration signals in white noise excited linear structures is considered in conjunction with a power spectrum blind sampling/estimation technique which retrieves/samples the power spectral density matrix from arrays of sensors directly from the sub-Nyquist measurements (i.e., in the compressed domain) without signal reconstruction in the time-domain and without posing any signal sparsity conditions. The frequency domain decomposition algorithm is then applied to the power spectral density matrix to extract natural frequencies and mode shapes as a standard OMA step. Further, the modal strain energy index (MSEI) is considered for damage localization based on the mode shapes extracted directly from the compressed measurements. The effectiveness and accuracy of the proposed approach is numerically assessed by considering simulated vibration data pertaining to a white-noise excited simply supported beam in healthy and in 3 damaged states, contaminated with Gaussian white noise. Good accuracy is achieved in estimating mode shapes (quantified in terms of the modal assurance criterion) and natural frequencies from an array of 15 multi-coset devices sampling at a 70% slower than the Nyquist frequency rate for SNRs as low as 10db. Damage localization of equal level/quality is also achieved by the MSEI applied to mode shapes derived from noisy sub-Nyquist (70% compression) and Nyquist measurements for all damaged states considered. Overall, the furnished numerical results demonstrate that the herein considered sub-Nyquist sampling and multi-sensor power spectral density estimation techniques coupled with standard OMA and damage detection approaches can achieve effective SHM from significantly fewer noisy acceleration measurements.

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.

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.

In FE modeling of port crane structures, errors may be caused due to the simplification of complicated components, unclear physical properties, local approximation and inaccurate boundary conditions. So it is very important to update the FE model accuracy according to the experiment model and results. However there are two major problems when solving the model updating of port crane structure, the one is the difficulty of collecting the structure modal parameters with traditional modal experiment method because of the large scale of port crane structure, the other is mismatching problem while updating processing due to most of the model updating method currently constructing the objective function only by frequency. In this paper, an operational modal analysis is adopted to estimate the modal parameters of port crane structure based on the operational condition. A parametric model updating methodology of finite element (FE) model is proposed to solve port crane structure model updating: a series of design parameters of the FE model of port crane structure are selected as the main factors based on the sensitivity analysis, the objective function is built by considering the frequency correlation and the modal assurance criterion (MAC) together, and finally a zero-order/first-order combination algorithm is utilized to get the final accurate FE model of port crane structure. Research results showed that methodology proposed in this paper has satisfied updating accuracy for port crane structure, which verifies that the proposed method is suitable for model updating problem of this kind of structure.

The vibrational effect is one of the primary considerations during construction due to the ambient excitation from the surrounding environment. Thus, it is necessary to find the most dependable Finite Element (FE) model for the actual structure to improve its quality and life span on site. In this context, the study included a 3-story building model that was analyzed using finite element analysis to determine its dynamic properties such as natural frequency and mode shape to avoid the resonance effect. It is due to the importance of obtaining a reliable and accurate model by verifying with the results of the Operational Modal Analysis (OMA). In general, the goal of this study is to update the FE model so that it is close to the actual scaled model based on the previous OMA study. As a result, an initial linear FE model of the 3-story building scaled model was created using CAD software based on actual geometry to describe the structure's physical properties. The FE model was then imported into CAE software, where the boundary condition and estimated material properties were assigned to determine the effect of random vibration. Through the pairing of Finite Element Analysis results and previous studies, 9 natural frequencies and 9 mode shapes were extracted. The Modal Assurance Criterion (MAC) was used to compare the mode shape of FE results against the OMA to determine the degree of consistency between paired mode shapes. A model updating process was carried out to reduce the discrepancy between the methods. The uncertainties arising from the initial conditions have been discussed in terms of the stiffness of the material used. The updated model allows for an evaluation of the structure's current actions as well as the development of models for a wide range of potential future research scenarios.

A thermal protection system (TPS) is designed and fabricated to protect a hypersonic vehicle from extreme conditions. Good condition of the TPS panels is necessary for the next flight mission. A loose bolted joint is a crucial defect in a metallic TPS panel. This study introduces an experimental method to investigate the dynamic characteristics and state of health of a metallic TPS panel through an operational modal analysis (OMA). Experimental investigations were implemented under free-free supports to account for a healthy state, the insulation effect, and fastener failures. The dynamic deformations resulted from an impulse force were measured using a non-contact three-dimensional point tracking (3DPT) method. Using changes in natural frequencies, the damping ratio, and operational deflection shapes (ODSs) due to the TPS failure, we were able to detect loose bolted joints. Moreover, we also developed an in-house program based on a modal assurance criterion (MAC) to detect the state of damage of test structures. In a damage state, such as a loose bolted joint, the stiffness of the TPS panel was reduced, which resulted in changes in the natural frequency and the damping ratio. The calculated MAC values were less than one, which pointed out possible damage in the test TPS panels. Our results also demonstrated that a combination of the 3DPT-based OMA method and the MAC achieved good robustness and sufficient accuracy in damage identification for complex aerospace structures such as TPS structures.

Asymptotic identification uncertainty of well-separated modes in operational modal analysis with multiple setups

<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>

Operational modal analysis (OMA) is a procedure that allows the modal parameters of a structure to be extracted from the measured response to an unknown excitation generated during operation. Nonlinearity is inevitably and frequently encountered in OMA. The problem: The traditional OMA method based on linear modal theory cannot be applied to a nonlinear oscillation system. The solution: This paper aims to propose a nonlinear OMA method for nonlinear oscillation systems. The new OMA method is based on the following: (1) a self-excitation phenomenon is caused by nonlinear components; (2) the nonlinear normal modes (NNMs) of the system appear under a single-frequency harmonic excitation; and (3) using forced response data, the symbolic regression method (SR) can be used to automatically search for the NNMs of the system, whose modal parameters are implicit in the expression structure expressing each NNM. The simulation result of a three-degree-of-freedom (3-DOF) nonlinear system verifies the correctness of the proposed OMA method. Then, a disc-rod rotor model is considered, and the proposed OMA method’s capability is further evaluated.

Rotating machines are key components in energy generation processes, and faults can lead to shutdowns or catastrophes encompassing economic and social losses. Structural Health Monitoring (SHM) of structures in operation is successfully performed via Operational Modal Analysis (OMA), which has advantages over traditional methods. In OMA, white noise inputs lead to the accurate extraction of modal parameters without taking the system out of operation. However, this excitation condition is not easy to attain for rotating machines used in power generation, and OMA can provide inaccurate information. This research investigates the applicability of machine learning as a pre-stage of OMA to differentiate adequate from inadequate excitations and prevent inaccurate extraction of modal parameters. Data from a rotor system was collected under different conditions and OMA was applied. In a training stage, measurements were characterized by statistical features and K-means was used to determine which features provided information about the excitation condition, that is, which excitation was adequate to extract the rotor’s modal parameters via OMA. In a testing stage, data were successfully classified as adequate or not adequate for OMA, achieving 100% accuracy and revealing the technique’s potential to support SHM of rotating machines. The technique is extendable to other monitoring systems based on OMA.

The investigation of dynamic behavior of aerospace structures is a challenging task due to the interaction of different types of aerodynamic and structural parameters. The identification of operational loads is essential for accurate investigation of the fatigue life especially in case of composite structures. In the present work, a mixed experimental and numerical approach is used for indirect identification of operational loads for composite structures using model correlation and updating technique. In the first step, the operational modal analysis using the response of accelerometers is performed for an anisotropic spar in cantilever position by using poly reference least square complex frequency (pLSCF) algorithm. Model correlation and updating of the FE model with the operational mode shapes is performed using Modal Assurance Criteria (MAC). The updated FE model is used for extraction of system matrices using Frequency Response Functions (FRFs). The physical response of accelerometers and transfer functions in the form of FRFs from the FE are used for identification of the operational loads at different locations of the structure. Finally, the proposed approach is validated by applying the extracted loads on the FE model and comparing the response at different locations.

The identification of modal parameters of a machine tool structure, in service, is important to ensure stability and productivity during machining operations. The characterization can be carried out through an Operational Modal Analysis (OMA). However, in the presence of strong harmonic excitation, the application of OMA is not straightforward. To overcome this difficulty, the Transmissibility Function-Based (TFB) method is proposed. The major advantage of this approach is its independence from the excitation nature and its ability to separate structural poles from spurious ones. The main novelty of this paper lies in the investigation of the TFB approach to identify the modal properties of a machine tool, during machining operations. Identified modal model through an Experimental Modal Analysis (EMA) of the considered machine tool, at rest, presents our reference modal base to validate results obtained through the TFB approach. For a comparison purpose, the modified Enhanced Frequency Domain Decomposition (EFDD) method is also investigated. Both methods enable the identification of the modal properties under operational conditions, with a clear advantage to the TFB approach due to its ability to eliminate all of the preponderant harmonic components from the measured data without the need of any additional selection criteria. The TFB method is thus a reliable technique for the identification of modal parameters of a machine tool in operational conditions.

Structural health monitoring through Operational Modal Analysis based on output-only methods has already a long term successful history of applications in conventional civil engineering structures and, more recently, onshore and offshore bottom fixed wind turbines. Although the former application has been proven as a reliable and relevant source of information regarding the dynamic response of the structure and ultimately in the definition of operation strategies, its application to floating offshore structures is yet to be explored, a gap that is tackled in this work. Since the introduction of a floating device, typically associated with very low frequency motions, impacts the relevant modal properties for the tower design, a preliminary and simplified analytical approach is presented, where these effects are properly characterised. The derived expressions, that are ready for design purposes, are validated based on numerical simulations in OpenFAST. After that, experimental data from a full scale floating wind turbine is used as input for Operational Modal Analysis for the first time. Here, not only some key features that are intrinsically associated with these type of structures are unveiled, but also the applicability of output-only methods for dynamic identification is confirmed.

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.