On the application of correlation function matrices in OMA

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.

A study and extension of second-order blind source separation to operational modal analysis

This research investigates the dynamic behavior of a railway bridge using both experimental and numerical methods. Field tests were conducted to capture the bridge response to live loading conditions with acceleration data collected via uniaxial accelerometers placed at critical locations along the structure. The dynamic characteristics, including the natural frequencies and mode shapes, were determined using two analytical techniques: Fast Fourier Transform (FFT) and Operational Modal Analysis (OMA). While FFT provides a frequency domain analysis, OMA enables the estimation of modal parameters, such as natural frequencies, mode shapes, and damping ratios, using the bridge's response to operational forces. The results revealed that the fundamental frequency obtained from the OMA (2.163 Hz) was higher than that obtained from the FFT (1.95 Hz) and the Finite Element Analysis (FEA) model (1.65 Hz). Additionally, the OMA produced mode shapes that were closely aligned with those predicted by the FEA, validating the accuracy of the numerical model. This study highlights the advantages of OMA over FFT, particularly the ability to capture mode shapes, and underscores the importance of integrating OMA with FEA for a comprehensive dynamic assessment of bridge structures. These findings contribute to the growing body of knowledge on structural monitoring and provide practical insights into improving bridge safety and performance.

A novel mode shape identification approach for structures having planes with rigid-like behavior

The problem of the optimal signal processing algorithm synthesizing in synthetic aperture radars equipped with a mode of power lines selection is solved by the maximum likelihood method. Polarization and Doppler differences are selected as the distinguishing features of power lines from the earth's surface. The useful signal at the output of the coherent receiver is specified in an analytical form with a complex envelope and high-frequency harmonic oscillations. The background radiation of the underlying surface is presented in stochastic form and is completely determined by the matrix of correlation functions. The internal noise of a multichannel receiver is described by a delta-correlated random Gaussian process with the same power spectral density in different observation channels. Due to the fact that the background radiation of the earth surface is correlated in time and at different polarizations the likelihood functional is recorded with the inverse matrix of inverse correlation functions. The obtained optimal algorithm consists in suppression of passive interference in decor-leveling filters, optimal coherent accumulation of trajectory signals for synthesizing the on-board antenna aperture and their optimal filtering in filters matched with the reference signal. Simultaneous use of Doppler and polarization differences between wanted signals and passive interference allows effectively allocate power lines against the background of reflections from the ground in multichannel synthetic aperture radars. The proposed sequence of operations on the received oscillations can be a functional addition to the operating modes of aerospace-based cognitive radars adapting in accordance with the covariance scattering matrix of the underlying surface known in advance or measured during remote sensing. The block diagram of a multichannel onboard polarization radar with synthesized antenna aperture is designed on the basis of the obtained results. The development of such a radar is relevant for a collision warning system for helicopter pilots especially in bad weather conditions and at night.

Nowadays Acoustic Modal Analysis often plays a significant role in the optimisation process of the acoustic parameters of all kinds of cavities, e.g. the interior of a car, a train, an airplane etc. A problem with the existing experimental Acoustic Modal Analysis techniques is that one has to use volume acceleration sources which interact with the acoustic system. The Operational Acoustic Modal Analysis (OAMA) technique doesn't suffer from this problem. An important disadvantage of the OAMA approach is that the estimated mode shapes are not correctly scaled. This scaling problem will be tackled in this paper. We will introduce a sensitivity based rescaling technique for the acoustic domain. Local volume changes of the cavity will make a correct estimate of the mode shapes possible. The technique will be validated by some simulations and experiments.

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.

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

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.

Cyclostationarity and the cepstrum for operational modal analysis of MIMO systems—Part II: Obtaining scaled mode shapes through finite element model updating

\nThis work presents a summary of the research study of operational modal analysis (OMA) application for condition monitoring of operating pipelines. Special focus is on the topicality of OMA for definition of the dynamic features of the pipeline (frequencies and mode shapes) in operation. The research was conducted using two operating laboratory models imitated a part of the operating pipeline. The results of finite-element modeling, identification of pipe natural modes and its modification under the influence of virtual failure are discussed. The work considers the results of experimental research of dynamic behavior of the operating pipe models using one of OMA techniques and comparing dynamic properties with the modeled data. The study results demonstrate sensitivity of modal shape parameters to modification of operating pipeline technical state. It is resumed about ability of operating pipeline condition monitoring by measuring dynamic deformations of the operating pipe and OMA techniques application for dynamic properties extraction.\n

Methods for scaling mode shapes determined by operational modal analysis (OMA) have been extensively investigated in the last years. A recent addition to the range of methods for scaling OMA mode shapes is the so-called OMAH technique, which is based on exciting the structure by harmonic forces applied by an actuator. By applying harmonic forces in at least one degree-of-freedom (DOF), and measuring the response in at least one response DOF, while using at least as many frequencies as the number of mode shapes to be scaled, the mode shape scaling (modal mass) of all modes of interest may be determined. In previous publications on the method the authors have proven that the technique is easy and robust to apply to both small scale and large scale structures. Also, it has been shown that the technique is capable of scaling highly coupled modes by using an extended multiple reference formulation. The present paper summarizes the theory of the OMAH method and gives recommendations of how to implement the method for best results. It is pointed out, as has been shown in previous papers, that the accuracy of the mode scaling is increased by using more than one response DOF, and by selecting DOFs with high mode shape coefficients. To determine the harmonic force and responses, it is recommended to use the three-parameter sine fit method. It is shown that by using this method, the measurement time can be kept short by using high sampling frequency and bandpass filtering whereas spectrum based methods require long measurement times. This means that even for structures with low natural frequencies, the extra measurement time for scaling the mode shapes can be kept relatively short.

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.

Modal identification based on the time–frequency domain decomposition of unknown-input dynamic tests

In order to identify the modal parameters of time invariant three-dimensional engineering structures with damping and small nonlinearity, a novel isometric feature mapping (Isomap)-based three-dimensional operational modal analysis (OMA) method is proposed to extract nonlinear features in this paper. Using this Isomap-based OMA method, a low-dimensional embedding matrix is multiplied by a transformation matrix to obtain the original matrix. We find correspondence relationships between the low-dimensional embedding matrix and the modal coordinate response and between the transformation matrix and the modal shapes. From the low-dimensional embedding matrix, the natural frequencies can be determined using a Fourier transform and the damping ratios can be identified by the random decrement technique or natural excitation technique. The modal shapes can be estimated from the Moore–Penrose matrix inverse of the low-dimensional embedding matrix. We also discuss the effects of different parameters (i.e., number of neighbors and matrix assembly) on the results of modal parameter identification. The modal identification results from numerical simulations of the vibration response signals of a cylindrical shell under white noise excitation demonstrate that the proposed method can identify the modal shapes, natural frequencies, and ratios of three-dimensional structures in operational conditions only from the vibration response signals.