Search

Operational modal analysis (OMA) identifies the modal properties, e.g., natural frequencies, damping ratios and mode shapes, of a structure using ‘output-only’ ambient vibration data. Instrument noise need not be negligible in ambient vibration data, and it is often modelled statistically. Simple noise models, e.g., independent and identically distributed (i.i.d.) among data channels, are often used and are found to give reasonable results in typical applications, although there may be concerns for data with, e.g., low signal-to-noise (S/N) ratio, large difference in noise intensities or significant correlation among data channels. This work aims at investigating the effect of noise models on OMA performed with a Bayesian approach in the frequency domain. In addition to modal identification results, noise models are also assessed from a Bayesian evidence perspective. To enable the study, algorithms for efficient calculation of Bayesian statistics (most probable value and covariance matrix) are developed to account for general noise models that have not been considered in existing algorithms. As a further contribution to OMA theory, it is shown that, by a suitable transformation of data, an OMA problem with general noise model can be converted to one with i.i.d. noise model. Based on this analogy, asymptotic formulae for identification uncertainty of modal parameters, i.e., ‘uncertainty law’, have been developed. The theory reveals a definition for the modal S/N ratio that is an intuitive yet nontrivial generalisation of the existing formula for i.i.d. noise. The proposed objectives and methodology are investigated in a comprehensive study through synthetic, laboratory and field data. Nanyang Technological University Submitted/Accepted version The research presented in this work is supported by grant SUG/4 (04INS000618C120) from the Nanyang Technological University (NTU). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not reflect the views of the funders. The first author would like to acknowledge the graduate research scholarship offered by NTU.

The Bayesian FFT method has gained attention in operational modal analysis of civil engineering structures. Not only the most probable value (MPV) of modal parameters can be computed efficiently, but also the identification uncertainty can be rigorously quantified in terms of posterior covariance matrix. A recently developed fast algorithm for general multiple (possibly close) modes was found to work well in most cases, but convergence could be slow or even fail in challenging situations. The algorithm is also tedious to computer-code. Aiming at resolving these issues, an expectation-maximization (EM) algorithm is developed by viewing the modal response as a latent variable. The parameter-expansion EM and the parabolic-extrapolation EM are further adopted, allowing mode shape norm constraints to be incorporated and accelerating convergence, respectively. A robust implementation is provided based on the QR and Cholesky decompositions, so that the computation can be done efficiently and reliably. Empirical studies verify the performance of the proposed EM algorithm. It offers a more efficient and robust (in terms of convergence) alternative that can be especially useful when the existing algorithm has difficulty to converge. In addition, it opens a way to compute the MPV in the Bayesian FFT method for other unexplored cases, e.g., multi-mode multi-setup problem. Accepted version The financial support from the UK Engineering and Physical Sciences Research Council, United Kingdom (EP/N017897/1) is gratefully acknowledged. The start-up fund from Zhejiang University (130000- 171207704/018) is also acknowledged.

Operational modal analysis (OMA) has received tremendous interest from engineering fields in recent years. This paper develops a compact-bandwidth regularization approach for OMA within the signal processing framework. The key ingredient lies in the fact that a structural mode is always compact in the frequency domain and this results in the compact-bandwidth constraint for structural modes. To implicitly enforce the compact-bandwidth constraint, the compact-bandwidth regularization is introduced to conventional blind modal separation. Then, the alternating minimization algorithm is used to iteratively get the solution in the immediately-at-once or one-by-one manner. In proceeding so, the spectrum-peak-based rule is adopted for choice of initial modal parameters and the L-curve method in conjunction with a theoretically derived bound is applied to estimate a proper regularization parameter. Numerical examples and an experimental test case are studied along with comparison to some usual OMA approaches to see the performance and advantages of the proposed approach in modal identification.

It is difficult to detect local damage of a structure based on modal analysis. This paper presents a novel local damage detection method of membranes under ambient excitation, which combines Bayesian operational modal analysis (BOMA) and three-dimensional digital image correlation (3D-DIC). It is a noncontact and full-field dynamic method with no sensors attached on a test membrane. Advantages of BOMA and 3D-DIC methods are integrated in this work. Anomalies caused by local damage in mode shapes and curvature mode shapes can be explicitly observed. Moreover, a mode shape damage index (MSDI) is used to improve local damage detection, and structural damage can be identified in neighborhoods with high values of MSDIs. The methodology is applied, as a demonstration, to detect damage introduced by razor cuts in circular and rectangular membranes with different boundary conditions.

Unstable vibrations in machining operations, known as chatter, may cause damages to the tool, workpiece, or the machine tool. Chatter is usually avoided by selecting the appropriate spindle speed and depth of cut determined by the dynamic models of the machining process. Due to modeling uncertainties or variations of the process dynamics, chatter may happen even when stable cutting conditions are used in the process planning stage. Therefore, it is critical to detect chatter during the process and take corrective actions in the shortest possible time. The existing chatter detection methods process various signals such as sound, vibration, and force to extract chatter indicators during the process, however these methods detect chatter only after vibrations become unstable. In this paper, a new chatter detection method is presented based on the identification of the dynamics of turning processes using Operational Modal Analysis (OMA). Because this new method estimates the stability margin of the process, rather than the stability/instability of vibrations, it determines the level of stability of the cut before vibrations become unstable. The performance of the presented chatter detection method is studied using numerical simulations and machining experiments.

This paper presents and experimentally validates an augmented Kalman filter extended with a fixed-lag smoother for solving joint state and input estimation problems. Sparse acceleration measurements from a truck side skirt excited by road-induced vibrations from a vibration test track are analysed. The system model is obtained experimentally from an operational modal analysis, reducing modelling errors and avoiding the need for a finite element model and it serves itself as a numerical model. The motion of the truck component is estimated and the results are compared to those of a joint input-state estimation filtering algorithm, in addition to the actual measured motion. Both algorithms are tuned according to a novel process based on minimal a priori information concerning the system states and inputs. The focus of this work is to assess the robustness, performance, and tuning of the algorithms. Two sensor configurations are studied: one where the number of response measurement sensors is high compared to the number of estimated motions and participating modes, and another where the number of response measurements is reduced. Both algorithms perform very well within the first configuration. With a reduced number of response measurements, the fixed-lag smoother is superior to the joint input-state filter in capturing the individual motion of each position on the side skirt.

The quality of the final product of machining is strongly dependent on our ability to understand, control and suppress the harmful interplay between the machine and machining which leads to chatter. Therefore extraction of the dynamic information of the machine tool is of utmost importance. Unfortunately, these dynamic parameters are known to be varying between static test condition and operational cutting condition. Operational Modal Analysis (OMA) is a suite of powerful output only analysis techniques which obviates the above problem by performing modal extraction during actual cutting operation. This work uses a novel workpiece design which when flat milled, produces the necessary broadband white exciting force input for the milling machine. The present work also proposes a novel iterative technique which discriminates between structural response modes and tooth passing harmonics which are inherently present due to the rotating spindle. The harmonic indicator does this detection by exploiting the separation in response of the feed and transverse directions to form a differenced mean power discriminator spectrum. Performance of this indicator against a few other well-known discriminants is compared. Thereafter, the complete automated modal decomposition methodology is applied on a legacy vertical milling machine to assess the efficacy of the proposal.

This paper investigates the uncertainty propagation properties of a two-stage fast Bayesian spectral density approach (two-stage fast BSDA) with separated modes proposed previously by the authors Yan and Katafygiotis (2015a,b) [1,2], deriving explicit formulas for the dependence of the posterior coefficients of variation (c.o.v.) of the identified modal parameters in terms of different parameters influencing the identification. For this, an approximation analysis strategy proposed in Au (2014a,b) [3,4] is adopted. Although the explicit closed-form approximation expressions are relatively complex, the expressions for the approximate dependence of uncertainty are simple and informative. The analysis reveals a strong correlation among the prediction error, the damping ratio and the power spectral density (PSD) of the modal excitation. While similar correlation trends have been observed in the posterior uncertainty analysis of the fast Bayesian FFT (fast BFFT) approach Au (2014a) [3], the present method shows that the identification results are more sensitive to modeling error. Note that this is not a contradicting result, as the uncertainty propagation properties of different methods may generally differ. Note that fast BFFT is a more fundamental method, in the sense that it processes FFT data directly, while the two-stage fast BSDA uses spectral density data in a manner that allows for decoupling of the mode shape data. Validation studies using synthetic data and field data measured from a laboratory model provide a practical verification of the rationality and accuracy of the theoretical findings.

A well known issue when performing Operational modal analysis in the time domain is the challenge of choosing the right order of the system matrix. Setting the order higher than the rank of the system results in over fitting, i.e. having more poles to place than modes of the system. This results in fitting poles to noise or non linearities of the system. In This paper a method to avoid over fitting by reducing the physical channels to a reduced number of pseudo channels condensing the relevant physical information and leveling the level of energy in each channel is suggested.

A novel multi-sensor power spectrum blind sampling (PSBS) approach is proposed supporting low-power wireless sensor networks (WSN) for Operational Modal Analysis (OMA) applications. The developed approach relies on arrays of wireless sensors, employing deterministic non-uniform in time multi-coset sampling to acquire structural response acceleration signals at sub-Nyquist sampling rates, treated as realizations of stationary random processes without making any assumption about the average signal frequency content and spectral support. The acquired compressed measurements are transmitted to a central server and collectively processed via a PSBS technique, herein extended to the multi-sensor case, to estimate the power spectral density matrix of an underlying spatially correlated stationary response acceleration random process directly from the compressed measurements. Structural modal properties are then extracted through standard frequency domain decomposition (FDD). The efficacy of the proposed approach to resolve closely-spaced modes is numerically tested for various data compression levels using noisy response acceleration signals of a white-noise excited finite element model of a space truss as well as field-recorded acceleration time-histories of an instrumented bridge under operational loading. It is shown that accurate mode shapes based on the modal assurance criterion can be obtained from as low as 89% less measurements compared to conventional non-compressive FDD at Nyquist sampling rate. Further, significant gains in energy consumption and battery lifetime prolongation of the order of years are estimated, assuming wireless sensors operating on multi-coset sampling at different data compression levels. It is, therefore, concluded that the proposed PSBS approach could provide long-term structural health monitoring systems with low-maintenance cost once wireless sensors with multi-coset sampling capabilities become commercially available.

Blind source separation receives increasing attention as an alternative tool for operational modal analysis in civil applications. However, the implementations on real-life structures in literature are rare, especially in the case of using limited sensors. In this study, an enhanced version of sparse component analysis is proposed for output-only modal identification with less user involvement compared with the existing work. The method is validated on ambient and non-stationary vibration signals collected from two bridge structures with the working performance evaluated by the classic operational modal analysis methods, stochastic subspace identification and natural excitation technique combined with the eigensystem realisation algorithm (NExT/ERA). Analysis results indicate that the method is capable of providing comparative results about modal parameters as the NExT/ERA for ambient vibration data. The method is also effective in analysing non-stationary signals due to heavy truck loads or human excitations and capturing small changes in mode shapes and modal frequencies of bridges. Additionally, closely-spaced and low-energy modes can be easily identified. The proposed method indicates the potential for automatic modal identification on field test data.

In the ambient vibration test (AVT) of structures, when the number of desired locations for mode shapes to be measured is larger than that of available sensors, a ‘multi-setup’ strategy is often adopted. That is, sensors are ‘roved’ (placed) at different locations in different setups while ‘reference’ sensors are placed at common locations to provide information for estimating the ‘global’ mode shape covering all desired locations from ‘local’ information in individual setups. Currently, multi-setup AVTs are often planned based on experience. Naturally, questions on how to judge the performance of the test configuration are relevant, e.g., number and location of reference/rover sensors. This paper proposes metrics to quantify a multi-setup configuration by comparing the identification uncertainty of mode shapes with its best achievable one. The latter is informed by recent theoretical findings, i.e., ‘uncertainty law’ of multi-setup in operational modal analysis. Two indices are proposed to measure the overall configuration and the placement of reference sensors. They are also simplified to aid the design of multi-setup AVT. Two examples, a laboratory shear-type building model and a (full-scale) suspension footbridge, are presented to illustrate the usage of the proposed indices for performance quantification. The theory and examples reveal that the placement of reference sensors has a decisive influence. Notably, it is found that as long as the reference sensors are in ‘good’ positions, the identification uncertainty of the global mode shape is insensitive to the choice of rover sensors. Practically, and somewhat consistent with conventional wisdom, this suggests that reference sensors should be placed with due consideration to signal-to-noise ratio and the modes that can be detected, but rover sensors can be planned primarily based on logistics constraints/considerations. Submitted/Accepted version This research is supported by the Zhejiang Provincial Natural Science Foundation (LY21E080025) and the National Natural Science Foundation of China (51908494). The first author is also partially supported by Fundamental Research Funds for the Central Universities (2021XZZX040). The second author is supported by the National Key R&D Program of China (2019YFB2102702). The above financial support is gratefully acknowledged.

In operational modal analysis, identified modal parameters, e.g., natural frequencies, damping ratios, and mode shapes, are subject to uncertainties due to effects such as limited data, measurement noise, modelling error and unknown excitations. It becomes relevant to quantify the associated uncertainty for downstream analyses, e.g., finite element model updating and damage detection. Fast computation of uncertainty lower bounds of modal parameters via the Cramér-Rao bound is addressed in this study for an (asymptotically) unbiased estimator of the stochastic state-space model (SSM). Starting with a modal-form SSM, the Fisher information matrix (FIM) of the SSM parameters can be obtained analytically. Direct evaluation of such FIM is computationally prohibitive for a high-dimensional parameter space and long data, however, rendering it infeasible in practical applications. Various approximation schemes are proposed to accelerate the computation of the FIM, including a re-parameterisation via the innovations form to remove the singularity of FIM, introducing stationarity assumption to eliminate recursive calculations and mode clustering for a further speedup. The proposed methodology is applied to synthetic and field data, and verified by direct Monte Carlo simulation. Although the methodology is demonstrated for the uncertainty analysis of modal parameters based on the maximum likelihood estimator of SSM, it can also be used to lower bound the identification uncertainty of any unbiased estimator of SSM. Accepted version The first author would like to thank the funding support from the National Natural Science Foundation of China (51408383), and Science and Technology Department of Sichuan Province (18GJHZ0111). The second author would like to thank the funding support from the National Natural Science Foundation of China (51908494) and the start-up fund (130000-541902/022) from the Zhejiang University.

The workpiece dynamics are an important factor in the planning of machining strategy. Usually, the structural dynamic parameters of workpiece are obtained by experimental modal analysis (EMA). However, the dynamics of thin-walled workpiece are varying due to material removal and tool-workpiece coupling. Operational modal analysis (OMA) provides a route to estimate the structural dynamic parameters during operation, but the input excitation of milling system is mainly periodic milling force, which violates the premise of OMA. In this paper, an output-only modal identification method for the dynamic parameters of thin-walled workpiece is proposed. The milling force is analyzed and the analysis results show the milling force contains white noise for OMA. However, strong harmonic components in milling force seriously interfere the identification of dynamic parameters. To remove the harmonic components in response signal, a revised least-squares (LS) method is proposed to fit harmonics with multiple fundamental frequencies. After the harmonic removal, OMA is conducted in frequency domain using the poly-reference least squares complex frequency (p-LSCF) method based on positive power spectral density (PSD). To verify the feasibility of the proposed method, the simulation case is conducted by exciting a 3-degree of freedom structure using measured milling force signal, and the results show a good agreement with the theoretical values both in frequencies and damping ratios. Furthermore, a series of milling tests under different cutting parameters are conducted on a thin-walled workpiece, where thin-film sensors are adopted to measure the structural response. The results show the method can be successfully applied to extract the workpiece dynamic parameters. Compared with results of EMA, the structural frequencies increase slightly, while the damping ratios show a large promotion due to the process damping.

Harmonics are signal components with energy concentrated at single frequencies and are often of special interest in vibration analysis. Depending on the application, harmonics are often either examined or removed. In the present paper, we present a novel automated method for detecting harmonic components in a signal, viz. the periodogram ratio detection (PRD) method. The method identifies each harmonic in a spectrum or in a frequency band, if desired. The frequency of each harmonic is obtained by the PRD method which may be utilized to keep or remove harmonic components. This provides a high degree of versatility, that may be utilized in several applications within vibration analysis. Often, the process of removing harmonics is desired to be automated, e.g. for condition monitoring. An automated removal method is also presented in the present paper, viz. the automated frequency domain editing (AFDE), which is a combination of the PRD method and an automated removal method. The AFDE method removes little of the energy of the non-harmonic part of the signal by editing only a small number of frequency bins at each harmonic. Smearing of the harmonics due to variable speed conditions may require the pre-processing step of resampling the signal to the angle domain for an optimal detection of the harmonics. The harmonics may then be removed by the AFDE method in the order domain. The methods developed in the paper are applied and validated on three signals, illustrating their use in a variety of applications. The performance of the PRD and AFDE method improves by increasing the length of the signal, i.e. increasing measurement time. This means that the method is applicable to signals with poor signal-to-noise-ratio.

Many efforts have been made in the last decade to define Automated Operational Modal Analysis (AOMA) procedures able to process large datasets from long-term monitoring systems. However, some issues are still open and further studies on this topic are needed; in particular following points should be better investigated: i) controlling the tuning parameters to minimize the modelling errors, ii) defining suitable automatization method for processing large datasets and iii) validating the extracted modal parameters. These points are investigated in this paper with the aim of enhancing the current AOMA procedures based on the Stochastic Subspace Identification (SSI) techniques, and the following novelties are introduced: i) a minimization approach for the tuning of the initial parameters in the SSI algorithm, ii) a statistical method to automatically define the cut-off threshold in the hierarchical clustering phase, and iii) a Modal Quality Index (MQI) – ranging from 0 to 1 – to validate the identified modes. The above novelties represent key aspects that allow a real-time check of the modal parameters provided by a monitoring system within a Continuous Structural Health monitoring (CSHM) framework.

Operational Modal Analysis allows to assess the modal model of rotating machinery. However, it is needed to pre-process the measured vibration data such that the influence of the harmonic content is suppressed. In this paper, such pre-processing techniques are discussed. The main focus is on evaluating the effects of amplitude modulation, originating for example from speed and load oscillations in rotating machines, on the harmonic removal. Both frequency and time domain editing techniques are considered. To illustrate the need of taking these modulations into account, their effect on the performance of a classical frequency domain editing technique is first discussed. Afterwards, different methods are developed that are able to cope better with frequency and amplitude modulated harmonics. To this end, it is opted to extract the harmonic response in the time domain using complex demodulation. This response can either be used to directly perform the system identification, or it can be used to design filters. Two different filters are proposed with distinct goals. The first one directly sets the harmonic response to the level of the white noise response based on the demodulated signal, whereas the second one is designed to remove the effects of amplitude modulation. For the latter, it is shown that classical editing methods can afterwards be applied in a more efficient manner.

Transmission error (TE) is recognised as one of the best measurements for gear condition monitoring and quality control, especially when measured at low speed. Geometric transmission error (GTE) is closely related to geometric tooth-profile anomalies. The difference between GTE and static transmission error (STE) is instead informative of the meshing stiffness of a gear pair. However, GTE and STE can only be obtained by measuring TE at low speed. When measured at moderate or high operating speeds, dynamic transmission error (DTE) is affected by a few resonances related to internal gearbox elements and/or encoders and their couplings, which distort the effects of geometric tooth-profile and meshing stiffness. Utilising a cepstrum-based operational modal analysis (OMA) method, this paper proposed a novel method to reconstruct equivalent STE or GTE from a single DTE measurement, i.e. from a moderate/high speed test. When measurements are obtained with two or more load cases, this method also allows the separation and detailed identification of the TE components related to geometric mismatch and gearmesh stiffness. The proposed method has been validated on a single-stage spur gear rig and can be useful in gear condition monitoring, gear quality control and as an aid to gearbox modelling.

In full-scale ambient vibration tests, multiple setups are often performed for measurement when it is demanded to obtain a detailed mode shape with more measured degrees of freedom than the available number of synchronous data channels. In a previous work, a Bayesian operational modal analysis framework for the general case of multiple modes identified with ambient data from multiple setups was developed, together with an Expectation-Maximisation algorithm for efficiently calculating the most probable value (MPV) of modal parameters. Complementing the previous effort, this work investigates the posterior uncertainty of the modal parameters in terms of their posterior covariance matrix. Mathematically, the posterior covariance matrix is equal to the inverse of the Hessian of negative log-likelihood function at the MPV. The computational issues are investigated and analytical expressions for the Hessian matrix are derived, allowing the covariance matrix to be determined efficiently and accurately without resorting to the finite difference method. The proposed algorithm is verified with synthetic data, where the Bayesian and frequentist statistics are compared, and the effect of reference location is investigated. The developed computational tools are applied to investigate identification uncertainty with field data, where associated practical issues are also discussed. Nanyang Technological University Accepted version This work is funded by the UK Engineering & Physical Sciences Research Council (EP/N017897/1). The majority of the work was performed during the PhD study of the first author supported by the Joint University of Liverpool/China Scholarship Council Scholarship. The second author is currently supported by grant SUG/4 (C120032000) at the Nanyang Technological University, Singapore. The financial supports are gratefully acknowledged. The authors would like to thank Prof. James Brownjohn at the University of Exeter for providing the Jiangyin Bridge field data.

As one of the most important dynamic characteristic of the widely-used rotating cantilever beams, the centrifugal hardening effect has been extensively studied through theoretical and Finite Element Method (FEM) modeling, but rarely through operational experiments due to the lack of proper measurement techniques. As a result, the previous studies have not been verified convincingly. With the help of Digital Image Correlation and the newly proposed Phase Mapping Method, direct measurements under operational conditions were carried out on three rotating blade-like cantilever beams under 20 different rotating speeds (0–16.25 rps). With the abundant data, full-field displacement responses of the beams were presented and several orders of natural frequencies under different speeds were obtained and compared to results from simulations based on a typical theoretical model and a FEM model. The test results show centrifugal hardening behaviors of different cantilever beams on the flap-wise direction in detail and proved the applied models to be accurate in the prediction of frequency hardening with an average error of 0.401 Hz for the theoretical model or 0.404 Hz for the FEM model. In addition, the mode shapes with dense measurement points (up to 168 points on one beam) are presented and found to be coincided with the result from simulation.

Damage-induced local singularities in structural characteristic deflection shapes (CDS’s) are widely used in non-model-based damage localisation. Despite substantial advances in this kind of methods, several issues must be addressed to boost their efficiency and practical applications. This study deals with two essential problems of CDS-based damage localisation: the noise robustness of CDS estimation and the criterion to properly weight damage information of several CDS’s. On the first problem, it is well known that CDS estimation is vulnerably compromised by various uncertainties such as measurement noise and computational errors, which will decrease the accuracy and increase the difficulties in damage localisation. A modified common eigenvector analysis (CEA) is proposed based on a bank of digital filters and a joint approximate diagonalisation technique, which statistically estimates the CDS’s as the common eigenvectors of a set of covariance matrices. On the second problem, a new robust damage index (DI) is proposed, which is comprised of damage-caused local shape distortions of several CDS’s weighted by their participation factors in the covariance matrix at zero-time delay. The advantage of doing this is to include fair contributions from changes of all CDS’s concerned and the proposed DI provides a measure of damage-induced changes in the covariance matrix. Then a numerical study is presented to demonstrate the noise robustness of the modified CEA method over proper orthogonal decomposition and second-order blind identification in CDS estimation. Moreover, a comparison of the proposed DI over some traditional damage localisation methods is conducted based on an experimental study. The results of numerical and experimental studies demonstrate that the proposed CDS estimation method is more robust to noise and the proposed DI is highly accurate for multi-damage localisation.

This paper explores the use of wireless Inertial Measurement Units (IMU) originally developed for bio-mechanical research applications for modal testing of civil engineering infrastructure. Due to their biomechanics origin, these devices combine a triaxial accelerometer with gyroscopes and magnetometers for orientation, as well as on board data logging capability and wireless communication for optional data streaming and to coordinate synchronisation with other IMUs in a network. The motivation for application to civil structures is that their capabilities and simple operating procedures make them suitable for modal testing of many types of civil infrastructure of limited dimension including footbridges and floors while also enabling recovering of dynamic forces generated and applied to structures by moving humans. To explore their capabilities in civil applications, the IMUs are evaluated through modal tests on three different structures with increasing challenge of spatial and environmental complexity. These are, a full-scale floor mock-up in a laboratory, a short span road bridge and a seven story office tower. For each case, the results from the IMUs are compared with those from a conventional wired system to identify the limitations. The main conclusion is that the relatively high noise floor and limited communication range will not be a serious limitation in the great majority of typical civil modal test applications where convenient operation is a significant advantage over conventional wired systems.

Transmissibility-based operational modal analysis is a recent and alternative approach used to identify the modal parameters of structures under operational conditions. This approach is advantageous compared with traditional operational modal analysis because it does not make any assumptions about the excitation spectrum (i.e., white noise with a flat spectrum). However, common methodologies do not include a procedure to extract closely spaced modes with low signal-to-noise ratios. This issue is relevant when considering that engineering structures generally have closely spaced modes and that their measured responses present high levels of noise. Therefore, to overcome these problems, a new combined method for modal parameter identification is proposed in this work. The proposed method combines blind source separation (BSS) techniques and transmissibility-based methods. Here, BSS techniques were used to recover source signals, and transmissibility-based methods were applied to estimate modal information from the recovered source signals. To achieve this combination, a new method to define a transmissibility function was proposed. The suggested transmissibility function is based on the relationship between the power spectral density (PSD) of mixed signals and the PSD of signals from a single source. The numerical responses of a truss structure with high levels of added noise and very closely spaced modes were processed using the proposed combined method to evaluate its ability to identify modal parameters in these conditions. Colored and white noise excitations were used for the numerical example. The proposed combined method was also used to evaluate the modal parameters of an experimental test on a structure containing closely spaced modes. The results showed that the proposed combined method is capable of identifying very closely spaced modes in the presence of noise and, thus, may be potentially applied to improve the identification of damping ratios.

To adaptively identify the transient modal parameters for linear weakly damped structures with slow time-varying characteristics under unmeasured stationary random ambient loads, this paper proposes a novel operational modal analysis (OMA) method based on the frozen-in coefficient method and limited memory recursive principal component analysis (LMRPCA). In the modal coordinate, the random vibration response signals of mechanical weakly damped structures can be decomposed into the inner product of modal shapes and modal responses, from which the natural frequencies and damping ratios can be well acquired by single-degree-of-freedom (SDOF) identification approach such as FFT. Hence, for the OMA method based on principal component analysis (PCA), it becomes very crucial to examine the relation between the transformational matrix and the modal shapes matrix, to find the association between the principal components (PCs) matrix and the modal responses matrix, and to turn the operational modal parameter identification problem into PCA of the stationary random vibration response signals of weakly damped mechanical structures. Based on the theory of “time-freezing”, the method of frozen-in coefficient, and the assumption of “short time invariant” and “quasistationary”, the non-stationary random response signals of the weakly damped and slow linear time-varying structures (LTV) can approximately be seen as the stationary random response time series of weakly damped and linear time invariant structures (LTI) in a short interval. Thus, the adaptive identification of time-varying operational modal parameters is turned into decompositing the PCs of stationary random vibration response signals subsection of weakly damped mechanical structures after choosing an appropriate limited memory window. Finally, a three-degree-of-freedom (DOF) structure with weakly damped and slow time-varying mass is presented to illustrate this method of identification. Results show that the LMRPCA algorithm, which is robustness to Gauss measurement noise disturbances, can well identify the transient modal parameters (transient natural frequencies & transient modal shapes) for weakly damped and slow LTV structures online only from non-stationary random response signals.

Operational Modal Analysis (OMA) is a common technique used to examine the dynamic properties of a system. Contrary to experimental modal analysis, the input signal is generated in object ambient environment. Operational modal analysis mainly aims at determining the number of pole pairs and at estimating modal parameters. Many methods are used for parameter identification. Some methods operate in time while others in frequency domain. The former use correlation functions, the latter - spectral density functions. However, while some methods require the user to select poles from a stabilisation diagram, others try to automate the selection process. Dual ant colony operational modal analysis parameter estimation method (DAC-OMA) presents a new approach to the problem, avoiding issues involved in the stabilisation diagram. The presented algorithm is fully automated. It uses deterministic methods to define the interval of estimated parameters, thus reducing the problem to optimisation task which is conducted with dedicated software based on ant colony optimisation algorithm. The combination of deterministic methods restricting parameter intervals and artificial intelligence yields very good results, also for closely spaced modes and significantly varied mode shapes within one measurement point.

A new method for estimating the modal mass ratios of buildings from unscaled mode shapes identified from ambient vibrations is presented. The method is based on the Multi Rigid Polygons (MRP) model in which each floor of the building is ideally divided in several non-deformable polygons that move independent of each other. The whole mass of the building is concentrated in the centroid of the polygons and the experimental mode shapes are expressed in term of rigid translations and of rotations. In this way, the mass matrix of the building can be easily computed on the basis of simple information about the geometry and the materials of the structure. The modal mass ratios can be then obtained through the classical equation of structural dynamics. Ambient vibrations measurement must be performed according to this MRP models, using at least two biaxial accelerometers per polygon. After a brief illustration of the theoretical background of the method, numerical validations are presented analysing the method sensitivity for possible different source of errors. Quality indexes are defined for evaluating the approximation of the modal mass ratios obtained from a certain MRP model. The capability of the proposed model to be applied to real buildings is illustrated through two experimental applications. In the first one, a geometrically irregular reinforced concrete building is considered, using a calibrated Finite Element Model for validating the results of the method. The second application refers to a historical monumental masonry building, with a more complex geometry and with less information available. In both cases, MRP models with a different number of rigid polygons per floor are compared.

Operational modal analysis (OMA) is used to identify the modal parameters of a structure under operational conditions, which reflects the real boundary conditions and state-dependent parameters. For rotating structures under operational conditions, in addition to broadband excitations, the harmonic excitation is also present and even dominates the responses in some cases. Since only the responses are recorded in OMA, the first problem lies in separating the resonant frequencies from the harmonic excitation frequencies. Moreover, if the frequency of the harmonic excitation is close to, or coincides with a resonant frequency of the system, classical modal analysis procedures will fail to identify the modal parameters accurately. These issues are addressed in this paper and a method is presented for distinguishing harmonic frequencies and removing the harmonic effect using correlation functions of the responses. Firstly, the correlation functions are deduced for the system under the combination of broadband and harmonic excitations, yielding the sum of free decay response of the system and sinusoids with the same frequencies as the harmonic excitations. Then, the principle of the method is presented using a numerical study on a two degree-of-freedom system. Both the noise resistance and method comparison studies are conducted in the simulation case. Subsequently, the effectiveness of the method is demonstrated through experiments on a light-damped beam, where cases of different harmonic intensity and different harmonic frequencies are considered. Lastly, the proposed method is applied to a rotating blade to extract modal parameters in its operational state.

Vibration monitoring from strain data is a promising alternative to the more conventional acceleration-based monitoring because a dense measurement grid can be achieved at a relatively low cost and because strain mode shapes are more sensitive to local stiffness changes than displacement mode shapes. However, the feasibility of monitoring strain mode shapes of full-scale civil structures, where the operational dynamic strain levels are of very low amplitude and temperature changes can influence the modal characteristics, has remained an open question. The present work provides a proof of concept in which the deck of a steel tied arch railway bridge is instrumented with eighty Fiber-optic Bragg Grating strain sensors, multiplexed in four fibers, that are interrogated with a technique that achieves high accuracy and precision. For more than a year, the natural frequencies and strain mode shapes of ten modes have been automatically identified from operational strain time histories, with typical root-mean-square values of 0.01 microstrain, on an hourly basis. Furthermore, using these modal data, the influence of temperature fluctuations and that of a retrofitting of the hangers connecting the bridge deck and the arches, which took place during the monitoring period, are extensively investigated. Both have an influence on the overall stiffness of the bridge and therefore they result in clear changes in the natural frequencies. They do not have an influence on the local stiffness and therefore they do not influence the strain mode shapes, except when the retrofitting induces an interaction between previously well-separated modes.

It is not widely realised that the first paper on cepstrum analysis was published two years before the FFT algorithm, despite having Tukey as a common author, and its definition was such that it was not reversible even to the log spectrum. After publication of the FFT in 1965, the cepstrum was redefined so as to be reversible to the log spectrum, and shortly afterwards Oppenheim and Schafer defined the “complex cepstrum”, which was reversible to the time domain. They also derived the analytical form of the complex cepstrum of a transfer function in terms of its poles and zeros. The cepstrum had been used in speech analysis for determining voice pitch (by accurately measuring the harmonic spacing), but also for separating the formants (transfer function of the vocal tract) from voiced and unvoiced sources, and this led quite early to similar applications in mechanics. The first was to gear diagnostics (Randall), where the cepstrum greatly simplified the interpretation of the sideband families associated with local faults in gears, and the second was to extraction of diesel engine cylinder pressure signals from acoustic response measurements (Lyon and Ordubadi). Later Polydoros defined the differential cepstrum, which had an analytical form similar to the impulse response function, and Gao and Randall used this and the complex cepstrum in the application of cepstrum analysis to modal analysis of mechanical structures. Antoni proposed the mean differential cepstrum, which gave a smoothed result. The cepstrum can be applied to MIMO systems if at least one SIMO response can be separated, and a number of blind source separation techniques have been proposed for this. Most recently it has been shown that even though it is not possible to apply the complex cepstrum to stationary signals, it is possible to use the real cepstrum to edit their (log) amplitude spectrum, and combine this with the original phase to obtain edited time signals. This has already been used for a wide range of mechanical applications. A very powerful processing tool is an exponential “lifter” (window) applied to the cepstrum, which is shown to extract the modal part of the response (with a small extra damping of each mode corresponding to the window). This can then be used to repress or enhance the modal information in the response according to the application.

Due to the independence of input spectra, the transmissibility-based operational modal analysis (OMA) methods have been developed rapidly over the last two decades. The response transmissibility (RT) methods can extract the modal parameters from multiple different load cases while the power spectral density transmissibility (PSDT) methods just use one single load case by combining multiple transfer outputs. Their ability to identify modal parameters under non-white excitations has been claimed, verified and applied by many researches. However, some problems, such as the robustness under different load conditions, the extent of noise influence and the application conditions, still arise when they are applied in the real field. Moreover, the comparison between different transmissibility-based methods on the identification accuracy and robustness is rare, which is significant for users to select the very method for their tasks. Four representative transmissibility-based OMA methods are considered and compared in this paper. The influence factors of RT and PSDT functions are analyzed first and the results reveal the requirements of the load case for four OMA methods. A numerical example is used to assess and compare the robustness of four methods to measurement noises, damping ratio of non-white noise excitations, non-fully-excited load cases and harmonic excitations, and their identification ability is further evaluated through a real field example. The results show that the transmissibility-based OMA methods are independent of input spectra only under certain conditions and some user-defined parameters play important roles in their identification ability.

A novel three-dimensional (3D) continuously scanning laser Doppler vibrometer (CSLDV) system that contains three CSLDVs and an external controller is developed to measure 3D vibration of a structure under random excitation, and a new operational modal analysis method is proposed to estimate its 3D modal parameters, including damped natural frequencies and undamped end-to-end mode shapes, by extending the conventional demodulation method. Calibration among three CSLDVs in the 3D CSLDV system based on the geometrical model of its scan mirrors is conducted to adjust their rotational angles and ensure that three laser spots can continuously and synchronously move along the same scan path on the structure. The extended demodulation method can estimate undamped mode shapes of the structure under random excitation by filtering raw response with a bandpass filter and multiplying the filtered response by sinusoidal signals with its damped natural frequencies obtained from the fast Fourier transform of raw response. Experimental investigation on one-dimensional (1D) and 3D CSLDV measurements for modal parameter estimation is conducted on a beam under white-noise excitation to validate the extended demodulation method and examine the accuracy of the 3D CSLDV system. Modal assurance criterion (MAC) values between the first four undamped mode shapes of the beam from 1D CSLDV measurement and corresponding damped mode shapes from 1D step-wise scanning measurement are larger than 95%. MAC values between the first four undamped mode shapes of the beam from 3D CSLDV measurement and corresponding damped mode shapes from 3D step-wise scanning measurement are larger than 90% in all the three directions of a specified measurement coordinate system.

Accurate knowledge of wind turbine tower vibration damping is essential for the estimation of fatigue life. However, the responses in the fore-aft and side-side directions are coupled through the wind-rotor interaction under operational conditions. This causes energy transfers and complicates aerodynamic damping identification using conventional damping ratios. Employing a reduced two-degree of freedom wind turbine model developed in this paper, this coupling can be accurately expressed by an unconventional aerodynamic damping matrix. Simulated time series obtained from this model were successfully verified against the outputs from the wind turbine simulation tool FAST. Based on the reduced system obtained, a matrix-based identification method is proposed to identify the aerodynamic damping for numerically simulated wind turbine tower responses. Applying harmonic excitations to the tower allowed the frequency response functions of the wind turbine system to be obtained and the aerodynamic damping matrix to be extracted. Results from this identification were compared to traditional operational modal analysis methods including standard and modified stochastic subspace identification. The damping in the fore-aft direction was successfully identified by all methods, but results showed that the identified damping matrix performs better in capturing the aerodynamic damping and coupling for the side-side responses.

The gear mesh vibration caused by planetary geartrains is modulated. Small planetary gearboxes, commonly used in servo drives, are examined experimentally and numerically. The goal is to link the modulation of the gearbox vibration to the deviations from the ideal geometry caused by manufacturing and assembly. Operational modal analysis shows that, while the dynamical behavior of nominally identical gearboxes is very similar, their modulation is not. The modulation also strongly depends on the operating conditions, such as speed and torque. Therefore, the modulation is aggregated to a vibration signature. In the vibration signature, distinct features are present. These can be linked to the eccentricity of the gears using numerical multi-body simulation and correlation analysis.

An optically based sensing system that can measure the displacement and strain over essentially the entire area of a utility-scale blade leads to a measurement system that can significantly reduce the time and cost associated with traditional instrumentation. This paper evaluates the performance of conventional three dimensional digital image correlation (3D DIC) and three dimensional point tracking (3DPT) approaches over the surface of wind turbine blades and proposes a multi-camera measurement system using dynamic spatial data stitching. The potential advantages for the proposed approach include: (1) full-field measurement distributed over a very large area, (2) the elimination of time-consuming wiring and expensive sensors, and (3) the need for large-channel data acquisition systems. There are several challenges associated with extending the capability of a standard 3D DIC system to measure entire surface of utility scale blades to extract distributed strain, deflection, and modal parameters. This paper only tries to address some of the difficulties including: (1) assessing the accuracy of the 3D DIC system to measure full-field distributed strain and displacement over the large area, (2) understanding the geometrical constraints associated with a wind turbine testing facility (e.g. lighting, working distance, and speckle pattern size), (3) evaluating the performance of the dynamic stitching method to combine two different fields of view by extracting modal parameters from aligned point clouds, and (4) determining the feasibility of employing an output-only system identification to estimate modal parameters of a utility scale wind turbine blade from optically measured data. Within the current work, the results of an optical measurement (one stereo-vision system) performed on a large area over a 50-m utility-scale blade subjected to quasi-static and cyclic loading are presented. The blade certification and testing is typically performed using International Electro-Technical Commission standard (IEC 61400-23). For static tests, the blade is pulled in either flap-wise or edge-wise directions to measure deflection or distributed strain at a few limited locations of a large-sized blade. Additionally, the paper explores the error associated with using a multi-camera system (two stereo-vision systems) in measuring 3D displacement and extracting structural dynamic parameters on a mock set up emulating a utility-scale wind turbine blade. The results obtained in this paper reveal that the multi-camera measurement system has the potential to identify the dynamic characteristics of a very large structure.

In this paper the theoretical solution for the correlation function matrix of the random response of a structural system is re-visited. It is shown that using the classical definition of the correlation functions, the row space is defined by the mode shapes of the system, whereas the column space is defined by the modal participation vectors. This means that only the rows can be used for unbiased modal identification in operational modal analysis and if the columns are used for identification, then bias will be introduced on the mode shape estimates. It is pointed out that the mode shape bias is strongly dependent on the frequency distance between the modes, i.e. bias will significantly increase in case of closely spaced modes. The identification errors on the estimated biased and unbiased mode shapes are studied in a simulation example.

The presence of damage in a civil structure alters its stiffness and consequently its modal characteristics. The identification of these changes can provide engineers with useful information about the condition of a structure and constitutes the basic principle of the vibration-based structural health monitoring. While eigenfrequencies and mode shapes are the most commonly monitored modal characteristics, their sensitivity to structural damage may be low relative to their sensitivity to environmental influences. Modal strains or curvatures could offer an attractive alternative but current measurement techniques encounter difficulties in capturing the very small strain (sub-microstrain) levels occurring during ambient, or operational excitation, with sufficient accuracy. This paper investigates the ability to obtain sub-microstrain accuracy with standard fiber-optic Bragg gratings using a novel optical signal processing algorithm that identifies the wavelength shift with high accuracy and precision. The novel technique is validated in an extensive experimental modal analysis test on a steel I-beam which is instrumented with FBG sensors at its top and bottom flange. The raw wavelength FBG data are processed into strain values using both a novel correlation-based processing technique and a conventional peak tracking technique. Subsequently, the strain time series are used for identifying the beam's modal characteristics. Finally, the accuracy of both algorithms in identification of modal characteristics is extensively investigated.

The field non-destructive vibration test plays an important role in the area of structural health monitoring. It assists in monitoring the health status and reducing the risk caused by the poor performance of structures. As the most economic field test among the various vibration tests, the ambient vibration test is the most popular and is widely used to assess the physical condition of a structure under operational service. Based on the ambient vibration data, modal identification can help provide significant previous study for model updating and damage detection during the service life of a structure. It has been proved that modal identification works well in the investigation of the dynamic performance of different kinds of structures. In this paper, the objective structure is a high-rise multi-function office building. The whole building is composed of seven three-story structural units. Each unit comprises one complete floor and two L shaped floors to form large spaces along the vertical direction. There are 56 viscous dampers installed in the building to improve the energy dissipation capacity. Due to the special feature of the structure, field vibration tests and further modal identification were performed to investigate its dynamic performance. Twenty-nine setups were designed to cover all the degrees of freedom of interest. About two years later, another field test was carried out to measure the building for 48 h to investigate the performance variance and the distribution of the modal parameters. A Fast Bayesian FFT method was employed to perform the modal identification. This Bayesian method not only provides the most probable values of the modal parameters but also assesses the associated posterior uncertainty analytically, which is especially relevant in field vibration tests arising due to measurement noise, sensor alignment error, modelling error, etc. A shaking table test was also implemented including cases with and without dampers, which assists in investigating the effect of dampers. The modal parameters obtained from different tests were investigated separately and then compared with each other.

Experimental or operational modal analysis traditionally requires physically-attached wired or wireless sensors for vibration measurement of structures. This instrumentation can result in mass-loading on lightweight structures, and is costly and time-consuming to install and maintain on large civil structures, especially for long-term applications (e.g., structural health monitoring) that require significant maintenance for cabling (wired sensors) or periodic replacement of the energy supply (wireless sensors). Moreover, these sensors are typically placed at a limited number of discrete locations, providing low spatial sensing resolution that is hardly sufficient for modal-based damage localization, or model correlation and updating for larger-scale structures. Non-contact measurement methods such as scanning laser vibrometers provide high-resolution sensing capacity without the mass-loading effect; however, they make sequential measurements that require considerable acquisition time. As an alternative non-contact method, digital video cameras are relatively low-cost, agile, and provide high spatial resolution, simultaneous, measurements. Combined with vision based algorithms (e.g., image correlation, optical flow), video camera based measurements have been successfully used for vibration measurements and subsequent modal analysis, based on techniques such as the digital image correlation (DIC) and the point-tracking. However, they typically require speckle pattern or high-contrast markers to be placed on the surface of structures, which poses challenges when the measurement area is large or inaccessible. This work explores advanced computer vision and video processing algorithms to develop a novel video measurement and vision-based operational (output-only) modal analysis method that alleviate the need of structural surface preparation associated with existing vision-based methods and can be implemented in a relatively efficient and autonomous manner with little user supervision and calibration. First a multi-scale image processing method is applied on the frames of the video of a vibrating structure to extract the local pixel phases that encode local structural vibration, establishing a full-field spatiotemporal motion matrix. Then a high-spatial dimensional, yet low-modal-dimensional, over-complete model is used to represent the extracted full-field motion matrix using modal superposition, which is physically connected and manipulated by a family of unsupervised learning models and techniques, respectively. Thus, the proposed method is able to blindly extract modal frequencies, damping ratios, and full-field (as many points as the pixel number of the video frame) mode shapes from line of sight video measurements of the structure. The method is validated by laboratory experiments on a bench-scale building structure and a cantilever beam. Its ability for output (video measurements)-only identification and visualization of the weakly-excited mode is demonstrated and several issues with its implementation are discussed.

The interest for robust automatic modal parameter extraction techniques has increased significantly over the last years, together with the rising demand for continuous health monitoring of critical infrastructure like bridges, buildings and wind turbine blades. In this study a novel, multi-stage clustering approach for Automated Operational Modal Analysis (AOMA) is introduced. In contrast to existing approaches, the procedure works without any user-provided thresholds, is applicable within large system order ranges, can be used with very small sensor numbers and does not place any limitations on the damping ratio or the complexity of the system under investigation. The approach works with any parametric system identification algorithm that uses the system order n as sole parameter. Here a data-driven Stochastic Subspace Identification (SSI) method is used. Measurements from a wind tunnel investigation with a composite cantilever equipped with Fiber Bragg Grating Sensors (FBGSs) and piezoelectric sensors are used to assess the performance of the algorithm with a highly damped structure and low signal to noise ratio conditions. The proposed method was able to identify all physical system modes in the investigated frequency range from over 1000 individual datasets using FBGSs under challenging signal to noise ratio conditions and under better signal conditions but from only two sensors.

Continuously identified natural frequencies of vibration can provide unique information for low-cost automated condition assessment of civil constructions and infrastructures. However, the effects of changes in environmental parameters, such as temperature and humidity, need to be effectively investigated and accurately removed from identified frequency data for an effective performance assessment. This task is particularly challenging in the case of historical constructions that are typically massive and heterogeneous masonry structures characterized by complex variations of materials' properties with varying environmental parameters and by a differential heat conduction process where thermal capacity plays a major role. While there is abundance of documented monitoring data highlighting correlations between environmental parameters and natural frequencies in the case of new structures, such as long-span bridges, similar studies for historical constructions are still missing, with only a few literature works occasionally reporting increments in natural frequencies with increasing temperature of construction materials due to the closure of internal micro-cracks in the mortar layers caused by thermal expansion. In order to gain some knowledge on the effects of changes in temperature and humidity on the natural frequencies of slender masonry buildings, the paper focuses on the case study of an Italian monumental bell tower that has been monitored by the authors for more than nine months. Correlations between natural frequencies and environmental parameters are investigated in detail and the predictive capabilities of linear statistical regressive models based on the use of several environmental continuous monitoring sensors are assessed. At the end, three basic mechanisms governing environmentally-induced changes in the dynamic behavior of the tower are identified and essential information is achieved on the optimal location and minimum number of environmental sensors that are necessary in a structural health monitoring perspective.

Experimental modal analysis aims at identifying the modal properties (e.g., natural frequencies, damping ratios, mode shapes) of a structure using vibration measurements. Two basic questions are encountered when operating in the frequency domain: Is there a mode near a particular frequency? If so, how much spectral data near the frequency can be included for modal identification without incurring significant modeling error? For data with high signal-to-noise (s/n) ratios these questions can be addressed using empirical tools such as singular value spectrum. Otherwise they are generally open and can be challenging, e.g., for modes with low s/n ratios or close modes. In this work these questions are addressed using a Bayesian approach. The focus is on operational modal analysis, i.e., with ‘output-only’ ambient data, where identification uncertainty and modeling error can be significant and their control is most demanding. The approach leads to ‘evidence ratios’ quantifying the relative plausibility of competing sets of modeling assumptions. The latter involves modeling the ‘what-if-not’ situation, which is non-trivial but is resolved by systematic consideration of alternative models and using maximum entropy principle. Synthetic and field data are considered to investigate the behavior of evidence ratios and how they should be interpreted in practical applications.

The authors have previously shown that many traditional approaches to operational modal analysis (OMA) struggle to properly identify the modal damping ratios for bridges under traffic loading due to the interference caused by the driving frequencies of the traffic loads. This paper presents a novel methodology for modal parameter estimation in OMA that overcomes the problems presented by driving frequencies and significantly improves the damping estimates. This methodology is based on finding the power spectral density (PSD) of a given modal coordinate, and then dividing the modal PSD into separate regions, left- and right-side spectra. The modal coordinates were found using a blind source separation (BSS) algorithm and a curve-fitting technique was developed that uses optimization to find the modal parameters that best fit each side spectra of the PSD. Specifically, a pattern-search optimization method was combined with a clustering analysis algorithm and together they were employed in a series of stages in order to improve the estimates of the modal damping ratios. This method was used to estimate the damping ratios from a simulated bridge model subjected to moving traffic loads. The results of this method were compared to other established OMA methods, such as Frequency Domain Decomposition (FDD) and BSS methods, and they were found to be more accurate and more reliable, even for modes that had their PSDs distorted or altered by driving frequencies.

The Modal Assurance Criterion (MAC) is a modal indicator designed to decide whether the mode shapes used in its computation are corresponding to the same mode. During structural monitoring, it can be applied to evaluate changes in the mode shapes. When the mode shapes are estimated from measurement data, the MAC inherits their statistical properties, thus is afflicted with statistical uncertainty. The evaluation of this uncertainty is particularly relevant when the MAC estimate is close to 1, where 1 indicates equal mode shapes. In structural monitoring, it can be used to assess changes in mode shapes after early damage. While the framework for uncertainty quantification of modal parameters is well-known and developed in the context of subspace-based system identification methods, uncertainty quantification for the MAC has not been developed yet. A particular challenge for its statistical characterization is its boundedness in the interval between 0 and 1. In this paper it is shown that this boundedness yields two different distributions of the MAC estimates. The MAC computed between estimates of different mode shapes is inside the interval ( 0 , 1 ) , and a Gaussian approximation of its distribution is obtained. When the MAC is computed between estimates of equal mode shapes, the resultant MAC estimate is close to 1, and the classical Gaussian approximation is inadequate. In this case it is shown that the MAC estimate is linked to a quadratic form of the mode shapes, whose distribution can be approximated by a scaled and shifted χ 2 distribution. For both cases, uncertainty bounds related to the MAC estimates are established. The proposed frameworks are validated by extensive Monte Carlo simulations and then applied to evaluate mode shape changes due to damage during monitoring of the S101 Bridge.

Operational Modal Analysis (OMA) is a widely used class of tools to identify the modal properties of existing civil engineering structures and mechanical systems under ambient vibrations or normal operating conditions by only employing measured responses. The OMA techniques, however, are confined to estimate modal properties of linear and time-invariant systems, which do not always comply with real life applications. Additionally, even though structures are found to be nonlinear and nonstationary (i.e., time-varying), it is often engineering practice to apply linear models due to their straightforward insight into the dynamic behaviour of the considered system and mathematical properties when compared to more time consuming consideration of nonlinearities and nonstationarities. With that point of view, the concept of evaluating a best linear approximation of a nonlinear response is well established within Experimental Modal Analysis, i.e., deterministic modal analysis where both the system excitation and the corresponding response are measured. Along these lines, the paper takes the first step towards extending the concept of a best linear approximation to correlation-driven OMA in terms of estimated natural frequencies, damping ratios, and mode shapes. This paper proves that the correlation function matrix calculated from the measured response of a nonlinear and nonstationary system can be approximated by a set of free decays of the best linear approximation. In other words, the paper derives and shows through appropriately designed numerical simulations that the linear modal model, estimated using the output-only framework of conventional correlation-driven (i.e., covariance-driven) OMA, leads to suboptimal minimisation of the difference between the true response of a nonlinear and/or nonstationary system and the response of the linear modal model in a least squares sense. Moreover, the paper demonstrates, using an additional numerical simulation case study, that the output-only framework causes an underestimation of the error that is inevitably involved when estimating linear modal properties from a nonlinear and nonstationary response.

A continuously scanning laser Doppler vibrometer (CSLDV) system is capable of rapidly obtaining spatially dense vibration measurement by continuously sweeping its laser spot along a path on a structure surface. This paper presents a new operational modal analysis (OMA) method for a rotating structure based on a rigorous rotating beam vibration theory, an image processing method, and a data processing method called the lifting method. A novel tracking CSLDV (TCSLDV) system was developed in this work to track and scan a rotating structure, and the real-time position of the rotating structure can be determined by image processing so that the TCSLDV system is capable of tracking a time-varying scan path on the rotating structure. The lifting method can transform raw TCSLDV measurement into measurements at multiple virtual measurement points as if they were measured by transducers attached to these measurement points. Modal parameters of the rotating structure with a constant speed, including damped natural frequencies, undamped mode shapes, and modal damping ratios, and operating deflection shapes (ODSs) of the structure with a constant or prescribed time-varying rotation speed can be determined by calculating and analyzing correlation functions with non-negative time delays among measurements at virtual measurement points. Experimental investigation is conducted using the TCSLDV system to study the OMA method with which modal parameters and an ODS of a rotating fan blade with different constant speeds, as well as an ODS of the rotating fan blade with a non-constant speed, are successfully estimated.

Industrial machinery is often equipped with proximity probes that measure continuously position and vibrations of the shaft, generally relative to the bearings. These measurements are used for surveillance of the machine, and furnish valuable information to be used in a diagnostic process, especially when run-up or run-down transients are recorded and processed. However, a question arises, during normal operating conditions at constant rotating speed can some additional information about the dynamical behavior of the rotating shaft be extracted from these measurements? Considering that in an industrial environment the shaft vibrations are generally affected by some noise transmitted through the supporting structure by other machines operating in the considered industrial plant, where also fluids are moving in pipes and vessels, all generating vibrations, operational modal analysis (OMA) could be used for identifying natural frequencies and modal damping. And which are the conditions of the vibration measurements required to get reliable results from processing the recorded signals? In this context, the present paper aims to analyze, with the aid of a test rig, the conditions in which OMA will result successful. These conditions include both the operating condition of the shaft and the amount and quality of the noise. The test rig, where also noise has been applied to the rotating shaft by means of a magnetic actuator, furnishes some test runs that are analyzed and used for tuning a model that can then be used to explore the different conditions.

A crucial aspect in system identification is the assessment of the accuracy of the identified system matrices. Stochastic Subspace Identification (SSI) is a widely used approach for the identification of linear systems from output-only data because it combines a high computational robustness and efficiency with a high estimation accuracy. Practical approaches for estimating the (co)variance of system matrices that are identified using SSI exist for the case where the system matrices are obtained from the shift-invariant structure of the extended observability matrix. However, in data-driven SSI, the system matrices are often obtained in a different way, using identified state sequences. This case is treated in the present work, for three common types of weighting. First, it is shown that the estimated system matrices depend entirely on sample output correlation estimates, the covariance of which can be straightforwardly estimated. Subsequently, a linear sensitivity analysis of the data-driven SSI algorithm is performed, such that the covariance of the identified system matrices can be also computed. A memory efficient implementation is obtained by computing the related Jacobian only implicitly. An extensive numerical validation, covering a range of parameter choices, demonstrates the accuracy of the estimated variance of the identified system description. Finally, the practical use of the method in the context of operational modal analysis is demonstrated in an experimental case study.

In this paper, broad-band phase-based motion magnification (BPMM) is used to improve the modal parameter estimation from high-speed video of a structure undergoing low amplitude vibration. The authors use a novel application of the phase-based motion magnification technique to obtain improved results in the presence of image noise. A numerical simulation is performed to demonstrate and quantify the effect of broad-band motion magnification. A clear correlation is found between the magnification factor and the error in the motion obtained from 2-dimensional point tracking (2DPT). In a laboratory experiment, operational modal analysis is performed on a metallic nozzle with a 5.3 in exit diameter and a constant wall thickness of 0.03 inches. Fluorescent markers are painted on the nozzle lip. The nozzle is attached to a test stand and excited by pressurized air. The induced vibrations are captured by a single high-speed camera, which takes images of the nozzle lip. Using a 2DPT algorithm, the displacement history of the markers in the nozzle exit plane is extracted. As is typical for small and stiff structures, the resonant frequencies are quite high and the resulting vibration amplitudes are fairly low. This leads to low signal-to-noise ratio in the higher frequencies, which makes the following operational modal analysis much harder. By using BPMM as a preprocessor, the authors demonstrate that the outcome of modal analysis is greatly improved. Comparison with Finite Element analysis shows that the mode shapes agree much better with the experimental results when motion magnification is used. The modal parameters of the first six modes are found in the frequency range 0–1400 Hz using the new methodology, whereas without motion magnification only the first five modes are found.

This work takes up the challenge of deriving the ‘uncertainty law’ for close modes, i.e., closed form analytical expressions for the remaining uncertainty of modal parameters identified using (output-only) ambient vibration data. In principle the uncertainty law can be obtained from the inverse of the Fisher information matrix of modal parameters. The key mathematical challenges stem from analytical treatment of entangled stochastic dynamics with a large number of modal parameters of different nature and the quest for closed form expressions for identification uncertainty, whose possibility is questionable. Fortunately the problem still admits insightful closed form solution under long data, high signal-to-noise ratio and wide resonance band for identification. Up to modelling assumptions and the use of probability, the uncertainty law dictates the achievable precision of modal properties regardless of the identification algorithm used. A companion paper discusses the insights, verification, scientific implications and recommendation for ambient test planning. Published version This work is part of a research project on ‘‘Uncertainty quantification and management in ambient modal identification” funded by the UK Engineering and Physical Sciences Research Council (grant EP/N017897/1 and EP/N017803) to understand ID uncertainty and provide a strong scientific basis for implementing and planning ambient vibration tests.