An approach to operational modal analysis using the expectation maximization algorithm

Cara, Javier; Carpio, Jaime; Ruiz, Jesús Juan; Alarcón, Enrique

doi:10.1016/j.ymssp.2012.04.004

Cited by 39 publications

(22 citation statements)

References 11 publications

(13 reference statements)

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“…SSI-EM is a combination of SSI and the EM algorithm. Using SSI for the initial estimate, the maximum-likelihood estimation requires iterations when using the EM algorithm (Cara et al 2012). SSI-EM introduces the EM algorithm (McLachlan and Krishnan 2007) to maximize the likelihood function, which optimizes the lack of optimal solution shown by Bauer (2005) and Chiuso and Picci (2004) and improves the SSI results but does not define criteria for choosing the pole in the stabilization diagram.…”

Section: Operational Modal Analysis Using Three Ssi Techniquesmentioning

confidence: 99%

Vibration Monitoring of a Steel-Plated Stress-Ribbon Footbridge: Uncertainties in the Modal Estimation

et al. 2016

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A low-cost vibration-monitoring system was developed and installed on an urban steel-plated stress-ribbon footbridge. The system continuously measures the acceleration [using 18 triaxial microelectromechanical system (MEMS) accelerometers distributed along the structure), the ambient temperature, and the wind velocity and direction. Automated output-only modal parameter estimation based on the stochastic subspace identification (SSI) was carried out to extract the modal parameters (i.e., the natural frequencies, damping ratios, and modal shapes). Thus, this study analyzed the time evolution of the modal parameters over data monitoring for 1 year. First, for similar environmental/ operational factors, the uncertainties associated with the SSI-based techniques used and to the acceleration records used were studied and quantified. Second, a methodology for tracking the vibration modes was established, because several of them with closely spaced natural frequencies were identified. Third, the modal parameters were correlated against external factors. It has been shown that this stress-ribbon structure is highly sensitive to temperature variations (frequency changes of more than 20%) with strongly seasonal and daily trends. Fairly simple dynamic multiple regression models for the lowest persistent vibration modes were derived, and excellent correlations for some of them were obtained. These correlations enable the influence of these uncertainties on modal estimates to be removed, thus facilitating their use as damage-sensitive features.

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Section: Operational Modal Analysis Using Three Ssi Techniquesmentioning

confidence: 99%

Vibration Monitoring of a Steel-Plated Stress-Ribbon Footbridge: Uncertainties in the Modal Estimation

et al. 2016

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“…Model (70) can be estimated using the EM algorithm: The corresponding equations can be easily derived from the ones presented in Section by deleting the terms related with the inputs u t . For example, the equations for the M‐step becomes

boldA = {boldS}_{x_{1} x} {boldS}_{x x}^{- 1},

boldQ = \frac{1}{N} ((), S_{x_{1} x_{1}} - S_{x_{1} x} A^{'} - boldA S_{x x_{1}} + boldA S_{x x} A^{'}),

boldC = S_{y x} {boldS}_{x x}^{- 1},

boldR = \frac{1}{N} ((), S_{y y} - S_{y x} C^{'} - boldC S_{x y} + boldC S_{x x} C^{'}) .

…”

Section: Numerical Example: Simulated Datamentioning

confidence: 99%

“…An interesting alternative is to consider the expectation–maximization (EM) algorithm because in modal analysis, the size of the estimated state space matrices is high, and the EM algorithm is better suited than Newton algorithms for these kinds of situations. () The main aspects of the algorithm are described in Section .…”

Section: Introductionmentioning

confidence: 99%

“…Summarizing, the main contributions of this paper are the following: •extension of the EM algorithm from OMA setting to OMAX setting (measured inputs in addition to ambient excitation); •analysis of the steady‐state formulation for the EM algorithm. This option reduces the computational burden of the algorithm; •application of the bootstrap method to estimate the variance of the identified modal parameters.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Modal identification of structures from input/output data using the expectation-maximization algorithm and uncertainty quantification by mean of the bootstrap

Cara

2018

Struct Control Health Monit

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Modal testing in civil engineering includes the possibility to apply measured forces in addition to the unmeasured ambient excitation. In these cases, it is necessary to consider mathematical models that account for both excitation sources, what explains the increasing interest in sophisticated system identification methods for modal analysis with input/output data. In this work, the maximum likelihood estimation of the state space model from input/output vibration data is investigated. This model can be estimated using different techniques: Among them, the maximum likelihood method has optimal statistical properties, so modal parameters computed using this approach will be optimum in a statistical point of view. The algorithm considered for maximizing the likelihood is the expectation-maximization algorithm. The quantification of modal parameters uncertainty is addressed using a Monte Carlo type approach called the bootstrap, which is based on resampling the residuals of the estimated model. Finally, the proposed techniques are applied to synthetic data and also to field data recorded on a stress-ribbon footbridge. KEYWORDS bootstrap, combined experimental-operational modal analysis, expectation-maximization algorithm, mass-normalized modal vectors, modal parameters, operational modal analysis with exogenous inputs, state space model, stress-ribbon footbridge Struct Control Health Monit. 2019;26:e2272.wileyonlinelibrary.com/journal/stc

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“…The frequency values of spurious mathematical modes, on the other hand, usually scatter around in the stabilization diagram. SSI certainly gains more popularity in the applications of civil structures with the help of stabilization diagram, but certain issues still need to be further explored for more complicated situations . Moreover, appropriate discrimination criteria based on theories like clustering analysis have to be established such that systematic extraction of effective physical modes from the stabilization diagram would become feasible to finally attain the goal of a totally automated SSI analysis .…”

Section: Introductionmentioning

confidence: 99%

Application of stochastic subspace identification for stay cables with an alternative stabilization diagram and hierarchical sifting process

Wang

Chen

et al. 2016

Struct. Control Health Monit.

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The modal parameters of numerous modes for a stay cable are usually required in engineering practice. The application of conventional stochastic subspace identification techniques without delicate cautions, however, has been found to be unsuccessful in this case. Aiming to attack such a difficulty, this study establishes a new methodology based on the covariance type of stochastic subspace identification for extensively identifying the modal parameters of a stay cable. Several details of choosing the parameters in performing stochastic subspace identification are first discussed. An important discovery is that the lower limit for setting the time lag parameter can be decided by the ratio of the fundamental period of cable to the sampling time increment for a valid identification with the conventional stabilization diagram. Inspired by the aforementioned criterion, an alternative stabilization diagram is further proposed to more conveniently distinguish stable modal parameters of cable. A hierarchical sifting process including three stages is then developed to systematically and automatically extract reliable modal parameters from the alternative stabilization diagram. Demonstrated by analyzing the ambient vibration measurements for three stay cables of Chi-Lu Bridge, the feasibility of this new approach is verified with successfully obtaining the modal frequencies, damping ratios, and mode shape ratios for almost all the cable modes in the examined frequency range. Another interesting finding is that the modal frequencies and damping ratios of bridge deck can also be effectively identified from the ambient vibration signals of cable.

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An approach to operational modal analysis using the expectation maximization algorithm

Cited by 39 publications

References 11 publications

Vibration Monitoring of a Steel-Plated Stress-Ribbon Footbridge: Uncertainties in the Modal Estimation

Vibration Monitoring of a Steel-Plated Stress-Ribbon Footbridge: Uncertainties in the Modal Estimation

Modal identification of structures from input/output data using the expectation-maximization algorithm and uncertainty quantification by mean of the bootstrap

Application of stochastic subspace identification for stay cables with an alternative stabilization diagram and hierarchical sifting process

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