An approach to operational modal analysis using the expectation maximization algorithm

This paper presents the Expectation Maximization algorithm (EM) applied to operational modal analysis of structures. The EM algorithm is a general-purpose method for maximum likelihood estimation (MLE) that in this work is used to estimate state space models. As it is well known, the MLE enjoys some optimal properties from a statistical point of view, which make it very attractive in practice. However, the EM algorithm has two main drawbacks: its slow convergence and the dependence of the solution on the initial values used. This paper proposes two different strategies to choose initial values for the EM algorithm when used for operational modal analysis: to begin with the parameters estimated by Stochastic Subspace Identification method (SSI) and to start using random points. The effectiveness of the proposed identification method has been evaluated through numerical simulation and measured vibration data in the context of a benchmark problem. Modal parameters (natural frequencies, damping ratios and mode shapes) of the benchmark structure have been estimated using SSI and the EM algorithm. On the whole, the results show that the application of the EM algorithm starting from the solution given by SSI is very useful to identify the vibration modes of a structure, discarding the spurious modes that appear in high order models and discovering other hidden modes. Similar results are obtained using random starting values, although this strategy allows us to analyze the solution of several starting points what overcome the dependence on the initial values used.

Maximum Likelihood Estimation of Modal Parameters in Structures Using the Expectation Maximization Algorithm

This paper presents a time-domain stochastic system identification method based on maximum likelihood estimation (MLE) with the expectation maximization (EM) algorithm. The effectiveness of this structural identification method is evaluated through numerical simulation in the context of the ASCE benchmark problem on structural health monitoring. The benchmark structure is a four-story, two-bay by two-bay steel-frame scale model structure built in the Earthquake Engineering Research Laboratory at the University of British Columbia, Canada. This paper focuses on Phase I of the analytical benchmark studies. A MATLAB-based finite element analysis code obtained from the IASC-ASCE SHM Task Group web site is used to calculate the dynamic response of the prototype structure. A number of 100 simulations have been made using this MATLAB-based finite element analysis code in order to evaluate the proposed identification method. There are several techniques to realize system identification. In this work, stochastic subspace identification (SSI)method has been used for comparison. SSI identification method is a well known method and computes accurate estimates of the modal parameters. The principles of the SSI identification method has been introduced in the paper and next the proposed MLE with EM algorithm has been explained in detail. The advantages of the proposed structural identification method can be summarized as follows: (i) the method is based on maximum likelihood, that implies minimum variance estimates; (ii) EM is a computational simpler estimation procedure than other optimization algorithms; (iii) estimate more parameters than SSI, and these estimates are accurate. On the contrary, the main disadvantages of the method are: (i) EM algorithm is an iterative procedure and it consumes time until convergence is reached; and (ii) this method needs starting values for the parameters. Modal parameters (eigenfrequencies, damping ratios and mode shapes) of the benchmark structure have been estimated using both the SSI method and the proposed MLE + EM method. The numerical results show that the proposed method identifies eigenfrequencies, damping ratios and mode shapes reasonably well even in the presence of 10% measurement noises. These modal parameters are more accurate than the SSI estimated modal parameters.

Modal analysis of buildings : quantification of uncertainties and model updating

Una estructura vibra con la suma de sus infinitos modos de vibración, definidos por sus parámetros modales (frecuencias naturales, formas modales y coeficientes de amortiguamiento). Estos parámetros se pueden identificar a través del Análisis Modal Operacional (OMA). Así, un equipo de investigación de la Universidad Politécnica de Madrid ha identificado las propiedades modales de un edificio de hormigón armado en Madrid con el método Identificación de los sub-espacios estocásticos (SSI). Para completar el estudio dinámico de este edificio, se ha desarrollado un modelo de elementos finitos (FE) de este edificio de 19 plantas. Este modelo se ha calibrado a partir de su comportamiento dinámico obtenido experimentalmente a través del OMA. Los objetivos de esta tesis son; (i) identificar la estructura con varios métodos de SSI y el uso de diferentes ventanas de tiempo de tal manera que se cuantifican incertidumbres de los parámetros modales debidos al proceso de estimación, (ii) desarrollar FEM de este edificio y calibrar este modelo a partir de su comportamiento dinámico, y (iii) valorar la bondad del modelo. Los parámetros modales utilizados en esta calibración han sido; espesor de las losas, densidades de los materiales, módulos de elasticidad, dimensiones de las columnas y las condiciones de contorno de la cimentación. Se ha visto que el modelo actualizado representa el comportamiento dinámico de la estructura con una buena precisión. Por lo tanto, este modelo puede utilizarse dentro de un sistema de monitorización estructural (SHM) y para la detección de daños. En el futuro, podrá estudiar la influencia de los agentes medioambientales, tales como la temperatura o el viento, en los parámetros modales. A structure vibrates according to the sum of its vibration modes, defined by their modal parameters (natural frequencies, damping ratios and modal shapes). These parameters can be identified through Operational Modal Analysis (OMA). Thus, a research team of the Technical University of Madrid has identified the modal properties of a reinforced-concrete-frame building in Madrid using the Stochastic Subspace Identification (SSI) method and a time domain technique for the OMA. To complete the dynamic study of this building, a finite element model (FE) of this 19-floor building has been developed throughout this thesis. This model has been updated from its dynamic behavior identified by the OMA. The objectives of this thesis are to; (i) identify the structure with several SSI methods and using different time blocks in such a way that uncertainties due to the modal parameter estimation are quantified, (ii) develop a FEM of this building and tune this model from its dynamic behavior, and (iii) Assess the quality of the model, the modal parameters used in this updating process have been; thickness of slabs, material densities, modulus of elasticity, column dimensions and foundation boundary conditions. It has been shown that the final updated model represents the structure with a very good accuracy. Thus, this model might be used within a structural health monitoring framework (SHM). The study of the influence of changing environmental factors (such as temperature or wind) on the model parameters might be considered as a future work.

A procedure to track vibration modes under changing external factors: application to a pedestrian bridge

A low-cost vibration monitoring system has been developed and installed on an urban steel- plated stress-ribbon footbridge. The system continuously measures: the acceleration (using 18 triaxial 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) is carried out in order to extract the modal parameters, i.e., the natural frequencies, damping ratios and modal shapes. Thus, this paper analyzes the time evolution of the modal parameters over a whole-year data monitoring. Firstly, for similar environmental/operational factors, the uncertainties associated to the time window size used are studied and quantified. Secondly, a methodology to track the vibration modes has been established since several of them with closely-spaced natural frequencies are identified. Thirdly, the modal parameters have been correlated against external factors. It has been shown that this stress-ribbon structure is highly sensitive to temperature variation (frequency changes of more than 20%) with strongly seasonal and daily trends