... linear superposition of the NLLF of fast. Bayesian FFT approach corresponding to different measurements shown in (Au, 2012c),. 2. 2. 1. 1. *. 1. FBSDA. FBFFT ...
Wireless Sensor Network based Structural Health Monitoring Accommodating Multiple Uncertainties
by Wangji Yan
A Thesis Submitted to The Hong Kong University of Science and Technology in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Civil Engineering
August 2013, Hong Kong
HKUST Library Reproduction is prohibited without the author’s prior written consent
Acknowledgements
I would like to express my deepest gratitude and appreciation to my supervisor, Prof. Lambros Katafygiotis, for his enthusiastic guidance, helpful suggestions and endless encouragement throughout my research. He initiated me into the area of uncertainty and reliability analysis. I benefit greatly from his philosophical perspectives in the course of my study. I heartily respect him for his integrity, wisdom and insight. Also, I gratefully acknowledge Professor Ka-Veng Yuen from University of Macau and Professor Siu-Kui Au from the University of Liverpool for their beneficial advices. I benefit a lot from their vision and persistence in researches related to uncertainty analysis using Bayesian approach. Special thanks go to my Thesis Supervision Committee Prof. Chih-Chen Chang and Prof. Ilias G. Dimitrakopoulos for their invaluable suggestions and continuous support. I sincerely appreciate Prof. Chih-Chen Chang for providing his laboratory models generously. I would like to thank Prof. Bill F. Spencer and his colleagues for releasing ISHMP Services Toolsuite to the public and providing technical supports patiently. The experiments would not be finished without their helps. I am deeply indebted to Prof. Wei-Xin Ren, for his unfailing support and insightful guidance, dating back to the days when I was pursuing a Master degree. His continuous encouragement and philosophical thinking have influenced me significantly throughout my graduate study. Also, high tribute shall be paid to all members from professor Lambros’ group, i.e., Zhouquan Feng, Jia Wang, and Konstantin Zuev for their kind supports. Moreover, I would like to thank Yashuai Li, Changli Yu, Xingyu Jiang, Fei Ding, etc. for their warm friendships. Last but not least, this thesis is dedicated to my family members, whose unconditional love and support cannot be overestimated. I am in debt to them for not being able to stay with them for too much time.
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Table of Contents Authorization.......................................................................................................................................................... i Acknowledgements............................................................................................................................................... iii Table of Contents ................................................................................................................................................. iv List of Figures ...................................................................................................................................................... iix List of Tables ....................................................................................................................................................... xii Abstract............................................................................................................................................................... xiii Chapter 1 Introduction ......................................................................................................................................... 1 1.1 Research Background and Motivation ......................................................................................................... 1 1.2 SHM using the Bayesian Statistical Framework .......................................................................................... 2 1.2.1 Ambient Modal Analysis ...................................................................................................................... 3 1.2.2 Vibration-based Damage Detection ...................................................................................................... 5 1.3 Outline of the Thesis ..................................................................................................................................... 7 Chapter 2 Background of WSN based SHM ...................................................................................................... 8 2.1 General Introduction to Wireless Sensors .................................................................................................... 8 2.1.1 Basic Components and Essential Features of Wireless Sensors ........................................................... 8 2.1.2 Wireless Sensor Platforms to Date ....................................................................................................... 9 2.1.3 Embedded Operational System........................................................................................................... 11 2.2 SHM using Wireless Sensors ...................................................................................................................... 12 2.3 Hardware and Software Employed in this Study ........................................................................................ 13 2.3.1 Hardware: Crossbow’s Imote2 Platform and SHM-H Sensor Board ................................................. 13 2.3.2 Software: ISHMP Services Toolsuite ................................................................................................. 17
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2.4 Summary..................................................................................................................................................... 19 Chapter 3 A Two-Stage Fast Bayesian Spectral Density Approach for Ambient Modal Analysis: Theory ... .............................................................................................................................................................................. 20 3.1 Introduction ................................................................................................................................................ 20 3.2 Revisiting Bayesian Spectral Density Approach ........................................................................................ 21 3.2.1 Formulation of Bayesian Spectral Density Approach (Katafygiotis and Yuen, 2001) ....................... 21 3.2.2 Computation Difficulties of BSDA .................................................................................................... 24 3.2.3 Solution Strategies .............................................................................................................................. 25 3.2.4 Formulation of Bayesian Spectral Trace Approach ............................................................................ 26 3.3 Formulation of Two-stage Bayesian Approach: Case of Separated Modes ............................................... 27 3.3.1 Stage One: Spectrum Variables Identification Using FBSTA ............................................................ 28 3.3.2 Stage Two: Spatial Variables Identification Using FBSDA ............................................................... 29 3.3.3 Summary of Procedure ....................................................................................................................... 33 3.4 Formulation of Two-stage Approach: Case of Closely Spaced Modes ...................................................... 33 3.4.1 Stage One: Spectrum Variables Identification by FBSTA ................................................................. 34 3.4.2 Stage Two: Spatial Variables Identification by FBSDA .................................................................... 38 3.4.3 Summary of Procedures...................................................................................................................... 43 3.5 Concluding Remarks .................................................................................................................................. 43 Appendix........................................................................................................................................................... 44 Chapter 4 A Two-Stage Fast Bayesian Spectral Density Approach for Ambient Modal Analysis: Mode Shape Assembly and Case Studies ..................................................................................................................... 49 4.1 Introduction ................................................................................................................................................ 49 4.2 Hierarchical Architecture of WSN ............................................................................................................. 49 4.3 Assembling Mode Shape from Multiple Clusters........................................................................................ 50 4.3.1 Bayesian Mode Shape Assembly Approach ....................................................................................... 51 4.3.2 Most Probable Values......................................................................................................................... 53
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4.3.3 Posterior Uncertainties ....................................................................................................................... 55 4.4 Test for Stationarity ................................................................................................................................ 57 4.5 Numerical Studies ...................................................................................................................................... 58 4.5.1 Case One: 2-D Shear Building ........................................................................................................... 58 4.5.2 Case Two: 3-D Torsional Shear Building........................................................................................... 63 4.6 Experimental Studies .................................................................................................................................. 68 4.6.1 Case One: 2-D Shear Building ........................................................................................................... 69 4.6.2 Case Two: 3-D Torsional Shear Building........................................................................................... 73 4.7 Concluding Remarks .................................................................................................................................. 77 Chapter 5 Bayesian Approach to Structural Model Updating Incorporating Modal Information from Multiple Clusters ................................................................................................................................................. 79 5.1 Introduction ................................................................................................................................................ 79 5.2 Formulation of Bayesian Model Updating Using Modal Data .................................................................. 79 5.2.1 Modal Data Available for Model Updating ........................................................................................ 80 5.2.2 Structural Model Class ....................................................................................................................... 81 5.2.3 Instrumental Variables ‘System Mode Shape’ ................................................................................... 82 ˆ r ,i ............................................................ 83 5.2.4 Probability Model for the Discrepancy between Φ r and ψ
5.2.5 Probability Model for the Eigenvalue Equation Errors ...................................................................... 84 5.2.6 Negative Logarithm Likelihood Function .......................................................................................... 85 5.3 Most Probable Parameters ........................................................................................................................ 86 5.3.1 Optimization for r ,i and r ,i ............................................................................................................... 88 5.3.2 Optimization for r ............................................................................................................................. 89 5.3.3 Optimization for Φ r .......................................................................................................................... 89 5.3.4 Optimization for ρ .............................................................................................................................. 89 5.3.5 Optimization for θ ............................................................................................................................. 90 5.4 Posterior Uncertainties Estimation ............................................................................................................ 90
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5.4.1 Analytical Derivation of Hessian Matrix ............................................................................................ 90 5.4.2 Suppressing Computational Complexity of Covariance Matrix ......................................................... 94 5.5 Probabilistic Damage Detection (Vanik et al., 2000) ................................................................................ 96 5.6 Numerical Study ......................................................................................................................................... 96 5.6.1 Case One: 2-D Shear Building ........................................................................................................... 96 5.6.2 Case Two: 3-D Torsional Shear Building......................................................................................... 100 5.7 Experimental Study .................................................................................................................................. 105 5.8 Concluding Remarks ................................................................................................................................ 107 Chapter 6 Use of Random Matrix Theory for Bayesian System Identification with Non-stationary Response Measurements Only ......................................................................................................................... 108 6.1 Introduction .............................................................................................................................................. 108 6.2 Problem Description ................................................................................................................................ 108 6.2.1 Model Class of a Dynamical System ................................................................................................ 108 6.2.2 Partition of the Measurements .......................................................................................................... 110 6.2.3 Transmissibility Matrix Relating Two Sets of Measurements.......................................................... 111 6.2.4 Random Matrix Theory .................................................................................................................... 113 6.3 Bayesian Model Updating Using Non-stationary Measurements ............................................................ 115 6.3.1 Negative Log-likelihood Function .................................................................................................... 115 6.3.2 Most Probable Values and Posterior Uncertainties .......................................................................... 116 6.3.3 Initial Guesses for the Parameters to be identified ........................................................................... 117 6.3.4 Summary of Procedures.................................................................................................................... 118 6.4 Damage Detection (Vanik et al., 2000) .................................................................................................... 119 6.5 Bayesian Substructuring Model Updating ............................................................................................... 120 6.6 Numerical Studies .................................................................................................................................... 122 6.6.1 Case one: 3-dof Spring-Mass System ............................................................................................... 122 6.6.2 Case Two: 20-Stoey Shear Building................................................................................................. 124
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6.7 Experimental Verification ........................................................................................................................ 129 6.8 Concluding Remarks ................................................................................................................................ 133 Chapter 7 Conclusions ...................................................................................................................................... 135 References .......................................................................................................................................................... 138
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List of Figures Figure 2.1: The components of a wireless sensor node (Akyildiz et al., 2002) Figure 2.2: Top side (left) and bottom side (right) of Imote2 main board Figure 2.3: An overview of sensor board and battery board stacked on Imote2 main board Figure 2.4: Top (left) and bottom (right) of SHM-H board (Jo et al., 2010) Figure 2.5: Software architecture of SHM application (Feng, 2010) Figure 4.1: Three-level hierarchical architecture of WSN Figure 4.2: Acceleration measurement (left) and auto-spectral density (right) of the top floor ) at f k 0.806 Hz (left) and f k 3.988 Hz (right) with different ns Figure 4.3: CDFs of tr (Ssum k
Figure 4.4: Conditional PDFs of f s (left) and s (right) for the 2-D shear building Figure 4.5: Contours of the marginal PDF of f s and s for the 2-D shear building Figure 4.6: Assembled global mode shape from multiple setups for the 2-D shear building Figure 4.7: Raw measurement of the top floor with respect to x-translational dof Figure 4.8: CDFs of tr (Ssum ) at f k 2.364 Hz , 6.958Hz and 11.146Hz with ns 20 k Figure 4.9: The plot of tr (Ssum ) with ns 20 k Figure 4.10: Conditional PDFs of f1 (left) and 1 (right) for the 3-D torsional shear building Figure 4.11: Contours of the marginal PDF of f1 and 1 for the 3-D torsional shear building Figure 4.12: Assembled mode shapes for the 3-D torsional shear building Figure 4.13: An overview of the gateway node connected to the laptop Figure 4.14: The tested shear building (left) and its simplified 3-dof model (right) Figure 4.15: Side view of sensor placement on the top floor Figure 4.16: Acceleration measured using the wireless sensor installed on the top storey Figure 4.17: Trace of spectral density matrix with 20 sets of measurements Figure 4.18: The tested torsional shear building Figure 4.19: Side view (left) and floor plan of sensor placement (right) for the torsional shear building
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Figure 4.20: Acceleration of the top floor with respect to x direction Figure 4.21: Trace of spectral density matrix of the torsional shear building Figure 4.22: Identified mode shapes for the torsional shear building Figure 5.1: Iteration histories of i for the 2-D shear building in healthy state (left) and damaged state (right) Figure 5.2: Comparison between ‘system mode shapes’ and exact mode shapes of the 2-D shear building Figure 5.3: Curves of damage probability for the 2-D shear building Figure 5.4: Iteration histories of i for the 3-D torsional shear building in healthy state (left) and damaged state (right) Figure 5.5: Comparison between ‘system mode shapes’ and exact mode shapes of the 3-D torsional shear building Figure 5.6: Curves of damage probability for the 3-D torsional shear building Figure 5.7: Iteration histories of model updating for four different scenarios Figure 5.8: Identified optimal ‘system mode shapes’ for four different scenarios Figure 6.1: Generic structure with some external forces Figure 6.2: Schematic plot of substructure I (left) and substructure II (right) Figure 6.3: Two-level hierarchical architecture of WSN Figure 6.4: Schematic plot of 3-dof spring-mass system Figure 6.5: Raw measurement (left) and spectral density (right) of the third dof Figure 6.6: Conditional PDFs of 1 (left) and 2 (right) Figure 6.7: Contours of the marginal PDFs of 1 and 2 Figure 6.8: Time history of ground motion Figure 6.9: Substructure including the first five floors of the shear building Figure 6.10: Acceleration measurement (left) and spectral density (right) of the fifth dof Figure 6.11: Conditional PDFs of 1 and 2 for the substructure in healthy state Figure 6.12: Conditional PDFs of 1 and 2 for the substructure in damaged state Figure 6.13: Probability of damage for each stiffness parameter of the substructure x
Figure 6.14: Acceleration (left) and spectral density (right) of the top floor under the first excitation condition Figure 6.15: Acceleration (left) and spectral density (right) of the middle floor under the second excitation condition Figure 6.16: Iteration histories for the full structural identification Figure 6.17: Iteration histories for the substructural identification Figure 6.18: Effect of time duration on the MPV (left) and c.o.v. values (right) for the full structural identification Figure 6.19: Effect of time duration on the MPV (left) and c.o.v. values (right) for the substructural identification
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List of Tables
Table 2.1: Main features of Imote2 (Crossbow Technology, 2007a) Table 2.2: Sampling rates and cutoff frequencies of ITS400C (Crossbow Technology, 2007b) Table 2.3: Characteristics of SHM-H sensor board (ISHMHP, 2009a) Table 4.1: Identified spectrum variables for the 2-D shear building Table 4.2: Setup information for the 2-D shear building Table 4.3: Identified spectrum variables for the 3-D torsional shear building. Table 4.4: Setup information for the 3-D torsional shear building Table 4.5: Identified spectrum variables for laboratory shear building model Table 4.6: Identified mode shapes for the laboratory shear building model Table 4.7: Identified spectrum variables for the torsional shear building Table 5.1: Setup information of incomplete measured dofs for the 2-D shear building Table 5.2: Identified modal properties for the 2-D shear building Table 5.3: Identified stiffness scaling factors i for the 2-D shear building Table 5.4: Setup information of incomplete measured dofs for the 3-D torsional shear building Table 5.5: Identified modal properties of the 3-D torsional shear building Table 5.6: Identified stiffness scaling parameters for the 3-D torsional shear building Table 5.7: Identified stiffness parameters for four different scenarios Table 6.1: Identified model parameters for the 3-dof spring-mass system Table 6.2: Identified stiffness parameters for the substructure considered Table 6.3: Identified results for different scenarios under different excitation conditions
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Abstract As a promising alternative to wired sensors, wireless sensor networks (WSN) have attracted widespread attention in the field of structural health monitoring (SHM) due to their unique features such as lower cost trend, wireless communication and onboard computation. Though there are many types of modeling and parametric uncertainties in a WSN based SHM system, the available results are usually restricted to deterministic optimal values while the uncertainties cannot be quantified. This work is therefore dedicated to the development of new advanced algorithms for WSN based SHM system with special attention to ambient modal analysis and structural damage detection using a Bayesian statistical framework. Firstly, a two-stage fast Bayesian spectral density approach is proposed to extract modal properties and their associated uncertainties for the cases of separated modes and closely spaced modes, respectively. A novel technique for variable separation is developed so that the interaction between spectrum variables (e.g., frequency, damping ratio as well as the amplitude of modal excitation and prediction error) and spatial variables (e.g., mode shape components) can be decoupled completely. As a result, these two kinds of variables can be identified separately. The spectrum variables can be identified through ‘fast Bayesian spectral trace approach’ (FBSTA) in the first stage, while the spatial variables can be estimated in a second stage by ‘fast Bayesian spectral density approach’ (FBSDA). This study also reveals the intrinsic relationship between FBSDA and fast Bayesian FFT approach when multiple sets of measurements are available. The proposed two-stage approach can be implemented in the environment of WSN through distributed computing strategy so that local mode shape components confined to different clusters can be identified. A Bayesian mode shape assembly methodology is herein proposed to form the overall mode shapes so that the weight for different clusters is accounted for properly according to their data quality. For the proposed method, there is no need to share the same set of reference dofs for all clusters to obtain proper scaling. Next, a novel Bayesian methodology based on the incomplete modal properties (e.g., natural frequencies and partial mode shapes of some modes) is developed for structural model updating. The model updating problem is formulated as one minimizing an objective function, xiii
which can incorporate the local mode shape components identified from different clusters automatically without prior assembling or processing. A fast analytic-iterative scheme is proposed to efficiently compute the optimal parameters so as to resolve the computational burden required for optimizing the objective function numerically. The posterior uncertainty of the model parameters can also be derived analytically and the computational difficulty in estimating the inverse of the high dimensional Hessian matrix required for specifying the covariance matrix is also properly treated. The proposed method can avoid the matching between the measured mode and the model mode, mode shape expansion and eigenvalue decomposition, which are frequently encountered in conventional model updating approaches. Finally, the problem of updating a structural model by utilizing non-stationary response measurements only is considered. A negative log-likelihood function utilized to determine the posterior most probable parameters and their associated uncertainties is formulated by incorporating random matrix theory and Bayes’ theorem. Instead of optimizing all the unknown parameters simultaneously, a numerically iterative coupled method involving the optimization of the parameters in groups is employed so as to reduce the dimension of the numerical optimization problem involved. The initial guess for the parameters to be optimized is also properly estimated. One novel feature of the proposed method is to avoid repeated time-consuming evaluation of the determinant and inverse of the covariance matrix during optimization due to exploring the statistical properties of the trace of Wishart matrix. Moreover, the proposed method allows the monitoring of some critical substructures rather than the entire structure, requiring no information about the model of the external input or interface forces. The efficiency and accuracy of all these methodologies are verified by numerical examples. Experimental studies are also conducted by employing laboratory shear building models installed with high-sensitivity accelerometer board (SHM-H sensor board) interfaced to the advanced wireless sensor node platform (the Crossbow Imote2). The software embedded on the Crossbow Imote2 is provided by the Illinois Structural Health Monitoring Project (ISHMP) Services Toolsuite. Successful validation of the proposed methods using measured acceleration demonstrates the potential for Bayesian approaches to accommodate multiple uncertainties for WSN based SHM systems. xiv
Chapter 1 Introduction 1.1 Research Background and Motivation Rapid economic development and technological advancements during past decades have allowed a huge amount of resources to be invested in civil infrastructures, such as bridges, buildings, pipelines, dams, and offshore platforms, etc. As the backbone of our society needs to function effectively, the civil infrastructures are always expected to serve human beings for a large number of years. However, their anticipated service life may decrease significantly due to hostile-loading environments and lack of timely maintenance. The decayed structures may cause catastrophic failures and threaten public safety seriously. In Hong Kong for example, one recent case was the sudden collapse of a five-storey building in the To Kwa Wan district in 2010, leading to four deaths and two injuries. Therefore, structural health monitoring (SHM) which aims to monitor the health condition of a structure has been an emerging field in civil engineering. Nowadays, it has been a common practice to install SHM systems to detect, locate, and assess structural damage by using measured responses. Since damage is intrinsically a local phenomenon, a dense array of sensors is required to be deployed on large-scale structures which often exhibit a wide variety of complex behaviour (Nagayama and Spencer, 2007). By taking the Stonecutters Bridge in Hong Kong as an example, it was equipped with a SHM system involving more than 1200 sensor channels. The number of sensors required to adequately monitor large-scale civil infrastructures is challenging to realize using traditional wired sensors with cables connected to a centralized data repository since installation of wired sensors often turn out to be prohibitively expensive and time-consuming (Sim and Spencer, 2009). For example, it took a long time to deploy 84 accelerometers in the Bill Emerson Memorial Mississippi River Bridge in USA, with the average cost per sensing channel exceeding ten thousand U.S. dollars (Çelebi et al., 2004). Wireless sensors, with their unique features such as on-board computational and wireless communication capabilities, etc. have
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been regarded as a promising alternative to the traditional wired sensors in SHM. They offer new opportunities for SHM of sizable structures due to their ability to address the issues of installation difficulties and enormous expense (Nagayama and Spencer, 2007). Therefore, wireless SHM presents plenty of opportunities for research, innovation and development. During recent years, many researchers have devoted their efforts to demonstrating the applicability of wireless sensors in SHM. However, these efforts have not resulted in fullfledged applications, and there is still significant room for improvement. One of the major challenges is that the robustness of SHM algorithms cannot be ensured and full-scale implementation of SHM approaches are still considered to be immature (Sim, 2011). Conventionally, SHM results are usually restricted to deterministic optimal values while the parametric uncertainties cannot be quantified. However, there are many kinds of uncertainties that deserve considering in SHM systems including uncertainties due to incomplete information, errors due to imperfect modeling of physical phenomena as well as measurement noise (Yuen, 2010a). To ensure the robustness of the identified results, SHM is best tackled as a statistical inference problem, which not only gives the optimal values for the various identified parameters but also provides a quantitative assessment of their accuracy (Beck and Katafygiotis, 1998). For example, the associated uncertainty information during damage detection is important for engineers when they make judgments about repairing works. Therefore, the primary focus of this thesis is to provide rigorous probabilistic algorithms for ambient modal analysis and structural damage detection using Bayesian statistics, and then apply the proposed methods in the environment of WSN. 1.2 SHM using the Bayesian Statistical Framework Generally speaking, there are two main broad approaches to statistical inference, e.g., the socalled ‘frequentist’ approach and Bayesian approach (Cox, 2005). The definition of probability in ‘frequentist’ statistics is a long-run limiting relative frequency (O’Hagan, 2003). One commonly-taught example shown in elementary statistics courses is counting a relative frequency of occurrences of ‘heads’ by tossing coins for a very large number of times. In Bayesian statistics, however, probability is viewed as a measure of the plausibility (a personal degree of belief in a proposition) of a proposition conditional on incomplete information, 2
where often we are not able to determine the truth or falsehood of the proposition (Zuev et al., 2012). Bayesian statistics was not accepted by most statisticians until late last century because of a widespread belief that probability can only be applied to aleatory uncertainty (inherent randomness in nature) while it cannot be applied to epistemic uncertainty (missing information). Due to the efforts of physicists Cox and Jaynes who presented a rigorous logic foundation for the Bayesian approach (Cox, 1946; Cox, 1961; Jaynes, 1983; Jaynes, 2003), Bayesian statistics has become a popular mathematical approach for statistical inference and uncertainty quantification in various fields. In the context of SHM, the procedure of statistical inference usually involves the probability of structural models, which are not repeatable events. The traditional interpretation of probability as a relative frequency of occurrences in the long run is not applicable for analyzing noisy data, selecting a model, or measuring uncertainty in conclusions. Moreover, since there are no true values of the model parameters whose estimation is often not unique, it is more reasonable to interpret the probability herein as a multivalued logic for plausible reasoning given incomplete information (Beck, 2010). A general Bayesian probabilistic model updating framework was presented in (Beck and Katafygiotis, 1998; Katafygiotis and Beck, 1998). Following this framework, there has been substantial development in recent years for different purposes such as propagating uncertainties for robust response and reliability predictions, updating the parameters of a finite element model, and selecting the model class from a family of competitive model classes (Yuen, 2010a; Yuen, 2010b). Some state-of-the-art work will be reviewed in section 1.2.1 and 1.2.2, mainly focusing on applications in ambient modal analysis and structural damage detection. 1.2.1 Ambient Modal Analysis Modal analysis has widespread applications in the fields of structural vibration control, structural health monitoring and structural damage detection. Primarily, modal properties include the natural frequencies, damping ratios and mode shapes. Remarkable progress has been made on experimental modal analysis formulated based on both input and output measurement data (Ljung, 1987; Ewins, 2000). These approaches, however, are not devoid of 3
potential problems for large-scale civil infrastructures in that they usually require special experiments which are often time consuming, costly, and obtrusive (Zong et al., 2005). Therefore, ambient modal analysis using output-only measured response without the knowledge of the input has aroused increasing interest in industrial applications since they can be carried out in a much more economical and efficient manner. An increasing number of ambient modal analysis approaches in frequency domain have been developed. Among others, the peak-picking method (Bendat and Piersol, 1993), frequency-domain decomposition (FDD) method (Brinker et al., 2001), PolyMAX method (Guillaume et al., 2003), etc. are popular techniques capable of quickly extracting the modal properties. More attention has been devoted to ascertaining the dynamic properties in time domain. Well-known examples include auto-regressive-moving-average (ARMA) method (Andersen, 1996), eigensystem realization algorithm (ERA) method (Juang, 1994) with data correlation, and stochastic subspace identification (SSI) method (Peeters and De Roeck, 2000), etc. Recent interest has arisen to calculate the uncertainties of modal parameters by using Bayesian approaches. In the context of ambient modal analysis, a number of Bayesian approaches including Bayesian spectral density approach (BSDA) (Katafygiotis and Yuen, 2001), Bayesian time domain approach (BTDA) (Yuen and Katafygiotis, 2001; Yuen et al., 2002), and Bayesian FFT approach (BFFTA) (Yuen and Katafygiotis, 2003) have been proposed. These methods have been compared comprehensively with each other by Yuen (1999). Compared to BTDA, BSDA and BFFTA working in the frequency domain are more promising candidates for ambient modal analysis since they can utilize data in selected resonance bandwidths so as to legitimately avoid using information in the entire frequency bands (Au, 2011b). These methods provide rigorous means for obtaining modal properties as well as their uncertainties. However, computational difficulty has severely hindered their wider applications even for a moderate number of measured dofs. To address the computational challenges of conventional Bayesian FFT approach, a breakthrough was made by Au recently (Au, 2011b; Au, 2012b; Au, 2012c). The most probable values as well as the posterior covariance matrix can be computed quickly by fast solutions, which allow the results to be obtained even on site within a few seconds. The relationship between Bayesian method and frequentist approach in statistical system identification were also investigated in 4
detail (Au, 2012a). Field applications in different engineering structures such as a coupled floor slab system (Au et al., 2012), a primary-secondary structure (Au and Zhang, 2012a), and a super-tall building under strong wind (Au and To, 2012) have demonstrated the efficiency of these approaches successfully. An overview of the newly formulated approach including issues of theoretical, computational and practical nature has also been presented more recently (Au et al., 2013). 1.2.2 Vibration-based Damage Detection A wide variety of structural damage detection methods have been produced over the past few decades, which culminates in various conference proceedings and journals. In particular, vibration-based damage detection has received enormous amount of attention since it is able to provide a global approach to evaluate the structural state (Ren and De Roeck, 2002a; 2002b). The basic premise of vibration-based damage detection is that the changes of physical properties will cause changes in the measured dynamic response of the structure (Farrar et al., 2001). A comprehensive historic development of vibration based damage assessment methodologies has been provided by some recent surveys (Doebling et al., 1996; Sohn et al., 2003; Fan and Qiao, 2011). Based on the dependency on an analytical model, vibration-based damage assessment methods are usually divided into model-free and model-dependent ones (Sun, 2003). For the model free damage detection methods, the core idea is to compare a selected damage index between the baseline model and the possibly damaged model to infer structural damage. The model free methods might be able to detect and locate structural damage, but they are not likely to establish structural damage severity. On the contrary, the model-dependent approaches are expected to achieve the goal of indicating structural damage existence, damage location, and damage degree. A common proposal for the model-based approaches is to determine the structural model parameters before and after a possible damage from measured dynamic responses. As a result, they usually require solving the inverse problem given some measured data by employing structural model updating methodologies (Vanik et al., 2000). Most of the model-dependent approaches ignore different kinds of uncertainties. For example, the structures under consideration are usually assumed to be well characterized by initial analytical models, while 5
a sufficiently large amount of measurements with low level of noise are assumed to be available for model updating. The approaches under these assumptions are more likely to produce poor results in practice. Therefore, the problem of how to treat the uncertainties explicitly always arises. One novel school of thoughts is the Bayesian approach which is able to find the plausible structural damage extents as well as their probabilities given a model of the structural system and the measured data. Within the Bayesian statistical framework, a probabilistic damage detection method to calculate the probability of damage was proposed using a sequence of identified modal parameter data sets (Vanik, 1997; Vanik et al., 2000; Beck et al., 2001). Yuen et al. (2004) applied this approach to the Phase I benchmark study sponsored by the IASC-ASCE Task Group on Structural Health Monitoring using measured structural response from the undamaged system and the possibly damaged system. Later on, Ching and Beck (2004b) proposed a new Bayesian model updating algorithm with incomplete mode shape information by employing the Expectation-Maximization algorithm to determine the most probable parameter values. The proposed method was then used to analyze the Phase II experimental benchmark studies sponsored by the IASC-ASCE Task Group on structural health monitoring (Ching and Beck, 2004a). Ching et al. (2006) also presented a new approach using a stochastic simulation method that decomposes the uncertain model parameters into three groups so that the direct sampling from any one group was possible conditioned on the other groups. Yuen et al. (2006) proposed a new Bayesian structural model updating approach using noisy incomplete modal data, which employs an iterative scheme involving a series of coupled linear optimization problems. Lam et al. (2006) formed a new practical SHM methodology by combing the pattern recognition method and the Bayesian ANN design method. Christodoulou and Papadimitriou (2007) formulated the damage detection problem as a weighted least square one based on the measured modal data and then used the Bayesian statistical approach to rationally select the optimal values of the weights. The Bayesian probabilistic approaches were also used for detection of cracks in beam-like structure (Lam and Ng, 2008) and railway ballast (Lam et al., 2012). Recently, a framework was presented by Papadimitriou et al. (2012) to integrate the component mode synthesis (CMS) technique and Bayesian finite element model updating formulations in order to reduce
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the time consuming operations due to reanalysis of structural models with a large number of degree of freedoms. 1.3 Outline of the Thesis Though significant progress has been achieved over the past decade, the application of Bayesian SHM approaches in the environment of WSN is still in its infancy. Therefore, the primary focus of this thesis is to develop new advanced Bayesian algorithms to fulfill the need of uncertainty quantification in WSN based SHM systems. This Thesis is organized as follows. Chapter 2 introduces the background of WSN based SHM. The hardware and software architecture to be adopted in this study are also described. Chapter 3 proposes a two-stage fast Bayesian spectral density approach to extract modal properties and their associated uncertainties for the cases of separated modes and closely spaced modes. Chapter 4 presents a Bayesian mode shape assembly methodology to assemble local mode shapes confined to different clusters. Simulated responses and in-situ measurements of shear building models are used to verify the accuracy of the proposed approaches for ambient modal analysis. Chapter 5 proposes a new Bayesian model updating methodology using incomplete modal properties identified through the two-stage fast Bayesian spectral density approach. The efficiency of the proposed method is investigated with numerical and experimental studies. Chapter 6 develops a Bayesian model updating technique by using non-stationary response measurements only. The theory described in this chapter is applied to system identification using synthetic data and laboratory testing data. Chapter 7 presents the conclusions of the thesis.
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Chapter 2 Background of WSN based SHM 2.1 General Introduction to Wireless Sensors 2.1.1 Basic Components and Essential Features of Wireless Sensors
Application dependent additional components
Sensing unit Sensor ADC
Processing unit Processor Storage
Transceiver unit
Power unit
Figure 2.1: The components of a wireless sensor node (Akyildiz et al., 2002) As autonomous data acquisition nodes, wireless sensors are usually viewed as platforms where the sensing transducer converges with wireless communication as well as mobile computing components (Lynch and Loh, 2006). As shown in Figure 2.1, a wireless sensor node is usually composed of a sensing unit, a processing unit, a transceiver unit, and a power unit as well as application dependent additional components, whose function has been well illustrated by Akyildiz et al. (2002). In particular, sensing units usually include sensors and analog to digital converter (ADC). In the ADC, the analog signal produced by the sensors can be converted to digital signals, which can then be stored, processed, and readied for communication in the processing unit. As the computational core, the processing unit is generally composed of a processor and a storage unit which can store measured data in random access memory (RAM) and data interrogation programs in read only memory (ROM) (Lynch and Loh, 2006). A transceiver unit is responsible for wireless communication so as to connect the node to the whole sensor network. The power unit may be supported by batteries or other renewable energy unit such as solar cells. 8
Compared with wired sensors, wireless sensors usually have several unique features (Spencer et al., 2004; Nagayama and Spencer, 2007) summarized as follows: (i) Wireless communication: Wireless sensors can communicate with each other through a wireless link. Radio frequency (RF) communication is still the most widely used transmission media for most of the wireless sensors to date. (ii) On-board computation: Programs can be embedded so as to facilitate the intelligence capabilities of saving data locally, performing computations, making decisions, sending results quickly, and coordinating with surrounding sensors, etc. (iii) Lower cost trend: With the increasing popularity of Micro-Electro-Mechanical System (MEMS) and microprocessors for a variety of applications, wireless sensors have the potential to be available at a low cost. Moreover, installation cost is also inexpensive compared with wired sensors, which enables numerous sensors to be densely distributed over large-scale civil infrastructures. (iv) Smaller size trend: Large scale integration technology for manufacturing MEMS brings the advantage that wireless sensors have the potential to be produced in small size. 2.1.2 Wireless Sensor Platforms to Date Numerous wireless sensor platforms well suited for SHM applications have emerged in recent decades. Spencer et al. (2004) provided an overview of the state-of-the-art of the wireless sensors technology and specified their opportunities and challenges in SHM. Lynch and Loh (2006) provided a literature review on the topic of wireless sensors for SHM by reviewing more than 150 papers conducted at over 50 research institutes worldwide in recent years. The platforms presented in these references were chronologically summarized into two broad categories, i.e., academic prototypes and commercially available platforms. For academic wireless sensor platforms, most of them were produced by individual research groups (e.g., Straser and Kiremidjian, 1998; Wang et al., 2003; Lynch, 2005, etc.) using commercial offthe-shelf (COTS) components. Though the academic prototypes improve the state-of-the-art of wireless sensors significantly, yet there are no widespread applications due to their proprietary nature. Due to the coordinated efforts between industrial and academic teams, some
commercial
wireless
sensor
platforms
such
as
platforms
from
Ember
(http://www.ember.com/) and Sensametrics (http://www.sensametrics.com/) have emerged in 9
recent years. Though well suited for SHM, these commercial wireless sensors still lack open sources, which has impeded their widespread application by the broader community. In contrast to the platforms not available to the public, researchers from the University of California at Berkeley developed in the late 1990s the first open source hardware and software which can be customized by users for particular applications. Funded by the Defense Advanced Research Projects Agency (DARPA), ‘smart dust’ or also so-called ‘mote’ was developed with the primary goal of creating massively distributed sensor networks to achieve autonomous, tiny, and low-power platforms (Hollar, 2000). Since the ‘smart dust’, a family of Berkeley Motes has been created. COTS Dust incorporating communications, processing, sensors and batteries into a package was usually regarded as the first generation of Berkeley Mote (Hollar, 2000). Following COTS Dust, Rene from the family of Berkeley Motes initially developed in 2000 was among one of the first prototypes to be commercialized. Hill and Culler (2002) released another well-known generation of Mote termed as ‘Mica’, whose memory capacity and processor speed were reported to be improved. Subsequent advancements to the Mica lead to the Mica2, Mica2Dot, and MicaZ platforms with their sensor boards separated from the main radio and processor board so that they can be used for a variety of purposes (Nagayama and Spencer, 2007). A next-generation Mote platform termed as Imote was released in 2003 by researchers from the University of California at Berkeley and the Intel Research Berkeley Laboratory (Kling, 2003). Later improvements lead to the second generation of Intel Mote termed as Imote2 which was released in 2005 (Adler et al., 2005). Compared with previous Motes, the Intel motes can improve the performance without increasing the overall power consumption significantly, while the sensor design for applications is much more flexible since there is no built-in ADC. The Intel motes with the enhanced computation, storage and communication capabilities have gained unparalleled popularity since they were made commercially available through Crossbow Technology (http://www.xbow.com/). Comparisons of these commercially available wireless sensor platforms are reported in (Nagayama and Spencer, 2007; Rice and Spencer, 2009).
10
2.1.3 Embedded Operational System The platforms introduced above cannot represent a complete wireless sensor since embedded software also occupies a decisive position in real applications. Lynch and Loh (2006) provided a comprehensive summary on the topic of software embedded in the wireless sensors. In their literature review, software was generally divided into two types, i.e., the operating system and engineering analysis software. The operating system located in the lowest layer can manage the operation of hardware. As an abstraction layer, it is able to hide the details of hardware from upper layers. Layers of engineering analysis software located above the operating system are responsible for automating wireless sensor operations so as to collect data, store data, process data and communicate to other wireless sensors collaboratively. In the context of vibration-based SHM, engineering analysis software embedded in wireless sensors usually includes algorithms of system identification. One of the most widely used operation systems is TinyOS, which was originally developed due to close research collaboration between the University of California at Berkeley and the Intel Research Berkeley laboratory. TinyOS is written in a new language named nesC, an extension to the C programming language (http://nescc.sourceforge.net/). As is illustrated in the official website (www.tinyos.net), TinyOS is an event-driven operating system, which utilizes a component-based architecture. The components are connected with each other through interfaces. The interfaces and components provided by TinyOS are used for sensing, routing and storage etc. Compared with other operation systems, TinyOS has many advantages (Lynch and Loh, 2006): (i) It is an open-source operation system readily available to the public for free use and modification; (ii) It is designed for sensor nodes with very limited resources such as severe memory and power constraints; (iii) It supports ad-hoc networking and multi-hop data transmission explicitly; (iv) In TinyOS, there are a number of basic services such as data collection, signal processing and wireless communication; (v) Numerous low-power modes of operation are included in the TinyOS to extend the service life of wireless sensors powered by portable batteries.
11
2.2 SHM using Wireless Sensors As a new emerging technology, wireless sensors have been installed upon a diverse set of laboratory structures or full-scale civil infrastructures to assess their performance. Straser and Kiremidjian were among the first researchers to detect the general structural state using wireless sensors (Straser and Kiremidjian, 1998; Kiremidjian et al., 1997). They developed a flexible instrumentation system, whose practical issues associated with remotely monitoring civil structures was also investigated through a highway bridge. Lynch et al. (2004a) designed a prototype wireless sensing unit, whose performance was validated with a series of laboratory and field tests including the Alamosa Canyon Bridge. They also embedded various modal identification or damage detection algorithms for local execution in their own wireless sensor prototype (Lynch et al., 2003; Lynch et al., 2004b; Lynch et al., 2006; Lynch, 2007). Eight Crossbow Mica Motes were deployed to measure the wind-induced response of the DiWang Tower in China (Ou et al., 2005). 64 MicaZ motes were installed on the Golden Gate Bridge by researchers from the University of California at Berkeley (Kim et al., 2007; Pakzad et al., 2008). With their customized accelerometer board, reliable communication protocol technique was implemented to achieve reliable communication. A centralized strategy was used for processing data and modal parameters were identified by autoregressive moving average model (ARMA) method (Pakzad and Fenves, 2009). Wang (2007) developed an academic prototype of hardware and software interfaces, which were successfully applied to both structural control (Wang et al., 2007) and structural health monitoring (Weng et al., 2008). An internationally well-known and esteemed research center is The Illinois Structural Health Monitoring Project (ISHMP) which aims to develop hardware and software for the reliable
monitoring
of
civil
infrastructure
systems
using
wireless
sensors
(http://shm.cs.uiuc.edu). Many critical issues of the Imote2 regarding time synchronization, synchronized sensing, model-based data aggregation and reliable communication were addressed by Nagayama and Spencer (2007). These issues were provided as middleware services of the ISHMP Services Toolsuite Version developed for the Imote2 platform. A decentralized data acquisition and processing approach termed as Distributed Computing 12
Strategy (DCS) was proposed by Gao and Spencer (2008) to ensure the scalability of WSNs, whose efficiency was experimentally verified by a 5.6 m long three-dimensional truss structure. Some important issues including decentralized modal analysis, efficient decentralized system identification and multimetric sensing were considered specifically by Sim and Spencer (2009) to increase performance, flexibility, and versatility of WSNs. Rice and Spencer (2009) designed a multi-metric Imote2 sensor board with onboard signal processing specifically designed for SHM applications. They also developed a flexible wireless sensor framework for full-scale, autonomous SHM integrating necessary software and hardware. Key implementation requirements were also well addressed. These accumulative efforts have led to the ISHMP Services Toolsuite, an open source library of services essential for developing SHM applications (http://shm.cs.uiuc.edu). Based on these efforts, full-scale applications such as monitoring the new Jindo Bridge (Jang et al., 2010) and a historic steel truss bridge in Mahomet (Jang et al., 2011) have been conducted, which demonstrate the strong potential for Imote2 to monitor large-scale civil infrastructures. 2.3 Hardware and Software Employed in this Study The hardware employed in this study mainly includes the Crossbow’s Imote2 platform and SHM-H sensor board developed by the ISHMP group. The software to operate the wireless sensors is composed of the operating system TinyOS and the engineering analysis software ISHMP Services Toolsuite. 2.3.1 Hardware: Crossbow’s Imote2 Platform and SHM-H Sensor Board Imote2 has been commercially available through Crossbow Technology since it was firstly developed by the University of California at Berkeley and the Intel Research Berkeley Laboratory. As an advanced wireless sensor platform, the Imote2 is built around the Intel’s low power XScale processor (PXA271), which is able to provide variable processing speeds according to application requirements so as to optimize its performance and overall power consumption (Intel Corporation Research, 2005). The onboard memory for Imote2 includes 32 MB of flash, 256 KB SRAM and 32 MB of SDRAM, which enables computationally intense data processing ideal for SHM employing ambient vibration. Also, it integrates an 13
802.15.4 radio (CC2420) with an external or onboard antenna for wireless communication. The wireless band for the radio is over 2.4 GHz, while its maximum transfer speed is around 250 Kbits/sec. Imote2 is more suitable for data intensive SHM applications due to its larger RAM space, larger flash memory size, low power radio and faster on-board processor. Some critical indices of the Imote2 are summarized in Table 2.1. For more details, one is referred to Imote2 Hardware Reference Manual (Crossbow Technology, 2007a). The top and bottom sides of Imote2 main board are shown in Figure 2.2. As can be seen from Figure 2.2, both the top and bottom sides of Imote2 are stackable with interface connectors. In particular, the top one is able to provide a standard set of I/O signals for basic expansion boards, while the bottom one is able to provide additional high-speed interfaces for application specific I/O (Crossbow Technology, 2007a). As a result, the sensor board and the battery board can be interfaced to the Imote2 main board via connectors in a stackable configuration, just as shown in Figure 2.3.
Figure 2.2: Top side (left) and bottom side (right) of Imote2 main board Sensor Board Imote2 Battery Board
Basic Connector
Advanced Connector
Figure 2.3: An overview of sensor board and battery board stacked on Imote2 main board
14
Table 2.1: Main features of Imote2 (Crossbow Technology, 2007a) Feature Processor Clock speed SRAM Memory SDRAM Memory FLASH Memory Transceiver Frequency Band (ISM) Data Rate Range (line of sight) Radio power Battery Voltage Weight Size (mm)
Value Intel PXA271 13MHz–416MHz 256 kB 32MB 32MB TI CC2420 2400.0 – 2483.5 MHz 250 kb/s ~30 m (with integrated antenna) 52mW (TX) 59mW (RX) 3.2- 4.5 V 12g 48mm 36mm 7mm
Table 2.2: Sampling rates and cutoff frequencies of ITS400C (Crossbow Technology, 2007b) Decimation factor 512 128 32 8
Cutoff frequency (Hz) 10 40 160 640
Sampling rate (Hz) 40 160 640 2560
As is seen from Figure 2.3, though there are no intrinsic sensing capabilities available for Imote2, it can be expanded with extension boards to meet a range of specific sensing applications. The initial commercially available accelerometer sensor board interfaced with the Imote2 is the Basic Sensor Board (ITS400C), which has a 3-axis digital accelerometer, a temperature and relative humidity sensor, a digital temperature sensor, a light sensor, and a general purpose 12-bit ADC (Crossbow Technology, 2007b). Rice and Spencer (2009) have pointed out some limitations of ITS400C in the context of SHM applications: (1) ITS400C lacks flexibility in selecting the sampling rate as well as cutoff frequency for data acquisition. As shown in Table 2.2, it only has four options for cutoff frequencies and sampling rates, which are determined by setting a decimation factor. (2) Another critical limitation of the Basic Sensor Board is that it has significant sampling rate errors, which indeed vary from sensor to sensor even with the same decimation factor. (3) The accelerometer has a range ±2g and a resolution around 0.98 mg/LSb (least significant bit), which is not able to measure the
15
ambient vibration with magnitude less than 1mg which is common for most civil structures. To address the problems of the ITS400C, Rice and Spencer (2009) developed a generalpurpose accelerometer board, the Structural Health Monitoring Accelerometer (SHM-A), which can implement three axes of acceleration measurement, temperature and relative humidity measurement. Through the use of the Quickfilter QF4A512 ADC and analog accelerometer with the OF4A512, the SHM-A board provides highly accurate user-selectable sampling rates and cutoff frequencies. The resolution that the ADC can achieve is around 0.43 mg if the full span of the accelerometer (±2g) is used, which is designed to meet the requirements of vibration-based SHM applications (Jo et.al, 2012). Though the SHM-A sensor board provides excellent resolution, it is not sufficient for lowlevel ambient vibration levels in the 1-2 mg range. As the extension of the SHM-A board, a new high-sensitivity sensor board, SHM-H sensor board, shown in Figure 2.4 was developed by Jo et al. (2010) for measuring low-level ambient vibrations of structures. The most significant difference between the SHM-H board and the SHM-A board is that the z-axis of the SHM-A board is replaced with a low-noise and high-sensitivity sensor due to its highest noise among the 3-axes. Therefore, it should include a single axis of high-sensitivity acceleration as well as two axes of general purpose acceleration and temperature/humidity measurements. As a result, a maximum resolution of 0.043mg sufficient to capture low-level acceleration in the range of 1-2 mg is achieved by limiting the measurement range of the sensor to ±0.2g for horizontal acceleration and 1±0.2g for vertical acceleration (Jo et al., 2012). In our study, the SHM-H sensor boards will be adopted for data acquisition. The main acceleration characteristics of SHM-H board are shown in Table 2.3. The following equation can be utilized to convert the raw outputs of the sensor board (ADC values) to units of acceleration (ISHMHP, 2009b): Acceleration
ADC values offset scale
(2.1)
where the scale and offset are known as the calibration constants, which can be determined in advance by using dynamic calibration or static approach. More detailed information regarding the SHM-H sensor board can be found in Jo et al. (2010) along with the calibration guide for wireless sensors (ISHMHP, 2009a). 16
Figure 2.4: Top (left) and bottom (right) of SHM-H board (Jo et al., 2010) Table 2.3: Characteristics of SHM-H sensor board (ISHMHP, 2009a) Axis
x & y axes: general purpose
z axis: high sensitivity
Parameters Acceleration Range Least significant bit (LSB) Sensitivity Zero-g offset Zero-g change vs. temperature, x & y axes Zero-g change vs. temperature, z axis RMS Noise level Maximum Frequency Acceleration Range Least significant bit (LSB) Sensitivity Zero-g offset Zero-g change vs. temperature, x & y axes Zero-g change vs. temperature, z axis RMS Noise level Maximum Frequency
Typical Values ±2 g 0.143mg 7000 LSB/g 14000 LSB ‐1.25 mg/C ‐2.75 mg/C 0.3 mg 1448 Hz ±0.2 g 0.0148 mg 67500 LSB/g 14000 LSB 0.4 mg 0.05 mg 400 Hz
2.3.2 Software: ISHMP Services Toolsuite Complicated programming such as network functionality and implementation of different algorithms are required for WSN based SHM systems. Though TinyOS has attracted widespread attention in various WSN based applications, it is still challenging for nonprogrammers without extensive expertise to develop engineering application software due to its potential complexity. A common and effective way to tackle the software complexity is using service-oriented architecture (SOA), with the term ‘service’ defined as self-describing software components (Avilés-López et al., 2009). As pointed by Rice and Spencer (2009), the application of SOA strategy can divide the software system into many smaller components 17
which are more manageable so that one can build an application from a set of well-defined services. Moreover, SOA can improve adaptability since services initially developed for a given application can be reused by many other applications, which provides a separation of concerns in real applications. In other words, service programmers from different fields can only focus on the services in their application domain since the middleware services can be provided by system programmers (Rice and Spencer, 2009). Figure 2.5 shows the software architecture of SHM application (Feng, 2010), which illustrates that application designers with limited knowledge of hardware-software interface can just focus on the higher-level SHM applications, while sharing the available middleware services such as reliable communication, time synchronization, data aggregation, etc.
Figure 2.5: Software architecture of SHM application (Feng, 2010) Based on the design principle of SOA, researchers of ISHMP in civil engineering and computer science at the University of Illinois at Urbana-Champaign have developed a software framework for structural health monitoring, which is termed as ISHMP Services Toolsuite. The ISHMP Services Toolsuite is composed of key middleware services necessary to provide high-quality sensor data and reliable data transmission across the sensor network, as well as a broad number of numerical algorithms (http://shm.cs.uiuc.edu/software.html). The components of the service-based framework can be divided into three primary categories, i.e., foundation services, application services, tools and utilities (Rice and Spencer, 2009; Sim, 2011): (1) Foundation Services provide commonly used wireless sensor functionalities that are required to support higher-level applications including basic communication and sensing functionalities
such
as
time
synchronization, 18
reliable
communication,
multi-hop
communication, etc. (2) The application services provide the numerical functionality necessary to implement the SHM algorithms such as correlation function estimate (CFE), eigensystem
realization
algorithm
(ERA),
covariance-driven
stochastic
subspace
identification (SSI) algorithm, output-only damage detection using the stochastic damage locating vector (SDLV) method etc. This part also includes application modules to test these algorithms on both the PC and the Imote2 platforms. (3) The tools and utilities are appealing for large scale or long-term WSN deployments since they are able to evaluate network conditions, adjust system parameters and assess power consumption issues. The utilities usually include services facilitating network development, while the tools provide services of gathering synchronized data from the network, performing damage detection, and testing radio communication quality. In addition, libraries of supporting numerical functions common to many SHM algorithms such as fast Fourier transform (FFT), singular value decomposition (SVD), Eigenvalue analysis, etc. are also provided by the ISHMP Services Toolsuite. Sharing the software framework available, researchers may only focus on the advancement of SHM approaches instead of concerning themselves with low-level networking, communication and numerical sub-routines. A more detailed explanation of the ISHMP Services Toolsuite can be found in the reports (Rice and Spencer, 2009; Sim and Spencer, 2009). 2.4 Summary As promising alternatives with potential to leverage off SHM techniques, wireless sensors have attracted widespread attention over the past decade. In this chapter, the background of SHM using WSN, including the basic components of wireless sensors, the main features over traditional wired sensors, the state-of-the-art of the wireless sensor platforms and the embedded operational systems has been introduced. Some successful applications of the WSN in SHM have also been reviewed briefly. Moreover, the hardware and software to be employed in the experimental studies have been specified. The hardware involves Crossbow Imote2 platform as well as the SHM-H sensor board, while the engineering analysis software employed is the ISHMP Services Toolsuite.
19
Chapter 3 A Two-Stage Fast Bayesian Spectral Density Approach for Ambient Modal Analysis: Theory 3.1 Introduction The Bayesian spectral density approach (BSDA) formulated previously (Katafygiotis and Yuen, 2001) is a promising candidate for being used in ambient modal analysis since it presents a strict way for obtaining optimal modal properties and their associated uncertainties. However, as will be explained in section 3.2, difficulties such as computational inefficiency and ill-conditioning have severely hindered the realistic application of BSDA. Motivated by the fast Bayesian FFT approach proposed recently (Au, 2011b; Au, 2012b; Au, 2012c), a twostage fast Bayesian spectral density approach is proposed for ambient modal analysis in this chapter so as to address the difficulties of conventional BSDA. Following the tactic of ‘divide and conquer’, one can divide the spectral density bandwidth into a series of frequency subbands, and each selected resonant frequency sub-band composed of one separated mode or several closely spaced modes can be conquered via two stages. In both cases, the interaction between spectrum variables (e.g., frequency, damping ratio as well as the magnitude of modal excitation and prediction error) and spatial variables (e.g., mode shape components) can be decoupled completely. The spectrum variables and their associated uncertainties can be identified by ‘fast Bayesian spectral trace approach’ (FBSTA) in the first stage based on the statistical properties of the trace of the spectral density matrix. Then, the spatial variables and their uncertainties can be extracted instantaneously through ‘fast Bayesian spectral density approach’ (FBSDA) in the second stage by using the statistical properties of the spectral density matrix. In this stage, the ill-conditioning issue of conventional BSDA can be well addressed by using matrix determinant and inversion manipulation algorithms. Moreover, this study reveals that FBSDA can be viewed as the linear superstition of fast Bayesian FFT approach incorporating multiple sets of measurements. The theoretical findings are to be
20
illustrated in chapter 4 using simulated and field data measured from laboratory models installed with wireless sensors. 3.2 Revisiting Bayesian Spectral Density Approach 3.2.1 Formulation of Bayesian Spectral Density Approach (Katafygiotis and Yuen, 2001) For a linear system with nd dofs, it is assumed that discrete acceleration responses are available for no ( nd ) measured dofs and the sampling time interval is assumed to be t . Assume that there are ns sets of independent and identically distributed time histories for no measured dofs. The j -th measured response denoted by y j (n) ( j 1, 2, , ns ) at the n-th time step nt (n 1, 2, , N ) is modeled as y j ( n) x j ( n) μ j ( n)
(3.1)
where x j (n) is the j -th model response, a function of the model parameters λ to be identified; μ j ( n)
is the j -th prediction error process, which can be adequately represented by a discrete
zero-mean Gaussian white noise vector process satisfying
E μ j (n)μTj (n) nn Σμ
(3.2)
where E () denotes the mathematical expectation; ()T denotes the transpose; nn is the Kronecker delta with the value of unit when n n , and zero otherwise; Σμ denotes the covariance matrix of μ j (n) . The Fast Fourier transform (FFT) of y j (n) at f k is defined as Yj (k ) YR , j (k ) iYI , j (k )
t 2 N
N 1
y n 0
j
(n) exp(i 2 f k nt )
(3.3)
where i 2 1 , f k k f , k 1, 2, , Int ( N 2) , and f 1 ( N t ) . In (3.3), YR , j (k ) and YI , j (k ) denote the real and imaginary part of Yj (k ) , respectively. In this work, ‘ k ’ shown in the bracket or in the
21
subscript denotes the frequency point f k . The discrete estimator of power spectral density matrix of y j (n) is defined as S y , j ( k ) Y j ( k ) Y j ( k )
(3.4)
where ‘ ’denotes the conjugate transpose of a complex vector. It has been proved by Yuen et al. (2002) that the random vector Yj (k ) follows Gaussian distribution as N , with mean and covariance matrix given by E (Yj (k )) 0 ; Ck E (S y , j (k ))
(3.5)
Given model parameters λ , Ck is given by, Ck (λ ) E (S x (k )) E (S μ (k ))
(3.6)
where E (S x (k )) denotes the mathematical expectation of power spectral density matrix of model response x j (n) , while E (Sμ (k )) denotes the mathematical expectation of spectral density matrix of the prediction error μ j (n) . E (S x (k )) is originally expressed in terms of discrete Fourier transform of the model correlation function, which is too tedious to derive (Yuen, 1999). E (Sμ (k )) is originally expressed as E (Sμ (k ))
t Σμ 2
(3.7)
where Σμ always includes a large number of parameters to be identified. For high sampling rate and long duration of data, Ck (λ ) over the specified frequency band can be replaced by a structured asymptotic form (Au, 2012b), Ck (λ ) ψΛ k ψ T s I no
(3.8)
where s is the spectral density of the prediction error; I n denotes the no no identity matrix; o
ψ ψ (1), , ψ (m), , ψ (nm ) no nm
is
the
modal
matrix
composed
of
nm
modes
with ψ (m) denoting the m-th mode shape over the specified frequency band. It is assumed that
22
the mode shapes identified are all normalized to unit norm, i.e., ψ (m) 1 . In (3.8), Λ k nm nm is the theoretical spectral density matrix of modal response given by Λ k diag (h k )S f diag (hk ) m nm
where Sf n
diag (h k ) nm nm
(3.9)
is the spectral density matrix of modal forces under white noise excitation; denotes a diagonal matrix and its m-th diagonal entry can be expressed as 2 hmk [( mk 1) i (2 mk m )]1
(3.10)
where mk f m f k ; here f m and m denote m-th natural frequency (in Hz) and damping ratio. Therefore, the (m, m) -entry of Λ k is given by Λ k (m, m) Sf , mm hmk hm k . Since ns sets of discrete FFT coefficients Yj (k ) are assumed to be independent and follow an identical complex normal distribution, then the sum of spectral density matrix estimator ns
Ssum Y j (k )(Yj (k )) k
(3.11)
j 1
follows the central complex Wishart distribution of dimension no with ns degrees of freedom (Goodman, 1963). It is assumed that the spectral density set Ssum {S sum , k k1 , , k2 } formed k over the frequency band k1f , k2 f are employed for ambient modal analysis. Based on Bayes’
theorem,
the
posterior
PDF
of
λ
is
proportional
to
the
likelihood
function p(Ssum λ ) given by k2
p(Ssum λ ) 0 Ck (λ )
ns
k k1
exp(tr (Ck 1 (λ )Ssum )) k
(3.12)
where 0 is a constant not depending on the model parameters λ ; tr () and denote the trace and determinant of a matrix, respectively. Under the assumption of non-informative prior distribution, it is convenient to write (3.12) in terms of the ‘negative log-likelihood function’ (NLLF) as k2
k2
k k1
k k1
L BSDA (λ ) ns ln Ck (λ ) tr (Ck 1 (λ )Ssum ) k
The term independent of λ is not included in (3.13). 23
(3.13)
As a result, the full set of modal parameters to be identified include λ [{ f m , m };{Sf , mm : m, m 1, , nm ; m m}; s ;{ψ (m)}]
(3.14)
The number of unknown parameters is equal to n p 2nm nm2 1 nm no . For the convenience of further illustration, the parameters λ spe { f m , m ,{Sf , mm }, s } independent of the spatial information are defined as ‘spectrum variables’, while the parameters λ spa {ψ (m) : m 1, , nm } dependent on spatial information are defined as ‘spatial variables’. Obviously, the dimension of ‘spatial variables’ grows along with the number of measured dofs and the number of modes significantly. The optimal parameter vector λˆ can be obtained by minimizing L BSDA (λ ) . Considering the second-order Taylor series about λ , one obtains (Au, 2011b) 1 L BSDA (λ ) L BSDA (λˆ ) (λ λˆ )T Γ BSDA (λˆ )(λ λˆ ) 2
(3.15)
where Γ BSDA (λˆ ) is the Hessian matrix of L BSDA (λ ) at the most probable value λˆ . Since the firstorder term of (3.15) vanishes due to the optimality of λˆ , the posterior PDF p(λ Ssum ) becomes a Gaussian distribution, 1 1 p(λ Ssum (λ λˆ )T CBSDA (λ λˆ )) ) exp( 2
(3.16)
1 (λˆ ) is the posterior covariance matrix of λˆ . Therefore, the updated PDF of where CBSDA ΓBSDA
the parameter λ can be well-approximated by a Gaussian PDF (λˆ , CBSDA ) with mean λˆ and covariance matrix CBSDA . 3.2.2 Computation Difficulties of BSDA BSDA is novel since it can accommodate different kinds of uncertainties and presents a strict way for quantifying the uncertainties useful for further risk assessment. However, there are some challenges which may outweigh its advantages: (1) BSDA requires solving a high-dimensional numerical optimization problem whose computational effort grows with the number of measured dofs no and the number of
24
modes nm . It is not likely for conventional BSDA to be implemented directly when the number of measured dofs is large. (2) When calculating the posterior covariance function, BSDA involves computing the inverse of Hessian matrix Γ BSDA (λˆ ) with the dimension of n p n p , whose computational effort and required memory space grow with the number of measured dofs no and the number of modes nm . Therefore, the problem of reducing the dimension of the Hessian matrix always arises. (3) BSDA involves repeated evaluations of the determinant and inversion of Ck (λ ) for different trial values of λ when numerically minimizing L BSDA (λ ) . Ck (λ ) can approximately be rank deficient (singular) for measurements with high signal-to-noise ratio due to the fact that Ck (λ ) over the resonant frequency band is dominated by the first term of (3.8) as the prediction error term is comparatively small. Therefore, the minimization problem may be ill-conditioned. (4) The dofs of interest are always measured and processed separately in different setups for large-scale civil infrastructures. Therefore, BSDA involves the difficulty on how to fuse the optimal parameters and their uncertainties identified from different setups together. 3.2.3 Solution Strategies To address all the issues aforementioned, a two-stage fast Bayesian spectral density approach is proposed in this study. To begin with, the entire frequency band can be divided into multiple sub-bands and one can only focus on a specified resonant frequency sub-band each time (Au, 2011b). There are two possible cases for each specified resonant frequency subband, i.e., the case of separated modes with a single mode to be identified, and the case of closely spaced modes with multiple modes to be identified. For both cases, the interaction between spectrum variables and spatial variables can be decoupled so that they can be conquered in two consecutive stages. In the first stage, the spectrum variables as well as their uncertainties can be identified through ‘fast Bayesian spectral trace approach’ (FBSTA) by 25
employing the statistical properties of the trace of spectral density matrix. The auto-spectral density of all measured dofs from different setups are collected together, thus FBSTA avoids the procedure of fusing spectrum variables identified from different setups. Once the spectrum variables are extracted, the spatial variables as well as their uncertainties can be identified by ‘fast Bayesian spectral density approach’ (FBSDA) using the statistical information of the spectral density matrix in the second stage. In this stage, the matrix determinant lemma and matrix inversion lemma are employed to avoid the ill-condoning of conventional BSDA. Then efficient optimization approach is employed to identify the mode shapes. As a result, the computation efforts of optimization and taking the inverse of Hessian matrix can be reduced significantly. The proposed method can deal with the practical difficulties mentioned above even when there are a large number of measured dofs. 3.2.4 Formulation of Bayesian Spectral Trace Approach As explained in section 3.2.3, a novel feature of this study is that FBSTA will be employed to decouple the interaction between spectrum variables and spatial variables. In this section, the general principle of ‘Bayesian spectral trace approach’ (BSTA) is formulated firstly. For an arbitrary Wishart matrix A with ns degrees of freedom and covariance matrix Σ , it can be proved that (tr ( A) nstr ( Σ) )
2ns tr ( Σ 2 )
asymptotically follows standard normal distribution
as ns is large (Mathai, 1980). As illustrated in section 3.2.1, the superposition of ns sets of spectral density matrix estimator, i.e., Ssum ) follows Wishart distribution, then the trace of Ssum should asymptotically k k follow normal distribution with mean ns tr (Ck (λ )) and covariance 2ns tr (Ck2 (λ )) when ns is large. Thus the PDF of tr (Ssum ) is given by k p(tr (Ssum ) λ) k
(tr (Ssum ) ns tr (Ck (λ ))) 2 k exp 4ns tr (Ck2 (λ )) 4 ns tr (Ck2 (λ )) 1
(3.17)
When the number of discrete data points N , Yj (k ) and Yj (k ) are uncorrelated as k k (Yuen, 1999). Therefore, tr (Ssum ) and tr (Ssum for k k are also independently normal k k ) 26
sum distributed. Conditioned on the set of trace tr (Ssum ) k k1 , k2 } formed over ) {tr (S k
[k1f , k2 f ]
, the updated PDF of λ is given by, according to the Bayes’ theorem, sum p(λ tr (S sum )) co p ( λ ) p (tr (S ) λ )
(3.18)
where co is a normalizing constant and the likelihood function p(tr (Ssum ) λ ) is equal to k2
sum p(tr (Ssum ) λ) ) λ ) p (tr (S k
(3.19)
k k1
In the case where a non-informative prior is used, the posterior PDF of λ is proportional to the likelihood function, which can be written in terms of NLLF L BSTA (λ ) as p(λ tr (S sum )) exp( L BSTA ( λ ))
(3.20)
with L BSTA (λ )
k2 [tr (Ssum ) ns tr (Ck (λ ))]2 1 k2 2 k ln(4 n tr ( C ( λ ))) s k 2 k k1 4ns tr (Ck2 (λ )) k k1
(3.21)
The most probable parameter vector λˆ can be obtained by minimizing L BSTA (λ ) , while the covariance matrix of λ can be obtained by taking the inverse of the Hessian matrix calculated at λ λˆ . As will illustrated in section 3 and section 4, the spectrum variables and spatial variables can be decoupled completely with the help of (3.21). The NLLF of BSTA shown in (3.21) is the foundation of (3.29) and (3.56), which are central to the first stage; while the NLLF of BSDA shown in (3.13) is the foundation of (3.42) and (3.85), which are the central to the second stage. 3.3 Formulation of Two-stage Bayesian Approach: Case of Separated Modes For the case of separated modes, there is only one mode to be identified over the specified frequency band k1f , k2 f . For clarity, the natural frequency, damping ratio, the amplitude of the modal excitation and the prediction error are denoted by f s , s , S fs , and s s , respectively. The mode shape is denoted by ψ s n . The subscript ‘ s ’ denotes ‘separated modes’. o
27
3.3.1 Stage One: Spectrum Variables Identification Using FBSTA 3.3.1.1 Negative Log-Likelihood Function For the case of separated modes, the covariance matrix Ck (λ ) in the vicinity of the resonant frequency shown in (3.8) reduces to Ck (λ ) k ψ s ψ Ts s s I no
(3.22)
k S fs Pk1
(3.23)
k in (3.22) is given by
where Pk [( sk2 1) 2 (2 sk s )2 ] with sk f s f k . As a result, the trace of Ck (λ ) can be written as tr (Ck ) k tr (ψ s ψ Ts ) tr ( s s I no )
(3.24)
tr (ψ s ψ Ts ) ψ Ts ψ s
(3.25)
tr ( s s I no ) no s s
(3.26)
It is a simple fact that
Under the unit norm normalization assumption, i.e., ψTs ψ s 1 , it is possible to arrange (3.24) as tr (Ck (λ )) k no s s
(3.27)
Similarly, tr (C2k (λ )) is given by tr (Ck2 (λ )) tr ( 2k (ψ s ψ Ts ) 2 2 k s s ψ s ψ Ts s2 s I no ) tr ( k2 ψ s ψ Ts 2 s s k ψ s ψ Ts s2 s I no )
(3.28)
2 s s k no s s 2 k
2
After substituting (3.27) and (3.28) into the NLLF of BSTA, i.e., (3.21), the NLLF of FBSTA can be obtained as L FBSTA (λ )
k2 [tr (Ssum ) ns ( k no s s )]2 1 k2 k 2 2 n s n s ln 4 ( 2 ) s k s k o s 2 2 2 k k1 k k1 4ns [ k 2 s s k no s s ]
28
(3.29)
3.3.1.2 Most Probable Values From (3.29), one can observe that the NLLF depends only on four spectrum variables, i.e., λ spe { f s , s , S fs , s s } .
Moreover, there is no complicated matrix operation (e.g., determinant,
inverse and singular value decomposition, etc.) in (3.29). Thus the optimal value can be obtained by unconstrained numerical optimization without much computational effort. The initial guess for the natural frequency can be picked from the peak of the trace of spectral density matrix, and the initial damping ratio can be assigned a nominal value around 0.01. It has been proved that (Au, 2011b) the maximum eigenvalue of the spectral density matrix S y (k )
at the resonant frequency can be approximately by S fs 4 s2 , which implies that one can
assign the initial guess of S fs as 4 m2 multiplied by the largest eigenvalue of S y (k ) with f k fixed at the initial guess of the natural frequency. 3.3.1.3 Posterior Uncertainties of Spectrum Variables With a sufficiently large amount of data, the updated PDF of λ spe can be well-approximated by a Gaussian PDF equivalent to second-order approximation of L FBSTA (λ spe ) . The Hessian matrix contains the second derivatives of L FBSTA (λ spe ) with respect to all the spectrum parameters λ spe arranged into a one-dimensional array. The derivatives of Γ FBSTA (λˆ spe ) can be calculated through analytical derivation shown in Appendix I or finite differentia approach (Yuen, 2010). 3.3.2 Stage Two: Spatial Variables Identification Using FBSDA 3.3.2.1 Negative Log-Likelihood Function In this stage, the core issue is to address the ill-conditioning of estimating Ck (λ ) and Ck 1 (λ ) numerically. According to matrix determinant lemma (Harville, 1997), if A is an arbitrary invertible square matrix while u , v are column vectors, then Α uvT (1 vT Α 1u) Α
29
(3.30)
Let A s s k 1I n and u v ψ s , then Ck (λ ) can be rearranged as o
Ck (λ ) k ( s s k 1I no ψ s ψ Ts ) nko (1 s1s k ψ Ts I no ψ s ) s s k1I no
(3.31)
It is easy to find that, s s k1I no ( s s k1 ) no
(3.32)
Ck (λ ) s( nso 1) ( s s k )
(3.33)
Substituting (3.32) into (3.31) leads to
Moreover, as illustrated in the Sherman-Morrison formula (Sherman et al., 1949), if Α is an arbitrary invertible square matrix, while u , v are column vectors with 1 vT Α 1u 0 , then ( Α uvT ) 1 Α 1 (1 vT Α 1u) 1 Α 1uvT Α 1
(3.34)
By setting A s s k 1I n and u v ψ s , according to (3.34), Ck 1 (λ ) can be rewritten as o
( s2s k2 )ψ s ψ Ts Ck1 (λ ) k1 ( s s k1I no ψ s ψ Ts ) 1 k1 s1s k I no (1 s1s k ψ Ts ψ s )
(3.35)
(3.35) can be rearranged as Ck1 (λ ) s1s I no s1s [1 s s k ]1 ψ s ψ Ts
(3.36)
After substituting (3.33) and (3.36) into the NLLF of BSDA, i.e., (3.13), the NLLF of FBSDA is given by k2
k2
k k1
k k1
L FBSDA (λ ) ns n f (no 1) ln s s ns ln( s s k ) s1s tr (Ssum ) k s
1 s
k2
[1 s
k k1
1
s
k ] tr (ψ s ψ S T s
sum k
(3.37)
)
where n f k2 k1 1 . One should note that (Brookes, 2005) tr (ψ s ψ Ts Ssum ) ψ Ts S sum k k ψs
(3.38)
After simplification, (3.37) can be written as k2
L FBSDA (λ ) ns n f (no 1) ln s s ns [ln( s s k ) s1s (d ψ Ts Ξψ s )] k k1
where
30
(3.39)
k2
d tr (Ssum ) k k k1
k2
and Ξ [1 s s k ]1Ssum k
(3.40)
k k1
(3.39) can be rearranged as the sum of ns terms incorporating ns different sets of measurements, ns
L FBSDA (λ ) L FBSDA , j (λ )
(3.41)
j 1
where L FBSDA , j (λ ) is given by k2
L FBSDA , j (λ ) n f (no 1) ln s s [ln( s s k ) s1s (d j ψ Ts Ξ j ψ s )]
(3.42)
k k1
with dj L FBSDA , j (λ )
k2
Y (k ) Y (k ) *
k k1
j
j
k2
and Ξ j [1 s s k ]1 Yj (k )Yj (k )*
(3.43)
k k1
is the NLLF corresponding to j -th set of measured time history. In the case of
separated modes, the spectral density matrix over the selected resonant frequency band satisfies Yj (k )Yj (k )* YR , j (k )YR , j (k )T YI , j (k )YI , j (k )T .
Therefore,
by
ignoring
the
term
independent of λ , the formulation of LFBSDA , j (λ ) is completely the same as NLLF of fast Bayesian FFT approach with separated modes using j -th measurement only (Au, 2011b). Fast Bayesian spectral density approach for the case of separated modes can be viewed as the linear superstition of fast Bayesian FFT approach incorporating different sets of measurement.
3.3.2.2 Most Probable Values In (3.39), one can observe that the mode shape has been expressed in an explicit manner. Since LFBSDA (λ ) depends on ψ s solely through the last quadratic term, the most probable values of ψ s can be obtained just by minimizing last quadratic term Min. L FBSDA (ψ s ) s1s ψ Ts ψ s S.t.
(3.44)
ψ Ts ψ s 1
(3.45)
(3.44) together with (3.45) belong to classical Rayleigh Quotient minimization problem, and the optimal ψˆ s is just the eigenvector of Ξ corresponding to the largest eigenvalue (Anstreicher
and
Wolkowicz,
2000;
Au,
2011b).
Since
the
optimal
spectrum
variables f s , s , S fs and s s have been obtained in the first stage by FBSTA described in section 31
3.3.3.1, the optimal mode shape ψˆ s can be obtained instantaneously by performing a singular value decomposition on Ξ after substituting optimal fˆs , ˆs , Sˆ fs and sˆ s into LFBSDA (ψ s ) ψTs ψ s .
3.3.2.3 Posterior Uncertainties The posterior covariance matrix of the mode shape can also be approximated by the inverse of the Hessian matrix of the LFBSDA (ψ s ) . The Hessian matrix Γ FBSDA (ψˆ s ) can be calculated through analytical derivation or finite differentia approach. It is worth noting that the norm constraint of each mode shape ψ s 1 needs to be taken into account when differentiating the NLLF of FBSDA in terms of ψ s . The constraint can be handled by expressing the mode shapes explicitly in unit vector form in LFBSDA (ψ s ) with ψ s being replaced by ψ s ψ s ψ s (Au, 2011b) so that the modified NLLF is given by L FBSDA (ψ s ) s1s
ψ Ts ψ s ψ Ts ψ s
(3.46)
The general formulation of the derivatives of the Rayleigh quotient with respect to ψ s can be found in Au (2011b). L FBSDA (ψˆ s ) is invariant to the scaling of the mode shapes by construction, indicating that Γ FBSDA (ψˆ s ) have zero eigenvalues with eigenvectors parallel to the mode shape directions (Au, 2011b). Let {1 , 2 , , n } arranged in ascending order be the eigenvalues o
of Γ FBSDA (ψˆ s ) , while their corresponding eigenvectors are assumed to be {υ1 , υ2 , , υn } . The o
covariance matrix CFBSDA (ψˆ s ) can be evaluated properly via its eigen-basis representation with the first zero eigenvalue ignored (Au, 2011b) no
ˆ s ) i1υi υTi CFBSDA (ψ
(3.47)
i2
As a result, the irrelevant contributions from the singular terms (zero curvature) can be excluded.
32
3.3.3 Summary of Procedure The procedures of two-stage fast Bayesian spectral density approach with separated modes can be summarized as follows: (i) Set initial guess for { f s , s , S fs , s s } ; (ii) Optimize { f s , s , S fs , s s } to minimize (3.29) till convergence; (iii) Calculate the covariance matrix of CFBSTA (λ spe ) with respect to the spectrum variables λ spe at the most probable value; (iv) Perform a singular value decomposition on Ξ to obtain the most probable value (MPV) of mode shape after substituting the most probable values of { f s , s , S fs , s s } into (3.44); (v) Calculate the covariance matrix CFBSDA (ψˆ s ) with respect to the spatial variables using (3.47).
3.4 Formulation of Two-stage Approach: Case of Closely Spaced Modes In section 3.3, a two-stage fast Bayesian spectral density approach has been developed to identify modal parameters instantaneously even for a large number of measured dofs when there is a single mode over the specified frequency band. However, such method is not suitable for the case of closely spaced modes commonly encountered in real application. As a sequel to section 3.3, this section aims to consider the case of closely spaced modes. Assume that there are nm modes over the specified resonant frequency band k1f , k2 f . Sharing the same symbols as those in section 3.2.1, the m-th natural frequency, m-th damping ratio, the amplitude of the modal excitation and the prediction errors are denoted by f m , m , Sf n
m nm
and s , while the mode shapes are denoted by ψ ψ (1), , ψ (m), , ψ (nm ) n n . o
m
To begin with, the mode shapes ψ n n are represented as a linear combination of mode o
m
shape basis B n n which is composed of orthonormal basis of mode shape subspace (Au, o
m
2012b), ψ Bα
33
(3.48)
m nm
where α n
is a condensed set of coordinates with m-th column α (m) containing the
coordinates of ψ (m) with respect to the basis B . The dimension nm mentioned above is equal to the rank of ψ with nm min(nm , no ) . Since ψ (m) 1 and BT B I n , α (m) should satisfy the m
following constraint α (m) α (m)T α (m) α (m)T BT Bα (m) ψ (m)T ψ (m) 1 2
(3.49)
One can observe that α is independent of the number of measured dofs (spatial information), thus it belongs to spectrum variables. Therefore, only B belongs to spatial variables. As a result, the natural target herein is to separate B from the full set of variables.
3.4.1 Stage One: Spectrum Variables Identification by FBSTA 3.4.1.1 Negative Log-Likelihood Function For the case of closely spaced modes, the covariance matrix is the same as the general formulation shown in (3.8). Substituting (3.48) into (3.8) results in Ck BΘ k BT s I no
(3.50)
where Θ k αΛ k αT . For an arbitrary matrix A of appropriate size, one has (Brookes, 2005) tr (BABT ) tr (BT BA)
(3.51)
Since B is orthonormal basis satisfying BT B I n , (3.51) can be further simplified as m
tr (BABT ) tr ( A)
(3.52)
tr (Ck (λ )) tr (BΘ k BT ) tr ( s I no ) tr (Θ k ) s no
(3.53)
As a result, tr (Ck (λ )) can be written as
Furthermore, tr (C2k (λ )) can be expressed as tr (Ck2 (λ )) tr (BΘ k BT BΘ k BT 2BΘ k BT s I no s2 I no ) tr (BΘ k2 BT ) tr (2s BΘ k BT ) tr ( s2 I no )
Using (3.51), (3.54) can be simplified as
34
(3.54)
tr (C2k ) tr (Θ 2k ) 2 s tr (Θ k ) no s2
(3.55)
By substituting (3.53) and (3.55) into the NLLF of BSDA (i.e., (3.21)), the most probable parameters can be obtained by minimizing the L FBSTA (λ ) given by L FBSTA (λ )
k2 [tr (Ssum ) ns (tr (Θ k ) s no )]2 1 k2 k 2 2 ln[4 n ( tr ( Θ ) 2 s tr ( Θ ) s n )] o s k k 2 2 2 k k1 k k1 4ns (tr (Θ k ) 2 s tr (Θ k ) s no )
(3.56)
As can be seen from (3.56), the spatial variables B have been eliminated completely from the full set of modal parameters by FBSTA so that the optimization dimension has been reduced dramatically. The minimization objective function does not include any time-consuming matrix operations such as determinant, inverse and singular value decomposition.
3.4.1.2 Most Probable Values One can observe from (3.56) that the NLLF of FBSTA depends only on the spectrum variables including λ spe [ f m , m ; α (m); U f , mm , Vf , mm : m, m 1, , nm , m m; s ]
(3.57)
where Uf ,mm and Vf ,mm denote the real and imaginary part of Sf ,mm . As a result, the first stage of the proposed method reduces to solving the following optimization problem, Min.
k2 [tr (S sum ) ns (tr (Θ k ) s no )]2 1 k2 k 2 2 ln[4 n ( tr ( Θ ) 2 s tr ( Θ ) s n )] o s k k 2 2 2 k k1 k k1 4ns (tr (Θ k ) 2 s tr (Θ k ) s no )
S.t.
2
α ( m) 1
(3.58) (3.59)
The optimal value of the spectrum variables λ spe can be obtained by numerical optimization approach with unit norm constraint of α(m) . To consider the constraints of α(m) automatically, an efficient parameterization scheme (Au, 2012b) is employed in this study to relax the constraint of α(m) . Following the parameterization scheme, the spectral density matrix of modal force can be written as Sf diag (Sf )Sf diag (S f )
where Sf is a dimensionless form of Sf whose (m, m) -entry is equal to
35
(3.60)
Sf , mm S f , mm
S f , mm S f , mm
(3.61)
As a result, Θ k can be rearranged as Θ k αΛ k αT αΛk αT
(3.62)
Λ k diag (h( f k ))Sf diag (h( f k ))
(3.63)
α S f ,11 α (1), , S f , nm nm α (nm )
(3.64)
where
As can be seen from (3.64), α are free parameters without norm constraints, while Sf are subjected to the following constraints: Sf , mm 1 ; Sf , mm [Sf , mm ] and Sf , mm 1
(3.65)
The parameterization scheme shows that, through introducing the angles umm , vmm , the conditions in (3.65) can be automatically satisfied Sf , mm sin(umm ) exp(ivmm )
(3.66)
As a result, the spectrum variables required to be identified reduce to λ spe [ f m , m ; α (m); umm , vmm : m, m 1, , nm , m m; s ]
(3.67)
Once the optimal values of α and {umm , vmm } are obtained, the optimal values of α and Sf can be recovered by the following expressions α (m) α(m) α(m)
Sf , mm α (m)
2
; Sf , mm Sf , mm and Sf , mm Sf ,mm Sf , mm sin(umm ) exp(ivmm )
(3.68) (3.69)
Common to fast Bayesian FFT approach, the initial guess for all parameters can be set approximately. The initial guess for the natural frequencies can be picked from the peaks of the trace of spectral density matrix. The initial guess for damping ratio can be assigned a nominal value of 0.01. The angles umm and vmm may be nominally assigned as 0.1. The maximum eigenvalue of the spectral density matrix S y ( f k ) at the resonant frequency can be approximately by Sf , mm 4 m2 , implying that one can assign the initial guess of Sf ,mm as
36
4 m2 multiplied by the largest eigenvalue of S y ( f k ) with f k being the initial guess of the natural
frequency (Au, 2012b). The first nm nm partition of α can be assumed to be an identity matrix with the remaining entries assigned randomly. Once α and Sf ,mm are assigned, α can be initially assigned by (3.64) .
3.4.1.3 Posterior Uncertainties With a sufficiently large amount of data, the updated PDF of the parameter λ spe can be wellapproximated by a Gaussian PDF. Its posterior covariance matrix can be approximated by the inverse of the Hessian matrix containing the second derivatives of L FBSTA (λ ) with respect to the spectrum parameters λ spe arranged into a one-dimensional array shown in (3.57). It is worth noting that the norm constraint of each mode shape α (m) 1 should be considered. When differentiating LFBSTA (λ ) with respect to α , α(m) should be replaced by α (m) α (m) α (m) . Then (3.56) can be modified as L FBSTA (λ )
k2 ) s n )]2 [tr (Ssum ) ns (tr (Θ 1 k2 o k k 2 ) 2 s tr (Θ ) s 2 n )] Θ ln[4 n ( tr ( s k k o 2 2 2 k k1 k k1 4ns (tr (Θ k ) 2 s c tr (Θ k ) s no )
(3.70)
k α T . The Hessian matrix Γ FBSTA (λˆ spe ) can be calculated through analytical where Θ k αΛ
derivation shown in appendix II or finite differentia approach. As can be seen from (3.70), Γ FBSTA (λ spe ) is
invariant to the scaling of α(m) by construction, which implies that the Hessian
matrix has zero curvature along the directions of α , and it has zero eigenvalues with eigenvectors parallel to α . Assume that {1 , 2 , , n } arranged in ascending order denotes the spe
eigenvalues
of
the
Hessian Γ FBSTA (λˆ spe ) ,
while
their
corresponding
eigenvectors
are {υ1 , υ2 , , υn } . The irrelevant contributions from the singular terms can be excluded by spe
CFBSTA (λ spe )
nspe
i nm 1
37
i1υi υTi
(3.71)
3.4.2 Stage Two: Spatial Variables Identification by FBSDA 3.4.2.1 Negative Log-Likelihood Function In this stage, the spatial variables and their uncertainties can be extracted instantaneously through ‘fast Bayesian spectral density approach’ (FBSDA). To address the ill-condition of conventional BSDA, the matrix determinant lemma and matrix inversion lemma are adopted to calculate the analytical expression of Ck (λ ) and Ck 1 (λ ) so as to avoid numerical estimation. For complex matrices A, U, W, V of appropriate size with A and W invertible, the matrix determinant lemma states that (Brookes, 2005) A UWVT W 1 VT A 1U W A
(3.72)
Using (3.72) with A s I n , U V B and W Θ k , one obtains that o
Ck (λ ) BΘ k BT s I no Θ k 1 BT ( s I no ) 1 B Θ k s I no Θ k 1 s1I nm Θ k s I no
Using (3.72) again with A Θ k1 , U V I n and W s1I n , the term Θ k 1 s1I n m
m
m
(3.73)
in (3.73) can
be further rearranged as, Θ k 1 s1I nm s I nm I nm Θ k I nm Θ k 1 s1I nm s I nm Θ k Θ k 1 s1I nm
(3.74)
Substituting (3.74) into (3.73) gives Ck (λ ) s( no nm ) s I nm Θ k
(3.75)
Furthermore, given complex matrices A, U, W and V of appropriate size with A and W invertible, the matrix inversion lemma can be expressed as (Brookes, 2005), ( A UWV ) 1 A 1 A 1U( W 1 VA 1U) 1 VA 1
(3.76)
Using (3.76) with A s I n , U B , V BT and W Θ k , one can obtain that o
Ck1 (λ ) ( s I no ) 1 ( s I no ) 1 B(Θ k1 s1BT I no B) 1 BT ( s I no ) 1
(3.77)
Above equation can be further simplified as Ck1 (λ ) s1I no s2 B(Θ k1 s1I nm ) 1 BT
(3.78)
Using (3.76) again with A s1I n , U V I n and W Θ k 1 , the term (Θ k 1 s1I n )1 can be m
m
rearranged as 38
m
(Θ k 1 s1I nm ) 1 s I nm s2 (Θ k s I nm ) 1
(3.79)
Substituting (3.79) into (3.78) gives Ck1 (λ ) s1I no s1B(I nm s (Θ k s I nm ) 1 )BT
(3.80)
Denoting Εk Θ k s I n and substituting (3.75) and (3.80) into the NLLF of BSDA, (3.13), m
result in k2
k2
k k1
k k1
L FBSDA (λ ) ns n f (no nm ) ln s ns ln Εk s1d s1 tr (BΩ k BT S sum ) k k2
where d tr (Ssum ) and Ω k I n k
m
k k1
(3.81)
s Εk 1 .
For a matrix A and complex vector u , one has (Brookes, 2005) tr ( Auu ) u Au and tr (uu ) uu
Then
sum tr (B k BT Ssum k ) and tr (S k ) can
(3.82)
be formulated as, ns
ns
j 1
j 1
tr (B k BT Ssum ) tr (B k BT Y j (k )Yj (k ) ) Y j ( k ) B k BT Y j (k ) k ns
tr (Ssum ) Y j ( k ) Y j ( k ) k
(3.83) (3.84)
j 1
Substituting (3.83) and (3.84) into (3.81) leads to ns
k2
j 1
k k1
ns
k2
L FBSDA λ ns n f (no nm )ln s s1 d j ns ln Ε k s1 Yj (k ) B k BT Y j (k )
(3.85)
j 1 k k1
k2
with d j Yj (k ) Yj (k ) . k k1
One can observe that (3.85) can be written as the summation of ns terms, ns
L FBSDA λ L FBSDA, j (λ )
(3.86)
j 1
where k2
k2
k k1
k k1
L FBSDA, j λ n f (no nm )ln s s1d j ln Ε k s1 Yj ( k ) B k BT Y j (k )
(3.87)
As can be seen from (3.87), by ignoring the constant term independent of λ , the form of L FBSDA, j (λ )
is the same as the NLLF of fast Bayesian FFT approach with closely spaced modes
(Au, 2012b), which again reveals that FBSDA can be viewed as the linear superposition of fast Bayesian FFT approach incorporating ns different sets of measurements.
39
3.4.2.2 Most Probable Values The spectrum variables have been available from the first stage, thus only the last term with mode shape basis B should be considered for identification. As a result, the second stage of the proposed method reduces to solving the following optimization problem with the spectrum variables fixed at their optimal values obtained in the first stage: ns
k2
Min. L FBSDA (B) s1 Yj (k ) B k BT Yj (k )
(3.88)
j 1 k k1
S.t.
BT B I nm
(3.89)
The feasible set defined by the constraint (3.89) is called the Stiefel manifold in differential geometry. It is not trivial to minimize (3.88) with respect to B due to the orthonormal constraint in (3.89) which is quadratic and matrix in nature. Au (2012b) proposed a general approach to optimize B under orthonormal constraints, in which B can be parameterized with a necessary and sufficient number of free parameters so that it can be automatically satisfied. Only the basic procedures are illustrated here. Interested readers are referred to Au (2012b). k2
The first no real eigen-basis of
S
k k1
sum k
arranged in descending order of eigenvalues can be
taken as the reference mode shape basis Bo n n . Then the orthonormal basis Bo can be o
o
divided two groups with one containing the first nm columns and the other containing the remaining no nm columns. It has been illustrated that B can be parameterized by nm (no nm ) free
angles aij ( i 1, , nm ; j 1, , no nm ) corresponding to the rotations within the
planes formed by one column vector from the first group and the other from the second group. Through the deviatoric rotations from reference basis Bo , a new basis B n n containing a o
o
better estimate for an orthonormal basis B in first nm columns can be obtained as B Bo R
(3.90)
where R ({aij :}) n n is the function of rotational angles {aij :} given by o
o
nm no nm
R ({aij :}) i 1
40
R j 1
ij
(aij )
(3.91)
Here R ij (aij ) denotes the rotation matrix in the standard Euclidean space rotating from the i -th axis to the j -th axis
by an angle aij (i 1, 2, nm ; j nm 1, , no ) , which is given by R ij (aij )
no
e
k i, j
kk
(eii e jj ) cos(aij ) (e ji eij ) sin(aij )
(3.92)
where eij n n denotes a no no matrix with the (i, j ) -entry equal to unity while all other o
o
entries are zero. The first nm columns of R is denoted as R n n satisfying o
m
B Bo R
(3.93)
Substituting (3.93) into (3.88), after rearrangement, gives {R :} L FBSDA (R ) {R :}T Q
(3.94)
where R : denotes the vectorization of R , and Q is a Hermitian matrix given by ns
k2
T T s 1 Q (I nm Y j ( k ) B o ) Ω k (I nm Y j ( k ) B o )
(3.95)
j 1 k k1
where ‘ ’ denotes Kronecker product. The
gradient
and
Hessian
(i 1, 2, nm ; j nm 1, , no )
of
L FBSDA (R )
with
respect
to
the
angles aij
can be derived analytically as, ij {R L FBSDA (R ) {R :}T Q
(a )
ij rs {R L FBSDA (R ) 2{R ( ars ) :}T Q
(a a )
( aij )
( aij )
(3.96)
:}
{R :} 2{R :}T Q
( aij ars )
(3.97)
:}
where {R ( a ) :} , {R ( a ) :} and {R ( a a ) :} denote the vectorization of R ( a ) , R ( a ) and R( a a ) . Here ij
R
( aij )
rs
ij rs
ij
rs
ij rs
and R ( a ) are the gradient R , while R( a a ) is the Hessian of R . ij rs
rs
Then Newton’s method can be used for optimizing the rotation matrix R efficiently, starting with initial guess aij 0 . Once the optimal rotation matrix Rˆ is obtained, the optimal mode shape basis can be calculated as ˆ Bˆ B o R
(3.98)
As a result, using αˆ calculated in the first stage, the optimal values of the mode shapes ˆ {ψ ˆ (1), , ψ ˆ (m), , ψˆ (nm )} can ψ
be obtained as ˆ Bˆ αˆ ψ
41
(3.99)
3.4.2.3 Posterior Uncertainties Since we are more interested in the uncertainty of ψ rather than B , the posterior covariance matrix of ψ rather than B ought to be obtained. The covariance of ψ can be approximated by the inverse of the Hessian of the (3.13). The mode shapes can be arranged as a onedimensional array with the number of components nspa no nm . It is worth noting that the norm constraint of each mode shape ψ (m) 1 needs to be taken into account when calculating the Hessian matrix Γ FBSDA (ψ ) , thus (3.13) should be modified as k k2 tr (C L BSDA (ψ ) ns k2 k ln C ) k k k 1Ssum k k 1
(3.100)
1
ψ (1) ψ (1) , , ψ (m) ψ (m) , , ψ (n) ψ (n) . According to the kψ T s I n with ψ where C k (λ ) ψΛ o
definition of Ssum , (3.100) can be further rearranged as, k ns
k2
k2
(λ ) L FBSDA (ψ ) ns ln C Y*j (k )C k 1 (λ )Yj (k ) k k k1
(3.101)
j 1 k k1
The Hessian matrix Γ FBSDA (ψˆ ) can be calculated through analytical derivation or finite differentia approach. (3.101) can be written as the linear superposition of the NLLF of fast Bayesian FFT approach corresponding to different measurements shown in (Au, 2012c), ns ns k k2 2 (λ ) 1 (λ )Y (k ) L FBSDA (ψ ) L FBFFT, j (ψ ) ln C Y*j (k )C k k j j 1 j 1 k k1 k k1
(3.102)
Thus ΓFBSDA (ψ ) is just equal to the linear superposition of Hessian matrix of L BFFT, j (ψ ) , which has been derived by Au (2012c). Assume that {1 , 2 , , n } arranged in ascending order is the spa
eigenvalues of the Hessian Γ FBSDA (ψˆ ) and their corresponding eigenvectors are assumed to be {υ1 , υ2 , , υnspa } which
form an orthonormal eigen-basis. Similar to the case of separated modes,
the inverse of Γ FBSDA (ψˆ ) can be evaluated properly via CFBSDA (ψˆ ) first nm zero eigenvalues .
42
nspa
i nm 1
i1υi υTi by ignoring the
3.4.3 Summary of Procedures The procedure of two-stage Bayesian spectral density approach with closely spaced modes can be summarized as follows: (i) Set initial guess for { f m , m } , α , {umm , vmm } and s ; (ii) Optimize { f m , m } , α , {umm , vmm } and sc to minimize (3.56); (iii) Calculate Γ FBSTA (λˆ spe ) to obtain the covariance of the spectrum variables using (3.71); (iv) Set reference basis Bo and the initial guess for angle aij 0 ; (v)Optimize B to minimize (3.94) with spectrum variables fixed at the optimal values obtained in the first stage so that ψˆ can be obtained using (3.99). (vi) Calculate CFBSDA (ψˆ ) using CFBSDA (ψˆ )
nspa
i nm 1
i1υi υTi .
3.5 Concluding Remarks To address the challenges of conventional BSDA, a two-stage fast Bayesian spectral density approach was proposed in this chapter for ambient modal analysis. The interaction between spectrum variables and spatial variables can be decoupled for the cases of separated modes and closely spaced modes so that the modal parameters involved can be identified independently via a two-stage approach. The spectrum variables can be identified instantaneously by FBSTA, while the spatial variables can be identified subsequently through FBSDA. The most probable variables can be extracted by fast numerical optimization approaches, while their uncertainties can be derived analytically. The dimension involved in solving the most probable values as well as taking the inversion of the Hessian matrix is reduced significantly after employing the proposed strategies. The issue of ill-condition was well addressed by employing the matrix determinant lemma and matrix inversion lemma to obtain analytical expression and avoid numerical estimation. There is no need to fuse the identified spectrum variables from different setups together since the FBSTA is able to incorporate information contained in all measured dofs. This study also reveals the
43
relationship between FBSDA and fast Bayesian FFT approach when there are multiple sets of measurements. The proposed theory is to be illustrated with applications to simulated and field test data, after discussing the problem of mode shape assembly from multiple clusters in chapter 4.
Appendix Appendix I: Derivatives of LFBSTA for case of separated modes
For the purpose of evaluating the Hessian, the NLLF L FBSTA for the case of separated modes is denoted as L FBSTA
1 k2 1 ln(4 ns LS1,k ) 4n 2 k k1 s
k2
L
k k1
2 S2,k
(3.103)
LS1,k
where LS1,k k2 2 s s k no s2 s
and LS2,k tr (Ssum ) ns ( k no s s ) k
(3.104)
A parenthesized variable is employed in the superscript to denote a derivative with respect to it. For any x or y denoting f s , s , S fs or s s , the derivatives of LFBSTA can be obtained as x) L(FBSTA
( x) 1 k2 LS 1, k 1 2 k k1 LS 1, k 4ns
2 LS 2, k L(Sx2,) k LS1, k L2S 2, k L(Sx1,) k L2S 1, k L2S1, k k k1 k2
(3.105)
and xy ) L(FBSTA
k2
L(Sxy1, )k LS 1, k L(Sx1,) k L(Sy1,) k
k k1
2 L2S 1, k
k2
( x) S 2, k
(4 L
( y) S 1, k
L
k2
( L(Sy2,) k L(Sx2,) k LS 2, k L(Sxy2,)k ) 2ns LS 1, k
k k1
2L
( y) S 2, k
( x) S 1, k 2 s S 1, k
L
LS 2, k L
( xy ) S 1, k
) LS 2, k
4n L
k k1
k2
L2S 2, k L(Sx1,) k L(Sy1,) k
k k1
2ns L3S 1, k
(3.106)
The derivatives of LS1,k and LS 2, k The derivatives of LS1,k and LS 2, k with respect to any variables x or y other than s s is given by, L(Sx1,) k 2 k (kx ) 2 s s (kx ) ; L(Sx2,) k ns (kx )
(3.107)
L(Sxy1, )k 2 (ky ) (kx ) 2 k (kxy ) 2 s s (kxy ) ; L(Sxy2,)k ns (kxy )
(3.108)
The derivatives of LS1,k and LS 2, k with respect to s s is given by, 44
LS 1, sk 2 k 2no s s ; LS 2,sk ns no
(3.109)
LS 1,sk s 2no ; LS 2, sks 0
(3.110)
(s )
(s )
(s s )
(s s )
The derivatives of L(Ss1, k) and L(Ss2, k) with respect to any variables x other than s s is given by, s
s
LS 1,sk 2 (kx ) ; LS 2,sk 0 (s x)
(s x)
(3.111)
The derivatives k with respect to f s , s or S fs k
( S fs )
Pk1
(3.112)
(k fs ) 4 S fs Pk2 f k1[ sk3 sk (1 2 s2 )]
(3.113)
(k s ) 8S fs Pk2 sk2 s
(3.114)
k
0
(3.115)
4 Pk2 f k1[ sk3 sk (1 2 s2 )]
(3.116)
( S fs S fs )
k
( S fs f s )
( S fs s )
k
8Pk2 s sk2
(3.117)
(k f s f s ) 4 f k2 S fs Pk2 {8Pk1[ sk3 sk (1 2 s2 )]2 [3 sk2 (1 2 s2 )]}
(3.118)
(k f s s ) 8S fs s Pk2 f k1{8Pk1 sk2 [ sk3 sk (1 2 s2 )] 2 sk }
(3.119)
(k s s ) 8S fs sk2 {2 Pk3 s 8 s sk2 Pk2 } 8S fs sk2 Pk2 {16 Pk1 s2 sk2 1}
(3.120)
Appendix II: Derivatives of LFBSTA for case of closely spaced modes
For the purpose of evaluating the Hessian, the NLLF L FBSTA for the case of closely spaced modes is expressed as L FBSTA
1 k2 1 ln(4 ns LC1,k ) 4n 2 k k1 s
k2
L
k k1
2 C 2, k
LC1,k
(3.121)
where 2 ) 2 s tr (Θ ) n s2 LC1, k tr (Θ k k o
)s n ) and LC 2, k tr (Ssum ) ns (tr (Θ k k o
k α T . where Θ k αΛ
For any variables x in λ spe , the derivatives of LFBSTA can be obtained as 45
(3.122)
x) (L(FBSTA )
xy ) (L(FBSTA )
( x) 1 k2 LC1, k 1 2 k k1 LC1, k 4ns
k2
L(Cxy1,)k LC1, k L(Cx1,) k L(Cy1,) k
k k1
2 L2C1, k
k2
( x) C 2, k
(4 L
( y) C1, k
L
k2
k k1
L2C1, k
( L(Cy2,) k L(Cx2,) k LC 2, k L(Cxy2,)k )
k k1
2ns LC1, k
( y) C 2, k
( x) C1, k 2 s C1, k
L
LC 2, k L(Cxy1,)k ) LC 2, k
4n L
k k1
L2C 2, k L(Cx1,) k L2C1, k
k2
2L
2 LC 2, k L(Cx2,) k LS 1, k
k2
L2C 2, k L(Cx1,) k L(Cy1,) k
k k1
2ns L3C1, k
(3.123)
(3.124)
The general expression for the derivatives of LC1, k and LC 2, k should be obtained firstly. For x , y denoting any variables other than s ,
Θ ( x ) ) 2 s tr (Θ ( x ) ) ; L( x ) n tr (Θ ( x) ) L(Cx1,) k 2tr (Θ k k k C 2, k s k
(3.125)
( y )Θ ( x) Θ Θ ( xy ) ) 2s tr (Θ ( xy ) ) ; L( xy ) n tr (Θ ( xy ) ) L(Cxy1,)k 2tr (Θ k k k k k C 2, k s k
(3.126)
Derivatives of LC1, k and LC 2, k with respect to s The derivatives of LC1, k and LC 2, k with respect to s are given by, (s ) ) 2n s and L( s ) n n LC1, k 2tr (Θ k o C 2, k o s
(3.127)
LC1, k 2no and LC 2, k 0
(3.128)
(s s )
(s s )
The derivatives of L(Cs1,)k and L(Cs2,)k with respect to any variables x other than s are given by,
( x ) ) and L 0 LC1, k 2tr (Θ k C 2, k (s x)
(s x)
(3.129)
Derivatives of Θ k For x , y denoting any variables other than s or ij , ( x ) αΛ (kx ) α T Θ k
(3.130)
( xy ) αΛ (kxy ) α T Θ k
(3.131)
(ij ) α (ij ) Λ α T αΛ k [α (ij ) ]T Θ k k
(3.132)
(ij x ) α (ij ) Λ ( x ) α T αΛ (kx ) [α (ij ) ]T Θ k k
(3.133)
(ij rs ) α (ij rs ) Λ α T α (αij ) Λ [α ( rs ) ]T α ( rs ) Λ [α (αij ) ]T αΛ k [α (αij rs ) ]T Θ k k k k
(3.134)
46
Derivatives of α with respect to α ij The derivative α (α ) is a nm nm matrix with all entries being zero expect for the j -th column ij
given by (α ) α ij ( j ) α ( j )
1
[ei ij α ( j )]
(3.135)
1
where ij ij α ( j ) ; ei is an nm 1 vector with the i-th entry equal to unit and all other entries equal to zero. The j -th column of the cross derivative of α between ij and rs is given by, [0]nm , j s (α α ) α ij rs ( j ) 2 α ( j ) [(3 ij rj ir )α ( j ) ij e r rj ei ],
js
(3.136)
Derivatives of Λ k among f m and m For xm , ym denoting either f m or m , ( xm ) ( xm ) Λ (kxm ) hmk diag(h k )Sf e mm [hmk diag(h k )S f e mm ]*
( xm ) ( ym ) ( xm ) ( ym ) Λ k ( xm ym ) S f , mm hmk hmk e mm [Sf , mm hmk hmk e mm ]* ( xm ym )* ( xm ym )* mm [hmk diag(h k )S f e mm (hmk diag(h k )Sf e mm )* ]
(3.137) (3.138)
Derivatives of Λ k among Sf ,mm , Uf ,mm and Vf ,mm (S
Λk Λk
( Uf ,mm )
( Vf ,mm )
Λ k f ,mm hmk hmk e mm
(3.139)
hmk hm k e mm [hmk hm k e mm ]*
(3.140)
ihmk hm k e mm [ihmk hm k e mm ]*
(3.141)
)
All second derivatives of with respect to any pair of Sf ,mm , Uf ,mm and Vf ,mm are zero.
Cross derivatives of Λ k involving f m , m with Sf ,mm , Uf ,mm , Vf ,mm For xm denoting either f m or m , ( x Sf ,mm )
Λk m ( x Uf ,mm )
Λk m
( xm ) ( xm ) hmk hmk e mm (hmk hmk e mm )*
( x Uf ,mm )
( xm ) ( xm ) hmk hmk e mm (hmk hmk e mm )* ; Λ k m
47
hmk hm(kxm ) e mm [hmk hm(kxm ) e mm ]*
(3.142) (3.143)
( x Vf ,mm )
Λk m
( x Vf ,mm )
( xm ) ( xm ) ihmk hmk e mm [ihmk hmk emm ]* ; Λ k m
ihmk hm(kxm ) e mm [ihmk hm(kxm ) e mm ]*
(3.144)
Derivatives of hmk involving f m , m ( fm ) 2 hmk 2 f k1hmk ( mk i m )
(3.145)
( m ) 2 hmk 2i mk hmk
(3.146)
( fm m ) 3 2 hmk 2if k1hmk (3 mk 1 2i mk m )
(3.147)
( fm f m ) 3 2 hmk 2 f k2 hmk (3 mk 1 4 m2 6i mk m )
(3.148)
( m m ) 2 3 hmk 8 mk hmk
(3.149)
48
Chapter 4 A Two-Stage Fast Bayesian Spectral Density Approach for Ambient Modal Analysis: Mode Shape Assembly and Case Studies 4.1 Introduction In chapter 3, a two-stage Bayesian spectral density approach was formulated for ambient modal analysis. The spectrum variables and spatial variables can be identified separately. As will be shown in section 4.2, the proposed method can be implemented in the environment of wireless sensor network through a distributed computing strategy so that a set of local mode shapes confined to different clusters can be identified. This study presents a theory to assemble the local mode shapes using the Bayesian statistical framework so that the data quality of different clusters can be accounted for automatically. The optimal global mode shape can be obtained by a fast iterative scheme, while the associated uncertainties can be derived analytically. There is no need to share the same set of reference dofs for all clusters to obtain proper scaling. The theory described in chapters 3 and 4 of this work is applied to modal identification using synthetic data with separated modes and closely spaced modes, respectively. Furthermore, field data measured from two laboratory models using wireless sensors are also employed to illustrate the efficiency and accuracy of the proposed approaches. It is demonstrated that the proposed method can obtain satisfactory results either for the numerical or the experimental studies.
4.2 Hierarchical Architecture of WSN The proposed two-stage fast Bayesian spectral density approach can be implemented through a distributed computing strategy. As shown in Figure 4.1, a three-level hierarchical architecture is a common choice to realize distributed computing strategy. Wireless sensors in the context of schematic network architecture can be divided into hierarchical communities, with each community composed of a cluster head node and several leaf nodes. All cluster head nodes can report to the manager node, while the manager node could report to base 49
station node directly connected to a PC. The ambient modal analysis can be implemented in two stages. In the first stage, the leaf nodes can process the measured signal so that the data of all measured dofs can be collected centrally in the manager node. FBSTA can be adopted to identify the spectrum variables and their associated uncertainties. Subsequently, based on the power spectral density matrix for each cluster collected at the cluster head nodes, a set of local mode shapes can be identified by FBSDA. From the optimal values and covariance matrix of the local mode shapes corresponding to each community, global mode shapes and their uncertainties can then be identified with the help of the overlapping nodes public for different clusters to be shown in next section.
Base Station Node
Manage Nodes
FBSTA
Cluster Head Nodes
FBSDA
FFT
Leaf Nodes
Figure 4.1: Three-level hierarchical architecture of WSN
4.3 Assembling Mode Shape from Multiple Clusters As can be seen from Figure 4.1, a group of local mode shapes sharing some reference sensors across different clusters are available. To address the issue that the reference dof is lack of modal contribution in some particular modes, there may be more than one reference dof. Moreover, there are no reference dofs shared by all the clusters in many cases. Therefore, the local mode shapes identified from different clusters may have different senses and scaling factors since they are normalized individually (Au, 2011a). The assembly of these local modes shapes to form a global mode shape is an important issue in modal analysis. Conventionally, the overall mode shape can be obtained by ‘local least square method’ which determines the scaling factors by best fitting the mode shapes with respect to a selected reference setup in a least square sense. The local least square method is simple, but its results are inevitably affected by the choice of the reference setup. To address the drawbacks of ‘local least square method’, a novel ‘global least square method’ with an automated procedure 50
for determining the global mode shape by minimizing a measure-of-fit function was proposed recently by Au (2011a). However, the global least square approach assigns the same weights for all setups and no uncertainty information contained in different setups is involved. There is still room for improvement since the quality of originally well-identified clusters may be corrupted by the quality of some more problematic setups (Au and Zhang, 2012b). Therefore, it is reasonable to try to incorporate the uncertainty information of different clusters into the mode shape assembly procedure. In particular, local mode shapes not well identified in particular clusters should be assigned less weight due to their relatively unreliable data quality. As a sequel to the global least square method, a Bayesian mode shape assembly approach is developed in this study so that the weight for different clusters can be accounted for properly according to the data quality.
4.3.1 Bayesian Mode Shape Assembly Approach For the r -th given mode, let ψˆ r ,i n and Cψ n n be the optimal values and covariance i
i
i
r ,i
matrix of the mode shape confined to the measured dofs of i - th cluster (i 1, 2, , nt ) ; ni is the number of sensors measured in the i - th cluster and nt is the total number of clusters included in the ambient vibration test. For the case with closely spaced modes, the correlations among different modes are not considered here for simplicity. Assume that nl is the total number of nt
distinct measured dofs from all clusters, which should satisfy nl ni since some dofs are i 1
shared by more than one cluster. Let φ r n be the r - th given global mode shape covering all measured dofs, while l
φ r ,i ni
be the components of φ r confined to the measured dofs in the i - th setup. The local
mode shape φ r ,i can be mathematically related to the global mode shape φ r as (Au, 2011a) φ r ,i L i φ r
(4.1)
where Li n n is a selection matrix, where Li ( p, q) 1 if the p - th sensor of the i - th setup gives i
l
the q - th dof of φ r and zero otherwise. 51
Conceptually, the mode shape assembly problem under the Bayesian framework is to determine the global mode shape so that it can best fit the identified counterparts by assigning different weight for various clusters according to their data quality. It is worth noting that ψˆ r ,i is normalized to unity when it is identified by FBSDA. Therefore, the measure-of-fit should be implemented based on the discrepancy between φ r ,i φ r ,i and ψˆ r ,i both of unit norms. As is shown in section 3.3 and 3.4, the i - th local mode shape can be well-approximated by a Gaussian distribution, thus the likelihood function expressing the contribution of {ψˆ r ,i , Cψˆ } is r ,i
given by, 1 ˆ r ,i , Cψ r ,i φ r ,i ) exp[ (φ r ,i φ r ,i ψ ˆ r ,i )T (Cψ1r ,i )(φ r ,i φ r ,i ψ ˆ r ,i )] p(ψ 2
(4.2)
The constant independent of φ r ,i is ignored in (4.2). It is assumed that local mode shapes identified from different clusters are statistically independent. Therefore, the likelihood function reflecting the contribution of local mode shapes {ψˆ r ,i , Cψˆ : i 1, , nt } is given by, r ,i
nt
ˆ r ,i , Cψ r ,i φ r ,i ) p({ψˆ r ,i , Cψˆ r ,i : i 1,, nt } φ r ) p(ψ
(4.3)
i 1
Within the Bayesian framework, the updated probabilities of the global mode shape given the measured local mode shapes should be p (φ r ) c0 p (φ r ) p ( φ r )
(4.4)
In the case where a non-informative prior is used, the posterior PDF of φ r is proportional to the likelihood function p ( φ r ) , which can be written in terms of NLLF Las
1 nt (φr ,i φr ,i ψˆ r ,i )T (Cψr ,i )1 (φr ,i φr ,i ψˆ r ,i ) 2 i 1
(4.5)
(4.5) can be explicitly written in terms of the global mode shape φ r , Las
1 nt (Li φr Li φr ψˆ r ,i )T (Cψr ,i )1 (Li φr 2 i 1
ˆ r ,i ) Li φ r ψ
(4.6)
2
To enforce the norm constraint of φ r , i.e. φ r 1 , (4.6) can be rearranged as, using the Lagrange multipliers approach (Au, 2011a) Las
1 nt (Li φr Li φr ψˆ r ,i )T (Cψ1r ,i )(Li φr Li φr ψˆ r ,i ) r (1 φTr φr ) 2 i 1
52
(4.7)
where r is Lagrange multiplier that enforce φ r 1 . As seen, (4.7) is not quadratic about φ r , 2
whose optimal value cannot be determined analytically. To avoid the above difficulty, the auxiliary variables r ,1 ,, r ,n similar to (Au, 2011a) are introduced t
r2,i 1 Li φ r
2
(4.8)
As a result, the objective function can be formulated as, nt nt 1 ˆ r ,i )T (Cψ1r ,i )( r ,i Li φ r ψ ˆ r ,i ) r (1 φTr φ r ) r ,i ( r2,i Li φ r Las ( r ,i Li φ r ψ i 1 2 i 1
2
1)
(4.9)
where r ,i are Lagrange multipliers that enforce (4.8).
4.3.2 Most Probable Values The full set of parameters to be identified includes λ as {φ r , r , r ,i , r ,i : i 1, 2 , nt } . Direct solution for the optimal values λˆ as from the objective function (4.9) is not trivial due to its high-dimensional as well as nonlinear features. Sharing some common features with ‘global least square method’, the minimization problem (4.9) can be solved by the iterative solution strategy. The optimal values of r ,i and r ,i in terms of φ r and r are derived analytically firstly, following which φ r and r given the remaining parameters can also be derived analytically. A sequence of iterations comprised of the following linear optimization problems can be implemented.
(1) Optimal r ,i and r ,i The gradient of Las with respect to r ,i is given by Las ˆ Tr ,i Cψ1r ,i L i φ r 2 r ,i r ,i Li φ r r ,i (Li φ r )T Cψ1r ,i L i φ r ψ r ,i
Setting
2
(4.10)
Las 0 and solving for r ,i gives r ,i
r ,i
ˆ Tr,i Cψ1r ,i L i φ r ψ (Li φ r )T Cψ1r ,i L i φ r 2 r ,i Li φ r
Substituting (4.11) into (4.8) leads to two cases as
53
2
(4.11)
ˆ Tr,i Cψ1r ,i Li φ r ψ (L i φ r ) C T
1 ψ r ,i
L i φ r 2 r ,i L i φ r
2
Li φ r
1
(4.12)
Solving (4.12) for r ,i gives two roots r ,i
(Li φˆ r )T Cψ1r ,i Li φ r 2 Li φ r
2
ˆ Tr ,i Cψ1r ,i Li φ r ψ
(4.13)
2 Li φ r
It is worth noting that the Hessian of Las with respect to r ,i is given by 2 Las (L i φ r )T Cψ1r ,i L i φ r 2 r ,i Li φ r r2,i
The minimum of Las occurs only when
2 Las 0, r2,i
r ,i
2
(4.14)
which implies that
(L i φ r )T Cψ1r ,i L i φ r 2 Li φ r
(4.15)
2
Therefore, it is obvious that the larger root should be taken for r ,i r ,i
(Li φ r )T Cψ1r ,i Li φ r 2 Li φ r
2
ˆ Tr ,i Cψ1r ,i Li φ r ψ
(4.16)
2 Li φ r
Substituting (4.16) into (4.11) leads to, r ,i
ˆ Tr,i Cψ1r ,i (L i φ r ) ψ ˆ Tr,i Cψ1r ,i L i φ r L i φ r ψ
ˆ Tr ,i Cψ1r ,i L i φ r ) Li φ r sgn(ψ
1
(4.17)
where sgn() denotes the signum function.
(2) Optimal φ r and r The gradient of Las with respect to φ r is given by nt ns nt Las ˆ r ,i 2 r φ r 2 r ,i r2,i LTi Li φ r r2,i LTi Cψ1r ,i Li φ r r ,i LTi Cψ1r ,i ψ φ r i 1 i 1 i 1
Setting
Las 0 and φ r
(4.18)
solving for φ r give Αr φr br r φr
(4.19)
Where r
nt 1 nt 2 T 1 r ,i Li Cψ r ,i Li r ,i r2,i (LTi )(Li ) 2 i 1 i 1
54
and br
1 nt r ,i LTi Cψ1r ,i ψˆ r ,i 2 i 1
(4.20)
2
The equation (4.19) and the constraint φ r 1 form a constrained eigenvalue problem different from conventional eigenvalue problem, which can be solved by constructing an augmented vector satisfying the standard eigenvalue equation. The reader is referred to the reference (Au, 2011a). As a result, the optimal parameters can be optimized in group given the remaining groups till convergence instead of optimizing the full set of parameters simultaneously.
4.3.3 Posterior Uncertainties The posterior uncertainty of the global mode shape can also be obtained by inversing the Hessian matrix of Las with respect to λ as . The uncertainty of λ as can be well approximated by a Gaussian distribution centered at the most probable parameter values. Its covariance matrix is equal to the inverse of the Hessian Γ as calculated at the optimal parameters λˆ as . This Hessian matrix is given by Lφr φr L χ r φr Γ as β φ L r r γ r φr L
L
φr χ r
L
φr βr
L
L
χr χr
L L
φr γ r
L
χ r βr
L
χr γr
βr χ r
L
βr β r
L
βr γ r
γr χr
L
γ r βr
L
γr γr
(4.21)
In (4.21), χ [χ1T ,, χ Tr ,, χTn ]T with the r -th block χ Tr [ r ,1 ,, r ,i ,, r ,n ]T ; β [β1T ,, βTr ,, βTn ]T m
t
m
with the r -th block βTr [ r ,1 ,, r ,i ,, r ,n ]T . Lφ φ denotes the second order derivatives of with r
r
t
respect to φ r . Similar explanation can be given to the other blocked members in (4.21). Γ as is a symmetrical matrix, and only the blocked members in the upper triangle, i.e., Lφ φ , Lφ χ , r r
L
φr βr
r r
, Lφ γ , L χ χ , L χ β , L χ γ , Lβ β , Lβ γ and L γ γ need to be computed analytically. Among r r
r r
r r
r r
r r
r r
r r
these blocks, L χ γ n , Lβ β n n , Lβ γ n and L γ γ 1 are all zero matrixes. The nonr r
t
r r
t
t
r r
t
r r
zero blocks can be derived analytically as follows:
(1) Derivatives of Lφ φ r r
L
φr φr
nl nl
can be obtained by taking the derivative of (4.21) with respect to φ r , which can be
formulated as: 55
L
φr φr
nt
nt
i 1
i 1
r2,i LTi Cψ1r ,i Li 2 r ,i r2,i LTi Li 2 r I nl
(4.22)
(2) Derivatives of Lφ χ r r
L
φr χ r
nl nt is
a matrix formulated as ( φ r r ,1 )
L( φr r ) [L
( φr r ,i )
, , L
( φ r r ,nt )
,, L
(4.23)
]nl nt
where Lφ χ can be expressed as r r ,i
L
φ r r ,i
ˆ r ,i 4 r ,i r ,i LTi Li φ r 2 r ,i LTi Cψ1r ,i Li φ r LTi Cψ1r ,i ψ
(4.24)
(3) Derivatives of Lφ β r r
L
φr βr
nl nt is
a matrix formulated as: ( φ r β r ,1 )
L( φr βr ) [L
( φr β r ,i )
,, L
( φ r β r ,nt )
, , L
(4.25)
]nl nt
In which the i-th block Lφ β n can be expressed as: r r ,i
l
φr βr ,i
L
2 r2,i LTi Li φ r
(4.26)
(4) Derivatives of L(φ γ ) : r r
L( φr γ r ) nl
is a vector which can be expressed as φr r ,i
L
2φ r
(4.27)
(5) Derivatives of L χ χ : r r
L
χr χr
nt nt
is a diagonal matrix shown as follows L
χr χr
( χ r ,i χ r ,i )
diag (L
(4.28)
)
whose i-th diagonal entry can be derived analytically as follows, L
χ r ,i χ r ,i
(Li φ r )T (Cψ1r ,i )(Li φ r ) 2 r ,i Li φ r
2
(4.29)
(6) Derivatives of L χ β : r r
L
χ r βr
nt nt
is a diagonal matrix shown as follows L
χ r βr
( χ r ,i β r ,i )
diag (L
56
)
(4.30)
where diag () is the diagonal matrix with its i-th diagonal entry can be derived analytically as follows, χ r ,i βr ,i
L
2 r ,i L i φ r
2
(4.31)
4.4 Test for Stationarity In the proposed modal identification method, it assumes that the excitation is a broad-band stochastic process which can be adequately modeled as Gaussian white noise. However, sometimes structures are subjected to non-stationary excitation in real applications, which may distort identified results significantly. Therefore, stationarity evaluation for ambient vibration testing has been an important issue. Bendat and Piersol (1986) introduced a statistical method to detect the non-stationarity of the measured response based on the assumption that non-stationarity can be revealed by time trends in the mean square value of the data, which is equal to the autocorrelation at 0 denoted by R(0) . The rationale behind their method is that R(0) is able to reveal a time-varying autocorrelation since it is not likely for non-stationary signal to have an autocorrelation function without varying at 0 . The proposed method is mainly composed of three steps: (i) The measured signal for one channel is divided into N segments with equal length; (ii) The mean root square values for each segment are calculated, which are then aligned in a sequence as x12 , x22 , x32 , , xN2 ; (iii) Finally, the sequence of mean root square values are employed to test the underlying non-stationary trend by using nonparametric test approaches such as run test (Bendat and Piersol, 1986). When using the run test, the sequence of mean root square values should be classified into two mutually exclusive categories simply denoted by plus (+) or minus (-). In real applications, let the mean root value xi2 greater than the median value of the mean root square values x0
1 N
N
x be identified by (+) and all with a value less than i 1
2 i
x0
be identified by (-). A
run is defined as a series of observations that is followed and preceded by a different observation. The next step is to count the number of runs in the data sequence, which provides information regarding whether the sequence is random or not. It is highly likely for a signal 57
containing few runs to be influenced by an underlying trend, while a signal with many runs is more likely to be random, from which one can judge the stationarity of the measurement concerned. More details regarding run test approach are referred to (Bendat and Piersol, 1986; Siegel and Castellan, 1988).
4.5 Numerical Studies 4.5.1 Case One: 2-D Shear Building 150
10
5
Acceleration (ms -2)
100
Sy(m2s-3)
50 0 -50
10
10
0
-5
-100 -150 0
100 200 300 400 500 Time(seconds)
10
-10
0
10 Frequency (Hz)
20
Figure 4.2: Acceleration measurement (left) and auto-spectral density (right) of the top floor Simulated data of a 15-storey shear building with separated modes are processed firstly to illustrate the accuracy of the proposed approach. It is assumed that the stiffness to mass ratio at each floor is 2500s 2 . Rayleigh damping is assumed with the damping ratios for the first two modes set to be 1%. The structure is excited by ground motion xg modeled as Gaussian white noise with auto-spectral intensity of 0.25m 2 s 3 , while the prediction error is also assumed to be Gaussian white noise with auto-spectral intensity of 0.001m 2 s 3 . The raw measurement and auto-spectral density of the 15th dof is shown in Figure 4.2. The nonparametric KolmogorovSmirnov test is employed to verify the accuracy of the Gaussian approximation for tr (Ssum ) when the number of data sets ns varies, i.e., ns 2, 5,10, 20,50,100 . For each k ns considered,
five hundred independent samples of tr (Ssum ) are generated to calculate the k 58
empirical cumulative distribution function (CDF). Figure 4.3 shows the comparisons between CDF of standard normal distribution (denoted by ‘Normal’ in Figure 4.3) and CDFs of tr (Ssum ) k
with different ns in the vicinity of the first and the third resonant frequencies,
respectively. It can be seen from Figure 4.3 that the curves approximate the standard normal distribution well. In this study, twenty sets of data (i.e., ns 20 ) are employed for ambient modal analysis with each data set lasting for 500 seconds. 1 0.9 0.8
1 Normal ns =2
0.9
ns =5
0.8
0.7
ns =20
0.7
0.6
ns =50
0.6
ns =100
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1 -2 0 2 4 sum Normalized Tr(Sk ) at f k =0.806 Hz
ns=10 ns=20
0 -5
6
ns=100
0.5
0.4
0 -4
ns=5
ns=50 CDF
CDF
ns =10
Normal ns=2
0 5 sum Normalized Tr(Sk ) at f k =3.988Hz
10
Figure 4.3: CDFs of tr (Ssum ) at f k 0.806 Hz (left) and f k 3.988 Hz (right) with different ns k To identify the spectrum variables, the auto-spectral densities of all measured dofs are ) . The resonant peaks of the sum of the auto-spectral density collected together to form tr (Ssum k
are used for locating the initial guess of the natural frequencies. The frequency bands for modal identification can also be selected in the vicinity of these peaks, which are shown in the second column of Table 4.1. Then the spectrum variables as well as their uncertainties can be estimated based on the procedures summarized in section 3.3. Each mode is identified separately and the identified results are shown in Table 4.1, which includes the exact values in the fourth column, the most probable values (MPV) ˆ in the fifth column, the c.o.v. (coefficients of variation= standard deviation / optimal values ˆ ) in the sixth column,
59
as well as the normalized distance | ˆ | / in the seventh column. The normalized distance represents the absolute value of the difference between the optimal and target value, normalized by the corresponding posterior standard deviation (Yuen, 2010). The time consumed by identifying the spectrum variables is shown in the last column of Table 4.1. The performance illustrated here is according to calculations using Matlab7.0 on Dell desktop computer (Intel(R) Q9650 and 3 GB RAM). It can be seen that, for each mode, the first stage can be finished within one second in a PC. As can be seen from Table 4.1, the most probable values identified by the proposed method and the exact values corresponding to the structural model that generates the ambient data agree very well with each other. Moreover, the results show that the c.o.v. values of the frequencies are much smaller than those of damping ratios, indicating that frequencies can be identified with less uncertainty than damping, which is consistent with the intuition. Figure 4.4 shows the conditional PDFs of the first natural frequency and the first damping ratio (keeping the remaining parameters fixed at their optimal values) calculated from Bayesian sum approach p(λ tr (Ssum and the Gaussian approximation (ˆ, ) . The )) co p ( λ ) p (tr (S ) λ )
conditional PDFs obtained from the Bayesian approach in solid lines and the Gaussian approximation in dashed lines are extremely close to each other, indicating that the Gaussian approximation is effective. The marginal updated PDF of the first natural frequency and the first damping ratio can be obtained by integrating out the trace of spectral density matrix. Based on the method introduced in Appendix B of (Yuen, 2010), the contours for the marginal updated PDF of f s and s can be drawn and shown in Figure 4.5. The solid line encloses the area of 50% probability, while the dashed line encloses the area of 90% probability. The distance between the exact values and the optimal values are reasonable in terms of calculated standard deviations. Therefore, the optimal values and their uncertainties are of high accuracy though only the frequency band around the resonant frequency is employed.
60
Table 4.1: Identified spectrum variables for the 2-D shear building Mode
1st
Band (Hz)
Variables
ˆ
c.o.v.
fs
0.8061 0.0100 1.7708 0.0316 2.4100 0.0100 0.5862 0.0316 3.9893 0.0139 0.3468 0.0316 5.5275 0.0183 0.2425 0.0316
0.8056 0.0100 1.6734 0.0318 2.4105 0.0105 0.5962 0.0323 3.9873 0.0143 0.3500 0.0321 5.5167 0.0187 0.2531 0.0336
0.0007 0.0697 0.0252 20.7620 0.0005 0.0755 0.0462 39.9620 0.0009 0.1435 0.1204 27.9450 0.0025 0.3374 0.3204 13.5460
0.9890 0.0013 2.3113 0.0002 0.3495 0.6019 0.3616 0.0005 0.5403 0.1761 0.0748 0.0005 0.7751 0.0667 0.1309 0.0044
s
[0.756,0.856]
S fs
s s fs
s
2nd
[2.360,2.460]
S fs
s s fs
s
3rd
[3.928,4.028]
S fs s s fs
s
4th
[5.486,5.586]
S fs
s s
900
800
700
700
600
600
500
500
400
300
200
200
100
100 0.804
0.806 f
0.808
0.015
0.015
0.016
0.81
s
Bayesian Approach Gaussian Approximation
400
300
0 0.802
0.016
900
Bayesian Approach Gaussian Approximation
PDF
PDF
800
Time (s)
0 0.006
0.008
0.01 s
0.012
0.014
Figure 4.4: Conditional PDFs of f s (left) and s (right) for the 2-D shear building 61
0.0115 Actual Value Optimal Point 50% 90%
0.011
s
0.0105
0.01
0.0095
0.009
0.0085 0.804
0.8045
0.805
0.8055
0.806
0.8065
0.807
0.8075
fs
Figure 4.5: Contours of the marginal PDF of f s and s for the 2-D shear building Once the spectrum variables are identified in the first stage, the mode shape can then be identified with f s , s , S fs and s s fixed at their optimal values. In real applications, they may be processed by multiple clusters as introduced in section 4.2. To verify the efficiency of the Bayesian mode shape assembly method, the sensors are divided into 3 setups with the setup information shown in Table 4.2. The most probable global mode shapes for the first four modes are denoted by squares and shown in Figure 4.6. The exact mode shapes and the mode shapes identified using one setup only are denoted by solid lines and asterisks in Figure 4.6, respectively. These three kinds of mode shapes almost coincide with each other. The modal assurance criterion (MAC) value between either two kinds of them is very close to unity. Therefore, one can conclude from this numerical study that the proposed method can identify the overall mode shapes with high quality. The time consumed by identifying the local mode shape confined to each cluster is within one second, which illustrates that FBSDA can be also performed very quickly. Table 4.2: Setup information for the 2-D shear building Setup 1 2 3
Measured dofs 1, 2, 3, 4, 5,6,7 6, 7, 8, 9, 10,11,12 11, 12, 13, 14, 15
62
Storey
15
15
15
15
14
14
14
14
13
13
13
13
12
12
12
12
11
11
11
11
10
10
10
10
9
9
9
9
8
8
8
8
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
0 -1
0 Mode 1
1
0 -1
0 Mode 2
1
0 -1
0 Mode 3
0 -1
1
0 Mode 4
1
Figure 4.6: Global mode shape assembled from multiple setups for the 2-D shear building
4.5.2 Case Two: 3-D Torsional Shear Building In this example, a six-storey three-dimensional torsional shear building with closely spaced modes is studied. It is square in plane with width a 10m . Four stiffness parameters used for modeling each storey are denoted by ki , x , ki , y , ki , x and ki , y (i 1, 2,,6) , where i is the storey number while ‘ x ’, ‘ y ’, ‘ x ’ and ‘ y ’ represent the direction of the outward normal of the face. To simulate the case of closely spaced modes, similar stiffness is assumed in both x and y principal directions with ki , x ki , x 400 kN m and ki , y ki , y 380 kN m . The element stiffness matrix corresponding to the lower floor and the upper floor of the i -th storey with respect to the stiffness center of i-th floor is given by (Yuen, 2010) 0 0 ( ki , y ki , y ) 0 ( ki , x ki , x ) 0 0 0 ki , t Ki ( ki , y ki , y ) 0 0 0 ( ki , x ki , x ) 0 0 0 ki , t
( ki , y ki , y )
0
0
0
( ki , x ki , x )
0
0
0
ki , t
0
0
( ki , y kl , y ) 0 0
where ki ,t is given by
63
( ki , x ki , x ) 0
0 ki , t
(4.32)
ki , t
a2 [ ki , x ki , y ki , x ki , y ] 4
(4.33)
The element mass matrix corresponding to the lower floor and the upper floor of the i -th storey is given by m0 0 0 Mi 0 0 0
0
0
0
0
0
m0
0
0
0
0
0 2m0 a 12 0
0
0
0
0
m0
0
0
0
0
0
m0
0
0
0
0
0 2m0 a 2
2
12
(4.34)
The floor mass is taken to be m0 200kg . The first three dofs and the last three dofs of (4.32) and (4.34) correspond to the x -translational, y -translational and torsional motion of the lower floor and the upper floor of the i-th storey, respectively. Rayleigh damping is assumed in this study, and the damping ratios for the first two modes are set to be 1%. The structure is assumed to be excited by random forces along the x -translational and y -translational motion modeled as Gaussian white noise. The prediction errors of the responses are taken to be i.i.d Gaussian white noise with the spectral density of 1 103 m 2 s 3 . Only accelerations with respect to x -translational and y -translational dofs are assumed to be measured, while accelerations with respect to the torsion dofs are not available. 15
Acceleration (ms -2)
10 5 0 -5 -10 -15 0
10
20 30 Time(seconds)
40
50
Figure 4.7: Raw measurement of the top floor with respect to the x-translational dof
64
1 0.9
Standard normal f k=2.364Hz
0.8
f k=6.958Hz
0.7
f k=11.146Hz
CDF
0.6 0.5 0.4 0.3 0.2 0.1 0 -4
-3
-2
-1 0 1 sum Normalized Value of Tr(S k )
2
3
4
Figure 4.8: CDFs of tr (Ssum ) at f k 2.364 Hz , 6.958Hz and 11.146Hz with ns 20 k 10
2
Tr(Ssum ) (m2s-3) k
[Mode 1 & 2] [Mode 4 & 5] [Mode 6 & 7]
10
10
1
0
0
2
4
6 8 10 Frequency (Hz)
12
14
16
Figure 4.9: The plot of tr (Ssum ) with ns 20 k Raw measurement of the top floor with respect to the x-translational dof is shown in Figure 4.7. To verify the accuracy of Gaussian approximation for tr (Ssum ) , its empirical CDF can be k obtained by generating 500 samples with each sample calculated by using ns sets of measurements. The obtained CDF can be compared with the CDF of standard normal ) at f k 2.364 Hz , 6.958Hz and 11.146Hz distribution. Figure 4.8 gives the CDFs of tr (Ssum k
with ns 20 . It can be seen that the normal distribution provides a good approximation
65
) . Figure 4.9 provides the plot of tr (Ssum ) with ns 20 , which includes three pairs of for tr (Ssum k k
closely spaced translational modes (i.e., TX1 and TY1, TX2 and TY2, TX3 and TY3) due to the similar interstorey stiffness along the two directions. Thus the two-stage fast Bayesian spectral density approach with closely spaced modes proposed in section 3.4 is employed here to identify modal properties. The frequency bands are shown in the second column of Table 4.3. Some spectrum variables (i.e., f m , and m ) identified by the proposed method is shown in Table 4.3, which includes the exact values , the most probable values (MPV) ˆ , the c.o.v. (coefficients of variation=standard deviation /optimal values ˆ ) and the normalized distance | ˆ | / . The most probable values are quite close to their exact values corresponding to the structural model that generated the ambient data. Consistent with common observations, the c.o.v. values of the natural frequencies are much smaller than those of the damping ratios. The last column in Table 4.3 shows the time consumed by identifying the spectrum variables. One can observe that the time consumed by identifying the spectrum variables is still within minutes. Figure 4.10 shows the conditional PDFs of the first natural frequency f1 and damping ratio 1 with the remaining parameters fixed at its optimal values. The conditional PDFs calculated
by using the fast Bayesian spectral trace approach and the Gaussian approximation are plotted with solid lines and dashed lines, respectively. The two sets of curves are on top of each other, indicating that the Gaussian approximation is accurate enough to represent the updated PDF. The elliptical contours of the marginal updated PDF of f1 and 1 , which enclose the region with probability 0.5 and 0.9, are shown in Figure 4.11, indicating that the actual parameters are at a reasonable distance from the identified optimal parameters. To observe the accuracy of the Bayesian mode shape assembly method proposed in this paper, it is assumed that the measured 12 dofs are divided into 3 setups as shown in Table 4.4. Based on the measured acceleration data, local mode shapes corresponding to three clusters can be identified. According to the Bayesian mode shape assembly method, the global mode shape can be assembled from three local mode shapes. Figure 4.12 shows the comparison between the 66
exact mode shape (solid line) and the overall mode shapes (square) assembled from multiple setups in x and y translation, respectively. The Modal Assurance Criteria (MAC) values between these two kinds of mode shapes are very close to unit, suggesting that the mode shapes are also identified with high quality when there are closely spaced modes. Table 4.3: Identified spectrum variables for the 3-D torsional shear building Mode
Band (Hz)
Variables f1
TX1 & TY1
[2.244, 2.546]
[6.838, 7.258]
1 2 f5
4 5 f6
TX3 & TY3 [11.026, 11.556]
c.o.v.
2.3652 2.3639 0.0007 0.7011 2.4266 2.4251 0.0006 1.0218 0.0100 0.0096 0.0806 0.4578 0.0100 0.0095 0.0674 0.7223 6.9580 6.9583 0.0023 0.0166 7.1388 7.1379 0.0014 0.0948 0.0162 0.0166 0.4777 0.0485 0.0166 0.0167 0.1718 0.0270 11.1470 11.3090 0.0058 2.4825 11.4360 11.3090 0.0056 1.9955 0.0243 0.0328 0.2323 1.1077 0.0249 0.0322 0.2247 1.0103
f2
f4
TX2 & TY2
ˆ
f7
6 7
Time (s) 15.703
31.094
116.2
Table 4.4: Setup information for the 3-D torsional shear building Setup 1 2 3
Measured dofs 1-x, 1-y, 2-x, 2-y, 3-x,3-y 3-x, 3-y, 4-x, 4-y, 5-x,5-y 5-x, 5-y, 6-x, 6-y
500 450
1400 Bayesian Approach Gaussian Approximation
1200
Bayesian Approach Gaussian Approximation
400 1000
350
PDF
PDF
300 250
800 600
200 150
400
100 200
50 0
2.36
2.365 f1
2.37
0 8
9
10 1
11 -3 x 10
Figure 4.10: Conditional PDFs of f1 (left) and 1 (right) for the 3-D torsional shear building 67
11.5 x 10
-3
11 10.5
1
10 9.5 9 Actual Value Optimal Point 50% 90%
8.5 8 7.5 2.36
2.361
2.362
2.363
2.364
2.365
2.366
2.367
2.368
2.369
f1
Figure 4.11: Contours of the marginal PDF of f1 and 1 for the 3-D torsional shear building
Storey
(a) Mode 1
(b) Mode 2
(d) Mode 5
(e) Mode 6
(f) Mode 7
6
6
6
6
6
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
3
3
3
2
2
2
2
2
2
1
1
1
1
1
1
0 -1
0
0 1 -1
x (translation)
Storey
(c) Mode 4
6
0
0 1 -1
x (translation)
0 1 -1
0 x (translation)
0
0 1 -1
x (translation)
0
0 1 -1
x (translation)
6
6
6
6
6
6
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
3
3
3
2
2
2
2
2
2
1
1
1
1
1
1
0 -1
0 y (translation)
0 1 -1
0 y (translation)
0 1 -1
0 1 -1
0 y (translation)
0 y (translation)
0 1 -1
0 y (translation)
0
1
x (translation)
0 1 -1
0
1
y (translation)
Figure 4.12: Assembled mode shapes for the 3-D torsional shear building
4.6 Experimental Studies To further verify the accuracy of the proposed two-stage fast Bayesian modal identification approaches, experimental studies were conducted by employing a three-storey shear building model and a three-storey torsional shear building. These models were kept in an airconditioned room to ensure a steady environment. The experimental equipment includes a shaker table and a WSN system composed of a laptop and Crossbow Imote2 platforms. A 68
gateway node mainly consisting of an interface board and an Imote2 main board was connected to the laptop directly, as shown in Figure 4.13. The gateway node works as a bridge communicating between the laptop and leaf nodes. The main hardware components of a leaf node include an Imote2 board, a high-sensitivity SHM-H sensor board, and a battery board. The gateway node uses power supplied by the laptop while the leaf nodes are powered by AAA batteries. The engineering analysis software employed in this study is the ISHMP Services Toolsuite provided by the Illinois SHM Project (http://shm.cs.uiuc.edu). RemoteSensing component included in ISHMP Services Toolsuite was employed to collect precisely synchronized data from the leaf nodes, and transfer them back to the gateway node, and write the output to the file. More details on how to install the ISHMP Services Toolsuite and acquire data using the Imote2 sensor platform interfaced with the SHM-H sensor board is referred to the user’s guidance (ISHMP, 2010).
Figure 4.13: An overview of a gateway node connected to a laptop
4.6.1 Case One: 2-D Shear Building The proposed modal identification method is firstly experimentally investigated with the three-storey building model. The model shown in Figure 4.14 was constructed by aluminum with dimensions of 401(width) 314 (depth) 1158 (height ) in mm, which can be simplified as a 3dof shear building. The mass coefficients were measured as m1 5.63 kg , m2 6.03 kg
69
and m3 4.66 kg , while the linear stiffness coefficients obtained from a static test were k1 20.88 kN / m , k2 22.37 kN / m and k3 24.21 kN / m (Chang and Poon, 2010). The building was fixed on a shake table which can generate ground motion horizontally. Random numbers following Gaussian distribution was imported into the shake table to generate ground motion. Each storey of the shear building was installed with a wireless sensor and a wired sensor. Figure 4.15 shows side view of the sensors placed on the top floor. The time histories measured from the wired sensors could be viewed on site and they were not used for modal analysis.
k3
C3
k2
C2
k1
C1
Figure 4.14: The tested shear building (left) and its simplified 3-dof model (right)
Figure 4.15: Side view of sensor placement on the top floor
70
Horizontal accelerations were measured by each wireless sensor node with a sampling rate of 100 Hz and a 40 Hz cut-off frequency. The measured acceleration time histories lasted 15 minutes with 90000 points, and 20 data sets were recorded for ambient modal identification. Figure 4.16 shows sample acceleration time history corresponding to the top storey. The trace of power spectral density matrix is shown in Figure 4.17, from which the initial guess of natural frequencies and frequency bands adopted for modal identification can be determined. The identified most probable values (MPV) of spectrum variables as well as their c.o.v. values are shown in the fifth and sixth column of Table 4.5. The frequencies identified by using recursive stochastic subspace identification method (Li, 2011) as well as the damping ratios identified by processing the free vibration response through Hilbert-Huang transform (Chang and Poon, 2010) are shown in the fourth column of Table 4.5. The frequencies identified using two different methods agree well with each other. However, the damping ratios identified using the proposed method is much smaller than those obtained from free vibration response. This coincides with the conclusion by (Magalhaes et al., 2009) that, under ambient vibration condition, the accuracy of the damping ratio estimation is dependent on the length of the collected time series. Therefore, time duration should be increased for the measurement to get more precise damping ratios. As is seen from Table 4.5, the c.o.v. values for the parameters except for the prediction errors increased with the modes, indicating that the lower modes can be identified with higher accuracy. The time consumed by the proposed method is shown in the last column of Table 4.5. Table 4.6 gives the identified optimal mode shapes as well as their coefficient of variance (c.o.v.) for three scenarios considered. For Scenario I, the sensors were divided into two local groups. Each group was composed of two sensor nodes, with the one in the middle storey being overlapping (reference) node. Local modal mode shapes could be estimated. For scenario II, the global mode shapes were assembled subsequently from the local mode shapes identified in scenario I. For scenario III, the mode shape components of all measured dofs were identified simultaneously in a single setup. The MAC between the mode shapes obtained in scenario II and III are shown in the last row of Table 4.6, which indicates that no significant difference is found between these two scenarios. Therefore, the two-stage Bayesian ambient modal analysis approach is reliable.
71
1
Accerlation(g)
0.5
0
-0.5
-1 0
50
100
150 200 Time(seconds)
250
300
Figure 4.16: Acceleration measured using the wireless sensor installed on the top storey 10
Tr(Ssum ) (m2s-3) k
10
6
4
12.63
10 10 10 10
18.6
4.177
2
0
-2
-4
0
5
10 15 Frequency (Hz)
20
25
Figure 4.17: Trace of spectral density matrix with 20 sets of measurements Table 4.5: Identified spectrum variables for the laboratory shear building model Mode
Band (Hz)
Variables Reference fs
1
[4.110, 4.243]
s S fs s s fs
s
2
[12.563, 12.697]
S fs s s fs
s
3
[18.533, 18.667]
S fs
s s
4.21 0.0239 12.62 0.0087 18.49 0.0065 72
Bayesian Time (s) MPV c.o.v. (%) 4.1677 0.027 0.0086 0.698 0.094 0.0103 5.087 0.0462 87.936 12.6450 0.032 0.0068 10.430 0.016 0.0158 8.486 0.0966 79.643 18.5740 0.032 0.0047 14.784 0.093 0.0093 16.600 0.2470 26.163
Table 4.6: Identified mode shapes for the laboratory shear building model Mode 1 Scenario dofs c.o.v. φˆ 1 (%) 1 0.515 0.341 2 0.857 0.123 I 2 0.637 0.185 3 0.771 0.126 1 0.355 0.343 II 2 0.595 0.049 3 0.721 0.049 1 0.357 0.372 III 2 0.595 0.192 3 0.720 0.137 MAC 0.9999
Mode 2 c.o.v. φˆ 2 (%) 0.946 0.057 0.323 0.482 -0.362 0.484 0.932 0.073 -0.726 0.177 -0.248 0.197 0.641 0.197 -0.733 0.119 -0.248 0.499 0.633 0.156 0.9999
Mode 3 c.o.v. φˆ 3 (%) -0.575 0.573 0.818 0.282 0.888 0.229 -0.459 0.844 0.526 0.044 -0.750 0.044 0.400 0.232 0.533 0.587 -0.751 0.328 0.390 0.874 0.9998
4.6.2 Case Two: 3-D Torsional Shear Building
0.26m
0.40m
0.50m
Figure 4.18: The tested torsional shear building A three-storey torsional shear building was also employed to verify the proposed modal identification approach with closely spaced modes experimentally. The tested structure is shown in Figure 4.18. The building was fixed on a shake table which can generate ground motion horizontally in two directions. The level of maximum acceleration responses were around 0.5 g. As can be seen from Figure 4.19, each storey of the structure was installed with three wireless sensors and a wired sensor. The corresponding schematic diagram of the sensor plan is also shown in Figure 4.19, from which one can observe that two of those sensors
73
located at the middle of the faces (1) and (3) of the structure records the lateral acceleration time-histories along the x-direction, while the other one located at the middle of the face (2) records the acceleration time-histories along the y-direction. The acceleration data from the wired sensors could be visualized on site to check the performance of the shake table, while they are not utilized for ambient modal analysis. y
0.3m
(1)
x
(2) O (3) a=0.3m
Figure 4.19: Side view (left) and floor plan of sensor placement (right) for the torsional shear building For each wireless sensor node, the accelerations were measured with a sampling rate of 100 Hz and a 40 Hz cut-off frequency. The measured acceleration time histories lasted 15 minutes with 90000 points, and 19 data sets were recorded for modal identification. Figure 4.20 shows the measured acceleration corresponding to the top floor. The trace of power spectral density matrix is shown in Figure 4.21. As can be seen from Figure 4.21, there are five clear peaks, indicating that the wireless sensors can acquire satisfactory measurements under current excitation level and conditions. Eight modes are labeled according to their nature confirmed from the identification results (see Table 4.7) and the engineering judgment of torsional shear building. The modes are identified in groups as follows, as is normally done: {mode 1 & 2 (x & y translation)}, {mode 3 (torsion)}, {mode 4&5 (x & y translation)}, {mode 6 (torsion)}, {mode 7&8 (x & y translation)}. That is, three pairs of x & y translation modes are identified using the two-stage Bayesian approach with closely spaced modes while the torsion modes are identified using two-stage Bayesian approach with separated modes. Moreover, one can pick the peaks for initial guesses of natural frequencies, and the frequency bands for modal identification can be determined correspondingly. The identified natural frequencies and 74
damping ratios are given in Table 4.7. The local mode shapes with respect to the dofs (1), (2) and (3) can be identified and assembled to form the global mode shapes o . As a result, the global mode shape denoted by 0 with respect to the x-translational, y-translational and torsional directions can be obtained using the transformation matrix, T 0 0 o 0 T 0 0 0 T
1
o
1 with T 0 1
0 1 0
a 2 a 2 a 2
(4.35)
where a 0.3m . The optimal x-translation, y-translation and rotational mode shape components are shown in Figure 4.22. As can be seen from Table 4.7, similar to the numerical studies, the damping ratios exhibit much larger variability compared to frequencies, which is also often reported in different literatures. Moreover, the c.o.v. values of the 4th, 5th, 7th, and 8th modes are smaller than those of 1st and 2nd modes, which is consistent with the phenomenon shown in Figure 4.21 that the magnitude of spectral density corresponding to the third and fifth peaks are much more significant. 1
Accerlation (g)
0.5
0
-0.5
-1 0
50
100
150 200 Time(seconds)
250
300
Figure 4.20: Acceleration of the top floor with respect to x direction
75
Tr(Ssum ) (m2s-3) k
10
10
4
[4 & 5]
2
[7 & 8] [1 & 2]
10
0
[3] [6]
10
10
-2
-4
0
5
10
15 20 Frequency (Hz)
25
30
35
Figure 4.21: Trace of spectral density matrix for the laboratory torsional shear building
Table 4.7: Identified spectrum variables for the laboratory torsional shear building model Mode
Band (Hz)
Variables f1
1&2
f2
[2.648, 2.758]
1
2
3
f3
[5.135, 5.224]
3 f4
4&5
[11.713, 12.047]
6
[19.666, 19.754]
f5
4 5 f6
6 f7
7&8
f8
[24.243, 24.577]
7 8
76
Bayesian MPV c.o.v. (%) 2.6308 0.066 2.7378 0.344 0.0165 16.968 0.0261 77.832 5.1605 0.121 0.0130 18.148 11.8440 0.045 11.8850 0.024 0.0049 11.542 0.0033 6.390 19.7030 0.020 0.0031 18.094 24.3730 0.046 24.4180 0.040 0.0029 9.334 0.0022 7.869
Height (m)
1.16
1.16
(b) Mode 2
1.16
(c) Mode 3
1.16
0.9
0.9
0.9
0.9
0.5
0.5
0.5
0.5
0 -1
1.16
Height (m)
(a) Mode 1
0 x (translation)
1
(e) Mode 5
0 -1
1.16
0 y (translation)
0 -1
1
(f) Mode 6
1.16
0
(torsion)
1
(g) Mode 7
0 -1
1.16
0.9
0.9
0.9
0.9
0.5
0.5
0.5
0.5
0 -1
0 y (translation)
1
0 -1
0
(torsion)
1
0 -1
0 x (translation)
1
0 -1
(d) Mode 4
0 x (translation)
1
(h) Mode 8
0 y (translation)
1
Figure 4.22: Identified mode shapes for the laboratory torsional shear building model
4.7 Concluding Remarks To assemble the local mode shapes confined to different clusters, a Bayesian assembly methodology was proposed in this chapter so as to account for the weights of different clusters properly according to their data qualities. The issue of mode shape assembly was formulated as an optimization problem within the Bayesian framework. A fast iterative scheme was employed to solve the optimal values for the overall mode shapes, while their associated uncertainties can also be derived analytically. The accuracy and efficiency of the theoretical methods proposed in chapter 3 and chapter 4 were verified by two numerical examples. Moreover, a 2-D shear building model and a 3-D torsional shear building subjected to ground motion were employed to demonstrate applicability of the proposed methods to responses measured by using wireless sensors. Results from numerical and experimental studies show that the proposed approaches can identify the modal properties as well as their uncertainties successfully, and the challenges of conventional BSDA can be well addressed. However, as can be seen from the numerical examples and experimental studies, the resonant frequency bands are picked by hand. Moreover, the number of modes within the specified frequency band is also required to be determined in advance. Automatic mode-picking and mode-order determination process without human intervention is a future endeavor. The proposed method requires that the power spectral density of the modal excitation and 77
prediction error should be flat within the selected band. Though the amount of spectral information used for identification increase with the frequency band, yet this will weaken the flat spectrum assumption. Therefore, how to trade off between the amount of spectral information used and modeling error in real applications is another big issue (Au, 2012c), which is not trivial but worth of further investigation.
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Chapter 5 Bayesian Approach to Structural Model Updating Incorporating Modal Information from Multiple Clusters 5.1 Introduction Modal properties such as natural frequencies and mode shapes have been frequently exploited for model updating during past decades. Full-scale ambient tests for large-scale civil infrastructure usually employs multiple clusters sharing some reference sensors to cover all the dofs of interest since it is not trivial to process the measured data synchronously in a single cluster (Au, 2011a). As a result, a group of natural frequencies and local mode shape components constrained to different clusters are available. When implementing structural model updating, how to fuse the ‘local’ modal properties together is an important issue required to be well addressed. In this chapter, a Bayesian structural model updating approach which is able to incorporate uncertain modal information from multiple clusters automatically without prior processing is proposed. The Bayesian model updating problem can be formulated as one minimizing an objective function. The optimization problem can be solved by a fast iterative scheme so as to resolve the computational burden required for optimizing the objective function numerically. The Hessian matrix of the identified parameters can be derived analytically. Moreover, the computational difficulty due to estimating the inverse of the high dimensional Hessian matrix required for specifying the covariance matrix is also properly addressed. Finally, numerical examples and experimental studies are presented to illustrate the efficiency of the proposed model updating method.
5.2 Formulation of Bayesian Model Updating Using Modal Data The underlying premise of structural model updating using modal data is to modify the model parameters so as to produce modal properties most consistent with the measured counterparts representing the structural current state. Most existing modal data-based model updating methods should minimize a goodness-of-fit function measuring the difference between 79
analytical predictions from a finite element model and those measured from dynamic tests, which has a generic form shown as follows (Yuen, 2010a): nm
nm
m 1
m 1
L( ) m [m ( ) ˆ m ]2 m m ( ) ˆm
2
(5.1)
where ˆ m and ˆm are the m-th measured frequency and mode shape; m ( ) and m ( ) are the m-th frequency and mode shape from the analytical model; m and m are weighting factors. During model updating using (5.1), the following steps should be implemented: (i) As is stated, it is necessary to determine the ‘global’ modal properties incorporating all information contained in different clusters; (ii) The incomplete mode shapes should be expanded to complete mode shapes through mode shape expansion approaches if only measurements of partial mode shapes are available; (iii) It is necessary to match the model mode with the measured mode; (vi) The weighting factors should be chosen according to the specific method or engineering judgments; (v) In each iteration of convergence process, eigenvalue decomposition of the structural dynamical model with updated parameters should be implemented to obtain the analytical frequencies and mode shapes, which is usually time-consuming. To address the issues aforementioned, the work in this study is based on the Bayesian framework presented by (Beck and Katafygiotis, 1998). First of all, instrumental variables ‘system mode shapes’ proposed previously are introduced. The discrepancies between ‘system mode shapes’ and the measured local mode shapes should be bridged firstly, in which the ‘system mode shape’ ought to best fit the measured local mode shapes by incorporating their posterior uncertainties. After that, the gaps among the ‘system mode shape’, the measured natural frequencies and the model parameters also should be connected through the Gaussian probability model for the eigenvalue equation errors. As a result, the Bayesian model updating problem can be formulated as one minimizing an objective function.
5.2.1 Modal Data Available for Model Updating As is illustrated from chapter 3, the natural frequencies can be estimated using FBSTA by collecting auto-spectral densities from all measured sensors, while a group of local mode shapes confined to different clusters can be identified by FBSDA. The modal data to be
80
utilized in structural model updating consist of {ˆ r , ψˆ r ,i , Cψ } with r 1, 2, , nm and i 1, 2, , nt . r ,i
Here nm and nt denote the number of modes and setups, respectively; ˆ r denotes the ˆ r ,i ni and Cψ ni ni denote the optimal values r - th optimal natural frequency (in rad/s); ψ r ,i
and covariance matrix of the local mode shape confined to the measured dofs of the i - th setup for the r - th given mode; ni is the number of dofs measured in the i - th setup. The total number of measured dofs denoted by nl should be equal to the number of distinct measured dofs from nt
all clusters. Moreover, it should satisfy nl ni since some dofs are shared by more than one i
setup.
5.2.2 Structural Model Class For the linear structural model with nd dofs, the stiffness matrix K n n and mass matrix d
M nd nd
d
are parameterized by θ {1 , 2 , , n } and ρ {1 , 2 , , n } shown as follows,
n
n
i 1
i 1
K (θ) K 0 i K i ; M (ρ) M 0 i M i
(5.2)
where K i and M i are nominal substructures contributing to the global stiffness matrix and global mass matrix, which can be obtained from the conventional finite-element model of the structure; i (i 1, 2, , n ) and i (i 1, 2, , n ) are scaling factors which enable to modify the nominal substructure stiffness and mass matrix so as to be more consistent with the real structure. It is assumed that classical normal modes are used for the structure to be updated, thus the damping matrix is not considered explicitly. The uncertainties in the mass parameters are explicitly treated to ensure that the identification results be more robust to modeling errors (Yuen et al., 2004) instead of assuming that a known mass matrix with sufficient accuracy is available. The prior PDF of θ and ρ are taken to be independent Gaussian with mean θ and ρ0 , while 0
the covariance matrix are taken to be diagonal matrixes Cθ and Cρ , respectively. Thus one has 1 p θ exp[ (θ θ0 )T Cθ1 (θ θ0 )] 2
81
(5.3a)
1 p ρ exp[ (ρ ρ0 )T Cρ1 (ρ ρ0 )] 2
(5.3b)
The choice for θ should be able to reflect that the nominal structural mode is the most 0
probable model, while the choice for Cθ should reflect the level of uncertainty in the nominal model in the absence of any data. It is worth noting that the variances of the prior covariance of ρ are taken to be small so as to avoid making the identification problem ill-posed when treating θ and ρ as uncertain variables simultaneously (Ching and Beck, 2004b; Ching et al., 2006).
5.2.3 Instrumental Variables ‘System Mode Shape’ In this study, instrumental variables ‘system mode shape’ Φr n (r 1, 2,, nm ) proposed d
previously (Vanik, 1997; Vanik et al., 2000) are introduced as extra parameters to be updated. ‘system mode shapes’ represent the actual mode shapes of the structure without being constrained to be eigenvectors of the structural model. By introducing the instrumental variables, there are some advantages (Beck et al., 2001; Ching and Beck, 2004b; Yuen, 2010a): (i) There is no need to do the mode-matching between the measured mode-shapes and those corresponding to the updated model; (ii) There is no need to implement eigenvalue decomposition, which often proves to be computationally inefficient in practice; (ii) The ‘system mode shapes’ provide extra flexibility in model updating in that it might not be always possible to produce theoretical mode shapes (eigenvectors) agreeing well with the experimental mode shapes due to the imprecise mathematical constraints for the structural model class adopted; (iv) It can skip the step of modal expansion from measured dofs to the unmeasured dofs, which is able to connect the full mode shape information with partial mode shape information. As will be found in the remaining part of this study, one appealing advantage not mentioned before is that the instrumental variables help to incorporate the local mode shape components identified from multiple clusters automatically without prior assembling or processing. Moreover, when performing an ambient vibration test, there is no need to share the same reference dofs for all clusters for the scaling purpose.
82
5.2.4 Probability Model for the Discrepancy between Φr and ψˆ r ,i Let Φr n be the components of Φr only confined to the measured dofs. Φr can be l
mathematically related to Φr by Φr Lo Φ r
(5.4)
where Lo n n is a selection matrix which picks the observed degrees of freedom from the l
d
‘system mode shape’ Φr . Let Φr ,i n be the components Φr only confined to the measured i
dofs in the i - th setup. Then Φ r ,i can be mathematically related to Φr by Φ r ,i Li Φr
(5.5)
while Li n n is another selection matrix with Li ( j, k ) 1 if the j - th data channel in the i
l
i - th setup gives the k - th dof of Φr and zero otherwise. According to (5.4) and (5.5), Φ r can be
related to Φ r ,i by Φ r ,i L i Φ r
(5.6)
where L i Li Lo . Similar to the mode shape assembly problem (Au, 2011a), the discrepancy between Φ r ,i Φ r ,i
and ψˆ r ,i both of unit Euclidian norms, instead of the discrepancy between
ˆ r ,i , Φ r ,i and ψ
ought to be bridged since the measured local mode shapes are normalized to unit
norm (i.e. ψˆ r ,i 1 ). Under the Bayesian framework, Φr ,i Φ r ,i should best fit the identified counterparts by assigning a weight according to the data quality. Since the identified local mode
shape
can
be
ˆ r ,i : r 1, , nm ; i 1, , nt } {ψ
well-approximated
by
a
Gaussian
PDF
with
mean
and covariance matrix {Cψ : r 1,, nm ; i 1,, nt } , the likelihood r ,i
function of the local mode shape measured from the i-th setup can be written explicitly in terms of Φ r ,i as
83
ˆ r ,i , Cψ r ,i Φ r ,i ) p(ψ
1 (2 )
nd
C ψ r ,i
1 ˆ r ,i )T (Cψ1r ,i )(Φ r ,i Φ r ,i ψ ˆ r ,i )] exp[ (Φ r ,i Φ r ,i ψ 2
(5.7)
According to (5.6), (5.7) can be written explicitly in terms of ‘system mode shape’ Φr as, ˆ r ,i , Cψ r ,i Φ r ,i ) p(ψ
1 (2 )
nd
Cψ r ,i
1 exp[ (L i Φ r 2
ˆ r ,i )T (Cψ1r ,i )(L i Φ r L i Φ r ψ
ˆ r ,i )] L i Φ r ψ
(5.8)
It is assumed that local mode shapes identified from different setups are statistically independent. Therefore, the likelihood function of {ψˆ r ,i , Cψ : i 1,, nt } is given by r ,i
nt
ˆ r ,i , Cψ r ,i Φ r ,i ) p({ψˆ r ,i , Cψ r ,i : i 1,, nt } Φ r ) p(ψ
(5.9)
i 1
So far, the discrepancy between the system mode shape and the measured local mode shapes has been well bridged. The probability model shown in the above is reasonable since the local mode shapes not well identified in particular setups should have less influence on determining the final ‘system mode shape’. As a result, less weight should be assigned for the unreliable clusters because their data quality is relatively poor. This is explicitly accounted for through the inverse of Cψ in (5.7) and (5.8). r ,i
5.2.5 Probability Model for the Eigenvalue Equation Errors For the purpose of model updating, structural model parameters should also be connected with the measured frequencies and ‘system mode shape’ through a mathematical model. Usually, the prediction-error between the Rayleigh quotient and the measured frequency is adopted as such connection: εr ΦTr K (θ) Φ r (ΦTr M (ρ)Φ r ) ˆ r2
(5.10)
where εr is always modeled as independent Gaussian variables. However, (5.10) will lead to unexpected optimization function which fall into certain forms which is nontrivial to be optimized (Ching and Beck, 2004b). In this study, (5.10) is replaced by the eigenvalue equation errors in that it can also measure how well the identified modal properties are matched with the counterparts from the updated structural model: ε r K (θ)Φ r ˆ r2 M (ρ)Φ r
84
(5.11)
Similar to εr , the prediction-error vector ε r (r 1, 2,, nm ) can also be modeled as independent Gaussian variables shown as follows, ε r (0, r I nd )
(5.12)
where I n is a nd by nd identity matrix and r denotes the variance of the prediction error which d
is unknown. (5.12) can be justified by the maximum entropy principle stating that the Gaussian PDF gives the largest uncertainty for any statistical distribution with specified means and variances (Jaynes, 1957). Therefore, the likelihood function for measured frequencies ˆ r is p(ˆ r ρ, θ, Φ r , r )
1 (2 )
(2 r )
nd
n d 2
r In
1 exp[ ( ηr Φ r )T ( r I nd ) 1 ( ηr Φ r )] 2
d
(5.13)
1 2 exp[ r1 ηr Φ r ] 2
where denotes the Euclidean norm and ηr K (θ) ˆ r2 M (ρ)
(5.14)
As a result, a relationship is established between the ‘system mode shape’, the measured natural frequencies and the model parameters connected through the Gaussian probability model for the eigenvalue equation errors.
5.2.6 Negative Logarithm Likelihood Function In this context, the full set of parameters to be identified is composed of λ {ρ, θ, Φ r , r : r 1, 2, , nm } .
Under the Bayesian framework, the updated probabilities of the
parameters λ given the measured data should be p (λ , M ) c0 p (λ M ) p ( λ , M )
(5.15)
where c0 is a normalizing constant; p(λ , M ) is the PDF of the model parameters given the modal data and the model assumptions M ; p(λ M ) is the initial (‘prior’) PDF of the model parameters based on engineering and modeling judgment; and p( λ , M ) is the PDF of the
85
modal data given the model parameters. Under the assumption that modal parameters of different modes are statistically independent, thus the likelihood function p( λ , M ) can be obtained as, nm
ˆ r ,i , Cψ r ,i : i 1, , nt } ρ, θ, Φ r , r ) p( λ , M ) p({ˆ r , ψ
(5.16)
r 1
According to the probability models illustrated in section 5.2.4 and 5.2.5, one obtains that nm
ˆ r ,i , Cψ r ,i : i 1, , nt } Φ r ) p ({ˆ r } ρ, θ, Φ r , r ) p( λ , M ) p({ψ
(5.17)
r 1
After substituting (5.9) and (5.13) into (5.17), the likelihood function p( λ , M ) can be obtained. Then substituting (5.17) and the prior information (i.e. (5.3a) and (5.3b)) into (5.15) result in the updated probabilities p(λ , M ) . The most probable values (MPV) of the unknown parameters based on the modal data are given by maximizing p( , M ) . For this optimization problem, it is more convenient to work with ‘negative logarithm likelihood function’ (NLLF), Lupd ( )
1 1 1 nm 2 (θ θ0 )T Sθ1 (θ θ0 ) (ρ ρ0 )T Sρ1 (ρ ρ0 ) {nd ln r r1 ηr Φ r } 2 2 2 r 1 1 nm nt (Li Φr Li Φr ψˆ r ,i )T Cψ1r ,i (L i Φr Li Φr ψˆ r ,i ) 2 r 1 i 1
(5.18)
It is worth noting that the constant terms independent of the parameters to be identified have been ignored here. To find the most probable values, one must minimize Lupd (λ ) . As can be seen from (5.18), there is no need to implement eigenvalue decomposition, mode-matching, and modal expansion. Moreover, the local mode shapes from different setups have been incorporated together automatically so that the need of assembling the local mode shapes prior to model updating has been avoided.
5.3 Most Probable Parameters Note that the function L upd (λ ) is not quadratic about the unknown parameters λ , and a numerical optimization algorithm needs to be employed for solving the desired optimize problem. However, the use of numerical optimization methods usually requires evaluation of gradients or Hessians with respect to λ , which is computationally challenging due to the high 86
dimensional and nonlinear features of the problem at hand. One effective way to solve (5.18) is to decompose the complicated optimization problem into several coupled simple optimization problems with analytical solutions. Then the objective function can be optimized iteratively through a sequence of linear optimization problems until satisfactory convergence is achieved. As can be seen from (5.18), the objective function is quadratic about ρ and θ . Moreover, the objective function in terms of r has the form of a log x b x . Therefore, it is easy to get the optimal analytical solutions for ρ , θ and r . However, the objective function is not quadratic about Φr , and does not fall into certain functional forms that are easy to optimize. To by-pass this difficulty, a Lagrange multiplier approach (Au, 2011a) is employed in this study. The auxiliary variables r ,i are introduced, defined as: r2,i 1 L i Φ r
2
(5.19)
This means that the objective function (5.18) can be re-formulated as Lupd ( )
1 1 1 nm 2 (θ θ0 )T Sθ1 (θ θ0 ) (ρ ρ0 )T Sρ1 (ρ ρ0 ) {nd ln r r1 ηr Φ r } 2 2 2 r 1
nm nt 1 nm nt ˆ r ,i )T Cψ1r ,i ( r ,i L i Φ r ψ ˆ r ,i ) r ,i ( r2,i L i Φ r ( r ,i L i Φ r ψ 2 r 1 i 1 r 1 i 1
2
(5.20)
1)
where r ,i are Lagrange multipliers that enforce the definition of (5.19). As a result, after the objective function is rearranged the full set of model parameters has become λ { r ,i , r ,i , Φ r , r , ρ, θ : r 1,, nm ; i 1,, nt }
(5.21)
The dimension of λ is nupd nm nd 2nm nt nm n n . By making use of the special properties of objective function (5.20), a sequence of iterations comprised of the following linear optimization problems can be finished until the prescribed convergence criteria is satisfied once the nominal values of the parameters λ are assigned in advance.
87
5.3.1 Optimization for r ,i and r ,i The analytical MPV of r ,i is first derived in terms of the remaining parameters. Partial differentiation of Lupd with respect to r ,i gives Lupd r ,i
Setting
Lupd r ,i
r ,i (L i Φ r )T Cψ1r ,i (L i Φ r ) ψˆ Tr ,i Cψ1r ,i (L i Φ r ) 2 r ,i r ,i L i Φ r
2
(5.22)
0 and solving for r ,i gives
r ,i
ˆ Tr,i Cψ1r ,i (L i Φ r ) ψ (L i Φ r )T Cψ1r ,i (L i Φ r ) 2 r ,i L i Φ r
(5.23)
2
Substituting (5.23) into (5.19) leads to two cases as ˆ Tr,i Cψ1r ,i (L i Φ r ) ψ (L i Φ r )T Cψ1r ,i (L i Φ r ) 2 r ,i L i Φ r
2
L i Φ r
1
(5.24)
Solving (5.24) for r ,i gives r ,i
(L i Φ r )T Cψ1r ,i (L i Φ r ) 2 L i Φ r
2
ˆ Tr ,i Cψ1r ,i (L i Φ r ) ψ
(5.25)
2 L i Φ r
It is worth noting that the second derivatives of Lupd with respect to r ,i is shown as follows, 2 Lupd
2 r ,i
(L i Φ r )T Cψ1r ,i (L i Φ r ) 2 r ,i L i Φ r
The minimum of Lupd occurs only when
2 Lupd r2,i
r ,i
2
(5.26)
0 , which implies that r ,i should satisfy
(L i Φ r )T Cψ1r ,i (L i Φ r ) 2 L i Φ r
(5.27)
2
Therefore, it is obvious that the larger root will be taken as the optimal solution for r ,i , r ,i
(L i Φ r )T Cψ1r ,i (L i Φ r ) 2 L i Φ r
2
ˆ Tr ,i Cψ1r ,i (L i Φ r ) ψ
(5.28)
2 L i Φ r
Substituting (5.28) into (5.23) leads to, r ,i
ψˆ Tr,i Cψ1r ,i (L i Φ r ) ˆ C ψ T r ,i
1 ψ r ,i
(L i Φ r ) L i Φ r
ˆ Tr ,i Cψ1r ,i L i Φ r L i Φ r sgn ψ
88
1
(5.29)
Here sgn denotes the signum function.
5.3.2 Optimization for r The gradient of Lupd with respect to r is given by Lupd r
Setting
Lupd r
1 2 (nd r1 r2 ηr Φ r ) 2
(5.30)
0 and solving for r gives:
r ηr Φ r
2
(5.31)
nd
5.3.3 Optimization for Φr The gradient of Lupd with respect to Φr is given by Lupd Φ r
Setting
Lupd Φ r
nt
nt
nt
i 1
i 1
i 1
ˆ r ,i 2 r ,i r2,i L iT Li Φ r r1 ηr2 Φ r r2,i L i T Cψ1r ,i L i Φ r r ,i L iT Cψ1r ,i ψ
(5.32)
0 and solving for Φ r gives 1
nt nt nt ˆ r ,i Φ r r2,i L Ti Cψ1r ,i L i 2 r ,i r2,i L Ti Li r1 η2r r ,i L iT Cψ1r ,i ψ i 1 i 1 i 1
(5.33)
5.3.4 Optimization for ρ
The gradient of Lupd with respect to ρ is given by Lupd ρ
Setting
Lupd ρ
0 and
nm
nm
r 1
r 1
( r1G M Tr G M r Sρ1 )ρ ( r1G M Tr g M r Sρ1ρ0 )
(5.34)
solving for ρ gives: 1
nm nm ρ r1G M Tr G M r S 1 ( r1G M Tr g M r S ρ1ρ 0 ) r 1 r 1
(5.35)
G M r ˆ r2 M1Φ r , ˆ r2 Μ 2 Φ r , , ˆ r2 M i Φ r , , ˆ r2 M n Φ r
(5.36)
where
89
g M r K (θ) ˆ r2 M o Φ r
(5.37)
5.3.5 Optimization for θ The gradient of Lupd with respect to θ is given by Lupd θ
Setting
Lupd θ
nm
nm
r 1
r 1
( r1G K Tr G K r Sθ1 )θ ( r1G K Tr g K r S θ1θ0 )
(5.38)
0 and solving for θ gives: 1
nm nm θ r1G K Tr G K r S θ1 ( r1G K Tr g K r S θ1θ 0 ) r 1 r 1
(5.39)
G K r K1Φ r , K 2 Φ r , , K i Φ r , , K n Φ r
(5.40)
g K r [ˆ r2 M (ρ) K o ]Φ r
(5.41)
where
5.4 Posterior Uncertainties Estimation 5.4.1 Analytical Derivation of Hessian Matrix The posterior PDF of λ can be well approximated by a Gaussian distribution cantered at the optimal parameters and with covariance matrix Cupd equal to the inverse of the Hessian matrix Γupd calculated
at the optimal parameters. The Hessian matrix is given by
Γupd
LΦΦ L χΦ βΦ L = δΦ L ρΦ L LθΦ
L
Φχ
χχ
L
βχ
L
L
Φβ
χβ
L
ββ
L
Φδ
χδ
L
βδ
L
Φρ
L
χρ
L
βρ
δρ
L
χθ
L
L
L
L L
L
δχ
L
δβ
L
δδ
L
L
ρχ
L
ρβ
L
ρδ
L
L
θχ
L
θβ
L
θδ
L
Φθ
βθ
δθ
ρρ
L
θρ
L
ρθ
θθ
(5.42)
In general, L xy denotes the derivatives of Lupd with respect to any two vectors x and y . Here x or y represents a one-dimensional array arranging from the following parameters: (1) Φ [Φ1T ,, ΦTr ,, ΦTn ]T ; (2) χ [χ1T ,, χ Tr ,, χ Tn ]T with the r -th block χ Tr [ r ,1 ,, r ,i ,, r ,n ]T ; m
m
90
t
(3) β [β1T ,, βTr ,, βTn ]T with the r -th block βTr [ r ,1 ,, r ,i ,, r ,n ]T ; (4) δ [1 ,, r ,, n ]T ; (5) m
ρ [ 1 , , i ,, n ]T
t
m
; (6) θ [1 ,,i ,, n ]T . Γupd is a symmetrical matrix, and only the blocked
members in the upper triangular LΦΦ , LΦχ , LΦβ , LΦδ , LΦρ , LΦθ , L χχ , L χβ , L χδ , L χρ , L χθ , Lββ , L , L , L , L , L , L , L , L βδ
βρ
βθ
δδ
δρ
δθ
ρρ
ρθ
and Lθθ need to be computed analytically. Among
these blocks, L χδ n n n , L χρ n n n , L χθ n n n , Lββ n n n n , Lβδ n n n , Lβρ n n n m t
m t
m
m t
m t
m t
m t
m
m t
and Lβθ n n n are all zero matrix. The non-zero blocks are derived analytically as follows. m t
(1) Derivatives of LΦΦ L
ΦΦ
nd nm nd nm is
a block diagonal matrix formulated as, L
ΦΦ
diag (L
Φr Φr
(5.43)
)
with the r -th diagonal block LΦ Φ n n equal to r
L
Φ r Φr
r
d
d
nt
nt
i 1
i 1
r2,i L i Cψ1r ,i L i 2 r ,i r2,i L iT L i r1 ηr2
(5.44)
(2) Derivatives of LΦχ L
Φχ
nd nm nm nt is
a block diagonal matrix formulated as L
Φχ
diag (L(Φr χ r ) )
(5.45)
with the r -th block equal to L(Φ χ ) , which is a nd nt matrix formulated as r r
( Φ r χ r ,1 )
L(Φr χ r ) [L
( Φ r χ r ,i )
,, L
( Φ r χ r ,nt )
,, L
]nd nt
(5.46)
Here L(Φ χ ) n denotes i-th column of L(Φ χ ) analytically given by r r ,i
d
r r
( Φ r χ r ,i )
L
ˆ r ,i 4 r ,i r ,i L iT L i Φ r 2 r ,i L iT Cψ1r ,i L i Φ r L i T Cψ1r ,i ψ
(5.47)
(3) Derivatives of LΦβ L
Φβ
nd nm nm nt is
a block diagonal matrix formulated as L
Φβ
diag (L
Φr β r
(5.48)
)
with the r -th block being equal to LΦ β , which is a nd nt matrix formulated as r r
( Φ r β r ,1 )
L(Φr βr ) [L
( Φ r β r ,i )
,, L
( Φ r β r ,nt )
,, L
]nd nt
Here L(Φ β ) n denotes the i-th column of LΦ β analytically derived as r r ,i
d
r r
91
(5.49)
2 r2,i L iT L i Φ r
( Φr β r ,i )
L
(5.50)
(4) Derivatives of LΦδ L
Φδ
nd nm nm
is a block diagonal matrix formulated as L
Φδ
diag (L
Φr δr
(5.51)
)
with the r -th block equal to LΦ δ . LΦ δ n is a vector which can be derived analytically as r r
r r
m
L
Φr r
r2 ηr2Φ r
(5.52)
[(L(Φ1ρ ) )T , ,(L(Φr ρ ) )T ,,(L(Φr ρ ) )T ]T
(5.53)
(5) Derivatives of LΦρ L
Φρ
nm nd n
is a block matrix formulated as L
Φρ
The r -th block L(Φ ρ ) n n is formulated as r
d
L
( Φ r n )
Φr ρ
[L(Φr 1 ) , , L(Φr i ) , , L
]nd n
(5.54)
where L(Φ ) n is a vector comprised of the following entries r i
d
L(Φr i ) 2 r1ˆ r2 ηr M i Φ r
(5.55)
(6) Derivatives of LΦθ L
Φθ
nm nd n
is a block matrix formulated as L
Φθ
[(L(Φ1θ ) )T ,,(L(Φr θ ) )T ,,(L(Φr θ ) )T ]T
(5.56)
In above equation, the r -th block L(Φ θ) n n is composed of d
r
L
Φr θ
( Φ r n )
[L(Φr1 ) , , L(Φri ) , , L
]nd n
(5.57)
where L(Φ ) n is a vector comprised of the following entries: r i
d
L(Φri ) 2 r1 ηr K i Φ r
(5.58)
(7) Derivatives of L χχ L
χχ
nm nt nm nt
is a block diagonal matrix formulated as: L
χχ
diag (L
χr χr
(5.59)
)
with the r -th block being equal to L χ χ . L χ χ n n is a diagonal matrix written as r r
r r
L
χr χr
t
t
( χ r ,i χ r ,i )
diag (L
92
)
(5.60)
whose i-th diagonal entry can be derived analytically as follows, χ r ,i χ r ,i
(L i Φ r )T Cψ1r ,i (L i Φ r ) 2 r ,i L i Φ r
L
2
(5.61)
(8) Derivatives of L χβ L
χβ
nm nt nm nt is
a block diagonal matrix formulated as L
χβ
diag (L( χ r βr ) )
(5.62)
with the r -th block equal to L χ β . L χ β n n is a diagonal matrix formulated as r r
r r
t
L
χ r βr
t
( χ r ,i β r ,i )
diag (L
(5.63)
)
whose i-th diagonal entry can be derived analytically as, ( r ,i r ,i )
L
2 r ,i L i Φ r
2
(5.64)
(9) Derivatives of Lδδ L
δδ
nm nm is a diagonal matrix formulated as L
δδ
diag (L(δr δr ) )
(5.65)
with its r -th diagonal entry L(δ δ ) derived analytically as r r
L
δr δr
1 nd r2 r3 ηr Φ r 2
2
(5.66)
(10) Derivatives of Lδρ L
δρ
nm n
is a matrix formulated as L
δρ
( nm ρ ) T T
[(L(1ρ ) )T (L( r ρ ) )T (L
) ]
(5.67)
L( r ρ ) r2 ρT G M Tr G M r r2 g M Tr G M r
(5.68)
where r -th row L( ρ ) 1n is given by
r
(11) Derivatives of Lδθ L
δθ
nm n
is a matrix formulated as L
δθ
( nm θ ) T T
[(L(1θ ) )T (L( r θ ) )T (L
) ]
(5.69)
L( r θ ) r2 θT G K Tr G K r r2 g K Tr G K r
(5.70)
where r -th row L( θ ) 1n is given by r
93
(12) Derivatives of Lρρ L
ρρ
n n
is a matrix with the analytical expression shown as follows, L
ρρ
nm
r1G M Tr G M r Sρ1
(5.71)
r 1
(13) Derivatives of Lρθ L
ρθ
n n
is a matrix with the analytical expression shown as follows, L
ρθ
nm
r1G MTr G K r
(5.72)
r 1
(14) Derivatives of Lθθ L
θθ
n n
is a matrix with the analytical expression shown as follows, L
θθ
nm
r1G K Tr G K r Sθ1
(5.73)
r 1
5.4.2 Suppressing Computational Complexity of Covariance Matrix Once the Hessian matrix Γupd is obtained, the covariance matrix Cupd can be calculated by inversing the Hessian matrix calculated at the optimal values of the full set of parameters. As can be seen from above derivation, Γupd is a nupd by nupd square matrix, which can be of high dimension. To suppress the growth of computational burden with the dimension of nupd , Γupd is partitioned into a block form Γ11 Γupd T Γ12
Γ12 Γ 22
(5.74)
Where LΦΦ L χΦ βΦ Γ11 = L δΦ L ρΦ L
L
Φχ
L
L
χχ
L
βχ
δχ
L
ρχ
L
Φβ
L
Φδ
L
χβ
L
χδ
L
ββ
L
δβ
L
ρβ
L
L
L
βδ
δδ ρδ
L
Φρ
L Φθ χρ L χθ L βθ θθ βρ L ; Γ12 = L ； Γ 22 = L δθ δρ L L ρθ ρρ L L
(5.75)
Based on the general formula of the matrix inversion in block form (Brookes, 2005), one can obtain, 1 T 1 T 1 Γ 1 Γ11 Γ12 (Γ 22 Γ12 Γ11 Γ12 ) 1 Γ12 Γ11 1 11 Γupd T T 1 1 (Γ 22 Γ12 Γ11 Γ12 ) 1 Γ12 Γ11
94
1 T 1 Γ11 Γ12 (Γ 22 Γ12 Γ11 Γ12 ) 1 T 1 (Γ 22 Γ12 Γ11 Γ12 ) 1
(5.76)
For the case in which only stiffness scaling parameters are employed for further processing, one can only be focused on calculating the block (Γ 22 Γ12T Γ111Γ12 ) 1 located in the bottom-right 1 . The dimension is then dramatically reduced from nupd to n . corner instead of the entire Γupd
Computational complexity of mathematical operations in calculating uncertainty of θ is reduced significantly. Before calculating the bottom-right block (Γ 22 Γ12T Γ111Γ12 ) 1 , the computational effort of 1 Γ11 can
be further reduced. Γ11 is also divided into four blocks, ΓA Γ11 = T ΓB
ΓB ΓD
(5.77)
where
Γ A = L
ΦΦ
L χχ L βχ Φρ L ] ; Γ D = δχ L ρχ L
; Γ B = [LΦχ LΦβ LΦδ
L
χβ
L
χδ
L
ββ
L
βδ
L
δβ
L
δδ
L
ρβ
L
ρδ
L
χρ
βρ L δρ L ρρ L
(5.78)
Similar to (5.76), (5.77) can be rearranged using the matrix inversion in block form again, Γ -1 + Γ -1A Γ B (Γ D - ΓTB Γ -1A Γ B )-1 ΓTB Γ -1A -1 Γ11 = A - (Γ D - ΓTB Γ -1A Γ B )-1 ΓTB Γ -1A
- Γ -1A Γ B (Γ D - ΓTB Γ -1A ΓB )-1 (Γ D - ΓTB Γ -1A Γ B )-1
(5.79)
As can be seen from (5.79), this strategy is particularly advantageous when it only requires the inversion of the block diagonal matrix Γ A and the comparatively small matrix (Γ D - ΓTB Γ -1A Γ B ) . Since L(ΦΦ ) is a block diagonal matrix, its inversion is also block diagonal,
Γ A1 diag L(Φr Φr )
1
(5.80)
1
where the r -th block of Γ A1 is equal to L(Φ Φ ) . As a result, (Γ D - ΓTB Γ-1A ΓB )-1 can be further r
r
reduced to a more simple formulation, nm 1 (Γ D - ΓTB Γ -1A Γ B )-1 Γ D ΓTB , r L(Φr Φr ) Γ B , r r 1
1
(5.81)
where Γ B,r [LΦ χ LΦ β LΦ δ LΦ ρ ] . Since only the diagonal blocks in Γ A1 are employed in r
r
r
r
(5.81), the zero entries of Γ A1 have been excluded in (5.81). Thus the required memory space
95
has also been reduced. After substituting (5.80) and (5.81) into (5.79), one can obtain -1 Γ11 easily.
Finally, the covariance of stiffness scaling factors (Γ 22 Γ12T Γ111Γ12 ) 1 can be obtained
by using Γ11-1 .
5.5 Probabilistic Damage Detection (Vanik et al., 2000) The Bayesian model updating methodology proposed in previous sections can be applied to identify the stiffness scaling factors as well as their uncertainties. Assume that θˆ udi and θˆ ipd denote
the most probable values of the stiffness scaling factors for the undamaged and
possibly damaged structure, respectively, while iud and ipd are their corresponding standard deviations. Based on the Gaussian approximation of stiffness scaling factors, one can calculate the probability of damage in terms of a fractional damage level d
Pi
dam
d
1 d θˆ udi θˆ ipd 1 d
2
iud ipd 2
2
(5.82)
where () is the standard Gaussian cumulative distribution function.
5.6 Numerical Study 5.6.1 Case One: 2-D Shear Building The fifteen-storey 2-D shear building introduced in chapter 4 is used again here. The substructure mass matrix are given by 0114 1 M1 ρ1 ; 0141 01414
0 i 215 01i 2 1 0 0115 i M i ρi 2500 01i 2 0 1 0115 i 015 i 15
(5.83)
The substructure stiffness matrix are given by 0114 1 K1 1 2500 ; 0141 01414
0 i 215 01i 2 1 1 0115 i K i i 2500 01i 2 1 1 0115 i 015 i 15
96
(5.84)
Here the stiffness scaling factor i (i 1, 2,3, ,15) reflects the damage extent (in percentage) of the substructure stiffness. Two scenarios have been considered for the structure: (i) no damage occurs in the structure; (ii) damages occurs in 1st, 3rd and 10th storey with damage extent taken to be 30%, 20% and 10%, respectively. The stiffness scaling factors i for the first scenario should be unit while i for the second scenario should be unit except for 1 0.70 , 3 0.80 and 10 0.90 .
Table 5.1: Setup information of incomplete measured dofs for the 2-D shear building Setup 1 2 3
Measured dofs 1, 2, 3, 4, 5, 6 5, 6, 7, 9, 10 9, 10, 11, 13, 15
Table 5.2: Identified modal properties for the 2-D shear building Undamaged State Mode 1 2 3
f1
1 f2
2 f3
3
Actual
MPV
0.806 0.010 2.410 0.010 3.989 0.014
0.806 0.011 2.410 0.010 3.993 0.016
Damaged State
c.o.v. (%) 0.068 7.109 0.046 7.270 0.095 15.751
Actual
MPV
0.772 0.010 2.317 0.010 3.892 0.014
0.772 0.011 2.317 0.011 3.895 0.014
c.o.v. (%) 0.070 7.021 0.050 7.445 0.073 13.357
It is assumed that no sensors are installed on 8th, 12th and 14th floors. The measured 12 dofs can be divided into three setups, and the setup information is shown in Table 5.1. Twenty sets of acceleration response are generated for the healthy state and damaged state, respectively. The measurement noise at different measured dofs is assumed to be Gaussian white noise with a spectral density of 0.001 m 2 s 3 . The two-stage fast Bayesian approach for the case of separated modes is employed to identify the modal properties as well as their uncertainties. The first three measured modes are used for model updating based on the approach proposed in sections 5.2, 5.3 and 5.4. The actual natural frequencies, the identified natural frequencies 97
and their coefficient of variance (c.o.v.) in healthy state and damaged state are shown in Table 5.2. The initial guess for the stiffness scaling factors are taken to be θ0 {10,10,,10} so that the nominal values are significantly overestimated. Figure 5.1 shows the iterative histories for the most probable values of i corresponding to the structure in healthy state and damaged state, respectively. This figure indicates that convergence occurs within 10 iterations for both scenarios. Figure 5.2 shows the comparison between the most probable ‘system mode shape’ (the square) and the exact mode shapes (the solid line) for the structure in undamaged state and damaged state, respectively. Results show that these two kinds of mode shapes agree well with each other after model updating. For the damaged state, the MAC between the ‘system mode shape’ and the exact mode shape for the first three modes are 1, 0.999, and 0.997. The exact stiffness scaling factors, most probable values of stiffness scaling factors as well as their c.o.v. values are shown in Table 5.3. Based on the most probable values and the standard deviation of the stiffness scaling factors, the curves associated with the damage probabilities for the shear building can be obtained and shown in Figure 5.3. The median values of the scaling factors can also be calculated based on the curves of damage probability, which are shown in the last column of Table 5.3. It can be clearly seen from Figure 5.3 that the 1st floor, the 3rd floor and the 10th floor have damage with probability of almost unity. Moreover, the median values shown in Table 5.3 are quite close to the actual healthy states. These evidences indicate that the proposed method is able to update the structural model parameters even though the modal information employed is not complete. Moreover, the associated uncertainties of the identified stiffness parameters are useful in further processing, such as calculating the probability of damage, risk assessment, etc.
98
9
9
8
8
7
7
6
6
5
5
i
10
i
10
4
4
3
3
2
2
1
1
0 0
0 0
50 100 Iterations (Undamaged State)
50 Iterations (Damaged State)
100
Figure 5.1: Iteration histories of i for the 2-D shear building in healthy state (left) and damaged state (right)
Table 5.3: Identified stiffness scaling factors i for the 2-D shear building
Parameter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Healthy state c.o.v Actual MPV (%) 1.000 0.997 0.278 1.000 0.998 0.281 1.000 0.998 0.286 1.000 0.998 0.294 1.000 0.997 0.305 1.000 0.998 0.318 1.000 0.996 0.334 1.000 0.999 0.354 1.000 0.997 0.378 1.000 0.996 0.408 1.000 0.996 0.447 1.000 0.997 0.499 1.000 0.995 0.576 1.000 1.002 0.706 1.000 0.982 1.000
99
Damaged state c.o.v Actual MPV (%) 0.700 0.699 0.280 1.000 0.998 0.283 0.800 0.799 0.289 1.000 0.999 0.297 1.000 0.999 0.307 1.000 0.998 0.320 1.000 0.996 0.344 1.000 1.020 0.970 1.000 0.977 1.038 0.900 0.898 0.411 1.000 0.999 0.452 1.000 0.999 3.222 1.000 0.996 4.264 1.000 1.005 6.412 1.000 0.980 12.517
Median (%) 30.426 0.781 20.568 0.684 0.600 0.720 0.722 -1.673 2.806 10.543 0.484 0.639 0.665 0.484 1.031
15
15
15
10
10
10
10
5
5
5
5
15
15
10
10
5
5
Storey
15
0 0
0.2 Mode 1
0 0.4 -0.5
0 Mode 2
0 0.5 -0.5
0 Mode 3
0 0
0.5
0 0.4 -0.5
0.2 Mode 1
(a) Healthy state
0 Mode 2
0 0.5 -0.5
0 Mode 3
0.5
(b) Damaged state
Figure 5.2: Comparison between ‘system mode shapes’ and exact mode shapes for the 2-D shear building 1 0.9
Probability of damage
0.8 0.7 P
0.6
dam 10
P
0.5
dam 3
0.4 P
0.3
dam 1
0.2 0.1 0 -1
-0.8
-0.6
-0.4
-0.2 0 0.2 Damage level
0.4
0.6
0.8
1
Figure 5.3: Curves of damage probability for the 2-D shear building
5.6.2 Case Two: 3-D Torsional Shear Building The six-storey three-dimensional torsional shear introduced in chapter 4 is used here as the second numerical case to verify the proposed model updating approach. The mass matrix can be parameterized as n
M (ρ) i M i i 1
100
(5.85)
The scaled mass parameters i are equal to unit, and n 18 . Similar stiffness is assumed in both x and y principal directions for the baseline model with ki0, x ki0, x 400 kN m and ki0, y ki0, y 380 kN m . To locate the damage, four scaled stiffness parameters are introduced for each storey to represent the stiffness of current state, which are denoted by ki , x 4( i 1) 1ki0, x ; ki , y 4( i 1) 2 ki0, y ; ki , x 4(i 1) 3 ki0, x ; ki , y 4( i 1) 4 ki0, y
(5.86)
As a result, there are twenty four stiffness scaling parameters to be identified. Since the stiffness matrix is not linear in the stiffness parameters, the relationship between the stiffness matrix and the stiffness parameters should be linearized as follows (Yuen, 2010) n
K (θ) K 0 i K i
(5.87)
i 1
where n 24 and Ki
K i
n
and K 0 K (θ) i K i
(5.88)
i 1
Two states have been considered for the 3-D torsional shear building: (i) No damage occurs in the structure so that the scaled stiffness parameters should be equal to unit ; (ii) Damage occurs in 2nd, 4th, and 6th floors with the stiffness parameters assumed to be 5 7 80% ; 14 16 85% , 22 24 90% , and unit otherwise.
Table 5.4: Setup information of incomplete measured dofs for the 3-D torsional shear building Setup 1 2 3
Measured dofs 1-x, 1-y, 2-x, 2-y, 3-x,3-y 3-x, 3-y, 4-x, 4-y, 5-x,5-y 5-x, 5-y, 6-x, 6-y
Only acceleration with respect to the x -translational and y -translational dofs are assumed to be measured, while the acceleration with respect to the torsional dofs are be not available. The prediction error level is taken to be 10%, i.e. the RMS of the noise for a particular channel is equal to 10% of the RMS of the noise-free response at the corresponding dof. As shown in Table 5.4, the measured dofs can be divided into three setups. As can be seen from Table 5.4, there are two ‘reference’ dofs across each setup. The 1st, 2nd, 4th and 5th modes 101
identified by the two-stage fast Bayesian spectral density approach are adopted for structural model updating. The natural frequencies for these modes are shown in Table 5.5. The nominal stiffness scaling factor are all taken to be 10 for undamaged state and damaged state. Figure 5.4 shows the iterative histories of the most probable scaled stiffness parameters corresponding to the undamaged state and the damaged state, respectively. For both scenarios, the stiffness parameters converge to the values around the exact values quickly. Figure 5.5 presents the most probable values of the ‘system mode shape’ (square) and the exact mode shape (solid line) of the torsional shear building in damaged state. The mode shape components with respect to x-translational dofs, y-translational dofs and torsional dofs are shown in the first row, second row and third row of Figure 5.5, respectively. It indicates that these two kinds of mode shapes agree well with each other after model updating. The exact values, the most probable values (MPV), and the c.o.v. values of the stiffness scaling parameters for both scenarios are presented in Table 5.6. Based on the most probable values and the standard deviation of the stiffness scaling parameters for both scenarios, the probability curves of damage are shown in Figure 5.6, from which one can observe that the second storey, the fourth storey and the six storey have possible damage with highest probability. The median values of the stiffness scaling parameters calculated according to the damage possibility curves are shown in the last column of Table 5.6. Table 5.6 indicates that the identified structural damage extents approximate the exact ones quite well, thus the proposed method is correct and efficient. Table 5.5: Identified modal properties for the 3-D torsional shear building
Modes 1st 2nd 4th 5th
f1 f2 f4 f5
Undamaged State Actual MPV c.o.v (Hz) (Hz) (%) 2.366 2.367 0.087 2.428 2.428 0.100 6.962 6.973 0.099 7.142 7.153 0.203
102
Damaged State Actual MPV c.o.v (Hz) (Hz) (%) 2.336 2.335 0.224 2.350 2.348 0.313 6.790 6.783 0.117 7.086 7.085 0.027
9
9
8
8
7
7
6
6
5
5
i
10
i
10
4
4
3
3
2
2
1
1
0 0
0 0
50 100 Iterations (Undamaged State)
50 Iterations (Damaged State)
100
Figure 5.4: Iteration histories of i for the 3-D torsional shear building in healthy state (left)
Storey
Storey
Storey
and damaged state (right)
6 5 4 3 2 1 0 -1 6 5 4 3 2 1 0 -1 6 5 4 3 2 1 0 -1
(a) Mode 1
0 x (translation)
0 y (translation)
0
(torsion)
1
6 5 4 3 2 1 0 -1
1
6 5 4 3 2 1 0 -1
1
6 5 4 3 2 1 0 -1
(b) Mode 2
0 x (translation)
0 y (translation)
0
(torsion)
1
6 5 4 3 2 1 0 -1
1
6 5 4 3 2 1 0 -1
1
6 5 4 3 2 1 0 -1
(c) Mode 4
0 x (translation)
0 y (translation)
0
(torsion)
1
6 5 4 3 2 1 0 -1
1
6 5 4 3 2 1 0 -1
1
6 5 4 3 2 1 0 -1
(d) Mode 5
0 x (translation)
1
0 y (translation)
1
0
1
(torsion)
Figure 5.5: Comparison between ‘system mode shapes’ and exact mode shapes of the 3-D torsional shear building
103
1 0.9
Probability of damage
0.8 0.7 P
0.6 0.5
dam 22
P
0.4
dam 14
P
0.3
(P dam ) 24 (P dam ) 16
dam 5
(P dam ) 7
0.2 0.1 0 -1
-0.8
-0.6
-0.4
-0.2 0 0.2 Damage level
0.4
0.6
0.8
1
Figure 5.6: Curves of damage probability for the 3-D torsional shear building Table 5.6: Identified stiffness scaling parameters for the 3-D torsional shear building Undamaged Case Parameter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Damaged Case
Actual
MPV
c.o.v (%)
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.002 1.006 1.002 1.006 0.992 1.030 0.992 1.030 1.010 0.986 1.010 0.986 1.011 0.996 1.011 0.996 0.999 0.997 0.999 0.997 1.004 1.001 1.004 1.001
2.026 1.718 2.026 1.718 1.862 1.156 1.862 1.156 2.507 1.347 2.507 1.347 2.345 0.773 2.345 0.773 0.932 3.021 0.932 3.021 0.806 0.553 0.806 0.553
Actual
MPV
c.o.v (%)
1.000 1.000 1.000 1.000 0.800 1.000 0.800 1.000 1.000 1.000 1.000 1.000 1.000 0.850 1.000 0.850 1.000 1.000 1.000 1.000 1.000 0.900 1.000 0.900
1.005 1.004 1.005 1.004 0.812 0.997 0.812 0.997 0.987 0.973 0.987 0.973 0.994 0.846 0.994 0.846 0.996 0.992 0.996 0.992 0.997 0.901 0.997 0.901
1.850 3.549 1.850 3.549 1.374 3.703 1.374 3.703 1.099 3.440 1.100 3.440 0.305 0.173 0.305 0.173 0.502 0.449 0.502 0.449 0.517 0.741 0.517 0.741
104
Median (%)
0.006 0.009 0.006 0.009 0.188 0.039 0.188 0.039 0.031 0.021 0.030 0.021 0.024 0.158 0.024 0.158 0.011 0.013 0.011 0.013 0.015 0.107 0.015 0.107
5.7 Experimental Study The performance of the proposed model updating method is experimentally investigated with the three-storey shear building introduced in chapter 4. The modal properties identified by the two-stage fast Bayesian spectral density approach are employed for model updating. The structure can be simplified as a 3-dof shear building, with its substructure stiffness given by 1 K1 1 021
01 2 ; 02 2
0 i 2 3 01i 2 1 1 01 3 i K i i 01i 2 1 1 01 3 i 0 3 i 3
(5.89)
Here i (i 1, 2,3) represents the stiffness factors of the substructure stiffness. The linear stiffness coefficients obtained from a static test were k1 20.88 kN / m , k2 22.37 kN / m and k3 24.21 kN / m (Chang and Poon, 2010), which were used as the initial guess for structural model updating. The mass coefficients are measured as m1 5.63 kg , m2 6.03 kg and m3 4.66 kg . The substructure mass matrix are given by
1 M1 ρ1 021
01 2 ; 02 2
0 i 2 3 01 i 2 1 0 013 i M i ρi 01 i 2 0 1 013 i 0 3 i 3
(5.90)
where 1 5.63 kg , 2 6.03 kg and 3 4.66 kg . Four scenarios are considered for the structural model updating problem. For scenario 1, the mode shape components of three measured dofs are identified simultaneously in a single setup and three modes are all involved in model updating. For Scenario 2, the sensors are divided into two local groups. Each group is composed of two sensor nodes, with the one in the middle storey being overlapping (reference) node. For scenario 3, the mode shape components of the top floor are not used for model updating. For scenario 4, the third mode shape is not employed for model updating. Figure 5.7 presents the convergence curves of the stiffness coefficients for four scenarios, which indicate that the stiffness factors converge quickly by using the proposed model updating method. The identified most probable stiffness
105
coefficients as well their associated c.o.v. for four scenarios are shown in Table 5.7. The results from the static test by Chang and Poon (2010) are also presented for comparison in Table 5.7. As is seen, the results given by the static test and the proposed technique seem to be consistent. Figure 5.8 shows the most probable ‘system mode shape’ for different scenarios. The system mode shapes identified from different scenarios also agree well with each other after model updating, indicating that the proposed method is able to treat the case of missing information and multiple setups. We make no position as to whether the experimental studies can give accurate uncertainty information or not since uncertainty itself is uncertain as it depends on the identification model, the environmental conditions, or even the nature of the parameter under question (Au, 2012a). Rather, the experimental observations serve to give estimation of the order of uncertainty magnitude in the test conditions stated.
22
2 3
20
22
1 2 3
26
22
10 20 30 40 50 (b) Iterations for Scenario 2
26 1 2 3
24
22
1 2 3
i
24
i
20 18 0
10 20 30 40 50 (a) Iterations for Scenario 1
(kN/m)
18 0
(kN/m)
24
i
24
26 1
(kN/m)
i
(kN/m)
26
20 18 0
20 18 0
10 20 30 40 50 (c) Iterations for Scenario 3
10 20 30 40 50 (d) Iterations for Scenario 4
Figure 5.7: Iteration histories of model updating for four different scenarios Table 5.7: Identified stiffness parameters for four different scenarios
Parameter 1 2 3
Static (kN/m) 20.88 22.37 24.21
Scenario 1 MPV c.o.v (kN/m) (%) 19.926 6.276 23.937 5.790 23.153 5.804
Scenario 2 MPV c.o.v (kN/m) (%) 20.014 7.417 24.446 5.969 23.209 5.665
106
Scenario 3 MPV c.o.v (kN/m) (%) 19.602 6.627 23.807 5.762 23.21 5.046
Scenario 4 MPV c.o.v (kN/m) (%) 18.951 6.721 22.902 6.226 23.062 5.663
3
3
3 Scenario Scenario Scenario Scenario
Scenario 1 Scenario 2 Scenario 3 Scenario 4
1 2 3 4
2
2
1
1
1
Storey
2
Scenario 1 Scenario 2 Scenario 3
0 -1
-0.5
0
Mode 1
0.5
1
0 -1
-0.5
0
Mode 2
0.5
1
0 -1
-0.5
0
0.5
1
Mode 3
Figure 5.8: Identified optimal ‘system mode shapes’ for four different scenarios
5.8 Concluding Remarks In this chapter, a Bayesian structural model updating approach accommodating multiple uncertainties and based on natural frequencies and partial mode shapes of some modes was proposed with application to structural health monitoring. The two-stage fast Bayesian spectral density approach for ambient modal analysis proposed in chapter 3 was employed to identify the optimal modal parameters as well as their uncertainties. Then an objective function to be minimized was formulated by using Bayesian approach. The optimization problem can be solved by a fast iterative scheme so as to resolve the computational burden required for optimizing the objective function numerically. Unlike most existing model updating approaches, the proposed method can avoid the steps of mode shape expansion, eigenvalue decomposition, mode matching and assigning weighting factors subjectively, etc. Another novel feature of the proposed method is that it can incorporate the local mode shape components identified from different clusters automatically without prior assembling or processing. The numerical and experimental examples confirm the effectiveness of the proposed approach, showing it to be both computationally efficient and accurate. Case studies also illustrate that the proposed method can treat incomplete modal data (that is, missing mode shape components and missing modes).
107
Chapter 6 Use of Random Matrix Theory for Bayesian System Identification with Non-stationary Response Measurements Only 6.1 Introduction Most damage detection approaches employing wireless sensors require the response to be stationary, rendering difficulties when dealing with non-stationary excitation. Moreover, most of these damage detection approaches should consider the entire structure so as to identify a large number of uncertain parameters. This definitely raises the computational burden as well as the numerical convergence difficulty in obtaining reasonably accurate results. To circumvent these practical limitations, this research attempts to propose a technique allowing for monitoring of some critical substructures subject to non-stationary excitation. Inspired by the work of Yuen and Katafygiotis (2005; 2006), the method proposed in this study makes full use of the statistical properties of the trace of Wishart matrix. Compared with Yuen and Katafygiotis’ approach (2005; 2006), the newly proposed approach can avoid repeated timeconsuming evaluation of the determinant and inverse of the high-dimensional covariance matrix involved. Instead of optimizing all model parameters synchronically in a single group, an iterative scheme optimizing the parameters in groups is employed so as to reduce the dimension of the numerical optimization problem involved. The proposed approach allows one to obtain the most probable values of the updated model parameters as well as their associated uncertainties. Numerical and experimental shear building models subjected to nonstationary excitation are used to verify the proposed method.
6.2 Problem Description 6.2.1 Model Class of a Dynamical System Consider a stable linear structure with nd dofs and its equation of motion with generalized coordinates x(t ) given by 108
(t ) Cx (t ) Kx(t ) To f (t ) Mx
(6.1)
where M n n , C n n and K n n are the mass, damping and stiffness matrices of the d
d
d
d
d
d
structure; T0 n n is a force distributing matrix, and f (t ) n is assumed to be unknown; n f is d
f
f
the number of independent force excitations driving the system. The mass matrix M n n is d
d
assumed to be known from structural drawings with sufficient accuracy. The stiffness matrix K can be parameterized by θ {1 , 2 ,, n } as
n
K (θ) i K i
(6.2)
i 1
where K i are nominal contributions of different substructures to the stiffness matrix of the structure; i are scaling factors of various substructure stiffness matrix contributions, which aims to make the overall stiffness more consistent with the actual structural model. Rayleigh damping is assumed here with C( 0 , 1 , θ) 0 M 1K (θ)
(6.3)
Taking the Fourier transform of (6.1), one can obtain X Η ( )T0 F
(6.4)
where Η( ) denotes the frequency response function (FRF) of the structure in terms of the displacement measurements given by Η ( ) [ 2 M iC K ]1
(6.5)
Here i 2 1 . According to (6.3), (6.5) can be rearranged as Η ( ) [(i 0 2 )M (1 i1 )K ]1
(6.6)
Correspondingly, the FRF of the structure in terms of the acceleration measurement can be expressed as Η a ( ) 2 Η( ) .
109
6.2.2 Partition of the Measurements Assume that ns sets of independent and identically distributed discrete response data are available for no ( n f ) measured dofs in the given structure. The sampling time step is assumed to be t . Due to the prediction error including measurement noise and modeling error, etc., the j -th measured response y j (n) at the n-th time step nt will differ from the model response x j (n) . In the context of Bayesian inference, the measured acceleration y j (n) can be modeled as y j ( n) x j ( n) μ j ( n)
(6.7)
In practice, y j (n) is usually accelerations and in this case Η( ) must be appropriately modified. It is assumed that the difference between the measured response y j (n) and the model response x j (n) can be adequately represented by a discrete zero-mean Gaussian white noise vector process which satisfies E(μ j (n)μ j (n)T ) nn Σμ
(6.8)
where E() denotes the mathematical expectation; nn denotes the Kronecker delta; Σμ denotes the covariance matrix of the prediction-error process μ j (n) . The measurements y j (n) can be partitioned into two groups, y j (n) [y A , j (n)T , y B , j (n)T ]T
(6.9)
where y A , j (n) n denote the j -th measurement corresponding to partition A (the first n f f
coordinates) and y B, j (n) n n denote the j -th measurement corresponding to partition B (the o
f
remaining no n f coordinates). It is worth noting that any element in y A , j (n) should not be a linear combination of other elements in y A , j (n) ) as well as their derivatives, while at least one element in y B, j (n) should be not a linear combination of the elements in y A , j (n) and their
110
derivatives (Yuen and Katafygiotis, 2005). Similarly, x j (n) and μ j (n) can be partitioned with the same manner as (6.9), x j (n) [x A , j (n)T , x B , j (n)T ]T
(6.10)
μ j (n) [μ A , j (n)T , μ B , j (n)T ]T
(6.11)
As a result, the relationship between the measurements y j (n) and the model response quantities x j (n) can be written as y A , j (n) x A , j (n) μ A , j (n) y B , j (n) xB , j (n) μ B , j (n)
(6.12)
Taking the FFT for [y A , j (0), , y A , j ( N 1)] and [y B , j (0), , y B, j ( N 1)] , one obtains YA , j (k ) YB , j (k )
ik nt N 1 t n 0 y A , j ( n ) e N 1 ik nt 2 N ( n ) e y , j B n0
(6.13)
where i 2 1 ; k k , k 1, 2,, Int N 2 ; and 2 N t . The FFT of x A , j (n) , xB, j (n) , μ A , j (n) and μ B, j (n) are denoted by X A , j (k ) , XB , j (k ) , N A , j (k ) and N B , j (k ) , respectively. Then the FFT for (6.12) is given by YA , j (k ) X A , j (k ) N A , j (k ) YB , j (k ) XB , j (k ) N B , j (k )
(6.14)
6.2.3 Transmissibility Matrix Relating Two Sets of Measurements The transmissibility matrix proposed by Ribeiro et al. (2000) is usually employed to formulate the relationship between two sets of outputs so as to eliminate the input information. This concept has been widely applied in different fields of engineering, and its basic concept will be reviewed briefly here. According to (6.4), X A , j (k ) and XB, j (k ) can be formulated as X A , j (k ) H A (k )T0 F j (k )
(6.15)
X B , j k H B (k )T0 F j (k )
(6.16)
111
where H A (k ) n
f
n f
and H B (k ) ( n n o
f
) n f
denote the FRF matrixes relating all applied forces of
the structure to the measurements from partitions A and B , respectively. Note that the matrix H A (k ) is invertible, thus the applied force can be expressed as T0 F j (k ) [H A (k )]1 X A , j (k )
(6.17)
As a result, XB, j (k ) can be expressed in terms of X A , j (k ) after substituting (6.17) into (6.16) X B , j (k ) H T (k ) X A , j (k )
(6.18)
where H T (k ) is defined as the transmissibility matrix for a multi-degrees-of-freedom system with its formula given by H T (k ) H B (k )[H A (k )]1
(6.19)
According to (6.18), YB, j (k ) in (6.14) is given by YB , j (k ) H T (k ) X A , j (k ) N B , j (k )
(6.20)
From (6.14), it is not difficult to find that X A , j (k ) YA , j (k ) N A , j (k ) , thus YB , j (k ) H T (k )(YA , j (k ) N A , j (k )) N B , j (k )
(6.21)
Above equation can be further rearranged as YB , j (k ) ηB , j (k ) H μ (k )N μ , j (k )
(6.22)
ηB , j (k ) H T (k )YA , j (k )
(6.23)
H μ (k ) [ H T (k ), I no n f ]
(6.24)
N μ , j (k ) [N A , j (k )T , N B , j (k )T ]T
(6.25)
where
where I n n is a (no n f ) (no n f ) unity matrix. As can be seen from (6.22), two sets of o
f
measurements have been connected properly with each other, and the input information acting on the structure has been eliminated completely after introducing the concept of transmissibility matrix.
112
6.2.4 Random Matrix Theory Since the prediction error vector μ j (n) is assumed to be Gaussian white noise with covariance matrix shown in (6.8), then Nμ , j (k ) follows multivariate Gaussian distribution with mean and covariance matrix as (Yuen, 2010) E Nμ 0 ; CNμ
t Σμ 2
(6.26)
Thus the probability density function of Nμ , j (k ) is shown as
p Nμ , j (k )
1
2
no
det CNμ
1 exp N*μ , j (k )CN1μ N μ , j (k ) 2
(6.27)
Assume that a continuous random vector u has probability density function pU (u) . If there exists an invertible matrix A and a vector d such that v = Au + d , then the probability density of the transformed random vector v is equal to, pV ( v )
pU ( A 1 ( v d)) det A
(6.28)
According to (6.28), the probability density of YB, j (k ) H μ (k )Nμ , j (k ) ηB, j (k ) is equal to p YB (k )
1
1
2
det H μ (k )
no
det CNμ
1 exp N*μ , j (k )CN1μ N μ , j (k ) 2
(6.29)
One can observe from (6.22) that
N μ , j (k ) H μ1 (k ) YB , j (k ) ηB , j (k )
(6.30)
Furthermore,
det H μ (k ) det H μ (k )H*μ (k )
(6.31)
According to (6.30) and (6.31), (6.29) can be rearranged as
(k ) p Y B
1 * 1 exp Y YB , j ( k ) B , j ( k )C Y B 2 det CY (k ) 1
2
no
(6.32)
B
where (k ) Y (k ) η (k ) Y B, j B, j B, j
CY B (k ) H μ (k )CNμ H*μ (k )
113
t H μ (k ) Σμ H*μ (k ) 2
(6.33) (6.34)
From (6.32), it is not difficult to find that the new random vector Y B, j (k ) follows multivariate normal distribution. Therefore, from the random matrix theory, the following multivariate complex matrix ns
* Ssum ( k ) YB , j ( k )YB , j ( k ) Y B
(6.35)
j 1
should follow central Wishart distribution with the covariance matrix CY (k ) (Goodman, 1963). B
Substituting (6.23) into (6.33), after rearrangement, leads to Y (k ) (k ) H (k ) A , j Y B, j μ YB , j (k )
(6.36)
By substituting (6.36) into (6.35), one can obtain that ns YA , j (k ) * * * k Ssum ( ) H μ (k ) YA , j (k ) YB , j (k ) H μ (k ) Y B k Y ( ) j 1 B , j
(6.37)
Rearranging (6.37) gives sum * Ssum ( k ) H μ ( k )S Y ( k ) H μ ( k ) Y B
(6.38)
n n is just the sum of power spectral density matrix corresponding to ns sets where Ssum Y (k ) o
o
of measurements, given by ns Y A , j (k ) * * Ssum ( ) k YA , j (k ) YB , j (k ) Y Y ( ) k j 1 B , j
(6.39)
It can be shown that the matrix Ssum ( k ) is a central Wishart matrix with the degrees of freedom Y B
of ns and its covariance matrix CY (k ) . Furthermore, it can be proved that when the degree of B
freedom ns is large, the trace of a central Wishart matrix A denoted by tr ( A) asymptotically follows a normal distribution with mean ns tr ( Σ) and variance 2ns tr ( Σ 2 ) , where Σ is the covariance matrix of (Mathai, 1980), i.e., tr ( A) (ns tr ( Σ), 2ns tr ( Σ 2 ))
(6.40)
Therefore, tr (Ssum ( k )) asymptotically follows normal distribution as ns is large, with the Y B
probability density function given by
114
p(tr (S
sum Y B
2 [tr (S sum ( k )) ns tr (C Y ( k ))] Y B B (k ))) [4 ns tr (C (k ))] exp 4ns tr (C2Y (k )) B 2 Y B
1 2
(6.41)
6.3 Bayesian Model Updating Using Non-stationary Measurements 6.3.1 Negative Log-likelihood Function When the number of discrete data points N , Y B, j (k ) and Y B , j (k ) are uncorrelated as k k , sum which states that Ssum ( k ) and S Y ( k ) are also independently central Wishart distributed Y B
B
sum as k k . As a result, their traces tr (Ssum ( k )) and tr (S Y ( k )) are independently normal distributed Y B
B
for k k , i.e., sum sum sum p(tr (Ssum ( k )), tr (S Y ( k ))) p (S Y ( k )) p (S Y ( k )) Y B B B B
(6.42)
A subset of {tr (Ssum ( k )) k k1 , k 2 } can be formed from the range of frequencies k1 , k 2 . Y B
Therefore, given the class of models M and the model parameters λ , the likelihood function p ( , M )
can be expressed as
2 [tr (Ssum ( k )) ns tr (C Y ( k ))] Y B B p( , M ) p (y B , y A , M ) [4 ns tr (C (k ))] exp B 4ns tr (CY2 (k )) k k1 B k2
2 Y
1 2
(6.43)
Using the Bayes’ theorem, the updated PDF of the model parameters λ given the measured response y [yTA , yTB ]T is given by (Beck and Katafygiotis, 1998) p ( y , M ) c0 p ( M ) p (y , M )
(6.44)
where c0 is a normalizing constant; p( M ) is the prior distribution based on previous knowledge, which can be absorbed into the normalizing constant c0 when no prior information is available. The likelihood function p(y , M ) can be expanded into a product of conditional probabilities (Yuen and Katatygitios, 2005) p(y , M ) p(y A , y B , M ) p(y B , y A , M ) p (y A , M ) p (y A , M ) can be absorbed into the constant
(6.45)
c0 if it is non-informative, i.e.,
p ( y , M ) c0 p (y B , y A , M )
115
(6.46)
where c0 c0 p ( M ) p (y A , M ) . As a result, the posterior PDF of parameters is proportional to the likelihood function p(y B , y A , M ) shown in (6.43), which can be written in terms of the log-likelihood function Ltrace ( ) as p ( y , M ) exp( Ltrace ( ))
(6.47)
where 2 [tr (Ssum ( k )) ns tr (C Y ( k ))] 1 k2 Y 2 B B Ltrace ( ) ln 4 ns tr (CY (k )) B 2 k k1 4ns tr (C2Y (k ))
(6.48)
B
Substituting (6.34) and (6.38) into (6.48) results in
tr H (k )(Ssum (k ) n C )H (k ) μ Y μ s Nμ 1 2 Ltrace ( ) ln 4 ns tr ([H μ (k )CNμ H μ (k )] ) 2 2 k k1 4ns tr ([H μ (k )CNμ H μ (k )] ) k2
2
(6.49)
6.3.2 Most Probable Values and Posterior Uncertainties Let ˆ be the most probable value that minimizes Ltrace ( ) and consider the second-order Taylor series about ˆ 1 Ltrace ( ) Ltrace (ˆ ) ( ˆ )T Γtrace (ˆ )( ˆ ) 2
(6.50)
where the first-order term vanishes due to optimality of ˆ ; Γtrace is the Hessian of Ltrace ( ) at the most probable value ˆ . Therefore, the updated PDF of the parameter λ can be well approximated by a Gaussian distribution 1 1 ( ˆ )) p( y, M ) exp( ( ˆ )T Ctrace 2
(6.51)
1 where Ctrace trace (ˆ ) is the posterior covariance matrix. Since the Gaussian PDF is completely
characterized by the most probable value ˆ and the Hessian Γtrace , the efficient computation of these quantities becomes the natural target in Bayesian system identification problems. The most probable parameter values λˆ can be calculated by minimizing the objective function Ltrace ( ) numerically, while the Hessian matrix Γtrace can be calculated through finite differentia approach (Yuen, 2010). 116
In
this
study,
the
model
parameters
required
to
be
identified
include [{i : i 1, , n }; ;{ 0 , 1}] . For the covariance matrix of prediction errors (i.e., ), the elements of the upper right triangle (diagonal inclusive) need to be identified. To reduce the dimension of optimization problem, it is reasonable to assume that the prediction errors for different measurements are independent. Therefore, reduces to be a diagonal matrix. The covariance of the prediction error corresponding to the i-th measured dof is denoted by i2 . As a result, the parameters required to be identified include [{i : i 1,, n };{ 12 , 22 ,, n2 };{ 0 ,1}]
(6.52)
o
The dimension for the numerical optimization problem can still be very high if one optimizes all the model parameters simultaneously, which may render the problem of converge. Numerical trials show that changing one group of the parameters does not significantly affect the optimal value of others, indicating that the full set of model parameters can be optimized by an iterative scheme. In other words, the parameters can be optimized in groups given the remaining variables till convergence so as to reduce the dimension of the numerical optimization problem (Au, 2012a). Different groupings have been investigated illustrates that optimizing {i : i 1,, n } , { 0 ,1} and { 12 , 22 ,, n2 } sequentially and iteratively lead to better o
results.
6.3.3 Initial Guesses for the Parameters to be identified The stiffness parameters from initial finite element model can be used as the initial guess for i . A nominal value of 1% may be assumed for the first two modes of damping ratios, which can be used to achieve the initial guess of the damping coefficients. To get the initial guess of the prediction errors, one can assume that the magnitudes of prediction errors corresponding to all measurements are the same, i.e., i2 2 . As a result, (6.49) can be simplified as
2 tr H μ (k )(Ssum Y (k ) ns I no ) H μ ( k ) 1 k2 4 2 Ltrace ( ) ln 4 ns tr ([H μ (k )H μ (k )] ) 4 2 2 k k1 4ns tr ([H μ (k )H μ (k )] )
117
2
(6.53)
For the case with high signal-to-noise ratio, it is reasonable to assume that 2 sum Ssum Y ( k ) ns I no S Y ( k )
(6.54)
Thus (6.53) can be simplified as k2
Ltrace ( ) n ln 4ns 4 [ln tr ([H μ (k )H μ (k )]2 )] k k1
1 4ns 4
k2
2 [tr (H μ (k )S sum Y (k ) H μ ( k ))]
tr ([H μ (k )H μ (k )]2 )
k k1
(6.55)
where n k2 k1 1 . The differentiation of Ltrace ( ) in (6.55) with respect to 2 are given by n Ltrace 2 8ns 2 4ns 4 4ns 6 2
Solving
Ltrace 0 2
k2
k k1
2 [tr (H μ (k )S sum Y ( k ) H μ ( k ))]
tr ([H μ (k )H μ (k )]2 )
(6.56)
for 2 gives 2
1 4ns n
k2
2 [tr H μ (k )S sum Y (k )H μ (k ) ]
tr ([H μ (k )H μ (k )]2 )
k k1
(6.57)
6.3.4 Summary of Procedures The procedure of Bayesian approach for model updating using non-stationary measurements only can be summarized as follows: (i) Set initial guess for {i : i 1,, n } , { 0 ,1} and { 12 , 22 ,, n2 } as illustrated in section 6.3.4; o
(ii) Optimize {i : i 1,, n } to minimize (6.49); (iii) Optimize { 0 ,1} to minimize (6.49); (iv) Optimize { 12 , 22 ,, n2 } to minimize (6.49); o
(v) Repeat step (ii) to step (iv) till convergence; (vi) Calculate Γtrace ( ) at ˆ to obtain the covariance matrix of the identified parameters. Compared with the Bayesian FFT approach (Yuen and Katatygitios, 2005), the proposed method can avoid repeated evaluation of the determinant and inverse of the following covariance matrix
118
1 Re CY B (k ) Cfft 2 Im C (k ) Y B
T Im CY (k ) B Re CY (k ) B
(6.58)
at different frequency points k during the process of optimization. Matrix determinant and inversion are time-consuming operations whose computation complexity nonlinearly increases with the size of matrix. For each trial of λ , the number of evaluation for the determinant and inverse is equal to n k2 k1 1 (i.e., the number of frequency points), which can be very large in that all the resonant peaks of the spectral density estimates are suggested to be included. Moreover, Cfft cannot be ensured to be well-conditioned, especially for response with high signal-to-noise ratio since Σμ can approximately be rank deficient (singular). However, using the trace of Wishart matrix can well avoid the problems aforementioned.
6.4 Damage Detection (Vanik et al., 2000) It has been proved by Vanik et al. (2000) that the probability of damage given a fractional damage level d can be approximated as ˆ pd (1 d )θˆ ud i θi Pi dam (d ) (1 d ) 2 ( iud ) 2 ( ipd ) 2
(6.59)
where () is the standard Gaussian cumulative distribution function; ˆudj and ˆjpd denote the most probable values of the stiffness parameters for the undamaged and possibly damaged structure, respectively; iud and ipd are the corresponding standard deviations of the stiffness parameters. ˆudj , ˆjpd , iud and ipd , and can be obtained from section 6.3.
119
6.5 Bayesian Substructuring Model Updating
F2(t)
F1(t)
Figure 6.1: Generic structure with some external forces
f int(t) F2(t)
g(t)
F1(t)
f int(t)
Figure 6.2: Schematic plot of Substructure I (left) and Substructure II (right) Base Station Node
Cluster Head Nodes Leaf Nodes
Figure 6.3: Two-level hierarchical architecture of WSN Most available damage detection approaches require considering the entire structure with a large number of uncertain parameters to be identified. To reduce the number of unknown model parameters, one can consider some part of the structure as independent target for analysis. The proposed model updating approach introduced in section 6.2, 6.3 and 6.4 can be
120
applied for a substructure easily just by taking the interface forces at the interface dofs as input excitations. For a generic structure with some external forces, there are two kinds of substructures (Yuen & Katafygitios, 2006). When the substructure below the interface is considered, the substructure shown in left part of Figure 6.2 by itself is a stable structure. The substructure is subjected to interface forces fint t from adjacent substructures besides the external excitation F1 t . When the substructure above the interface shown in right part of Figure 6.2 is considered, an artificial support at an arbitrary point with support motion g (t ) should be enforced since the substructure by itself is not a stable structure. In this case, the uncertain input vector for the substructure is composed of the external excitation F2 (t ) , interface forces fint (t ) and the support motion g (t ) . The proposed damage detection algorithm can be implemented by a distributed computing strategy, which can be formed as a two-level hierarchical architecture shown in Figure 6.3. Wireless sensors are grouped into several clusters with each cluster responsible for a substructure considered. In the context of schematic network architecture, each cluster is a community composed of a cluster head node and several leaf nodes. All of the leaf nodes can report to cluster head nodes directly, while the cluster head nodes can report to base station node directly which are connected to a computer. The first step of implementation in WSN is data acquisition. The time history can be divided into ns segments. Subsequently, take FFT for each acquired segment in the leaf nodes. The FFT coefficients in the selected frequency band including all the dominant peaks should be sent to the cluster node from the leaf nodes. Then in the cluster node, FFT coefficients are used to calculate the spectral density matrix. Once one obtains the spectral density matrix, Bayesian approach introduced in section 6.2 and 6.3 can be employed to identify the model parameters.
121
6.6 Numerical Studies 6.6.1 Case one: 3-dof Spring-Mass System
y(t)
k
k
1
k
2
m y
m y
1
2
1
2
3
m y
3
3
Figure 6.4: Schematic plot of 3-dof spring-mass system 30
10
2
20 1
10 2 -3
S y (m s )
-2
Acceleration (ms )
10
0
10
0
-10 10
-1
-20
-30 0
10
20 30 40 Time(seconds)
50
10
-2
0
5 10 Frequency (Hz)
15
Figure 6.5: Raw measurement (left) and spectral density (right) of the third dof To illustrate the accuracy of the proposed approach, simulated data of a simple 3-dof springmass system shown in Figure 6.4 is processed firstly. Here the excitation is set to be a stochastic process modeled as Gaussian white noise with spectral density 0.05 m 2 s 3 multiplied by an envelope function v(t ) 0.1te0.1t . The measurement noise is taken to be white noise with the spectral density 0.001 m 2 s 3 . The parameters used for generating the data are m j 1kg and k j 300 N / m ( j 1, 2,3 ).
Rayleigh’s damping is assumed, and the damping for the first two
modes is taken to be 2%. The scaling parameters j are introduced so that the identified stiffness can be scaled as k j j k j . Twenty sets of acceleration time histories are measured for 122
all dofs with each segment lasting 200s, and the sampling frequency is set to be 200Hz. Raw measurement and its spectral density of the third dof are shown in Figure 6.5. Model updating is carried out using the spectral density of these measurements up to 8Hz including all significant responses. The measurements are partitioned as y A { y1} and y B { y2 , y3 } . The exact value of the parameters λ , the most probable value (MPV) λˆ , the standard deviation , the coefficient of variation c.o.v. ˆ and the ‘normalized distance’ ˆ are shown in Table 6.1. The ‘normalized distance’ represents the absolute value of the difference between the identified optimal value and the exact value, normalized with respect to the corresponding calculated standard deviation. Figure 6.6 shows the conditional PDFs of the stiffness scaling factors 1 and 2 . The solid lines are obtained by using the Bayesian approach with all other parameters fixed at their optimal values, while the dashed lines are obtained by employing the Gaussian approximation. It can be seen that the two sets of curves agree very well with each other, indicating that the proposed Gaussian approximation is accurate. Therefore, the inverse of the Hessian matrix can be used to represent the covariance matrix. Figure 6.7 shows contours in the (1 , 2 ) plane of the marginal updated PDF. It is seen that the exact values are located at a reasonable distance from the optimal point. Moreover, 1 and 2 are considered to be slightly correlated since the elliptical contours are rotated a bit. From this numerical example, one can conclude that the proposed method is able to identify the model parameters when considering the entire system. Table 6.1: Identified model parameters for the 3-dof spring-mass system Parameters
λ
λˆ
c.o.v.
1 2 3 0 1
1.0000 1.0000 1.0000 0.2272 0.0014
1.0075 1.0022 0.9930 0.2033 0.0014
0.0070 0.0022 0.0034 0.0366 0.0001
0.0069 0.0022 0.0034 0.1799 0.0507
1.0755 0.9718 2.0521 0.6539 0.7014
123
300
350 Bayesian Approach Gaussian Approximation
300
250
Bayesian Approach Gaussian Approximation
250 200 PDF
PDF
200 150
150 100 100 50
50
0 1
1.005
1.01
0 0.99
1.015
0.995
1
1
1.005
1.01
2
Figure 6.6: Conditional PDFs of 1 (left) and 2 (right) 1.01 1.008 1.006
Optimal Point Actual Point 50% 90%
2
1.004 1.002 1 0.998 0.996 0.99
0.995
1
1.005
1
1.01
1.015
1.02
1.025
Figure 6.7: Contours of the marginal PDFs of 1 and 2
6.6.2 Case Two: 20-Stoey Shear Building The second numerical example uses simulated response data from a 20-storey shear building. It is assumed that this building has a uniformly distributed mass and stiffness at each floor and the stiffness to mass ratio is chosen to be 1500s 2 . The damping ratios of the first two modes are assumed to be 1.0% so that the damping coefficients in this case are o 0.0386 s 1 and 1 0.00194 s . The excitation is taken to be ground motion xg which can be
adequately modeled as stochastic process with its frequency content modeled by the Clough124
Penzien spectrum, multiplied by an envelope function v(t ) 100te0.1t . The Clough-Penzien spectrum is shown as follows S g ( ) So
1 2 g ( g ) 2
( f ) 4
(6.60)
[1 ( g ) 2 ]2 (2 g g ) 2 [1 ( f ) 2 ]2 (2 f f ) 2
with parameters g 15.7 rad s , f 0.1g 1.57 rad s , f g 0.6 and So 0.001m 2 s 3 . The way to simulate the random process with Clough-Penzien spectrum can be referred to (Wang J., 2010). One time history of the ground motion is shown in Figure 6.8.
Ground Motion (ms-2)
10
5
0
-5
-10 0
10
20
30 40 Time(seconds)
50
60
70
Figure 6.8: Time history of ground motion f int (t )
k5
C5
k4
C4
k3
C3
k2
C2
k1
C1
xg
Figure 6.9: Substructure including the first five floors of the shear building 125
The lowest floors are more susceptible to damage since they are subjected to larger shear forces than the upper stories. Therefore, a substructure comprising the first five stories shown in Figure 6.9 is considered for identification. In this case, n f 2 (ground motion xg and shear force fint (t ) at the top of the substructure). Two scenarios have been considered for the substructure including the first five floors: (i) no damage occurs in the substructure; (ii) damages occur in the first and fourth floor with the damage extent of 20% and 10%, respectively. Acceleration measurements of the first five floors (i.e., no 5 ) are assumed to be available with a sampling interval t 0.01s . The measurement noise of the responses is taken to be Gaussian white noise. Twenty sets of data with each set lasting 200 seconds are generated for the substructure in undamaged state and damaged state, respectively. Figure 6.10 shows a sample acceleration time history of the fifth floor as well as its corresponding auto-spectral density. The measurements of the first two floors are grouped into y A while the measurements of the third to fifth floor are grouped into y B . Model identification is carried out using the spectral density up to 10 Hz, and there are 8 modal frequencies exceeding 10 Hz. Figure 6.11 and Figure 6.12 show typical plots of the conditional PDFs of 1 and 2 for the substructure in healthy state and damaged state. As is seen, the curves calculated from Bayesian approach (solid line) with all other parameters fixed at their optimal values and Gaussian approximation (dashed line) coincide with each other, indicating that the Gaussian approximation is reliable. The identified stiffness factors for the substructure in healthy state and possibly damaged state are shown in Table 6.2. The second column and the fifth column of Table 6.2 are exact values corresponding to the structural model that generated the time histories. The third column and sixth column correspond to the most probable parameters, while the fourth column and seventh column present the coefficient of variance (c.o.v.) of the identified stiffness factors. Based on the most probable parameters as well as the standard deviation of the identified stiffness factors, the probability of damage for each stiffness factor can be obtained. The probability damage curves are shown in Figure 6.13, from which one can clearly observe that the first floor and the third floor are possibly damaged. The median values of these probability 126
curves are shown in the last column of Table 6.2. As is seen from Table 6.2, the median values corresponding to the first floor and the third floor are around 14.5% and 9.8%, respectively, which agree well with the exact damage extents. Therefore, the proposed approach can successfully locate and quantify damage within the confidence levels. From this numerical example, one can conclude that the proposed method is applicable for substructuring identification. 4
250
10
200 2
10
100 0
2 -3
Sy (m s )
-2
Acceleration (ms )
150
50 0
10
-2
10
-50 -100
-4
10
-150 -200 0
-6
50
100 150 Time(seconds)
10
200
0
5
10 15 Frequency (Hz)
20
Figure 6.10: Acceleration measurement (left) and spectral density (right) of the fifth dof 900 800
1100 Bayesian Approach Gaussian Approximation
1000
Bayesian Approach Gaussian Approximation
900 700 800 700
500
PDF
PDF
600
400
600 500 400
300
300 200 200 100 0 0.992
100 0.994
0.996
0.998
0 0.996
1
1
0.997
0.998
0.999
1
1.001
2
Figure 6.11: Conditional PDFs of 1 and 2 for the substructure in healthy state
127
900
900 Bayesian Approach Gaussian Approximation
800
700
700
600
600
500
500
PDF
PDF
800
400
400
300
300
200
200
100
100
0 0.78
0.782
0.784
0.786
Bayesian Approach Gaussian Approximation
0.788
0 0.982
0.984
1
0.986
0.988
0.99
2
Figure 6.12: Conditional PDFs of 1 and 2 for the substructure in damaged state Table 6.2: Identified stiffness parameters for the substructure considered Healthy state Parameters 1 2 3 4 5
Exact
MPV
1.000 1.000 1.000 1.000 1.000
0.996 0.999 0.999 1.039 1.025
Damaged state c.o.v. (%) 2.276 1.948 0.799 4.553 0.948
Exact
MPV
0.800 1.000 0.850 1.000 1.000
0.784 0.986 0.853 1.022 1.017
Median (%)
c.o.v. (%) 1.344 1.424 0.943 4.263 1.733
21.821 2.063 15.293 2.010 1.178
1 0.9
Probability of damage
0.8 0.7 0.6 P
0.5
dam 3
0.4 P
0.3
dam 1
0.2 0.1 0 -1
-0.8
-0.6
-0.4
-0.2 0 0.2 Damage level
0.4
0.6
0.8
1
Figure 6.13: Probability of damage for each stiffness parameter of the substructure 128
6.7 Experimental Verification The performance of the proposed Bayesian model updating approach was also experimentally investigated with the three-storey shear building. The shear building model has been described in detail in Chapter 4 and thus is omitted here. Two kinds of excitations were considered for the experimental study. First of all, non-stationary random numbers were imported into the shake table to generate ground motion. The stiffness parameters of the entire structure were required to be identified. Secondly, only the top floor of the structure was excited by manual impacts with unmeasured random speed and magnitude. In this case, only the substructure including the first two floors was required to be considered for identification. Wireless sensors were installed on each floor of the shear building to measure the acceleration time history. Horizontal accelerations were measured by each wireless sensor node with a sampling rate of 100 Hz and a 40 Hz cut-off frequency. Three sets of acceleration time histories were recorded for these two different excitation conditions, with each set lasting 15 minutes with 90000 points. The measured response of top floor for the first excitation condition and its corresponding auto-spectral density are shown in Figure 6.14. Figure 6.15 presents the measured response of the middle floor and its corresponding auto-spectral density for the second excitation case. The level of maximum acceleration responses is within 1g and 2
g.
For
the
first
excitation
case,
the
measurements
were
partitioned
as
y A {y1} and y B {y 2 , y 3 } , while y A {y1} and y B {y 2 } for the second excitation case.
To observe the effect of time duration of measured data on the identified results, the measured three data sets are divided into different number segments with each segment lasting 60s (denoted as scenario 1), 100s (denoted as scenario 2), 150s (denoted as scenario 3), and 300s (denoted as scenario 4), respectively. For the first excitation condition, the convergence curves of different scenarios are shown in Figure 6.16, while those corresponding to the second excitation condition are presented in Figure 6.17. Table 6.3 illustrates the identified most probable values (MPV) as well as their coefficient of variances (c.o.v.) values for different scenarios. The identified most probable stiffness factors are also compared with the results from static test (Chang and Poon, 2010) shown in the third column of Table 6.3. The results identified from the proposed technique seem to be consistent with 129
the results of static test, without significant differences. The variations of the identified values (MPV and c.o.v.) for the full structure and the substructure with the increase of time duration are shown in Figure 6.18 and Figure 6.19, respectively. The variations of the MPV and c.o.v. are not significant for different time durations. This indicates that the identified results are not sensitive to the time duration and the number of data sets. Table 6.3: Identified results for different scenarios under different excitation conditions Duration Parameters (s)
Scenario
1
1
60
2 3 1
2
100
2 3
150
3
1 2 3 1
4
300
2 3
Full structure MPV c.o.v.(%) 20.349 0.38 22.811 0.35 22.873 0.64 20.406 0.36 22.848 0.34 22.915 0.63 20.370 0.34 22.871 0.33 23.200 0.61 20.240 0.344 22.904 0.342 24.074 0.662
Static 20.88 22.37 24.21 20.88 22.37 24.21 20.88 22.37 24.21 20.880 22.370 24.210
3
10
Substructure MPV c.o.v.(%) 19.549 0.17 22.189 0.11 19.528 0.08 22.222 0.07 19.507 0.07 22.230 0.06 19.503 0.062 22.269 0.050 -
5
2 0
1 2 -3
S y (m s )
Acceleration (g)
10
0
10
-5
-1 10
-10
-2
-3 0
50 Time(seconds)
100
10
-15
0
50 100 150 Frequency (rad/s)
200
Figure 6.14: Acceleration (left) and its spectral density (right) of the top floor under the first excitation condition
130
2
10
2
1.5 10
0
0.5
2 -3
S y (m s )
Acceleration (g)
1
0 -0.5
10
10
-1 10
-2
-4
-6
-1.5 -2 0
50 Time(seconds)
10
100
-8
0
50 100 150 Frequency (rad/s)
200
Figure 6.15: Acceleration (left) and spectral density (right) of the middle floor under the second excitation condition
25
2 3
i
20 15 0
5 10 15 20 (a) Iterations for Scenario 1
40
35
2 3
2 3
30
5 10 (b) Iterations for Scenario 2
40 1
1
15
35
30
1 2 3
i
30
40
20 0
25
i
(kN/m)
50 1
(kN/m)
(kN/m)
i
(kN/m)
30
25 20 0
5 10 (c) Iterations for Scenario 3
15
25 20 0
5 10 (d) Iterations for Scenario 4
Figure 6.16: Iteration histories for full structural identification
131
15
25
2
22
1 2
21
i
20 15 0
5 10 15 20 (a) Iterations for Scenario 1
20 19 0
25
23
23
22
22
21
1 2
10 20 30 (b) Iterations for Scenario 2
40
1 2
21
i
i
(kN/m)
23 1
(kN/m)
(kN/m)
i
(kN/m)
30
20 19 0
5 10 15 (c) Iterations for Scenario 3
20 19 0
20
20 40 (d) Iterations for Scenario 4
60
Figure 6.17: Iteration histories for the substructure identification 50
45
40
2 1
1.8
2
1.6
3
1 2 3
1.4 c.o.v.
1.2
30
1
i
(kN/m)
35
0.8
25
0.6 20 0.4 15 10 0
0.2 100 200 Time duration (s)
0 0
300
100 200 Time duration (s)
300
Figure 6.18: Effect of time duration on the MPV (left) and c.o.v. values (right) for the full structural identification
132
50
45
1 1
0.9
2
0.8
1 2
40 0.7 c.o.v.
0.6
30
0.5
i
(kN/m)
35
0.4
25
0.3 20 0.2 15 10 0
0.1 100 200 Time duration (s)
300
0 0
100 200 Time duration (s)
300
Figure 6.19: Effect of time duration on the MPV (left) and c.o.v. values (right) for the substructural identification
6.8 Concluding Remarks In this chapter, a Bayesian system identification approach based on noisy non-stationary response measurements was proposed by employing the concept of transmissibility matrix and the random matrix theory. The model updating problem can be formulated as one minimizing a negative log-likelihood function, which can be used to determine the most probable values by an effective iterative coupled method and their associated uncertainties by taking the inverse of its Hessian matrix. The updated PDF of the parameters can be accurately approximated by a multivariate Gaussian distribution. One novel feature of the proposed method is to avoid repeated time-consuming manipulations of the determinant and inverse of the covariance matrix during convergence process due to exploiting the statistical properties of the trace of Wishart matrix. Moreover, the proposed approach allows one to monitor a small critical substructure. It requires no information about the spectral density model of the unmeasured input. However, the proposed method requires that the number of independent measurements is larger than the number of independent inputs driving the system. The theory described in this chapter was employed to identify the stiffness factors of different systems
133
using synthetic data and laboratory testing data. Results from experimental studies show that the proposed method has the potential to be applied in real applications.
134
Chapter 7 Conclusions
Uncertainty is the only certainty there is, and knowing how to live with insecurity is the only security. by John Allen Paulos To fulfill the need of uncertainty quantification in the environment of WSN, this work is focused on developing new advanced statistical algorithms for ambient modal analysis and structural damage detection using the Bayesian statistical framework. In this chapter, we briefly summarize the main results achieved in the present work. In chapter 2, the background of WSN based SHM was introduced. Namely,
Some important issues such as the components of wireless sensors, the main features and advantages over traditional wired sensors, the state-of-the-art of the wireless sensor platforms, and the embedded operational systems as well as some typical applications over the past decade were reviewed briefly.
The hardware (e.g., Crossbow Imote2 platform as well as the SHM-H sensor board) and the engineering analysis software (e.g., ISHMP Services Toolsuite) employed in the experimental studies were also described briefly.
In chapter 3 and chapter 4, we developed a two-stage fast Bayesian spectral density approach for ambient modal analysis under stationary excitation,
Following the tactic of ‘divide and conquer’, the spectral density bandwidth can be divided into several frequency sub-bands. The selected resonant frequency sub-band either composed of one separated mode or multiple closely spaced modes can be conquered individually to extract the modal properties and the associated uncertainties.
The interaction between spectrum variables (e.g., frequency, damping ratio as well as the amplitude of modal excitation and prediction error) and spatial variables, i.e., mode 135
shape, can be decoupled completely for the case of separated modes and closely spaced modes.
The spectrum variables can be identified through ‘fast Bayesian spectral trace approach’ (FBSTA) in the first stage, while the spatial variables, i.e., mode shape components, can be estimated in a second stage by ‘fast Bayesian spectral density approach’ (FBSDA). The intrinsically simple relationship between FBSDA and fast Bayesian FFT approach was also revealed. The challenges of conventional BSDA are well addressed by employing the two-stage Bayesian approach.
A Bayesian mode shape assembly methodology was proposed to assemble the local mode shapes confined to different clusters so that the weight for different clusters is accounted for properly according to their data quality. The optimal global mode shape as well as their associated uncertainties can be obtained by effective schemes.
In chapter 5, a Bayesian structural model updating methodology based on incomplete modal properties (e.g., natural frequencies and partial mode shapes of some modes) was developed.
An objective function able to incorporate the identified modal information from multiple clusters automatically without prior assembling or processing was formulated for the structural model updating problem. The most probable modal parameters and their covariance matrix can be determined from the formulated objective function.
To resolve the computational burden required for optimizing the objective function numerically, a fast analytic-iterative scheme was proposed to efficiently compute the optimal parameters effectively. The Hessian matrix required for specifying the covariance matrix was also derived analytically. Moreover, the computational difficulty in estimating the inverse of high dimensional Hessian matrix was also properly treated.
The steps frequently encountered in conventional model updating approaches involving assigning weighting factors subjectively, matching between the measured mode and the model mode, mode shape expansion and eigenvalue decomposition were skipped by employing the proposed method.
Finally, in chapter 6, 136
A Bayesian model updating approach using non-stationary response measurements only was formulated by employing the concept of transmissibility matrix, random matrix theory and Bayes’ theorem. The proposed method provides a rigorous means for obtaining structural model parameters as well as their associated uncertainties by operating in the frequency domain.
To reduce the dimension of the numerical optimization problem involved, an iterative coupled method involving the optimization of the parameters in groups instead of optimizing all the unknown parameters simultaneously was employed. The initial guess for the parameters to be estimated was also properly estimated.
The proposed method allows for monitoring of some critical substructures rather than the entire structure, requiring no information about the model of the external input or interface forces. By exploiting the statistical properties of the trace of Wishart matrix, repeated time-consuming evaluations of the determinant and inverse of the covariance matrix in the optimization problem were avoided in the proposed method.
Numerical examples were presented to illustrate the accuracy and efficiency of the methods proposed in chapter 3, 4, 5 and 6. Moreover, the Crossbow Imote2 platforms interfaced with SHM-H sensor boards were installed in laboratory shear building models to verify the proposed method experimentally. Successful completion of ambient modal analysis and model updating using measured acceleration responses from wireless sensors demonstrates the feasibility and potential they possess to monitor structural health accommodating multiple uncertainties.
137
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