Structural systems often show nonlinear behavior under severe excitations generated by natural hazards. In that condition, the restoring force becomes highly nonlinear showing significant hysteresis. The hereditary nature of this nonlinear restoring force indicates that the force cannot be described as a function of the instantaneous displacement and velocity. Accordingly, many hysteretic restoring force models were developed to include the time dependent nature using a set of differential equations. This survey contains a review of the past, recent developments and implementations of the Bouc-Wen model which is used extensively in modeling the hysteresis phenomenon in the dynamically excited nonlinear structures.

The Hysteresis Bouc-Wen Model, a Survey

Preface. List of Figures. List of Tables. 1. Introduction 1.1 Objective and contents of the book 1.2 The Bouc-Wen model: origin and literature review 2. Physical consistency of the Bouc-Wen model 2.1 Introduction 2.2 BIBO stability of the Bouc-Wen model 2.2.1 The model 2.2.2 Problem statement 2.2.3 Classi ~ cation of the BIBO stable Bouc-Wen models 2.2.4 Practical remarks 2.3 Free motion of a hysteretic structural system 2.3.1 Problem statement 2.3.2 Asymptotic trajectories 2.3.3 Practical remarks 2.4 Passivity of the Bouc-Wen model 2.5 Limit cases 2.5.1 The limit case n = 1 2.5.2 The limit case (r) = 1 2.5.3 The limit case (r) = 0 2.5.4 The limit case ~ + - = 0 2.6 Conclusion 3 Forced limit cycle characterization of the Bouc-Wen model 3.1 Introduction 3.2 Problem statement 3.2.1 The class of inputs 3.2.2 Problem statement 3.3 The normalized Bouc-Wen model 3.4 Instrumental functions 3.5 Characterization of the asymptotic behavior of the hysteretic output 3.5.1 Technical Lemmas 3.5.2 Analytic description of the forced limit cycles for the Bouc-Wen model 3.6 Simulation example 3.7 Conclusion 4 Variation of the hysteresis loop with the Bouc-Wen model parameters 4.1 Introduction 4.2 Background results and methodology of the analysis 4.2.1 Background results 4.2.2 Methodology of the analysis 4.3 Maximal value of the hysteretic output 4.3.1 Variation with respect to 4.3.2 Variation with respect to 4.3.3 Variation with respect to n 4.3.4 Summary of the obtained results 4.4 Variation of the zero of the hysteretic output 4.4.1 Variation with respect to 4.4.2 Variation with respect to 4.4.3 Variation with respect to n 4.4.4 Summary of the obtained results 4.5 Variation of the hysteretic output with the Bouc-Wen model parameters 4.5.1 Variation with respect to 4.5.2 Variation with respect to 4.5.3 Variation with respect to n 4.5.4 Summary of the obtained results 4.6 The four regions of the Bouc-Wen model 4.6.1 The linear region Rl 4.6.2 The plastic region Rp 4.6.3 The transition regions Rt and Rs 4.7 Interpretation of the normalized Bouc-Wen model parameters 4.7.1 The parameters and 4.7.2 The parameter 4.7.3 The parameter n 4.8 Conclusion 5 Robust identification of the Bouc-Wen model parameters 5.1 Introduction 5.2 Parameter identi ~ cation for the Bouc-Wen model 5.2.1 Class of inputs 5.2.2 Identi ~ cation methodology 5.2.3 Robustness of the identi ~ cation method 5.2.4 Numerical simulation example 5.3 Modeling and identi ~ cation of a magnetorheological damper 5.3.1 Some insights into the viscous + Bouc-Wen model for shear mode MR dampers 5.3.2 Alternatives to the viscous + Bouc-Wen model for shear mode MR dampers 5.4 Identi ~ cation methodology for the viscous + Dahl model . . 5.4.1 Numerical simulations 5.5 Conclusion 6 Control of a system with a Bouc-Wen hysteresis 6.1 Introduction and problem statement 6.2 Control design and stability analysis 6.3 Numerical simulation 6.4 Conclusion A Mathematical background A.1 Existence and uniqueness of solutions A.2 Concepts of stability A.3 Passivity and absolute stability A.3.1 Passivity in mechanical systems A.3.2 Positive realness A.3.3 Sector functions A.3.4 Absolute stability A.4 Input-output properties References. Index.

Systems with Hysteresis: Analysis, Identification and Control Using the Bouc-Wen Model

In this paper, a thermodynamic analysis of Bouc-Wen models endowed with both strength and stiffness degradation is provided. It is based on the relationship between the flow rules of these models and those of the endochronic plasticity theory with damage, discussed in a companion paper (Erlicher and Point, 2008). Using the theoretical framework of that extended endochronic theory, it is shown that an elastic Bouc-Wen model with damage, i.e. without plastic strains, can be formulated. Moreover, a proper definition of the dissipated energy of these Bouc-Wen models with degradation is given and some thermodynamic constraints on the parameters defining the models behavior are emphasized and discussed. In particular, some properties of the energetic linear stiffness degradation rule as well as the so-called pivot rule, well-known in the seismic engineering field, are illustrated and commented upon. An improved energetic stiffness degradation rule and a new stiffness degradation rule are proposed.

Bouc-Wen-type models with stiffness degradation: thermodynamic analysis and applications

For reliable and practical application of structural health monitoring approaches in conjunction with dense sensor arrays deployed on 'smart' systems, there is a need to develop and evaluate alternate strategies for efficient problem decomposition to rapidly and accurately determine the occurrence, location and level of small changes in the underlying structural characteristics of a monitored system based on its vibrational signature. Furthermore, there is also a need to quantify the level of uncertainties in the identified system characteristics so as to have a measurable level of confidence in the parameters to be relied on for the detection of genuine changes (damage) in the monitored system. This study presents the results of two time-domain identification techniques applied to a full-scale 17-story building, based on ambient vibration measurements. The Factor building is a steel frame structure located on the UCLA campus. This building was instrumented permanently with a dense array of 72-channel accelerometers, and the acceleration data are being continuously recorded. The first identification method used in this study is the NExT/ERA, which is regarded as a global (or centralized) approach, since it deals with the global dynamic properties of the structure. The second method is a time-domain identification technique for chain-like MDOF systems. Since in this method the identification of each link of the chain is performed independently, it is regarded as a local (or decentralized) identification methodology. For the same reason, this method can be easily adopted for large-scale sensor network architectures in which the centralized approaches are not feasible due to massive storage, power, bandwidth and computational requirements. To have a statistically meaningful results, 50 days of recorded data are considered in this study. The modal parameter and chain identification procedures are performed over time windows of 2 h each and with 50% overlap. Using the NExT/ERA method, 12 dominant modes of the building were identified. It was observed that variations in the frequency estimation are relatively small; the coefficient of variation is about 1–2% for most of the estimated modal frequencies. Chain system identification was successfully implemented using the output-only data acquired from the Factor building. Probability distributions of the estimated coefficients of displacement and velocity terms in the interstory restoring functions (which are the mass-normalized local stiffness and damping values) that were found based on the chain system identification are presented. The variability of the estimated parameters due to temperature fluctuations is investigated. It is shown that there is a strong correlation between the modal frequency variations and the temperature variations in a 24 h period.

http://iopscience.iop.org/article/10.1088/0964-1726/17/2/025006/pdf

A novel approach for the structural identification and monitoring of a full-scale 17-story building based on ambient vibration measurements

The presence of mechanical joints--typified by the lap joint--in otherwise linear structures has been accommodated in structural dynamics via ad hoc methods for a century. The methods range from tuning linear models to approximate non-linear behavior in restricted load ranges to various methods which introduce joint dissipation in a post-processing stage. Other methods, employing constitutive models for the joints are being developed and their routine use is on the horizon.

Modeling joint friction in structural dynamics.

Building on the basic idea behind the Restoring Force Method for the non-parametric identification of non-linear systems, a general procedure is presented for the direct identification of the state equation of complex non-linear systems. No information about the system mass is required, and only the applied excitation(s) and resulting acceleration are needed to implement the procedure. Arbitrary non-linear phenomena spanning the range from polynomial non-linearities to the noisy Duffing–van der Pol oscillator (involving product-type non-linearities and multiple excitations) or hysteretic behavior such as the Bouc–Wen model can be handled without difficulty. In the case of polynomial-type non-linearities, the approach yields virtually exact results for sufficiently rich excitations. For other types of non-linearities, the approach yields the optimum (in least-squares sense) representation in non-parametric form of the dominant interaction forces induced by the motion of the system. Several examples involving synthetic data corresponding to a variety of highly non-linear phenomena are presented to demonstrate the utility as well as the range of validity of the proposed approach.

J.P. Caffrey

Papers

Identification of the state equation in complex non-linear systems