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

Changes in the performance of bearings can significantly vary the distribution of internal forces and moments in a structure as a result of environmental or operational loads. The response of a bearing has been traditionally idealized using a linear model but a non-linear representation produces a more accurate picture at the expense of modelling complexity and computational time. In this article, a lead rubber bearing is idealized using the hysteretic Bouc–Wen model. The Hilbert–Huang transform is then employed to characterize the features of the non-linear system from the instantaneous frequencies of the bearing response to a time-varying force. Instantaneous frequencies are also shown to be a useful tool in detecting sudden damage to the bearings simulated by a reduction in the effective stiffness of the force-deformation loop.

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Characterization of non-linear bearings using the Hilbert-Huang transform

Equivalent linearization and Monte Carlo simulation in stochastic dynamics

Many civil and mechanical structures exhibit hysteresis with degradation and/or pinching when subject to severe cyclic loadings such as earthquakes, wind, or sea waves. The modeling and identification of non-linear hysteretic systems with degradation and pinching is therefore a practical problem encountered in the engineering mechanics field. On-line identification of degrading and pinching hysteretic systems is quite a challenging problem because of its complexity. A recently developed technique, the unscented Kalman filter (UKF) which is capable of handling any functional non-linearity, is applied to the on-line parametric system identification of hysteretic differential models with degradation and pinching. Simulation results show that the UKF is efficient and effective for the real-time state estimation and parameter identification of highly non-linear hysteretic systems with degradation and pinching.

Real-time parameter estimation for degrading and pinching hysteretic models

This study focuses upon SDOF systems having non-linear, hysteretic stress-strain behavior which is asymmetric about the initial linear elastic relationship Differences between tensile and compressive strength witnessed in various materials, particularly in the unique stress-strain properties of the shape-memory alloy Nitinol, motivate the need for mathematical models capable of simulating asymmetric response Two models are presented, and the specific capabilities of each are explored Both models follow the rate-type format of the Bouc-Wen model, and offer previously unavailable stress-strain relationships Response statistics under Gaussian White Noise input are obtained using the method of equivalent linearization For asymmetric systems, a need for non-zero-mean analysis is evident, even under zero-mean input

Modeling and random vibration analysis of SDOF systems with asymmetric hysteresis

The first and second moments of response variables for SDOF systems with hysteretic nonlinearity are obtained by a direct linearization procedure. This adaptation in the implementation of well-known statistical linearization methods, provides concise, model-independent linearization coefficients that are well-suited for numerical solution. The method may be applied to systems which incorporate any hysteresis model governed by a differential constitutive equation, and may be used for zero or non-zero mean random vibration. The implementation eliminates the effort of analytically deriving specific linearization coefficients for new hysteresis models. In doing so, the procedure of stochastic analysis is made independent from the task of physical modeling of hysteretic systems. In this study, systems with three different hysteresis models are analyzed under various zero and non-zero mean Gaussian White noise inputs. Results are shown to be in agreement with previous linearization studies and Monte Carlo Simulation.

S. Dobson

Papers

Modeling and random vibration analysis of SDOF systems with asymmetric hysteresis

Direct implementation of stochastic linearization for SDOF systems with general hysteresis