Nonlinear modeling of hysteretic systems with double hysteretic loops using position and acceleration information

This paper presents two new dynamic hysteresis models obtained from the Bouc–Wen model by incorporating position and acceleration information. On the one hand, the model employing position information is rate-independent and it is able to reproduce some kind of double hysteretic loops unable to be reproduced with the original Bouc–Wen model. On the other hand, the model employing acceleration information is insensitive to linear time-scale variations. Moreover, a classification of the BIBO-stable models has been derived for both position and acceleration cases. Double hysteretic loops have been experimentally reported in shape-memory alloys, reinforced concrete structures, wood structures and lightweight steel shear wall structures. The proposed hysteretic models represent a prominent use in the field of structural dynamics and earthquake engineering because they can capture the nonlinear dynamics of the materials and structures presented earlier when they are subjected to dynamic loads as earthquake excitations.     

An hysteretic quadrilateral plane stress element

In this work, a new plane stress element is proposed for the nonlinear static and dynamic analysis of plane stress/plane strain problems. The four node quadrilateral element formulation for the elastic case is extended by introducing a novel hysteretic constitutive relation, based on the Bouc–Wen model of hysteresis. The hysteretic model introduced is directly derived from the governing equations of classical plasticity based on the flow rule and specific hardening law. The stiffness matrix of the element is formulated using the principle of virtual displacements, where the elastic stress–strain relation is substituted by the hysteretic relation proposed. The derived stiffness matrix is expressed as a smooth function of the internal stress field both in the elastic and inelastic regime. The efficiency of the proposed element in the simulation of the cyclic behavior in plane structures is presented through illustrative examples.     

Identification of nonlinear hysteretic parameters by enhanced response sensitivity approach

Hysteresis is a ubiquitous phenomenon describing the special nonlinear memory-based relation between the input and the output in many physical systems. Identifying the hysteretic parameters is the first step towards practical application of hysteretic models. In this paper, a general framework for parameter identification of nonlinear hysteretic models is developed based on the enhanced response sensitivity approach. To do so, three typical hysteretic models—Bouc–Wen model, bilinear model with kinematic hardening and bilinear model with equal yielding force are analyzed at first and the general way to model a structure with such hysteretic components is established thereafter. Then, the enhanced response sensitivity approach is presented for inverse parameter identification where the key lies in the sensitivity analysis and the trust-region constraint. Particularly, smoothing procedure is introduced to overcome the non-differentiability of bilinear hysteretic functions for sensitivity analysis of bilinear models. Numerical examples are studied to testify the feasibility and performance of the proposed approach.     

Novel identification approach for nonlinear systems with hysteresis

Nonlinear Dynamics - A new identification approach for a nonlinear system with hysteresis, namely a cascading Bouc–Wen hysteresis model with linear dynamics, is proposed in this study. The...     

Nonclassical Responses of Oscillators with Hysteresis

The responses and codimension-one bifurcations in Masing-type andBouc–Wen hysteretic oscillators are investigated. The pertinent statespace is formulated for each system and the periodic orbits are soughtas the fixed points of an appropriate Poincaré map. The implementedpath-following scheme is a pseudo-arclength algorithm based on arclengthparameterization. The eigenvalues of the Jacobian of the map, calculatedvia a finite-difference scheme, are used to ascertain the stability andbifurcations of the periodic steady-state solutions. Frequency-responsecurves for various excitation levels are constructed consideringrepresentative hysteresis loop shapes generated with the two models inthe primary and superharmonic frequency ranges. In addition to knownbehaviors, a rich class of solutions and bifurcations, mostly unexpectedfor hysteretic oscillators – including jump phenomena,symmetry-breaking, complete period-doubling cascades, fold, andsecondary Hopf – is found. Complex (mode-locked) periodic andnonperiodic responses are also investigated thereby allowing to draw amore comprehensive picture of the dynamical behavior exhibited by thesesystems.     

Backstepping-based Lyapunov redesign control of hysteretic single degree-of-freedom structural systems

This paper considers the problem of active control design for a hysteretic single-degree-of-freedom (SDOF) structural system which is exposed to an earthquake excitation. First, backstepping-based control is used to design a controller for the structural system neglecting the effect of the earthquake disturbance. Then, Lyapunov redesign is utilized to design a robust controller for the system in the presence of the earthquake excitation. The hysteretic part of the structural system is modeled by the well-known Bouc–Wen equation, and this equation is directly utilized in the controller design. The controller is proposed for two cases: (a) when the parameters of the structure and the Bouc–Wen model are known, and (b) when these parameters are uncertain. A Lyapunov function is introduced for the closed-loop system, which guarantees the stability of the system equilibrium point. Since the controllers use the nominal and/or minimum and maximum values of the system parameters, the proposed methods are model based. Numerical evaluations are conducted to show the effectiveness of the proposed method. Seven different earthquakes are considered as the external excitations. Simulation results show that the displacement, velocity, and acceleration responses of the controlled structure are reduced significantly compared to the uncontrolled structure.     

Active control for a flexible beam with nonlinear hysteresis and time delay

A new active control method is presented to attenuate vibrations of a flexible beam with nonlinear hysteresis and time delay. The nonlinear and hysteretic behavior of the system is illustrated by the Bouc—Wen model. By specific transformation and augmentation of state parameters, we can convert the motion equation of the system with explicit time delay to the standard state space representation without any explicit time delay. Then the instantaneous optimal control method and Runge—Kutta method in fourth-order are applied to the controller design with time delay. Finally, in order to verify the effectivity of the time-delay controller proposed, numerical simulations are implemented. It is indicated by the simulation results that the control performance will deteriorate if neglect the time delay in process of the controller design and proposed time delay controller works well with both small and large time delay problems.     

Dynamical behavior of a Bouc–Wen type oscillator coupled to a nonlinear energy sink

We try to enlighten the time multi-scale energy exchange between a main oscillator of Bouc–Wen type and a nonlinear energy sink. A general methodology is presented to detect the invariant manifold of the system at fast time scale and then to search equilibrium points and fold singularities at the slow time scale. A numerical example is enclosed to show the application of the coupled nonlinear energy sink in passive control of Bouc–Wen type main structures.     

Variation of the hysteresis loop with the Bouc–Wen model parameters

The Bouc–Wen model for smooth hysteresis has received an increasing interest in the last few years due to the ease of its numerical implementation and its ability to represent a wide range of hysteresis loop shapes. This model consists of a first-order nonlinear differential equation that contains some parameters that can be chosen, using identification procedures, to approximate the behavior of given physical hysteretic system. Despite a large body of literature dedicated to the Bouc–Wen model, the relationship between the parameters that appear in the differential equation and the shape of the obtained hysteresis loop is little understood. The objective of this paper is to fill this gap by analytically exploring this relationship using a new form of the model called the normalized one. The mathematical framework introduced in this study formalizes the vague notion of “loop shape" into precise quantities whose variation with the Bouc–Wen model parameters is analyzed. In light of this analysis, the parameters of Bouc–Wen model are re-interpreted.     

A hysteretic model of localized frictional contacts with instrumental stiffness

The steady-state dynamic response of a single-degree-of-freedom system comprising both a hysteretic element and a spring is considered. The Hertz–Cattaneo–Mindlin theoretical framework for modeling of local tangential contact with friction is applied in conjunction with the Masing model of hysteresis to describe the hysteretic behavior of the multiple localized frictional contact interface. The steady-state tangential displacement amplitude of a rigid body under harmonic tangential force excitation is approximately determined by means of the equivalent linearization technique, based on the harmonic balance principle. A special attention is paid to the evaluation of the frictional damping and the determination of the backbone curve of the Masing model from the dissipation-amplitude relation.

     

Thermodynamic admissibility of Bouc–Wen type hysteresis modelsAdmissibilité thermodynamique de modèles d'hystérésis de type Bouc–Wen

Starting from the relationship between the Bouc model and the endochronic theory and by adopting some new intrinsic time measures, the thermodynamic admissibility of the Bouc–Wen model is proved, in the univariate case as well as in the tensorial one. Moreover, the proposed proof encompasses the cases where a strength degradation term appears. To cite this article: S. Erlicher, N. Point, C. R. Mecanique 332 (2004).     

Trapping vibratory energy of main linear structures by coupling light systems with geometrical and material non-linearities

Cubic potential and hysteresis behavior (Bouc–Wen type) of a non-linear energy sink are used to localize the vibratory energy of a linear structure. A general methodology is presented to deal with time evolutionary energy exchanges between two oscillators. Invariant manifold of the system and its stability borders are detected at fast time scale while traced equilibrium and singular points at slow time scale let us predict possible behaviors of the system during its pseudo-stationary regime(s). The paper is followed by an example that considers the Dahl model for representing the hysteresis behavior of the non-linear energy sink. All analytical developments and results are compared with those obtained by direct integration of system equations. Obtained analytical developments can be endowed for designing non-linear energy sink devices with hysteresis behavior to localize vibratory energy of main structures for the aim of passive control, energy harvesting and/or both of them.     

Dual input–output pairs for modeling hysteresis inspired by mem-models

We call attention to a dual-pair concept for modeling hysteresis involving instantaneous switching: Specifically, there are two input–output pairs for each hysteresis model under one specific input, namely a differential pair and an integral pair. Currently in engineering mechanics, only one pair is being recognized and utilized, not the other. Whereas this dual-pair concept is inherent in the differential and algebraic forms of memristors and memcapacitors, the concept has not been carried over to memristive system theory, nor to memcapacitive system theory. We show that the “zero-crossing” feature in memristors, memcapacitors, and memristive/memcapacitive models (i.e., the “mem-models”) is also a feature of the differential pairs of well-known non-mem-models, examples of which are Ramberg–Osgood, Bouc–Wen, bilinear hysteresis, and classical Preisach. The dual-pair concept thus connects mem-models and non-mem-models, thereby facilitating the modeling of hysteresis, and raising a set of scientific questions for further studies that might not otherwise come to awareness.     

Elastic cables–rigid body coupled dynamics: asymptotic modeling and analysis

An elastic cables–rigid body coupled model is proposed for investigating dynamic interactions between cables’ nonlinear transversal vibrations and boundary tower’s torsional dynamics, arising in large transmission line–tower systems and suspended cable–bridge tower systems. By introducing a weak torsion assumption and a large moment of inertia for the tower, an asymptotic expansion of cables–tower coupled dynamics is conducted in a weakly nonlinear framework, and a cables–tower reduced coupled model is eventually established. After model’s validations using direct numerical simulations, two distinct kinds of coupled dynamics are fully investigated. The first is that an external torque is applied to the tower and the two cables would both be indirectly excited, asymmetrically, by the torsional/oscillating tower. The two cables’ responses are the same in this case. The second is that only one of the two cables, i.e., the leader cable, is directly excited, and the other cable, i.e., the follower one, is only indirectly excited through cables–tower dynamic interactions. In such kind of leader–follower dynamics, the leader cable is quite different from the follower one. Nonlinear coupled frequency response diagrams for both systems are constructed using numerical continuation algorithms, mainly focused on the coupled steady solutions’ stabilities and bifurcations. Furthermore, the dynamic effects of tower’s moment of inertia, wing span and damping are thoroughly investigated.

     

Design and dynamic performance research of MR hydro-pneumatic spring based on multi-physics coupling model

Aiming at the problem that the damping coefficient of the traditional hydro-pneumatic spring cannot be adjusted in real-time, the magnetorheological (MR) damping technology was introduced into the traditional hydro-pneumatic spring with single gas chamber. A new shear-valve mode MR hydro-pneumatic spring was proposed. And its dynamic performance was analyzed based on multi-physical coupling simulation and mechanical property test. Firstly, a structural scheme of MR hydro-pneumatic suspension was proposed to ensure the original height adjustment function based on the working principle of traditional hydro-pneumatic suspension with single gas chamber. Secondly, based on the design requirements, the parameter of MR hydro-pneumatic spring damping structure was designed by using MR damper design method. Thirdly, the multi-physical coupling dynamic performance of the MR hydro-pneumatic spring damping structure was analyzed based on the electromagnetic field analysis theory, flow field analysis theory and thermal field analysis theory. The analysis results showed that the designed MR hydro-pneumatic spring has reasonable magnetic circuit structure and excellent working performance. Then, the mechanical properties of MR hydro-pneumatic spring were tested. The results showed that the maximum damping force can reach 20 kN, and the dynamic adjustable multiple can reach 6.4 times. It has good controllability and meets the design requirements. Finally, a nonlinear model of MR hydro-pneumatic spring was established based on the elastic force calculation model of the gas and the Bouc–Wen model. The simulation results of the established model agree well with the experimental results, which can accurately describe the dynamic properties of the hydro-pneumatic spring. The proposed design and modeling method of the MR hydro-pneumatic spring can provide a theoretical basis for the related vibration damping devices.

     

Wave propagation in nonlinear and hysteretic media––a numerical study

This paper considers the problem of one dimensional wave propagation in nonlinear, hysteretic media. The constitutive law in the media is prescribed by an integral relationship based on the Duhem model of hysteresis. It is found that the well known nonlinear elastic stress–strain relationship is a special case of this integral relationship. It is also shown that the stress–strain relationship from the McCall and Guyer model of hyesteretic materials can also be derived from this integral stress–strain relationship. The first part of this paper focuses on a material with a quadratic stress–strain relationship, where the initial value problem is formulated into a system of conservation laws. Analytical solutions to the Riemann problem are obtained by solving the corresponding eigenvalue problem and serve as reference for the verification and illustration of the accuracy obtained using the applied numerical scheme proposed by Kurganov and Tadmor. The second part of this research is devoted to wave propagation in hysteretic media. Several types of initial excitations are presented in order to determine special characteristics of the wave propagation due to material nonlinearity and hysteresis. The results of this paper demonstrate the accuracy and the robustness of this numerical scheme to analyze wave propagation in nonlinear materials.     

Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

     

论迟滞与时滞

迟滞和时滞是自然科学、工程科学、乃至社会科学中常见的两种现象,但在我国学术界常被混淆. 从两种现象的定义和本质出发,阐述两者的共性、个性及其联系. 通过多个例子说明:迟滞现象反映两个相关参变过程周期变化时彼此间的相位滞后关系;而时滞现象则反映两个相关动态过程任意变化时彼此间的时间滞后关系.在某些特定的情况下, 它们可以等同; 但在一般情况下,它们是具有不同性质的两类现象, 尤其在描述记忆特性方面, 两者有本质的差异.      

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton's principle. A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method. A high-dimensional model of the buckled beam is derived, concerning nonlinear coupling. The incremental harmonic balance (IHB) method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve, and the Floquet theory is used to analyze the stability of the periodic solutions. Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited. Bifurcations including the saddle-node, Hopf, perioddoubling, and symmetry-breaking bifurcations are observed. Furthermore, quasi-periodic motion is observed by using the fourth-order Runge-Kutta method, which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.     

Fixed-interval smoothing of an aeroelastic airfoil model with cubic or free-play nonlinearity in incompressible flow

Fixed-interval smoothing, as one of the most important types of state estimation, has been concerned in many practical problems especially in the analysis of flight test data. However, the existing sequential filters and smoothers usually cannot deal with nonlinear or high-dimensional systems well. A state-of-the-art technique is employed in this study to explore the fixed-interval smoothing problem of a conceptual two-dimensional airfoil model in incompressible flow from noisy measurement data. Therein, the governing equations of the airfoil model are assumed to be known or only partially known. A single objective optimization problem is constructed with the classical Runge–Kutta scheme, and then estimations of the system states, the measurement noise and even the unknown parameters are obtained simultaneously through minimizing the objective function. Effectiveness and feasibility of the method are examined under several simulated measurement data corrupted by different measurement noises. All the obtained results indicate that the introduced algorithm is applicable for the airfoil model with cubic or free-play structural nonlinearity and leads to accurate state and parameter estimations. Besides, it is highly robust to Gaussian white and even more complex heavy-tailed measurement noises. It should be emphasized that the employed algorithm is still effective to high-dimensional nonlinear aeroelastic systems.