Hysteresis Operator Research Articles

Finite Element Formulation for Ferroelectric Hysteresis of Piezoelectric Materials

For the numerical simulation of non-linear piezoelectric material behavior, we use a constitutive relation that is based on a decomposition of the physical quantities dielectric displacement and mechanical strain into a reversible and an irreversible part. Therein, we set the irreversible part of the dielectric displacement equal to the irreversible electric polarization and express the irreversible mechanical strain by a polynomial ansatz of the irreversible electric polarization. The reversible parts of mechanical strain and dielectric displacement are further described by the linear piezoelectric constitutive law. We apply a Preisach hysteresis operator to compute the irreversible polarization from the history of the driving electric field. Furthermore, the entries of the piezoelectric modulus tensor are assumed to be functions of the electric polarization. To efficiently solve the non-linear system of partial differential equations, we have developed a quasi-Newton scheme and use the finite element (FE) method for the numerical solution. This FE scheme has been applied to numerically calculate the dynamic behavior of a piezoelectric disc and a stack actuator. The obtained results compare well to measured data.

Well-posedness of a thermo-mechanical model for shape memory alloys under tension

We present a model of the full thermo-mechanical evolution of a shape memory body undergoing a uniaxial tensile stress. The well-posedness of the related quasi-static thermo-inelastic problem is addressed by means of hysteresis operators techniques. As a by-product, details on a time-discretization of the problem are provided.

Uniqueness and decay estimates for a class of parabolic partial differential equations with hysteresis and convection

Abstract We deal in detail with the question of existence, uniqueness and asymptotic behaviour of solutions to a parabolic equation with hysteresis and convection. This equation is part of a model system which describes the magnetohydrodynamic (MHD) flow of a conducting fluid between two ferromagnetic plates. The result of this paper complements the content of a previous paper of the first author, where existence of the solution has been proved under fairly general assumptions on the hysteresis operator and the uniqueness was only obtained for a restricted class of hysteresis operators.

Tracking with Prescribed Transient Performance for Hysteretic Systems

Tracking of reference signals (assumed bounded with essentially bounded derivative) is considered for a class of single-input, single-output, nonlinear systems, described by a functional differential equation with a hysteresis nonlinearity in the input channel. The first control objective is tracking, by the output, with prescribed accuracy: determine a feedback strategy which ensures that, for every reference signal and every system of the underlying class, the tracking error ultimately satisfies the prescribed accuracy requirements. The second objective is guaranteed output transient performance: the graph of the tracking error should be contained in a prescribed set (performance funnel). Under a weak sector boundedness assumption on the hysteresis operator, both objectives are achieved by a memoryless feedback which is universal for the underlying class of systems.

Elastoplastic reaction of a container to water freezing

The paper deals with a model for water freezing in a deformable elastoplastic container. The mathematical problem consists of a system of one parabolic equation for temperature, one integrodifferential equation with a hysteresis operator for local volume increment, and one differential inclusion for the water content. The problem is shown to admit a unique global uniformly bounded weak solution.

On a class of scalar variational inequalities with measure data

We consider a class of variational inequalities defining the so-called hysteresis play operator. We propose a new approach to discontinuous BV-solutions based on measure theoretical arguments, which enable us to infer the existence of solutions as a simple consequence of the classical theory. In this way, we generalize a recent result where only continuous BV-solutions were studied. We also provide a representation formula which allows to deduce the continuity of the play operator from general theorems on hysteresis operators.

Parabolic problems with the Preisach hysteresis operator in boundary conditions

Parabolic initial boundary-value problems coupled (via the boundary condition) with ordinary differential equations whose right-hand side contains the Preisach hysteresis operator are considered. In particular, these problems model thermocontrol processes in chemical reactors, climate-control systems, biological cells, etc. For the Preisach operator with and without time delay, solvability, periodicity of solutions, and global B-attractors are studied.

Physical modeling and numerical computation of magnetostriction

PurposeMagnetostrictive alloys are widely used in actuator and sensor applications. The purpose of this paper is to developed a realistic physical model and a numerical computational scheme for their precise computation.Design/methodology/approachThe main step in the physical modeling is the decomposition of the mechanical strain and the magnetic induction into a reversible and an irreversible part. For the efficient solution of the arising coupled nonlinear partial differential equations the authors apply the finite element method.FindingsIt can be demonstrated, that the hysteresis operators can be fitted by appropriate measurements. Therewith, the developed physical model and numerical simulation scheme is applicable for the design of magnetostrictive actuators and sensors.Originality/valueThe decomposition of the mechanical strain and the magnetic induction into a reversible and an irreversible part. The reversible part is described by the linear magnetostrictive constitutive equations, where the entries of the coupling tensor depend on the magnetization. The irreversible part of the magnetic induction is modeled by a Preisach hysteresis operator, and the irreversible part of the mechanical strain by a polynomial function depending on the magnetization.

A strong hysteretic model of Okun’s Law: theory and a preliminary investigation

This paper presents a ‘strong hysteretic’ version of Okun’s Law, that is, a version of the law in which ‘history matters.’ In this version of the link between fluctuations in unemployment and growth, the most important past growth shock exerts an influence on the current unemployment rate. A theoretical framework is proposed in order to lay the foundations of this version of Okun’s Law. In this framework, the hysteresis property arises because a large number of heterogeneous firms discontinuously adjust their activity levels in response to fluctuations in the rate of growth. The foundations having been laid, a method for empirically testing our hysteretic Okun’s Law is presented. An algorithm permits construction of a hysteresis operator, which synthesizes, for every moment, the growth shocks that have remained in the memory bank of the unemployment rate. Empirical tests are conducted to assess the empirical relevance of this version of Okun’s Law, as compared to the more familiar linear relationship. Empirical results consistent with hysteresis are found for several of the countries in our sample.

Experimental characterization and modeling of rate-dependent hysteresis of a piezoceramic actuator

Laboratory experiments were performed to characterize the rate-dependent hysteresis properties of a piezoceramic actuator under harmonic, complex harmonic and triangular excitations in the 0.1–500 Hz frequency range. The measured data were analyzed to describe the major and minor hysteresis loops as functions of frequency, magnitude and bias of the input voltage. The results revealed considerably larger hysteresis loop width and lower displacement response amplitude under frequencies above 10 Hz. A rate-dependent Prandtl–Ishlinskii model is developed for describing the rate-dependent hysteresis behaviour of the actuator. This model integrates rate-dependent play operator and density functions formulated on the basis of the rate of change of input and experimentally observed behaviors. The fundamental properties of the proposed rate-dependent play and stop hysteresis operators are also investigated. The model results attained under harmonic, complex harmonic and triangular inputs at different frequencies in the 0.1–500 Hz were compared with the corresponding experimental data to demonstrate model validity over the wide range of inputs. Very good agreements were observed between the model results and the measured data, irrespective of the type and frequency of excitation.

Homogenization of the Prager model in one-dimensional plasticity

We propose a new method for the homogenization of hysteresis models of plasticity. For the one-dimensional wave equation with an elasto-plastic stress-strain relation we derive averaged equations and perform the homogenization limit for stochastic material parameters. This generalizes results of the seminal paper by Franců and Krejci. Our approach rests on energy methods for partial differential equations and provides short proofs without recurrence to hysteresis operator theory.

Sobolev and strict continuity of general hysteresis operators

AbstractThe most natural and important topologies connected with hysteresis operators are those induced by uniform convergence, W1, 1‐convergence, and strict convergence. Indeed the supremum norm and the variation are invariant under reparameterization. We prove a general result that implies that if a hysteresis operator is continuous with respect to the topology of W1, 1, then it is continuous with respect to the strict topology. Copyright © 2009 John Wiley & Sons, Ltd.

Input-to-State Stability of Differential Inclusions with Applications to Hysteretic and Quantized Feedback Systems

Input-to-state stability (ISS) of a class of differential inclusions is proved. Every system in the class is of Lur'e type: a feedback interconnection of a linear system and a set-valued nonlinearity. Applications of the ISS results, in the context of feedback interconnections with a hysteresis operator or a quantization operator in the feedback path, are developed.

Magnetohydrodynamic Flow with Hysteresis

We consider a model system describing the two-dimensional flow of a conducting fluid surrounded by a ferromagnetic solid under the influence of the hysteretic response of the surrounding medium. We assume that this influence can be represented by the Preisach hysteresis operator. Existence and uniqueness of solutions for the resulting system of PDEs with hysteresis nonlinearities is established in the convexity domain of the Preisach operator.

Tracking control of hysteretic piezoelectric actuator using adaptive rate-dependent controller

With the increasing popularity of actuators involving smart materials like piezoelectric, control of such materials becomes important. The existence of the inherent hysteretic behavior hinders the tracking accuracy of the actuators. To make matters worse, the hysteretic behavior changes with rate. One of the suggested ways is to have a feedforward controller to linearize the relationship between the input and output. Thus, the hysteretic behavior of the actuator must first be modeled by sensing the relationship between the input voltage and output displacement. Unfortunately, the hysteretic behavior is dependent on individual actuator and also environmental conditions like temperature. It is troublesome and costly to model the hysteresis regularly. In addition, the hysteretic behavior of the actuators also changes with age. Most literature model the actuator using a cascade of rate-independent hysteresis operators and a dynamical system. However, the inertial dynamics of the structure is not the only contributing factor. A complete model will be complex. Thus, based on the studies done on the phenomenological hysteretic behavior with rate, this paper proposes an adaptive rate-dependent feedforward controller with Prandtl–Ishlinskii (PI) hysteresis operators for piezoelectric actuators. This adaptive controller is achieved by adapting the coefficients to manipulate the weights of the play operators. Actual experiments are conducted to demonstrate the effectiveness of the adaptive controller. The main contribution of this paper is its ability to perform tracking control of non-periodic motion and is illustrated with the tracking control ability of a couple of different non-periodic waveforms which were created by passing random numbers through a low pass filter with a cutoff frequency of 20 Hz.

BV ‐extension of rate independent operators

AbstractRate independent operators naturally arise in the mathematical analysis of hysteresis. Among rate independent operators, the locally monotone ones are those better suited for the study of PDE's with hysteresis. We prove that a rate independent operator R: Lip (0, T) → BV (0, T) ∩ C (0, T) which is locally monotone and continuous with respect to the strict topology of BV admits a unique continuous extension R̃: BV (0, T) → BV (0, T). This general result applies to several concrete hysteresis operators. For many of these operators the existence of a continuous extension was previously known at most to the space BV (0, T) ∩ C (0, T). (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Wellposedness Results for a Class of Parabolic Partial Differential Equations with Hysteresis

In this paper we deal with the mathematical investigation of a class of parabolic partial differential equations containing a continuous hysteresis operator arising in the context of magnetohydrodynamics. The main result we achieve is the existence of a weak solution by means of a technique based on an implicit time discretization scheme. Uniqueness is obtained under some suitable monotonicity assumptions on the hysteresis operator; finally we also analyse the dependence of the solution on the data.

Self-sensing force control of a piezoelectric actuator

This paper describes an approach to controlling the force generated by a piezoelectric actuator (PEA) accurately without using any force sensor. A model PEA is proposed that includes a new asymmetric hysteresis operator and that takes the external force into account. A detection model is deduced that allows computation in real time of the PEA elongation and generated force starting from the measurement of the driving voltage and current. This detection model is used to replace a force sensor for closed-loop force control of a PEA. Experiments are carried out using 2 PEAs in an experimental setup. The first actuator is the controlled actuator, and the second one is used as a dynamic controllable mechanical load. It is shown that a good control performance can be obtained whatever the mechanical loading conditions.

Outward pointing properties for vectorial hysteresis operators and some applications

In a number of papers, the concept of outward pointing hysteresis allowed the application of the method of invariant regions to derive uniform bounds for solutions to evolution equations involving scalar hysteresis operators and input perturbations.The aim of this note is to generalize the ‘pointing outward’ properties to vectorial hysteresis operators, and to show how these properties can be used to derive uniform estimates for differential equations involving vectorial hysteresis operators and input perturbations.

On a parabolic equation with hysteresis and convection: A uniqueness result

A uniqueness result for a parabolic partial differential equation with hysteresis and convection is established. This equation is a part of a model system which describes the magnetohydrodynamic (MHD) flow of a conducting fluid between two ferromagnetic plates. The result of this paper complements the content of [6], where existence of the solution has been proved under fairly general assumptions on the hysteresis operator and the uniqueness was obtained only for a restricted class of hysteresis operators