Rate-independent Hysteresis Research Articles

Energy-based modeling of rate-independent hysteresis and viscoelastic effects in dielectric elastomer actuators

Dielectric elastomer (DE) transducers are known to exhibit a rate-dependent hysteresis in their force-displacement response, which is commonly attributed to the viscoelastic behavior of elastomer materials and compliant electrodes. In the case of DE materials characterized by low mechanical losses, such as silicone, the mechanical hysteresis often turns out to be practically rate-independent in the low frequency range (sub-Hz), whereas rate-dependent hysteretic effects only become relevant at higher deformation rates. Most of the existing literature focuses on describing DE hysteretic losses using viscoelasticity theory. This approach results in relatively simple dynamic models, which are not capable of describing rate-independent hysteretic behaviors. In this work, we propose a control-oriented modeling framework for both rate-dependent and rate-dependent hysteresis occurring in uniaxially loaded DE actuators. To this end, classic thermodinamically-consistent modeling approaches for DEs are combined with a new energy-based Maxwell-Lion formalization of the hysteretic losses. The resulting dynamic model comprises a set of nonlinear ordinary differential equations, and is capable of simultaneously describe geometric dependencies, large deformation nonlinearities, electro-mechanical coupling, and rate-independent and rate-dependent hysteretic effects. To deal with the large number of involved parameters, a novel systematic identification algorithm based on quadratic programming is also proposed. After presenting the theory, the model is validated based on experiments conducted on a silicone-based rolled DE actuator. Its superiority compared to classic DE viscoelastic models is quantitatively assessed.

Rate-independent hysteretic energy dissipation in collagen fibrils.

Nanoindentation cycles measured with an atomic force microscope on hydrated collagen fibrils exhibit a rate-independent hysteresis with return point memory. This previously unknown energy dissipation mechanism describes in unified form elastoplastic indentation, capillary adhesion, and surface leveling at indentation velocities smaller than 1 μm s-1, where viscous friction is negligible. A generic hysteresis model, based on force-distance data measured during one large approach-retract cycle, predicts the force (output) and the dissipated energy for arbitrary indentation trajectories (input). While both quantities are rate independent, they do depend nonlinearly on indentation history and on indentation amplitude.

Major and minor hysteresis loops in the enthalpy-temperature and phase fraction-temperature diagrams of solid/liquid phase change materials

A rate-independent hysteresis model originally proposed by Ivshin and Pence (1994) is extended to describe thermal hysteresis in enthalpy, specific heat capacity and phase fraction of solid/liquid phase transitions. It turns out that, even for smooth cyclic temperature variations, major and minor loops in the enthalpy-temperature diagram are not necessarily closed. Jump discontinuities may occur upon switches between heating and cooling. This issue is highly problematic: the model does not obey the law of conservation of energy. This problem particularly concerns the frequently found setting that solid and liquid heat capacities do not coincide and the phase transition enthalpy is assumed constant, i.e. independent of temperature. Two possible solutions are proposed. They follow the idea of Kirchhoff’s Law and relate temperature variations of the phase transition enthalpy to the difference in specific heat capacities of the distinct phases. Conditions for path reversibility and for the existence of closed major and minor loops (limit cycles) are given. The models show good agreement with experimental data generated by differential scanning calorimetry (DSC) of three types of paraffin-based solid/liquid phase change materials (PCM). These materials are increasingly adopted for storing thermal energy in building applications and show thermal hysteresis for complete and partial melting/solidification cycles.

Analytical and differential reformulations of the Vaiana–Rosati model for complex rate-independent mechanical hysteresis phenomena

A recent analytical model of hysteresis, developed by the authors to simulate a great variety of complex uniaxial rate-independent mechanical hysteresis phenomena, is reformulated in a twofold manner. First, it is proposed an analytical reformulation that, compared to the current version of the model, does not require the evaluation of any internal variable. Second, the closed form expressions provided by the analytical reformulation are expressed in rate form to foster its use especially in the field of nonlinear dynamics. To verify their accuracy, both formulations are first employed to reproduce four different complex hysteresis phenomena and the related results are compared in terms of generalized force–displacement hysteresis loops, tangent stiffness functions, and work-displacement relations. Subsequently, nonlinear time history analyses are carried out on four single degree of freedom mechanical systems for three different types of external generalized forces. To this end, two solution strategies are adopted. The former combines the model analytical reformulation with an explicit structure-dependent time integration method used to directly integrate the second-order ordinary differential equation of motion. The latter employs the model differential reformulation in conjunction with the Runge–Kutta method implemented to solve the equivalent system of three first-order ordinary differential equations. The comparison of the numerical results obtained by the two solution procedures shows an excellent agreement.

Underwater dynamic hysteresis modeling and feedforward control of flexible caudal fin actuated by macro fiber composites

Underwater bionic robotics and vehicles that use flexible structures as power sources have attracted considerable attention. Thus, the underwater dynamic behaviors of the flexible structure of a variable cross-section actuated by smart actuators must be further investigated. In this study, underwater dynamic hysteresis modeling and feedforward compensation of a flexible caudal fin driven by macro fiber composites (MFC) at different frequencies were investigated. The MFC-actuated flexible caudal fin is considered a Euler–Bernoulli beam. A revised hydrodynamic function governed by the thickness of the viscous layer and the gap-to-width ratio of the cross-section was developed. Then, a characteristic-frequency-dependent extended transform function model is proposed for the hydrodynamic response of a non-uniform beam. An asymmetric hysteresis model is presented to characterize the rate-independent hysteresis behavior of the MFC actuators. Further, the underwater dynamic equation of the MFC-actuated caudal fin with hysteresis is obtained by connecting the two established models in series. Experiments demonstrate the dynamic behavior between the underwater responses of the caudal fin and the input voltages have an ellipse-like shape and vary in shape and size with oscillating frequency. The ellipse-like dynamic hysteresis loop first grows, reaching its maximum at the resonant frequency, and then decreases, because of the frequency-dependent dynamic behavior. Meanwhile, a phase shift of approximately 90°between the input voltage and underwater dynamic response was observed at the resonant state. Furthermore, the experimental results demonstrate that the proposed model can describe the nonlinear underwater dynamic hysteresis behavior of the proposed caudal fin at different frequencies. The trajectory tracking performances are remarkably improved over a broad frequency range with the proposed compensator. Consequently, the effectiveness of the proposed dynamic model and compensation method was demonstrated. These results are helpful for robotic fish or autonomous underwater vehicles propelled by oscillating foils actuated by smart actuators, including MFC.

Diagonal Recurrent Neural Network-Based Hysteresis Modeling.

The Preisach model and the neural networks are two of the most popular strategies to model hysteresis. In this article, we first mathematically prove that the rate-independent Preisach model is actually a diagonal recurrent neural network (dRNN) with the binary step activation function. For the first time, the hysteresis nature and conditions of the classical dRNN with the tanh activation function are mathematically discovered and investigated, instead of using the common black-box approach and its variants. It is shown that the dRNN neuron is a versatile rate-dependent hysteresis system under specific conditions. The dRNN composed of those neurons can be used for modeling the rate-dependent hysteresis and it can approximate the Preisach model with arbitrary precision with specific parameters for rate-independent hysteresis modeling. Experiments show that the classical dRNN models both kinds of hysteresis more accurately and efficiently than the Preisach model.

The Development of a Built-In Shoe Plantar Pressure Measurement System for Children.

There is a rapid increase in plantar pressure from the infant to toddler stage, yet little is known about the reasons for this change. More information about plantar pressure distribution can help clinicians identify early-stage foot-related diseases that may occur during transitions from childhood to adulthood. This information also helps designers create shoes that adapt to different needs. This research describes the development of a low-cost, built-in shoe plantar pressure measurement system that determines foot pressure distribution in toddlers. The study aimed to improve and provide data on pressure distribution during foot growth. This was accomplished by implementing a plantar pressure capacitive measurement system within shoes. The capacitive sensors were laminated using a copper tape sheet on plastic backing with adhesive, elastomer layers, and a combination of conductive and non-conductive fabrics. Constructed sensors were characterized using compression tests with repeated loads. Results demonstrated that the sensors exhibited rate-independent hysteresis in the estimation of pressure. This enabled a calibration model to be developed. The system can mimic more expensive plantar pressure measurement systems at lower fidelity. This emerging technology could be utilized to aid clinicians, researchers, and footwear designers interested in how pressure distribution changes from infants to toddlers.

Reformulation of a phenomenological model for symmetric rate-independent hysteresis

The present paper enhances, both theoretically and numerically, a recent uniaxial phenomenological model aiming to reproduce complex hysteretic behaviors of seismic isolation devices, including steel- and fiber-reinforced elastomeric bearings, presenting smooth hysteresis loops with hardening and/or softening regions. The investigated constitutive model is based on five parameters that can be directly related to the experimentally observed hysteresis loops. Moreover, because of its analytical nature, the presented formulation is capable of computing in closed form the generalized response relevant to any displacement increment starting from a known equilibrium state. Hence, differently from the original differential formulation of the investigated model as well as from popular materials such as the celebrated Bouc–Wen, the presented formulation does not require any iterative strategy for computing the hysteretic force. Finally, the investigated model is validated by numerical simulations compared with experimental tests selected from the literature.

Application of Least-Squares Support-Vector Machine Based on Hysteresis Operators and Particle Swarm Optimization for Modeling and Control of Hysteresis in Piezoelectric Actuators

Nanopositioning systems driven by piezoelectric actuators are widely used in different fields. However, the hysteresis phenomenon is a major factor in reducing the positioning accuracy of piezoelectric actuators. This effect makes the task of accurate modeling and position control of piezoelectric actuators challenging. In this paper, the learning and generalization capabilities of the model are efficiently enhanced to describe and compensate for the rate-independent and rate-dependent hysteresis using a kernel-based learning method. The proposed model is inspired by the classical Preisach hysteresis model, in which a set of hysteresis operators is used to address the problem of mapping, and then least-squares support-vector machines (LSSVM) combined with a particle swarm optimization (PSO) algorithm are used for identification. Two control schemes are proposed for hysteresis compensation, and their performance is evaluated through real-time experiments on a nanopositioning platform. First, an inverse model-based feedforward controller is designed based on the LSSVM model, and then a combined feedback/feedforward control scheme is designed using a classical control strategy (PID) to further enhance the tracking performance. For performance evaluation, different datasets with a variety of hysteresis loops are used during the simulation and experimental procedures. The results show that the proposed method is successful in enhancing the generalization capabilities of LSSVM training and achieving the best tracking performance based on the combination of feedforward control and PID feedback control. The proposed control scheme outperformed the inverse Preisach model-based control scheme in terms of both positioning accuracy and execution time. The control scheme that uses the LSSVM based on nonlinear autoregressive exogenous (NARX) models has significantly less computational complexity compared to our control scheme but at the expense of accuracy.

Viscosity solutions for doubly nonlinear evolution equations

We extend the theory of viscosity solutions to treat scalar-valued doubly nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontinuous viscosity solutions in this setting, we show that Perron's construction is still available, i.e. we prove an existence result. Moreover, we will prove comparison principles and stability results for these problems. The theoretical considerations are accompanied by several examples, e.g. we prove the existence of a solution to a rate-independent level-set mean curvature flow. Finally, we discuss in detail a rate-independent ordinary differential equation stemming from a problem with non-convex energy. We show that the solution obtained by maximal minimizing movements and the solution obtained by the vanishing viscosity method coincide with the upper and lower Perron solutions and show the emergence of a rate-independent hysteresis loop.

Tuned mass systems with the Bouc-Wen and a hybrid hysteresis

Tuned-mass-dampers are extensively used in physical systems in order to reduce the dynamic response of the primary structures. The damping in a tuned-mass-damper system plays a significant role. In many engineering systems, we see damping is hysteretic in nature. Tuned-mass systems with nonlinear hysteretic damping are mathematically challenging. In this paper, tuned mass systems with rate-independent hysteresis are studied. In particular, first we study tuned mass systems with the famous Bouc-Wen model, and then we study tuned mass systems with the Bouc-Wen model and a recently developed scalar hysteresis model in parallel. Numerical approaches are used to study the dynamic responses of the systems. The sine sweep responses show two resonance peaks. The sine sweep responses indicate the amplitude versus frequency of the systems. The numerical results obtained for two different hysteretic dampers are compared.

Parametric description of power loss in soft magnetic materials with unbiased, rate-independent hysteresis loops

This paper develops a parametric model describing power loss per unit volume, PV, on unbiased minor loops, in the low frequency limit, using the corresponding remanence Br, coercivity Hc, saturated field value BS and power loss per unit volume PVS, from the major loop. In the saturating regime where Hmax>Hc, PV scales as BSHc, and depends on BrHmax/(BSHc), with Hmax the maximum value H on the minor loop. In the intermediate regime where Hmax<Hc, PV scales as μ0μiHc2, and depends on Hmax/Hc. For composite magnetic materials, a major loop may be inaccessible, in which case an expression is obtained for the low field regime, where PV scales as Hmax2, and depends on the initial relative permeability μi. The models are in good agreement with published experimental data. Observed deviations are explained through experimental uncertainties and limitations of the model.

Hysteresis in the Thermo-Responsive Assembly of Hexa(ethylene glycol) Derivative-Modified Gold Nanodiscs as an Effect of Shape.

Anisotropic gold nanodiscs (AuNDs) possess unique properties, such as large flat surfaces and dipolar plasmon modes, which are ideal constituents for the fabrication of plasmonic assemblies for novel and emergent functions. In this report, we present the thermo-responsive assembly and thermo-dynamic behavior of AuNDs functionalized with methyl-hexa(ethylene glycol) undecane-thiol as a thermo-responsive ligand. Upon heating, the temperature stimulus caused a blue shift of the plasmon peak to form a face-to-face assembly of AuNDs due to the strong hydrophobic and van der Waals interactions between their large flat surfaces. Importantly, AuNDs allowed for the incorporation of the carboxylic acid-terminated ligand while maintaining their thermo-responsive assembly ability. With regard to their reversible assembly/disassembly behavior in the thermal cycling process, significant rate-independent hysteresis, which is related to their thermo-dynamics, was observed and was shown to be dependent on the carboxylic acid content of the surface ligands. As AuNDs have not only unique plasmonic properties but also high potential for attachment due to the fact of their flat surfaces, this study paves the way for the exploitation of AuNDs in the development of novel functional materials with a wide range of applications.

A Fractional-Order Normalized Bouc–Wen Model for Piezoelectric Hysteresis Nonlinearity

This article presents a new fractional-order normalized Bouc–Wen (BW) (FONBW) model to describe the asymmetric and rate-dependent hysteresis nonlinearity of piezoelectric actuators (PEAs). In view of the fact that the classical BW (CBW) model is only efficient for the symmetric and rate-independent hysteresis description, the FONBW model is devoted to characterizing the asymmetric and rate-dependent behaviors of the hysteresis in PEAs by adopting an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> th-order polynomial input function and two fractional operators, respectively. Different from the traditional modified BW models, the proposed FONBW model also eliminates the redundancy of parameters in the CBW model via the normalization processing. By this way, the developed FONBW model has a relatively simple mathematic expression with fewer parameters to simultaneously characterize the asymmetric and rate-dependent hysteresis behaviors of PEAs. Model parameters are identified by the self-adaptive differential evolution algorithm. To validate the effectiveness of the proposed model, a series of model verification and inverse-multiplicative-structure-based feedforward control experiments are carried out on a PEA system. Results show that the proposed model is superior to the CBW model and traditional modified BW model in modeling accuracy and hysteresis compensation.

A unified non-linear system model view of hyperelasticity, viscoelasticity and hysteresis exhibited by rubber

Rubber modeling is an old subject and so many models exist that it is difficult to have a clear vision of what exists and is more appropriate. Rather than attempting a standard review, this paper proposes classification using the traditional system modeling strategy, where raw measurements are either processed to obtain non-parametric models, or used to identify parametric models, whose accuracy can be controlled by order selection or by numerical implementation considerations. A full test campaign, including multi-step relaxation, low speed triangular and sine tests, on a large deformation compression sample is used to illustrate the need to model and combine the base behaviors known as hyperelasticity, viscoelasticity, and rate independent hysteresis. The equivalence between linear viscoelasticity and linear time invariant systems is used to clarify the link between order selection and accuracy of a generalized Maxwell model. Rate independent hysteresis is analyzed using a convolution product like the one used for viscoelastic transients by introducing a relaxation modulus. Measurements of the hysteretic relaxation modulus are used to propose strategies to measure the asymptotic hyperelastic modulus and discriminate between different hysteretic model forms. A parallel between Iwan and Maxwell models is detailed, and non-parametric models are used to show that the two overlap in the low frequency small deformation regime. Regularized rate independent hysteresis and non-linear viscoelasticity are finally shown to lead to a similar view allowing a transition between the rate independent and linear relaxation models. The instantaneous ratio analytic force and displacement signals, or instant complex modulus, is introduced as novel non-parametric estimation of sine measurements and shown to be a powerful tool to analyze and validate the fact that a force rate relaxation with non-linear relaxation frequencies is most appropriate to represent the non-linear coupling of all three effects.

Experimental characterization and modular modeling of hystereses for smart material actuators

In this article, we present a novel modular modeling approach to describe the hystereses for piezoelectric, magnetostrictive and shape memory alloy (SMA) actuators. For the above actuators, the output vs. input loops exhibit varying hystereses under the input signals of the different frequencies and amplitudes. To this end, the experimental characterization is conducted and hysteresis modeling approach is studied. Two characteristic indexes, i.e. loop relative width, loop asymmetry coefficients, are quantitatively analyzed according to the open-loop experiments for the three actuators. Based on the hysteresis phenomenon analyses, different submodels are selected to describe those phenomena. The Prandtl-Ishlinskii submodel is applied for symmetry rate-independent hysteresis identification; the arctangent-polynomial modified Prandtl-Ishlinskii submodel is proposed for asymmetry rate-independent hysteresis identification; infinite impulse response submodel is used for rate-dependent hysteresis identification. Those submodels are selected to construct a cascaded overall model to describe the hysteresis of piezoelectric, magnetostrictive, and SMA actuators. The hysteresis experimental identification results show that, with the proposed phenomenon-based hysteresis modular modeling approach, better performance can be obtained in terms of modeling accuracy and computation time than some other approaches.

Identification of Hammerstein–Wiener models with hysteresis front nonlinearities

This paper deals with the identification of Hammerstein–Wiener models. The novelty lies in the fact that the front nonlinear block is allowed to be the memory of hysteresis type. The latter is any rate-independent hysteresis operator, that is characterised by closed limit cycles when excited by periodic input signals called ‘loading-unloading’ (or ‘increasing-decreasing’). Designing appropriate ‘loading-unloading’ periodic inputs will prove to be crucial in the proposed identification method. By using such signals, the intermediary linear subsystem is made transparent and the hysteresis front block is equivalent to two static nonlinearities (i.e. the lateral borders of the limit cycle). Then, the input and output signals turn out to be related by two composed functions. A decomposition algorithm is then resorted to extract the back nonlinearity and both lateral borders of the front hysteresis operator. Finally, the impulse response of the intermediary linear subsystem is determined using a least-squares like estimator. The quality of the estimated model is checked using a theoretical analysis and simulation results.

Wave propagation in a one-dimensional bar with rate-independent hysteresis

A single frequency ultrasonic wave propagation in nonlinear materials generates higher harmonics. The generation of higher harmonics depends on the type of material nonlinearity considered, like odd and even harmonics in quadratically nonlinear materials, and only odd harmonics in cubically nonlinear material. The objective of this study is to present numerical study of one dimensional wave propagation in rate-independent hysteretic media by considering one dimensional chain of spring masses. Preisach-Mayergoyz (1994) and Hodgdon hysteretic models (1988) are commonly used in the theoretical and numerical study of nonlinear wave propagation studies. Here we implement the famous scalar Bouc-Wen model (1976) and a recently developed two-state hysteresis model by Biswas and Chatterjee (2015) to consider hysteresis in the large spring mass system. Number of spring masses are decided based on the spatial resolution needed for the nonlinear wave propagation to capture higher harmonics. The study presents interesting comparison between the wave propagation through an aluminum bar damaged under low cycle fatigue modeled as two different rate-independent hysteretic models. Both models show higher harmonics in responses. However, being a scalar model, the Bouc-Wen model cannot capture small minor loops due to partial reversals, whereas the two-state model captures the minor loops well.

A comprehensive dynamic model for pneumatic artificial muscles considering different input frequencies and mechanical loads

The pneumatic artificial muscle (PAM) actuated with different input frequencies and mechanical loads suffers from complex dynamic asymmetric hysteresis behaviors, leading to inaccurate positioning performance of the PAM. In order to predict the dynamic hysteresis behaviors with both rate-dependent and load-dependent effects, a comprehensive dynamic model is developed which includes two components: the first component is used to describe the rate-independent hysteresis nonlinearity with un-parallel Prandtl-Ishlinskii operators, and the cascaded second component is applied to represent the load-dependent dynamic behavior of the PAM using a nonlinear autoregressive moving average with exogenous input model implemented by recurrent fuzzy neural networks. To validate the proposed model, experiments are conducted with different input frequencies and mechanical loads. The experimental results demonstrate that the dynamic model shows a good agreement with the dynamic behaviors of the PAM under input conditions with different mechanical loads and input frequencies. In addition, the inverse model is designed to predict the mechanical load exerted to the PAM, and shows good prediction ability.

Pinning of interfaces by localized dry friction

We consider a model for the evolution of an interface in a heterogeneous environment governed by a parabolic equation. The heterogeneity is introduced as obstacles exerting a localized dry friction. Our main result establishes the emergence of a rate-independent hysteresis for suitable randomly distributed obstacles, i.e., interfaces are pinned by the obstacles until a certain critical applied driving force is exceeded. The treatment of such a model in the context of pinning and depinning requires a comparison principle. We prove this property and hence the existence of viscosity solutions. Moreover, under reasonable assumptions, we show that viscosity solutions are equivalent to weak solutions.