Ralph C. Smith

Smart Material Systems: Model Development

A curtain having an integral tie-back arrangement includes a piece of flexible material, part of which is fixed to the curtain for providing a channel from one of its vertical edges to the other of its vertical edges. An elongated flexible member which extends through the channel is fixed at one end to one of the vertical edges. As a result, if the free end of the elongated flexible member is pulled, the curtain is shirred along its width. The flexible material supports a device having a circular loop and an S-shaped section to which the elongated flexible member may be frictionally tied so as to support the shirred curtain. After the curtain is hung, the loop may be coupled to a nail on its window, thereby supporting the tie-back arrangement.

Energy-based hysteresis model for magnetostrictive transducers

This paper addresses the modeling of hysteresis in magnetostrictive transducers in the context of control applications that require an accurate characterization of the relation between input currents and strains output by the transducer. This relation typically exhibits significant nonlinearities and hysteresis because of inherent properties of magnetostrictive materials. The characterization considered here is based on the Jiles-Atherton mean field model for ferromagnetic hysteresis in combination with a quadratic moment rotation model for magnetostriction. As demonstrated by comparison with experimental data, the magnetization model very adequately quantifies both major and minor loops under various operating conditions. The combined model can then be used to accurately characterize output strains at moderate drive levels. The advantages of this model lie in the small number (six) of required parameters and its flexibility under a variety of operating conditions.

Structural magnetic strain model for magnetostrictive transducers

This paper addresses the modeling of strains generated by magnetostrictive transducers in response to applied magnetic fields. The measured strains depend on both the rotation of moments within the material in response to the field and the elastic properties of the material. The magnetic behavior is characterized by considering the Jiles-Atherton mean field theory for ferromagnetic hysteresis in combination with a quadratic moment rotation model for magnetostriction. Elastic properties must be incorporated to account for the dynamics of the material as it vibrates. This is modeled by force balancing, which yields a wave equation with magnetostrictive inputs. The validity of the resulting transducer model is illustrated by comparison with experimental data.

A Free Energy Model for Hysteresis in Ferroelectric Materials

This paper provides a theory for quantifying the hysteresis and constitutive nonlinearities inherent to piezoceramic compounds through a combination of free energy analysis and stochastic homogenization techniques. In the first step of the model development, Helmholtz and Gibbs free energy relations are constructed at the lattice or domain level to quantify the relation between the field and polarization in homogeneous, single crystal compounds which exhibit uniform effective fields. The effects of material nonhomogeneities, polycrystallinity, and variable effective fields are subsequently incorporated through the assumption that certain physical parameters, including the local coercive and effective fields, are randomly distributed and hence manifestations of stochastic density functions associated with the material. Stochastic homogenization in this manner provides low-order macroscopic models with effective parameters that can be correlated with physical properties of the data. This facilitates the identification of parameters for model construction, model updating to accommodate changing operating conditions, and control design utilizing model-based inverse compensators. Attributes of the model, including the guaranteed closure of biased minor loops in quasistatic drive regimes, are illustrated through examples.

Free energy model for hysteresis in magnetostrictive transducers

This article addresses the development of a free energy model for magnetostrictive transducers operating in hysteretic and nonlinear regimes. Such models are required both for material and system characterization and for model-based control design. The model is constructed in two steps. In the first, Helmholtz and Gibbs free energy relations are constructed for homogeneous materials with constant internal fields. In the second step, the effects of material nonhomogeneities and nonconstant effective fields are incorporated through the construction of appropriate stochastic distributions. Properties of the model are illustrated through comparison and prediction of data collected from a typical Terfenol-D transducer.

A domain wall model for hysteresis in piezoelectric materials

This paper addresses the modeling of hysteresis and nonlinear constitutive relations in piezoelectric materials at moderate to high drive levels. Hysteresis and nonlinearities are due to the domain structure inherent to the materials and both aspects must be addressed to attain the full potential of the materials as sensors and actuators in high performance applications. The model employed here is based on previously developed theory for hysteresis in general ferroelectric materials. This theory is based on the quantification of the reversible and irreversible motion of domain walls pinned at inclusions in the material. This yields an ordinary differential equation (ODE) model having five parameters. The relationship of the parameters to physical attributes of the materials is detailed and algorithms for determining estimates of the parameters using measured values of the coercive field, differential susceptibility and saturation properties of the materials are detailed. The accuracy of the model and its capability for the prediction of measured polarization at various drive levels is illustrated through a comparison with experimental data from PZT5A, PZT5H and PZT4 compounds. Finally, the ODE model formulation is amenable to inversion which facilitates the construction of an inverse compensator for linear control design.

Reduced-order model feedback control design: numerical implementation in a thin shell model

Reduced-order models employing the Lagrange and popular proper orthogonal decomposition (POD) reduced-basis methods in numerical approximation and feedback control of systems are presented and numerically tested. The system under consideration is a thin cylindrical shell with surface-mounted piezoceramic actuators. Donnell-Mushtari equations, modified to include Kelvin-Voigt damping, are used to model the system dynamics. Basis functions constructed from Fourier polynomials tensored with cubic splines are employed in the Galerkin expansion of the full-order model. Reduced-basis elements are then formed from full order approximations of the exogenously excited shell taken at different time instances. Numerical examples illustrating the features of the reduced-basis methods are presented. As a first step toward investigating the behavior of the methods when implemented in physical systems, the use of reduced-order model feedback control gains in the full order model is considered and numerical examples are presented.

Inverse compensation for hysteresis in magnetostrictive transducers

This paper addresses the development of inverse compensation techniques for a class of ferromagnetic transducers including magnetostrictive actuators. If unaccommodated, the hysteresis and nonlinear dynamics can produce severe loss of control authority and potential instabilities when the actuators are incorporated in control design. In this work, hysteresis is modeled through the domain wall theory originally proposed by Jiles and Atherton [1]. This model is based on the quantification of the energy required to translate domain walls pinned at inclusions in the material with the magnetization at a given field level specified through the solution of an ordinary differential equation. A complementary differential equation is then employed to compute the inverse which can be used to compensate for hysteresis and nonlinear dynamics in control design. The performance of the inverse compensator and its employment in LQR control design are illustrated through numerical examples.

Domain Wall Theory for Ferroelectric Hysteresis

This paper addresses the modeling of hysteresis in ferroelectric materials through consideration of domain wall bending and translation. The development is considered in two steps. First, dielectric constitutive relations are obtained through consideration of Langevin, Ising spin and preferred orientation theories with domain interactions incorporated through mean field relations. This yields a model for the anhysteretic polarization that occurs in the absence of domain wall pinning. Second, hysteresis is incorporated through the consideration of domain wall dynamics and the quantification of energy losses due to inherent inclusions or pinning sites within the material. This yields a model analogous to that developed by Jiles and Atherton for ferromagnetic materials. The viability of the model is illustrated through comparison with experimental data from a PMN-PT-BT actuator operating at a temperature within the ferroelectric regime.

A homogenized energy framework for ferromagnetic hysteresis

This paper focuses on the development of a homogenized energy model which quantifies certain facets of the direct magnetomechanical effect. In the first step of the development, Gibbs energy analysis at the lattice level is combined with Boltzmann principles to quantify the local average magnetization as a function of input fields and stresses. A macroscopic magnetization model, which incorporates the effects of polycrystallinity, material nonhomogeneities, stress-dependent interaction fields, and stress-dependent coercive behavior, is constructed through stochastic homogenization techniques based on the tenet that local coercive and interaction fields are manifestations of underlying distributions rather than constants. The resulting framework incorporates previous ferromagnetic hysteresis theory as a special case in the absence of applied stresses. Attributes of the framework are illustrated through comparison with previously published steel and iron data