Review of Modeling and Control of Magnetostrictive Actuators

Abstract

Magnetostrictive actuators play an important role in the perception of usefulness of smart materials and devices. Their applications are potentially wider than that of piezoelectric actuators because of the higher energy density and intrinsic robustness. However, the non-negligible hysteresis and complexity of their characteristics make the design and control quite difficult and has limited their diffusion in industrial applications. Nevertheless, the scientific literature presents a wide offer of results in design and geometries, modeling and control that may be exploited for applications. This paper gives a reasoned review of the main results achieved in the literature about design, modeling and control of magnetostrictive actuators exploiting the direct effects of magnetostriction (Joule and Wiedemann). Some perspectives and challenges about magnetostrictive actuators development are also gathered.

1. Introduction

Magnetostrictive actuators (MA) play an important role in the perception of usefulness of smart materials and devices because they represent one of the applications of magnetostriction, one of the first discovered smart behaviors of materials.

Magnetostriction has been known since the first observations of James P. Joule, over iron, nickel and other ferromagnetic materials, in 1842 [1]. It consists in the change of length of a ferromagnetic body due to the magnetization processes taking place in the material. The deformation, as a response to magnetization experienced by the sample, is universally referred to as Joule or direct magnetostriction. Joule experienced feeble but still observable deformations (<1 ppm), which were, however, too small for realistic applications. The inverse effect, discovered by Emilio Villari [2] in 1865 and bearing his name (Villari effect) [3], is the rise of magnetization due to a mechanical stress applied to the material and can be exploited for sensing or harvesting tasks. The direct (Joule) effect implies a conversion from magnetic to mechanical energy and is exploited for actuation purposes, subject of this review. Other manifestations named apart include the Wiedemann effect [4], which is the twisting of a magnetostrictive cylinder when simultaneous longitudinal and circumferential magnetic fields are applied to the material [5], while the less known Matteucci effect [6] may be considered the inverse of Wiedemann and is the arise of a helical magnetization vector, i.e., the superposition of an axial and a circular vector, when a torsion is applied to the magnetostrictive sample and is widely exploited in amorphous wires [7,8]. These effects are sketched in

Table 1. The magnetostrictive effects with available reviews in the literature. The sketches are simplified for an immediate visualization. H is the magnetic field, M is the magnetization, $\Delta$x is the longitudinal deformation, $\sigma$ is the longitudinal stress, $H_l$ and $H_c$ are a longitudinal and a circular magnetic field respectively, green arrows show an applied or induced torque.

Effect:	Joule (Direct)	Villari (Inverse)	Wiedemann	Matteucci
Application:	Actuation	Sensing/Harvesting	Actuation/Torque	Sensing
Sketch:
Reviews:	This review [9,10,11]	[12,13,14,15]	This review [5]	[5,8]

with references to the main review papers about the respective effect.

Magnetostriction rested aside for about one century without any effective technological fall-out, until new rare earth-based materials came into play. In the late 1950s and early 1960s, a consistent attention was devoted to the study of rare earth elements [16], observing that rare earths, such as single crystals Dysprosium (Dy) [17] and Terbium (Tb) [18], exhibited large magnetostriction (≈8000 ppm) [19]. Unfortunately, due to the low Curie temperature (88 K for Dy and 237 K for Tb), such large magnetostriction was attainable only at cryogenic or low temperatures, with no observable effects at room temperature. Later developments led off the design of compounds bonding Fe atoms to rare earth elements, able to exhibit large magnetostriction at room temperature [20], thanks to higher Curie temperatures. However, the large magnetocrystalline anisotropy required quite high fields to drive the magneto-induced deformation of the sample and gave rise to the development of quaternary and ternary rare earth Fe compounds [21]. The achieved better performances yielded to Terfenol-D, i.e., a Tb-Dy-Fe compound [22], showing up to 2000 ppm magnetostriction at room temperature, with a significantly reduced magnetocrystalline anisotropy. The era of giant magnetostriction was born. A quite exhaustive review on the physics of giant magnetostriction can be found in [23].

Another route in magnetostriction investigations was the study of binary, iron-based alloys, such as NiFe, SiFe, and CoFe [24,25], which evolved until the development of specific Iron–Gallium and Iron–Aluminum binary alloys (known as Galfenol and Alfenol, respectively) with noticeable magnetostrictive coupling (≈300 ppm for Galfenol) and improved mechanical properties that allowed a better workability and strength [26,27], with respect to Terfenol-D and similar rare earth-based compounds. While in principle, all of them are suitable for actuation or sensing scopes, Terfenol was mostly employed for actuation in rod or prismatic geometries, due to its very large deformations and forces and limited workability and brittleness [28]. Fe-Ga, Fe-Al and Fe-Co, due to the higher magnetization saturation, appeared more suitable for sensing and harvesting purposes. Furthermore, although Galfenol shows smaller deformations than Terfenol, its mechanical properties allow sample shapes in optimized geometries, for specific actuation tasks [29].

The availability of these compounds helped the flowering of design proposals of new actuators, when the direct magnetostriction is concerned (Joule effect) or sensors when, conversely, the Villari effect is exploited, and dating back since the mid-1970s. In [30,31], the stronger Villari effect shown by Fe${}_{60}$Co${}_{10}$B${}_{20}$ alloy was evidenced while, conversely, the perhaps first attempt to apply the magnetostriction of rare earth-based compounds for actuation purposes was proposed in [32,33].

Actuators based on magnetostrictive materials promised to fill a range of applications, which available devices, e.g., piezo-actuators, were unable to guarantee. Micro-actuation offering high strokes with high energy density, as well as reasonable frequency ranges, stimulated the spread of application proposals of Terfenol-D based actuators until the end of the 1990s [34,35,36]. At the beginning of the 2000s, the new Fe-Ga binary alloy showing lower strokes but much better mechanical performances came into play, providing a new boost to the issue of smart device design [37].

The application of magnetostrictives for actuation purposes can be classified according to these groups, are detailed in the following:

• Micropositioning. The micrometric strokes of both Terfenol and Galfenol alloys, along with high exerted forces, is exploited for micropositioning task, normally with low or very low working frequencies. Micrometric or sub-micrometric actuations are normally attained. The actuation precision represents the basic performance parameter of such application, which could be compromised by rate-independent memory effects, i.e., hysteresis, mainly shown by rare earth-based ternary compounds [38,39,40,41].
• Motors. The relatively high strokes provided by magnetostrictive materials allows conceiving micro- or inchworm-motors [38,42,43] with promising characteristics in terms of forces/torques and resolution at low speed. Several solution have been proposed in the last decades, such as friction motors, e.g., Flex-M1 by Cedrat Researches [44] or other prototypes invented in different research centers. An interesting and quite exhaustive review was provided by Claeyssen et al. [11]. The availability of Fe-Ga alloys allowed inventing micro-motors where the active material was suitably shaped due to its good mechanical characteristics [45], which demonstrated the applicability of lower-stroke magnetostrictives for actuation purposes.
• Active Vibration Control. This application has seen piezo-actuators as the leading solution in high frequency application of smart materials. However, due to their high strokes and higher energy density (1.4–2.5 J/m${}^3$), magnetostrictive materials attracted the interests for that kind of high frequency actuation [46,47]. During the early 2000s, EU projects facilitated the rise of research consortiums from industries and research institutions to seek new solutions to the vibration control for aeronautical applications (MESA and MESEMA projects), in which Terfenol-D was widely investigated [48].
• Miscellanea. The framework of application for magnetostrictive materials is not limited to the above issues, but offers specific solutions to specific actuation problems. Among them, the needle actuation for fuel injectors is one of the most interesting applications of MA [10], which were also exploited for micro-pumps [49] or acoustic applications [50].

Magnetostrictive actuators are presented in several designs and solutions and, generally speaking, they consist of two macro elements. The information unit provides all the algorithms for the control system and exploits all the smart features of magnetostrictives and the hardware unit, as sketched in Figure 1, and has following the main components:

• The active material, such as Terfenol or Galfenol, is in the shape of cylindrical rods, beams, cymbals, etc.
• The magnetic circuit is necessary to guide the magnetic flux lines and strengthen the field within the active material. Permanent magnets can be added to impart a magnetic bias to the active material.
• The power coil is required to apply the input current and to induce the magnetic field.
• The structural frame has the function of applying a mechanical prestress through the use of springs, to mechanically decouple the active rod from external forces and to make the displacement available outside by using a steel cylinder or a threaded bar.
• Sensor(s) is required to detect displacement, force, strain, current, etc. and is exploited for monitoring purposes or to provide the feedback signal in the feedback control loop.

Such components are suitably designed and realized in connection to the specific actuator’s task and affects the overall behavior of the final device. They are normally dimensioned through a careful analysis on physical ground, taking into account the material’s characteristic, and the coupled phenomena taking place in the modeled system (i.e., electromagnetic coupled to thermal or mechanical phenomena, etc.). To this aim, several contributions focused on those analysis, with particular emphasis on the modeling at the macro-scale of the active material [51,52,53,54,55,56] the numerical modeling of the device [57,58,59], or the analysis of loss phenomena taking place in the material [60,61,62,63].

The performances of the task demanded to the actuator are linked both to the quality of the hardware unit components, and to the accuracy and reliability of all control algorithms employed, i.e., the software. Such processes require a feedback signal of good quality and an as reproducible as possible behavior of the active material. The latter task is not trivial for magnetostrictives because, being ferromagnetic in nature, they display rate-independent memory phenomena, worsening the overall behavior and performance of the system. Concerning this issue, remarkable work has been carried out to compensate hysteresis through different modeling approaches [56,64,65,66]. A thorough discussion of these points is carried out in the rest of the paper.

This review is, to the best of our knowledge, the first one on MA in general. While some are present about sensing, few reviews have been published on magnetostrictives materials and devices. Nevertheless, it is worth noting the above-mentioned one from A. Flatau’s group on Galfenol alloys [37] with a deep experimental presentation of the material’s properties that can be exploited for actuators design. Deng and Dapino [9] recently published a review on the use of magnetostrictives to damp mechanical vibrations for several applications. Of course, the damping of vibration makes use of MA. About a specific application, the review in [10] is about the state-of-the-art design and modeling of MA for fuel injection. Some reviews are present about magnetostrictive effects. In [12], a thorough review of magnetostrictives, magnetoelectrics, and PZT, for energy harvesting applications, i.e., based on the Villari effect, is found. Another wide review on energy harvesting, including the magnetostrictive based [14], is found in [13]. A short review of magnetostrictive transducers based on the (reversed) Wiedemann effect, specially intended for Non Destructive Evaluations (NDE), is presented in [5], while a review of magnetostrictive patch transducers for ultrasonic nondestructive testing is presented in [67]. A review of applications related to Matteucci effect is presented in [8]. Finally, an examination of the various principles of sensing based on magnetostrictives is presented in [68]. Some further technical information on search queries are provided in Appendix A.

2. Geometries and Design

Research on MA has pointed out a deep interest in developing devices with peculiar geometrical configurations. Starting from early works considering only the simple linear actuator configuration, the proposal of the binary alloys, such as Galfenol, has prompted the study of new geometries, thanks to the improvements in terms of workability and machinability of the new materials. The principal geometrical configurations for magnetostrictive actuators proposed in past and recent years are described in this section and summarized in

Table 2. Summary of the main geometrical configurations found for MA and related references.

Linear	Cantilever	Amplified	Inchworm	Others
[36,47,69,70,71,72,73,74,75,76,77,78]	[29,39,79,80,81,82,83,84,85,86]	[50,87,88,89,90,91,92,93,94,95,96,97,98]	[38,40,42,43,88,89,99,100,101,102]	[103,104,105,106]

, along with references and example sketches for each geometry.

2.1. Linear

The simplest way to use a giant magnetostrictive material for actuation purposes consists in the exploitation of its axial magneto-induced strain. In other words, a linear magnetostrictive actuator can be achieved by simply transferring to the moving tip of the actuator the strain induced by a magnetic field on the active material. It is for this reason that linear actuators have been developed since the 1970s [69], through the 1980s and 1990s [36,47], up to recently [70,71,72,73,74].

Most of the studies deal with Terfenol-D based linear actuators. A typical example is given in [75] where the authors presented the design and fabrication of a linear magnetostrictive actuator prototype aimed to structural vibrations control. A thorough modeling of the magnetic circuit is proposed, followed by the static and dynamic analysis of the prototype device, as shown in Figure 2. A similar device is designed in [76] for a CNG Fuel Injector.

Particular configurations of linear actuators are also used for motors. An example is presented in [77], where a hybrid linear motor based on a self-moving principle is proposed. The motor is made up of elementary actuating cells connected in series and placed into a guide-way.

The working principle of the linear motor is described in Figure 3a. Conversely, the elementary cell is shown in Figure 3b. It is a magnetostrictive actuator exploiting a Terfenol-D rod placed in a ring-shaped shell structure. The speed is proportional to the stroke produced by the actuator, and the frequency and amplitude of the excitation current.

Finally, despite the lower maximum magnetostriction, Galfenol samples have also been used in linear actuation systems. For example, a Galfenol micro-actuator is developed in [78]. The system is shown in Figure 4 and includes a $F{e}_{81.6}G{a}_{18.4}$ rod (1 mm in diameter and 5 mm in length), a 270-turn excitation coil and a ferromagnetic holder, which, together with the end cap, allows the closure of the magnetic path. Total dimensions of the actuator are 2 mm in diameter and 11 mm in length. Despite the low stroke attainable by Galfenol, and the small dimensions, the actuator shows good mechanical properties. Furthermore, the relatively low magnetic field needed to achieve the maximum magnetostriction allows the microactuator to be driven by a portable music player and to be used as an acoustic speaker.

2.2. Cantilever

Another geometry used for magnetostrictive actuators is the standard cantilever configuration [29,79,80,107]. An example is given in [80], in which a Galfenol cantilever actuator has been developed. The device is based on a Galfenol and stain-less steel (SUS, non-magnetic) to improve mechanical performances. An improvement of this prototype was proposed by the same authors in [29].

In particular, a Galfenol–Nickel bilayer has been developed to exploit the positive (Galfenol) and negative (Nickel) magnetostriction of the two materials. A complete dynamic characterization is carried out in the paper. The effect of the Nickel negative magnetostriction allow obtaining a $70\%$ increase of the displacement with respect to the Galfenol-SUS cantilever.

Furthermore, a huge effort in decreasing the device dimensions has been carried out in the framework of cantilever-type magnetostrictive actuation, leading to the exploitation of actuators based on magnetostrictive thin-films. Most of works mainly use $TbFe$-based thin films coupled in multilayers with $SmFe$ alloys [81,82]. In particular, the actuation response of $TbFe$, $TbDyFe$, $SmFe$ and $SmFeB$ films are studied in both cantilever-type and membrane-type mode on $Si$ substrates in [81]. Conversely, $TbFe$-$FeCo$ multilayers are studied in [39,83]. In addition, $FeGa$ thin-films have been used for integrating a magnetostrictive material in MEMS actuators. An example is proposed in [84]. The authors proposed a capacitance measurement to estimate the thin-film magnetostriction. Finally, they also integrated Tungsten/Galfenol beams on a Si wafer and studied their magnetic-induced motion using an optical microscope. Other commonly used materials for cantilever actuators are Metglas^TM (type 2826 MB) [85], and $FeSiB$ [86].

2.3. Amplified Configurations

Several works dealing with MA present the attempt to mechanically amplify the total stroke, spanning from the crossbow geometry [44,87,88,89] to hydraulic amplification through fluids [90,91,92], lever mechanisms [50,93,94,95] and other more peculiar configurations [96,97,98].

For example, a magnetostrictive actuator using Terfenol-D with differential displacement amplification system is proposed in [93]. In this amplification system, both the micro-actuator and the output end are on the same axis. The amplification mechanism used is based on a differential lever system combined in a $60S{i}_{2}Mn$ flexible hinge structure and it is shown in Figure 5 (left). The magnetostrictive induced displacement of about $22\mathsf{\mu }$m was amplified to about $160\mathsf{\mu }$m, giving a gain of $7.14$.

Another example of non-ordinary amplifying configurations can be found in [96,97], where an amplification system for a linear actuator is developed for the valve opening mechanism of a common-rail proportional fuel injector. In particular, three Terfenol-D rods are placed in a rod-holder equally spaced at $120°$. Then, two sets of rods are combined serially in a Z-shaped holder, which allows amplifying the total stroke. A sketch of this configuration, called Tandem arrayed magnetostrictive actuator, is represented in Figure 5 (right). The total stroke achieved with such amplifying configuration is about 50 $\mathsf{\mu }$m.

Among the crossbow (or shell-like) configurations, there is the one shown in Figure 6, already proposed in commercial applications by [44] for magnetostrictives and piezostack. The relatively simple geometry can be specialized to high forces (larger than 20 kN) or large strokes (up to 200 $\mathsf{\mu }$m).

2.4. Inchworm Actuators

A particular geometry widely used for magnetostrictive actuators is also the one known as inchworm configuration. It is mainly exploited in linear [40,42,88,99,100] and rotary motors [43,89]. Interesting examples of hybrid linear motors exploiting the combination between piezoelectric elements and a magnetostrictive rod can be found in [101,102]. In Figure 7, a sketch of the hybrid inchworm mechanism is shown.

In particular, the multistack PZT actuators operate as clamping device on the moving rod. On the other hand, the Terfenol-D rod acts as push device of the linear motor. In [101], the capacitive behavior of piezoelectrics and the inductive one of magnetostrictive rod along with its excitation coil, are exploited to build up a resonant circuit, too. This configuration allows obtaining a maximum speed of about 33 mm/s.

The inchworm motion through MA also attracted research because of the possibility of developing robots with contactless excitation, by using an external magnetic field. An example is proposed in [38]. In particular, two kinds of micro mobile robots based on MA are developed: a macro model (21 mm in diameter) and a micro model (6 mm in diameter). A magnifying mechanism of the displacement exploiting suitable structures of legs allows the robots to move in the inchworm mode. Legs structure for the macro model is shown in Figure 8. The MA in the robot makes the legs vibrate, moving the robot. The movement is reversed by changing the inclination of the legs in the opposite direction. Prototypes are tested in different conditions obtaining velocities of about 1 mm/s at 100 Hz of the exciting current.

2.5. Others Geometries

There are other designs in the literature that do not fall in any of the previous categories. An example is given in [103]. The authors presented the optimized design of a fish-like underwater robot. In particular, two $Al$–$TbF{e}_2$ tails allow the motion in two directions by exploiting the frequency modulation of the external driving magnetic field.

Another actuator configuration is based on the Wiedemann effect [104,105,106]. According to Smith and Overshott [108], the application of a helical magnetic field allows the magnetic domains to rotate, creating a shear strain in the material. For example, a spring-type actuator is proposed in [105] to develop a linear motor. Conversely, a magnetostrictive torsional transducer is designed and fabricated in [106] with the aim of developing a vibrational drilling tool. In particular, a permendur hollow cylinder is used as magnetostrictive transducer. A sketch of the working principle is shown in Figure 9.

The helical field is created by the superposition of two magnetic fields: an axial component generated by a coaxial excitation coil and a circular component produced by the wire passing through the sample hole. Both FEM analysis and experimental tests are described and a comparison with conventional drilling systems is also carried out.

2.6. Magnetoelectric Coupling in Magnetostrictive Actuators

Among the different geometries exploited for magnetostrictive actuators, it is worth mentioning that class of devices based on the coupling between magnetostrictive and piezoelectric materials. Since the past century, research has been investigating such a coupling, known as magnetoelectric effect (ME), from both material science and engineering viewpoints, with a huge explosion in the last decades [109,110].

Although this effect can also be shown in single-phase materials [110], mechanically mediated multi-phase composites, also referred to as multiferroic composites [111], are usually considered in this framework. As described in the seminal paper [112], the ME is the product of piezoelectricity and magnetostriction, taking place in the ferroelectric phase and ferromagnetic phase, respectively, connected by the elastic response:

$$\begin{array}{c}{\mathrm{ME}}_{\mathrm{H}}=\frac{\mathrm{magnetic}}{\mathrm{mechanical}}\times \frac{\mathrm{mechanical}}{\mathrm{electric}}\end{array}$$

(1)

$$\begin{array}{c}\hfill {\mathrm{ME}}_{\mathrm{E}}=\frac{\mathrm{electric}}{\mathrm{mechanical}}\times \frac{\mathrm{mechanical}}{\mathrm{magnetic}}\end{array}$$

(2)

where ME_H and ME_E are known as direct and indirect magnetoelectric effects, respectively [113]. In the former case, the magnetostrictive phase works as an actuator. Indeed, an applied magnetic field induced strain is mechanically transferred to the piezoelectric phase, which experiences a change in electrical polarization. In most cases, this effect is exploited for sensing purposes [114,115,116].

Different materials have been used in magnetoelectric composites. The most common among magnetostrictives is Terfenol-D [114,115,117,118,119] coupled with different piezoelectric alloys, mainly lead zirconate titanate (PZT) and lead magnesium niobate-lead titanate (PMN-PT). For example, a typical magnetoelectric laminate composite is proposed in [119]. It consists of a PZT disk sandwiched by two Terfenol-D disks, as shown in Figure 10.

The two magnetostrictive disks experience a shrink when an external magnetic field is applied along the cylinder axis. The strain is then transferred to the PZT sample with a consequent change in its polarization. A similar structure is proposed in [117], but in a cantilever-like shape.

A completely different geometry is developed in [118], concerning a different method for the mechanical interaction. A revised sketch of the device is shown in Figure 11. In particular, a Terfenol-D rod is exploited to actuate a brass cymbal structure containing a PMN-PT piezoelectric sample. The output magnetoelectric voltage (V_ME) is therefore measured.

Galfenol has also been used in magnetoelectric composites (e.g., in [120]), while a review on polymeric magnetoelectric composites is given in [113].

Finally, it is worth mentioning that a strong interest has been shown in the last decade in thin film magnetoelectric devices [116,121,122]. For example, a comparison between different thin film composites used as magnetic field sensors is given in [116].

3. Modeling

MA are devices composed of an active material (Terfenol or Galfenol in most cases), a structural frame, a magnetic circuit (that can have structural aims too), a power coil and tools to transfer the displacement to the external plant. In the literature, most of the modeling is aimed at the bulky active material because, as pointed out in Section 2, the actuator structure is dominated by the material and its properties, and several models have been built in the last decades with the aim of describing the magnetostrictive phenomena—from either a microscopic or a macroscopic point of view—and optimize, develop, control and improve the performance of MA. The literature including some modeling of MA may be classified with respect to one or more of the following aspects of modeling:

• Physics-based models or phenomenological models: In the first category, models relate some basic physical properties of magnetostrictive or magnetic behavior to macroscopic quantities, while, in the second category, the models simply treat the material as a black-box and relate the input and output from a phenomenological point of view.
• Linear, nonlinear or hysteretic models.
• One-input one-output models, two-input one-output models or two-input two-output models: These consider among the two mechanical local variables (strain and stress) and two magnetic ones (field and induction), or among the macroscopic variables (displacement and force, current and magnetic flux).
• Low frequency (rate-independent) or dynamic modeling: This considers whether dynamic phenomena within the active material, the structural frame (inertia or elastic effects) or the magnetic circuit (eddy currents, etc) are neglected.
• Dimension of modeling: The model may be composed by lumped elements or allow a spatial variability of fields in 1D, 2D or 3D.

The simplest phenomenological approach to the magneto-mechanical modeling of a magnetostrictive material/device is the linear one, as described in [23,28,123]. The linear modeling can describe quite well small magnetostrictive deformations induced by variable magnetic field or stress superposed to a constant magnetic bias and/or a mechanical pre-stress. Indeed, in most cases, it is applied to a magnetic bias, by means of permanent magnets to get a bipolar magnetostriction in response to a bipolar magnetic field [124]. Moreover, the magnetic bias may be exploited to reduce the effect of non-linearity. In this case, it can be assumed that the material behavior is reversible and almost linear. The most general linear representation, with fields applied in all three directions, is the following:

$$\mathit{\epsilon }={\mathit{s}}^{\mathit{H}}\mathit{\sigma }+\mathit{d}\mathit{H}$$

(3)

$$\mathit{B}=\mathit{d}\mathit{\sigma }+{\mathit{\mu }}^{\mathit{\sigma }}\mathit{H}$$

(4)

where $\mathit{\epsilon }$ and $\mathit{\sigma }$ are the strain and stress tensors, respectively, while $\mathit{B}$ and $\mathit{H}$ are the induction and magnetic field vectors, respectively. ${\mathit{s}}^{\mathit{H}}$ and ${\mathit{\mu }}^{\mathit{\sigma }}$ represent the pure linear elastic compliance matrix at constant magnetic field and the pure linear magnetic permeability matrix at constant applied stress, respectively. Finally, the magneto-mechanical coupling is taken into account by the matrix $\mathit{d}$, which is called magnetostrictive or piezomagnetic matrix.

As discussed in Section 2, most geometries employed for actuation are rods. Therefore, they can be modeled along the longitudinal axis of the active material, leading to the use of 1D models, or even with linear lumped parameters, as in [23,125,126]. For example, the magnetic field to bias the active material is provided by permanent magnets and a magnetic yoke. Then, under suitable working conditions, the magnetic circuit can be modeled as quasi-static through the Hopkinson’s law:

$$Ni=\oint \mathit{H}·d\mathit{l}=\sum _k{H}_k{l}_k+\mathcal{F}({\varphi }_{mm})=\sum _k{\mathcal{R}}_k{\varphi }_k+\mathcal{F}({\varphi }_{mm})$$

(5)

where N is the power coil turn-number, i is the applied current, and $H_k$, ${\varphi }_k$ and ${\mathcal{R}}_k$ are, respectively, the magnetic field, flux and reluctance of the magnetic path of length $l_k$. Moreover, $\mathcal{F}({\varphi }_{mm})$ represents the magneto-motive force contribution of the magnetostrictive material, which can be modeled as a linear or nonlinear function of the magnetic flux ${\varphi }_{mm}$ [127].

The mechanical dynamic behavior can occur in several applications, as in the vibrations control, and need to be taken into account in the overall system’s model. From a mechanical point of view, the actuating system is governed by Newton’s second law, which in axisymmetric geometry is usually simplified to an equivalent single DOF mass–spring–dumper equation, as follows:

$$F=m_e\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+c_e\frac{\mathrm{d}x}{\mathrm{d}t}+k_{e}x$$

(6)

where F is the output force of the magnetostrictive material; $m_e$, $c_e$ and $k_e$ are the equivalent mass, damping coefficient and stiffness, respectively; and x is the output displacement, as sketched in Figure 12. Moreover, the equivalent mass $m_e$ is often considered equal to $m_{MA}/3+m_l$ [10,128], where $m_{MA}$ is the mass of the MA rod and $m_l$ is the mass of the load.

In [70,72,75,129,130], linear models of the active material coupled with magnetic circuits lumped elements and mass–spring–dumper linear ODE equations have been used. In [35,131] thermal–magnetic–mechanical coupled linear models are proposed and compared with experimental tests to perform an adaptive vibration control, while in [132] a dynamic simulation model has been used to study the performance of a magnetostrictive actuator, which takes into account electrical, eddy current, mechanical and hysteresis losses with an input frequency of 21 kHz. Aiming to a deeper characterization of the MA, in [133], the authors modified a commercial actuator to characterize the structural and electromagnetic dynamic behavior.

Linear modeling is inadequate when fields with large variations are applied to the material and magnetic and magnetostrictive saturation, as well as hysteresis, come into play. Thus, many papers have considered the above nonlinear effects. As examples, in [134,135], analytical nonlinear constitutive models to simulate the active vibration control of Terfenol-D are studied, while, in [60,61], eddy currents losses are reconstructed from a hysteretic characteristic and a finite difference representation of the diffusion equation for the active rod. Several studies about the use of Finite Element Method (FEM) to solve nonlinear electro-magneto-mechanical equations and analyze the characteristics of a MA have been published. By modeling, meshing, loading, and defining boundary conditions of the whole structure, FEM codes are able to numerically solve the magnetic field overall the actuator. In particular, in [57,58,59,136], the authors implemented nonlinear FEM in static or quasi-static conditions, while dynamic regimes are investigated in [137] using Hamilton principle minimal potential energy linear model and in [138], where the MA is modeled to study fast transients (lower than 0.1 ms).

About hysteresis modeling, many efforts have been made with the aim to model and compensate magnetic hysteresis and its effects because of the detrimental effects of hysteresis in the control system. Generally speaking, hysteresis models can be classified into two types: the physics-based and phenomenological models [10,139]. The models can be defined with respect to local inputs/outputs (magnetic field, induction, strain and stress) or, via suitable hypothesis, can be extended to macroscopic variables (current, magnetic flux, displacement and force). The most adopted in the physics-based category is the Jiles–Atherton (J-A) model [140]. It considers the total magnetization in magnetic materials as sum of an irreversible part which take into account the domain wall motion and a reversible part due to domain wall bowing and it is defined by the following set of equations [141]:

$$\left\{\begin{array}{c}{H}_{eff}=H+\alpha M\hfill \\ M_{an}=M_s\left[coth\left(\frac{H_{eff}}{a}\right)-\frac{a}{H_{eff}}\right]\hfill \\ M=M_{irr}+M_{rev}\hfill \\ M_{rev}=c(M_{an}-M)\hfill \\ \frac{\mathrm{d}{M}_{irr}}{\mathrm{d}H}=\frac{1}{\delta k/{\mu }_0-\alpha (M_{an}-M)}(M_{an}-M)\hfill \end{array}\right.$$

(7)

where H is the external applied field, $H_{eff}$ is the effective magnetic field, $M_{an}$ is the anhysteretic magnetization, $M_{irr}$ is irreversible magnetization, $M_{rev}$ is reversible magnetization and M is the total magnetization. Moreover, $\alpha$ is the domain interaction quantifier, $M_s$ is the saturation magnetization, c is the reversibility coefficient, a is the anhysteretic shape parameter, k is the energy to break pinning sites, and $\delta$ is the sign of the time variation in H (being 1 when $\mathrm{d}H/$$\mathrm{d}t>0$ while −1 when $\mathrm{d}H/$$\mathrm{d}t<0$).

To couple magnetic and magnetostrictive hysteresis and consider the effect of mechanical stress on both types of hysteresis, the Jiles–Atherton–Sablik (J-A-S) model for magneto-mechanical hysteresis has been reviewed. In particular, under an applied stress, the effective field of classical J-A model is modified as follows [51]:

$$H_{eff}=H+\alpha M+H_{\sigma }$$

(8)

where $H_{\sigma }$ is a contribution related to the stress. The magnetization and the magnetostriction ($\lambda$) are coupled through a suitable derivative:

$$H_{\sigma }=\frac{3}{2}\frac{\sigma }{{\mu }_0}{\left(\frac{\partial \lambda }{\partial M}\right)}_T$$

(9)

where T is the temperature, while nonlinear constitutive relationship between magnetostriction and magnetization was generally given by the quadratic domain-rotation model, as [123]:

$$\lambda =\frac{3}{2}\frac{{\lambda }_s}{M_s^2}{M}^2$$

(10)

where ${\lambda }_s$ and $M_s$ are, respectively, the saturation magnetostriction and magnetization. In [94], a hybrid model of mechanically amplified magnetostrictive actuator is provided. In particular, the J-A-S model is related to the input stiffness of the mechanical amplifier when quantifying the magneto-mechanical effects, including stress-dependent magnetization, stress dependent magnetostriction and Young’s modulus variation effect. The J-A model has been considered recently in a modified version with model parameters identified through a data mining technique [142].

The J-A model shown in the following set of equations (Equation (7)) is capable of considering the rate-independent energy losses due to hysteretic effect but, unfortunately, because of its quasi-static nature, it is not effective when describing dynamic conditions. In [143], a frequency-dependent J-A model is presented.

$$\begin{array}{c}\left(\frac{{\mu }_0{d}^2}{2\rho \beta }\frac{\mathrm{d}H}{\mathrm{d}t}\right){\left(\frac{\mathrm{d}M}{\mathrm{d}H}\right)}^2+{\left(\frac{Gdw{\mu }_0{H}_o}{\rho }\right)}^{1/2}{\left(\frac{\mathrm{d}H}{\mathrm{d}t}\right)}^{1/2}{\left(\frac{\mathrm{d}M}{\mathrm{d}H}\right)}^{3/2}+\hfill \\ \left[k\delta -\alpha \left(M_{an}-M+k\delta c\frac{\mathrm{d}{M}_{an}}{\mathrm{d}{H}_e}\right)\right]\left(\frac{\mathrm{d}M}{\mathrm{dH}}\right)-\left(M_{an}-M+k\delta c\frac{\mathrm{d}{M}_{an}}{\mathrm{d}{H}_e}\right)=0\hfill \end{array}$$

(11)

where $\beta$ is a geometrical factor, $\rho$ is the electrical resistivity, G is a dimensionless constant, w and d are the width of lamination and the thickness, and $H_o$ is the parameter that represents the internal potential experienced by domain walls. It should be noted that Equation (11) consists in a power density-balance equation; in particular, the first term is the eddy current power loss per unit volume and the second term represents the anomalous loss results from the changes in the domain configuration. In [128,144], mathematical models are established according to J-A dynamic hysteresis model, nonlinear quadratic domain model and structure dynamics equation in term of Partial Differential Equation (PDE). In [145], the authors built a general model constituted by three main part: J-A mean field theory for ferromagnetic hysteresis; quantification of magnetostriction through consideration of a quadratic model posed in terms of the magnetization, similarly to Equation (10); and, finally, a force balancing providing a set of PDEs that quantifies material displacements due to the magnetostriction. In [64], hysteresis in a magnetostrictive transducer is modeled through J-A model and a complementary differential equation provides the inverse transforms for the actuators; then, a compensator is numerically built and employed in LQR control design. The response of a Terfenol-D based actuator under step-input conditions at different preloads are investigated in [146] by coupling a numerical code for flux density, the J-A model for magnetization and a magneto-mechanical coupled model for magnetostriction.

Besides the J-A-S based models, some other physical models have been presented for magnetostriction, such as the Armstrong model [147,148] that is an hysteretic magneto-elastic constitutive theory of pseudo-cubic ferro-magnetostrictive alloys. The theory can qualitatively predict the magneto-elastic response of these materials under generally applied magnetic field and stress. Other models [135,149,150,151] are based on the concept of free energy (Helmholtz and Gibbs) of the polycrystal composing the material. Those model can give a qualitative behavior of the materials that is useful for the materials design and development but they are seldom applied to MA modeling.

Recent contributions to physics-based modeling include a model of the magnetostrictive hysteresis loop based on a differential equation describing magnetostriction due to the domain wall movement as well as domain magnetization rotation [152].

The phenomenological models of hysteresis deal with output variables, either magnetization or strain, from a macroscopic point of view. The most widely employed model is the Preisach model (PM), presented in 1935 [153]. The general expression of this model is [154]:

$$f(t)=\widehat{\mathsf{\Gamma }}u(t)={\int \int }_{\alpha \ge \beta }\mu (\alpha ,\beta ){\widehat{\gamma }}_{\alpha \beta }[u(t)]\mathrm{d}\alpha \mathrm{d}\beta$$

(12)

where $f(t)$ and $u(t)$ are, respectively, the output and input of the hysteretic system, for example the strain and the magnetic field in the material. ${\widehat{\gamma }}_{\alpha \beta }$ is the ideal relay, a hysteresis operator which takes the values of $\pm 1$ and $\mu (\alpha ,\beta )$ represents the weighting function of ${\widehat{\gamma }}_{\alpha \beta }$, i.e., a probability density function or Preisach Distribution Function (PDF). $\alpha$ and $\beta$ are the “up” and “down” relay switching values. In applications to real hysteresis problems, in principle, the PDF can be determined as follows [154]:

$$\mu (\alpha ,\beta )=-\frac{{\partial }^{2}E(\alpha ,\beta )}{\partial \alpha \partial \beta }$$

(13)

where $E(\alpha ,\beta )$ is the so-called Everett function directly related to a particular set of measured outputs, the first order reversals (FOR) curves [154]. In such a relatively simple way, the PM is able to efficiently mimic the magnetization, or the strain, under static or quasi-static conditions, and it has been widely studied and used in magnetostrictive transducers modeling and control. In particular, much effort has been focused on PDF identification methods and on the inversion of PM, with the aim to obtain compensator or pseudo-compensator working in different operative conditions [52,65,155,156,157,158]. In [53], a phenomenological hysteresis model, constituted by a suitable composition of Preisach-like operators, for magnetostrictive materials is proposed, while, in [54], the authors presented an identification of the Preisach model through fuzzy approximators and a feed-forward neural network, by introducing the concept of pseudo-compensator. A comparison of PM and JA models performances about a MA based on Terfenol-D is shown in [159], while Li et al. [160] firstly developed a PM inverse compensation approach by using the inverse multiplicative structure and obtained good results with the advantage of avoiding complicated approximate inverse algorithms.

The Prandtl–Ishlinskii (PI) is another phenomenological model that has been considered together with its generalized version (GPI). Hysteresis theory establishes that PI and GPI are subsets of Preisach. However, even though PM is a powerful tool to model and characterize the hysteresis, it is considered difficult and computationally expensive to invert, in order to perform a control system. On the other hand, PI and GPI models provide an inverse analytical expression of the hysteresis operator, which could be easily implemented in a feed-forward compensator controller to mitigate nonlinear effects [161]. In particular, when the PDF can be expressed in the form of

$$\tilde{\mu }(\alpha ,\beta )=\frac{1}{2}\rho (\xi (\alpha )-\xi (\beta )){\xi }^{\prime }(\alpha ){\xi }^{\prime }(\beta )$$

(14)

where $\rho$ is a non-negative function and $\xi$ an increasing function, it is the so-called GPI operator as proposed in [66]. Al Janaideh et al. [162] proposed the application of a generalized play operator based upon hyperbolic-tangent envelope functions, leading to a GPI model that can reproduce asymmetric, as well as symmetric, saturated minor and major hysteresis loops. The prediction error is lower than 3% with respect to the measured data of a MA. Moreover, in [163], an asymmetric shifted PI model is proposed, being composed by a PI operator, a shift operator and a Lipschitz continuous function. A first preliminary study on the representation error of GPI with respect to a general PM is presented and evaluated through numerical test in [164]. Conversely, in [56], the so-called modified PI model, defined by a serial combination of conventional PI operator and a memory-free non-linearity with an asymmetrical graph, is achieved to model, identify and compensate complex hysteretic behaviors with non-convex branches and asymmetrical hysteresis loops. Aljanaideh et al. [165] proposed a dynamic hysteresis model for MA, based on the generalized rate-dependent Prandtl–Ishlinskii model that incorporates a rate-bias-dependent threshold and a memoryless function.

Some studies have been conducted to add dynamic effects to the classical Preisach model. In [55,166], researchers coupled the Preisach operator to an ordinary differential equation (ODE), and an efficient inversion algorithm is then introduced. Contextually to the previous mentioned papers, in [167], the authors proposed a two-stage model able to describe the dynamic behavior of a magnetostrictive actuator for fast actuation purposes: the first stage is a rate-independent model of hysteresis, while the second one is a linear dynamic model. In [168,169], the authors dynamically modeled a Terfenol-D based magnetostrictive actuator by coupling the Preisach theory with a FEM electromagnetic field solution, while rate-dependent losses in magnetostrictive rods with Preisach modeling are discussed in [61,62]. The GPI model has been considered as the combination of a rate-dependent PI model with a function of deadband operators [170]. It has been tested as an inverse of the rate-dependent PI model with the inverse of the deadband functions with a peak percent positioning error of nearly 3.7% over a 1–250 Hz frequency range.

The magnetostriction can be modeled by considering characteristics with two input and/or two outputs, among the two mechanical variables (strain and stress) and the two magnetic ones (field and induction). Indeed, typical magnetostrictive plots show a combined effect of those variables [124,171]. The problem of two-input modeling for MA has been tackled in some papers. In [172], a general model of magneto-elastic coupling is proposed while a simplification of such a model is proposed for real-time control purposes in varying stress conditions [173]. In [174], a large class of stress-dependent Preisach-like models, based on the employment of a genera effective field that is a nonlinear function of magnetic field and stress is presented. In [175,176], a particular application of that general class, limiting the nonlinear function to a ratio of magnetic field and stress, is presented. This assumption is demonstrated to be effective when compared with experiments [176]. In [177], mechanical stress is considered as further input in a tri-node Hopfield Neural Network hysteresis operator giving a good agreement with respect to experiment in both magnetostrictive and magnetic curves. Finally, in [178], a class of phenomenological model for magneto-elastic interactions in materials with hysteresis, where both mechanical and magnetic variables are fully coupled, is presented. The approach makes use of the hysteresis potential concept and takes into account magneto-mechanical hysteresis dissipation, within the framework of thermodynamic compatibility. Comparisons with experimental data confirmed that complicated magneto-mechanical paths can be reconstructed by that model. The model has been further considered with three mathematical models of Galfenol, namely a non-hysteretic model, a non-hysteretic model with a feedback loop, and a model where the hysteresis is represented by the Preisach operator with a simplified Preisach density function. The output of the non-hysteretic model with a feedback loop best fits the measured data up to minor features of the magneto-elastic curves [179]. Another model making use of thermodynamic theory and having the stress, frequency and ambient temperature as input parameters, is reported in [180]. This model can describe the curves of magnetostriction and magnetization under quasi-static and dynamic conditions. As a summary,

Table 3. Main papers, found in the literature, identified according to the geometry (rows) and type of modeling (columns). 0-D is equivalent to a lumped model.

	Linear	Nonlinear	Mechanical Dynamics	Electro-Magnetic Dynamics	Hysteresis
0-D	[23,70,72,75,125,126,129,130]	[10,23,34,51,135,139,140]	[23,132,135]	[10,23,55,75,132,135,138,148,166]	[10,51,52,53,54,55,56,60,61,62,64,65,66,94,128,132,139,140,141,142,143,144,145,146,148,149,151,154,155,156,157,158,159,161,162,163,164,166,167,168,169,170,173,178,179,180]
1-D	[23,70]	[23,57,131,134]	[10,34,64,72,128,129,130,131,139,144,145]	[61]
2-D	[23]	[23,34,60,137,138,146]	[60,137,151]	[60,136,168]
3-D	[23]	[58,59]	[94]	[62,169]

places the research activities found in the literature according to the model properties presented above.

Characterization

Experimental characterization plays a key-role in the study of a magnetostrictive material, especially when it is used as active core in actuators [146,181]. A large part of the papers found in the literature about modeling and control of MA starts with, and/or includes, an experimental characterization of the input–output variable relationships and magneto-mechanical parameters [182]. These variables could be considered from local or macroscopic points of views, as commented in Section 3. In particular, the characterization of the magnetostrictive material is performed through the local variables and it is useful to describe and model the relations between mechanical and magnetic variables [124]. Conversely, the characterization of the MA is obtained with the macroscopic variables, which take into account mechanical amplification factors, non-linearities or dynamics due to the external mechanical or magnetic systems [133]. However, the setup used to characterize the local and macroscopic variables are similar and some differences occur to measure the proper physical variables [37]. Generally speaking, the experimental setup to characterize the magnetostrictive material has the same elements reported in Figure 1. The magnetostrictive sample (1) is placed in a magnetic path (5), made of bulky iron elements or iron sheets (for better dynamic performances), and is wrapped by a pickup coil (2). The magnetic circuit provides the field in the sample axis direction with an excitation coil (3), which could be wounded around the pickup coil or suitably placed on one or more branches of the magnetic path. Some permanent magnets (4) could be placed in the magnetic circuit to bias the sample. Often, the magnetic path is a mechanical support too and helps to accommodate the external applied force and the sensor to measure the displacement (10). In particular, the magnetostrictive sample is rigidly fixed at one extremity while is connected, through some supports and springs (8), to a test-machine able to provide a force, which is measured with a load cell. The springs allow mitigating the direct magnetostrictive effect in order to have a constant applied force [183]. A target is placed on the output shaft (9) to have a reference for the displacement sensor. In this case, a possible amplification mechanism of the actuator is considered too.

When the characterization concerns the active material, the magnetic field, H, along the sample axis is measured with a Hall sensor (11) or flux-meter placed in contact to the rod and transversely to its axis [127]. Moreover, the strain variable is measured with one or more strain gauge (electrical resistance based, non-inductive, semiconductor, etc.) suitably attached to the sample [152,184]. Conversely, when the characterization concerns the overall MA, the input could be the current, which is directly measured with amperometer, or the magnetic field, which could be approximated as [23,146]:

$$H=G_{ex}{N}_{ex}{I}_{ex}\sqrt{\frac{2\pi }{l_s{\alpha }_1}\frac{(\alpha +1)}{(\alpha -1)}}$$

(15)

where $G_{ex}$ is a geometrical factor [23], ${\alpha }_1$ and ${\alpha }_2$ are the inner and outer radii of coil, $\alpha ={\alpha }_2/{\alpha }_1$, $N_{ex}$ is the excitation coil turn number, $I_{ex}$ is the input current and $l_s$ is the length of the sample. Finally, in cases of local or macroscopic characterization, the magnetic induction, B, is measured with the time integration of the pickup voltage ($v_{pk}$) as:

$$B=-\frac{1}{N_{pk}S}{\int }_t{v}_{pk}\mathrm{d}t$$

(16)

where $N_{pk}$ is the pickup coil turns while S is the cross section.

It is worth noting that the characterization process of the active material is a quite standard operation because it concerns the intrinsic magneto-mechanical properties of the magnetostrictive sample, while the characterization of the MA is more related to the geometry and to the end-use of the actuator. As a consequence, the MA’s characterization setup may require more specific tools.

4. Control

The characteristic of magnetostrictives materials shows strong non-linearities, hysteretic behavior and saturation. For micro-positioning and for almost all practical applications, those side effects ask for a control system to mitigate their influence on the actuator behavior and give an input–output relationship, i.e., current–displacement, that is as linear as possible.

The employed techniques make use of a closed loop or feedback scheme, while open loop or feedforward schemes are rarely employed alone. Feedback control is usually preferred because it is a mandatory solution in presence of disturbances, or modeling uncertainties. In those cases, the feedback allows obtaining performances otherwise unachievable by open loop solutions. However, the feedforward control has some important advantages with respect to the feedback control. For instance, it is easy to analyze control systems with feedforward controllers since this action (supposed to be implemented through a stable system) does not affect the stability of the overall control system and so its design is easier than the feedback control that, as well known, might make unstable a control system made of stable systems. This issue may be particularly severe for MA [185]. The feedforward control is faster than the feedback control since its action (the input signal to the process) depends only on the feedforward controller dynamics and not also on the process dynamics. For these reasons, the control of MA often employs both actions. Such a configuration is called a two-degrees-of-freedom (2DoF) control system since, in this way, it is possible to design a controller that weights independently the reference signal and the measurements coming from the process [186,187,188].

As sketched in Figure 13, both controllers can be composed by different blocks with the first one as simple as a Proportional–Integral–Derivative (PID) controller or a more sophisticated one (${\mathcal{H}}_{\infty }$, for example). Moreover, the controller can have a compensator of the hysteretic non-linearity [54,55,157,163,166,170,189]. Then, the sequence of the compensator and the hysteretic MA acts as a linear system, within the limit of the compensation [170], and this makes the overall control system easier to design. The control scheme is completed by a current amplifier feeding the power coil and a displacement sensor, needed in the feedback loop.

The use of compensation and 2DoF control is presented in [55,166], where the authors achieved relevant results in terms of performance and stability of the control scheme. The compensation of hysteresis with an efficient, non-iterative, algorithm and a PID feedback control is presented in [189]. The approach proposes the employment of a classical and well behaved and reliable Preisach model of hysteresis in such a way to easily compute, through a lookup table, the inverse of a hysteretic transducers. Then, it is possible to embed the control algorithm directly in inexpensive micro-controllers [190].

Control schemes are widely employed to control and attenuate structural vibrations of beams, plates and structures in general, with a single or several actuators (6DoF [70,191,192]), or with a feedback control system [47,72,75,98,193]. Control of vibrations through a MA and a nonlinear compensation in a feedback system was numerically simulated in [134].

Robust control designs (${\mathcal{H}}^2$ and ${\mathcal{H}}_{\infty }$) have been applied also in the context of a magnetostrictive transducer used for high-speed, high-accuracy milling applications [194]. The high output force of MA allows integrating them within a complex control system with a dual stage actuation system and on feedforward hysteresis cancellation (with a dynamic Preisach model) and a feedback correction, with a sliding mode controller design [195].

The effectiveness of the feedback control is demonstrated in [151] where it is presented the development of a nonlinear control design for attenuating structural vibrations using MA. A thin plate is exposed to vibrations and controlled via a Terfenol-D transducers at the plate edges. The device is modeled by a homogenized energy model and the resulting nonlinear constitutive relations are used to construct a PDE representation of the structural system. A nonlinear open loop controller accommodate the hysteresis model but it is not robust enough with regard to unmodeled dynamics or disturbances. Robustness is incorporated by linear feedback laws acting on measured disturbances [135,196].

It is worth mentioning that neural networks (NN) have been exploited to design the controller in feedback control systems of MA [197,198,199] or to develop a 2DoF system where the feedforward action is compensating the hysteresis through a NN and the feedback is a proportional–derivative controller [200].

Smart Self-Sensing

The use of MA can be defined smart when the magnetostrictive characteristics with two mechanical variables (strain and stress) and two magnetic ones (field and induction) are fully exploited in the control system. Indeed, within the framework of thermodynamic compatibility, those four variables are connected through two characteristics [178]. This approach has been exploited in applications by using a magnetic flux coil wrapped around the active material and a suitable system processing the measured voltage, leading to the self-detection of the displacement [125,201], velocity [202] and force [203,204], without the use of any bulky external sensor.

In [125], the smart use of the magnetostrictive characteristics in a MA application as a self-sensing ability of the transducer to sense its own motion as it is being driven by means of a bridge circuit to extract a signal proportional to transducer motion is reported. The paper makes use of coupled electromechanical linear equations, the concept of the transducer’s “blocked” electrical impedance and motional impedance are developed, and then a bridge-based design is proposed. However, the presented results show that magnetostrictive transduction is inherently nonlinear, and does not, therefore, lend itself well to the traditional bridge circuit approach to self-sensing. Then, the possibility to use the inductivity of the excitation coil of the MA as a method to estimate the mechanical stress experienced by the actuator is theorized in [205]. Finally, a method for solving the nonlinear problems in MA self-sensing is proposed in [206]. A dynamic equivalent circuit model of MA is analyzed within the concept of an active Kelvin bridge and the experimental results show that the Kelvin bridge is more effective than the conventional Wheatstone bridge while extracting the self-sensing signal.

In [201], a self-sensing MA by measuring the magnetic flux in the active material is described. An Hall sensor is integrated into the casing of the actuator. The approach makes use of a separation of the sensing from the actuation information contained in the magnetic flux measurement by modeling the two effects with hysteresis operators. This method allows the compensation of hysteresis in real-time, while the displacement reconstruction allows the implementation of an integral feedback controller for the additional compensation of force-dependent variations of the displacement due to the finite stiffness of the magnetostrictive material.

A recent contribution presents a procedure for the estimation of the real-time stress experienced by a MA by using a pickup coil around the magnetostrictive rod [204]. The paper exploits the fully coupled model of hysteresis presented in [178] and gives a nonlinear estimation of the stress with an overall error lower than 12%.

5. Applications

The applications of MA exploit the higher energy density of magnetostrictives and can be found often when high forces and micrometric precision in displacements, in both quasi-static and dynamic conditions, are needed. MA are found as standalone elements, often related to micropositioning tasks, or to the damping of vibrations [47,72,75,98,193]. Quite a few applications exploit MA as part of more complex systems [195]. In vibrations control, MA are exploited in systems with more elements, 6-DOF systems [70,191,192] with reductions up to one-tenth of the floor accelerations, and kept less than $0.008$ cm/s${}^{-2}$. About vibrations suppression, another application is related to the improvement of performances for tooling machines [106], such as for lathes [207] or boring bars [208]. One of the first applications of MA is in sonar systems, as for an ultrasonic-vibration-assisted microforming system [209], or to develop a guided wave generation system for nondestructive tests [210]. In [11], several applications are reported, such as a Tripode Tonpilz-type sonar transducer that is one of the first and most successful applications of MA to show their high power capabilities. Indeed, with three rods 100 mm long and 20 mm in diameter, the device is able to achieve 208 dB of sound level at $f=1$ kHz, with an output power of 4 kW.

MA can be part of hydraulic systems, acting as hydraulic valves (see, for example, [90,211,212]). Another popular application that, apparently, has not yet found wide spread commercial application is the use of MA in fuel injectors [10,96,97]. Figure 14 shows a drawing of a fuel injector with the MA on top. The latter makes use of an amplification mechanism, based on a tandem-type configurations with 2 × 3 rods, as sketched in Figure 5 (right). It is also worth mentioning a commercial application of MA as audio speakers [78], with the actuator mechanically connected to large flat surfaces, such as glasses or thin walls. The whole surface generates sound pressure [50], driven by the MA, and this can be exploited for clean rooms, or even for shops windows.

6. Conclusions

The analysis of the literature shows that magnetostrictive actuators have been partially exploited with respect to their best features, such as the easiness of hardware design and the high density of stored energy that allow compact and powerful devices.

About geometries and design, the brittleness of Terfenol-D initially limited the design of actuators, resulting in bulky geometries aimed to isolate the active rod from unwanted mechanical stresses. The development of magnetostrictive materials with better mechanical properties (workability and welding), such as Galfenol, and in perspective other iron alloys FeCo and FeAl, allowed more geometries and applications and it has been important for the diffusion of MA.

About modeling, from the literature analysis, in view of a wide spread of MA to industrial applications, it is apparent that some approaches suffer from one or more of the following problems: the identification procedure is not robust enough; the accuracy of experimental reproduction is not enough; and too many numerical resources are needed. On the other hand, recent control techniques show encouraging results that may help to mitigate some of the previous issue.

It is true that magnetostrictive actuators are still some of the best solutions to those applications where micrometric displacements and high forces are needed and cost is not an issue.