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Sensors and Actuators A 141 (2008) 536–547

Nonlinear Hamiltonian modelling of magneticshape memory alloy based actuators

Jean-Yves Gauthier a,∗, Arnaud Hubert a, Joel Abadie a,Nicolas Chaillet a, Christian Lexcellent b

a Laboratoire d’Automatique de Besancon, UMR CNRS 6596, ENSMM, UFC, 24 rue Alain Savary, 25000 Besancon, Franceb Institut Femto-st, UMR CNRS 6174, ENSMM, UFC, UTBM, 24 chemin de l’Epitaphe, 25000 Besancon, France

Received 23 May 2007; received in revised form 5 September 2007; accepted 3 October 2007Available online 16 October 2007

bstract

This paper proposes an application of the Lagrangian formalism and its Hamiltonian extension to design, model and control a mechatronicystem using magnetic shape memory alloys. In this aim, an original dynamical modelling of a magnetic shape memory alloy based actuator isresented. Energy-based techniques are used to obtain a coherent modelling of the magnetical, mechanical and thermodynamic phenomena. Theagrangian formalism, well suited in such a case, is introduced and used to take into account the dynamical effects. Hamilton equations are deduced

nd used for the computation of the theoretical behaviour of this actuator. These numerical simulations are compared with some experimentaleasurements permitting the validation of the proposed modelling. Beyond the work presented here, these results will be used to design an energy

haping nonlinear control well adapted for a strongly nonlinear active material.2007 Elsevier B.V. All rights reserved.

ic sha

bodIppilmsnbltu

eywords: Micro-mechatronic; Nonlinear dynamics; Active materials; Magnet

. Introduction

In engineering fields, the current trend is to design smallernd smaller components. The field of mechatronics is not anxception and leads to the emergence of the micro-mechatroniceld. In this case sensors and actuators design is more and moreominated by the use of active materials. Most of these materi-ls permit simultaneously actuation and sensing functions in anntegrated and distributed way. This integration makes the imple-

entation easier at the mini- and microscopic scale as comparedith the classical design of the multi-components mechatronics

ystems.Among the current offer of active materials (see [1] for a

ecent review), magnetic shape memory alloys (MSMAs) arenteresting candidates because their performances are situated

etween piezoelectric materials (high frequencies but smalltrain) and classical shape memory alloys (SMAs) (large strainut low frequencies). As shown in numerous works performed

∗ Corresponding author.E-mail address: jygauthi@ens2m.fr (J.-Y. Gauthier).

esptbi[

924-4247/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.sna.2007.10.012

pe memory alloys; Hamiltonian and Lagrangian modelling

y the material research community, the strongly nonlinearityf the MSMAs behaviour will be undoubtedly an issue for theesign and the control of innovative systems based on MSMAs.n previous works the authors of the present paper have pro-osed a quasi-static modelling for MSMAs with a reasonablerediction [2]. This model is based on the thermodynamics ofrreversible processes because it can model hysteretic and non-inear material behaviours. In our case, this model gives the

echanical strain of the MSMA as a function of the mechanicaltress and the magnetic field. Nevertheless, in order to designew actuators and their efficient control laws, this model has toe extented to the dynamic case. In this way, a relevant approachies in the use of an energetic and variational point of view. Inhis paper, we propose to model a complete MSMA actuator bysing the Lagrangian formalism. This formalism permits a rel-vant and coherent approach to model a complete mechatronicystem. Beyond the work presented here, this modelling willermit in future works to explore some recent nonlinear con-

rol techniques based on energy functions such as Lyapunovased design [3], passivity based control [4], energy shap-ng and damping assignment [5] and port-Hamiltonian control6].

nd Ac

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2

2

pSsossa[cartrnt

ed

F

F(t

pfTtaitiMhvdsifittap

tnkph

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2

J.-Y. Gauthier et al. / Sensors a

A description of MSMA properties will be first reported.hen the Lagrangian and Hamiltonian formalisms will be shortlyescribed. Then, the modelling of an MSMA based actuatorncluding its mechanical load is proposed. Simulations andxperimental results are finally compared with some conclusionsnd perspectives.

. Magnetic shape memory alloys (MSMA)

.1. MSMA properties and characteristics

The value of the maximum reachable strain is an importantroperty to design actuators using active materials. ClassicalMA are one of the active materials that provide the largesttrains. Nevertheless, they present a low response time becausef the energy conversion involving a heat transfer. A new SMAensitive to the magnetic field was obtained in 1995/1996 byome research teams in the USA, at the MIT for the Ni–Mn–Galloy [7] and at the University of Minnesota for the FePd alloy8]. Since then these materials knew some improvements con-erning mainly the working temperature range and the maximumvailable strain. When actuating by magnetic fields, these mate-ials permit now to obtain a large strain (6–10%) with a responseime shorter than the classical SMA. Today this time is in theange of the millisecond. The MSMA used in this paper is aon-stoechiometric Ni–Mn–Ga monocrystal corresponding tohe most currently used MSMA material.

In this alloy, the martensite phase can appear in three differ-

nt variants corresponding to the three possible crystallographicirections in the sample (see Fig. 1(a)).

The martensitic reorientation principle is presented inig. 1(b): at high temperature, the MSMA sample is in austenitic

ig. 1. MSMA behaviour: (a) austenite phase and the three martensite variants;b) martensitic reorientation: effects of mechanical stress, magnetic field andemperature.

tbbmtip

ssv(AtTavmvad

2

pFf

tuators A 141 (2008) 536–547 537

hase (A). After a cooling process, the austenite phase is trans-ormed into a martensite phase without any favoured variants.he resulting sample contains therefore martensite variants into

hree equal portions (M1, M2 and M3). If a mechanical stress ispplied in a specific direction, then the fraction of variant withts short axis in this direction grows. If this stress is high enoughhen the sample will only contain this variant (for example M1n Fig. 1(b)). If the stress decreases, the volume fraction of the

1 variant does not completely decrease due in part to a largeysteresis. In the same way, if a magnetic field is applied, theariant which has its easy magnetization direction in the fieldirection is favoured. As the easy magnetization direction is theame as the short axis of the M2 variant, this variant fractionncreases as shown in Fig. 1(b) by the application of a magneticeld perpendicular to the stress field. The distribution between

he magnetic field and the mechanical stress permits to controlhe macroscopic strain. With a pre-stress, one can then obtain anctuator driven by the magnetic field only. By heating, austenitehase can also be recovered.

Beyond these interesting properties, it must be noticed thathis material has also some important drawbacks like the brittle-ess of the single-crystal, the high required magnetic field (400A/m), the large dependence of the material parameters on tem-erature changes, a small blocking stress (2–3 MPa) and a largeysteretical thermo-magneto-mechanical behaviour.

.2. MSMA quasi-static modelling

.2.1. IntroductionThe quasi-static modelling of the MSMA is based on the

hermodynamics of irreversible processes with internal variablesecause this subject can take into account the strong nonlinearehaviour of this kind of materials. The resulting model per-its to express the behaviour of the material including all the

hermo-magneto-mechanical couplings. More details concern-ng the modelling approach can be found in [2] and only the keyoints of this modelling are reported in this paper.

As the active material is used in this paper in a two dimen-ional actuation way (in the xy plane shown in Fig. 1), the MSMAample can be considered as constituted by only two martensiteariants M1 and M2. The internal variable z and its complement1 − z) are respectively the M1 and the M2 volume fractions.

magnetic field H and a mechanical stress σ are applied tohe MSMA sample as described in Section 2.1 and in Fig. 1(b).he total strain ε of the MSMA sample can be separated inton elastic strain σ/E and reorientation strain of the martensiteariants γ · z. E is the Young modulus of the MSMA, and γ is theaximum strain due to the reorientation process of martensite

ariants. In addition, an interaction between these two vari-nts takes place. The energy corresponding to this interactionepends on martensite fractions and on a specific parameter K12.

.2.2. Helmholtz free energy

An Helmholtz free energy F of the MSMA sample was pro-

osed in [2]. It can be divided into four parts: F = Fchem +therm + Fmech + Fmag. In this paper, because we are mainly

ocused on the reorientation process (M1�M2) and not on the

538 J.-Y. Gauthier et al. / Sensors and Ac

Fn

tlfak

toMz

F

F

F

2

ivtaiinmifiht

M

wt

0

wiTtb

2

tvfCt

d

d

wd

π

iσ

fic

π

i

i

Ffafrearrangement starts, the behaviour can be represented by the

ig. 2. Representative elementary volume, arrows represent local domains mag-etizations [10].

ransformation process (A �M), the chemical energy Fcheminked to a phase transition does not change and can be neglectedor the variational computations. Moreover there is also no vari-tion of the thermal energy Ftherm because the temperature isept constant in the actuator principle described in this paper.

The Helmholtz energy will then not take into account thesewo contributions. Fmech takes into account the elastic energyf the MSMA, the energy due to the reorientation of M1 and2 martensite variants and the interaction energy Fint = K12 ·· (1 − z) between these two variants [9]:

mech(ε, z) = E

2(ε − γz)2 + K12 · z · (1 − z) (1)

mag is the MSMA magnetic energy:

mag(B, z) =∫ B

0H(b) db (2)

The latter is detailed in the following section.

.2.3. MSMA magnetic behaviourFor the MSMA magnetic modelling using the thermodynam-

cs of irreversible processes with internal variables, two internalariables are considered. Fig. 2 shows the representative elemen-ary volume as proposed in [10]. α(H) is the Weiss domain widthnd θ(H) is the magnetization rotation angle. If no magnetic fields applied, α and θ are equal to zero. When a small magnetic fields applied in the x direction, α increases corresponding to a mag-etization in the M1 easy magnetization direction. When a largeragnetic field is applied, θ increases to obtain a magnetization

n the same direction as the direction of the applied magneticeld. This process corresponds to a magnetization in the M2ard magnetization direction. From Fig. 2, the magnetization inhe x direction can be expressed as

(z, H) = MS[(2α(H) − 1)z + sin(θ(H))(1 − z)] (3)

We choose to use a linear with saturation behaviour of M(H)hen z is kept constant. Then, in this case, the following func-

ions for α(H) and θ(H) can be used:

≤ α = χaH

2MS+ 1

2≤ 1, −π

2≤ θ = arcsin

(χtH

MS

)≤ π

2(4)

f

π

tuators A 141 (2008) 536–547

here χa and χt are respectively the magnetic susceptibilitiesn the easy and hard magnetization directions of the martensite.he anisotropic energy is taken into account by the way of these

wo parameters. Moreover, the saturation behaviour is renderedy the use of the magnetisation saturation level MS.

.2.4. Clausius–Duhem inequalityAs seen in the scientific literature concerning MSMA,

he mechanical behaviour of this material is highly irre-ersible. In the thermodynamics of irreversible processesramework, the irreversibility is described and expressed by thelausius–Duhem inequality. In [2], we propose for the MSMAs,

he following inequality:

D = −ρ · dF(σ, H, z) + H dB + σ dε ≥ 0 (5)

This expression can be reduced to

D = πf∗ dz ≥ 0 (6)

here πf∗ is the thermodynamic force associated with z and isefined as

f∗ = −∂F∂z

= σγ − K12(1 − 2z) + πfmag(α, θ) (7)

This equation express the distribution between the mechan-cal force associated with the mechanical compressive stress< 0 and the mechanical force associated with the magnetic

eld πfmag(α, θ). The expression of π

fmag(α, θ) is more easily

omputed using a Legendre transformation between B and H:

fmag(α, θ) = − ∂

∂z

(∫ B

0H(b) db

)= ∂

∂z

(∫ H

0B(h, z) dh

)

= ∂

∂z

(∫ H

0μo(M(h, z) + h) dh

)

= ∂

∂z

(∫ H

0μoM(h, z) dh

)

= μoM2S

[(1 − 2α) sin θ

χt

+ (2α − 1)2

2χa+ sin2 θ

2χt

](8)

In addition, in order to satisfy the Clausius–Duhem inequal-ty, one have to ensure that:

f πf∗ ≥ 0, z ≥ 0; if πf∗ ≤ 0, z ≤ 0 (9)

It indeed permits to represent some hysteretical processes.or example, in a major loop, i.e. a complete rearrangementrom z = 0 to z = 1 (path a) and from z = 1 to z = 0 (path b)s reported in Fig. 3, the rearrangement begins when πf∗ ≥ πcr

or the path a and when πf∗ ≤ −πcr for the path b. After the

ollowing kinetic equation:

˙ f∗ = λCz (10)

In this case, it corresponds to a linear piecewise discretization.

J.-Y. Gauthier et al. / Sensors and Ac

Fh

3

3

opazt

S

te

L

dp

wgnka

•

•

•

δ

L

fi

3

ftat

H

ig. 3. The thermodynamic force πf∗ according to the internal variable z for anysteretical behaviour.

. Lagrangian and Hamiltonian formalisms

.1. The Lagrangian formalism

The lagrangian formalism is a modelling technique basedn some energy functions used in addition with the Hamiltonrinciple [11,12]. This principle postulates that the variation ofn action S between two times on a real path is always equal toero. This action is the lagrangian L(q, q, t) integrated betweenhe two times t1 et t2 (see Fig. 4).

=∫ t2

t1

L dt ⇒ δS = 0 on a real path (11)

In the case of non relativist systems, the lagrangian function ishe difference between a kinetic co-energy T∗(q) and a potentialnergy V(q) [13]:

(q, q) = T∗(q) − V(q) (12)

In the case of conservative systems (i.e. closed and non-issipative:L = L(q, q)), a variational calculus on the Hamiltonrinciple leads to the set of n Lagrange equations:

∂L∂qi

− d

dt

(∂L∂qi

)= 0, i ∈ [1, n] (13)

Fig. 4. Variation of the action in the Hamilton principle.

cm

H

i

q

saH

q

tuators A 141 (2008) 536–547 539

here (∂L/∂qi) are generalized forces and pi = (∂L/∂qi) areeneralized momentums. This formalism can be extended to theon-conservative case (open and dissipative systems) includinginematic constraints using an extented Lagrangian function L′nd the Lagrange multipliers technique:

the external generalized forces fext(q, t) are taking intoaccount in the variation of L′ by adding their virtual worksδWext = fext(q, t) · δq;dissipations by static and viscous frictions are taking intoaccount by adding their dissipated energies variations δQs(q)and δQv(q). The dissipation by viscous friction Qv(q) is cal-culated with a Rayleigh dissipation function R(q): Qv(q) =∫ t2t1

R(q) dt;the holonomic kinematic constraints c(q) = 0 are taking intoaccount with a Lagrange multipliers technique by adding theterm −λ · δc(q) to the variation δL′.

Finally, we have:

L′ = δL + fext · δq + δQs + δQv − λ · δc (14)

The Hamilton principle using δL′ gives the followingagrange equations:

∂L∂qi

− d

dt

(∂L∂qi

)− ∂R

∂qi

+ ∂Qs

∂qi

+ fext,i − λi · ∂ci

∂qi

= 0 (15)

More details concerning these modelling techniques can beound in [13] for electrical and electromechanical systems andn [14,15] for mechatronic and distributed systems.

.2. The Hamilton formalism

The Hamilton formalism is an extension of the lagrangianormalism which uses a Legendre transformation to substitutehe time rate functions q in the lagrangian L(q, q) for the gener-lized momentum p = (∂L/∂q) in a new energy function calledhe Hamiltonian function H(q, p):

(q, p) = p · q − L(q, q) (16)

In the case of non relativist systems, the Hamiltonian functionorresponds to the total energy expressed with coordinates q andomentums p instead of coordinates q and velocities q:

(q, p) = T(p) + V(q) (17)

Then the n second order Lagrange equations are transformednto a set of 2n first order Hamilton equations:

˙ i = ∂H∂pi

, pi = −∂H∂qi

, i ∈ [1, n] (18)

The extented Lagrangian function for a controlled dissipativeystem with kinematic constraints can also be transformed inton extented Hamiltonian function. This leads to the following

amilton equations:

˙ i = ∂H∂pi

, pi = −∂H∂qi

− ∂R∂qi

+ ∂Qs

∂qi

+ fext,i − λi · ∂ci

∂qi

(19)

540 J.-Y. Gauthier et al. / Sensors and Actuators A 141 (2008) 536–547

tcc

krcst

4

pFcaemMtaasi

ts

susiae

5

edeeFmke

5

aFarc

5

ba

Fig. 5. General scheme of the MSMA based actuator.

The recent control techniques called port-Hamiltonian usehis set of equations in a matrix form. In addition, it uses a spe-ific writing to characterize the connections between differentoordinates. For a conservative system, this gives [5]:

d

dt

(qi

pi

)=(

0 I

−I 0

)︸ ︷︷ ︸

J

·

⎛⎜⎜⎝

∂H∂qi

∂H∂pi

⎞⎟⎟⎠ (20)

The extension to a controlled dissipative system withinematic constraints is also possible by using a matrix Aepresenting the kinematic constraints between the differentoordinates qi, a dissipation matrix R including the viscous andtatic frictions and finally a matrix B and a control input u(t) toake into account the external forces:

d

dt

(q

p

)=(

0 I

−I −R

)·

⎛⎜⎜⎝

∂H∂q∂H∂p

⎞⎟⎟⎠+

(0

A

)· λ

+(

0

B

)· u(t) (21)

. Description of the MSMA actuator

The mechatronic system considered in this paper is a sim-le actuator including an MSMA sample. It is presented inig. 5. A magnetic circuit including a coil and a ferromagneticore permits to create a magnetic field inside an airgap wheren MSMA sample is inserted. This sample is attached at onextremity to the fixed support and at the other extremity to aobile load. The weight of the load permits to pre-stress theSMA sample to obtain a motion in both directions. Gravi-

ational and inertial effects of the load have to be taken into

ccount. The coil is supplied by an home-made switching powermplifier (200 V, 2 A). The displacement of the load is mea-ured with a laser sensor (Keyence LK-152) and the controls performed using a DSP board (dSpace). A PC is used for

atta

Fig. 6. Photograph of the actuator test bench.

he displacement signal acquisition and to control the completeystem.

A picture of this actuator is presented in Fig. 6. As it can beeen in the experimental system, a fulcrum and a arm lever aresed to amplify the inertial effects. The working of the completeystem is equivalent to the system depicted in Fig. 5 if we takento account two different masses mg and mi for the gravitynd the inertial effects. Large white arrows represent all forcesxerted on the bar.

. Energies expressions

In this section the different energy expressions for the consid-red system are presented. A powerfull property of an energeticescription is the property of additivity. It states that the totalnergy of one system is equal to the sum of the energy ofach sub-system (magnetic circuit, MSMA, and driven load).or each sub-system, all generalized coordinates qi, generalizedomentums pi, Hamiltonian function, dissipation matrix R, and

inematics constraints matrix A will be detailed. The differentnergies are sketched in Fig. 7.

.1. Magnetic circuit modelling

In order to model the core and the magnetic circuit, we usen electrical network including resistances and inductances (seeig. 8). This lumped-parameters model permits to take intoccount the magnetic leakage in the surrounding air and the fer-omagnetic saturation without any time-consuming numericalomputation such as the finite elements method.

.1.1. Definition of magnetic circuit coordinatesA Lagrangian electrostatic convention (see [16]) is chosen

ecause it permits to consider the voltage u(t) applied to the coils an external generalized force fext(t). Inductances LFe, Ll, La

nd L are respectively associated with the magnetic flux acrosshe Fe–Si core, the leakage magnetic flux in the surrounding air,he magnetic flux across the airgap, and lastly the magnetic fluxcross the MSMA. It must be noticed that both inductances LFe

J.-Y. Gauthier et al. / Sensors and Actuators A 141 (2008) 536–547 541

Fig. 7. Different energies

Fn

atc

cgmual

trmat

5

ar

R

TGt

CFAAM

5

e

ame

5

t

f

5

at

N

ig. 8. Lumped-parameter model used for the modelling of electrical and mag-etical quantities.

nd L are not constant parameters but they take into accounthe nonlinear magnetic behaviours of these two materials. Theorresponding coordinates are reported in Table 1.

For the coil’s quantities, a global form is used: the electricharge qc is the generalized coordinate q1, the current I is theeneralized velocity q1, the magnetic flux φ is the generalizedomentum p1. For the other magnetic quantities, a local form is

sed: the magnetic excitation field Di corresponds to the gener-lized coordinate, the curvilinear integral Hi · li along the pathi corresponds to the generalized velocity, the flux Bi · Si acrosshe surface Si is the generalized momentum. lFe, ll, la and l areespectively the ferromagnetic core, leakage, airgap and MSMAean flux path lengths. The dissipation due to eddy-currents

re neglected because the laminated Fe–Si magnetic core limitshem drastically.

.1.2. Dissipation potential of the magnetic circuitJoule effect losses in the coil are taken into account with

dissipation potential (Rayleigh function R ). If r is the full

1esistance of the coil, we have:

1 = 1

2r · q2

c = 1

2r · I2 (22)

able 1eneralized coordinates, velocities and momentums used in the modelling of

he magnetic circuit of Fig. 8

i qi qi pi

oil 1 Charge qc I φ

e–Si core 2 DFe HFe · lFe BFe · SFe

ir leakage 3 DL HL · lL BL · SL

irgap 4 Da-g Ha-g · la-g Ba-g · Sa-g

SMA 5 D H · l B · S

H

5

5

Mε

aeMt

of the full system.

.1.3. Magnetic circuit energies and Hamiltonian functionThe magnetic energy depends on the magnetic fields Bi (gen-

ralized momentum pi) in the volume V:

H = Wmag =∫

V

∫ Bi

oHi(b) db · dV

= WL2 + WFe + WL + Wa-g,

WL2(φ) = φ2

2LL2, WFe(BFe) = VFe ·

∫ BFe

oHFe(b) db,

WL(BL) = VL · 1

2μoB2

L, Wa-g(Ba-g) = Va-g · 1

2μoB2

a-g

(23)

The magnetic energy WFe stored in the Fe–Si core takes intoccount the nonlinear saturated magnetic behaviour of the Fe–Siaterial with an arctan shape function. The value of WL2 is

xtremely smaller than these of the three other terms.

.1.4. External force of the magnetic circuitThe generalized external force applied to this sub-system is

he voltage applied to the coil:

ext = u(t) (24)

.1.5. Kinematic constraints of the magnetic circuitAmpere’s law (Kirchhoff’s current law in Fig. 8) gives some

lgebraic relations between coordinates resulting in the defini-ion of two kinematic constraints:

· I = HFe · lFe + HL · lL

∫dt⇒ c1(q) = DFe + Dl − N · qc=0,

L · lL = Ha-g · la-g + H · l

∫dt⇒ c2(q) = DL − Da-g − D = 0

.2. MSMA modelling

.2.1. Definition of MSMA coordinatesTwo types of generalized coordinates are considered in the

SMA modelling. The first ones, the temperature T, the strainand the magnetic field H are classical thermodynamic vari-

bles. They are associated with three thermodynamic forces, thentropy s, the mechanical stress σ and the magnetization of theSMA M. The coordinate T is not used afterwards because of

he isothermic working of the actuator and the constant value of

542 J.-Y. Gauthier et al. / Sensors and Actuators A 141 (2008) 536–547

Table 2Generalized coordinates, velocities and momentums used in the modelling ofthe MSMA

i qi qi pi

MSMA 5 Field D H · l B · S

MM

FiwcntMrM

5

tsoibc

H

wtctoccm0i

5

ssC

wst

π

λ

n

Table 3Generalized coordinate, velocity and momentum used in the modelling of theload

i qi qi pi

8

d

P

=

b

5

(efafiT

aT

l

6

t

H

d

SMA 6 Fraction z z pz

SMA 7 Strain ε ε pε

therm. The second type of generalized coordinates appears onlyn the frame of the thermodynamics of irreversible processesith internal variables. An internal variable is a generalized

oordinate characterizing an internal working of the materialot directly linked to any external forces. This variable permitso take into account the memory effect of the material. For the

SMA, the volume fraction z is an internal variable. All theelevant generalized coordinates used for the modelling of the

SMA are reported in Table 2.

.2.2. MSMA energies and Hamiltonian functionThe Hamiltonian function of the MSMA sample corresponds

o its total energy. Because of the size and weight of the MSMAample compared to the size and weight of the load, the influencef the potential and the kinetic energies of MSMA is quite lown the complete device energy and therefore these two terms cane neglected. The Hamiltonian function of the MSMA samplean be expressed as

MSMA = VMSMA · (Fmech + Fmag) + p2z

2mz

+ p2ε

2mε

(25)

here mz and mε are inertial parameters corresponding respec-ively to the z and ε variables. We consider them in the generalase, but they will be neglected in the computation because (i)he mass of the MSMA material is very lower than the massf the load and (ii) the process of martensite reorientation isonsidered as nearly instantaneous compared to the other timeonstants (electrical and mechanical). In the Hamiltonian for-alism, it is equivalent to say that pε and pz are nearly equal to

. The Helmholtz free energies Fmech and Fmag are expressedn Section 2.2.2.

.2.3. Dissipation potential of the MSMAThe internal variable z was introduced to model the dis-

ipative hysteretical behaviour of the material. In order toatisfy the second thermodynamic law, z is used to define thelausius–Duhem inequality:

(26)

ith πf∗(z, z) the thermodynamic force associated with z. Aimplified expression is chosen corresponding to a static hys-eresis without inner loops:[

sign(z) 1]

f∗(z, z) = λC z +2

−2

+ πcr · sign(z) (27)

C corresponds to a constant kinetic parameter. πcr is an inter-al friction parameter. The dissipation power Physt is then

m

Displacement x x px

educed:

hyst = VMSMA · D = VMSMA · πf∗(z, z) · z

VMSMA ·(

λC · z ·[z + sign(z)

2− 1

2

]+ z · πcr · sign(z)

)

This power can be incorporated in the dissipation matrix Ry adding the following term as reported in [5]:

(∂Physt/∂z)

z(28)

.3. Load modelling

The driven load energy includes a kinetic energy (Tload =1/2m)p2

x with m the mass of the load) and a gravity potentialnergy (Vload = mgx with g the gravity constant). The viscousriction of the load in the surrrounding air is modelled usingdissipation potential (Rayleigh function R2 = (f/2)x2 withthe viscous friction coefficient). The only relevant general-

zed coordinate for the load is its position x as reported inable 3.

Moreover the load attached to the MSMA sample gives anlgebraic relation between the strain ε and the displacement x.his leads to an other kinematic constraint:

0 · ε = x ⇒ c3(q) = x − l0 · ε = 0

. Hamilton equations

By adding all the previous sub-system energies, the Hamil-onian function of the full system can be written as

= φ2

2LL2+ VFe

∫ BFe

oHFe(b) db + VL

1

2μoB2

L

+ Va-g1

2μoB2

a-g + VMSMA

·(

E

2(ε − γz)2 + K12 · z · (1 − z) +

∫ B

oH(b) db

)

+ p2z

2mz

+ p2ε

2mε

+ 1

2mp2

x + mgx (29)

By using the port-Hamiltonian representation, we have toefine different vectors such as the generalized coordinates q,

omentums p and Lagrange multipliers vector λ:

qT = [qc, DFe, DL, Da-g, D, z, ε, x],

pT = [φ, BFeSFe, BLSL, Ba-gSa-g, BS, pz, pε, px],

nd Ac

λ

mi

A

1coc

•

•

•

tpfleLpt

•

•

•

•

•

7

af

7

enaw

The second experiment is the measurement of the currentdynamical response versus time for a 100 V voltage step. Thismeasurement is reported in Fig. 10 compared with the simula-tions resulting from the identification procedure.

J.-Y. Gauthier et al. / Sensors a

T = (λ1 λ2 λ3) (30)

We also have to define different matrix such as the dissipationatrix R, the kinematic constraints matrix A and the external

nput matrix B:

R =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

r 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0(∂Physt/∂z)

z0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 f

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

T =

⎛⎜⎝−N 1 1 0 0 0 0 0

0 0 1 −1 −1 0 0 0

0 0 0 0 0 0 −lo 1

⎞⎟⎠ ,

BT = (1 0 0 0 0 0 0 0

)(31)

The expression of the Hamiltonian function permits to obtain6 Hamilton equations. Among these 16 equations, 8 are asso-iated with the time rate of coordinates and 8 with the time ratef momentums. The 8 equations associated with the time rate ofoordinates gives 8 definitions of variables:

the first Hamilton equation is the definition of the inductanceLL2:

LL2qc = φ (32)

the four following Hamilton equations are the definition ofthe magnetic fields Hi for i ∈ {Fe,L,a-g,Ø}:

Di = liHi (33)

the three following Hamilton equations are the definition ofthe relations between the momentum pi and the velocities qi

for qi ∈ {z, ε, x}:

pqi = mqi qi (34)

The eight equations associated with the time rate of momen-ums give eight relations that can be rewritten to obtain fourhysical equations (one voltage Kirchoff’s law, two magneticux conservations and one Newton’s law), one constitutivequation for the MSMA and finally the value of the threeagrange multipliers (as denoted before, due to some sim-lifications, pz, pε and φ can be considered as nearly equalo 0):

Fowu

tuators A 141 (2008) 536–547 543

the dynamic electrical equation (Voltage Kirchoff’s Law):

u = rI + NBFeSFe (35)

two equations for the conservation of magnetic fluxes in themagnetic circuit (see Fig. 8):

BLSL = BFeSFe − Ba-gSa-g, BS = Ba-gSa-g (36)

the dynamic equation of the load (Newton’s law):

mx = −mg − f x − Soσ, (37)

the quasi-static behaviour of the MSM material (the constitu-tive equation):

πf∗(z, z) = −γz + K12(1 − 2z) + ∂∫ B

o H(b) db

∂z. (38)

the values of the three Lagrange multipliers:

λ1 = BFeSFe, λ2 = −Ba-gSa-g, λ3 = −Soσ (39)

. Parameters identification

In order to determine some parameters of the MSMA materialnd some of the complete system, experimental measurementsollowed by an identification procedure have been performed.

.1. Magnetic circuit characteristics

The magnetic circuit parameters were identified by twoxperiments. The first one is the measurement of the airgap mag-etic field versus coil current without any MSMA sample in theirgap. These measurements are reported in Fig. 9 comparedith the simulations resulting from the identification procedure.

ig. 9. First experiment for the magnetic circuit identification procedure (with-ut MSMA): magnetic field versus coil current (experimental measurements: ‘+’hen the current is increasing and ‘×’ when the current is decreasing; simulationsing identified parameters: solid line).

544 J.-Y. Gauthier et al. / Sensors and Actuators A 141 (2008) 536–547

F(m

7

rhawstutbtoata

Fz

(

Fc

MtwicTBfip

7

ig. 10. Second experiment for the magnetic circuit identification procedurewithout MSMA): coil current versus time for a 100 V voltage step (experimentaleasurements: cross, ×; simulation using identified parameters: solid line).

.2. MSMA quasi-static characteristics

In order to find the χa and χt parameters, the dynamicalesponse of the magnetic circuit including the MSMA sampleas been measured. A 100 V voltage step is applied to the coilnd the current is measured. In a first experiment, the MSMAas blocked to keep the martensite volume fraction z = 0. In a

econd experiment, the MSMA was free of stress and initializedo z = 1. Fig. 11 shows experimental results compared with sim-lations using the identified parameters. These results show thathe response changes according to the z value. This property cane exploited in a self-sensing application. Indeed, by measuringhe current change, the volume fraction z and the displacement

f the load can be estimated. An example of such an applications sensor has already been studied in [17]. It uses the change ofhe inductance value at low magnetic field as an information forposition sensor application.

ig. 11. Self-sensing feasibility: current versus time for a 100 V voltage step for= 1 (modelling: solid line; experimental results: cross points) and for z = 0

simulation: dashed line; experimental results: circles).

m

s

8

Smdda1

apvTa

dv

ig. 12. Mechanical stress versus strain with and without a magnetic field for ayclic charging (simulation: solid line; experimental results: cross points).

A compression test was performed in order to identify allSMA mechanical parameters. Fig. 12 compares experimen-

al and simulation results for the mechanical stress versus strainith and without a magnetic field. For a mechanical cyclic charg-

ng without a magnetic field, the mechanical behaviour is notlosed (the ending point of the curve is not the starting point).his is due to the very large hysteresis of the MSMA material.ut for a mechanical cyclic charging with a sufficient magneticeld (600 kA/m), the curve is closed (the ending and the startingoints coincide).

.3. Load characteristics

The parameter f was identified thanks to a dynamical responseeasurement reported in Fig. 14.Table 4 summarizes all the parameters values directly mea-

ured or identified and used afterwards.

. Experimental validation of the model

Numerical computations were conducted with the Matlabimulink® software. Our simulation uses a finite differencesethod to compute the dynamical response of the complete

evice. Some S-functions programmed in C language wereeveloped to increase the computing speed. The solver usesclassical Euler integration scheme with a sampling time of

0 �s.In a first time, a sufficiently slow voltage ramp was applied

s an input to verify the quasi-static model behaviour. Fig. 13resents the voltage, current and displacement of the actuatorersus time for the experimental and the simulation values.he maximum reachable displacement is about 550 �m after

current ramp saturated at 1 A.

In a second time, a voltage step is applied to extract theynamical behaviour of this system. Fig. 14 reports also theoltage, current and displacement versus time (this graph uses

J.-Y. Gauthier et al. / Sensors and Ac

Table 4Summary of the device parameters directly measured or identified using someexperiments

Measured la-g = 0.65 mmlMSMA = 3.2 mmmgrav = 1.44 kgminertia = 2.32 kgS = 5 mm × 20 mmSair = S

r = 61.8 �

N = 1500 turnslo = 20 mm

Identified using Fig. 9 and Fig. 10 SFe = 5 mm × 23 mmSL = 30 × S

lL = 27 mmlFe = 3 mm

Identified using Fig. 11 χt = 1χa = 40

Identified using Fig. 12 γ = 0.055λC = 130, 000 J/m3

πcr = 0 J/m3

K12 = 35, 000 J/m3

μoMs = 0.65 TE = 5e8 Pa

Identified with dynamical results f = 100

Iron parameters χFe = 7700

an

trsso

am

9

Fs

9

eitebiaeta

claortrcVlgies (kinetic Tload and potential Vload) are quite small compared

μoMsFe = 2.03 T

smaller time range). The maximum reachable displacement isow about 750 �m for a 1 A current step.

One have to note that the dynamical effect combined withhe nonlinear MSMA behaviour permits to obtain a larger

eachable strain in the dynamic mode than in the quasi-tatic mode. This is due to the decrease of the compressivetress applied to the MSMA sample when the accelerationf the load appears in dynamic mode. The simulation results

ttb

Fig. 13. Quasi-static behaviour of the system: voltage, current and displacem

tuators A 141 (2008) 536–547 545

ppear reasonably accurate compared with the experimentaleasurements.

. Discussion on the energy distribution

The different energies previously computed are reported inig. 15 first column for the quasi-static mode and in Fig. 15econd column for the dynamic mode.

.1. Quasi-static mode

The first top left graph of Fig. 15 presents the electricalnergy Wext supplied to the device. As the energy exchanges dWext = u(t) · dqc, the energy supply rate corresponding tohe the electrical power is therefore Wext = u(t) · I(t). The heatxchange was not measured on the bench but the dissipationy Joule effect can be computed as Qjoule = ∫ t

0 R1(q1) dt ands reported in the same figure. As it can be seen, the two plotsre nearly superposed and therefore the main part of the supplynergy is dissipated as heat losses into the coil. This confirmshat MSMA as well as classical shape memory alloys are notttractive materials from the efficiency point of view.

The rest of the available energy is divided into the coil andore magnetic energies (recoverable energies, see the second topeft graph of Fig. 15) and into an energy transfer to the MSMAnd to the load (see the third top left graph of Fig. 15). A partf the MSMA energy is lost in the hysteretical loop of the mate-ial behaviour Qhyst = ∫ t

0 Physt dt when the other is convertedhrough the electromechanical energy conversion process. Theesult of this energy conversion is then distributed as a vis-ous friction process Qviscous = ∫ t

0 R2(x) dt, a potential energyload(x) and a kinetic energy Tload(px). We can see in the bottom

eft graph of Fig. 15 that the practical available mechanical ener-

o the input energy. The MSMA elastic energy Fmech − Fint,he kinetic energy Tload, and viscous losses Qviscous are constantecause of the quasi-static mode.

ent vs. time (simulation: dotted line; experimental results: solid line).

546 J.-Y. Gauthier et al. / Sensors and Actuators A 141 (2008) 536–547

aceme

9

ilqiw

MTet

Fig. 14. Dynamic behaviour of the system: voltage, current and displ

.2. Dynamic mode

The first top right graph of Fig. 15 shows also clearly that,n this system, the supplied energy is mainly dissipated by heat

osses into the coil. The differences between dynamic mode anduasi-static mode are now discussed. In Fmech, the martensitenteraction energy Fint = K12 · z · (1 − z) reaches a maximumhen the z value is equal to 0.5. An energy transfer between the

cscc

Fig. 15. Computations of the different energies (J) vs. time: in the stat

nt vs. time (simulation: dotted line; experimental results: solid line).

SMA elastic energy (E/2) · (ε − γ · z)2 and the kinetic energyload exists. Actually, this elastic energy increases when somenergy is supplied to the system, then decreases to a lower valuehan beginning because of kinetic energyTload. At this time,Vload

an increase to a higher value than in quasi-static mode. The dis-ipation is more important for Qhyst and Qviscous in the dynamicase than in the quasi-static but Qjoule is lower in the dynamicase because the time range is smaller than in quasi-static mode.

ic mode (left column) and in the dynamic mode (right column).

nd Ac

9

peeppdmod

1

epoacnmescna

R

[

[

[

[

[

[[

[

[

B

JmiFaidb

A1iFeodra

J11vm2tFms

N‘tSsdviec

CFFaB

J.-Y. Gauthier et al. / Sensors a

.3. Energy efficiency

Less than 8–11 mJ are recovered by the load for 5–20 J sup-lied to the actuator. These results clearly show a poor actuatorfficiency. Because the main energetic losses are due to the Jouleffect, an efficient actuator is an actuator which can hold a dis-lacement value without keeping a current into the coil. Thisroblem can be partially solved by using a Push–Pull actuatoresign working by voltage pulses: two MSMA samples and twoagnetic circuits are used in an antagonistic way in order to

btain a multi-stable actuator. This kind of actuator was alsoesigned and studied by the authors in [18].

0. Conclusion and perspectives

In this paper, a Lagrangian formalism and its Hamiltonianxtension combined with the thermodynamics of irreversiblerocesses were used to obtain the complete dynamical modellingf a mechatronic device including magnetic shape memorylloys. This model permits to simulate the complete dynami-al behaviour of this system even if the MSMA behaviour isonlinear and hysteretic. The self-sensing feasibility and an esti-ation of the different energies were given. All these results are

xperimentally validated and a satisfactory accuracy betweenimulations and experimental results is observed. The modelan be used confidently in the near future in order to design aew controller using some energy-based control techniques suchs it is proposed in [3–6].

eferences

[1] J.L. Pons, Emerging Actuator Technologies: A MicromechatronicApproach, John Wiley and Sons, 2005.

[2] J.Y. Gauthier, C. Lexcellent, A. Hubert, J. Abadie, N. Chaillet, Modelingrearrangement process of martensite platelets in a magnetic shape memoryalloy Ni2MnGa single crystal under magnetic field and (or) stress action,J. Intell. Mater. Syst. Struct. 18 (3) (2007) 289–299.

[3] R. Sepulchre, M. Jankovic, P. Kokotovic, Constructive Nonlinear Control,Springer-Verlag, 1997.

[4] R. Ortega, A. Loria, P.J. Nicklasson, H.S. Ramirez, Passivity-Based Controlof Euler-Lagrange Systems, Springer-Verlag, 1998.

[5] A.V. der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control,Springer-Verlag, 2000.

[6] A. Astolfi, A.J.V. der Schaft (Eds.), Lagrangian and Hamiltonian meth-ods for nonlinear control, Special Issue of Eur. J. Control 10 (5) (2007),Hermes.

[7] K. Ullakko, J.K. Huang, C. Kantner, R.C. O’Handley, V.V. Kokorin, Largemagnetic-field-induced strains in Ni2MnGa single crystals, Appl. Phys.Lett. 69 (1996) 1966–1968.

[8] R.D. James, M. Wuttig, Magnetostriction of martensite, Philos. Mag. A 77(1998) 1273–1299.

[9] S. Leclercq, C. Lexcellent, A general macroscopic description of the ther-momechanical behavior of shape memory alloys, J. Mech. Phys. Solids 44(6) (1996) 953–980.

10] L. Hirsinger, C. Lexcellent, Modelling detwinning of martensite plateletsunder magnetic and (or) stress actions in Ni–Mn–Ga alloys, J. Magn. Magn.Mater. 254–255 (2002) 275–277.

11] C. Lanczos, The Variational Principe of Mechanics, 4th ed.,Dovers Publications, Reprint of 1970 University of Toronto Press,1986.

12] H. Goldstein, Classical Mechanics, 2nd ed., Addison Wesley,1980.

(hota

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13] S.H. Crandall, D.C. Karnopp, E.F. Kurtz, Jr., D.C. Pridmore-Brown,Dynamics of Mechanical and Electromechanical Systems, Krieger Pub-lishing, Malabar, Reprint of 1968 McGraw Hill, 1982.

14] A. Preumont, Mechatronics: Dynamics of Electromechanical and Piezo-electric Systems, Springer Verlag, 2006.

15] B. Nogarede, Electrodynamique Appliquee, Dunod, 2005.16] P. Hammond, Energy Methods in Electromagnetism, Clarendon

Press/Oxford University Press, 1981.17] I. Suorsa, E. Pagounis, K. Ullakko, Position dependent inductance based

on magnetic shape memory materials, Sensors Actuators A 121 (2005)136–141.

18] J.Y. Gauthier, A. Hubert, J. Abadie, C. Lexcellent, N. Chaillet, Multistableactuator based on magnetic shape memory alloy, in: ACTUATOR 2006,10th International Conference on New Actuators, Bremen, Germany, 2006,pp. 787–790.

iographies

ean-Yves Gauthier graduated in electrical engineering from the Ecole Nor-ale Superieure, Cachan, France in 2003 and received the DEA (MSc) degree

n Applied Informatics and Automation from the University of Franche-Comte,rance in 2004. He is currently PhD student in the University of Franche-Comtend works at the Laboratoire d’Automatique de Besancon (LAB). His researchnterests are the study of active materials in micro-robotic field and he is nowa-ays focused on the modelling and the control of magnetic shape memory alloyased actuators.

rnaud Hubert received the engineer diploma in mechanical engineering in996, the DEA (MSc) degree in system control in 1997 and the PhD degreen 2000, all from the Compiegne University of Technology (UTC, Compiegne,rance). His PhD research fields was about the vibrations and noise control inlectrical drives. Since 2002, he has been an Associate Professor at the Universityf Franche-Comte, Besancon, France working at the Laboratoire d’Automatiquee Besancon (LAB). His current research interests are the design of micro-obotic and micro-mechatronic systems especially focused on the modellingnd control of actuators using active materials.

oel Abadie received the engineer diploma in mechanical engineering in994 from the ENIT, Tarbes, France, the DEA (MSc) and PhD degrees in997 and 2000 both in applied informatics and automation from the Uni-ersity of Franche-Comte, France. His PhD research fields was about theodelling and design of microactuator using shape memory alloys. Since

002, he has been an Research engineer at the French research National Insti-ute (CNRS) working at the Laboratoire d’Automatique de Besancon (LAB),rance. His current research interests are the design of micro-robotic and micro-echatronic systems especially focused on the design of micro- and nano-forces

ensors.

icolas Chaillet received his BSE degree in Electrical Engineering from theEcole Nationale Superieure de Physique’, Strasbourg, France, in 1990, andhe PhD degree in Robotics and Automation from the University Louis Pasteur,trasbourg, France, in 1993. In 1995 he became associate professor at the Univer-ity of France-Comte in Besancon working at the ‘Laboratoire d’Automatiquee Besancon’, LAB UMR CNRS 6596. Since 2001, he is professor at the Uni-ersity of Franche-Comte and the head of the LAB since 2007. His researchnterests are in microrobotics and more generally in micromechatronics fields,specially in micromanipulation, microgrippers, smart materials, modelling andontrol of microactuators.

hristian Lexcellent received the engineer diploma from the ENSMA, Poitiers,rance, the AEA (MSc) in thermodynamics from the University of Poitiers,rance, both in 1972. He received his PhD and his DSc in thermodynamicsnd material sciences in 1977 and 1987 from the University of Franche-Comte,esancon, France. He became Assistant Professor (1984), Associate Professor

1986) then Professor in 1988 at the ENSMM, Besancon, France. He is at theead of the material sciences research group in the Mechanical Departmentf FEMTO-ST, CNRS, Besancon, France. His research interests concern thehermodynamic modelling of material and he is a specialist of the shape memorylloys behaviour.