and structures nonlinear model for galfenol cantilevered ... · tions. the linear euler–bernoulli...

Original Article

Journal of Intelligent Material Systemsand Structures2014, Vol 25(2) 187–203� The Author(s) 2013Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X13489600jim.sagepub.com

Nonlinear model for Galfenolcantilevered unimorphs considering fullmagnetoelastic coupling

Liang Shu1,2, Leon M Headings2, Marcelo J Dapino2, Dingfang Chen3 andQuanguo Lu4

AbstractThis article presents a fully coupled, nonlinear model for the dynamic response of Galfenol-driven unimorph actuators ina cantilever configuration. The hysteretic magnetomechanical behavior of Galfenol is modeled using a discrete energy-averaged model, and the structural behavior of the unimorph is modeled using the finite element method. The weakform equations that describe bending of the unimorph are obtained using the principle of virtual work. Since the localstrain and stress are nonlinearly coupled with both the vertical and horizontal displacements, a nonlinear solver is devel-oped to approximate the coupled finite element equations. The nonlinear solver is verified against the analytical solutionand experimental data for the case of a passive beam. The analytical solution is obtained using beam theory for free andharmonic responses. The analytical model and experimental data verify that the nonlinear solver correctly quantifies thefirst natural frequency of the composite beam. The numerical simulations match the analytical solutions for both freeand harmonic responses. Finally, the dynamic response of the nonlinear magnetoelastic model is investigated and experi-mentally validated from 0.1 to 500 Hz, the range in which the model is accurate without the need for adjustableparameters.

KeywordsCantilevered unimorph actuator, laminated bender actuator, Galfenol, dynamic response, nonlinear coupling, finite ele-ment analysis

Introduction

Galfenol is a magnetostrictive material which can besafely operated in tension, compression, and shear(Evans and Dapino, 2008). Due to their ability to oper-ate in large deformation regimes without special treat-ments, laminated bender structures utilizing Galfenoldrivers provide an attractive alternative to brittle mate-rials like lead zirconate titanate (PZT) and Terfenol-D(Baillargeon and Vel, 2005; Shu et al., 2010). Wanget al. (2010) proposed a magnetoelectric cantilever com-posed of a layer of Galfenol and a layer of PZT-5H.This configuration offers advantages for potential usein microsurgical ablation tools and cutting tools formachining. Ueno and Higuchi (2008) presented a2-degree-of-freedom (DOF) microbending actuatordriven with Galfenol. An arrangement of orthogonal–parallel Galfenol beams creates bending deformation inboth the x- and y-directions. Basantkumar et al. (2006)developed a microelectromechanical system (MEMS)actuator by integrating thin-film Galfenol with glasscover slides. Using a capacitance technique, the

saturation magnetostriction of the film was measuredto be 147 ppm (parts per million). Downey and Flatau(2005) investigated the magnetoelastic bending ofGalfenol for sensor applications. When a sinusoidalload was applied to both polycrystalline and single-crystal rods, the magnetic induction was found toincrease with applied magnetic bias.

1The Key Laboratory of Low-Voltage Apparatus Intellectual Technology

of Zhejiang, Wenzhou University, Wenzhou, China2Department of Mechanical and Aerospace Engineering, The Ohio State

University, Columbus, OH, USA3School of Logistics Engineering, Wuhan University of Technology,

Wuhan, China4School of Mechanical and Electrical Engineering, Nanchang Institutite of

Technology, Nanchang, China

Corresponding author:

Marcelo J Dapino, Department of Mechanical and Aerospace Engineering,

The Ohio State University, E307 Scott Laboratory, 201 West 19th

Avenue, Columbus, OH 43210, USA.

Email: [email protected]

To investigate the structural and magnetoelastic cou-pling in laminated Galfenol benders, advanced modelsare required. Models for magnetostriction-induced rep-resent one class of advanced models in which tip deflec-tion is investigated in terms of magnetostriction,thickness ratio, and elastic properties (Du Tremolet deLacheisserie and Peuzin, 1994; Gehring et al., 2000;Guerrero and Wetherhold, 2003; Quandt and Ludwig,2000). Du Tremolet de Lacheisserie and Peuzin (1994)investigated the deformation of a bimorph consisting ofa nonmagnetic substrate and a magnetic thin film. Theirmodel is formulated for a thin film deposited on a non-magnetic substrate. Since the film thickness is assumedto be much smaller than the substrate thickness, theinternal stress in the film changes very little throughoutthe thickness, and the mechanical energy in the activelayer can be assumed to be constant. Models that canbe applied to any thickness ratio have been presented byGehring et al. (2000) and Guerrero and Wetherhold(2003) based on total internal energy minimization.However, these models assume that the magnetostric-tion of the active layer is known a priori. The nonlinearstress and field dependency of the magnetoelastic pro-cesses in the structural bending cannot be quantified.

Other models aim to determine magnetomechanicalcoupling from magnetostriction-induced bending. A non-linear optimal control strategy is presented by Smith(1998) in which a nonlinear magnetostrictive actuatormodel is developed to attenuate transverse beam vibra-tions. The linear Euler–Bernoulli equation is employedto model the structural bending of the cantilever. Jia etal. (2006) developed a nonlinear model to characterizethe magnetization and magnetostriction processes ofbimorph films. However, the change of internal stressduring magnetization and magnetostriction is neglected,making the model accurate only for low magnetic fields.Datta et al. (2008) developed a magnetomechanicalmodel for sensors in laminated plates. This model can beused to analyze the response of the active Galfenol layerto quasi-static axial and shear forces and bendingmoments. An actuation model was proposed by Datta etal. (2009) to describe magnetostrictive strains and stressin laminated plates under quasi-static magnetic fields.The contribution of this work was the combination ofthe magnetomechanical material model with the platestructural model for laminated structures; however, itcannot be used to examine the dynamic response of thesystem. A bidirectionally coupled magnetoelastic modelwas developed by Mudivarthi et al. (2008) in which theArmstrong model was combined with finite element elas-tic and magnetic models. The model describes materialresponses considering the spatial variation in magneticfield and stress. To investigate the dynamics of laminatedstructures with intrinsic magnetomechanical coupling, anonlinear model that accounts for dynamic responsetogether with the nonlinear stress and field dependencyof magnetostriction is required.

This article addresses the dynamic response ofGalfenol-driven cantilevered unimorphs. A discreteenergy-averaged model is employed to describe the hys-teretic magnetomechanical behavior of Galfenol, andthe finite element method is used to model the structuralbehavior of the beam. Since the finite element model istime dependent, this model offers the advantage of sol-ving dynamic problems in active laminated structures.The structural model is implemented in two dimensions(2D) by expressing the strain in terms of horizontal andvertical displacements. A discrete energy-averaged modelis used to quantify the magnetostriction as a function ofinduced stress and field, both of which are simultane-ously coupled with output displacements. In order tosolve the nonlinear coupled equations, a numerical solveris developed to solve for vertical and horizontal displace-ments. The nonlinear stress and field dependency of mag-netostriction is investigated, and experimental validationis conducted over the frequency range of 0.1–500 Hz.Simulation and experimental results show that the pro-posed model can predict the first natural frequency ofthe composite beam.

Composite beam model

Geometric considerations

A composite beam consisting of a Galfenol layerbonded to a nonmagnetic substrate is clamped at oneend, while the other end is free (Figure 1). The x� y

plane of the coordinate system is set on the midplaneof the cantilever; the z-axis is perpendicular to theactive layer. The beam has length L and width b; theGalfenol layer has thickness tg, modulus Eg, and den-sity rg and is assumed to be perfectly bonded to thesubstrate of thickness ts, modulus Es, and density rs.

Principle of virtual work

When a magnetic field H is applied along x, the magne-tostriction induces bending of the cantilever. The stressin the substrate ss is assumed to behave in a linearlyelastic fashion relative to the strain es, ss =Eses. Thetotal strain induced in the Galfenol layer eg is assumedto consist of a magnetostrictive component superim-posed on the elastic response of the material. The inter-nal stress in the Galfenol layer can thus be expressed as

sg =Eg eg � l sg,H� ��

ð1Þ

where the magnetostriction l sg,H� �

depends on bothinternal stress and magnetic field. For the magnetostric-tion-induced bending, the total strain is the superposi-tion of the extensional and bending strains

e=∂u t, xð Þ

∂x� z

∂2v t, xð Þ∂x2

: ð2Þ

188 Journal of Intelligent Material Systems and Structures 25(2)

Here, u t, xð Þ denotes the horizontal displacement of themidplane and v t, xð Þ denotes the vertical displacement.The principle of virtual work is considered in order torelate the magnetostriction to the horizontal and verticaldisplacements. Recognizing that the external virtual workis zero when no external load is applied, one obtains

� dWi � dWe = � dWi = 0: ð3Þ

The subscripts i and e denote internal and externalcomponents of the virtual work, respectively. The inter-nal virtual work arises from internal stress, dynamicinertia, and damping effects. The virtual work fromstress has the following form (see Appendix 1)

dWs =

ðL0

ðA

sdedAdx

=

ðL0

ðAg

sgdegdAgdx+

ðL0

ðAs

ssdesdAsdx

=EgIg

ðL0


d∂2v t, xð Þ∂x2

dx� EgQg

ðL0


d∂u t, xð Þ

∂xdx

� EgQg

ðL0

∂u t, xð Þ∂x


dx+EgAg

ðL0

∂u t, xð Þ∂x

d∂u t, xð Þ

∂xdx

+Eg

ðL0

ðAg

ld∂2v t, xð Þ∂x2

zdAgdx� Eg

ðL0

ðAg

ld∂u t, xð Þ

∂xdAgdx

+EsIs

ðL0



dx� EsQs

ðL0


d∂u t, xð Þ

∂xdx

� EsQs

ðL0

∂u t, xð Þ∂x


dx+EsAs

ðL0

∂u t, xð Þ∂x

d∂u t, xð Þ

∂xdx

ð4Þ

In order to calculate the internal virtual work fromdynamic inertia, the d’Alembert force, which is afunction of the horizontal and vertical accelerations isconsidered. Also, Kelvin–Voigt damping, a functionof the horizontal and vertical velocities, is employedto calculate the internal virtual work from dampingeffects

dWr=

ðL0

ðA

r∂2u t, xð Þ

∂t2dudAdx+

ðL0

ðA

r∂2v t, xð Þ

∂t2dvdAdx

dWc=

ðL0

ðA

c∂u t, xð Þ

∂tdudAdx+

ðL0

ðA

c∂v t, xð Þ

∂tdvdAdx

ð5Þ

where the subscripts r and c denote virtual work frominertia and damping, respectively. The weak form equa-tion of the magnetostriction-induced bending is there-fore given as follows

EgIg

ðL0



dx� EgQg

ðL0


d∂u t, xð Þ

∂xdx

�EgQg

ðL0

∂u t, xð Þ∂x


dx+EgAg

ðL0

∂u t, xð Þ∂x

d∂u t, xð Þ

∂xdx

+EsIs

ðL0



dx� EsQs

ðL0


d∂u t, xð Þ

∂xdx

�EsQs

ðL0

∂u t, xð Þ∂x


dx+EsAs

ðL0

∂u t, xð Þ∂x

d∂u t, xð Þ

∂xdx

Figure 1. Beam geometry and coordinate system used to describe the cantilevered unimorph. An axial magnetic field applied tothe Galfenol layer induces bending of the cantilever.

Shu et al. 189

+

ðL0

ðA

r∂2u t, xð Þ

∂t2dudAdx+

ðL0

ðA

r∂2v t, xð Þ

∂t2dvdAdx

+

ðL0

ðA

c∂u t, xð Þ

∂tdudAdx+

ðL0

ðA

c∂v t, xð Þ

∂tdvdAdx

= � Eg

ðL0

ðAg


zdAgdx+Eg

ðL0

ðAg

ld∂u t, xð Þ

∂xdAgdx

ð6Þ

It can be seen from equation (6) that both u t, xð Þ andv t, xð Þ depend on l sg,H

� �. From equations (1) and (2),

the internal stress s can be seen to also depend on theoutput displacements u t, xð Þ and v t, xð Þ. These equationsdescribe the nonlinear coupling in magnetostriction-induced. In order to solve for u t, xð Þ and v t, xð Þ fromequation (6), the weak form equation is discretized inthe spatial domain.

Finite element discretization

The beam is discretized into Ne elements of length le,each with two nodes, for a total of Nn =Ne + 1 nodes(see Figure 2(a)). Each node has 3 DOFs. The first twoare vertical displacement v t, xð Þ and rotation ∂v t, xð Þ=∂x,which contribute a total of Nv

q = 2Nn DOFs to the dis-cretized beam; the other nodal DOF is horizontal dis-placement u t, xð Þ, which contributes a total of Nu

q =Nn

DOFs. The values of the DOFs associated with the ver-tical displacement for a single element are denoted asqv

e, in which the first two components are the verticaldisplacement and rotation of the left node and the sec-ond two are the vertical displacement and rotation ofthe right node. The global notation is Qv. The values ofthe DOFs associated with the horizontal displacementfor a single element are denoted as qu

e , where the firstentry is the horizontal displacement of the left node and

the second entry is the horizontal displacement of theright node. The global notation isQu.

Hermite shape functions are chosen to ensure thecontinuity of v t, xð Þ and ∂v t, xð Þ=∂x. The local spatialcoordinate is taken as j, ranging from 21 to 1, and thevertical displacement is interpolated as follows

ve =H � qve = H1,

le

2H2,H3,

le

2H4

� �qv

e, 1, qve, 2, qv

e, 3, qve, 4

h i>ð7Þ

where H is the shape function vector. The horizontaldisplacement requires only continuity of the zeroth deri-vative. Linear shape functions are chosen, and the hori-zontal displacement is interpolated as follows

ue =N � que = ½N1,N2� qu

e, 1, que, 2

h i>ð8Þ

where N is the linear shape function vector. The com-ponents of H and N are shown in Appendix 2. The dis-cretized weak form equation can be obtained bysubstituting equations (7) and (8) into equation (6) (seeAppendix 2 for details)

Xe

qve

� > � kve

� �dqv

e

� � qv

e

� > � kuve

� �dqu

e

� � qu

e

� > � kuve

� �>dqv

e

� + qu

e

� > � kue

� �dqu

e

� +X

e

€que

� > � mue

� �dqu

e

� + €qv

e

� > � mve

� �dqv

e

� +X

e

_que

� > � cue

� �dqu

e

� + _qv

e

� > � cve

� �dqv

e

� =X

e

� fl, ve

� �dqv

e

� + fl, u

e

� �dqu

e

� ð9Þ

Applying variational principles, the assembled sys-tem to be solved is

M½ �f €Quvg+ C½ �f _Q

uvg+ K½ �fQuvg= Fuvl

� �ð10Þ

where

M½ �=mu

e 0

0 mve

� �, C½ �=

cue 0

0 cve

� �,

K½ �= kue � kuv

e

� �>� kuv

e

� �kv

e

" #, Fuv

l

� �=

fl, ue

�fl, ve

" #,

fQuvg=qu

e

qve

� �

It is seen in equation (10) that the global massmatrix M½ � and the global damping matrix C½ � are diag-onal, which is consistent with the d’Alembert force andKelvin–Voigt damping considered here since inducedforces are proportional to the mass and damping. Thestiffness matrix K½ � is nondiagonal because the internalstress is coupled with both the vertical and horizontal

(a)

(b)

1 2 3 Ne

e,4

Figure 2. (a) Discretization of the beam and (b) degrees offreedom of an element.


displacements. Array fQuvg is the generalized displace-ment, while f _Quvg and f €Quvg denote the generalizedvelocity and acceleration, respectively. The excitationvector Fuv

l

� �is a function of the magnetostriction, which

depends on the induced strain and stress in the actuator.This creates nonlinear coupling in the magnetoelasticprocess. Furthermore, since inertial and damping effectsare considered, this model can be used to solve dynamicproblems in laminated structures. The general frame-work developed here can also be applied to otherdynamic magnetomechanical coupling problems.

Three-dimensional hysteretic constitutive law

The magnetostriction induced by the Galfenol elementmust be known to solve for the state variable fQuvg. Inorder to account for field and stress dependency, a dis-crete energy-averaged model (Evans and Dapino, 2010)is employed to describe the hysteretic magnetomechani-cal behavior of Galfenol. This model improves recentwork (Zhou and Zhou, 2007), in which only anhystere-tic responses were considered in the active layer of alaminated structure. This work was limited to one-dimensional (1D) analysis and did not account formagnetocrystalline anisotropy. In the discrete energy-averaged model, local anisotropy energy is defined, andthree-dimensional (3D) magnetomechanical rsponsesare described through minimization of the total freeenergy of Galfenol. Hysteresis is considered by formu-lating the evolution equation of volume fraction interms of both stress and magnetic field. For a materialcomposed of a collection of Stoner–Wohlfarth (SW)particles (Stoner and Wohlfarth, 1948) in thermody-namic equilibrium and having r possible orientations,the bulk magnetostriction l is the sum of the magne-tostriction li in each direction, weighted by the volumefraction zi of particles in each orientation

l=Pr

i= 1

li mið Þzi; ð11Þ

where mi represents the magnetic orientations of theSW particles. Rather than use a globally defined energyexpression and sum over all possible orientations(Atulasimha et al., 2008), Evans and Dapino (2010)defined the energy around local energy minima corre-sponding to the easy crystallographic directions. Onlythe local minima are included in the summation. Forcubic materials, these directions are typically the 6100h i directions or 8 111h i directions or both setstogether for a total of 14 directions. The anisotropyenergy for these directions is analogous to a mechanicalspring with initial configuration mi

0 (corresponding tothe easy crystallographic directions) and stiffnessrepresented by the anisotropy coefficient Ki. In cubicmaterials, all the 100h i directions have the same coeffi-cient, Ki =K100, as do the 111h i directions, Ki =K111.

The local energy for each minimum mi therefore hasthe following form

Ei = 12

Ki mi �mi0

2 � l mið Þ � T� m0MSmi �H ð12Þ

where T is the internal stress tensor, H is the appliedfield, and MS is the saturation magnetization. The mag-netic orientations mi can be calculated by minimizationof the total free energy (12) with the constraintC = mij j � 1= 0 (since mi is a unit vector). In order tosolve for mi analytically, the constraint is relaxedthrough linearization about the easy direction mi

0 byassuming mi

0 �mi’mi �mi = 1. The solution for themagnetic orientations is

mi = Ki� ��1

Bi +1�mi

0� Kið Þ�1

Bi

mi0� Kið Þ�1

mi0

� �mi

0

� �; ð13Þ

where Ki is the magnetic stiffness matrix and Bi is aforce vector

Bi = Kimi +m0MSH;

Ki =Ki � 3l100T1 �3l111T4 �3l111T6

�3l111T4 Ki � 3l100T2 �3l111T5

�3l111T6 �3l111T5 Ki � 3l100T3

24

35:

This solution assumes that domains rotate a small angleaway from the easy directions. The error associatedwith this assumption is very small since particles thathave rotated far from the easy axes have much smallervolume fractions than particles that have not rotatedfar. Thus, the magnetostriction calculated from equa-tion (11) mostly depends on the particles that are rela-tively close to the easy directions. In order to calculatethe magnetostriction, the volume fraction zi must bequantified.

Armstrong (2003) proposed an evolution law for aunidirectional applied field

Dzi = 1k

zian � zi

� �DHj j; ð14Þ

in which hysteresis arises from energy losses as domainwalls overcome pinning sites. This model only explainshysteresis when the magnetic field is varied at constantstress and only applies to one-dimensional problems.Since both stress and field change the domain volumefractions through domain wall rotation, the irreversiblechanges of volume fractions due to 3D magnetic fieldand stress are

Dziirr =

1

kzi

an � zi� ��

m0MS

X3

i= 1

DHi +3

2l100

X3

i= 1

DTi + 3l111

X6

i= 4

DTi

�; ð15Þ

where k is a pinning site density constant that charac-terizes the energy loss associated with domain wall

Shu et al. 191

rotation. The anhysteretic value of the volume fractionzi

an can be calculated using an energy-weighted average,

zian =

e�Ei=OsPr

i= 1

e�Ei=Os

; ð16Þ

where Os is a smoothing factor and Ei is the free energyassociated with orientation mi. Following the theory byJiles and Atherton (1986), the total increment in vol-ume fraction is the sum of reversible and irreversiblecomponents

Dzi = cDzian + 1� cð ÞDzi

irr; ð17Þ

where c is a nondimensional constant which quantifiesthe reversible processes during domain wall motion. Ifc= 1, the total volume fraction corresponds to reversi-ble processes (Dzi =Dzi

an), whereas if c= 0, the domainprocesses are fully irreversible (Dzi =Dzi

irr).

Unidirectional, hysteretic constitutive law

The magnetostriction expression (11) is 3D. In order toapply equation (11) to unidirectional bending, a unidir-ectional constitutive law is needed. Assume that a mag-netic field or stress is applied in the direction u withrespect to the crystal frame. Then, the auxiliary vectoruT = ½u2

1 u22 u2

3 u1u2 u2u3 u3u1�> relates the 3D magne-tostriction to the longitudinal component li in theapplication direction

li = us � li =1

2mi � R �mi ð18Þ

where

R= 3

l100us, 1 l111us, 4 l111us, 6

l111us, 4 l100us, 2 l111us, 5

l111us, 6 l111us, 5 l100us, 3

24

35

so that the bulk unidirectional magnetostriction is rep-resented as follows

l=Pr

i= 1

lizi : ð19Þ

Simulation results of the hysteresis model are shownin Figure 3 for both completely reversible and irreversi-ble processes. When c= 0, there is a stress-dependenthysteresis between strain and the applied field (Figure3(a)). Since the whole process is completely irreversible,this hysteresis is quantified by the pinning site density.When c= 1, the behavior between strain and fieldbecomes anhysteretic (Figure 3(b)), and the magnetos-triction process is fully reversible. The induced straindue to varying stress is investigated in Figure 3(c) and(d), in which volume fractions are calculated fromequations (14) and (15), respectively. It can be seen

from Figure 3(c) that the induced strain is linear withstress, with the slope representing the material’s com-pliance. The model represented by equation (14) doesnot account for hysteresis when stress is varied at aconstant field. Furthermore, the four plot lines repre-senting different constant bias fields overlap, meaningthat this model also does not account for the influenceof bias field. Hysteresis is observed in Figure 3(d) forthe model represented by equation (15). It can be seenthat the slopes in the hysteretic regions are different atdifferent bias fields, consistent with the DE effect. Thelinear regions are due to the dead zone and saturationof magnetization. When the magnetization does notchange, the strain versus stress relationship is only gov-erned by Hooke’s law.

Numerical approximation

Since equation (10) is nonlinearly coupled, Newmarkintegration is employed to solve the system. The finiteelement equation can be discretized in the time domainas follows

M½ � €Qt+Dt

� + C½ � _Qt+Dt

� + K½ � Qt +Dtf g= Ft +Dtf g;

ð20Þ

where Dt represents the time increment. The generalizeddisplacement Qf g is calculated from the followingequation

Qt +Dtf g= K� ��1

F�

; ð21Þ

where

K� �

= K½ �+a0 M½ �+a1 C½ �; ð22Þ

F�

= Ft +Dtf g+ M½ � a0 Qtf g+a2_Qt

� +a3

€Qt

� � �+ C½ � a1 Qtf g+a4

_Qt

� +a5

€Qt

� � �; ð23Þ

where C½ � is the damping matrix and ai, i= 0, 1, ..., 7ð Þ arethe parameters of the Newmark solver. It can be seenfrom equations (21), (22), and (23) that the generalizeddisplacement Qt +Dtf g at the next time step is depen-dent on the excitation vector Ft +Dtf g at the next timestep and the generalized displacement Qtf g, generalizedvelocity _Qt

� , and generalized acceleration €Qt

� at the

current time step. Once the excitation vector at timestep t+Dt and initial conditions are known, the gener-alized displacement can be calculated from equation(21). Velocity and acceleration at the next time step canbe approximated by

€Qt +Dt

� =a0 Qt +Dtf g+ Qtf gð Þ � a2

_Qt

� � a3

€Qt

� ;

_Qt +Dt

� = _Qt

� +a6

€Qt

� +a7

€Qt +Dt

� :

ð24Þ


The parameters in the Newmark integration are asfollows

a0 =1

bDt2, a1 =

g

bDt, a2 =

1

bDt, a3 =

1

2b� 1,

a4 =g

b� 1, a5 =

Dt

2

g

b� 2

� �, a6 =Dt 1� gð Þ,a7 =Dtg

ð25Þ

where g and b are the solver coefficients. In the pro-posed cantilever system, the left end is clamped suchthat the displacements and rotations are zero at thatend. These boundary conditions can be applied usingthe elimination approach, namely, the entries in thematrices of equation (20) related to the clamped nodecan be eliminated. Furthermore, the excitation vector½Fl, u Fl, v�> at time step t +Dt contains the magnetos-triction, which is a function of the strain and therefore

of displacement. As a result, the excitation vector isnonlinearly coupled with the generalized displacementQt+Dtf g. This must be solved using an initial guessand iteration. From equation (19), it can be observedthat the unidirectional magnetostriction l is the sum ofthe magnetostrictions li mið Þ due to each orientation,weighted by the volume fractions zi. Hysteresis isincluded through the evolution of volume fractions(15). In order to calculate the evolution, we start theinitial guess at increments of displacement dQt +Dtf g,rather than Qt +Dtf g. The initial conditions areassumed to be zero. The procedure for approximatingthe nonlinear system is detailed in Table 1.

Verification of the solver

In order to verify the numerical solution from equa-tions (21) and (24), the numerical solution from the

−20 −10 0 10 200

0.5

1

1.5

2

2.5

3x 10−7

Magnetic Field / kA/m

Stra

in /

ppm

−10MPa−20MPa−30MPa−40MPa−50MPa

Increasing stress

c=0

−20 −10 0 10 200

0.5

1

1.5

2

2.5

3x 10−7

Magnetic Field / kA/m

Stra

in /

ppm

−10MPa−20MPa−30MPa−40MPa−50MPa

Increasing stress

c=1

(a) (b)

−60 −50 −40 −30 −20 −10 0−1.2

−1

−0.8

−0.6

−0.4

−0.2

0x 10−6

Stress / MPa

Stra

in /

ppm

H=4 kA/mH=6 kA/mH=8 kA/mH=10 kA/m

−60 −50 −40 −30 −20 −10 0−14

−12

−10

−8

−6

−4

−2

0

2x 10−7

Stress / MPa

Stra

in /

ppm

Increasingmagnetic field

(c) (d)

Figure 3. Simulation results of hysteresis model: (a) strain versus magnetic field at different constant stresses when c= 0, (b) strainversus magnetic field at different constant stresses when c= 1, (c) strain versus stress at different constant fields when volumefraction is calculated from equation (14) and (d) strain versus stress at different constant fields when volume fraction is calculatedfrom equation (15).

Shu et al. 193

Newmark solver is compared with the analytical solu-tion obtained using beam theory. Both free vibrationand harmonic responses are investigated.

Free vibration. The governing equation for a passivebeam subjected to a load density f (t, x) can be expressedas

EI∂4w t, xð Þ

∂x4+ c

∂w t, xð Þ∂t

+ rA∂2w t, xð Þ

∂t2= f (t, x) ð26Þ

where c is Kelvin–Voigt damping. For the case of freevibration, there is no external excitation applied to thebeam. As detailed in Appendix 3, if the initial verticaldisplacement is y0(x) and the initial velocity is zero, thenthe analytical solution of equation (26) is given as

y t, xð Þ=X‘

n= 1

Wn xð Þqn tð Þ

=X‘

n= 1

Wn xð Þ vn

vdn

1ÐL0

rAW 2n xð Þdx

3

ðL0

rAWn xð Þy0 xð Þdxe�znvnt cos vdnt � cð Þ;

ð27Þ

where c= tan�1 znvn=vdnð Þ and Wn(x) denotes the nthorder mode shape. In order to compare the results ofthe Newmark solver with the analytical solution (27),the finite element model of the passive beam, subjectedto the same initial conditions, is rewritten as

mue 0

0 mve

� �€qu

e

€qve

� �+

cue 0

0 cve

� �_qu

e

_qve

� �

+ku

e � kuvð Þ>

� kuvð Þ kve

" #qu

e

qve

� �=

0Nu

q 31ð Þ0

Nvq 31ð Þ

" #:

ð28Þ

The dynamic response of equation (28) only dependson the initial conditions of the generalized displace-ments ½qu

e qv

e�>. Assuming that the initial deformation is

caused by a concentrated load F0 applied at the freeend of the beam, the vertical displacement can beexpressed as

v xð Þ= F0x2

6EI3L� xð Þ: ð29Þ

The exact expression for the curvature of the beam is

k=1

~r=

d2v xð Þdx2

1+ dv xð Þdx

�2� �3=2

=8E2I2F0 L� xð Þ

4F20 L2x2 � 4F2

0 Lx3 +F20 x4 + 4E2I2

� �3=2:

ð30Þ

The horizontal displacement can be obtained by inte-grating the axial strain ex along the x-axis. If ~z is the dis-tance between the midplane and neutral plane, then theaxial strain of the midplane is ex = k~z. Hence, the hori-zontal displacement can be calculated as

u xð Þ= �ðx0

k tð Þ~zdt

= �ðx0

8E2I2F0 L� tð Þ4F2

0 L2t2 � 4F20 Lt3 +F2

0 t4 + 4E2I2� �3=2

~zdt;

ð31Þ

in which the position of the neutral plane needs to beknown. This neutral plane position can be determinedfrom a force balance equation

F =

ðAs

ssdAs +

ðAg

sgdAg

= � Es

ðh�tg

h�tg�ts

ðb0

kzð Þdydz� Eg

ðhh�tg

ðb0

kzð Þdydz

= � 1

2Eskb 2ts h� tg

� �� t2

s

� �� 1

2Egkb 2htg � t2

g

�= 0;

ð32Þ

where h denotes the distance from the top of the beamto the neutral line. Solving equation (32) for h gives

h=

12

Est2s +Egt2

g

�+Eststg

Ests +Egtg

: ð33Þ

The initial horizontal displacement is therefore cal-culated as

Table 1. Steps for approximating the nonlinear system.

(a) Give an initial guess of dQt+Dtf g.(b) Calculate the increments of strain det+Dt,

magnetostriction dl, and stress ds.

(c) Calculate the magnetostriction lt+Dt =lt + dl

and the excitation vector Fl, ut+Dt, Fl, v

t+Dt

� �T.

(d) Calculate the displacement Qt+Dtf g usingequation (21) and the velocity and accelerationusing equation (24).

(e) Compute the difference between the kth and(k+ 1)th iteration, D�h= Qk+ 1

t+Dt �Qkt+Dt

�� ;repeat the above steps until the conditionD�h= Qk+ 1

t+Dt �Qkt+Dt

�� \d is satisfied.(f) Calculate the volume fraction zt+Dt = zt + dz

and stress sg, t+Dt =sg, t + dsg , which areneeded for the next time step.


u0 xð Þ= �ðx0

k tð Þ h� tg + ts

2

� �dt

= �ðx0

8E2I2F0 L� tð Þ4F2

0 L2t2 � 4F20 Lt3 +F2

0 t4 + 4E2I2� �3=2

12

Est2s +Egt2

g

�+Eststg

Ests +Egtg� ts + tg

2

0@

1Adt:

ð34Þ

In order to verify the numerical solver, the numericalsolution calculated using equations (20) to (25) with ini-tial deformation (34) needs to be compared with the ana-lytical solution (27). Simulation results using analyticaland numerical methods are illustrated in Figure 4(a) and(c), respectively. It can be seen that the numerical solu-tion is consistent with the analytical solution. The FastFourier Transform of the decaying oscillations in Figure4(a) and (c) is presented in Figure 5. Both solutions pre-dict the same fundamental frequency. An experimentaltest (Figures 4(b) and 5) was conducted to verify thesolutions. It is observed that the numerical and analyti-cal solutions for displacement are consistent with theexperimental data and that the finite element model (28)correctly predicts the system’s fundamental frequency.

Forced vibration. We assume that a harmonic point forceFd tð Þ=F0 sin vitð Þ is applied at the free end of the can-tilever and that the initial conditions for the verticaldisplacement and velocity are zero. As detailed inAppendix 3, the governing equation (26) can be

analytically solved for the forced dynamic response interms of vertical displacement as

y t, xð Þ=X‘

n= 1

Wn xð Þqn tð Þ

=X‘

n= 1

Wn xð Þ F0Wn(L)

rAÐL0

W 2n xð Þdx

1

vdn

ðt0

sin vi t � tð Þð Þe�znvnt sinvdntdt:

ð35Þ

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−200

0

200

Time / sD

ispl

acem

ent /

µm

Analytical Solution

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−200

0

200

Time / s

Dis

plac

emen

t / µm

Experimental Data

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−200

0

200

Time / s

Dis

plac

emen

t / µm

Numerical Solution

(a)

(b)

(c)

Figure 4. Free vibration of the passive beam: (a) analytic solution, (b) experimental data, and (c) numerical solution.

500 550 600 650 700 750 800 850 9000

1

2

3

4

5

6

7

8

9

10 x 105

Frequency / Hz

Mag

nitu

de

Analytical SolutionExperimental DataNumerical Solution

Figure 5. Fast Fourier Transform (FFT) of the decay (analyticalsolution, experimental data, and numerical solution).

Shu et al. 195

The finite element model for the passive beam, sub-jected to the same initial conditions and harmonic exci-tation, can be written as follows

mue 0

0 mve

� �€qu

e

€qve

� �+

cue 0

0 cve

� �_qu

e

_qve

� �

+ku

e � kuvð Þ>

� kuvð Þ kve

" #qu

e

qve

� �=

0Nuq 31

0ðNvq�2Þ31

Fd131

0131

26664

37775;

ð36Þ

in which the concentrated vertical load is only appliedto the end node. The loads on the other two DOFs ofthe node are zeros. Figure 6 compares (in the timedomain) the prediction from the numerical solver withthe analytical solution (35) at different frequenciesranging from 50 to 500 Hz. Figure 7 compares the fre-quency response of the prediction from the numericalsolver with the analytical solution (35) at different fre-quencies ranging from 10 to 1000 Hz. The plot showsthat the numerical calculations are consistent with theanalytical solution in both magnitude and phase

0 0.05 0.1 0.15 0.2−300

−200

−100

0

100

200

300

Time / s

Dis

plac

emen

t / µ

m

300Hz

0 0.05 0.1 0.15−200

−100

0

100

200

Time / s

Dis

plac

emen

t / µ

m

150Hz

0 0.05 0.1 0.15−200

−100

0

100

200

Time / sD

ispl

acem

ent /

µm

50Hz

0 0.05 0.1 0.15 0.2−1000

−500

0

500

1000

Time / s

Dis

plac

emen

t / µ

m

500Hz

Analytical Solution Numerical Solution Error

Figure 6. Analytical and numerical solutions for forced vibration of the passive beam.

0 200 400 600 800 100020

40

60

80

Mag

nitu

de

/ d

B

Numerical Solution Analytical Solution

0 200 400 600 800 1000−200

−150

−100

−50

0

Frequency / Hz

Frequency / Hz

Pha

se/

Deg

rees

Figure 7. Frequency response of analytical and numerical solutions for forced vibration of the passive beam.


responses. Calculation errors at the high frequenciesare larger than the errors at low frequencies due to the

limitation of the sampling frequency used in the numer-ical calculation. Higher sampling frequencies areneeded at higher excitation frequencies to achieve thesame calculation error.

Experimental results and discussion

Static and dynamic experiments were conducted in orderto validate the nonlinear model. The applied magneticfield was assumed to be uniform along the length of thecantilever and throughout its thickness. Accordingly, alinear model was used for the field–current relationship,H =NI , with N = 3300 turns in the solenoid coil. Theexperimental setup, shown in Figure 8, used a laser sensorto measure tip displacement of the Galfenol unimorphand dSPACE ControlDesk for real-time manipulation. Asampling frequency of 10 kHz was used for the dSPACEsystem. Geometric and elastic parameters for both theGalfenol and substrate layers are shown in Table 2, and

Figure 8. Experimental setup used for model validation andanalysis.

−30 −20 −10 0 10 20 300

20

40

60

80

100

120

Magnetic Field kA / m

Dis

plac

emen

t / µ

m

0.1Hz

−4 −2 0 2 4 6−50

0

50


Dis

plac

emen

t / µ

m

50Hz

−5 −2.5 0 2.5 5−40

−20

0

20

40


Dis

plac

emen

t / µ

m

100Hz

ModelExperiment

ModelExperiment

ModelExperiment

−2 −1 0 1 2−30

−20

−10

0

10

20

30


Dis

plac

emen

t / µ

m

200Hz

−2 −1 0 1 2−30

−20

−10

0

10

20

30


Dis

plac

emen

t / µ

m

250Hz

−3 −2 −1 0 1 2 3−40

−20

0

20

40


Dis

plac

emen

t / µ

m

350Hz

−1.5 −1 −0.5 0 0.5 1 1.5−30

−20

−10

0

10

20

30


Dis

plac

emen

t / µ

m

400Hz

−1 −0.5 0 0.5 1

−20

−10

0

10

20


Dis

plac

emen

t / µ

m

450Hz

−0.6 −0.3 0 0.3 0.6−20

−10

0

10

20


Dis

plac

emen

t / µ

m

500Hz

ModelExperiment

ModelExperiment

ModelExperiment

ModelExperiment

ModelExperiment

ModelExperiment

Figure 9. Comparison of nonlinear model calculations with static and dynamic experimental data for frequencies from 0.1 to 500Hz.

Shu et al. 197

the material parameters used to model the Galfenol con-stitutive behavior are shown in Table 3.

Figure 9 compares the finite element beam modelwith quasistatic and dynamic experimental results. Inthe quasistatic case, no bias magnetic field is appliedand a full hysteresis loop is attained, demonstratingbutterfly-type nonlinearity. Since deflection is an evenfunction of the input field, no negative tip displacementis observed. In the dynamic case, a direct current (DC)bias current is applied and a sign change is observed inthe deflection. The model accurately describes thechanging shape of the hysteresis loop and peak-to-peakdeflection with increasing frequency. While the hystere-tic behavior of the material constitutive model is fre-quency independent, it can be seen that the hysteresisloop becomes wider as the frequency increases. Thisadditional lag at high frequencies, which results mainlyfrom the system vibrations and dynamic magneticlosses, is accurately described by the dynamic nonlinearmodel.

Concluding remarks

Galfenol’s steel-like structural properties motivate itsapplication in laminated devices. A fully coupled magne-toelastic model was developed to describe the nonlineardynamic response of a Galfenol unimorph actuator. Thehysteretic behavior of Galfenol was modeled by a dis-crete energy-averaged model, and the structuraldynamics were incorporated using a finite elementmodel. The magnetic hysteresis model was found todescribe magnetic hysteresis for both field and stressinputs. A numerical solver was developed to calculatethe nonlinear coupling between the magnetostrictionand output displacements. The solver was verified for apassive beam, with the numerical solution shown to beconsistent with both the analytical solution using beamtheory and with the experimental data. Finally, the non-linear magnetoelastic model was experimentally verifiedwith the beam actuator excited between 0.1 and 500 Hz.Both the experimental data and model results show thatthe hysteresis loop becomes wider as the frequencyincreases. This dynamic nonlinearly coupled model

accurately describes the frequency-dependent hysteresisloop shape and the peak-to-peak deflections, withoutthe need for adjustable parameters.

Acknowledgements

The authors acknowledge ETREMA Products, Inc. for sup-plying the Galfenol samples used in this study.

Funding

Support for L.S. was provided by the National NaturalScience Foundation of China (grant nos 51175395 and51205293), the PhD Programs Foundation of Ministry ofEducation of China (grant no. 20090143110005), and theChina Scholarship Council. We acknowledge the financialsupport by the Office of Naval Research, MURI grant No.N000140610530.

References

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Layers L (mm) b (mm) t (mm) E (GPa) r (kg/m3)

Galfenol 25 6.35 0.381 60 7870Substrate 25 6.35 0.381 100 8400

Table 3. Material parameters for the Galfenol layer.

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Appendix 1

Derivation of internal virtual work

The internal virtual work due to stress is

dWs =

ðL0

ðA

sdedAdx

=

ðL0

ðAg

sgdegdAgdx+

ðL0

ðAs

ssdesdAsdx

= �ðL0

ðAg

Eg


z+l� ∂u t, xð Þ∂x

� �d

∂u t, xð Þ∂x

� ∂2v t, xð Þ∂x2

z

� �dAgdx

�ðL0

ðAs

Es


z� ∂u t, xð Þ∂x

� �d

∂u t, xð Þ∂x

� ∂2v t, xð Þ∂x2

z

� �dAsdx:

ð37Þ

Substituting equation (2) and expanding equation (37)gives the following

dWs =Eg

ðL0

ðAg


zd∂2v t, xð Þ∂x2

zdAgdx

� Eg

ðL0

ðAg


zd∂u t, xð Þ

∂xdAgdx

� Eg

ðL0

ðAg

∂u t, xð Þ∂x


zdAgdx

+Eg

ðL0

ðAg

∂u t, xð Þ∂x

d∂u t, xð Þ

∂xdAgdx

+Eg

ðL0

ðAg


zdAgdx

� Eg

ðL0

ðAg

ld∂u t, xð Þ

∂xdAgdx

+Es

ðL0

ðAs


zd∂2v t, xð Þ∂x2

zdAsdx

� Es

ðL0

ðAs


zd∂u t, xð Þ

∂xdAsdx

� Es

ðL0

ðAs

∂u t, xð Þ∂x


zdAsdx

+Es

ðL0

ðAs

∂u t, xð Þ∂x

d∂u t, xð Þ

∂xdAsdx:

ð38Þ

Collecting z in equation (38) gives

dWs =Eg

ðL0



ðAg

z2dAg

0B@

1CAdx

� Eg

ðL0


d∂u t, xð Þ

∂x

ðAg

zdAg

0B@

1CAdx

� Eg

ðL0

∂u t, xð Þ∂x


ðAg

zdAg

0B@

1CAdx

+Eg

ðL0

∂u t, xð Þ∂x

d∂u t, xð Þ

∂x

ðAg

dAg

0B@

1CAdx

Shu et al. 199

+Eg

ðL0

ðAg

ld∂2n t, xð Þ

∂x2zdAgdx

� Eg

ðL0

ðAg

ld∂u t, xð Þ

∂xdAgdx

+Es

ðL0



ðAs

z2dAs

0B@

1CAdx

� Es

ðL0


d∂u t, xð Þ

∂x

ðAs

zdAs

0B@

1CAdx

� Es

ðL0

∂u t, xð Þ∂x


ðAs

zdAs

0B@

1CAdx

+Es

ðL0

∂u t, xð Þ∂x

d∂u t, xð Þ

∂x

ðAs

dAs

0B@

1CAdx:

ð39Þ

The cross section of the cantilever is assumed to beuniform. Integrating equation (39) along the y and z

directions, the internal virtual work due to stress can bewritten as

dWs =EgIg

ðL0



dx

� EgQg

ðL0


d∂u t, xð Þ

∂xdx� EgQg

ðL0

∂u t, xð Þ∂x


dx

+EgAg

ðL0

∂u t, xð Þ∂x

d∂u t, xð Þ

∂xdx+Eg

ðL0

ðAg


zdAgdx

� Eg

ðL0

ðAg

ld∂u t, xð Þ

∂xdAgdx+EsIs

ðL0



dx

� EsQs

ðL0


d∂u t, xð Þ

∂xdx� EsQs

ðL0

∂u t, xð Þ∂x


dx

+EsAs

ðL0

∂u t, xð Þ∂x

d∂u t, xð Þ

∂xdx;

ð40Þ

where

Ig =Ð

Ag

z2dAg, Qg =Ð

Ag

zdAg

Is =ÐAs

z2dAs, Qs =ÐAs

zdAs:

Appendix 2

Discretization of weak form equation

The Hermite shape function vector is chosen as

H1 =1

41� jð Þ2 2+ jð Þ, H2 =

1

41� jð Þ2 1+ jð Þ

H3 =1

41+ jð Þ2 2� jð Þ, H4 =

1

41+ jð Þ2 j � 1ð Þ:

ð41Þ

The linear shape function is chosen as

N1 =1�j

2, N2 =

1+ j2: ð42Þ

Since the local spatial coordinate j varies from �1 to1, the coordinate transformation is interpolated as

x=1� j

2x1 +

1+ j

2x2

=x1 + x2

2+

x2 � x1

2j:

ð43Þ

Since le = x2 � x1 is the length of the element, wehave

dx= le2

dj : ð44Þ

Thus, the derivatives in equation (4) can be rewrittenas


=4

l2e

∂2v t, xð Þ∂j2

=4

l2e

d2H

dj2qv

e

=4

l2e

3

2j,

le

4�1+ 3jð Þ, � 3

2j,

le

41+ 3jð Þ

� �qv

e

∂u t, xð Þ∂x

=2

le

∂u t, xð Þ∂j

=2

le� 1

2,

1

2

� �qu

e =B � que :

ð45Þ

Substituting equation (45) into equation (4), the virtualwork due to stress can be written as


dWs =X

e

EI

ð1�1

4

l2e

d2H

dj2qv

ed4

l2e

d2H

dj2qv

e

� �le

2dj

� EQ

ð1�1

4

l2e

d2H

dj2qv

ed B � que

� � le

2dj

� EQ

ð1�1

B � qued

4

l2e

d2H

dj2qv

e

� �le

2dj

+EA

ð1�1

B � qued B � qu

e

� � le

2dj

+Eg

ð1�1

ðAg

l H , q,sð Þzd4

l2e

d2H

dj2qv

e

� �le

2dAgdj

� Eg

ð1�1

ðAg

l H , q,sð Þd B � que

� � le

2dAgdj

=X

e

qve> � 8EI

l3e

ð1�1

d2H

dj2

� �>d2H

dj2dj

24

35dqv

e

� qve> � 2EQ

le

ð1�1

d2H

dj2

� �>Bdj

24

35dqu

e

� que> � 2EQ

le

ð1�1

Bd2H

dj2dj

24

35dqv

e + que> � EAleB

> � B� �

dque

+2Egb

le

ð1�1

ðtg

l H , q,sð Þz d2H

dj2dzdj

264

375dqv

e

� Egble

2

ð1�1

ðtg

l H , q,sð ÞBdzdj

264

375dqu

e

=X

e

qve> � kv

edqve � qv

e> � kuv

e dque � qu

e> � kuv

e

� �>dqv

e

+ que> � ku

edque + fl, v

e dqve � fl, ue dqu

e :

ð46Þ

From (5), the virtual work components due to iner-tial and damping effects are

dWr =

ðL0

ðA

r∂2u t, xð Þ

∂t2dudAdx+

ðL0

ðA

r∂2v t, xð Þ

∂t2dvdAdx

dWc =

ðL0

ðA

c∂u t, xð Þ

∂tdudAdx+

ðL0

ðA

c∂v t, xð Þ

∂tdvdAdx:

ð47Þ

Substitution of equation (7) and equation (8) into equa-tion (47) gives

dWr =X

e

ð1�1

ðA

r N � €que

� �>d N � qu

e

� � le

2dAdj

+

ð1�1

ðA

r H � €qve

� �>d H � qv

e

� � le

2dAdj

=X

e

€que> � lerA

2

ð1�1

N> �Ndj

24

35dqu

e

+ €qve> � lerA

2

ð1�1

H> �Hdj

24

35dqv

e

=X

e

€que

� > � mue

� �dqu

e

� + €qv

e

� > � mve

� �dqv

e

� ð48Þ

dWc =

ðL0

ðA

c _ududAdx+

ðL0

ðA

c _vdvdAdx

=X

e

ð1�1

ðA

c N � _que

� �>d N � qu

e

� � le

2dAdj

+

ð1�1

ðA

c H � _qve

� �>d H � qv

e

� � le

2dAdj

=X

e

_que> � lecA

2

ð1�1

N> �Ndj

24

35dqu

e

+ _qve> � lecA

2

ð1�1

H> �Hdj

24

35dqv

e

=X

e

_que

� > � cue

� �dqu

e

� + _qv

e

� > � cve

� �dqv

e

� :

ð49Þ

Setting the sum of the discretized virtual work due tothe internal stress (46), inertia (48), and damping effects(49) equal to zero, the discretized weak form equationcan be written as

Xe

qve

� > � kve

� �dqv

e

� � qv

e

� > � kuve

� �dqu

e

� � qu

e

� > � kuve

� �>dqv

e

� + qu

e

� > � kue

� �dqu

e

� +X

e

€que

� > � mue

� �dqu

e

� + €qv

e

� > � mve

� �dqv

e

� +X

e

_que

� > � cue

� �dqu

e

� + _qv

e

� > � cve

� �dqv

e

� =X

e

� fl, ve

� �dqv

e

� + fl, u

e

� �dqu

e

� :

ð50Þ

Shu et al. 201

Appendix 3

Analytical solution of passive beam

The response of a passive beam is determined by first sol-ving for the general solution. Then, solutions for the casesof free vibration and forced harmonic response are obtainedby applying different loads and initial conditions to the gen-eral solution. The governing equation for a passive beam,subjected to load density f (t, x), can be written as

EI∂4w t, xð Þ

∂x4+ c

∂w t, xð Þ∂t

+ rA∂2w t, xð Þ

∂t2= f (t, x) ð51Þ

where c is Kelvin–Voigt damping. Using modal analy-sis, the solution of equation (51) can be expressed as

w t, xð Þ=X‘

n= 1

Wn xð Þqn tð Þ: ð52Þ

Here, Wn xð Þ denotes the nth mode shape of the cantile-ver, which can be expressed as

Wn xð Þ=Cn sinbnx� sinhbnx� an cosbnx� coshbnxð Þ½ �ð53Þ

where an = sinbnl + sinhbnlð Þ= cosbnl+ coshbnlð Þand the frequency equation is cosbnl � coshbnl= � 1.Substitution of equation (52) into equation (51) gives

EIX‘

n= 1

d4Wn xð Þdx4

qn tð Þ+ rAX‘

n= 1

Wn xð Þ d2qn tð Þdt2

+ cX‘

n= 1

Wn xð Þ dqn tð Þdt

= f t, xð Þ:ð54Þ

Since

EId4Wn xð Þ

dx4=v2

nrAWn xð Þ; ð55Þ

(54) can be rewritten as

X‘

n= 1

v2nrAWn xð Þqn tð Þ+ rA

X‘

n= 1

Wn xð Þ d2qn tð Þdt2

+ cX‘

n= 1

Wn xð Þ dqn tð Þdt

= f t, xð Þ:ð56Þ

By multiplying equation (56) throughout by Wm xð Þ,integrating from 0 to L, and using the orthogonalitycondition, we obtain

d2qn tð Þdt2

+c

rA

dqn tð Þdt

+v2nqn tð Þ

=1

rAÐL0

W 2n xð Þdx

ðL0

f x, tð ÞWn xð Þdx:ð57Þ

The problem investigated here is for a point loadapplied at the end of the passive beam. The integral inequation (57) can thus be simplified as

ðL0

f x, tð ÞWn xð Þdx=

ðL0

Wn(x)F0u(t)d(x� L)dx=F0u(t)Wn(L)

ð58Þ

where F0 is the magnitude of the force, d( � ) is theDirac delta function, and u(t) is the trajectory of theload in the time domain. Substituting equation (58)into equation (57) and solving analytically, one obtainsthe modal displacement

qn tð Þ= 1

rAÐL0

W 2n xð Þdx

1

vdn

F0Wn(L)

ðt0

u t � tð Þe�znvnt sinvdntdt +qn 0ð Þvn

vdn

e�znvnt cos vdnt � cð Þ+ _qn 0ð Þvdn

e�znvnt sinvdnt

ð59Þ

where vn is the nth order natural frequency, vdn is thedamped natural frequency, and qn(0) and _qn(0) are theinitial modal displacement and velocity, respectively.To express the initial modal displacement in terms ofthe actual initial displacement function y0 xð Þ, we lett = 0 in equation (52) and write

y 0, xð Þ=X‘

n= 1

Wn xð Þqn 0ð Þ= y0 xð Þ: ð60Þ

Then, multiplying equation (60) by rAWn xð Þ, inte-grating over the length of the beam, and using theorthonormality relations, we obtain

qn 0ð Þ= 1ÐL0

rAW 2n xð Þdx

ðL0

rAWn xð Þyo xð Þdx: ð61Þ

Using the same procedure, the initial modal velocity_qn 0ð Þ can be related to the actual initial velocity v0 xð Þ as

_qn 0ð Þ= 1ÐL0

rAW 2n xð Þdx

ðL0

rAWn xð Þvo xð Þdx: ð62Þ

Free vibration. For free vibration, the dynamic responseof the passive beam depends only on the initialconditions. Since F0 = 0, the first term in the right-hand side of equation (59) becomes zero. In addition,because the initial velocity v0 xð Þ is zero, equation (62)and the third term in equation (59) are zero as well.


Thus, the analytical solution for the displacement canbe written from equation (52) using the modaldisplacement equation (59), where the initial modaldisplacement is given by equation (61)

y t, xð Þ=X‘

n= 1

Wn xð Þqn tð Þ

=X‘

n= 1

Wn xð Þ vn

vdn

1ÐL0

rAW 2n xð Þdx

ðL0

rAWn xð Þy0 xð Þdxe�znvnt cos vdnt � cð Þ

ð63Þ

where c= tan�1 znvn=vdnð Þ.

Forced vibration. Assume that a concentrated harmonicforce Fd(t)=F0 sin (vit) is applied at the free end of thecantilever. If the initial conditions for the vertical dis-placement and velocity are assumed to be zero, then thesecond and third terms on the right-hand side of equa-tion (59) become zero. Substituting u(t)= sin (vit) intoequation (59) gives

qn tð Þ= 1

rAÐL0

W 2n xð Þdx

1

vdn

F0Wn(L)

ðt0

sin vi t � tð Þð Þe�znvnt sinvdntdt: ð64Þ

The harmonic response of the passive beam in termsof vertical displacement can be calculated from equa-tion (52), for the modal displacement (64), as

y t, xð Þ=X‘

n= 1

Wn xð Þqn tð Þ

=X‘

n= 1

Wn xð Þ F0Wn(L)

rAÐL0

W 2n xð Þdx

1

vdn

ðt0

sin vi t � tð Þð Þe�znvnt sinvdntdt: ð65Þ

Shu et al. 203

and structures nonlinear model for galfenol cantilevered ... · tions. the linear euler–bernoulli...

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