Finite Elements in Analysis and Design 49 (2012) 35–51

Contents lists available at SciVerse ScienceDirect

Finite Elements in Analysis and Design

0168-87

doi:10.1

n Corr

E-m

olivier.t

journal homepage: www.elsevier.com/locate/finel

Finite element reduced order models for nonlinear vibrations of piezoelectriclayered beams with applications to NEMS

A. Lazarus a,b, O. Thomas a, J.-F. Deu a,n

a Structural Mechanics and Coupled Systems Laboratory, Cnam, 2 rue Conte 75003 Paris, Franceb Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

a r t i c l e i n f o

Article history:

Received 23 December 2010

Received in revised form

24 June 2011

Accepted 14 July 2011Available online 20 October 2011

Keywords:

Piezoelectric materials

Nanoelectromechanical systems

Reduced order models

Nonlinear vibrations

Continuation methods

4X/$ - see front matter & 2011 Elsevier B.V.

016/j.finel.2011.08.019

esponding author.

ail addresses: alazarus@mit.edu (A. Lazarus),

homas@cnam.fr (O. Thomas), jean-francois.de

a b s t r a c t

This article presents a finite element reduced order model for the nonlinear vibrations of piezoelectric

layered beams with application to NEMS. In this model, the geometrical nonlinearities are taken into

account through a von Karman nonlinear strain–displacement relationship. The originality of the finite

element electromechanical formulation is that the system electrical state is fully described by only a

couple of variables per piezoelectric patches, namely the electric charge contained in the electrodes and

the voltage between the electrodes. Due to the geometrical nonlinearity, the piezoelectric actuation

introduces an original parametric excitation term in the equilibrium equation. The reduced-order

formulation of the discretized problem is obtained by expanding the mechanical displacement unknown

vector onto the short-circuit eigenmode basis. A particular attention is paid to the computation of the

unknown nonlinear stiffness coefficients of the reduced-order model. Due to the particular form of the

von Karman nonlinearities, these coefficients are computed exactly, once for a given geometry, by

prescribing relevant nodal displacements in nonlinear static solutions settings. Finally, the low-order

model is computed with an original purely harmonic-based continuation method. Our numerical tool is

then validated by computing the nonlinear vibrations of a mechanically excited homogeneous beam

supported at both ends referenced in the literature. The more difficult case of the nonlinear oscillations of

a layered nanobridge piezoelectrically actuated is also studied. Interesting vibratory phenomena such as

parametric amplification or patch length dependence of the frequency output response are highlighted in

order to help in the design of these nanodevices.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Nanoelectromechanical systems or NEMS are systems consist-ing of integrated electromechanical resonators of nanometer-scale dimensions [1]. Thanks to the outstanding mechanicalproperties of the resonant nanostructure, such as its qualityfactor, NEMS response can exceed the quality of electrical signalfrom purely electronic devices. Recently, they have been proposedfor use in ultrasensitive mass sensor [2], ultra high frequencyelectronic filters [3], bit storage systems [4] and radio-frequencytelecommunication devices [5]. As researchers orient towardsfrequencies reaching the GHz, one of the key issue in theoptimization of NEMS becomes the scaling down and the electro-mechanical efficiency of the actuation and detection process.Comparing to the more conventionally employed electrostaticexcitations [6] or even the more original self-oscillation of

All rights reserved.

u@cnam.fr (J.-F. Deu).

nanowires in field emission [7,8], NEMS based on piezoelectricactuation appears to be particularly advantageous due to thelinearity of forcing, their intrinsic integrability, high-efficiencyand low power consumption [9]. While the experimental realiza-tion of such NEMS with high electromechanical coupling remainsan actual challenge [10], the predictive simulations of the non-linear oscillations of these piezoelectrically laminated, and gen-erally prestressed, slender nanostructures is no less a mandatorystep in the future production of these devices.

In the scope of NEMS applications, the present study proposesa predictive model for nonlinear vibrations of piezoelectriclayered beams. The nonlinear steady-state response is simulatedwith an original finite element reduced order model in order toenable every beam geometrical configurations for a reasonablecomputational time. Starting from a nonlinear finite elementformulation of the considered laminated piezoelectric beam, thediscrete problem is expanded on its truncated linear modal basisresulting in a low-order model. Only the latter needs then to besolved to compute the periodic beam oscillations either by aclassical time integration method or in our case, by an originalpurely harmonic-based continuation tool [11,12].

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–5136

Since first investigations in micro- and nanoelectromechanicalsystems have been conducted by experimental physicists, most ofthe theoretical models associated with the nonlinear vibrations ofthe resonant nanostructures are still heuristic models [13,14].These approaches, although very convenient for the identificationof electromechanical parameters of the nanosystems, are uselesswhen dealing with the design or optimization of the resonator asheuristic models are not predictive.

One possibility to predict the behavior of nonlinear vibratingsystems is the use of analytical models [15,16]. The nonlinear forcedvibration of plates or shallow shells has been investigated in [17,18]by expanding the displacements of the structure on the linearnatural modes and solving the associated low-order analyticalmodel with a perturbation method. The case of a piezoelectricallyexcited plate at the nanoscale has been treated in [19] where theanalytical steady-state solution is computed thanks to an asymptoticnumerical continuation method. The method of averaging is used in[6] for determining the nonlinear dynamics of nanomechanicalbeam resonators electrostatically excited. Numerous papers, basedon similar analytical methods and dedicated to the modeling ofnonlinear vibration of piezoelectric beams can be found in theliterature. Nonlinear free and forced oscillations of piezoelectricclamped–clamped microresonators have been studied in [20] and in[21] when lying in an electrostatic environment. The influence ofimperfections on the nonlinear vibratory behavior of doubly sup-ported sandwich piezoelectric beam has been treated in [22,23].Analytical models of piezoelectrically actuated microcantilevers maybe found in [24–26]. All these methods, although predictive andoften relevant suffer a critical drawback: solving analytically thenonlinear equilibrium equations is only reasonable when dealingwith structures with simple geometries.

A solution to free ourselves from this awkward geometrydependence is the use of finite element methods. The computa-tion of large oscillations of homogeneous structures by coupling anonlinear finite element analysis with a direct time integrationprocedure [27,28] or a harmonic balance method [29,30] isrelatively well known today. The finite element modeling ofnonlinear vibrations of piezoelectric structures is less commonbut has been treated in [31] or [32]. However, the vibratoryresponses of beams at the nanoscale are characterized by possiblehighly anharmonic steady-state response and very long transientresponse due to their outstanding quality factors. Consequently,NEMS are typical applications where the use of classical nonlinearfinite element method lead to non-reasonable computationaltime and reduced order models become naturally relevant. Manyreduction techniques for nonlinear dynamics have been devel-oped by the mechanical engineering community. A finite elementbased nonlinear modal reduction may be found in [33] for thenonlinear vibration of rotating beams and in [34] when coupledwith a temporal continuation method. The well-known properorthogonal decomposition techniques are developed in [35] for

b

x,1

z,3

y,2

y,2

Fig. 1. Piezoelectric l

random excitation and in [36] for the particular case of electro-statically actuated MEMS. The reduction on a classical linearmodal basis by determining the nonlinear modal stiffness coeffi-cients for an arbitrary finite element model is explained in [37,38]and in [39] for uncertain geometrically nonlinear structures withstochastic methods. Finally, the enrichment of the linear vibrationmodes with their modal derivatives is described in [40,41] and in[42] for the MEMS nonlinear dynamic analysis.

In this paper, we choose to reduce our nonlinear finite elementmodel on a linear modal basis following the method presented in[38]. Several reasons can explain this choice. One advantage of thismethod is the physical meaning of the linear eigenmodes whichlead to criteria to select important eigenmodes. Also, this naturalbasis seems to be the most relevant to use for non-experiencedcomputational scientists such as experimental physicists. Numeri-cally, one needs only to compute the reduced order model once for agiven geometry, the latter being valid for different external loading.Finally, our work is not limited to the description of the reductiontechnique implementation and has also some real practical applica-tions to discuss the method. We will see in this paper that, evenwhen dealing with a relatively simple one-dimensional piezoelectriclayered model, questions relative to the choice and truncature ofeigenvectors, or to the meaning of the nonlinear modal stiffnesscoefficients, are definitively not obvious and deserve to be treatedfor future application of this method.

The present article is divided into five main parts. After thepresent introduction, Section 2 exposes the details of a nonlinearfinite element discretization of a laminated elastic beam withpiezoelectric patches. This finite element model originality resultsmainly from a von Karman nonlinear strain–displacement relation-ship and some relevant electrical assumptions leading to only onecouple of variables per piezoelectric patches [43]. The modalexpansion of the general finite element formulation is explained inSection 3. Details of the computation of the nonlinear modalstiffness coefficients as well as description of the original purelyfrequential continuation procedure used to compute the reducedorder model are given in this section. In the last sections, we applythe above method to two numerical examples. In Section 4, wecompute the nonlinear vibration of a simply supported homoge-neous beam with different numerical methods in order to validateour approach. In Section 5, we perform the numerical simulations ofa piezoelectric layered nanobeams in real geometrical configura-tions. We compare the numerical results obtained with our reducedorder model with a classical full nonlinear finite element model.

2. An elastic beam with piezoelectric patches

This section is devoted to the general formulation of thenonlinear governing equations of a laminated beam composedof elastic and piezoelectric layers (Fig. 1). In this one-dimensional

hk h

z0z1

z2

zk−1

zk

zN−1

x−x+L

x,1

z,3

21

k

K zk

aminated beam.

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–51 37

model, we assume a von Karman nonlinear strain–displacementrelationships for taking into account the geometric nonlinearitiesassociated with large amplitude oscillations of nanobeams. Usingthe finite element method with the electrical hypotheses of anelastic structure with piezoelectric patches described in [43], weare able to express the nonlinear variational formulation in termsof discrete variables such as the displacements at each nodes andthe electrical potential of each patch.

2.1. Mechanical and electrical field assumptions

The beam is modeled using the classical laminate theory basedon Euler–Bernoulli assumptions (Fig. 2 (left)). We limit our studyto the vibration in the ðx2zÞ plane. The mechanical displacementfield can then be written as

uxðx,z,tÞ ¼ uðx,tÞþzyðx,tÞ, ð1aÞ

uzðx,z,tÞ ¼wðx,tÞ, ð1bÞ

where ux and uz are the axial and transverse displacements; u isthe axial displacement of the center line of the beam, w thetransverse displacement, and y the fiber rotation defined by

y¼�@w

@x: ð2Þ

From the von Karman nonlinear strain–displacement relation-ships and the above described kinematics, axial strain can bewritten as follows:

e1 ¼ eþzk, ð3Þ

where membrane strain e and curvature k are defined by

e¼@u

@xþ

1

2

@w

@x

� �2

and k¼� @2w

@x2: ð4Þ

Concerning the electrostatic aspects, we consider that the piezo-electric beams under study satisfy all assumptions expounded in[43]. In particular, the electric field vector, of components Ek, isnormal to the electrodes and its magnitude is uniform in thepiezoelectric patch p, so that for all pAf1, . . . ,Pg, the only non-vanishing component of the electric field is

EðpÞ3 ¼�V ðpÞ

hðpÞn3, ð5Þ

where V ðpÞ is the potential difference of the p-th patch and n3 is thecomponent of the normal unit vector to the surface of the electrodes(see Fig. 1). In this case, two main coupling can be obtained: a ‘‘33’’coupling between the transverse electric field and the stress/straincomponents in the same direction and a ‘‘31’’ coupling between thetransverse electric field and the membrane stresses/strains (seeFig. 2 (right)). We recall from [43] that the present hypothesis isvalid as long as the piezoelectric patches are thin, with a constantthickness, denoted hðpÞ for the p-th patch, smaller than its char-acteristic longitudinal length. Note that the piezoelectric patch

u(x)

w(x)

∂w∂x

∂w∂x

z,3

x,1

�1 �1

E3 V = −Ehp

Fig. 2. Mechanical and electrical assumptions. (Left) Euler–Bernoulli hypothesis.

(Right) ‘‘31’’ piezoelectric effect.

materials are transverse isotropic and the electric end effects forthe piezoelectric patches are neglected.

2.2. Piezoelectric constitutive equations

Following the hypotheses of [43], the piezoelectric layers ofthe laminated beam are poled in the thickness ðz,3Þ-direction withan electrical field applied parallel to this polarization. As saidabove, such a configuration is characterized in particular by theelectromechanical coupling between the axial strain e1 and thetransverse electrical field E3. For sake of simplicity, we assumethe classical plane-stress assumptions of beam theories ðs2 ¼ s3 ¼ 0Þ.The classical linear piezoelectric constitutive equations are restrictedto the mechanical axial components only and reads:

sk1 ¼ Yke1�ek

31E3,

Dk3 ¼ ek

31e1þEk33E3,

8<: ð6Þ

where sk1 and Dk

3 are the axial stress and the transverse electricaldisplacement within the k-th layer, Yk is the k-th layer Young’smodulus in the ð1;2Þ plane at constant electric field, ek

31 is thepiezoelectric constant and Ek

33 is the dielectric permittivity atconstant strain. Note that Eq. (6) is a simple electromechanicalapproximation for constitutive behavior of one-dimensionalbeams, nevertheless fully sufficient in the scope of this paper.A more accurate Euler–Bernoulli model of laminated piezoelectricbeam can be found in [44] where the influence of one-dimen-sional distribution of mechanical stresses and strains is takeninto account through corrected electromechanical constitutiveequations.

2.3. Variational formulation

Considering a particular axial part of the laminated beam(of length L¼ xþ�x� and with a total of K layers includingP piezoelectric patches as in Fig. 2), the variational formulationassociated with the local equilibrium equation of our electro-mechanical system is given in [43] and can be expressed in thefollowing form:

XK

k ¼ 1

ZOkrkð €uxduxþ €uzduzÞ dOþ

XK

k ¼ 1

ZOk

Yke1de1 dO

þXP

p ¼ 1

V ðpÞ

hðpÞ

ZOðpÞ

eðpÞ31de1 dO

¼XK

k ¼ 1

ZGk

t

ðtkxduxþtk

zduzÞ dSþXK

k ¼ 1

ZOkðf k

xduxþ f kzduzÞ dO, ð7aÞ

�XP

p ¼ 1

dV ðpÞ

hðpÞ

ZOðpÞ

eðpÞ31e1 dOþXP

p ¼ 1

V ðpÞCðpÞdV ðpÞ ¼XP

p ¼ 1

Q ðpÞdV ðpÞ, ð7bÞ

where rk and Ok are the mass density and the domain occupiedby the k-th layer and the p-th patch capacitance is

CðpÞ ¼blðpÞ

hðpÞEðpÞ33 ð8Þ

with b and lðpÞ are the width of the beam and the p-th patchlength. Notice that the beam is subjected to surface axial andtransversal forces at the boundaries of each face sublayer (tx

k, tzk)

and to body ones (fxk, fz

k). The originality of the formulation is thatthe electrical state is fully described by only a couple of discretevariables per piezoelectric patches, namely the electric charge Q ðpÞ

contained in the electrode and the voltage V ðpÞ between theelectrodes. The electromechanical problem now consists in find-ing the admissible displacements ui and the P electric charges Q ðpÞ

(pAf1, . . . ,Pg), such that the associated potential differences V ðpÞ

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–5138

are prescribed, satisfying Eqs. (7a) and (7b) with appropriateinitial conditions.

2.4. Finite element discretization

Using any finite element procedure to discretize the mechanicalpart of Eqs. (7a) and (7b) leads to introduce u, the vector of nodalvalues of ui. By introducing Q ð1ÞQ ð2Þ . . .Q ðPÞ and V ð1ÞV ð2Þ . . .V ðPÞ, theelectric charges and potential differences of the p-th patch, onefinally obtains the general nonlinear finite element formulation ofthe laminated piezoelectric beam:

Mm €uþKmuþf nlðuÞþXP

p ¼ 1

½f ðpÞc þK ðpÞc u�V ðpÞ ¼ f e, ð9aÞ

CðpÞV ðpÞ�Q ðpÞ�½f ðpÞc þK ðpÞc u�T u¼ 0 8pAf1 . . . Pg: ð9bÞ

In the above equation, Mm and Km are the mechanical mass andstiffness matrices, of size N�N, if N is the number of mechanicalunknowns. Due to the assumed von Karman strain–displacementrelationships, the nonlinear part of the internal energy is repre-sented by the N-dimensional column vector of nonlinear stiffnessforces f nlðuÞ. f e is the column vector of mechanical forcing, of lengthN. All details that lead to matrices Mm, Km, f nlðuÞ and f e are writtenin Appendix A. f ðpÞc and K ðpÞc u are the electromechanical couplingvectors, of size N, that couple the mechanical d.o.f. to the potentialdifference of the p-th patch only. In particular, f ðpÞc is the linearcoupling vector found in the formulation of [43] and K ðpÞc is theN�N electromechanical matrix due to the nonlinear membrane/flexural coupling of the beam. Because of the von Karman geome-trical nonlinearities, a piezoelectric patch may induce some para-metric excitations in the equilibrium equation (9a) through thevector K ðpÞc u. For an applied sinusoidal voltage V ðpÞ with a frequency2o, the nanobeam is likely to oscillate with a fundamentalfrequency o (see Section 5). Note that one single patch is generallydiscretized by more than one finite element such that f ðpÞc and K ðpÞc

are simply obtained by assembling the elementary vectors f eðpÞc of

size (6�1) and elementary matrices KeðpÞc of size (6�6), whose

explicit expressions are given in Appendix A.

3. Reduced order model

In this section, a reduced-order formulation of the discretizedproblem obtained in Section 2 is introduced, by expanding themechanical displacement unknown vector onto the short-circuiteigenmode basis. The main motivation of choosing this particularbasis is that, beyond the physical meaning of these entities, it canbe computed with a classical elastic mechanical problem, once fora given configuration. The unknown nonlinear stiffness coeffi-cients in modal space are found by solving a set of prescribeddisplacements problems. Finally, the well-defined low-ordermodel is recast in a quadratic dimensionless form for computingthe periodic oscillation of our electromechanical system with anoriginal harmonic-based continuation method.

3.1. Short-circuit normal modes

The system short circuit normal modes are solutions of Eq. (9a)with V ðpÞ ¼ 0 for all p, f e ¼ 0 and f nlðuÞ ¼ 0. The natural frequenciesoi and mode shapes Ui (each Ui is a column vectors of length N)are the N eigensolutions of the following problem:

KmU�o2MmU¼ 0, ð10Þ

which depend only on the mechanical properties of the system.

These modes verify the following orthogonality properties:

8ði,jÞ, UTi MmUj ¼ dij, UT

i KmUj ¼o2i dij, ð11Þ

where dij is the Kronecker delta and the Fi (i¼ 1, . . . ,N) have beennormalized with respect to the mass matrix:

8i, UTi MmUi ¼ 1: ð12Þ

One can note that the above equation imposes that the modeshapes Ui units are kg�1/2 for the elements linked to d.o.f. u and w

and kg�1/2 m�1 for the elements linked to d.o.f. y (see Eq. (A.4) forthe units of Mm).

3.2. Modal expansion of the general problem

The displacement vector is sought as

uðtÞ ¼XN

i ¼ 1

UiqiðtÞ: ð13Þ

By inserting the above equation in Eqs. (9a) and (9b), multiplyingthe first obtained equation by UT

i and using the orthogonalityproperties of Eqs. (11), the problem writes, for all iAf1, . . . ,Ng:

€qiþ2xi _qiþo2i qiþ

XN

j ¼ 1

XN

kZ j

bijkqjqkþ

XN

j ¼ 1

XN

kZ j

XN

lZ j

gijklqjqkql

þXP

p ¼ 1

wðpÞi V ðpÞ þXP

p ¼ 1

XN

j ¼ 1

YðpÞij qjVðpÞ¼ Fi, ð14aÞ

CðpÞV ðpÞ�Q ðpÞ�XN

j ¼ 1

wðpÞj qj�XN

j ¼ 1

XN

k ¼ 1

YðpÞjk qjqk ¼ 0: ð14bÞ

In Eq. (14a), wðpÞi is the linear electromechanical coupling coefficientof the i-th mode and the p-th piezoelectric patch, that is defined by

wðpÞi ¼UTi f ðpÞc 8iAf1 . . .Ng, 8pAf1 . . . Pg ð15Þ

and YðpÞij is the nonlinear electromechanical coefficient introducingthe parametric excitation of the i-th mode, that is defined by

YðpÞij ¼UTi K ðpÞc Uj, 8i,jAf1 . . .Ng, 8pAf1 . . . Pg: ð16Þ

xi and Fi ¼UTi f e are respectively the modal viscous damping and

the modal external forcing of the i-th mode. The quadratic and cubicpolynomial stiffnesses of Eq. (14a) result from the modal expansionof the nonlinear internal forces UT

i f nlðuÞ. Assuming von Karmangeometrical nonlinearities, these nonlinear internal forces are atmost cubic in u (see Eq. (A.16)) and the governing equation (14a) istherefore exact.

The initial finite element formulation introduced in Section 2 hasbeen replaced by the modal formulation of Eq. (14), whoseunknowns are the N modal coordinates qi and the P voltage/chargepairs ðV ðpÞ,Q ðpÞÞ associated to the P piezoelectric patches. Thereduced order model is then obtained by truncated the modalequation (14) to a number of M modes where M is generally muchsmaller than N. Its major interest, and especially the choice of theshort-circuit eigenmodes as the expansion basis, is that the abovecomputations of the parameters necessitates only a single modalanalysis of the elastic problem. This operation can thus be done byany standard finite-elements code. To summarize, in order to obtainthe finite element reduced order model of a given laminatedpiezoelectric beam, one has to:

�

Compute all matrices and vectors of the initial problem (9),that is to say Mm, Km, f ðpÞc and K ðpÞc . � Calculate the M natural frequencies oi under interest in short

circuit as well as the associated mode shapes Ui, by solving theclassical eigenvalues problem (10).

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–51 39

�

Apply the matrix products (15) and (16) to obtain wðpÞi and YðpÞij .i � Determine the unknown nonlinear stiffness coefficients bjk

and gijkl thanks to the following method.

3.3. Computation of nonlinear coefficients

The nonlinear modal stiffness coefficients bijk and gi

jkl arecomputed with the method explained in [38]. Assuming thevon Karman relationship (4), the nonlinear stiffness forces f nlðuÞwritten in Appendix A are at most cubic in u and (14) isconsequently exact. The procedure is based on the restoration ofnodal applied forces by relevant prescribing nodal displacementsin nonlinear static solution settings.

To illustrate the technique, one can begin by prescribing thedisplacement fields u¼U1a and u¼�U1a where a is an arbitraryconstant scalar. In the modal formulations (13) and (14), this isequivalent to impose respectively:

q1 ¼ a and qi ¼ 0 8ia1, UT1f nlðU1aÞ ¼ b1

11a2þg1111a3, ð17aÞ

q1 ¼�a and qi ¼ 0 8ia1, UT1f nlð�U1aÞ ¼ b1

11a2�g1111a3:

ð17bÞ

The expanded nonlinear stiffness forces UT1f nlðU1aÞ and

UT1f nlð�U1aÞ can be computed using (A.16) since a is a prescribed

scalar and U1 is known from the classical eigenvalues problem(10). From then, the coefficients b1

11 and g1111 can be determined

from the resulting system (17) of two linear equations. Contrarilyto [38], the relation (17) is exact in our case and theoreticallyindependent on a. However, for numerical purposes, this constantscalar has to be consistent with the dimension of the finiteelement model to avoid bad conditioning. In the following, wesimplify the computations and the explicit expressions by choos-ing a¼1. By using an analogous manner to determine thecoefficients bi

ii and giiii for ia1, one obtains the general formulae

for the nonlinear stiffness coefficients with equal lower indices:

2biii ¼UT

i ðf nlðUiÞþf nlð�UiÞÞ, ð18aÞ

2giiii ¼UT

i ðf nlðUiÞ�f nlð�UiÞÞ ð18bÞ

8iAf1 . . .Mg where M is the number of retained eigenvectors.A similar technique can be employed to determine stiffnesscoefficients with two unequal lower indices. Prescribing thedisplacement fields in the form u¼ 7Uj7Uk and summing thecomputed associated nonlinear stiffness forces f nlðuÞ, one obtainsthe relations:

2bijk ¼UT

i ðf nlðUjþUkÞþf nlð�Uj�UkÞÞ�2bijj�2bi

kk, ð19aÞ

2gijjk ¼�UT

i ðf nlð�Uj�UkÞþf nlðUj�UkÞÞþ2bijjþ2bi

kk�2gikkk, ð19bÞ

2gijkk ¼UT

i ðf nlðUjþUkÞþf nlðUj�UkÞÞ�2bijj�2bi

kk�2gijjj ð19cÞ

8i,j,kAf1 . . .Mg and jok. Note that coefficients of this typeappear only if MZ2. Finally, for cases where the number ofretained eigenvectors is greater than or equal to three ðMZ3Þ,coefficients with three unequal lower indices may be determinedby prescribing the displacement fields u¼UjþUkþUl andread

gijkl ¼UT

i f nl UjþUkþUl

� ��bi

jj�bikk�b

ill�b

ijk�b

ijl�b

ikl

�gijjj�g

ikkk�g

illl�g

ijjk�g

ijjl�g

ikkl�g

ijkk�g

ijll�g

ikll ð20Þ

8i,j,k,lAf1 . . .Mg and joko l. With the nonlinear coefficients (18),(19) and (20), we have completely defined the upper triangularform of the nonlinear modal governing equations (14). Their

solutions can be undertaken through a large variety of techniques.In this paper, we will use an original purely frequential continua-tion procedure.

Remark. For practical purposes, an important number of coeffi-cients bi

jk and gijkl are equal to zero. For an obvious time

computation issue, it is relevant to discard these negligible termsfrom the reduced order model. Numerically, a solution is to sortout the non-negligible terms by comparing the ratio between thecoefficients and the average of the natural frequencies involvedby the latters (the coefficient gi

jkl involves the natural frequenciesoi, oj, ok and ol).

3.4. Continuation procedure

The nonlinear system of differential equations (14) is solvedwith the harmonic-based continuation software MANLAB [45]. Thisnumerical tool, based on the classical harmonic balance method(HBM) and asymptotic numerical method (ANM), allowed us tocontinue the harmonic contributions of periodic solutions [11]. Ateach continuation step, the stability is assessed by computing andsorting the eigenvalues of the Hill matrix, which is simply theharmonic state operator of the computed periodic Jacobian of thedynamical system (14) as explained in [12].

Dealing with structures at the nanoscale and with first-orderderivatives, the first mandatory step in the continuation proce-dure is to transform the finite element reduced order model (14)in a dimensionless form. We define L0 as a characteristic length ofthe modeled structure and ai as a scalar such that ai ¼maxðUiÞ.Using T0 ¼ 1=o1 as a characteristic time constant for the mechan-ical oscillator (14) and by normalizing the eigenvectors Ui suchthat U i ¼ aiUi, we define the dimensionless variables

t ¼o1t, u ¼u

L0, qi ¼

qi

L0ai, o i ¼ T0oi, F i ¼

T20

L0aiFi,

wðpÞi ¼T2

0

L0aiwðpÞi , Y

ðpÞ

ij ¼T2

0aj

aiYðpÞij ,

bi

jk ¼ T20L0

ajak

aibi

jk, gijkl ¼ T2

0L20

ajakal

aigi

jkl: ð21Þ

At this stage, we do not consider the electrical equation (14b)which may be of interest if the patches are connected to anexternal electrical circuit. After simple calculations, the mechan-ical equilibrium (14a) may be rewritten in the first order dimen-sionless form of 2M equations (for a sake of clarity, variableswritten with our without bars will be considered dimensionlessin the following):

_qi ¼ pi,

_pi ¼�2xioipi�o2i qi�

XMj ¼ 1

XMkZ j

bijkqjqk�

XMj ¼ 1

XMkZ j

XMlZk

gijklqjqkql

�XP

p ¼ 1

wðpÞi V ðpÞ�XP

p ¼ 1

XMj ¼ 1

YðpÞij qjVðpÞþFi:

8>>>>>>>><>>>>>>>>:

ð22Þ

The leading idea of the harmonic-based continuation methodused in [11,12] is to systematically recast the dynamical system(22) into a quadratic polynomial form. This procedure is fullyexplained in [11], which also shows that this quadratic recast canbe applied to a large class of smooth systems with a few algebraicmanipulations and a few additions of auxiliary variables. Toillustrate the technique, we give an example of quadratic recastwhen the structure is excited by a mechanical harmonic forcingf eðtÞ ¼ f 0 cosðotÞ (Section 4) and by several sinusoidal voltagesV ðpÞ ¼ V ðpÞ0 cosðotÞ (Section 5). In that case, the new quadratic

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–5140

system can be written as follows:

_qi ¼ pi,

_pi ¼ Fi0 cosðotÞ�2xioipi�o2

i qi�XP

p ¼ 1

wðpÞi V ðpÞ0 cosðotÞ

�XP

p ¼ 1

XMj ¼ 1

YðpÞij qjVðpÞ0 X�

XMj ¼ 1

XMkZ j

bijkqjqk�

XMj ¼ 1

XMkZ j

XMlZk

gijklqjRkl,

0¼ Rkl�qkql,

0¼ cosðotÞ�X,

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð23Þ

where the new unknown vector z¼ ½q1 . . . qL p1 . . . pL R11 R12 . . .

RLL X�T contains the original components of the state vector½q1 . . . qL p1 . . . pL�

T and some new variables added to get thequadratic form. A periodic solution of (23), of minimal periodT ¼ 2p=o, is expanded into the Fourier series

z0ðtÞ ¼ z0þXH

k ¼ 1

zkc cosðkotÞþ

XH

k ¼ 1

zks sinðkotÞ ¼

XH

k ¼ �H

zk0eikot : ð24Þ

Using relation (24) in the governing equation (23) leads to analgebraic nonlinear system of ð2Hþ1Þ � ð2Mþ1þðMþM2

Þ=2Þequations. From then, the ANM continuation method enables tocompute the zk

0, k¼�H, . . . ,H for various contiguous values ofcontrol parameter l¼o, detect possible bifurcations andcompute different branches of solutions. To assess the stabilityof these solutions, one needs to express the 2M � 2M Jacobianof the dynamical system (22). To keep the computation efficiencyof the continuation software [12], it is written in the quadraticform:

JðtÞ ¼0 I

�K0 �C0

" #þ

0 0

�KcðtÞ�KLðtÞ 0

" #þ

0 0

�KQ ðtÞ 0

" #, ð25Þ

where K0, C0, KcðtÞ, KLðtÞ and KQ ðtÞ are expressed in Appendix B.In the above definitions, all matrices except J are of size M�M, 0being filled with zeros and I being the identity matrix. Thequadratic first order dynamical system (23) and its Jacobian(25) are implemented in the MANLAB opensource softwareprogram [45]. MANLAB is an interactive software program forthe continuation and bifurcation analysis of algebraic systems,based on ANM continuation. The latest version is programmed inMatlab using an object-oriented approach enhanced by fortranacceleration. It has a graphical user interface (GUI), with buttons,online inputs and graphical windows for generating, displayingand analyzing the bifurcation diagram and stability of the system.In the following, the nonlinear vibrations of two supported beamsis treated by solving their finite element reduced order modelswith this software.

P1(t)

L/4 L/2 L/4

P2(t)

Fig. 3. Periodic vibration of geometrically nonlinear structures from R. Lewandowski [

curve for c=2o1 ¼ 0:005.

4. Test example: N.L. vibration of homogeneous beam

In this section, we validate the previous theoretical develop-ments by computing the nonlinear oscillations of a simplysupported homogeneous beam studied in [29,30]. This examplebeing relatively simple, we are able to compare the resultsobtained with some classical analytical or finite element methodswith our reduced order model.

4.1. System under study

The problem is to determine the nonlinear resonance curvefor the simply supported beam subjected to external mech-anical forces shown in Fig. 3 (left). The axial displacement isalso prevented at each end. It is assumed a viscous dampingmatrix Dm proportional to the mass matrix so that Dm ¼ cMm

where c=2o1 ¼ 0:005 and o1 is the first natural frequency. Theharmonic excitation forces read P1ðtÞ ¼ 13:63EIr=l3 cos lt andP2ðtÞ ¼ 9:62EIr=l3 cos lt where EI denotes the bending rigidity ofthe beam and r¼

ffiffiffiffiffiffiffiI=A

pis the radius of gyration. The results, first

computed in [29], are presented in Fig. 3 (right) where themodulus of non-dimensional amplitude of the first harmonic atx¼ 0:75L versus the non-dimensional frequency l=o1 is shown.In this work, the author computed the matrix amplitude equationof the discretized beam (eight elements) thanks to a Galerkinmethod. The resonance curve was obtained with an incremental-iterative continuation method and the stability was assessed withFloquet theory. Only the second and third harmonics were takeninto account in the periodic solution but that was sufficient toqualitatively predict the internal resonance emerging from thesystem (visible in Fig. 3 (right) as the small slope).

We model the system of R. Lewandowsky with the methodpresented in Sections 2 and 3. In this purely mechanical problem,the electromechanical finite element formulation (9) reads simply:

Mm €uþDm _uþKmuþf nlðuÞ ¼ f e, ð26Þ

where each matrix has been previously defined. In the following, weuse a rectangular beam with a length l¼1 m, a width b¼0.1 m, athickness h¼0.001 m, a Young modulus E¼100 MPa and a densityr¼ 2000 kg=m3.

4.2. Modal basis truncation

The eigenmodes of (26) are computed with the classicaleigenvalue problem (10). Fig. 4 represents the first flexural modesUf and axial modes Ua computed with Matlab for a simplysupported beam discretized with 512 elements. The beam beinghomogeneous, the classical membrane and bending modes aretotally decoupled.

29]. (Left) Simply supported beam under the excitation forces. (Right) Resonance

0 0.5 10

1

2

3

4

Flexural mode 10 0.5 1

−4

−2

0

2

4

Flexural mode 20 0.5 1

−4

−2

0

2

4

Flexural mode 3

wu

0 0.5 1−2

0

2

4

Axial mode 10 0.5 1

−4

−2

0

2

4

Axial mode 20 0.5 1

−4

−2

0

2

4

Axial mode 3

Fig. 4. Normal modes of the simply supported beam computed with 512 elements. First row: flexural modes. (Left) of 1 ¼ 3:21 Hz. (Middle) of 2 ¼ 12:82 Hz. (Right)

of 3 ¼ 28:86 Hz. Second row: axial modes. (Left) oa1 ¼ 3536 Hz. (Middle) oa2 ¼ 7072 Hz. (Right) oa3 ¼ 10 609 Hz.

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

0.5

1

1.5

2

2.5

Frequency λ/ω1

Am

plitu

de W

/r at

0.7

5L

ANM FE reductionANM analyticFE upFE down

SN1 SN2

NS

Fig. 5. Nonlinear frequency resonance around mode 1 obtained with different numerical method: ANM continuation of finite element reduced order model with three

flexural modes, six axial modes and H¼10, ANM continuation of classical analytical reduced order model with three modes and H¼10 simulated harmonics, direct

integration with 32 finite elements (Newmark average acceleration method þ Newton–Raphson). Inset: Floquet multipliers of the perturbed solution represented in the

unit circle. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–51 41

Expanding Eq. (26) on its modal basis ½Uf Ua� leads to thefollowing purely mechanical reduced order model:

€qiþ2xioi _qiþo2i qiþ

XMj ¼ 1

XMkZ j

bijkqjqkþ

XMj ¼ 1

XMkZ j

XMlZk

gijklqjqkql ¼ Fi,

ð27Þ

8i,j,k,lAf1 . . .Mg where M is the number of retained eigenvectors.All terms in Eq. (27) are computed according to Section 3 andxi ¼ c=2oi.

4.3. Nonlinear vibratory behavior

Following the continuation procedure described in Section 3.4,we compute the harmonic contributions of each modal coordinatesqi(t) of Eq. (27) from the quadratic dynamical system (23) with

V ðpÞ0 ¼ 0 8p. The stability of the periodic solution is determinedthrough the quadratic Jacobian (25) with V ðpÞ0 ¼ 0 8p for eachcontrol parameter l. The nonlinear oscillations of the beam areobtained by reconstructing the displacement vector with themodal recombination (13). The modulus of the maximal non-dimensional amplitude w=r of the beam oscillation at x¼ 0:75L

versus the non-dimensional excitation frequency l=o1 is shown inFig. 5 ðo1 ¼of 1 ¼ 3:21 Hz is given in Fig. 4). The resonance curveand its stability have been obtained with the MANLAB continuationsoftware when keeping the three first flexural modes and six firstaxial modes of the beam in the reduced order model. The numberof simulated harmonics in the periodic solution is H¼10. Stablesolutions are plotted with full lines when unstable ones arecharacterized by dotted lines.

The resonance curve of Fig. 5 is in good agreement with theone obtained by R. Lewandowski, the qualitative differences

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–5142

between Figs. 5 and 3 (right) being only due to the choice ofplotted amplitudes. The stability regions of the saddle-nodebifurcations are well predicted thanks to our original purelyfrequential stability analysis [12]. Besides the T-periodic instabil-ity regions of solutions with period T ¼ 2p=l predicted by theFloquet multipliers (inset of Fig. 5), we detect a small quasiper-iodic instability region leading to a Neimark–Sacker bifurcationnear the internal resonance (point NS in Fig. 5). This instability isfound with both analytical and finite element reduced ordermodels and was not observed in [29]. The crosses in Fig. 5 areobtained with a direct integration of the complete finite elementmodel with 32 elements by mixing a Newmark average accelera-tion algorithm and a Newton–Raphson method. The continuousfull and dotted blue lines result from the continuation of ananalytical reduced order model with 3 modes and 10 harmonicswhich similar formulations can be found in [19] for the nonlinearvibrations of a MEMS biosensor and in [16,46] for the nonlinearoscillations of a forced string. As visible, the results obtained byeach method are very close. However, one has to keep in mindthat the analytical model is only applicable because of the simplegeometry of the studied structure and the direct finite elementmethod do not compute every solutions of the problem (only thestable ones) since Eq. (26) is numerically integrated from theappropriately chosen initial condition up to time when thesteady-state is reached. Moreover, the computation time of onesteady-state solution using the direct finite element time integra-tion method is greater than the time needed for the continuation

0 0.5 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

W/h

X/L0 1 2

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time t/T

Fig. 6. Nonlinear vibratory response at the SN1 point of Fig. 5. (Left) Vibratory behavio

flexural and axial modes.

0 0.5 1−1

−0.5

0

0.5

1

W/h

X/L0 1 2

−1

−0.5

0

0.5

1

Time t/T

Fig. 7. Nonlinear vibratory response at the SN2 point of Fig. 5. (Left) Vibratory behavio

flexural and axial modes.

of our reduced order models. It is however difficult to emphasizethe amount of time saved with our method in every cases sincethe necessary quadratic recast of Section 3.4 mixed with theharmonic balance method may significantly increase the numberof equations to be solved and the MANLAB 1.0 continuationsoftware used in this paper had some slowness due to theobject-oriented approach in the Matlab language. In the futureapplications, the computation time should not be an issue any-more since the MANLAB 2.0 version enhanced computation usingfortran acceleration [45].

Figs. 6 and 7 give an insight in the vibratory behavior of thebeam at two critical points. The use of the short-circuit normalmodes of Section 3.1 in our reduced order model is very relevantin our physical understanding of the problem. In particular, thebar chart of Fig. 6 shows that the SN1 point of the resonance curvedisplays a one-to-three internal resonance, i.e. a nonlinear cou-pling between the first harmonic of the first flexural mode and thethird harmonic of the second flexural mode. Note that thiscomplex nonlinear behavior could have been deduced from thelinear analysis of Fig. 4 where of 2 � 3of 1. The SN2 point of theresonance curve is simply the primary resonance of the firstflexural mode as shown in Fig. 7. Another advantage of thisnumerical method is that the convergence, in terms of harmonicsnumber and modal truncation, can be precisely checked. For3 flexural modes, 6 axial modes and H¼10 simulated harmonics,the solution is well converged since the bar graphs of Figs. 6 and 7show that only the first harmonics of the two first flexural modes

H0 H1 H2 H3 H4 H5 H6 H7 H8 H9H10−0.5

0

0.5

Flex

ural

mod

es

H0 H1 H2 H3 H4 H5 H6 H7 H8 H9H10−2

0

2 x 10−4

Axi

al m

odes

r of the supported beam. (Right) Amplitude of the harmonic contributions of the

H0 H1 H2 H3 H4 H5 H6 H7 H8 H9H10−0.5

0

0.5

Flex

ural

mod

es

H0 H1 H2 H3 H4 H5 H6 H7 H8 H9H10−2

0

2 x 10−4

Axi

al m

odes

r of the supported beam. (Right) Amplitude of the harmonic contributions of the

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–51 43

and the four first axial modes have a significant amplitude in theexcitation frequency range. Note that both flexural modes andaxial modes exist in the periodic oscillations at resonance.

4.4. Finite element and modal convergence

The finite element convergence is investigated in Fig. 8. As aclassical result, the smaller is the considered computed naturalfrequency, the faster it converges to the exact analytical solution.An Euler–Bernoulli beam model with 10 elements is sufficient toapproximate the sixth first natural frequencies at less than 1%from their exact values. Fig. 8 (right) shows the convergence ofthree nonlinear cubic coefficients gi

jkl towards their analyticalcounterparts. As in the frequency convergence, the higher is themode involved in the coefficient calculation, the slower is theconvergence. But in this case, we need much more finite elementsto compute the nonlinear stiffness coefficients at less than 1%from their exact values. Moreover, the convergence is not smoothcontrary to the one of Fig. 8 (left). The reason is that thecomputation of nonlinear coefficients is based on the restorationof nodal applied forces (see Section 3.3). The modeling accuracy ofthese internal nodal forces depends on the number of elements,and the same is true for the convergence. As a consequence, theconvergence has a ‘‘sawtooth’’ shape where the period dependson the considered coefficient. The higher is the mode involved,

10 15 20 25 30

1

1.05

1.1

Number of elements

ωFE

/ωan

alyt

ic

ω1

...ω8

Fig. 8. Finite element convergence. (Left) On the natural frequenci

0 20 40 60 800

0.5

1

1.5

2

2.5x 104

Mode number

ωk [

MH

z]

Flexural modesAxial modes

Fig. 9. Modal convergence. (Left) Localization of the retained eigenmodes in the mo

resonance curves (with two flexural modes and H¼10 simulated harmonics).

the larger is the period and the slower is the convergence. Thisawkward finite element dependence is not a real issue in our casesince the computation of the coefficients is made only once andthe number of retained eigenmodes is relatively small. However,this feature may explain the difficult convergence encounteredwhen dealing with more complicated two-dimensional or three-dimensional structures such as in [38].

Fig. 9 shows the influence of modal truncation on the con-vergence of the resonance curve of the excited beam. As pre-viously seen in Figs. 6 and 7, the axial modes are present in theperiodic oscillation. Actually, one needs to retain some flexuralmodes in the solution to represent the vibratory behavior but it isalso essential to keep sufficient axial modes to correctly modelthe flexural/membrane coupling of the system. The chosenmethod consists in retaining the first flexural modes and axialmodes of the system up to the convergence. Fig. 9 (left) illustratesthe location of the retained decoupled eigenmodes in the modalspectrum. In this simple test example, two flexural modes andfour axial modes are sufficient to quantitatively compute theactual resonance curve of the beam (Fig. 9 (right)).

5. A piezoelectric layered nanobridge

The goal of this section is to predict the nonlinear oscillatorybehavior of the piezoelectric layered nanobridges built by our

0 20 40 60 80 1000.9

1

1.1

1.2

1.3

1.4

Number of elements

Γ FE/Γ

anal

ytic

Γ1111

Γ2222

Γ3333

es. (Right) On the nonlinear modal stiffness cubic coefficients.

1 1.2 1.4 1.6 1.80

0.5

1

1.5

2

2.5

Frequency λ/ω1

Am

plitu

de W

/r at

0.7

5L

6 axial modes4 axial modes2 axial modes0 axial modes

dal spectrum. (Right) Influence of the number of axial modes on the computed

Piezoelectric patch7 layers

Silicon beam2 layers

lblp

500

nm

10 �m

PZT

TiPt

SiSiO2

Fig. 10. A piezoelectric layered nanobeam. (Left) Scanning electron microscope image of several silicon nanobridges after liberation. Picture courtesy of LAAS, France.

(Right) Sketch of the studied sample.

Table 1Mechanical and geometrical properties of each layer of the nanobridge [48].

Material Thickness (nm) r (kg m�3) E (GPa) n

Si 340 2500 169 0.3

SiO2 10 2150 70 0.17

Ti 10 4510 110 0.32

Pt 80 21 450 145 0.35

PZT 110 7800 96 0.45

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–5144

physicist colleagues, from the LAAS laboratory in France, in orderto help in the design of these upcoming devices. The beam ismodeled with the finite element reduced order model explainedin Sections 2 and 3 and validated in Section 4. The piezoelectricactuation of the slender layered structure lead to original andinteresting vibratory phenomena dependent on the patch length.

5.1. System under study

The scanning electron microscope image of the devices understudy is shown in Fig. 10 (left). The nanobridges are fabricated byemploying stepper UV lithography on silicon on insulator (SOI)wafers [47]. From then, the technical challenge will be to depositseveral activated electric layers at the nanoscale in order topiezoelectrically actuated the structure. For the numerical com-putations, we consider the slightly simplified sketch of theexpected structures shown in Fig. 10 (right). Notably, the layerscomposing the piezoelectric patch have the same length andwidth and their thicknesses are constant along the patch. In thatcase, the patch is defined by only one parameter, its length lpwhich is a fraction of the beam length lb. We assume perfectlyclamped boundary conditions and no slidings between each layer.

The mechanical and electrical properties of each layer is givenin Table 1. The length of the beam is lb ¼ 10 mm but the patchlength is not defined yet. Having no electrical external circuitconnected to the patch and no mechanical external forces, theelectromechanical finite element formulation of our model issimply

Mm €uþKmuþf nlðuÞþ½f cþKcu�V ¼ 0, ð28Þ

where each matrix has been previously defined and with V

prescribed.

5.2. Linear analysis

The eigenmodes of (28) are computed with the classicaleigenvalue problem (10). Contrarily to the test example ofSection 4, this eigenvalue problem is dependent on the patchlength lp. Fig. 11 shows the computed eigenfrequencies as

function of the patch length ratio lp=lb for a beam discretizedwith 512 elements. As visible, the influence of the patch geometryis important and some of the configurations may lead to non-linear internal resonances if the frequency ratio oi=o1 is close toan integer value. This parametric linear analysis is a veryconvenient way to have a first insight in the intrinsic behaviorof the design nanobridges.

Fig. 12 represents the first ‘‘flexural modes’’ Uf and ‘‘axialmodes’’ Ua computed with Matlab for a laminated nanobridgewith lp=lb ¼ 0:75 discretized with 512 elements. Due to thepresence of layers, the axial and transverse motions are linearlycoupled and it is no more possible to refer to classical membraneand bending modes such as in Section 4. Since axial contributionis fundamental to model the flexural/membrane coupling of thenonlinear oscillations, we need to include some ‘‘axial modes’’ inthe modal response. In this paper, we discriminate between themodes U by comparing the integral of u(x) and w(x) along thelength of the beam L. If the area below w(x) is superior to the onebelow u(x), the mode is a ‘‘flexural mode’’, otherwise, the mode isan ‘‘axial mode’’. Different visual methods have been tried to sortthe Uf and Ua modes, though always based on the u(x) and w(x)contributions, but the present method was the most efficient.Expanding Eq. (28) on its modal basis ½Uf Ua� leads to theelectromechanical reduced order model:

€qiþ2xioi _qiþo2i qiþ

XMj ¼ 1

XMkZ j

bijkqjqkþ

XMj ¼ 1

XMkZ j

XMlZ j

gijklqjqkql

þwiVþXMj ¼ 1

YijqjV ¼ 0, ð29Þ

8iAf1 . . .Mg where M is the number of retained eigenvectors andfor one patch only. All terms in Eq. (29) are computed accordingto Section 3 and the modal damping xi and applied voltage V arefixed in adequacy to the experimental expectations.

5.3. Nonlinear frequency response for lp=lb ¼ 3=4

In order to avoid possible nonlinear internal resonances (seeFig. 11), we compute the nonlinear vibrations of a nanobridgewith lp=lb ¼ 3=4. We choose a modal damping xi ¼ 0:01�o1=oi

corresponding to a relatively small quality factor Q ¼ 1=2x¼ 50,and a harmonic applied voltage V ¼ V0 cosðOtÞ with V0 ¼ 5 V.The nonlinear coefficients are calculated with 512 elements.Following the continuation procedure described in Section 3.4,we compute the harmonic contributions of each modal coordi-nates qi(t) of Eq. (29) and their stability by using Eqs. (23) and (25)with Fi

0 ¼ 08i (no mechanical external forces). For each controlparameter O, the nonlinear oscillations of the nanobridge can

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

300

lp/lb

ωi [

MH

z]

ω1

ω2

ω3

ω4

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

lp/lb

ωi/ω

1

Fig. 11. Four first natural frequencies of the nanobridge as a function of patch length ratio lp=lb . (Left) Physical values. (Right) Dimensionless ratio oi=o1.

0 5 10−10

−5

0

5 x 10−3

Flexural mode 10 5 10

−0.01

−0.005

0

0.005

0.01

Flexural mode 20 5 10

−0.015−0.01

−0.0050

0.005

Flexural mode 3

wu

0 5 10−10

−5

0

5 x 10−3

Axial mode 10 5 10

−0.015−0.01

−0.0050

0.005

Axial mode 20 5 10

−0.01

−0.005

0

0.005

0.01

Axial mode 3

Fig. 12. Normal modes of the layered nanobridge with lp=lb ¼ 0:75 (computed with 512 elements). First row: ‘‘flexural modes’’. (Left) of 1 ¼ 23:5 MHz. (Middle)

of 2 ¼ 67:7 MHz. (Right) of 3 ¼ 142:1 MHz. Second row: ‘‘axial modes’’. (Left) oa1 ¼ 203:7 MHz. (Middle) oa2 ¼ 459:2 MHz. (Right) oa3 ¼ 741:5 MHz.

0.9 1 1.1 1.2 1.30

0.2

0.4

0.6

0.8

1

Frequency Ω/ω1

Am

plitu

de W

/hS

i at 0

.75L

nw = 3, nu = 0nw = 3, nu = 8nw = 4, nu = 12nw = 6, nu = 16FE upFE down

0 10 20 300

1

2

3x 104

Mode number

ωk

[Mhz

]

Flexural modesAxial modes

H0 H1 H2 H3 H4 H5−5

0

5x 104

Flex

ural

mod

es

H0 H1 H2 H3 H4 H5−200

0

200

Axi

al m

odes

SN1

Fig. 13. Nonlinear frequency response around mode 1 at x¼ 0:75lb for lp=lb ¼ 3=4. (Left) Influence of modal truncation on frequency response. (Right) Amplitude of the

H¼5 simulated harmonics with 6 ‘‘flexural modes’’ and 16 ‘‘axial modes’’ at the SN1 point of Fig. 13 (left). Localization of the retained eigenmodes in the modal spectrum.

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–51 45

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–5146

be obtained by reconstructing the displacement vector withthe modal recombination (13). The nonlinear frequency responsearound the first ‘‘flexural mode’’ is shown Fig. 13 when H¼5harmonics are retained in the solution. The modulus of themaximal non-dimensional amplitude (amplitude divided by thesilicon thickness hSi) of the beam oscillation at x¼ 0:75lb versusthe non-dimensional excitation frequency O=o1 (o1 ¼of 1 ¼

23:5 MHz is given in Fig. 12) displays a classical resonance curveof hardening type whatever the number of retained ‘‘axial modes’’.However, one needs an important number of these ‘‘axial modes’’ tofind a quantitative convergent solution with a correct stabilityregion. For nw¼6 and nu¼16, the resonance curve is almostconverged in terms of harmonics and tends to the steady-stateresponse computed with the direct finite element time integrationmethod used in the previous section (crosses in Fig. 13 (left)). In thishighly layered case, the contribution of what we define as ‘‘axialmodes’’ is not as efficient as in the homogeneous case and we needmuch more of these modes to harden the nonlinear resonance curveto the exact solution. Note that the maximum number of truncatedmodes was limited by the computational time of the first version ofthe MANLAB software.

1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequ

Am

plitu

de W

/hS

i at 0

.75L

Mode 1 externally forcedMode 1 parametrically forcedMode 2 externally forced

SN1

Fig. 14. Maximum amplitude of the vibratory response of the nanobridge with lp=lb

0 5 10

−1

0

1

w/h

Si

lb

0 1 2

−1

0

1

Time t/T

w/h

Si a

t 0.7

5L

0 5−2

0

2

w/h

Si

l

0 1−2

0

2

Time

w/h

Si a

t 0.7

5L

Fig. 15. Nonlinear vibratory response at resonances of Fig. 14. (Le

The computed extended nonlinear frequency response of thepiezoelectric layered beam is given Figs. 14 and 15. Notably, Fig. 14represents the modulus of the maximal non-dimensional amplitudeof the nanobridge oscillation at x¼ 0:75L versus the non-dimen-sional excitation frequency O=o1. An insight of the vibratorybehavior at resonances is shown in Fig. 15. Due to the applied ACvoltage, the system displays some nonlinear hardening resonanceswhen the excitation frequency is close to the natural frequency of a‘‘flexural mode’’. At the SN1 point, the beam vibrates mostly on itsfirst mode with a fundamental frequency O (Fig. 15 (left)). At theSN3 point, the beam still oscillates with a period 2p=O but mostlyon its second flexural mode (Fig. 15 (right)). Note that these primaryresonances are slightly nonlinear or anharmonic. Several smallsecondary resonances can be observed. The second ‘‘flexural mode’’vibrates with a fundamental frequency 2O around O¼ 1:4o1 whenthe secondary resonance of the third ‘‘flexural mode’’ emergesaround O¼ 3o1. Because of our chosen geometrical ratio lp=lb andthe relatively small excitation amplitude, no internal resonancesappear in the system.

As expected in Section 2, Fig. 14 shows that the T-periodicsolution becomes unstable around O¼ 2o1 and bifurcates to a

2 2.5 3ency Ω/ω1

SN2

SN3

¼ 3=4 at x¼ 0:75lb for 5 harmonics, 6 ‘‘flexural modes’’ and 16 ‘‘axial modes’’.

10b

2 t/T

0 5 10

−0.5

0

0.5

w/h

Si

lb

0 1 2

−0.5

0

0.5

Time t/T

w/h

Si a

t 0.7

5L

ft) At point SN1. (Middle) At point SN2. (Right) At point SN3.

3.2 3.4 3.6 3.8 4 4.20

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency Ω/ω1

Am

plitu

de W

/hS

i at 0

.75L

0 5 10

−0.4

−0.2

0

0.2

0.4

0.6

w/h

Si

lb0 1 2

−0.4

−0.2

0

0.2

0.4

0.6

Time t/T

NS

Fig. 16. Nonlinear vibratory response of the nanobridge with lp=lb ¼ 0:5 for 6 harmonics, 5 ‘‘flexural modes’’ and 15 ‘‘axial modes’’. (Left) Maximum amplitude of the

vibratory response at x¼ 0:75lb . Inset: Floquet multipliers of the perturbed solution represented in the unit circle. (Right) Vibratory behavior at point NS.

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–51 47

2T-periodic solution. At the SN2 point, the structure vibrates mostlyon its first ‘‘flexural mode’’ with a fundamental frequency O=2,i.e. a period 2T, as shown in Fig. 15 (middle). Due to the parametricnature of the excitation, the amplitude of the 2T-periodic steady-state response exceeds the amplitude of the classic primaryresonances. This original parametric feature has been experimen-tally investigated and exploited in [49]. Indeed, as the readout ofmechanical motion becomes increasingly difficult as NEMS deviceare miniaturized, this parametric amplification (amplitude at SN2point) can be useful to amplify the output signal in the mechan-ical domain. Finally, note that, the bifurcated 2T-periodic branchis numerically continued by adding the O=2 harmonic contribu-tions in the T-periodic solution. As a consequence, the computa-tion time becomes longer and due to the continuation softwarelimitations, this branch has been computed with eight harmonics,three ‘‘flexural modes’’ and eight ‘‘axial modes’’. This relativelysmall modal and frequential truncature may explained the cur-ious behavior at the maximal value of the 2T-periodic branch.

5.4. Influence of patch length on the frequency response

Fig. 16 shows the nonlinear resonance curve of the doubledclamped piezoelectric beam with lp=lb ¼ 0:5 for H¼6, nw¼5 andnu¼15. The modal damping is still xi ¼ 0:01�o1=oi but we apply aharmonic voltage V ¼ V0 cosðOtÞ with V0 ¼ 2 V. The system displaysa two-to-four internal resonance, i.e. a nonlinear coupling betweenthe first harmonic of the second ‘‘flexural mode’’ and the fourthharmonic of the fourth flexural mode. As a consequence, the beamvibrates mostly on its second flexural mode with a frequencyO�of 2 but there is a non-negligible oscillation on the fourthflexural mode with a frequency of 4 � 3of 2. Note that this complexnonlinear behavior could have been deduced from the linear analysisof Fig. 11 where of 4 � 3of 2. Moreover, for some control parameterO, the Floquet multipliers tend to leave the unit circle with a non-zero imaginary part, predicting a Neimark–Sacker bifurcation of thesteady-state solution. This means that during this resonance, a newfrequency could appear in the solution. It is beyond the scope of thisarticle to seek the new bifurcation branches and solutions. But thissmall example shows us an interesting mechanical behavior for thedesign of the upcoming devices. We may want to avoid theseprimary, parametric and internal resonances for the integrity of thestructures but we may also make the most of these nonlinearphenomena. Indeed, these piezoelectric nanobridges behave aselectronic high-frequency multipliers where the relation betweenthe output and output signal is simply defined by the length of thepatch. By playing with several patches, we should be able to realizeeven more difficult input–output operations.

6. Conclusions and discussions

An original finite element reduced order model for thenonlinear vibrations of piezoelectric layered beams has beenpresented. In this model, the geometrical nonlinearities are takeninto account through a von Karman nonlinear strain–displace-ment relationship. The originality of the finite element electro-mechanical formulation is that the system electrical state is fullydescribed by only a couple of variables per piezoelectric patches,namely the electric charge contained in the electrodes and thevoltage between the electrodes. Due to the geometrical nonli-nearity, the piezoelectric actuation introduces an original para-metric excitation term in the equilibrium equation. A reduced-order formulation of the discretized problem is obtained byexpanding the mechanical displacement unknown vector ontothe short-circuit eigenmode basis. Beyond its physical meaning,this basis is computed with the classical linear elastic mechanicalproblem, only once for a given geometrical configuration.A particular attention is paid to the computation of the unknownnonlinear stiffness coefficients of the reduced-order model. Dueto the particular form of the von Karman nonlinearities, thesecoefficients are computed exactly, once for a given geometry, byprescribing relevant nodal displacements in nonlinear staticsolutions settings. Finally, the previously defined low-ordermodel is recast in a quadratic dimensionless form for computingthe periodic oscillations (and theirs stability) of the electrome-chanical system with an original harmonic-based continuationmethod, particularly well-adapted for the analysis of NEMSgenerally characterized by very high quality factor. The presentedmethod has been validated by computing the nonlinear vibrationsof a mechanically excited homogeneous beam supported at bothends referenced in the literature. The more difficult case of thenonlinear oscillations of a layered nanobridge piezoelectricallyactuated has also been studied. Interesting vibratory phenomenasuch as parametric amplification or patch length dependence ofthe frequency output response have been highlighted in order tohelp in the design of these nanodevices.

Although the presented self-sufficient method appears tobe a very promising tool in the design of nonlinear oscillatingbeam structures such as piezoelectrically actuated NEMS, furtherinvestigations should be considered to improve the numericalmethod. Beyond the time computation inconveniences due to theemployed continuation software version, it should be possible toaccelerate the computation by discarding the negligible nonlinearmodal stiffness coefficients. A simple method has been used forthe moment but this is not sufficient and the term ‘‘negligible’’needs to become clearer. This problem deserves a deeper

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–5148

investigation to develop an efficient automatic numerical methodof sorting. Another important theoretical issue remains thediscrimination between the axial and flexural modes in the caseof layered structures. More generally, a crucial question in thepresent reduction method is the choice of normal modes toinclude in the reduced order model. For our one-dimensionalmodels, this modal convergence issue may be relegated to thebackground just by keeping an important number of modes in thesolution. When extended the method to two-dimensional orthree-dimensional structures where the coupling between axialand flexural modes is even more complicated, this modal trun-cature issue becomes a fundamental question. A future directioncould be the use of modal derivatives or static modes, containingmuch beneficial reduction information, to enhance our solution.

Acknowledgments

This research is part of the NEMSPIEZO project, under fundsfrom the French National Research Agency (Project ANR-08-NANO-015-04), for which the authors are grateful.

Appendix A. Piezoelectric layered beam finite elementdiscretization

Details of the layered beam finite element discretization ofSection 2 are specified here, by starting from the variationalformulation of Eqs. (7a) and (7b). In the following, a superscript ora subscript e refers to elementary quantities. The generalizeddisplacements are discretized using linear (axial displacement)and cubic (transverse displacement) shape functions. Thus, axialand transverse displacements are related to the elementaryd.o.f.’s vector Ue by

ue ¼Nuue and we ¼Nwue, ðA:1Þ

where

ue ¼ ðu1 w1 y1 9 u2 w2 y2ÞT ,

u1, w1 and y1 (respectively u2, w2 and y2) corresponding to thefirst (respectively second) node of the beam element.

In Eq. (A.1), the interpolation vectors are defined as follows:

Nu ¼ ðN1 0 0 9 N2 0 0Þ

and

Nw ¼ ð0 N3 N4 9 0 N5 N6Þ

with, for all xA ½0,Le�, where Le is the elementary length,

N1 ¼ 1�x

Le, N2 ¼

x

Le,

N3 ¼ 1�x

Le

� �2

1þ2x

Le

� �,

N4 ¼ x 1�x

Le

� �2

, N5 ¼x2

L2e

3�2x

Le

� �,

N6 ¼x2

Le

x

Le�1

� �:

Note that Eqs. (2), (4) and (A.1) lead to the following expres-sion of elementary rotation, membrane strain and curvature

ye¼�N 0wue; ee ¼N 0uueþ1

2ðN0wueÞ

2 and ke ¼�N 00wue, ðA:2Þ

where ð�Þ0 ¼ @ð�Þ=@x¼ ð�Þ,x.The various terms appearing in the variational formulation of

Eqs. (7a) and (7b) are now successively discussed.

The kinetic energy variation is

dT ¼XN

k ¼ 1

ZOkrkð €uxduxþ €uzduzÞ dO

¼

Z xþ

x�½I0ð €uduþ €wdwÞþ I1ð €udyþ €yduÞþ I2

€ydy� dx

¼

Z xþ

x�ðdu dw dyÞ

I0 0 I1

0 I0 0

I1 0 I2

0B@

1CA

€u

€w€y

0B@

1CAdx, ðA:3Þ

where the zero, first, and second order moments of inertia aredefined by

½I0,I1,I2� ¼XN

k ¼ 1

rk½Jk0,Jk

1,Jk2�,

with the zero, first and second moment of area of each sublayerrespectively defined by

½Jk0,Jk

1,Jk2� ¼ bk

Z zk

zk�1

½1,z,z2� dz¼ bk hk,z2

k�z2k�1

2,z3

k�z3k�1

3

" #,

where bk and hk are the width and the thickness of the beamk-th layer. Applying the virtual work principle and assumingthe discretized variation of kinetic energy is dT e

¼ dueT Mem€ue

, theelementary mass matrix reads:

Mem ¼

Z Le

0ðNT

u NTw �N 0Tw Þ

I0 0 I1

0 I0 0

I1 0 I2

0B@

1CA

Nu

Nw

�N 0w

0B@

1CAdx,

whose explicit expression is

Mem ¼

I0Le

420

140 0 0 70 0 0

0 156 22Le 0 54 �13Le

0 22Le 4L2e 0 13Le �3L2

e

70 0 0 140 0 0

0 54 13Le 0 156 �22Le

0 �13Le �3L2e 0 �22Le 4L2

e

26666666664

37777777775

þI1

12

0 6 �Le 0 �6 Le

6 0 0 6 0 0

�Le 0 0 Le 0 0

0 6 Le 0 �6 �Le

�6 0 0 �6 0 0

Le 0 0 �Le 0 0

26666666664

37777777775

þI2

30Le

0 0 0 0 0 0

0 36 3Le 0 �36 3Le

0 3Le 4L2e 0 �3Le �L2

e

0 0 0 0 0 0

0 �36 �3Le 0 36 �3Le

0 3Le �L2e 0 �3Le 4L2

e

26666666664

37777777775: ðA:4Þ

The mechanical contribution to the internal energy variation is

dU ¼XN

k ¼ 1

ZOk

Yke1de1 dO¼Z xþ

x�½AedeþBðkdeþedkÞþDkdk� dx

¼

Z xþ

x�ðde dkÞ

A B

B D

� �e

k

� �dx, ðA:5Þ

where A is the extensional stiffness, D is the bending stiffness, andB is the bending–extensional coupling stiffness defined as

½A,B,D� ¼XN

k ¼ 1

Yk½Jk

0,Jk1,Jk

2�:

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–51 49

Following Eq. (A.2), the discretized formulation of the variation ofmembrane strain and curvature is given by

dee ¼N 0udueþN 0wueN 0wdue, ðA:6Þ

dke ¼�N 00wdue: ðA:7Þ

Replacing de, dk, e and k by their expressions and applying thevirtual work principle to the weak form (A.5) leads to thediscretized variation of the mechanical contribution to the inter-nal energy variation

dUe ¼ dueT KemueþdueT f e

nlðueÞ, ðA:8Þ

where the elementary linear stiffness matrix reads

Kem ¼

A

Le

1 0 0 �1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

�1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

2666666664

3777777775þ

B

Le

0 0 �1 0 0 1

0 0 0 0 0 0

�1 0 0 1 0 0

0 0 1 0 0 �1

0 0 0 0 0 0

1 0 0 �1 0 0

2666666664

3777777775

þD

L3e

0 0 0 0 0 0

0 12 6Le 0 �12 6Le

0 6Le 4L2e 0 �6Le 2L2

e

0 0 0 0 0 0

0 �12 �6Le 0 12 �6Le

0 6Le 2L2e 0 �6Le 4L2

e

26666666664

37777777775

ðA:9Þ

and where the explicit expression of the nonlinear internalelementary forces f e

nlðueÞ is given at the end of this Appendix.

Work done by the external mechanical forces

dW ¼XK

k ¼ 1

ZGk

t

ðtkxduxþtk

zduzÞ dSþXK

k ¼ 1

ZOkðf k

xduxþ f kzduzÞ dO

¼XK

k ¼ 1

ZSkðtk

xduþtkzdw�ztk

xdyÞ dS

� �xþ

x�

(þ

ZOkðf k

xduþ f kzdw�zf k

xdyÞ dO

¼ ½NduþTdw�Mdy�xþ

x� þ

Z xþ

x�ðnduþtdw�mdyÞ dx, ðA:10Þ

where Sk is the cross-section area of the layer k and where theboundary and distributed (i) normal resultant, (ii) shear resultantand (iii) bending moment are respectively defined by

½N,T,M� ¼XK

k ¼ 1

ZSk

tkx dS,

ZSk

tkz dS,

ZSk

ztkx dS

� �

and

½n,t,m� ¼XK

k ¼ 1

ZSk

f kx dS,

ZSk

f kz dS,

ZSk

zf kx dS

� �:

The elementary vector of generalized mechanical forces is then

f ee ¼ ½NNT

uþTNTwþMN 0Tw �

Le

0 þ

Z Le

0ðnNT

uþtNTwþmN 0Tw Þ dx,

whose explicit expression is

f ee ¼

N

T

M

0

0

0

2666666664

3777777775

x ¼ 0

þ

0

0

0

in

T

M

2666666664

3777777775

x ¼ Le

þnLe

2

1

0

0

1

0

0

2666666664

3777777775þ

tLe

12

0

6

Le

0

6

�Le

26666666664

37777777775þm

0

�1

0

0

1

0

2666666664

3777777775:

ðA:11Þ

The piezoelectric contributions of the internal energy varia-tion, related to the inverse effect, is

dUc ¼XP

p ¼ 1

V ðpÞ

hðpÞ

ZOðpÞ

eðpÞ31de1 dO¼XP

p ¼ 1

Z xþ

x�GðpÞ0 V ðpÞde dxþ

Z xþ

x�GðpÞ1 V ðpÞdk dx

" #

¼XP

p ¼ 1

Z xþ

x�ðde dkÞ

GðpÞ0

GðpÞ1

0@

1AV ðpÞ dx, ðA:12Þ

where GðpÞ0 and GðpÞ1 are defined by

½GðpÞ0 ,GðpÞ1 � ¼eðpÞ31

hðpÞ½JðpÞ0 ,JðpÞ1 �:

Thanks to the virtual work principle, the discretized variation ofthe piezoelectric contributions to the internal energy variationcan be written as

dUec ¼ dueT f eðpÞ

c þdueT KeðpÞc ue ðA:13Þ

leading to the elementary coupling vector

f eðpÞc ¼

Z Le

0ðN 0Tu �N 00Tw Þ

GðpÞ0

GðpÞ1

0@

1Adx¼ GðpÞ0

�1

0

0

1

0

0

2666666664

3777777775þGðpÞ1

0

0

1

0

0

�1

2666666664

3777777775

ðA:14Þ

and elementary coupling matrix

KeðpÞc ¼

Z Le

0N 0TwGðpÞ0 N 0w dx¼

GðpÞ0

30L

0 0 0 0 0 0

0 36 3Le 0 �36 3Le

0 3Le 4L2e 0 �3Le �L2

e

0 0 0 0 0 0

0 �36 �3Le 0 36 �3Le

0 3Le �L2e 0 �3Le 4L2

e

26666666664

37777777775:

ðA:15Þ

The same vectors and matrices are obtained for the direct effect.

f enlðu

eÞ ¼A

L3

L30 ð�18w2

1�3w1y1Lþ36w1w2�3w1y2L�2y21L2Þ

1280 ð�864w2

1w2þ864w1w22�288w3

2þ288w31

�y32L3�y3

1L3�216w1w2y2Lþ28L2u2y1þ28L2u2y2Þ

L840 ð18w1y1L2y2�18y1L2w2y2�324w2

1w2þ324w1w22

�108w32þ108w3

1�3y32L3þ24y3

1L3Þ

L30 ð18w2

1þ3w1y1L�36w1w2þ3w1y2Lþ2y21L2Þ

1280 ð864w2

1w2�864w1w22þ288w3

2�288w31þy

32L3þy3

1L3

þ216w1w2y2L�28L2u2y1�28L2u2y2Þ

L840 ð18w1y1L2y2�18y1L2w2y2�324w2

1w2þ324w1w22

�108w32þ108w3

1þ24y32L3Þ

2666666666666666666666664

3777777777777777777777775

þA

L3

L30 ð3y1Lw2þy1L2y2�18w2

2þ3w2y2L�2y22L2Þ

1280 ð336Lu2w1�336Lu2w2þ108w2

2y2L�36w2y22L2

�28L2u1y1þ108y1Lw22þ3y1L3y2

2�216w1y1Lw2�28L2u1y2Þ

L840 ð112L2u2y1�28L2u2y2þ84Lu2w1�84Lu2w2�9w2y

22L2

�112L2u1y1þ108y1Lw22þ6y1L3y2

2�216w1y1Lw2Þ

L30 ð�3y1Lw2�y1L2y2þ18w2

2�3w2y2Lþ2y22L2Þ

1280 ð�336Lu2w1þ336Lu2w2�108w2

2y2Lþ36w2y22L2

þ28L2u1y1�108y1Lw22�3y1L3y2

2þ216w1y1Lw2þ28L2u1y2Þ

L840 ð�3y3

1L3�216w1w2y2L�28L2u2y1þ112L2u2y2

þ84Lu2w1�84Lu2w2þ108w22y2Lþ9w2y

22L2þ28L2u1y1�9y1L3y2

2Þ

2666666666666666666666664

3777777777777777777777775

A. Lazarus et al. / Finite Elements in Analysis and Design 49 (2012) 35–5150

þA

L3

01

280 ð108w21y1Lþ108w2

1y2Lþ36w1y22L2þ36w1y

21L2

�36y21L2w2þ3y2

1L3y2�336Lu1w1þ336Lu1w2Þ

L840 ð28L2u1y2þ108w2

1y1Lþ9w1y22L2�9w1y

21L2

þ9y21L2w2�9y2

1L3y2�84Lu1w1þ84Lu1w2Þ

01

280 ð�108w21y1L�108w2

1y2L�36w1y22L2�36w1y

21L2

þ36y21L2w2�3y2

1L3y2þ336Lu1w1�336Lu1w2Þ

L840 ð�112L2u1y2þ108w2

1y2L�9w1y22L2þ9w1y

21L2

�9y21L2w2þ6y2

1L3y2�84Lu1w1þ84Lu1w2Þ

26666666666666666666664

37777777777777777777775

þB

0

0

1=2y21

0

0

�1=2y22

26666666664

37777777775: ðA:16Þ

Appendix B. Quadratic Jacobian of the reduced order model

Details of the quadratic Jacobian of Section 2.4 are specifiedhere:

JðtÞ ¼0 I

�K0 �C0

" #þ

0 0

�KcðtÞ�KLðtÞ 0

" #þ

0 0

�KQ ðtÞ 0

" #, ðB:1Þ

K0 ¼

o21 � � � 0

^ & ^

0 � � � o2L

264

375, C0 ¼ 2

x1o1 � � � 0

^ & ^

0 � � � xLoL

264

375,

KcðtÞ ¼

XP

p ¼ 1

YðpÞ11V ðpÞ0 X � � �XP

p ¼ 1

YðpÞ1L V ðpÞ0 X

^ & ^XP

p ¼ 1

YðpÞL1 V ðpÞ0 X � � �XP

p ¼ 1

YðpÞLL V ðpÞ0 X

2666666664

3777777775

,

KLðtÞ ¼

XL

k ¼ 1

B11kqk � � �

XL

k ¼ 1

B1Lkqk

^ & ^XL

k ¼ 1

BL1kqk � � �

XL

k ¼ 1

BLLkqk

266666664

377777775

with

Bijk ¼ bi

kj if j4k,

Bijk ¼ bi

jk if jok,

Bijk ¼ 2bi

jk if j¼ k,

8>>><>>>:

KQ ðtÞ ¼

XL

k ¼ 1

XL

lZk

G11klqkql � � �

XL

k ¼ 1

XL

lZk

G1Lklqkql

^ & ^XL

k ¼ 1

XL

lZk

GL1klqkql � � �

XL

k ¼ 1

XL

lZk

GLLklqkql

2666666664

3777777775

with

Gijkl ¼ 3Gi

jkl if j¼ k¼ l,

Gijkk ¼Gi

jkk if jok,

¼Gikkj if j4k,

Gijjk ¼ 2Gi

jjk if jok,

¼ 2Gikjj if jok,

8>>>>>>>>><>>>>>>>>>:

and

Gijkl ¼Gi

jkl if joko l,

¼Gijlk if jo lok,

¼Gikjl if ko jo l,

¼Giklj if ko lo j,

¼Gilkj if loko j,

¼Giljk if lo jok:

8>>>>>>>>>>>><>>>>>>>>>>>>:

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