ets

Aviv

vers

Article history:

Received 16 June 2011

Accepted 1 July 2011Available online 23 September 2011

Keywords:

Large-displacement actuator

Finite element analysis

Multistability

Multistep

In this work we report on a nite element modeling and design methodology, fabrication and

characterization of a large-displacement low voltage multistable micro actuator with an integrated

structural mechanics [15].

withfullyeamsuallyctro-

independent on the actuators displacement and the nonlinearity

Contents lists available at SciVerse ScienceDirect

.e

Finite Elements in An

Finite Elements in Analysis and Design 49 (2012) 5869may exhibit both mechanical snap-through and electrostatic (soE-mail address: vadis@eng.tau.ac.il (S. Krylov).was purely of a mechanical nature. Signicant attention was paidto the theoretical and experimental analysis of static and dynamicbehavior of fully compliant bistable micro beams [2433]. Notethat recently reported electrostatically actuated bistable devices

0168-874X/$ - see front matter & 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.nel.2011.08.021

n Corresponding author.structures liable to snap-through buckling, mainly arches, frames,cylindrical panels and spherical caps, is a well-established topic in

static comb drive [15,20,21,25] or magnetic [8,26] transducers.Note that in all cases listed above the actuation force wasture resulting in a non-monotonous stiffnessdisplacement char-acteristic. One of the most common examples is a exible archloaded by a transverse force [13], Fig. 1(a). This structure isbistable in the interval of the force between the snap-back(release) and snap-through values (see Fig. 1(b)). The analysis of

typically realized as chevron-shaped rigid links combinedcompliant pseudo-hinges [7,1517]. Designs incorporatingcompliant suspensions realized as initially curved or tilted bwere reported as well ([6,8,1823]). Actuation was done manby probe [16,22,24] or provided by thermal [18,19,23,24] eleBistability and multistability, namely, the existence of two orseveral different stable congurations at the same loading, is anintrinsic feature of many mechanical structures. This behaviortypically originates from the geometric nonlinearity of the struc-

including electrical [6,7] and optical [8] switches, optical attenua-tors [9], inertial sensors [10], light processing devices, tactiledisplays [11] and nonvolatile memories [1214]. A large variety ofarchitectures and operational principles of bistable micro deviceswere reported. Elastic suspensions in bistable micro devices wereSnap-through buckling

Pull-in

Comb drive

1. Introductionelectrostatic comb drive transducer. The compliant suspension of the device incorporates multiple

serially connected bistable arch-shaped beams and exhibits controllable sequential snap-through

buckling under an increasing actuation force. The device can be considered therefore as an example of a

compliant multistep structure. The device is also distinguished by its ability to remain in several

different stable congurations at the same actuation voltage while the forcedisplacement character-

istic of the suspension can be tailored by changing the geometry parameters of the exures. A model

built using the shallow arch approximation along with a nonlinear nite element analysis were used in

order to study the inuence of the suspension architecture on the stability limits of the structure and

for evaluation of design parameters of the actuator. Bistable and multistable devices were fabricated by

a Deep Reactive Ion Etching (DRIE) based process using silicon-on-insulator (SOI) wafers. Experimental

results, which are consistent with the model predictions, demonstrate that the compliant multistep

devices exhibit improved lateral stability and consequently larger stable displacements compared to

the conventional comb drive actuators. Stable displacements up to 80 mm at a voltage of 30 V wereregistered in the experiments while three snap-through and snap-back events took place during

loading and unloading, respectively. Our computational and experimental results show that the

suggested device has clear functional advantages and can be efciently used in applications including

switches, threshold inertial sensors, variable optical attenuators as well as in micro-and nanomecha-

nical logical elements.

& 2011 Elsevier B.V. All rights reserved.

In microsystems, bistability is benecial in many applicationsReceived in revised form

30 June 2011Design considerations of a large-displacwith serially connected bistable elemen

Y. Gerson a, S. Krylov a,n, B. Ilic b, D. Schreiber a

a School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Ramatb School of Applied and Engineering Physics and Cornell Nanoscale Facility, Cornell Uni

a r t i c l e i n f o a b s t r a c t

journal homepage: wwwment multistable micro actuator

69978, Tel Aviv, Israel

ity, Ithaca, NY, United States

lsevier.com/locate/finel

alysis and Design

modeling and design aspects of the device development. In thenext section, the model of the generic device based on a shallowcurved beam serving as a single bistable element of the suspen-sion is considered. Main features of the device stability behaviorare illustrated and the applicability of the shallow beam model isdiscussed. Next, several design congurations of the device areintroduced and results of nite element analysis of these cong-urations are presented. We show that the lateral (pull-in)

con

seria

Fig. 2. Model of a curved beam.

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 5869 59called pull-in) instabilities [2833]. The reason is that thesedevices combine both geometric mechanical nonlinearity origi-nating in an initially curved shape of the beams and electrostaticsoftening nonlinearity associated with the electrostatic force thatreduces the effective stiffness of the structure.

The concept of the device considered in this work is based on aserial connection of multiple mechanically bistable curved beams,each attached to a rigid frame, Fig. 1(c). Since different elementsof this chain of bistable elements are designed to exhibit adissimilar snap-through force, a sequence of snap-through eventstakes place under an increasing force applied to the last element,as shown in Fig. 1(d). The forcedisplacement curve of thestructure contains several stable branches and the device isactually a fully compliant multistep structure. By adjusting thegeometrical parameters of the curved beams forming the com-pliant suspension, the shape of the limit point buckling curve canbe tailored in a wide range. For appropriately chosen parameters,the device may remain in several different stable congurationsat the same actuation voltage. The ability to tailor the stability

Fig. 1. An arch loaded by a transverse force in a pre-buckling and post-bucklingOperational principle of the device-schematics of a device incorporating multipleproperties of the actuator is one of the distinguishing features ofthe device under consideration.

It should be noted that the idea to obtain a multistable behaviorby means of serial connection of bistable elements is not new.Results of theoretical investigation of the static and dynamicbehavior of chains of bistable elements as well as wave propaga-tion in these systems (often viewed as waves of phase transition)were largely reported in applied mechanics literature (e.g. see[3440]). Possible design realizations, design methodology andsynthesis of multistable compliant mechanisms using combina-tions of bistable elements were discussed in [41]. In microsystems,reported multistable devices mainly incorporated mechanicallatching (ratchet-type) elements (e.g., see [9,42]). Tri-stable micro-fabricated device based on a bi-directional (double tensural)operation was reported in [43]. The device included an assemblyof oblique beam-like suspension springs and was operatedmechanically by a micro manipulator. A tri-stable mechanism withbi-directional operation actuated by a electroactive polymericactuator (aritical muscle) was reported recently in [44]. Thefully compliant multistable device with the suspension incorpor-ating serially connected bistable elements and with integratedelectrostatic actuation was rst reported in [45].

In this work we present the design, fabrication and character-ization of the device. The main focus is on the nite elementguration (a) and schematics of a corresponding limit point buckling curve (b).

lly connected bistable beams (c) and a generic limit point buckling curve (d).instability of the electrostatically actuated structure representsthe main design challenge in this kind of device and requirescareful design and nite element modeling. Finally, we presentthe results of the device fabrication and characterization illus-trating the feasibility of the suggested approach. Conclusionssummarize the main ndings of the work.

2. Computational model

2.1. Curved beam

In order to provide an insight into the inuence of differentparameters on the stability properties of a curved beam andchoose the design parameters, the most suitable for the control ofthe multistable behavior, we rst consider a model of a singleinitially curved beam, Fig. 2.

We consider a exible, initially curved, prismatic micro beamof length L, of a rectangular cross-section of area Abd andsecond moment of the area Iyy bd3=12. The initial shape of thebeam is described by the function z0x hc0x (for convenienceit is considered positive in the negative direction of the z-axis,Fig. 2) where h is the initial elevation of the central point of the

beam about its ends and c0x is a non-dimensional function such

that max0oxoL

fc0xg 1. Hereafter we consider a beam of a circularshape and adopt

c0x 1

218

L

h

2 12

1 L

2h

2 !2 L

h

2 2xL1

2vuut 1We emphasize that the initial curved shape of the beam isprovided by lithography rather than by a pre-buckling. As aresult, the beam is stress-free in its initial conguration. Thebeam is assumed to be made of homogeneous isotropic linearelastic material with Youngs modulus E. Both ends of the beamare clamped. The beam is actuated by a concentrated force Facting at the midpoint of the beam in the z-direction (see Fig. 2).

We describe the behavior of the beam using two approaches.In the framework of the rst approach, the beam is considered

approach is implemented (see [49] for the case of conguration-dependent electrostatic force). The force F is considered as anunknown parameter while the midpoint deection of the beam isprescribed, i.e., wL=2 wM .

In addition, the stability of the beam was analyzed using thenite element method by means of the commercially availablesoftware. The planar straight beam element with an extensibleaxis, three nodal degrees of freedom (two translations and onerotation) and Hermitian polynomials as interpolation functionswas used. The element could also account for the shear deforma-tion of the beam. Note however that the inuence of the sheardeformation on the behavior of very slender beams considered inthis work is minimal. To enforce clamped boundary conditions,the translation in the x and z directions as well as the rotation ofthe end nodes of the nite element model were precluded. The

idpo

l n

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 586960using the EulerBernoulli theory combined with the shallow archapproximation. This simple model is convenient for the evalua-tion of the preliminary design parameters of the suspension. Inaddition, the results provided by this model will be used for thecomparison with the nite element results. We assume thatd5L, h5L and that the deections, while comparable with thethickness of the beam, are small with respect to the beamslength. The equilibrium of the beam is described by the system oftwo differential equations (e.g., [3,46], see also [30]).

EA u0hc00w0 1

2w02

0 0

EIwIVEA hc00w0 u0hc00w0 1

2w02

0 Fd x L

2

2

here w(x) is the lateral displacement; u(x) is the axial displace-ment, dx is the Dirac delta and 0 d=dx. Eq. (2) is completed bythe boundary conditions corresponding to the clamped ends ofthe beam. Note that in all the devices considered in this work, theanstisymmetric buckling is precluded by the design means.Namely, the beams are used in pairs such that two identicalbeams are connected at their midpoints by a rigid link (see [22]).For this reason in this section we consider only the half of thebeam and enforce symmetry conditions w0 0, EIyyw000 F=2 atthe midpoint of the beam (see [30] for the details).

The system of Eq. (2) was solved numerically. The solution isbased on the collocation method [47] and is obtained using thetwo-point boundary value problem solver bvp4c [48] integratedinto the Matlab package. The system (2) is written in the form ofsix rst order differential equations

y0 fy,F 3where y fu,u0,w,w0,w00,w000gT is the vector of unknown functionsand F is considered as a parameter. In order to describe theunstable branches of the buckling curve, the displacement control

Fig. 3. (a) Limit point buckling curves of an arch-shaped beam for different initial mcorresponds to the shallow beam model, Eq. (2); markers represent the numericah6 mm.calculations were performed using the large deection analysis.The unstable branches of the limit point buckling curves weredescribed using force control combined with the arc-lengthcontinuation method (e.g. see [50]) implemented in the commer-cially available software. The parameters of the arc-length pro-cedure were chosen by trial and error in such a way that theentire buckling curve was obtained. A total number of 200 forceincrements in the nonlinear solution was used. The mesh wasrened until convergence. The results presented hereafter corre-spond to the convergent solution and to the beam subdivided into80 elements. Hereafter in this section the width and the thicknessof the beam used in calculations were b30 mm and d3 mm,respectively, Youngs modulus was E169 GPa.

The results of calculations are shown in Fig. 3. Comparisonbetween the shallow beam model, Eq. (2) and the nite elementsolution is shown in Fig. 3(a) for three different elevations. Excellentagreement between the two models is observed. For h10 mm therelative error in the snap-through value of the force was 0.38%. Weattribute the certain discrepancy mainly to the approximate char-acter of the shallow beam model, which disregards the nonlinearcurvature of the beam. A graphical representation of the curvedbeam during the loading is shown in Fig. 3(b).

In accordance with Fig. 3(b), each of the elements of themultistable suspension should exhibit bistable behavior whilethe value of the critical force corresponding to the snap-throughinstability should be different for each of the beams. Generallyspeaking, for the prescribed initial shape and material of theclamped arch, the forcedisplacement characteristic of the beamcan be controlled by three parametersthe initial elevation, thethickness and the length of the beam. It is well known thatthe beam described by Eq. (2) is bistable when the ratio betweenthe initial elevation of the beam and its thickness is higher than acertain value. In accordance with [2], in the case of a beam with arectangular cross-section, the snap-through takes place when the

int elevations h (numbers, in mm). The length of the beam is L1000 mm. Solid lineite element solution. (b) Snapshots of the beam at different actuation forces for

ratio m d=h3

po0:42, which corresponds to h4.13 mm in the

case of 3 mm thick beam (for the case of an initially sinusoidalarch the value of mo0.4 was obtained in [51] using the two-termmodal expansion solution). Since the bistability criterion isindependent of the length and is very sensitive to the width andthickness, the buckling behavior of the arch can be controlled bychoosing appropriate values of h (see Fig. 3(a)) and/or d. However,for microstructures, both h and especially d can be very uncertaindue to low tolerances of micromachining. The structures consid-ered in this work are fabricated from single crystal silicon usingdeep reactive ion etching (DRIE). In the framework of this process,thin (typically a few micrometers) beams are surrounded by largeopen area and may suffer from signicant over etch. As a result,the actual thickness of the beams is usually smaller than thenominal value. Although corrections (bias) of the nominal dimen-

the beams. However, since the beams are of dissimilar length, thevalues of the forces corresponding to these displacements aredifferent for each beam. On the other hand, within the suspensionthe beams are connected serially and, from the equilibriumconsiderations, the force acting on each of the bistable elementsof the chain is the same. To overcome this difculty, the tablelook up approach was used and the forcedisplacements char-acteristic for each of the beams was approximated using a

polynomial t Fi Pin wiM . Here n is the order of the polynomial(in most cases seventh order polynomial was used) and

wiM , i 1::N is the midpoint deection of the ith beam. Next, inthe framework of the displacement control approach, the dis-placement wA of the end point of the suspension (hereafterreferred to as an actuator displacement) was prescribed, and

i

L(n

widt

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 5869 61sions could be made at the design stage in order to account forpossible over etch, in view of sensitivity of the buckling force tothe thickness, the uncertainty originates from the lack of repeat-ability and uniformity of the process is still high. Uncertainty inthe initial elevation is related to small residual stress or stressgradients appearing in the bonded silicon-on-insulator (SOI)wafers as well as to possible variations in temperature (see [21]and references therein). In view of the aforementioned, in thiswork we keep the thickness and the elevation of each of thebeams forming the suspension to be constant and use thelength of the beam to tailor the forcedeection characteristicof the beam.

2.2. Multistable suspensiona chain of curved beams

The solution for a single beam was used as a building block forthe description of the multistable suspension incorporatingmultiple serially connected bistable elements. Hereafter we refera suspension as multistable or multistep if the correspondinglimit point buckling curve contains several different (not adja-cent) stable branches. A nite element solution was obtainedsimilarly to the case of the single beam, namely using a nonlinearlarge deection analysis combined with the arc-length procedure.To enforce boundary conditions, translation in the x-direction aswell as the rotation of the end nodes were precluded whereas thetranslations in the z-direction were released. The beams wereconnected by rigid links in such a way that the compliance of thesystem was associated solely to the compliance of the beams.

The solution based on the shallow arch model was obtainedusing the following procedure. First, the dependence between theforce and the midpoint deection was obtained separately foreach of the beams distinguished by different length. The displace-ment control procedure was used and the prescribed incrementsof the midpoint displacements are taken to be identical for each of

Fig. 4. (a) Limit point buckling curves of an arch-shaped beam for different lengthcurve of the multistable suspension assembled from four bistable beams. Nominalbeam of d3.3 mm and d2.7 mm, respectively.the midpoint displacements wM , i 1::N of each of the beamsalong with the force were found as the solutions of the system ofN1 nonlinear algebraic equationsPin wiMF 0XNi 1

wiMwA 0 4

The limit point buckling curves of the separate beams are shownin Fig. 4(a), the limit point buckling curve of the suspensionassembled from four beams is shown in Fig. 4(b). The length ofthe beams is L700 mm, 800 mm, 900 mm and 1000 mm, the initialelevation of all the beams is h8 mm and the width of the beams isd2.7 mm. One observes that, as a result of the serial connection ofthe beams, the snap-through values of the chain are identical tocritical values of the individual beams whereas the correspondingdisplacements are larger in the chain. Fig. 4(b) illustrates also therobustness of the suspension to the uncertainty in the beams width.One observes that while the critical values of the forces are stronglyaffected by the beams width, the bucking behavior is qualitativelypreserved and the sequential snap-through can be achieved in thesuspensions with uncertain geometric parameters. Note that thedevice can be viewed as multistable in a sense that it may haveseveral overlapping or non-overlapping bistability regions. A com-parison between the results obtained using the shallow beammodeland the nite element analysis revealed very good agreementbetween the two. In the considered example, the error in thesnap-through value of the force corresponding to the highest limitpoint of the multistable chain was 0.4%.

One of the central advantages of the suspension congurationconsidered in the present work is the ability to control the forcedisplacement curve in a very large range by choosing the appro-priate values of the beams parameters. Examples of the limitpoint buckling curves corresponding to different geometricalparameters of the beams are presented in Fig. 5 where the limit

umbers, in mm) and the midpoint elevation h8 mm. (b) The limit point bucklingh of the beam is d3 mm; dashed and dotted lines correspond to the width of the

instability, which is often the main factor limiting the stabledisplacement range of the comb drive actuator (e.g., see [53].).

In order to illustrate the approach used in this work for theestimation of the stability range of the devices, we rst consider asimplied model of the actuator. The model is shown schemati-cally in Fig. 6 and incorporates a rigid shutter connected to thesubstrate by the elastic suspension, a set of moveable electrodesattached to the shutter and a set of xed electrodes anchored tothe substrate. The device is constrained to move in the z and x

fou

the s

0 mm

Fig. 6. Schematics of a comb drive transducer model.

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 586962point buckling curves corresponding to a four-beam suspensionare shown. One observes that in the case of relatively smallelevations, slightly above the snap-through criterion, the beamparameters can be chosen in such a way that the characteristic,which is close to an effectively linear dependence, is achieved inthe interval of the forces between the snap-through values of thebistable elements incorporated into the suspension, Fig. 5(a). Onthe other hand, the choice of the higher elevation beams distin-guished by smaller difference between their lengths results inmultistability of the structure. In this case, the suspensionincorporating four beams can be in ve different stable stateswithin the interval of the forces limited by the highest releaseforce (corresponding to the shortest and consequently stiffestbeam of the chain) and the lowest snap-through force, associatedwith the longest beam, Fig. 5(b). Note that since the (tangent)stiffness of the bistable beam decreases in the vicinity of thesnap-through point and is signicantly higher prior to andespecially after the snap-through collapse (see Fig. 4(a)), mostof the compliance of the suspension in the intermediate deformedconguration is localized in one of the suspension elements,which is closest to the snap-through state. When the forceexceeds the snap-through value corresponding to the shortestbeam, all elements of the suspension are in a post-buckled stateand a further increase of the loading results in a stiffening of thestructure. In a sense, the structure can be effectively viewed as acompliant displacement limiter distinguished by low stiffnesswithin a certain interval of the displacements/forces and muchhigher stiffness when the displacement/force exceeds a certainvalue. This feature can be benecial in MEMS applications where

Fig. 5. (a) Limit point buckling curves of the multistep suspension assembled from820 mm, 900 mm, 1000 mm and the midpoint elevation is h5 mm. Inset illustratesmultistable suspension. The length of the beams is L850 mm, 900 mm, 950 mm, 100the bistability region in terms of the force.the realization of the displacement limiters based on contact isoften challenging from the reliability point of view and friction/stiction related problems. The effective forcedisplacementcharacteristic of this compliant limiter can be tailored in a verywide range.

2.3. Actuator model

The structures considered in this work are actuated by anintegrated comb drive transducer [52]. This kind of transducer ischosen since it allows, in contrast to the transducers based on aclose-gap conguration, for relatively large (with respect to thedistance between the electrode) displacements of the actuator. Oneof the distinguishing benecial features of the comb drive is thatthe force provided by the transducer is independent of the actuatordisplacement. This simplies the design and operation of thedevice and eliminates the undesired electrostatic pull-in instabilityin the direction of the actuation. However, the structures actuatedby a comb drive are still prone to the lateral (side) pull-inr bistable beams-nite element solution. The lengths of the beams is L750 mm,uspension stiffness increase at larger forces. (b) Limit point buckling curves of the

and the midpoint elevation is h15 mm. Dashed lines illustrate the boundaries ofdirections and is considered as a two degrees of freedom system.Consequently, the elastic suspension is represented by twosprings with the stiffness kz and kx. Note that in the actual devicesthe high stiffness in the out of plane (y) direction is provided dueto the high aspect ratio between the beams width, b (the heightof the SOI device layer) and the thickness d of the beams.Rotational degrees of freedom are eliminated by the designmeans, as will be specied in the next section. The equilibriumof the actuator is described by the system of two coupledalgebraic equations

kxuA ne0bw0wAV2

2g0uA2ne0bw0wAV

2

2g0uA2

kzwA ne0bV2

2g0uA ne0bV

2

2g0uA5

where uA and wA are the displacements of the actuator in thelateral (x) and axial (z) directions, respectively; g0 and w0 are theinitial distance and the initial overlap between the electrodes; n is

the number of the moveable electrodes; e08.8541012 F/m isthe permittivity and V is the applied voltage.

The electromechanical behavior and stability of the modelshown in Fig. 6 and described by Eq. (5) was analyzed in [53]. Itwas found that the stable displacement in the z-direction isbounded by the value

wMAXA g0

kx2kz

w02g0

2sw0

26

Note in passing that expressing wA in terms of uA from the secondpart of Eq. (5) and substituting the result into the rst equation,

one obtains a homogeneous nonlinear equation in terms of thelateral displacement. This equation may have three differentsolutions two unstable and one stable trivial solution uA0 or only one unstable trivial solution uA0, depending uponwhether the actuation voltage is higher or lower than the pull-in value corresponding to the subcritical pitchfork bifurcation.A stability analysis of this equation linearized in the vicinity of thetrivial solution leads to Eq. (6). The pull-in voltage is thenobtained using the second part of Eq. (5) for uA0 and wA wMAXA .

In the case of the geometrically nonlinear multistable suspen-sion considered in the present work and as a result of a structuralcoupling the stiffness kz and kx are not constant and are functionsof the actuator displacement. In addition, they are also affected bythe secondary compliances of the actuators structure (mainlycompliances of the shutter and of the connecting frames that thesuspension beams and the comb drive electrodes are attached to).In order to verify that the maximally achievable stable displace-ment is larger than the designed displacement range of the

Fig. 7. The limit point buckling curve (solid line) of the multistable suspensionassembled from bistable beams of four different length L700 mm, 800 mm,900 mm, 1000 mm, the midpoint elevation of h8 mm and the width of d3 mmand corresponding tangent stiffness kTz (dashed line).

Fig. 8. An artist view of a truss-like structure of the actuator.

in pa

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 5869 63Fig. 9. (a) Design #1bistable device containing a pair of curved beams connected

(b) Finite element analysis results of the lateral displacements (in mm) of the structureactuator, we used the following approximate approach. First,since the stability is analyzed in the vicinity of uA0 and thecongurations of the beams of the suspension are uniquelyrelated to the actuator displacement wA (see Figs. 35), weassume that both axial and lateral stiffness are decoupled andare solely functions of the axial displacements, i.e., kzkz(wA) andkxkx(wA). Next, the values of the stiffness appearing in Eq. (6)are replaced by the values of the tangent stiffness calculated inthe actual deformed conguration of the device. The axial (in thez-direction) compliance of the structure is associated mainly withthe exibility of the bistable beams and can be calculated usingeither nite element analysis or the shallow beam model. Incontrast, kx is strongly affected by the secondary compliances andshould be calculated only for the actual deformed geometry of thedevice. It was evaluated numerically using the nite elementmethod for multiple points within the actuator traveling range.Namely, a small probing force was applied to the shutter in thelateral (x) direction and the stiffness was obtained using the ratiobetween the increment of the force and the calculated displace-ment of the forces application point (e.g., see [54]). Note that thepoint of this probing force application was different in variousdesigns and was chosen to reect the location of the forcestransferred to the structure by a comb drive transducer. The axialand lateral tangent stiffness were calculated at several deformedcongurations corresponding to the highest kz and the lowest kx.It was found that kz has local maxima in the congurations aftereach of the snap-through jumps (see Fig. 7). The largest value of kzis in the conguration corresponding to the fully stretchedgeometry of the suspension when all bistable beams are in the

rallel. The length and the initial elevation of the beams are L1160 mm, h25 mm.

under a lateral force of 100 mN applied to the shutter.

buckled state (see Fig. 7). The congurations corresponding to thesmallest value of kx differ for the various actual designs of thestructure. For these values of the stiffness, the result provided byEq. (6) represents therefore the worst case scenario and thecorresponding value of wMAXA can be viewed as the lower boundestimation of the stable displacement range of the device. Thisvalue was required to be larger than the designed actuatordisplacement and was used as a preliminary estimation of thestability range during the design. In addition, the stability of thedevice was directly veried using Eq. (6) with locally minimal kx,or locally maximal kz and w0 corresponding to the actual overlapbetween the electrodes in these deformed congurations.

It should be noted that for technological reasons (namely, toallow a wet release of the suspended structures in a hydrouoric(HF) acid) and in order to reduce the area and the possibility ofstiction of the device to the substrate, the parts of the device, whichshould be ideally rigid were designed as a truss-like structure, Fig. 8.Finite element analysis of these kinds of structures using solid oreven structural beam elements could be computationally intensive,especially in the framework of the large deection nonlinear incre-mental solutions used for the analysis of the devices. In order tosimplify modeling and reduce the computational time, an approachbased on the use of equivalent structures was implemented whilecalculating the lateral stiffness, kx. The complex truss-like structureswere replaced by simple, effectively equivalent, beams with thelateral (x-direction) stiffness equal to that of the complex structure.The truss-like structure modeled as an assembly of beams using aplanar straight beam element served as a reference and was replaced

by a single planar beam with an equivalent cross-section and secondcross sectional moments of area identical to that of the truss-likestructure. This equivalent beam (planar beam element) was thenused in all nite element analyses of the designed structures.

3. Designed congurations

Multistable devices of four different congurations incorpor-ating one or two (connected in parallel) chains of three or fourserially connected curved beams and actuated electrostatically bya comb drive transducer were designed. In addition, a simplebistable device incorporating a pair of curved beams connected inparallel was designed as well. In all cases, the nominal width andthe thickness of the beams were b30 mm and d3 mm, respec-tively. In all multistable designs, the curved beams of differentlengths were attached by their ends to a relatively stiff framerealized as a truss-like structure. The midpoints of the shortestbeams in each of two chains of bistable elements were connectedto a central beam (hereafter referred as a shutter) with themovable part of the electrostatic transducer attached to it. Themidpoints of the longest beams were anchored to the substrate.For each design, the shallow beam model was used for thepreliminary evaluation of the design and operational parameters.Next, the nite element method was used at the stage of thedetailed analysis and design to obtain the forcedeection char-acteristics and estimate the stability of the device. Note that in allcases two-dimensional (planar) nite element models were usedand all the components of the structures were modeled using

ams

len

disp

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 586964Fig. 10. Limit point buckling curve of the bistable device shown in Fig. 9(a)(design #1)nite element analysis result.

Fig. 11. (a) Design #2device incorporating three serially connected bistable be1500 mm, 1300 mm, 1100 mm, the initial elevation of all the beams is h19 mm. Therespectively, the inclination is 3.51. (b) Finite element analysis results of the lateral

to the shutter.and supported by a tilted folded exure. The lengths of the curved beams are

gth and thickness of the beams of the folded suspension are 1100 mm and 3.5 mm,lacements (in mm) of the structure under two lateral forces of 100 mN each appliedFig. 12. Limit point buckling curve of the multistable device incorporating threeserially connected bistable beams and supported by a tilted folded exure

(Fig. 11(a), design #2)nite element analysis result.

planar beam elements. In all cases the curved beams incorporatedinto the suspensions were subdivided into 100 elements. Notethat in contrast to the shallow beam models described in Section2, no symmetry conditions at the midpoint of the beams wereused in the nite element models. The rigid parts of the device the shutter and the connecting frames were represented usingequivalent planar beams. The goal was to compare differentdesigns and to estimate, using the model, feasibility of themultistable operation and expected performance of the devices.

3.1. Design #1bistable device

The simplest bistable actuator containing a pair of identicalcurved beams, which are connected in parallel to a rigid shutter isshown in Fig. 9(a). The required actuation force is provided by abi-directional electrostatic comb drive transducer. Due to itssimplicity, the device represents a convenient platform for theinvestigation of stability properties of this kind of device and forcomparison between experimental and model results (e.g., see[20,21]). By connecting two beams in parallel, the (in-plane)

conventional actuators operated by a comb drive transducer. Thelateral deection of the shutter under two forces of 100 mN each,which were applied to the device in the initial, the most laterallycompliant, conguration, is shown in Fig. 11(b). One observes thatdue to the relatively high lateral compliance of the suspension, thedeection in the x-direction is signicantly higher than in thebistable device (see Fig. 9(b)). However, the device was found toexhibit a stable displacement of at least 100 mm. The limit pointbuckling curve is shown in Fig. 12. In this device the comb drivetransducer contained 340 electrodes with the nominal distance of5 mm between the electrodes. The actuation voltage correspondingto the maximal deection of 85 mm in Fig. 12 is 40 V.

3.3. Design #3device with curved double beams

The device is attached to the substrate by two multistable chains.Each chain incorporates three serially connected curved doublebeams (Fig. 13(a)). In this design, each bistable beam was replacedby a pair of identical, closely located, curved beams connected to eachother at the midpoint (see insert in Fig. 13(a)). This architectureprevents a possibility of the antisymmetric buckling of the beams (see[22,23]), precludes the rotation of the device around an axis perpen-dicular to the substrate and increases the lateral stiffness of theactuator. Fig. 13(b) illustrates the lateral compliance of the device.One observes that the compliance of the relatively long shuttercannot be disregarded and should be included into the analysis inaddition to the compliance of the suspensions. The nite element

leng

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 5869 65rotation of the shutter about an axis perpendicular to thesubstrate is precluded, the possibility of the antisymmetricbuckling of the beams is eliminated (see [22]) and the lateralstiffness of the suspension and consequently the stable stroke ofthe device are increased. Fig. 9(b) illustrates the nite elementresults for the lateral deection of the shutter (described asequivalent beam) under the lateral probing force of 100 mN.The computational limit point buckling curve is shown in Fig. 10.The device was found to exhibit a stable displacement of at least60 mm. The comb drive transducer contained 180 electrodes whilethe nominal distance between the electrodes was 4 mm. In thiscase the actuation voltage corresponding to the maximal deec-tion of 42 mm in Fig. 10 is 70 V.

3.2. Design #2device supported by a tilted folded exure

The second design is a multistable device incorporating a chainof three serially connected bistable beams and supported inaddition by a tilted folded exure [55,56] (Fig. 11(a)). In thisstructure, the folded exure is acting as a spring, which isconnected in parallel to the three serially connected bistable beams.The tilted folded exure is shown in [55,56] to increase the lateralstiffness of the structure and to enlarge the stable stroke of the

Fig. 13. (a) Design #3device with two chains of three double bistable beams. The

the beams is h13 mm. (b) Finite element analysis results of the lateral displacementsths of the curved beams are 1100 mm, 1000 mm, 900 mm, the initial elevation of all

Fig. 14. Limit point buckling curve of the multistable device incorporating twochains of three serially connected double beams (Fig. 13(a), design #3)nite

element analysis result.(in mm) of the structure under the lateral force of 100 mN applied to the shutter.

ths

ts (

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 586966Fig. 15. (a) Design #4device with curved beams connected by frames. The lengbeams is h15 mm. (b) Finite element analysis results of the lateral displacemenouter frame.

Fig. 16. Limit point buckling curve of the multistable device with curved beamsconnected by frames (Fig. 15(a), design #4)nite element analysis result.analysis shows that the displacement of the device is stable up to atleast 75 mm. The limit point buckling is shown in Fig. 14. The combdrive transducer contained 330 electrodes with the nominal distanceof 4.5 mm between the electrodes. The actuation voltage correspond-ing to the maximal deection of 60 mm in Fig. 14 is 65 V.

3.4. Design #4device with curved beams connected by frames

Similar to design #3, the device of design #4 is suspendedusing two multistable chains, each containing three curvedbistable beams. However, in contrast to design #3, each of thebeams incorporated into one chain is connected by a rigid frameto a beam of the same length, which is a part of the second chain(Fig. 15(a)). This arrangement of the beams prevents in-planerotation of the structure and improves its lateral stability.Fig. 15(b) illustrates the lateral deection of the frame underthree lateral forces of 100 mN each applied at the locations of theattachment of the comb drive transducer. One observes that theframe structure exhibits much higher lateral stiffness whencompared to design #3 (Fig. 13(b)). The stable displacement ofthe device was estimated to be 100 mm. The computational limitpoint buckling curve is shown in Fig. 16. The comb drivetransducer attached to the outer frame and located outside ofthe suspension area (see Fig. 15(a)) contained 330 electrodes withthe nominal distance of 3.5 mm between the electrodes. Theactuation voltage corresponding to the maximal deection of80 mm shown in Fig. 16 is 55 V.

3.5. Design #5device with curved beams connected by outer

frames

The device of design #5 is suspended using two chains, eachincorporating four curved beams. Similar to design #4, each of the4. Experiment

Using the detailed nite element analysis designs #4 and #5were found to exhibit the largest stable displacement combinedwith a relatively small footprint and the possibility to integraterelatively large number of electrodes. Fabrication and character-ization efforts were focused mainly on these devices. Design #1(bistable device) was fabricated as well due to its simplicity,robustness and convenience of operation and for comparison withthe model results.

The rst step in the fabrication of MEMS devices is preparationof the detailed layout of the structure. For the sake of compat-ibility with the mask making tools, the layout should be accom-plished using dedicated layout software originally developed forthe needs of the semiconductor industry. On the other hand,beams is connected by a rigid frame to its counterpart of the samelength in another chain (Fig. 17(a)). Similar to design #4, thedevice was found to manifest excellent lateral stability, Fig. 17(b).The stable stroke of the device was estimated to be least 115 mm.However, in contrast to the design #4, this device incorporates acentral shutter connecting the pair of the shortest curved beamsand with the comb drive transducer attached to it. This arrange-ment reduces the moment applied by the transducer, which mayresult in the undesired in-plane rotation of the structure. Thelimit point buckling curve is shown in Fig. 18. The comb drivetransducer contained 250 electrodes with the nominal distance of4 mm. In accordance with the nite element model results thedeection of 75 mm shown in Fig. 18 can be achieved by applyingthe actuating voltage of 70 V.

of the curved beams are 1250 mm, 1150 mm, 1050 mm, the initial elevation of thein mm) of the structure under three lateral forces of 100 mN each applied to themechanical design is typically carried out using three-dimen-sional CAD tools allowing easy visualization and parameterizationof complex geometries. In addition, the geometries created by themechanical CAD tools can be conveniently imported into niteelement software and then meshed and analyzed. In this work,the three-dimensional geometry of the devices was rst builtusing mechanical CAD tools (SolidWorks [57]), see Fig. 8, andnite element analysis of the devices was performed, as wasdescribed previously in Section 3, using the imported geometry.Then, the mechanical geometry was converted into the GDSIIformat compatible with the standard layout tools [57] (seeFig. 19). Note that mask generation was simplied due to thesingle layer architecture of the SOI devices.

The devices were fabricated from highly doped single crystal Siusing silicon-on-insulator (SOI) wafers with (1 0 0) surface orien-tation as a starting material and etched using a deep reactive ionetching (DRIE) based process. The patterning of the photoresistspun on top of the 30 mm thick device layer of the SOI wafer was

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 5869 67followed by reactive ion etching (RIE) of the silicon dioxide layerfor the formation of a hard mask. DRIE of the device layer wasstopped at the 2 mm thick buried silicon dioxide (BOX) layerfollowed by the device release using hydrouoric (HF) acid anddrying in a vacuum oven. An example of the fabricated device isshown in Fig. 20. Note that the truss-like structure of the shutter

Fig. 17. (a) Design #5device with two chains of four curved beams connected by framelevation of all the beams is h11 mm. (b) Finite element analysis results of the lateral dshutter.

Fig. 18. Limit point buckling curve of the multistable device with two chains offour curved beams connected by frames (Fig. 17(a), design #5)nite element

analysis result.

Fig. 19. Layout preparation owchart.es. The lengths of the beams are 1000 mm, 900 mm, 800 mm, 700 mm, the midpointisplacements (in mm) of the structure under a lateral force of 100 mN applied to theand of the frames simplies the release and decreases the areaprone to stiction.

The structures were mounted on a wafer prober Karl SussPSM6 located on an anti-vibration table (Kinetic systems, vibro-plane), and were operated at room temperature and underambient air conditions. The actuation voltage provided by avoltage source was applied to the unmovable electrodes of thecomb drive transducer while the movable parts of the device andthe substrate were connected to ground. The in-plane motion wascaptured by a CCD camera mounted on an optical microscopeMitutoyo FS70 (0/100, switchable microscope with long workingdistance objectives). The displacements of the actuator weremeasured using an analysis of captured images, each correspond-ing to specic values of the actuation voltage.

Preliminary experimental results demonstrating the feasibilityof the suggested approach are shown in Figs. 21 and 22.Fig. 21(a) shows a bistable device suspended using a pair ofcurved beams connected in parallel by a shutter (Design #1).Corresponding experimental and calculated limit point bucklingcurves are shown in Fig. 21(b). Note that actual geometryparameters of the device (mainly the thickness of the beamsand the distance between the electrodes), which were measuredby high magnication optical microscope, were used in themodel. The device is bistable in the interval of the voltagesbetween 14 and 30 V. It should be noted that all the designsincorporate relatively large (a few mm in size) connecting frameswith limited out of plane/tilting stiffness. For these reasons thedevices of designs #4 and #5 were found to be prone to stiction tothe substrate. In this perspective, device #4 demonstrated betterfunctionality when compared with design #5. Optical microscope

Fig. 20. Scanning Electron Microscope micrographs of the fabricated devicedesign #5.

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 586968micrograph (Fig. 22(a)) and experimental limit point bucklingcurve of the device #4 are shown in Fig. 22(b) along with thenumerical results. The snapshots of the deformed curved beamscorresponding to the different points on the curve are shown inFig. 23. One observes that, consistently with the nite elementmodel prediction, with increasing voltage the device manifestssequential snap-through buckling. We attribute the discrepancy

Fig. 21. (a) An optical microscope micrograph of the bistable device (Design #1). The ledrive transducer contains 180 electrodes with a gap of 3.5 mm. Blue arrow illustrates tnite element model results (dashed line) of the device. Arrows illustrate the direction of loading/unloading (for interpretation of the references to color in this gure

legend, the reader is referred to the web version of this article).

Fig. 22. (a) An optical microscope micrograph of the multistable device (design #4). Bcurves (markers) and nite element model results (solid line) of the device incorpora

(design #4). The lengths of the beams are 1250 mm, 1150 mm, 1050 mm and the initiainterpretation of the references to color in this gure legend, the reader is referred to

Fig. 23. Snap shots of the suspension (design #4) at different actuation voltages:(1) 16 V, (2) 22 V, (3) 25 V, (4) 32 V.ngth of the beam is 1160 mm, initial elevation of the midpoint is 25 mm. The combhe actuation direction. (b) Experimental limit point buckling curve (markers) andbetween the experimental and the model results to the differ-ences in the device geometry, mainly in the thickness ofthe beams, which is highly uncertain due to low fabricationtolerances of micromachining. The device exhibits stable totaldisplacement of 80 mm at the relatively low actuation voltage of33 V.

5. Conclusions

We presented a design approach of a multistable long displace-ment micro actuator. The device incorporates serially connectedbistable elements realized as shallow curved beams. One of theadvantages of the suggested design approach is that a desired limitpoint buckling curve can be achieved by changing geometricalparameters of each of the bistable elements. We found that in viewof high uncertainty in the device geometry due to low tolerances ofmicromachining the length of the beams is the most suitable forthe tailoring of the device forcedisplacement characteristics.Several design congurations were considered and the feasibilityof the suggested approach was demonstrated using the detailednite element model. We show that low lateral compliance of thedevice actuated by a comb drive transducer and containing severalserially connected bistable beams may make the device to be proneto the lateral pull-in instability. Direct numerical evaluation,by means of nite element modeling, of the axial and lateral

lue arrow illustrates the actuation direction (b) Experimental limit point buckling

ting two chains of three serially connected bistable beams connected by frames

l elevation of the midpoint is 14 mm. The device is actuated by 330 combs (forthe web version of this article).

mechanism, J. Microelectromech. Syst. 12 (2003) 273280.[20] J. Casals - Terre, A. Fargas - Marques, A.M. Shkel, Snap-action bistable

Y. Gerson et al. / Finite Elements in Analysis and Design 49 (2012) 5869 69micromechanisms actuated by nonlinear resonance, J. Microelectromech.Syst. 17 (2008) 10821093.

[21] Y. Gerson, S. Krylov, B.R. Ilic, Electrothermal bistability tuning in a largedisplacement micro actuator, J. Micromech. Microeng. 20 (2010) 112001.

[22] J. Lang, A. Slocum, A curved-beam bistable mechanism, J. Microelectromech.Syst. 13 (2004) 137146.

[23] J. Qiu, A bulk-micromachined bistable relay with U-shaped thermal actua-tors, J. Microelectromech. Syst. 14 (2005) 10991109.

[24] Y. Backlund, A lateral symmetrically bistable buckled beam, J. Micromech.Microeng. 8 (1998) 2932.

[25] M.T.A. Saif, On a tunable bistable MEMS-theory and experiment, J. Micro-electromech. Syst. 9 (2000) 157170.compliances of the designed structure allowed for the estimationof the stability range of the devices and serves as a basis for thefeasibility and comparative study between different designs.Several design congurations were suggested and analyzed whichallow achieving large stable displacements of up to 100 mm. Thedevices fabricated from SOI wafers using the DRIE based processdemonstrated stable displacements of 80 mm travels and threesnap-through and snap-back events. The experimental results wereconsistent with the computational model predictions.

Acknowledgments

This work was supported by the Israel Science Foundation(Grant no. 1426/08) and the National Science Foundation (GrantECS-0335765). A preliminary version of this work initially appearedin [45].

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Design considerations of a large-displacement multistable micro actuator with serially connected bistable elementsIntroductionComputational modelCurved beamMultistable suspension--a chain of curved beamsActuator model

Designed configurationsDesign #1--bistable deviceDesign #2--device supported by a tilted folded flexureDesign #3--device with curved double beamsDesign #4--device with curved beams connected by framesDesign #5--device with curved beams connected by outer frames

ExperimentConclusionsAcknowledgmentsReferences