-
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:
[email protected] (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].
References
[1] S. Timoshenko, Theory of Elastic Stability, 2nd ed.,
McGraw-Hill, New York,1961.
[2] C.L. Dym, Stability Theory and its Applications to
Structural Mechanics,Noordhoff Pub., Groningen, 1974.
[3] G.J. Simitses, D.H. Hodges, Fundamentals of Structural
Stability, BostonElsevier/Butterworth-Heinemann, Amsterdam,
2006.
[4] J. Singer, J. Arbocz, T. Weller, Buckling Experiments:
Experimental Methods inBuckling of Thin-walled Structures, Wiley,
Chichester, New York, 1998.
[5] A.P. Seyranian, I. Elishakoff (Eds.), Modern Problems of
Structural Stability,Springer, Vienna, New York, 2004.
[6] T. Gomm, In-plane linear displacement bistable microrelay,
J. Micromech.Microeng. 12 (2002) 257264.
[7] R.A.M. Receveur, C.R. Marxer, R. Woering, V.C.M.H. Larik,
N.-F. de Rooij,Laterally moving bistable MEMS DC switch for
biomedical applications, J.Microelectromech. Syst. 14 (2005)
10891098.
[8] J. Ko, Y. Kim, B. Kwak, Parametric study and optimization of
a micro-opticalswitch with a laterally driven electromagnetic
microactuator, J. Micromech.Microeng. 12 (2002) 939947.
[9] R.R.A. Syms, H. Zou, J. Stagg, Multistate latching MEMS
variable opticalattenuator, IEEE Photonics Technol. Lett. 16 (2004)
191193.
[10] J. Zhao, J. Jia, H. Wang, W. Li, A novel threshold
accelerometer withpostbuckling structures for airbag restraint
systems, IEEE Sensors J. 7(2007) 11021109.
[11] R. Vitushinsky, S. Schmitz, A. Ludwig, Bistable thin-lm
shape memoryactuators for applications in tactile displays, J.
Microelectromech. Syst. 18(2009) 186194.
[12] B. Charlot, W. Sun, K. Yamashita, H. Fujita, H. Toshiyoshi,
Bistable nanowirefor micromechanical memory, J. Micromech.
Microeng. 18 (2008) 045005.
[13] B. Halg, On a micro-electro-mechanical nonvolatile memory
cell, IEEE Trans.Electron Dev. 37 (1990) 22302236.
[14] J. Rubin, S. Tiwari, An electronic nonvolatile memory
device based onelectrostatic deection of a bistable mechanical
beam, in: Proceedings ofthe Technical Digest of MRS Fall Meeting,
Boston, 27 November1 December2006 (oral presentation) O310.
[15] Y. Shim, J. Lee, Modeling and experimental characterization
of the chevron-type bi-stable microactuator, J. Micromech.
Microeng. 13 (2003) 948954.
[16] j. Tsay, S. Liang-Qing, S. Cheng-Kuo, Design of a linear
micro-feeding systemfeaturing bistable mechanisms, J. Micromech.
Microeng. 15 (2004) 6370.
[17] D.A. Wang, H.T. Pham, Y.H. Hsieh, Dynamical switching of an
electromagne-tically driven compliant bistable mechanism, Sensors
Actuators A: Phys. 149(2009) 143151.
[18] S.M. Baker, L.L. Howell, On-chip actuation of an in-plane
compliant bistablemicromechanism, J. Microelectromech. Syst. 11
(2002) 566573.
[19] N.D. Masters, L.L. Howell, A self-retracting fully
compliant bistable micro-[26] S. Park, D. Hah, Pre-shaped
buckled-beam actuators: theory and experiments,Sensors Actuators A:
Phys. 148 (2008) 186192.
[27] S.M. Baker, L.L. Howell, Design criteria for bi-stable
behavior in a buckledmulti-layered MEMS bridge, J. Micromech.
Microeng. 16 (2006) 20342043.
[28] K. Das, R.C. Batra, Symmetry breaking, snap-through and
pull-in instabilitiesunder dynamic loading of
microelectromechanical shallow arches, SmartMater. Struct. 18
(2009) 115008.
[29] S. Krylov, S. Seretensky, Pull-in and multistability
analysis of an initiallycurved beam, Digest Tech. Papers APCOT
2006, Singapore, June 2528 2006,paper D-27.
[30] S. Krylov, B.R. Ilic, D. Schreiber, S. Seretensky, H.
Craighead, The pull-inbehavior of electrostatically actuated
bistable microstructures, J. Micromech.Microeng. 18 (2008)
055026.
[31] S. Krylov, N. Dick, Dynamic stability of electrostatically
actuated initiallycurved shallow micro beams, Continuum Mech.
Thermodyn. 22 (2010)445468.
[32] H.M. Ouakad, The static and dynamic behavior of MEMS arches
underelectrostatic actuation, in: Proceedings of the ASME
International DesignEngineering Technical Conferences and Computers
and Information inEngineering Conference 2009, DETC2009, vol. 6,
2010, p. 607.
[33] Y. Zhang, Y. Wang, Z. Li, Y. Huang, D. Li, Snap-through and
pull-in instabilitiesof an arch-shaped beam under an electrostatic
loading, J. Microelectromech.Syst. 16 (2007) 684693.
[34] A. Balk, A. Cherkaev, L. Slepyan, Dynamics of chains with
non-monotonestressstrain relations. I. Model and numerical
experiments, J. Mech. Phys.Solids 49 (2001) 131148.
[35] A. Balk, A. Cherkaev, L. Slepyan, Dynamics of chains with
non-monotonestressstrain relations. II. Nonlinear waves and waves
of phase transition,J. Mech. Phys. Solids 49 (2001) 149171.
[36] L.I. Slepyan, M.V. Ayzenberg-Stepanenko, Localized
transition waves inbistable-bond lattices, J. Mech. Phys. Solids 52
(2004) 14471479.
[37] L. Slepyan, A. Cherkaev, E. Cherkaev, Transition waves in
bistable structures.I. Delocalization of damage, J. Mech. Phys.
Solids 53 (2005) 383405.
[38] L. Slepyan, A. Cherkaev, E. Cherkaev, Transition waves in
bistable structures.II. Analytical solution: wave speed and energy
dissipation, J. Mech. Phys.Solids 53 (2005) 407436.
[39] L. Truskinovsky, Rate independent hysteresis in a bi-stable
chain, J. Mech.Phys. Solids 50 (2002) 165187.
[40] L. Truskinovsky, Mechanics of a discrete chain with
bi-stable elements,J. Mech. Phys. Solids 48 (2000) 127.
[41] Y.S. Oh, S. Kota, Synthesis of multistable equilibrium
compliant mechanismsusing combinations of bistable mechanisms, J.
Mech. Des. 131 (2009)021002.
[42] J. Oberhammer, M. Tang, A.Q. Liu, G. Stemme, Mechanically
tri-stable, truesingle-pole-double-throw (SPDT) switches, J.
Micromech. Microeng. 16(2006) 22512258.
[43] G. Chen, D.L. Wilcox, L.L. Howell, Fully compliant double
tensural tristablemicromechanisms (DTTM), J. Micromech. Microeng.
19 (2009) 025011.
[44] R. Mutlu, G. Alici, A multistable linear actuation
mechanism based onarticial muscles, J. Mech. Des. 132 (2010)
111001.
[45] Y. Gerson, S. Krylov, B.R. Ilic, D. Schreiber, Large
displacement low voltagemultistable micro actuator, Proc. IEEE
Micro Electro Mech. Syst. (2008)463466.
[46] P. Villaggio, Mathematical Models for Elastic Structures,
Cambridge Univer-sity Press, Cambridge, New York, 1997.
[47] L. Shampine, A BVP solver based on residual control and the
MATLAB PSE,ACM Trans. Math. Software 27 (2001) 299316.
[48] L. Shampine, M.W. Reichelt, J. Kierzenka, Solving Boundary
Value Problemsfor Ordinary Differential Equations in MATLAB with
bvp4c, Available at:/www.mathworkscom/bvp tutorialS.
[49] O. Bochobza-Degani, O. Elata, Y. Nemirovsky, An efcient
DIPIE algorithm forCAD of electrostatically actuated MEMS devices,
J. Microelectromech. Syst. 11(2002) 612620.
[50] M.A. Criseld, Non-linear Finite Element Analysis of Solids
and Structures,John Wiley & Sons, London, UK, 1991.
[51] G.J. Simitses, Dynamic Stability of Suddenly Loaded
Structures, Springer-Verlag, New York, 1990.
[52] W.C. Tang, T.-C.H. Nguyen, M.W. Judy, R.T. Howe,
Electrostatic-comb drive oflateral polysilicon resonators, Sensors
Actuators A: Phys. A21A23 (1990)328331.
[53] A. Groeneveld, M. Elwenspoek, Comb-drive actuators for
large displacements,J. Micromech. Microeng. 6 (1996) 320329.
[54] D.G. Smith, G.T. Larsen, L.L. Howell, Design optimization
of a linear-motionlarge-displacement micromechanism for high
off-axis stiffness, in: Proceed-ings of the ASME International
Design Engineering Technical Conferencesand Computers and
Information in Engineering Conference 2009, DETC2009,DETC 2009
87301, pp. 459464.
[55] J.D. Grade, Design of large deection electrostatic
actuators, J. Microelec-tromech. Syst. 12 (2003) 335343.
[56] P. Dowd, Tilted folded-beam suspension for extending the
stable travel rangeof comb-drive actuators, J. Micromech. Microeng.
13 (2003) 178183.
[57] Z. Changfu, J. Zhuangde, L. Dejiang, R. Taian, 3D MEMS
design method viaSolidWorks, in: Proceedings of the Nano/Micro
Engineered and MolecularSystems, 2006 NEMS 06 1st IEEE
International Conference, 2006, pp. 747751.
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