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A tristable compliant micromechanism with two serially

connected bistable mechanisms

Dung-An Wang a, *, Jyun-Hua Chen a and Huy-Tuan Pham b

aGraduate Institute of Precision Engineering, National Chung Hsing University, Taichung40227, Taiwan, ROC bFaculty of Mechanical Engineering, University of Technical Education, Ho Chi MinhCity, Vietnam.

Abstract

A tristable micromechanism with a bistable mechanism embedded in asurrounding bistable mechanism is developed. Three stable equilibrium positions arewithin the range of the linear motion of the mechanism. The proposed mechanism hasno movable joints and gains its mobility from the deflection of flexible members. Thetristability of the mechanism originates from the different actuation loads of the two bistable mechanisms. Finite element analyses are used to characterize the tristable behavior of the mechanism under static loading. An optimal design formulation is proposed to find the geometry parameters of the mechanism. Prototypes of themechanism are fabricated by a simple electroforming process. The characteristics of themechanism are verified by experiments. The force versus displacement curve of themechanism exhibits the tristable behavior within a displacement range of 260 m .

Keywords : Tristable micromechanism; bistable

____________ * Corresponding author: Tel.:+886-4-22840531 ext. 365; fax:+886-4-22858362

E-mail address : [email protected] (D.-A. Wang).

1. Introduction

Multiple passive stable equilibrium configurations enable the design of systemswith both power efficiency and kinematic versatility while the actuators and control staysimple [1]. For example, multistable mechanisms can be used for multiple switching andoptical networking [2]. With the concept of multistable mechanisms, a wide range of operating regimes or novel mechanical systems without undue power consumption can becreated [1]. Substantial interest has focused on design of bistable [3-10], tristable, [11-16], and quadristable mechanisms [1, 2, 17].

In the regime of tristable mechanisms (TMs), Ohsaki and Nishiwaki [11] used ashape optimization approach to generate a truss-like TM. Due to the random nature of


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their design method, the number of structural members of the generated mechanismmight be large. Su and McCarthy [12] synthesized a compliant four-bar linkage withthree equilibrium configurations. A successful design relies on the fact that bothkinematic and static constraints of their compliant mechanisms can be modeled in polynomial equations. Oberhammer et al. [13] proposed tristable mechanism actuated byelectrostatic actuators. A large electrostatic force is required to avoid contact stiction between the structural members of their mechanism. Based on geometric symmetry,Pendleton and Jensen [14] demonstrated a tristable four-link mechanism. Their designhas three mechanically stable positions gained through storage and release of elasticenergy, not through friction or detents. Chen et al. [15] developed a tristablemicromechanism based on the operation of certain bistable compliant mechanism withsoft spring-like behavior. When pulled in the opposite direction from the fabricated position, their mechanism exhibits the three stable equilibrium positions. Chen et al. [16] proposed a tristable mechanism which employs orthogonal compliant mechanisms toachieve tristability. Nonsymmetric designs may be needed to replace the symmetricconfiguration of their mechanism in order to reach a desired equilibrium position betweenthe two possible deflected positions of the end-effector.

This paper describes a design of a compliant TM. The proposed TM has acurved-beam bistable mechanism embedded in another curved-beam bistable structure.Multi-stability is provided by buckling of curved-beam structures of the mechanism. Thedesign concept of combining two bistable mechanisms has been reported by Han et al. [2],Chen et al. [17] and Oh and Kota [18], where the multistability originates from bistable behaviors of the mechanism along two orthogonal directions [2, 17] or of a combinedmotion of two bistable rotational mechanisms [18]. The motion of the proposed TM istranslational in a one-dimensional manner. Finite element analyses are carried out toevaluate the mechanical behaviors of the design. Prototypes of the device are fabricatedusing an electroforming process. Experiments are carried out to demonstrate theeffectiveness of the TM.

2. Design2.1 Operational principle

A schematic of the TM is shown in Fig. 1(a). A Cartesian coordinate system isalso shown in the figure. The z axis completes the right handed orthogonal set. Themechanism consists of a shuttle mass, a guide beam, inner curved beams and outer curved beams. The inner curved beams clamped at one end by the shuttle mass and fixedat the other end by the guide beam acts similar to a bistable mechanism of curved beamtype. The outer curved beams with one end clamped at the guide beam and the other endfixed at the anchor also behave similar to a bistable mechanism of curved beam type.The shuttle mass and the guide beam are employed to prevent the mechanism fromtwisting during operation, and are designed to be stiff. Furthermore, curved beams withlarge thickness in the z-direction could also be used to prevent twisting of the mechanism.Upon the application of a force F to the shuttle mass in the y direction, the outer curved beams deflect initially, increasing the strain energy. The compression energy inthe outer curved beams increases to a maximum at a certain displacement of themechanism, but then decreases as the mechanism snaps towards its second stable position,as shown in Fig. 1(b). As the TM deflects further, the bending energy in the outer and


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inner curved beams increases. While the compression energy in the inner curved beamsincreases to a maximum at a certain displacement of the mechanism, but then decreasesas the mechanism snaps towards its third stable position, as shown in Fig. 1(c).

Fig. 1 Operational principle of the TM.

Vangbo [19] treated the snap-through behavior of a double-clamped curved beamusing Eulers beam buckling theory [20]. He evaluated the bending and compressionenergy terms of his analytical solution, and found that bending energy is larger thancompression energy when the beam is loaded initially; as the displacement of the beamincreases, compression energy increases rapidly and bending energy decreases; after theevent of snap-through of the beam, bending energy starts to increase again while thecompression energy remains constant due to a constant stress normal to the cross-sectionof the beam.


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Fig. 2 A typical force versus displacement curve of the TM and the correspondingconfigurations at displacement a , displacement b , and displacement c , shown in the

inlets.

Fig. 2 shows a typical reaction force versus displacement (f-d) curve of the TM.The configurations of the TM in its three stable positions are shown in the inlets. When aforce is applied to the mechanism through the shuttle mass, the value of the reaction forceof the TM increases initially, and reaches a local maximum,

max1 F , the critical force for

outer beams of the TM to buckle. When the force applied to the mechanism is greater than

max1 F , the outer beams of the TM buckle and the reaction force decreases, reaches a

local minimum,min1

F , then increases and attains a value of 0, where the TM is in its

second stable position b . As the shuttle mass is displaced further, the reaction forceincreases, reaches a local maximum,

max2 F , the critical force for inner beams of the TM to

buckle. . When the force applied to the mechanism is greater thanmax2

F , the inner beams

of the TM buckle and the reaction force decreases, reaches a local minimum,min2

F , then

increases again and attains a value of 0, where the TM is in its third stable position c .


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Fig. 3 (a) A schematic of a quarter model. (b) Dimensions of the guide beam and theshuttle mass.

2.2 DesignThe shape of the outer and inner curved beams of the TM is based on cosine

curves. Due to symmetry, only a quarter model of the mechanism is considered. Fig. 3(a)is a schematic of the quarter model. The shape of the curved beams is

i

r ir L

xh y

cos1

2(1)

where ),( r r y x is the position vector with the reference origin at the left end of the curved beams. L and h are the span and apex height of the curved beams, respectively. Thesubscript i = 1 and 2 refer to the outer and inner curved beams, respectively. The widthsof the outer and inner curved beams, 1W and 2W , respectively, are indicated in Fig. 3(a).The design of the TM is based on an optimization procedure where the geometry parameters of the TM is optimized via the parameters of 1 L , 1h , 1W , 2 L , 2h and 2W . The

nondominated sorting genetic algorithm [21] is applied to the optimization of the shapeof outer and inner curved beams. In the optimization process, number of generations andnumber of populations are specified as 10 and 20, respectively. The objective functionsof the optimization problem are

1min

max

1

1

F F

Min (2)

1min

max

2

2

F F

Min (3)

wheremax1

F ,min1

F ,max2

F , andmin2

F are the reaction forces of the TM indicated in Fig. 2.

The objective function of Equation (2) is selected for assurance of a high level of snap-


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through behavior of the TM so that the mechanism can settle down to its second stable position easily. The objective function of Equation (3) is also formulated for assurance of a high level of snap-through behavior of the TM to have the mechanism settle down to itsthird stable position and to eliminate the possibility of returning to its second stable position under influence of small disturbance.

The widths of the outer and inner curved beams, 1W =8 m and 2W =7 m ,respectively, are indicated in Fig. 3(a). The outer curved beams have the span 1 L =1252m , and the apex height 1h =75 m . The inner curved beams have the span 2 L =1105m , and the apex height 2h =74 m . The z-direction thickness of the outer and inner curved beams is taken as 10 m . The guide beam and the shuttle have their dimensionsindicated in Fig. 3(b). The z-direction thickness of the guide beam and the shuttle mass is20 m .

Fig. 4 A mesh for the finite element model.

Due to the geometry complexity, finite element analysis is utilized to obtain the f-d curve of the TM. Fig. 4 shows a mesh for a two-dimensional finite element model. ACartesian coordinate system is also shown in the figure. As shown in Fig. 4, a uniformdisplacement is applied in the y direction to the right end of the inner curved beam,and the displacements in the x , directions and the rotational degree of freedom at theanchors are constrained to represent the clamped boundary conditions in the experiment.The displacement in the direction and the rotational degree of freedom of thesymmetry plane are constrained to represent the symmetry conditions due to the loadingconditions and the geometry of the model. In this investigation, the material of thedevice is assumed to be nickel. For the linear elastic and isotropic model, the Youngsmodulus E is taken as 205 GPa, and the Poissons ratio p is taken as 31.0 . The

commercial finite element program ABAQUS is employed to perform the computations.The finite element model has 124 2-node beam elements. The width and the z-directionthickness of the beam element B21 employed in the finite element analyses are specifiedaccording to the dimensions of the TM. A mesh convergence study is performed toobtain accurate solutions of displacement solutions.


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Fig. 5 (a) f-d curve and maximum stress versus displacement curve; (b) strain energycurve.

2.3 AnalysisFig. 5(a) shows the f-d curve of the TM when the shuttle mass is displaced in the

direction, where a = 0 m , b = 139 m , c = 296 m ,max1

F = 584 N ,min1

F = -

168 N ,max2

F = 514 N , andmin2

F = -260 N . As seen in the figure, when the

displacement of the shuttle mass increases from 0 (the first stable equilibrium position),the reaction force increases initially, then reaches a local maximum value,

max1 F . As the

displacement increases further, the reaction force decreases; in the event of snap-through

of the outer beams of the mechanism, where the strain energy of the outer beams is near alocal maximum

max1U shown in Fig. 5(b), the reaction force reaches a value of 0, then

decreases and attains a local minimum,min1

F . With the increasing displacement of the

shuttle mass, the reaction force increases again and reaches a value of 0 , where themechanism is in its second stable equilibrium position b . As the shuttle mass isdisplaced further, the reaction force increases, reaches a local maximum value,

max2 F . As

the displacement increases further, the reaction force decreases and reaches a value of 0;then the inner beams of the TM buckle, where the strain energy of the inner beams is near a local maximum

max2U shown in Fig. 5(b), and the reaction force decreases, attains a


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local minimum,min2

F . Next the reaction force increases again, reaches a value of 0 , and

the mechanism reaches its third stable equilibrium position c . It can be said that the

tristability of the mechanism originates from the different actuation loads of the two bistable mechanisms. Fig. 5(a) also shows the von Mises stress as a function of thedisplacement based on the finite element computations. The highest stress, 602 MPa,occurs in the event of the second snap-through of the TM. In order to avoid yielding of the TM under loading, the stress in the mechanism should not exceed the typical yieldstrength, 700 MPa, of the nickel material used for the TM in this investigation. As seenin the figure, the value of the highest stress is less than the yield strength of the material.

As shown in Fig. 5(b), when the displacement of the mechanism increases fromits first stable position, the strain energy of the outer beams increases, and the strainenergy of the inner beams increases slightly. As the displacement of the mechanismincreases further before the mechanism snaps to the second stable equilibrium position,

b , the majority of the strain energy is absorbed by the outer beams. In the event of snap-through of the outer beams of the mechanism, the strain energy is near a local maximum

max1U , then decreases and attains a local minimum. As the TM deflects further, the strain

energy in the outer and inner curved beam increases. As the displacement of themechanism increases beyond a certain displacement, where the inner beams of the TM buckle, the strain energy of the inner beams drops abruptly from a local maximum,

max2U ,

and the strain energy of the outer beams decreases gradually. While the strain energy of the inner beams decreases towards a local minimum; the mechanism settles in its thirdstable equilibrium position, c .

As shown in Fig. 5(a), the f-d curve of the TM is highly nonlinear. Thenonlinearities can be attributed to the post-buckling behavior, geometric nonlinearity anddamping effects. Geometry nonlinearities due to large deflections are commonlyencountered in compliant mechanisms [22]. Large strain of the TM causes significantchanges in its geometry. Modeling of force-deflection characteristics of multistablecompliant mechanisms can be performed by the pseudo-rigid-body model (PRBM) [15].However, in order to accurately describe the behavior of compliant mechanisms usingPRBM, where to place the added springs and what value to assign their spring constantsare important. Chen et al. [15] have shown that a pseudo-rigid-body model can be usedto identify tristable configurations. The link lengths and spring constants of the modelneed to be calculated. An elaborate system of virtual work and kinematic equations issolved numerically to obtain the f-d curve of their TM. Theory of static Euler bucklingof a double-clamped slender beam can be used to model the force-deflection and snap-through behavior of the compliant beams [23]. This classical treatment has been used for the analysis of compliant bistable mechanisms with curved double-clamped beams withfixed ends [20, 23, 24].

In order to analyze and design the tristable compliant mechanisms, an analyticalmodel should be provided. Because the TM includes initially curved beams andundergoes a large deformation, we presented an analytical model to provide an insightinto the influence of the design parameters on the behaviors of the TM. The model is based on a curved beam model reported by Gerson et al. [25], where arch-shaped beamsare connected serially and exhibit sequential buckling. The TM considered hereexperiences multiple snap-through between its stable equilibrium positions. A


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multistable mechanism shall require no force to keep the mechanism in its stableequilibrium position. Multiple snap-through behavior of the serially connected bistable beams of Gerson et al. [25] is observed. However, external force is needed to keep their mechanism in some of its stable positions. The model presented below follows the basic procedure described by Gerson et al. [25] for their curved beams. It is assumed that theapex height h and the width W are much smaller than the span L of the curved beamsof the TM. The equilibrium of the beams is described by the two differential equations as[25]

02

)( 2

w

yu EA r (4)

)(2

)( 2 L x F

ww yuw y EA EIw r

IV

(5)

where )( xu and )( xw are the axial displacement and the lateral displacement,respectively, )( x is the Dirac delta function, F is the force applied to the midpoint of the TM, and dxd /)( . E , A and I are the Young s modulus, the cross-sectional areaand the second moment of the area, respectively. The proposed TM is composed of double-clamped beams with fixed ends, where the embedded bistable mechanism hassliding clamped boundary conditions. Due to symmetry, only a half TM is consideredand the symmetry conditions 0u , 0w and 2/ F w EI at the midpoint of the TMare enforced. Equations (4) and (5) are solved numerically with bvp4c, a two-point boundary value problem (BVP) solver integrated in the software Matlab. In order tosolve the three-point BVP of the TM with two serially connected beams using bvp4c, thisBVP is solved as two problems. One set on ],0[ 1 L and the other on ],[ 21 L L . By definingan independent variable )/()( 1211 L L L x L , ranges from 0 to 1 L in the secondinterval like in the first interval. Equations (4) and (5) must be written as a system of first order ordinary differential equations for each set. By prescribing a deflection at themidpoint of the TM, the force F considered as an unknown parameter can be calculated.

Fig. 6 shows the f-d curve obtained by the analytical model. The f-d curve basedon the finite element model is also shown in the figure. The trend of the f-d curve predicted by the analytical model agrees with that of the finite element model. Therelative errors in the first stable position and the second stable position are 6.5% and0.3%, respectively. The discrepancy can be attributed to the approximate nature of the

analytical model due to the high geometry nonlinearities and the large deflectionsexhibited by the TM. It is assumed that h L and the deflections, while comparablewith the thickness of the beam, are small with respective to L in order to obtain accurate predictions of the f-d curve based on the analytical model [25]. In this investigation, thedeflection 310 m is relatively large compared to the spans 12 L and 22 L , which valuesare indicated in Fig. 3(a). As described by Gerson et al. [25], this simple analyticalmodel is convenient for the evaluation of the preliminary design parameters of curved beams. Accurate predictions of the f-d curves of curved beams, e.g. the tristablecompliant mechanisms considered here, should resort to more elaborate models.


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Fig. 6 f-d curves based on the analytical model and the finite element model.

Fig. 7 Fabrication steps.


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Table 1 Chemical composition and operation conditions for the low-stress nickelelectroplating solution.

Chemical/plating parameter Amount/value Nickel sulfamate( O4H)SO Ni(NH 2232 )

600 g/L

Boric acid ( 33BOH ) 40 g/L

Nickel chloride ( OH6 NiCl 22 ) 5 g /LStress reducer 15 ml/LLeveling agent 15 ml/LWetting agent 2 ml/LBath temperature 45 C

Plating current type dc current pH of the solution 4.3Plating current density 1.5 2A/dmDeposition rate 0.26 m/minAnode-cathode spacing 100 mmAnode type titanium

3. Fabrication and testingIn order to prove the tristability of the TM design, prototypes of the mechanisms

are fabricated by a simple electroforming process on glass substrates. Fig. 7 shows thefabrication steps, where three masks are used. First, a titanium metallization layer isdeposited on the whole glass substrate. Next, a 5 m-thick photoresist (AZ4620) iscoated and patterned to prepare a mold for electrodeposition of a copper sacrificial layer.Then, the photoresist is stripped and a 10 m-thick photoresist (AZ4620) is coated and patterned on top of the copper sacrificial layer. Into this mold, a 10 m-thick nickel layer is electrodeposited using a low-stress nickel sulfamate bath with the chemicalcompositions listed in Table 1. Next, a 10 m-thick photoresist (AZ4620) is coated and patterned to prepare a mold for nickel electroplating on top of the shuttle, the rigid link and the anchors. Then, a 10 m-thick nickel is electrodeposited onto the mold.Following that, the photoresist and copper sacrificial layers are removed to release thenickel microstructures. Finally, the titanium layer outside the anchor regions is wetetched. Fig. 8(a) shows an optical microscope (OM) photo of a fabricated device. Fig.8(b), (c) and (d) shows the close-up views of the TM near the guide beam; the shuttlemass and the anchor, respectively.


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Fig. 8 (a) A photo of a fabricated TM. Close-up views of the TM near (b) the guide

beam; (c) the shuttle mass; (c) the anchor.

Fig. 9 (a) Experimental setup. (b) Dimensions of the probe. (c) A close-up view near the tip of the probe.

Fig. 9(a) is a photo of the experimental setup for measurement of the f-d curve of the TM. A Cartesian coordinate system is also shown in the figure. The substrate withspecimens for testing is mounted on a rotation stage. The rotation stage can rotate withrespect to the z axis. A probe for pushing the micromechanism is attached to a load cell


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which is fixed to a translation stage with z translation degree of freedom. The z -translation stage is placed on top of a translation stage with translational degree of freedoms in and y directions. The dimensions of the probe are indicated in Fig. 9(b).In order to facilitate the translation of the probe parallel to the substrate and to maintaincontact of the probe tip with the micromechanism, the tip of the probe is machined tohave a 135 slant angle to the probe axis as shown in Fig. 9(c). The translation stageshave a resolution of 10 m . The resolution of the rotation stage is

6/1 . The load cell(LVS-5GA, Kyowa Electronic Instruments Co., Ltd.) has a rated capacity of 50 mN , anda resolution of 10 N .

The alignment of the probe axis to the loading axis of micromechanisms isadjusted by the rotation stage and the translation stages so that the micromechanism doesnot twist during loading. Initially the mechanism is in its first stable equilibrium position.The probe tip is pushed slowly against the edge surface of the shuttle mass of the

mechanism. The pressing force applied to the mechanism is increased until it snapstoward its other stable equilibrium positions. The displacement of the shuttle mass andthe reading of the load cell are recorded. A CCD camera is used for capturing thesuccessive images of the motion of the mechanism.

Fig. 10 Snapshots for motion of the TM.

4. Results and discussionsUsing the experimental setup, the tristable behavior of the TM is demonstrated.

The experimental f-d curves of the motion of the mechanism are also obtained. Fig. 10shows sequences of snapshots from experiments. As shown in Fig. 10(a-d), a force isapplied on an edge surface of the shuttle mass. When the magnitude of the force isincreased, the shuttle mass moves forward. As the force reaches a certain maximumvalue, the probe tip loses contact with the top surface of the shuttle mass, and themechanism snaps into its second stable position (Fig. 10(b)). Then the probe tip ismoved to contact with the edge surface of the shuttle mass again and pushed slowly untilthe TM snaps into its third stable position (Fig. 10(c)). As the probe tip is pulled backward, the TM stays in its third stable position (see Fig. 10(d)).


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Fig. 11 shows the f-d curves of the micromechanism bases on experiments andfinite element computations. As the shuttle mass is displaced, the reaction forceincreases initially. In the event of snap-through, the probe tip loses contact with theshuttle mass, and the TM snaps into its second stable position as seen by thediscontinuous curve of the experiments. A displacement controlled approach is adoptedin the finite element analyses, where the snap-through behavior is signaled by the 0 valueof the reaction force, and the negative values of the reaction force are obtained while theshuttle mass is displaced further towards the second stable position of the TM. As theshuttle mass is displaced further, a local maximum of the reaction force is reached, then areduction in the reaction force indicates the second snap-through of the TM. At thesecond snap-through, the probe tip loses contact with the shuttle mass and the mechanism jumps to its third stable position. As shown in the figure, the micromechanism exhibitsthe tristable behavior within a displacement range of 260 m . The general trend of theexperimental results agrees with that based on the finite element model with the designeddimensions of the TM.

Fig. 11 f-d curves of the fabricated TM.

The discrepancies between the experiments and finite element analyses of thedevice can be attributed to uncertainties in material properties, geometry and loadingconditions of the experiments. It is revealed by optical microscopic inspections that thewidth measured on the top surface of the beam is larger than that of the design. Thewidths of the outer and inner curved beams measured by an Olympus Bx60 confocal

microscope are 9 m and 9 m , respectively, which are larger than the designed widthsof the outer and inner curved beams, 1W =8 m and 2W =7 m , respectively. As shownin Fig. 11, the finite element model with the measured dimensions does not improvemuch in terms of the prediction of the stable positions, compared to those based on themodel with the designed dimensions. Measurement errors of force and displacement alsocontribute to the discrepancies. The contact of the probe tip for force measurement withthe surface of the shuttle mass is not fixed, where sliding may occur, and the alignment of the longitudinal axis of the probe with the symmetry plane of the TM may not be perfectduring the experiments, where twisting of the TM may happen.


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Fig. 12 Effects of the geometry parameters on the tristability of the TM.


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The effects of the geometry parameters of the TM on the characteristics of thedesign are investigated using the finite element model. Fig. 12(a) illustrates how the apexheights

1h and

2h affect the tristability of the TM. The TMs preserving its tristability are

marked in the figure. The TM with the values of the geometry parameters causing theTM to lose its tristability are also marked in the figure. For the TM with large values of

1h and 2h , the reaction forces max1 F , min1 F , max2 F , and min2 F , and the deflection ranges are

significantly larger than those with small values of 1h and 2h . Fig. 12(b) shows theeffects of the spans 1 L and 2 L on the tristability of the TM. For the TM with large valuesof 1 L and 2 L , the range of the spans to preserve its tristability is large. However, thetristability of the TM with large values of the spans degrades, which means it is difficultto keep the TM in its stable equilibrium positions due to the relatively small values of thereaction forces

max1 F ,

min1 F ,

max2 F , and

min2 F (in the order of 1 N ).

The effects of the widths of the outer and inner curved beams, 1W and 2W ,respectively, are illustrated in Fig. 12(c). The reaction forces of the TM with smallvalues of the widths are much smaller than those with large values of the widths. For example, the reaction forces of the TM with 1W =1 m and 2W =1 m are in the order of 1 N , and the smallest reaction force of the TM with 1W =12 m and 2W =13 m is inthe order of 10 N . The values of 1W and 2W of the TM of the optimal design, indicatedin Fig. 12(c), fall in about the middle of the ranges of 1W and 2W which preserve thetristability of the TM. The designed beam width of the TM has a large margin of error toensure its tristability. Obviously, if the geometry parameters are uncertain due tomicrofabrication processes, the device may lose the tristability.

Both the outer curved beams and inner curved beams of the TM buckle under loading. In the current design, the outer beams buckle first and then inner beams buckle(See Fig. 10). With the dimensions of 1W =8 m , 1 L =1110, 1h =79 m , 2W =6 m ,

2 L =1164 m , and 2h =77 m (these symbols are illustrated in Fig. 3(a)), the inner beams buckle first and then outer beams buckle based on the analytical model. Thegeometric parameters can be changed to achieve a different design. The proposed TM provides tristability in a linear, sequential manner. A potential application of the TMcould be a microrobot system on chip for control of droplet dispensing in microchannelsin the field of chemical engineering and bioengineering which requires a sequentialoperation with multistability for efficient reaction and productivity [26]. Another possible application is the control of cascaded rubber-seal valves for gas regulation inmicrochannels, where rectifying of gas flow can be achieved through the deflection of valve diaphragms [27].

5. ConclusionsA TM has been designed, fabricated and validated by experiments. The TM

consists of beams with profiles of cosine curves. The combination of two bistablemechanisms provides the tristability of the TM. Prototypes of the TM are fabricated by asimple electroforming process. A device for obtaining the in-plane force-displacementcharacteristics of the micromechanism is developed. The observed force versusdisplacement curve of the micromechanism exhibits a well-defined tristability. The


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micromechanism exhibits the tristable behavior within a displacement range of 260 m .The presented design provides a means of attaining a tristable, planar micromechanismwith its motion in a linear manner.

AcknowledgementThe computing facilities provided by the National Center for High-Performance

Computing (NCHC) are greatly appreciated. The authors are thankful for financialsupport from the National Science Council, R.O.C., under Grant No. NSC 96-2221-E-005-095.


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[17] G. Chen, Y. Gou, A. Zhang, Systehsis of compliant multistable mechanismsthrough use of a single bistable mechanism, Journal of Mechanical Design 133(2011) 081007.

[18] Y.S. Oh, S. Kota, Synthesis of multistable equilibrium compliant mechanism usingcombinations of bistable mechanisms, Journal of Mechanical Design 131 (2009)021002.

[19] M. Vangbo, An analytical analysis of a compressed bistable buckled beam, Sensorsand Actuators A 69 (1998) 212-216.

[20] J.M. Gere, S.P. Timoshenko, Mechanics of Materisls, Second edition, WadsworthInc., Belmont, California, USA, 1984.

[21] K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiobjectivegenetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation 6(2002) 182-197.

[22] S. Park, D. Hah, Pre-shaped buckled-beam actuators: Theory and experiments,Sensors and Actuators A 148 (2008) 186-192.

[23] J. Qiu, J.H. Lang, A.H. Slocum, A curved-beam bistable mechanism, Journal of Microelectromechanical Systems 13 (2004) 137-146.

[24] P. Cazottes, A. Fernandes, J. Pouget, M.Hafez, Bistable buckled beam: Modeling of actuating force and experimental validations, Journal of Mechanical Design 131(2009) 101001.

[25] Y. Gerson, S. Krylov, B. Ilic, D. Schreiber, Design considerations of a large-displacement multistable micro actuator with serially connected bistable elements,Finite Elements in Analysis and Design 49 (2012) 58-69.

[26] F. Arai, On-chip robotics: Technical issues and future direction, 2010 IEEEInternational Conference on Robotics and Automation Workshop on Micro-NanoRobotics and New Evolution, Anchorage, Alaska, May 3-8, 2010.

[27] O.C. Jeong, S. Konishi, Pneumatic gas regulator with cascaded PDMS seal valves,Sensors and Actuators A 143 (2008) 84-89.


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List of figuresFig. 1 Operational principle of the TM.Fig. 2 A typical force versus displacement curve of the TM and the corresponding

configurations at displacement a , displacement b , and displacement c , shown inthe inlets.

Fig. 3 (a) A schematic of a quarter model. (b) Dimensions of the guide beam and theshuttle mass.

Fig. 4 A mesh for the finite element model.Fig. 5 (a) f-d curve and maximum stress versus displacement curve; (b) strain energy

curve.Fig. 6 f-d curves based on the analytical model and the finite element model.Fig. 7 Fabrication steps.Fig. 8 A photo of a fabricated TM.Fig. 9 (a) Experimental setup. (b) Dimensions of the probe. (c) A close-up view near

the tip of the probe.Fig. 10 Snapshots for motion of the TM.Fig. 11 f-d curves of the fabricated TM.Fig. 12 Effects of the geometry parameters on the tristability of the TM.

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