Mechanism and Machine Theory 71 (2014) 27–39

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Mechanism and Machine Theory

j ourna l homepage: www.e lsev ie r .com/ locate /mechmt

A tristable compliant micromechanism with two seriallyconnected bistable mechanisms

Dung-An Wang a,⁎, Jyun-Hua Chen a, Huy-Tuan Phamb

a Graduate Institute of Precision Engineering, National Chung Hsing University, Taichung 40227, Taiwan, ROCb Faculty of Mechanical Engineering, University of Technical Education, Ho Chi Minh City, Viet Nam

a r t i c l e i n f o

⁎ Corresponding author. Tel.: +886 4 22840531x36E-mail address: daw@dragon.nchu.edu.tw (D.-A. W

0094-114X/$ – see front matter © 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.mechmachtheory.2013.08.

a b s t r a c t

Article history:Received 20 March 2013Received in revised form 27 August 2013Accepted 30 August 2013Available online 27 September 2013

A tristable micromechanism with a bistable mechanism embedded in a surrounding bistablemechanism is developed. Three stable equilibrium positions are within the range of the linearmotion of the mechanism. The proposed mechanism has no movable joints and gains itsmobility from the deflection of flexible members. The tristability of the mechanism originatesfrom the different actuation loads of the two bistable mechanisms. Finite element analyses areused to characterize the tristable behavior of the mechanism under static loading. An optimaldesign formulation is proposed to find the geometry parameters of the mechanism. Prototypesof the mechanism 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 μm.

© 2013 Elsevier Ltd. All rights reserved.

Keywords:Tristable micromechanismBistable

1. Introduction

Multiple passive stable equilibrium configurations enable the design of systems with both power efficiency and kinematicversatility while the actuators and control stay simple [1]. For example, multistable mechanisms can be used for multipleswitching and optical networking [2]. With the concept of multistable mechanisms, a wide range of operating regimes or novelmechanical systems without undue power consumption can be created [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 a shape optimization approach to generate atruss-like TM. Due to the random nature of their design method, the number of structural members of the generated mechanismmight be large. Su and McCarthy [12] synthesized a compliant four-bar linkage with three equilibrium configurations. A successfuldesign relies on the fact that both kinematic and static constraints of their compliant mechanisms can be modeled in polynomialequations. Oberhammer et al. [13] proposed tristable mechanism actuated by electrostatic actuators. A large electrostatic force isrequired to avoid contact stiction between the structuralmembers of theirmechanism. Based on geometric symmetry, Pendleton andJensen [14] demonstrated a tristable four-link mechanism. Their design has three mechanically stable positions gained throughstorage and release of elastic energy, not through friction or detents. Chen et al. [15] developed a tristable micromechanism based onthe operation of a certain bistable compliant mechanism with soft spring-like behavior. When pulled in the opposite direction fromthe fabricated position, their mechanism exhibits the three stable equilibrium positions. Chen et al. [16] proposed a tristablemechanism which employs orthogonal compliant mechanisms to achieve tristability. Nonsymmetric designs may be needed toreplace the symmetric configuration of their mechanism in order to reach a desired equilibrium position between the two possibledeflected positions of the end-effector.

This paper describes a design of a compliant TM. The proposed TM has a curved-beam bistable mechanism embedded in anothercurved-beam bistable structure. Multi-stability is provided by buckling of curved-beam structures of the mechanism. The design

5; fax: +886 4 22858362.ang).

ll rights reserved.018

28 D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

concept of combining two bistable mechanisms has been reported by Han et al. [2], Chen et al. [17] and Oh and Kota [18], where themultistability originates from bistable behaviors of themechanism along two orthogonal directions [2,17] or of a combinedmotion oftwo bistable rotationalmechanisms [18]. Themotion of the proposed TM is translational in a one-dimensionalmanner. Finite elementanalyses are carried out to evaluate the mechanical behaviors of the design. Prototypes of the device are fabricated using anelectroforming process. Experiments are carried out to demonstrate the effectiveness of the TM.

2. Design

2.1. Operational principle

A schematic of the TM is shown in Fig. 1(a). A Cartesian coordinate system is also shown in the figure. The z axis completes theright handed orthogonal set. The mechanism 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 shuttlemass and fixed at the other end by the guide beamacts similar to a bistablemechanism of curved beam type. The outer curved beams with one end clamped at the guide beam and the other end fixed at theanchor also behave similar to a bistable mechanism of curved beam type. The shuttle mass and the guide beam are employed toprevent themechanism from twisting during operation, and are designed to be stiff. Furthermore, curved beamswith large thicknessin the z-direction could also be used to prevent twisting of themechanism. Upon the application of a force F to the shuttlemass in the−y direction, the outer curved beams deflect initially, increasing the strain energy. The compression energy in the outer curvedbeams increases to a maximum at a certain displacement of the mechanism, but then decreases as the mechanism snaps towards itssecond stable position, as shown in Fig. 1(b). As the TM deflects further, the bending energy in the outer and inner curved beamsincreases. While the compression energy in the inner curved beams increases to a maximum at a certain displacement of themechanism, but then decreases as the mechanism snaps towards its third stable position, as shown in Fig. 1(c).

Vangbo [19] treated the snap-through behavior of a double-clamped curved beam using Euler's beam buckling theory [20]. Heevaluated the bending and compression energy 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 beam increases, compression energy increasesrapidly and bending energy decreases; after the event 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-section of the beam.

Anchor

Shuttlemass

Outer curvedbeam, BS2

Inner curvedbeam, BS1

Guidebeam

x

y

F

a

F

b

F

c

z

Fig. 1. Operational principle of the TM.

29D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

Fig. 2 shows a typical reaction force versus displacement (f–d) curve of the TM. The configurations of the TM in its three stablepositions are shown in the inlets.When a force is applied to themechanism through the shuttlemass, the value of the reaction force ofthe TM increases initially, and reaches a local maximum, F1max , the critical force for outer beams of the TM to buckle. When the forceapplied to the mechanism is greater than F1max , the outer beams of the TM buckle and the reaction force decreases, reaches a localminimum, F1min

, then increases and attains a value of 0, where the TM is in its second stable position b. As the shuttlemass is displacedfurther, the reaction force increases, reaches a local maximum, F2max , the critical force for inner beams of the TM to buckle. When theforce applied to themechanism is greater than F2max, the inner beamsof the TMbuckle and the reaction force decreases, reaches a localminimum, F2min

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

2.2. Design

The shape of the outer and inner curved beams of the TM is based on cosine curves. Due to symmetry, only a quarter model ofthe mechanism is considered. Fig. 3(a) is a schematic of the quarter model. The shape of the curved beams is

wherefor asspositioTM to

Fig. 2. Athe inle

yr ¼hi2

1− cosπxrLi

� �ð1Þ

(xr,yr) is the position vector with the reference origin at the left end of the curved beams. L and h are the span and apex

whereheight of the curved beams, respectively. The subscript i = 1 and 2 refer to the outer and inner curved beams, respectively. Thewidths of the outer and inner curved beams, W1 and W2, respectively, are indicated in Fig. 3(a). The design of the TM is based onan optimization procedure where the geometry parameters of the TM is optimized via the parameters of L1, h1,W1, L2, h2 andW2.The nondominated sorting genetic algorithm [21] is applied to the optimization of the shape of outer and inner curved beams. Inthe optimization process, number of generations and number of populations are specified as 10 and 20, respectively. The objectivefunctions of the optimization problem are

Min F1max

.−F1min

−1

���������� ð2Þ

Min F2max

.−F2min

−1

���������� ð3Þ

F1max, F1min

, F2max, and F2min

are the reaction forces of the TM indicated in Fig. 2. The objective function of Eq. (2) is selectedurance of a high level of snap-through behavior of the TM so that the mechanism can settle down to its second stablen easily. The objective function of Eq. (3) is also formulated for assurance of a high level of snap-through behavior of thehave the mechanism settle down to its third stable position and to eliminate the possibility of returning to its second stablen under influence of small disturbance.

positio

The widths of the outer and inner curved beams, W1 = 8 μm and W2 = 7 μm, respectively, are indicated in Fig. 3(a).The outer curved beams have the span L1 = 1252 μm, and the apex height h1 = 75 μm. The inner curved beams have the spanL2 = 1105 μm, and the apex height h2 = 74 μm. The z-direction thickness of the outer and inner curved beams is taken as 10 μm.

Displacement, δ

For

ce, f

a b c

F max1

F min1

F max2

F min2

x

y

typical force versus displacement curve of the TM and the corresponding configurations at displacement a, displacement b, and displacement c, shown ints.

1

x

y 1

a

b

2

2

1

2

W

W 7

75

741105

1252

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

30 D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

The guide beam and the shuttle have their dimensions indicated in Fig. 3(b). The z-direction thickness of the guide beam and theshuttle mass is 20 μm.

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 atwo-dimensional finite element model. A Cartesian 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, y directionsand the rotational degree of freedom at the anchors are constrained to represent the clamped boundary conditions in theexperiment. The displacement in the x direction and the rotational degree of freedom of the symmetry plane are constrained torepresent the symmetry conditions due to the loading conditions and the geometry of the model. In this investigation, thematerial of the device is assumed to be nickel. For the linear elastic and isotropic model, the Young's modulus E is taken as205 GPa, and the Poisson's ratio νp is taken as 0.31. The commercial finite element program ABAQUS is employed to perform thecomputations. The finite element model has 124 2-node beam elements. The width and the z-direction thickness of the beamelement B21 employed in the finite element analyses are specified according to the dimensions of the TM. A mesh convergencestudy is performed to obtain accurate solutions of displacement solutions.

2.3. Analysis

Fig. 5(a) shows the f–d curve of the TM when the shuttle mass is displaced in the −y direction, where a = 0 μm, b = 139 μm,c = 296 μm, F1max = 584 μN, F1min

= −168 μN, F2max = 514 μN, and F2min= −260 μN. As seen in the figure, when the

AnchorSymmetry plane

NodeBeam element

x

y

Fig. 4. A mesh for the finite element model.

0 50 100 150 200 250 300-300

0

200

400

600

800

1000

Displacement (μm)

For

ce (

μN)

Fmax1

Fmin1

Fmin2

a b c

Fmax2

Max

imum

str

ess

(MP

a)

σy

-300

0

200

400

600

800

1000a

Outer beams

Inner beams

Total

0 50 100 150 200 250 300

Displacement (μm)

0

20

40

60

80

100

120

Str

ain

ener

gy (

ρJ)

b

Umax1

Umax2

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

31D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

displacement of the shuttle mass increases from 0 (the first stable equilibrium position), the reaction force increases initially, thenreaches a local maximum value, F1max . As the displacement increases further, the reaction force decreases; in the event ofsnap-through of the outer beams of themechanism, where the strain energy of the outer beams is near a localmaximumU1max shownin Fig. 5(b), the reaction force reaches a value of 0, then decreases and attains a local minimum, F1min

. With the increasingdisplacement of the shuttle mass, the reaction force increases again and reaches a value of 0, where the mechanism is in its secondstable equilibrium position b. As the shuttle mass is displaced further, the reaction force increases, and reaches a local maximumvalue, F2max. As the displacement increases further, the reaction force decreases and reaches a value of 0; then the inner beams of theTM buckle, where the strain energy of the inner beams is near a local maximum U2max shown in Fig. 5(b), and the reaction forcedecreases, attains a local minimum, F2min

. Next the reaction force increases again, reaches a value of 0, and themechanism reaches itsthird stable equilibrium position c. It can be said that the tristability of themechanism originates from the different actuation loads ofthe two bistable mechanisms. Fig. 5(a) also shows the vonMises stress as a function of the displacement based on the finite elementcomputations. The highest stress, 602 MPa, occurs in the event of the second snap-through of the TM. In order to avoid yielding of theTM under loading, the stress in themechanism should not exceed the typical yield strength, 700 MPa, of the nickel material used forthe TM in this investigation. As seen in 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 from its first stable position, the strain energy of theouter beams increases, and the strain energy 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

32 D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

absorbed by the outer beams. In the event of snap-through of the outer beams of the mechanism, the strain energy is near a localmaximum U1max , then decreases and attains a local minimum. As the TM deflects further, the strain energy in the outer and innercurved beam increases. As the displacement of the mechanism increases beyond a certain displacement, where the inner beamsof the TM buckle, the strain energy of the inner beams drops abruptly from a local maximum, U2max , and the strain energy of theouter beams decreases gradually. While the strain energy of the inner beams decreases towards a local minimum; the mechanismsettles in its third stable equilibrium position, c.

As shown in Fig. 5(a), the f–d curve of the TM is highly nonlinear. The nonlinearities can be attributed to the post-buckling behavior,geometric nonlinearity and damping effects. Geometry nonlinearities due to large deflections are commonly encountered in compliantmechanisms [22]. Large strain of the TM causes significant changes in its geometry. Modeling of force–deflection characteristics ofmultistable compliant mechanisms can be performed by the pseudo-rigid-body model (PRBM) [15]. However, in order to accuratelydescribe the behavior of compliant mechanisms using PRBM, where to place the added springs and what value to assign their springconstants are important. Chen et al. [15] have shown that a pseudo-rigid-body model can be used to identify tristable configurations.The link lengths and spring constants of themodel need to be calculated. An elaborate systemof virtualwork and kinematic equations issolved numerically to obtain the f–d curve of their TM. Theory of static Euler buckling of a double-clamped slender beam can be used tomodel the force–deflection and snap-through behavior of the compliant beams [23]. This classical treatment has been used for theanalysis of compliant bistable mechanisms with curved double-clamped beams with fixed ends [20,23,24].

In order to analyze and design the tristable compliant mechanisms, an analytical model should be provided. Because the TMincludes initially curved beams and undergoes a large deformation, we presented an analytical model to provide an insight intothe influence of the design parameters on the behaviors of the TM. The model is based on a curved beam model reported byGerson et al. [25], where arch-shaped beams are connected serially and exhibit sequential buckling. The TM considered hereexperiences multiple snap-through between its stable equilibrium positions. A multistable mechanism shall require no force tokeep the mechanism in its stable equilibrium position. Multiple snap-through behavior of the serially connected bistable beams ofGerson et al. [25] is observed. However, external force is needed to keep their mechanism in some of its stable positions. Themodel 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 beams of the TM. The equilibrium of the beams isdescribed by the two differential equations as [25]

whereforce amomebistab

EA u′ þ y′r þw′ð Þ22

!′

¼ 0 ð4Þ

EIwIV þ EA −y′r−w′� �

u′ þ y′w′ þ w′ð Þ22

!" #′

¼ Fδ x−Lð Þ ð5Þ

u(x) and w(x) are the axial displacement and the lateral displacement, respectively, δ(x) is the Dirac delta function, F is thepplied to the midpoint of the TM, and ()′ = d/dx. E, A and I are the Young's modulus, the cross-sectional area and the secondnt of the area, respectively. The proposed TM is composed of double-clamped beams with fixed ends, where the embeddedle mechanism has sliding clamped boundary conditions. Due to symmetry, only a half TM is considered and the symmetry

Displacement (μm)

For

ce (

μN)

0 50 100 150 200 250 310−800

−600

−400

−200

0

200

400

600

800

1000AnalyticalFEM

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

conditbvp4cthe TMBy defEqs. (4

33D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

ions u′ = 0, w′ = 0 and EIw‴ = F/2 at the midpoint of the TM are enforced. Eqs. (4) and (5) are solved numerically with, a two-point boundary value problem (BVP) solver integrated in the software Matlab. In order to solve the three-point BVP ofwith two serially connected beams using bvp4c, this BVP is solved as two problems. One set on [0,L1] and the other on [L1,L2].

ining an independent variable τ = L1(x − L1)/(L2 − L1), τ ranges from 0 to L1 in the second interval like x in the first interval.) and (5)must bewritten as a systemof first order ordinary differential equations for each set. By prescribing a deflection at theint of the TM, the force F considered as an unknown parameter can be calculated.

midpo

Fig. 6 shows the f–d curve obtained by the analytical model. The f–d curve based on the finite element model is also shown in thefigure. The trend of the f–d curve predicted by the analytical model agrees with that of the finite elementmodel. The relative errors inthe first stable position and the second stable position are 6.5% and 0.3%, respectively. The discrepancy can be attributed to theapproximate nature of the analyticalmodel due to the high geometry nonlinearities and the large deflections exhibited by the TM. It isassumed that h ≪ L and the deflections, while comparable with the thickness of the beam, are small with respective to L in order toobtain accurate predictions of the f–d curve based on the analytical model [25]. In this investigation, the deflection 310 μm isrelatively large compared to the spans 2L1 and 2L2, whose values are indicated in Fig. 3(a). As described by Gerson et al. [25], thissimple analytical model is convenient for the evaluation of the preliminary design parameters of curved beams. Accurate predictionsof the f–d curves of curved beams, e.g. the tristable compliant mechanisms considered here, should resort to more elaborate models.

3. Fabrication and testing

In order to prove the tristability of the TM design, prototypes of the mechanisms are fabricated by a simple electroformingprocess on glass substrates. Fig. 7 shows the fabrication steps, where three masks are used. First, a titanium metallization layer is

(a) Sputter 2μm titanium

(c) Electroplate 5μm copper

(e) Electroplate 10μm nickel

(g) Electroplate 10μm nickel(b) Pattern 5μm AZ4620

(d) Pattern 10μm AZ4620

(f) Pattern 10μm AZ4620

(h) Copper wet etching

(i) Titanium wet etching

Titanium Copper Nickle PhotoresistGlass

Fig. 7. Fabrication steps.

Table 1Chemical composition and operation conditions for the low-stress nickel electroplatingsolution.

Chemical/plating parameter Amount/value

Nickel sulfamate (Ni(NH2SO3)2 ⋅ 4H2O) 600 g/LBoric acid (H3BO3) 40 g/LNickel chloride (NiCl2 ⋅ 6H2O) 5 g/LStress reducer 15 mL/LLeveling agent 15 mL/LWetting agent 2 mL/LBath temperature 45 °CPlating current type dc currentpH of the solution 4.3Plating current density 1.5 A/dm2

Deposition rate 0.26 μm/minAnode-cathode spacing 100 mmAnode type Titanium

34 D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

deposited on the whole glass substrate. Next, a 5 μm-thick photoresist (AZ4620) is coated and patterned to prepare a mold forelectrodeposition of a copper sacrificial layer. Then, the photoresist is stripped and a 10 μm-thick photoresist (AZ4620) is coatedand patterned on top of the copper sacrificial layer. Into this mold, a 10 μm-thick nickel layer is electrodeposited using alow-stress nickel sulfamate bath with the chemical compositions listed in Table 1. Next, a 10 μm-thick photoresist (AZ4620) iscoated and patterned to prepare a mold for nickel electroplating on top of the shuttle, the rigid link and the anchors. Then, a10 μm-thick nickel is electrodeposited onto the mold. Following that, the photoresist and copper sacrificial layers are removed torelease the nickel microstructures. Finally, the titanium layer outside the anchor regions is wet etched. Fig. 8(a) shows an opticalmicroscope (OM) photo of a fabricated device. Fig. 8(b), (c) and (d) shows the close-up views of the TM near the guide beam; theshuttle mass and the anchor, respectively.

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 alsoshown in the figure. The substrate with specimens 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 which is fixed to a translation stage with ztranslation degrees of freedom. The z-translation stage is placed on top of a translation stage with translational degrees offreedom in x and y directions. The dimensions of the probe are indicated in Fig. 9(b). In order to facilitate the translation of theprobe parallel to the substrate and to maintain contact of the probe tip with the micromechanism, the tip of the probe is machined

b b c d d

a a

200 μm 200 μm 200 μm

500 μm

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.

35D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

to have a 135° slant angle to the probe axis as shown in Fig. 9(c). The translation stages have a resolution of 10 μm. The resolutionof the rotation stage is 1/6 °. The load cell (LVS-5GA, Kyowa Electronic Instruments Co., Ltd.) has a rated capacity of 50 mN, and aresolution of 10 μN.

The alignment of the probe axis to the loading axis of micromechanisms is adjusted by the rotation stage and the translationstages so that the micromechanism does not 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 themechanism is increased until it snaps toward its other stable equilibrium positions. The displacement of the shuttle mass and thereading of the load cell are recorded. A CCD camera is used for capturing the successive images of the motion of the mechanism.

4. Results and discussions

Using the experimental setup, the tristable behavior of the TM is demonstrated. The experimental f–d curves of the motion ofthe mechanism are also obtained. Fig. 10 shows 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 is increased, the shuttle mass moves forward. Asthe force reaches a certain maximum value, 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 is moved to contact with the edge surface of theshuttle mass again and pushed slowly until the TM snaps into its third stable position (Fig. 10(c)). As the probe tip is pulledbackward, the TM stays in its third stable position (see Fig. 10(d)).

Fig. 11 shows the f–d curves of the micromechanism bases on experiments and finite element computations. As the shuttlemass is displaced, the reaction force increases initially. In the event of snap-through, the probe tip loses contact with the shuttlemass, and the TM snaps into its second stable position as seen by the discontinuous curve of the experiments. A displacementcontrolled approach is adopted in the finite element analyses, where the snap-through behavior is signaled by the 0 value of thereaction force, and the negative values of the reaction force are obtained while the shuttle mass is displaced further towards thesecond stable position of the TM. As the shuttle 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 the second snap-through, the probe tip losescontact with the shuttle mass and the mechanism jumps to its third stable position. As shown in the figure, the micromechanismexhibits the tristable behavior within a displacement range of 260 μm. The general trend of the experimental results agrees withthat based on the finite element model with the designed dimensions of the TM.

b

a c

d

500μm

500μm

500μm

500μm

Fig. 10. Snapshots for motion of the TM.

36 D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

The discrepancies between the experiments and finite element analyses of the device can be attributed to uncertainties inmaterial properties, geometry and loading conditions 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. The widths of the outer and inner curved beamsmeasured by an Olympus Bx60 confocal microscope are 9 μm and 9 μm, respectively, which are larger than the designed widthsof the outer and inner curved beams, W1 = 8 μm and W2 = 7 μm, respectively. As shown in Fig. 11, the finite element modelwith the measured dimensions does not improve much in terms of the prediction of the stable positions, compared to those basedon the model with the designed dimensions. Measurement errors of force and displacement also contribute to the discrepancies.The contact of the probe tip for force measurement with the surface of the shuttle mass is not fixed, where sliding may occur, andthe alignment of the longitudinal axis of the probe with the symmetry plane of the TMmay not be perfect during the experiments,where twisting of the TM may happen.

The effects of the geometry parameters of the TM on the characteristics of the design are investigated using the finite elementmodel. Fig. 12(a) illustrates how the apex heights h1 and h2 affect the tristability of the TM. The TMs preserving its tristability aremarked in the figure. The TM with the values of the geometry parameters causing the TM to lose its tristability are also marked inthe figure. For the TM with large values of h1 and h2, the reaction forces F1max , F1min

, F2max , and F2min, and the deflection ranges are

significantly larger than those with small values of h1 and h2. Fig. 12(b) shows the effects of the spans L1 and L2 on the tristabilityof the TM. For the TM with large values of L1 and L2, the range of the spans to preserve its tristability is large. However, the

Displacement (μm)

For

ce (

μN)

0 50 100 150 200 250 290

-200

-400

0

200

400

600

800

1000

1200

1400FEM (Measured dimensions)

FEM (Designed dimensions)

Experiment

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

h 2(10

2 μm)

h1(102μm)

L 2(10

2 μm)

L1(102μm)

W2(

μm)

W1(μm)

10 15 20 25 30 35

2

4

6

8

10

12

14

16

18

20W

1=8 μm, L

1=1252 μm

W2=7 μm, L

2=1105 μm

Non tristableTristableOptimized design

30 80

2

4

6

8

10

12

14

16

18

20

22

W1=8 μm, h

1=74 μm

W2=7 μm, h

2=75 μm

Non tristableTristableOptimized design

0 5

0 10 20 40 50 60 70 90

0 2 4 6 8 10 1412 16

2

4

6

8

10

12

14

16

18

20L

1=1252 μm, h

1=74 μm

L2=1105 μm, h

2=75 μm

Non tristableTristableOptimized design

a

b

c

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

37D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

38 D.-A. Wang et al. / Mechanism and Machine Theory 71 (2014) 27–39

tristability of the TM with large values of the spans degrades, which means it is difficult to keep the TM in its stable equilibriumpositions due to the relatively small values of the reaction forces F1max , F1min

, F2max , and F2min(in the order of 1 μN).

The effects of the widths of the outer and inner curved beams, W1 and W2, respectively, are illustrated in Fig. 12(c). Thereaction forces of the TM with small values of the widths are much smaller than those with large values of the widths. Forexample, the reaction forces of the TMwithW1 = 1 μm andW2 = 1 μm are in the order of 1 μN, and the smallest reaction force ofthe TM with W1 = 12 μm and W2 = 13 μm is in the order of 10 μN. The values of W1 and W2 of the TM of the optimal design,indicated in Fig. 12(c), fall in about the middle of the ranges ofW1 andW2 which preserve the tristability of the TM. The designedbeam width of the TM has a large margin of error to ensure its tristability. Obviously, if the geometry parameters are uncertaindue to microfabrication 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 beamsbuckle first and then inner beams buckle (see Fig. 10). With the dimensions ofW1 = 8 μm, L1 = 1110, h1 = 79 μm,W2 = 6 μm,L2 = 1164 μm, and h2 = 77 μm (these symbols are illustrated in Fig. 3(a)), the inner beams buckle first and then outer beamsbuckle based on the analytical model. The geometric parameters can be changed to achieve a different design. The proposed TMprovides tristability in a linear, sequential manner. A potential application of the TM could be a microrobot system on a chip forcontrol of droplet dispensing in microchannels in the field of chemical engineering and bioengineering which requires asequential operation with multistability for efficient reaction and productivity [26]. Another possible application is the control ofcascaded rubber-seal valves for gas regulation in microchannels, where rectifying of gas flow can be achieved through thedeflection of valve diaphragms [27].

5. Conclusions

A TM has been designed, fabricated and validated by experiments. The TM consists of beams with profiles of cosine curves. Thecombination of two bistable mechanisms provides the tristability of the TM. Prototypes of the TM are fabricated by a simpleelectroforming process. A device for obtaining the in-plane force–displacement characteristics of the micromechanism isdeveloped. The observed force versus displacement curve of the micromechanism exhibits a well-defined tristability. Themicromechanism exhibits the tristable behavior within a displacement range of 260 μm. The presented design provides a meansof attaining a tristable, planar micromechanism with its motion in a linear manner.

Acknowledgment

The computing facilities provided by the National Center for High-Performance Computing (NCHC) are greatly appreciated.The authors are thankful for financial support from the National Science Council, R.O.C., under Grant No. NSC 96-2221-E-005-095.

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