Luận án tiến sĩ: DESIGN AND ANALYSIS OF MULTISTABLE COMPLIANT MECHANISMS

DESIGN AND ANALYSIS OF MULTISTABLE COMPLIANT MECHANISMS DESIGN AND ANALYSIS OF MULTISTABLE COMPLIANT MECHANISMS DESIGN AND ANALYSIS OF MULTISTABLE COMPLIANT MECHANISMS DESIGN AND ANALYSIS OF MULTISTABLE COMPLIANT MECHANISMS

國立中興大學精密工程研究所 (National Chung Hsing University, Institute of Precision Engineering)

博士學位論文 (Ph.D Dissertation)

Design and Analysis of Multistable Compliant Mechanisms

Huy-Tuan Pham Graduate Institute of Precision Engineering

Doctor of Philosophy

(ABSTRACT)

Multistable mechanisms, which provide multi stable equilibrium positions within its operation range, can be adopted to design systems with power efficiency and kinematic versatility, oftentimes two conflicting goals Multistable compliant mechanisms have attracted more and more attention in recent years Two new specified multistable compliant mechanisms are developed in this dissertation: a compliant quadristable mechanism and a constant-force bistable mechanism Finite element analyses are used to characterize the behavior of these multistable mechanisms under static loading A design formulation is proposed to synthesize the shape and size

of these specified compliant mechanisms Millimeter scale polyoxymethylene prototypes of them are fabricated and tested The characteristics of these mechanisms predicted by theory are verified by experiments The design examples presented in this investigation demonstrates the effectiveness of the optimization approach for the design of the multistable compliant mechanism The proposed mechanisms have no movable joint and gain their mobility from the deflection of flexible members These compliant mechanisms have the ease of miniaturization and offer a significant advantage in the fabrication of micro actuators, micro sensors and microelectromechanical systems

Keywords: bistable, quadristable, constant-force bistable, multistable mechanisms,

vibration

ACKNOWLEDGEMENTS

First and foremost, I would like to send my deep gratitude to two universities, Ho Chi Minh City Nong Lam University, Vietnam where I have been working and National Chung Hsing University, Taiwan for providing me this valuable scholarship for Ph.D degree

Many people deserve thanks for their contributions to this dissertation and my life

It is my pleasure to thank my Taiwanese labmates for their help and friendship to overcome my initial culture and language obstacles I must thank Cheng-Hao Ciou, a master student at Micro/Nano Machining Laboratory in NCHU, who helped me with meticulous fabrication by using their CNC milling machine Helpful discussions with Professor Chao-Chieh Lan of National Cheng Kung University, Taiwan, ROC are greatly appreciated

Special thanks to Assoc Prof Dung-An Wang, a man I am honored to know as a teacher, an advisor, and a friend He has also spent a considerable portion of his time giving guidance and helping me in countless ways I have learned much more than engineering from him

Finally, I am very grateful to my parents, my sister and my girlfriend for their love, for their support and encouragement of my academic pursuits, and for always expressing confidence in my abilities

This dissertation was supported by the National Science Council (NSC) ROC, under grant no NSC 96-2221-E-005-095

TABLE OF CONTENTS

ABSTRACT i

ACKNOWLEDGEMENTS ii

TABLE OF CONTENTS iii

LIST OF FIGURES v

LIST OF TABLES viii

NOMENCLATURE ix

CHAPTER 1 INTRODUCTION 1

1.1 Motivation 1

1.2 Contributions 2

1.3 Literature Review 3

1.3.1 Compliant Mechanism 3

1.3.2 Bistable micromechanism 4

1.3.3 Multistable mechanism 5

1.3.4 Contant-force bistable mechanism 6

1.3.5 Actuation methods of multistable mechanisms 8

1.4 Dissertation Layout 9

CHAPTER 2 DESIGN OF A BISTABLE MECHANISM 11

2.1 Design .12

2.1.1 Operational principle 12

2.1.2 Modeling 13

2.1.3 Analysis 15

2.2 Fabrication and Testing 19

2.2.1 Fabrication 19

2.2.2 Testing .20

2.3 Results and Discussions 20

2.4 Summary 23

CHAPTER 3 DESIGN OF A QUADRISTABLE COMPLIANT MECHANISM 41

3.1 Design .42

3.1.1 Operational principle 43

3.1.2 Design 46

3.1.3 Optimization 49

3.2 Fabrication and Testing 53

3.3 Results and Discussions 54

3.4 Summary 58

CHAPTER 4 DESIGN OF A CONSTANT-FORCE BISTABLE MECHANISM 75

4.1 Design 75

4.1.1 Operational principle 77

4.1.2 Design 78

4.1.3 Optimization 83

4.2 Fabrication and Testing 85

4.3 Results and Discussions 85

4.4 Summary 88

CHAPTER 5 CONCLUSIONS AND FUTURE WORK 100

5.1 Conclusions 100

5.2 Future work 101

Bibliographies 103

Publications during Ph.D studies 115

LIST OF FIGURES Fig 2.1 (a) A schematic of a BM and a permanent magnet served to actuate

the mechanism (b) Length l and the angle θ with respect to the

y axis 25

Fig 2.2 Two stable equilibrium states of the mechanism 26

Fig 2.3 Four-step operation of the mechanism 27

Fig 2.4 A schematic of a BM 28

Fig 2.5 A typical f-d curve of a BM 29

Fig 2.6 A schematic of a quarter model 30

Fig 2.7 A mesh for the quarter model 31

Fig 2.8 F-d curve and potential energy curve based on the finite element model 32

Fig 2.9 Time responses of the mechanism (a) Switching from FSP to SSP; (b) switching from SSP to FSP 33

Fig 2.10 Fabrication steps 34

Fig 2.11 An array of fabricated devices 35

Fig 2.12 An OM photo of a fabricated device 36

Fig 2.13 A schematic of the experimental setup 37

Fig 2.14 Experimental apparatus placed under a high-speed camera 38

Fig 2.15 Snapshots for forward motion 39

Fig 2.16 Snapshots for backward motion 40

Fig 3.1 A ‘ball-on-the-hill’ analogy for a QM, similar to a figure presented by Chen et al [2009] 61

Fig 3.2 Operational principle 62 Fig 3.3 A typical force versus displacement curve of the QM and the

corresponding configurations at displacement a, displacement b,

displacement c, and displacement d, shown in the inlets 63

Fig 3.4 (a) A schematic of a quarter model (b) Dimensions of the guide beam and the shuttle mass 64

Fig 3.5 Flowchart of the optimization procedure 65

Fig 3.6 A mesh for the finite element model 66

Fig 3.7 Distribution of the population of several generations in the optimization

process 67

Fig 3.8 (a) f-δ curve and maximum stress versus displacement curve for forward motion; (b) strain energy curve for forward motion; (c) f-δ

curve and maximum stress versus displacement curve for backward motion; (d) strain energy curve for backward motion 68

Fig 3.9 Photos of a fabricated QM 69

Fig 3.10 A photo of the experimental setup 70

Fig 3.11 Snapshots for forward motion (a-c) and backward motion (d-f) 71

Fig 3.12 f-δ curves of the fabricated QM for (a) forward, (b) backward motion 72

Fig 3.13 (a) Schematic of the inner bistable structure, BS1 (b) Schematic of the outer bistable structure, BS2 (c) f-δ curves of BS1 and BS2 based on their individual finite element models 73

Fig 3.14 f-δ curve and maximum stress versus displacement curve of the microscale version of the QM for (a) forward motion; (b) backward motion 74

Fig 4.1 Schematic of a CFBM and its operational principle 90

Fig 4.2 (a) A typical force versus displacement curve of the CFBM and the corresponding positions at displacement a (b), displacement c (c), displacement d (d), and displacement g (e) 91

Fig 4.3 (a) A schematic of a quarter model (b) Dimensions of the guide

beam and the shuttle mass 92 Fig 4.4 Flowchart of the optimization procedure 93 Fig 4.5 A mesh for the finite element model 94 Fig 4.6 (a) A f-δ curve and maximum stress versus displacement curve;

(b) strain energy curve based on a finite element model of the

optimized solution 95 Fig 4.7 (a) A photo of a fabricated CFBM (b) A close-up view of the

flexible hinge 96 Fig 4.8 A photo of the experimental setup 97 Fig 4.9 Snapshots for forward motion (a-c) and backward motion (d-f) 98 Fig 4.10 f-δ curves of the fabricated CFBM for (a) forward,

(b) backward motion 99

LIST OF TABLES Table 2-1 Values of the coefficients of the nonlinear spring stiffness function 24 Table 2-2 Chemical composition and operation conditions for the low-stress nickel

electroplating solution 24 Table 3-1 Lower and upper bounds on the design variables 60 Table 3-2 The values of the design variables of the optimum design 60 Table 3-3 The values of the design variables of the optimum design of the

microscale QM 60 Table 4-1 Lower and upper bounds on the design variables 89 Table 4-2 The values of the design variables of the optimum design 89

F electromagnetic force on each beam

F electromagnectic force exerted to the mechanism

t

t

F ∆ effective force

)41(i=

c f f

f , , output forces of the CFBM

h out-of-plane thickness

)2,1(i=

h i apex height of the curved beams

i

I, current

i amplitude of the applied AC current

k,K ( X) nonlinear spring stiffness

l length of the hinged beam

s

L L

L1, 0, length of fexural segment, rigid segment and side beam

L span of the curved beams

P position vector of the control point B ui

S specified ratio of the output force f d to f c

(x r y r position vector

X design space

Greek symbols

a amplitude of the response

β phase of the response

CFM constant-force mechanisms

FSP first stable position

MEMS Micro Electro-Mechanical Systems

SMA shape memory alloy

SSP second stable position

From the last decade, many studies and researches have been committed on bistable and tristable compliant mechanisms However, few mechanisms with four or more mechanically stable positions have been reported The key challenge in developing multistable mechanisms is the difficulty in their synthesizing and analyzing

Geometry nonlinearities due to large deflections are commonly encountered in compliant multistable mechanisms Modeling of force-deflection characteristics of compliant mechanisms can be performed by the pseudo-rigid-body model (PRBM) (Howell 2002) However, in order to accurately describe the behavior of compliant mechanisms using PRBM, where to place the added springs and what value to assign their spring constants are important

Not only is it essential to synthesize new configurations of multistable mechanisms but also is it crucial to develop new activating methods So far switching methods of multistable mechanisms can be classified into two main categories: static and dynamic method The former method has been widely studied by many reseachers, while few publications are found on the later

An effort to develop a new quadristable compliant mechanism and a novel constant-force bistable mechanism with more versatile synthesis approach and a new method to switch a bistable mechanism has motivated the present dissertation

1.2 Contributions

Firstly, a new method to switch a bistable mechanism (BM) by electromagnetic actuation is presented in this dissertation Vibration is utilized to drive the BM on a substrate between its stable positions The feasibility of using vibration to achieve dynamical switching is confirmed by the derived analytical model and experiments as well The presented switching method provides a simple and efficient means of activating a BM without on-substrate driving mechanisms such as electrostatic or electrothermal actuators

Secondly, a design approach using multiobjective genetic algorithm optimization

to design compliant multistable mechanisms is proposed The effectiveness of the

methodology is verified by the design of the two new specified compliant multistable mechanisms A new quadristable mechanism (QM) with a bistable structure embedded in a surrounding beam structure is developed Three stable equilibrium positions are within the range of the forward motion of the mechanism, and the fourth stable equilibrium position can only be reached on the backward motion The characteristics of the mechanism predicted by theory are verified by experiments This compliant mechanism has the ease of miniaturization and offers a significant advantage

in the fabrication of micro actuators, micro sensors and microelectromechanical systems

Thirdly, a novel constant-force bistable mechanism (CFBM) allowing constant contact force and overload protection is developed This mechanism allows constant contact force and overload protection The feasibility of using the mechanism to achieve constant output force and bistability is confirmed by finite element analyses Using the CFBM, sophisticated sensors and control system for force regulation of machining systems can be eliminated

1.3 Literature Reviews

1.3.1 Compliant mechanism

Compliant mechanisms are devices which achieve at least some or all of their functionality from deflection of flexible members The use of compliant mechanisms has the advantages of part-count reduction, simple and inexpensive manufacturing processes and increase performance by increasing precision, reducing wear (Howell 2002) Compliant mechanisms can be classified as partially and fully mechanism These two types are all composed of rigid and flexible segments While partially compliant mechanism makes use of combination of traditional joints and compliant

segments to obtain motion, the latter type relies completely on the deflection of flexible segments The motion of compliant mechanisms causes their flexible members to deflect and store elastic strain energy This energy can then be released to execute the function of the device

The commonly used design methods for compliant mechanisms are PRBM and topology optimization methods While PRBM uses rigid body components which has the equivalent force-deflection characteristics to model the deformation of flexible components of conpliant mechanisms (Wilcox and Howell 2005), topology methods use continuum models (Larsen et al 1997) The latter method often requires further procedure to modify the optimum contour to eliminate the “point hinges” problem from the final continuum design (Hull and Canfield 2006)

Due to above specified characteristics of compliant mechanisms, certain groups

of mehanisms are favored for appropriate applications Some special-purpose compliant mechanisms studied so far include bistable mechanisms, constant-force mechanisms, and parallel guiding mechanisms This section is also intended to review some recently developed multistable mechanisms and their actuation methods as well

1.3.2 Bistable micromechanism

A BM has three locations where no input energy is required to maintain the device’s position, including two stable equilibria and an unstable “transition state” Input energy is only required to change from one stable point to another (Casals-Terre and Shkel 2005) Bistable micromechanisms are gaining attention in MEMS applications such as accelerometers (Hansen et al 2007), memory cells (Hälg 1990), switches (Freudenreich et al 2003; Masters and Howell 2003; Lee and Wu 2005; Yang

et al 2007; Chen et al 2009; Liao et al 2010), relays (Kruglick and Pister 1998; Gomm

et al 2005; Qiu et al 2005), valves (Wagner et al 1996), feeding systems (Tsay et al 2005), microassembly (Wang and Pham 2008), and energy harvesting (Ando et al 2010; Stanton et al 2010) Low actuation force and power, high cycle life, and predictable, repeatable motion are required for a BM in MEMS applications

BMs can also be partially compliant, where the device consists of one or more flexible segments as well as one or more traditional joints, or fully compliant, where the mechanism achieves all of its motion and function from the motion of compliant segments Configurations of compliant mechanisms which exhibit bistable behavior have been classified and presented in Jensen (1998)

1.3.3 Multistable mechanism

Multistable mechanisms, which provide multiple stable equilibrium positions within their operation range, can be used to design systems with both power efficiency and kinematic versatility, oftentimes two conflicting goals With the concept of multistable mechanisms, a wide range of operating regimes or novel mechanical systems without undue power consumption can be created (King et al 2004) Substantial interest has focused on design of bistable mechanisms (Gomm et al 2005; Qiu et al 2005; Hansen et al 2007; Yang et al 2007; Wang et al 2009) and tristable mechanisms (Ohsaki and Nishiwaki 2005; Su and McCarthy 2005; Oberhammer et al 2006; Pendleton and Jensen 2007; Chen et al 2009; Chen et al 2010)

Few mechanisms with four or more mechanically stable positions have been reported Han et al (2007) developed a planar QM using two pairs of curved beams to achieve quadristability with two stable positions in each of two orthogonal directions The sequence of switching between stable positions can be altered by selectively actuating the mechanism in one of the two orthogonal directions Oh and Kota (2009)

have proposed a mathematical approach to synthesize multistable compliant mechanisms based on a combination of multiple bistable rotational mechanisms A design case study of a QM with four stable rotational orientations is presented However a prototype of this QM with an appropriate actuator to validate this design is still needed King et al (2004) proposed a QM consisting of a rotating compliant beam with an armature magnet attached to it and an array of stator magnets Fields of strain energy, gravitational energy and magnetic energy are all involved in the stability modes of their mechanism The inherently nonlinear nature of the energy storage elements have proposed complexity for the definition of differential equations in order

to synthesize multiple equilibria in their optimization solution This may also require a significant effort to obtain a feasible design

Hafez et al (2003) proposed a robotic device with a large number of degrees of freedom, which can be taken as a large number of stable positions/orientations The discrete nature of their mechanism is ensured by the use of bistable mechanisms These kind of mechanisms can perform precise, discrete motions without need for sensing, complex electronics or feedback control With complexity of the assembly of modular parallel platforms and a network of flexible and rigid members with embedded actuators, their mechanism can be used for tasks which require a robot to operate in a three-dimensional space, such as camera placement and light positioning Despite of their promising characteristics, actuator technology is still a key challenge for the implementation of large degrees of freedom binary mechatronic systems

1.3.4 Contant–force bistable mechanism

Responding to the increasing needs in many systems where a variable output force is undesirable, mechanisms which provide a near–constant output force over a

prescribed deflection range have been developed and are defined as constant–force mechanisms (CFMs) These mechanisms have been gaining more and more attention

in recent years (Jenuwine and Midha 1994; Howell and Magleby 2006; Berselli et al 2009; Meaders and Mattson 2010; Lan et al 2010) CFMs can be designed for concrete testing equipment (Jenuwine and Midha 1994), exercise machines (Howell and Magleby 2006) and electrical contacts (Meaders and Mattson 2010) A constant–force actuator based on dielectric elastomers is proposed by Berselli et al (2009) for applications in robotics and mechatronics Lan et al (2010) developed a compliant CFM for force regulation of robot end–effectors operating in an unknown environment

CFMs can be utilized in systems to reduce the need for complex control algorithms and feedback loops (Erlbacher 1995; Bossert et al 1996) Without sophisticated control systems, CFMs made of passive mechanisms may minimize control effort and reduce cost without losing precision CFMs have been developed by various investigators (Howell 2002; Boyle et al 2003; Nahar and Sugar 2003; Pedersen

et al 2006; Weight et al 2007) Pedersen et al (2006) used topology and size optimization to design a transmission mechanism which converts the constant stiffness

of an actuator into constant output force CFM can also be explored for the currently mass using of electrical contact since providing sufficient contact force is a crucial method to minimize the contact resistance (Jang et al 2008) Weight et al (2007) proposed a CFM composed of a bent beam and a cam for electrical contacts in personal digital assistant (PDA) dock stations to improve the reliability of high–cycle electrical connectors Their beam and cam combination provides the strain energy storage device necessary for constant force behavior

Boyle et al (2003) presented a CFM consisting of a rigid link, a flexible segment and a slider His compliant CFM offers the possibility of a new type of spring

element for a variety of applications Nahar and Sugar (2003) designed a double-slider CFM, where two springs are attached to the sliders They also proposed the design of

a micro compliant slider–crank CFM However, in order to facilitate these mechanisms in MEMS devices, monolithic compliant mechanisms which can be fabricated by microfabrication technologies are favored over the macro devices With fewer movable joints, the use of compliant mechanisms results in reduced wear, reduced need for lubrication, and an increased mechanism precision (Howell 2002)

1.3.5 Actuation methods of multistable mechanisms

Various actuation methods of multistable mechanisms have been proposed including electrothermal actuation (Qiu et al 2005; Wilcox and Howell 2005; Yang et al 2007), electrostatic actuation (Hwang et al 2003; Freudenreich et al 2004; Casals-Terre and Shkel 2004; Receveur et al 2005; Kwon et al 2005; Krylov et al 2008), electromagnetic actuation (Ko et al 2006; Cao et al 2007), optical actuation (Sulfridge

et al 2002), piezoelectric actuation (Giannopoulos et al 2007), and shape memory alloy (SMA) actuation (Barth et al 2010) Depending on different application requirements,

an appropriate actuation method should be selected

Built-in driving mechanisms are usually required for electrothermal and electrostatic actuations as reported previously (Hwang et al 2003; Freudenreich et al 2004; Casals-Terre and Shkel 2004; Kwon et al 2005; Qiu et al 2005; Wilcox and Howell 2005; Yang et al 2007; Krylov et al 2008) Sulfridge et al (2002) reported an optical switch for a MEMS bistable beam using a laser light The laser being used is

200 mW and a load of 0.6 nN is generated It might not be convenient for micro devices requiring actuation forces on the µN scale The meso-scale piezoelectric bistable beam described by Giannopoulos et al (2007) needs a high driving voltage up

to 120 V and a precompression of the beam is needed for snap-through action

Cao et al (2007) demonstrated a bi-directional MEMS actuator based on electrothermal buckling and electromagnetic Lorenz force These type of actuators require relatively low voltages and high currents to operate and can exert large forces on the mN scale Electromagnetic actuation can be used in MEMS applications that require high displacement, high force and bi-directional actuation An external vibration can be utilized to move a BM between its bistable positions based on an investigation carried out by Kreider and Nayfeh (1998) for a buckled beam under harmonic excitation A harmonic or intermittent snap-through behavior of their prebuckled beam is observed The harmonic excitation by electromagnetic forcing provides a way for motion control of BMs that require high displacement output and high force actuation This kind of actuation will be exploided and investigated in this dissertation

1.4 Dissertation Layout

This chapter has given brief introduction on compliant mechanisms and reviewed researches on two types of special-purpose compliant mechanisms: multistable mechanisms which include bistable mechanisms, tristable mechanisms, and quadristable mechanism, and constant-force bistable mechanisms

Chapter 2 investigates a compliant chevron-type bistable mechanism A new vibration-actuated method to switch this BM by electromagnetic actuation is presented

An analytical model of the BM is derived in order to analyze its dynamic behavior Prototypes of the device are fabricated using an electroforming process Experiments are carried out to demonstrate the effectiveness of the dynamical switching of the BM

Chapter 3 describes a design of a new compliant quadristable mechanism An optimization method is proposed to design the QM Finite element analyses are carried out to evaluate the mechanical behaviors of the design obtained by the optimization procedure Prototypes of the device are fabricated using a milling process Experiments are carried out to demonstrate the effectiveness of the QM

Chapter 4 proposes a novel design of a compliant CFBM An optimization method is used to design the CFBMs Finite element analyses are carried out to evaluate the mechanical behaviors of the design obtained by the optimization procedure Prototypes of the device are fabricated using a milling process Experiments are performed to demonstrate the effectiveness of the CFBM

Chapter 5 summarizes the work of the dissertation Future researches for the dissertation are also recommended

Chapter 2

DESIGN OF A BISTABLE MECHANISM

This chapter describes a design of a vibration-actuated bistable micromechanism (BM) The vibration exploited to switch the on-substrate BM is provided by an electromagnetic Lorenz force, requiring no need for built-in driving mechanisms such

as electrothermal or electrostatic actuators The electromagnetic actuation is based on the actuation method reported by Cao et al (2007) and Ko et al (2006), where a precompressed beam is moving through the full range of its bi-directional motion (Cao

et al 2007) or a bistable beam is actuated statically (Ko et al 2006) Dissimilar to their design, the BMs are switched dynamically by shaking the device with an alternating electromagnetic force An analytical model of the BM is derived in order

to analyze its dynamic behavior Prototypes of the device are fabricated using an electroforming process Experiments are carried out to demonstrate the effectiveness

of the dynamical switching of the BM

2.1 Design

2.1.1 Operational principle

A schematic of a BM and a permanent magnet served to actuate the mechanism

is shown in Fig 2.1(a) A Cartesian coordinate system is also shown in the figure The mechanism is a compliant chevron-type mechanism consisting of a shuttle mass, flexible hinges, hinged beams and lateral springs The flexible hinges facilitate the

rotation of the hinged beams Upon the application of an actuation force F to the

hinged beams, the flexible hinges and lateral springs deflect, storing the strain energy while the mechanism moves towards the other stable equilibrium state Fig 2.2 shows the two stable equilibrium states of the mechanism

An electromagnetic effect is utilized to perform the bi-directional, in-plane

motions As shown in Fig 2.1(a), the mechanism is placed in a magnetic field B As

a current I passes through the contact pads, each beam experiences a force perpendicular

to both the directions of the current and magnetic field The force exerted on each beam can be expressed as

B l

the beam makes an angle θ with the y axis, see Fig 2.1(b), the resultant force F

exerted to the mechanism is given by

θcos

4F m

When the current direction is switched, the direction of F is reversed and the

bi-directional motion of the mechanism can be controlled

Fig 2.3 illustrates a four-step operation of the mechanism First, an AC current

1

I of frequency ω1 is passed though the hinged beams through the two contact pads The vibration induced by the electromagnetic actuation resonates the mechanism (see Fig 2.3(a)), causing it to move towards its second stable position (SSP) (see Fig 2.3(b)) Next, an AC current I2 of frequency ω2 is applied This induced vibration resonates the mechanism (see Fig 2.3(c)), causing it to move towards its first stable position (FSP) (see Fig 2.3(d)) This dynamical switching of the BM via an external magnetic field and an applied AC current provides with a simple means of controlling motion of the mechanism in the absence of on-chip driving mechanisms

When exciting the mechanism dynamically, the actual vibration of the mechanism is always a combination or mixture of all the vibration modes If the in-plane mode is favored, out-of-plane modes of vibration cause the mechanism to deform perpendicular to the substrate which is not desired for efficient operation of the

BM Therefore, the mechanism should have high stiffness in the out-of-plane direction and the exciting frequency should be carefully selected to avoid undesired vibration modes

2.1.2 Modeling

An equation of motion of a lumped parameter model of the BM shown in Fig 2.1(a) is derived to analyze its dynamic behavior Fig 2.4 shows a schematic of the mechanism and a Cartesian coordinate system It is assumed that the out-of-plane motion of the mechanism is small and does not affect its in-plane motion The

equation of motion for a simple mass-damper-spring system with excitation F from

electromagnetic actuation has the form

F kx x x

where m, c, and k are the mass, damping coefficient, and spring stiffness of the mechanism, respectively Assuming Stoke’s damping between the substrate and the mechanism, c can be expressed as Ye et al 2003 and Lee et al 2007

=

d d

d d

A

c

ββ

ββ

βµ

2cos2

cosh

2sin2

sinh

ν

ωβ

2

where µ and ν are the dynamic and kinematic viscosity of the fluid in the

environment, respectively A and d are the planar area of the mechanism and the gap between the substrate and the mechanism, respectively ω is the frequency of the

AC current passing through the mechanism When the mechanism is actuated by an

AC current I =i sin( tω ), using Eqs (2.1) and (2.2), the electromagnetic force F

exerted to the mechanism is

)sin(

mechanism, Qiu et al (2004) used a mode superposition method, where the first three buckling modes for a straight beam were used as the superposition basis for deflection shape of an initially curved beam Applying the theory of the PRBM (Howell 2002), Casals-Terre and Shkel (2004) obtained a f-d relation for a compliant BM In order to accurately predict the f-d relation by the PRBM, careful decision of the positions of equivalent springs and the values of their spring constants should be made Here, we use a ninth-order polynomial to model the nonlinear spring stiffness of the mechanism

This ninth-order stiffness function allows for the detailed modeling of the highly nonlinear f-d relation of the mechanism Third, fifth, seventh, eighth and ninth-order functions have been used to model the highly nonlinear force-displacement curve The curves generated by the polynomials with order less than nine do not fit the force-displacement curve well

2.1.3 Analysis

The equations of motion defined in Eqs (2.3)-(2.7) can be numerically integrated to predict the dynamic behaviors of the mechanism In order to obtain the nonlinear spring stiffness of the mechanism, finite element analyses are carried out Due to symmetry, only a quarter model of the mechanism is considered Fig 2.6 shows a schematic of a quarter model with L1 =300 µm, L0 =900 µm, L s =200

µm, t1 =5 µm, t0 =40 µm, t s =15 µm, and θ =2.25o The thickness h of the

device is 5 µm A Cartesian coordinate system is also shown in the figure The

displacement in the y direction at y=0 is constrained to represent the symmetry condition due to the mechanism geometry and the loading conditions Clamped boundary conditions are applied to the fixed ends of the mechanism A displacement

is applied in the +x direction at y=0 Fig 2.7 shows a mesh for the quarter model

A mesh sensitivity analysis is performed to assure convergence of the solutions In the analyses, the material of the device is assumed to be nickel For the linear elastic and

isotropic model, the Young’s modulus E is taken as 207 GPa, and the Poisson’s ratio

p

ν is taken as 0.31 The commercial finite element program ABAQUS (Hibbitt et al

2001) is employed to perform the computations The material properties of electroplated nickel can differ from that of bulk materials because it is highly dependent upon the composition of the plating bath, current density and temperature Fritz et al (2003) reported that the Young’s modulus of electroplated nickel ranges from 205 GPa for a mean current density of 0.2 A/dm to 165 GPa for 2.0 2 A/dm , respectively 2The material properties of bulk nickel are used in the finite element analyses in order to understand the f-d relation of the mechanism in the initial design stage No attempt is made to adjust the material properties to fit the experimental results in this investigation

A f-d curve and a potential energy curve based on the finite element model are shown in Figs 2.8(a) and (b), respectively We selected a ninth order function, Eq (2.7), to fit the simulation data in Fig 2.8(a) The values of the coefficients of Eq (2.7) are listed in Table 2.1 As shown in Fig 2.8(a), when the displacement of the shuttle mass increases from 0 (the first stable equilibrium position), the force increases initially, reaches its maximum value, then decreases to 0, which is an unstable equilibrium position When the displacement increases further, the force decreases, attains its minimum value, then increases again and reaches 0, the second stable

equilibrium position As shown in Fig 2.8(b), the two local minimums of the energy curve correspond to the two stable equilibrium positions of the mechanism The local maximum of the curve corresponds to the unstable equilibrium position of the mechanism The ninth-order stiffness function of Eq (2.7) allows for the detailed modeling of the highly nonlinear f-d relation of the mechanism For polynomials with order less than nine, the maximum actuation force, unstable and stable equilibrium positions of the mechanism are not close to those based on the finite element analyses Therefore, simulated amplitudes and frequencies of the currents passed through the hinged beams of the mechanism for dynamical switching can be different according to the polynomials selected

In order to predict the dynamic behaviors of the mechanism, Park stiffly stable method (D’Souza and Garg 1984) is used to solve the governing nonlinear differential equations, Eqs (2.3)-(2.7) The nonlinearities due to the post-buckling behavior, geometric nonlinearity and Stoke’s type damping introduce numerical instability to the integration of the equations Park method is a robust algorithm available to achieve stable solutions and frequently used for nonlinear dynamic problems with low- and high-frequency range input of the excitation Using Park method, Eq (2.3) can be rewritten as

t t t

m t

100

t t

t t

t t

t t t

t

m t x

m t x

m t x m t F

∆

6

1536

1506

11

6

15

2 2

&

&

&

t t t

t

m t x

c t

101

6

10

(2.10)

where t is time However, the Park method is not self-starting and requires special

starting procedure (Fung 1998) It is observed that the calculation of x t ∆t involves

x and x& at t , t−∆t, and t−2∆t The initial displacement and velocity can be taken as the displacement x and velocity x& at t Following the procedure

presented by Lee et al (2007), the Wilson Theta method (D’Souza and Garg 1984) is used to obtain the displacement x and velocity x& at t−∆t and t−2∆t

The values of L1, L0, L s, t1, t0, t s, d , and h used in the numerical integration are 300 , 900 , 200 , 5 , 40 , 15 , 5 , and 5 µm, respectively The value of θ is set to be 2.25o Many researchers have used the density of bulk nickel, nearly 8900 kg / m3, for electroplated nickel (Geear et al 2004; Li and Chen 2005; Huang et al 2006) Teh et al (2005) used a value of 9040 kg / m3 for the density of their electroplated Ni cantilever In this investigation, the value of the density of the beam material ρ, electroformed nickel, is taken as 8908 kg / m3 The values of the dynamic viscosityµ and kinematic viscosity ν of the environmental fluid, air, at 1 atm and 25 oC are taken to be 1.81034×10−5 Pa⋅s and 1.59528×10−5 m2/s, respectively These values are interpolated from the viscosity values at different

temperatures of air at 1 atm (White 1986) The magnetic field strength B of the

permanent magnet is 0.29 T The mechanism is assumed to be driven by an electromagnetic force with a sinusoidal current applied at different frequencies and amplitudes Simulations are carried out over a frequency range of 2-10 kHz with a supply current range of 60-90 mA peak-to-peak Fig 2.9 shows the time responses of the mechanism based on the numerical integration of Eqs (2.8)-(2.10) As shown in

Fig 2.9(a), when a current with frequency ω/2π =8.8 kHz and amplitude i=80

mΑ peak-to-peak is applied through the contact pads, the mechanism initially oscillates around its FSP, and switches to its SSP within t=0.5 msec Fig 2.9(b) shows that when a current with frequency ω/2π =9.02 kHz and amplitude i=80 mΑpeak-to-peak is applied, the mechanism is found to switch from its SSP to its FSP within

a 5 µm-thick nickel layer is electrodeposited using a low-stress nickel sulfamate bath with the chemical compositions listed in Table 2.2 Following that, the photoresist and copper sacrificial layers are removed to release the nickel microstructures Finally, the Cr/Au layer outside the anchor regions is wet etched for electrical isolation of the electromagnetic actuation Figs 2.11 and 2.12 show an optical microscope (OM) photo of an array of the fabricated mechanisms, and a close-up view of one mechanism, respectively Each mechanism in the array is designed for a different driving

frequency of the applied current

2.2.2 Testing

The fabricated mechanisms are tested using the experimental apparatus shown schematically in Fig 2.13, where a permanent magnet of NdFeB (ND-36, Magtech Magnetic Products Co., Taiwan) is attached at the bottom of the glass substrate containing the device array The substrate and the magnet are held in an acrylic fixture The magnet generates a magnetic field strength B=0.29 T on the glass substrate in the out-of-plane direction of the substrate With an AC current passes through the mechanism, an in-plane force is induced to drive the mechanism into vibration The input sinusoidal AC current is supplied by a function generator (WW5062, Tabor Electronics Ltd., Israel) Experiments are carried out over a frequency range of 2-10 kHz with a supply current range of 60-100 mA peak-to-peak Fig 2.14 is a photo of the experimental apparatus placed under a microscope

2.3 Results and Discussions

Using the experimental setup, the dynamical switching of the mechanism is successfully realized by the electromagnetic actuation Figs 2.15 and 2.16 show sequences of snapshots from experiments in atmospheric pressure for forward and backward motions, respectively A sinusoidal current of 96 mA peak-to-peak with

no dc offset is passed through the contact pads at frequency ω/2π =2.3 kHz to dynamically switch the mechanism forward, see Fig 2.15 In order to dynamically switch the mechanism backward, a sinusoidal current of 72 mA peak-to-peak with no

dc offset is passed through the contact pads at frequency ω/2π =3.3 kHz, see Fig

2.16 Based on the measurement from the video images, the maximum displacements

of the shuttle mass are nearly 90 µm for both forward and backward motion They are marked in Figs 2.15(e) and 2.16(a) for forward and backward motion, respectively The snapshots of the forward and backward motions are taken using a high-speed camera (SpeedCam MiniVis e2, Weinberger, Germany)

The experimental frequencies of the input AC currents for dynamic switching are not close to those based on the simulations The discrepancy between the experimental and simulated frequencies might come from uncertainties in material properties, geometry and loading conditions used in the simulations Geometry uncertainty due to fabrication process and Joule heating effects can not be appropriately modeled in simulations The resonance frequency of the test device measured by a MEMS motion analyzer (MMA G2, Umech Technologies, USA) is 5.1 kHz, which is much lower than that, 7.9 kHz, predicted by the finite element model based on the design parameters Lee et al (2007) pointed out that the decrease of spring stiffness of

a beam structure caused by Joule heating may reduce the resonance frequency Based

on OM observations, the fabricated device is bent upward slightly A residual tensile stress might have been induced in the beam structure during the fabrication The residual tensile stress in the beam structure can cause the resonance frequency to increase It is also revealed by OM inspections that the width measured on the top surface of the beam is less than that of the design and the cross section of the beam is trapezoidal Due to the non-square cross section of the beam structure, the stiffness of the device is decreased and that might also cause the experimental resonance frequency

to be different from the design value

It is known that Young’s modulus of electroplated nickel can be much smaller than that of bulk nickel Fritz et al (2003) reported Young’s modulus of electroplated

nickel ranging from 165 GPa for a current density of 20 A/dm during electroplating

to 205 GPa for a current density of 0.2 A/dm The current density used in the 2electroforming process in this investigation is 0.113 A/dm Therefore, the Young’s 2modulus of bulk nickel of 207 GPa is used in the simulations In order to understand the effects of the Young’s modulus on the simulated switching frequency, a f-d relation

of the mechanism with a Young’s modulus of 177 GPa is obtained by finite element analyses Using the f-d relation and the numerical integration scheme, the mechanism switches from its FSP to SSP when a current with frequency ω/2π =7.2 kHz and amplitude i=80 mΑ peak-to-peak is applied When a current with frequency

As described above, due to the uncertainties in material properties, geometry and loading conditions of the experiments, the frequencies and amplitudes obtained from the simulations for dynamic switching are used only as the guidelines to find the switching frequencies and amplitudes in the experiments

In order to evaluate the efficiency of the dynamical switching strategy, experiments of static switching the mechanism is also carried out For static testing,

DC currents are passed through the contact pads using the function generator The minimum currents needed for static switching of the mechanism forward and backward are found to be 169 and 131 mA, respectively, which are larger than those for dynamical switching, 96 and 72 mA peak-to-peak, respectively

In the present experimental setup, the volume of the permanent magnet is much

larger than the fabricated single device The sheer volume of the magnet will impose a challenge in the engineering applications of the device Future work will be the incorporation of a uniform and reliable magnetic field applied by a microcoil to the fabricated devices

2.4 Summary

A new method to switch a BM by electromagnetic actuation is presented An electric current passing through the BM placed on top of a permanent magnet creates an electromagnetic Lorenz force Vibration is utilized to drive the BM on a substrate between its stable positions The frequencies and amplitudes of the supply current for the dynamic switching are found by solving the nonlinear equation of motion of the BM The feasibility of using vibration to achieve dynamical switching is confirmed by the derived analytical model In order to confirm the effectiveness of this approach, prototypes of the device are fabricated on glass substrates using a simple electroforming process Dynamical switching of the BM at the current of 96 mA and 72 mA peak-to-peak for forward and backward motion, respectively, has been demonstrated Whereas when the BM is driving statically, higher currents of 169 mA and 131 mA for forward and backward motion, respectively, are needed for switching The experimental operating parameters for dynamic switching are not close to those based

on the simulations The discrepancy could be accounted for uncertainties in material properties, geometry and loading conditions and Joule heating effects which have not been appropriately modeled in simulations However the presented switching method still provides a simple and efficient means of activating a BM without on-substrate driving mechanisms such as electrostatic or electrothermal actuators

Table 2.1 Values of the coefficients of the nonlinear spring stiffness function

0

10055

1

10412

3

10702

5

10054

Fig 2.1 (a) A schematic of a BM and a permanent magnet served to actuate the

mechanism (b) Length l and the angle θ with respect to the y axis

Fig 2.2 Two stable equilibrium states of the mechanism