Design of a bistable mechanism with b spline profiled beam for versatile switching forces

Design of a bistable mechanism with B spline profiled beam for versatile switching forces Accepted Manuscript Title Design of a bistable mechanism with B spline profiled beam for versatile switching f[.]

Accepted Manuscript

Title: Design of a bistable mechanism with B-spline profiled

beam for versatile switching forces

Authors: I-Ting Chi, Tien Hoang Ngo, Pei-Lun Chang, Ngoc

Dang Khoa Tran, Dung-An Wang

DOI: https://doi.org/10.1016/j.sna.2019.05.028

To appear in: Sensors and Actuators A

Received date: 1 April 2019

Revised date: 29 April 2019

Accepted date: 16 May 2019

Please cite this article as: Chi I-Ting, Ngo TH, Chang P-Lun, Tran NDK,Wang D-An, Design of a bistable mechanism with B-spline profiled beam for

versatile switching forces, Sensors and amp; Actuators: A Physical (2019),

https://doi.org/10.1016/j.sna.2019.05.028

This is a PDF file of an unedited manuscript that has been accepted for publication

As a service to our customers we are providing this early version of the manuscript.The manuscript will undergo copyediting, typesetting, and review of the resulting proofbefore it is published in its final form Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers thatapply to the journal pertain

Design of a bistable mechanism with B-spline profiled

beam for versatile switching forces

I-Ting Chi1, Tien Hoang Ngo1, Pei-Lun Chang1, Ngoc Dang Khoa Tran2, Dung-An

Wang1*

1Graduate Institute of Precision Engineering, National Chung Hsing University, Taichung

40227, Taiwan, ROC

2Faculty of Mechanical Engineering, Industrial University of Ho Chi Minh City, 12

Nguyen Van Bao, Ward 4, Go Vap District, Ho Chi Minh City, Vietnam

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

E-mail address: daw@dragon.nchu.edu.tw (D.-A Wang).

Graphical Abstract

Highlights

 ‧ A bistable mechanism composed of B-spline curved beams for versatile switching forces in the forward and backward directions is proposed

 ‧An analytical model to solve for the nonlinear force-displacement characteristics

of the B-spline profiled beams is developed

 ‧Smoother force-displacement curve is achieved compared to existing design

ACCEPTED MANUSCRIPT

Abstract

A compliant bistable mechanism composed of B-spline curved beams for design of switching forces in the forward and backward directions is developed The parametric B-spline curve has five control points to give a high design freedom in the output force of the beam-type compliant mechanisms An analytical model is developed to provide an efficient tool to obtain the force-displacement characteristics of the B-spline profiled bistable mechanism B-splined profiled bistable mechanisms with various ratios of the switching forces in the forward and backward motions are designed The results are confirmed by experiments The developed bistable mechanism with high force versatility has applications in devices where precise geometric activation and quantifiable load bearing capacity are desired

Keywords: compliant mechanism; B-spline curve; switching force

1 Introduction

A compliant bistable mechanism (CBM) with desired characteristics of force and displacement output is desired in its industrial applications Aerospace industries need a reliable release mechanism to launch deployable structures, such as antennae, satellites and solar panels CBM’s high reliability, low vibration sensibility in force response, and less susceptibility to temperature variation have led them to space application [1] Precise geometric activation and quantifiable load bearing capacity of CBMs make them suitable for emerging applications in automotive, building and biomedical industries [2] In automotive industry, CBMs can be employed in trunk lid design to compensate to lid

ACCEPTED MANUSCRIPT

weight during opening/closing operation [3] Transition of a robotic end effector can be achieved with controllable bistable positions of CBMs [4] Precise control of the force-displacement behavior of CBMs is a prerequisite for their successful applications in switches [5], projection displays [6], and nonvolatile memory elements [7], etc For applications in nonvolatile memory devices, nearly equal switching forces in back and forth directions of CBMs may facilitate two logical levels “1” and “0” corresponding to the two stable states The equal switching force of CBMs can see its use in threshold accelerometers to achieve two sensing directions along one sensing axis [8]

Design for force/displacement output of CBMs has attracted attention in recent years Li and Chen [9] proposed a rational function to model the force-displacement (F-d) characteristics of several CBMs Their function with a cubic polynomial numerator and quadratic polynomial denominator can capture key features of the F-d curve, where regression analyses are required to obtain the polynomials to describe the relations between design parameters and the F-d characteristics of the CBMs Huang et al [10] developed

an optimization based method for design of CBMs with specific switching forces The presented CBMs have cosine curved beams with multiple reinforced segments By modifying the length, width, thickness, element number and position of the reinforced segments, the desired switching forces of the CBMs can be achieved The F-d curves of their designs with nearly equal switching forces in the forward and backward directions might not be smooth Smooth force spectrum may be advantageous to avoid chatter vibration during operation Palathingal and Ananthasuresh [11] proposed a shape optimization approach to obtain the switching forces and the distance between equilibrium states of CBMs numerically For improved accuracy of their method, several mode shapes

ACCEPTED MANUSCRIPT

should be used for approximation of the beam profile of the CBMs Han et al [12] proposed a CBM with both tensural segments and compresural segments The combination

of the tensural and compresural segments tailors their CBM for different design requirements, such as switching forces and equilibrium positions Gao et al [13] presented

a design process for a CBM with required switching forces and the distance between equilibrium states Pre-compression and reinforced segments are key design parameters of their design A lookup table representing the variation of snap-through properties with the design parameters, such as pre-compression length and length of reinforced segment, needs

to be constructed in their approach Finite element analyses are required to solve for the F-d characteristics for their pre-compressed bistable structures Li and Hao [14] investigated a generic double-slider four-bar linkage with spring and established its F-d formulation for design of a mechanism with expected nonlinear characteristics, including

a bistable behavior with desired switching forces

The beam profiles of traditional CBMs can be straight line [15,16,17] or cosine curve [11,18,19] The commonly used straight line and cosine curve profiles may restrict the degree of freedom in design of CBMs A pre-load operation may bring the beams of CBMs into desired initial configuration to control their switching forces [20] Design complexity can be introduced by the additional control scheme for the pre-compression of the beams Antagonistic pre-shaped beams can be utilized to obtain desired force output

of CBMs [21] A preload is still necessary to operate their device Beams of CBMs with parametric curve profiles may offer higher degree of freedom in design

In this investigation, we develop a double beam type CBM with a nonrational spline profile for design of switching forces and the distance between equilibrium positions

B-ACCEPTED MANUSCRIPT

The beams of the CBM have a reinforced segment at its center A model is developed to evaluate the F-d characteristics of the CBMs An optimal design of the CBM is sought by

a multiobjective optimization algorithm Experiments are carried out to verify the F-d output of the CBM Several designs of the CBMs with various design targets are obtained

to demonstrate the high design freedom of the B-spline profiled CBM

2 Design

2.1 Design

Fig 1(a) schematically shows a CBM It is comprised of a shuttle mass and four segmented beams The center of the shuttle mass is located at the symmetric axis of the CBM The beams have a nonrational B-spline profile [22] and are divided into three segments A Cartesian coordinate system is also shown in the figure It is assumed that the CBM remains parallel to the underlying plane Due to the chevron configuration of the CBM, the stiffness in X direction is much larger than that in the Y direction Therefore, the salient direction of the motion of the CBM is along the Y direction Due to geometry symmetry, a quarter model can be utilized to analyze the F-d characteristics of the CBM Fig 1(b) shows a quarter model of the CBM The left end of the beam of the quarter model can be represented by a fixed boundary condition The symmetric plane of the quarter model can be represented by a roller boundary condition The B-spline profile of the beam

is indicated by a center line in the figure The beam is dissected into three segments as numbered in the figure The width of segment 2, v , is much larger than the widths of 2

segment 1 and segment 3, v and 1 v , respectively The span of segment 1, segment 2 and 3

segment 3 are denoted by D , 1 D and 2 D , respectively The span and apex height of the 3

ACCEPTED MANUSCRIPT

B-spline curved beam are represented by D and H , respectively The shuttle mass has a

width of D and a height of m v (see Fig 1(a)) The CBM has a thickness of T m

A typical F-d curve of a CBM is shown in Fig 2 When the shuttle mass is displaced

in the  Y direction, see Fig 1(a), the CBM moves from its first stable equilibrium position

Huang et al [10] presented a cosine curved beam with multiple reinforced segments

to achieve desired force output of a CBM Dissimilar to the existing segment-reinforced CBM design of Huang et al [10], the beam of the CBM has a parametric curve as its profile and one reinforced segment to achieve high design freedom in switching forces and the distance between equilibrium positions In this investigation, an open uniform nonrational B-spline (OUNBS) curve is selected as the beam profile of the CBM The OUNBS curve

is determined by a five-point polygon B1B2B3B4B5 as shown in Fig 3 The OUNBS curve defined by a nonglobal basis allows the degree of the curve to be changed without changing the number of the defining polygon vertices The OUNBS curve is given by [22]

1

1 ,

)(

)()

( n

i k i

n

i

k i B

t N

t N P t

of the point B i N i,k(t) is the i th normalized B-spline basis function of order k , and is

if1)

1

,

i i

i

X t X t

and

n i n k i

k i n

k i n

i n k i

k i n i k

i

X X

t N t X X

X

t N X t t

N

1

1 , 1 1

, ,

)()

()()(

)(

2

11

10

n X

n i k k

i X

k i X

of the point B1 is fixed at the origin of a Cartesian coordinate system as shown in Fig 3 The position of the point B5 is given by the span D and apex height H of the B-spline

curved beam B2 and B4 are constrained to be below and above a dotted line connecting

approach During the optimization design process, values of D , H, T, v , 1 v and 2 v3 are specified Force outputs, Fmax and Fmin, and the second stable equilibrium position S can 2

be adjusted by varying the design variables of coordinates of B2, B3, B4, H , D2 and the

Y coordinate of the point B5 Note that the X coordinate of the point B5,

Y , is the apex height H

of the B-spline curved beam, where 3 /110 /10 3 /110

Y

D  B   and

5 4 2

where f1 and f2 are the target values of the output forces of the CBM, and d is the target

value of the second equilibrium position

2.2 Model

An analytical model that can accurately compute the F-d curve of the B-spline profiled CBM is essential for the optimization design process F-d curves of straight beam based CBMs can be obtained by the chained beam constraint model (CBCM) developed

by Ma and Chen [23] Based on the formulation of Awtar and Sen [24], Chen et al [25] developed discretization schemes to extend their CBCM for compliant mechanisms with initially curved beams In this investigation, a model to obtain the F-d relation of the CBM

ACCEPTED MANUSCRIPT

is developed based on the works of Chen and Ma [26] and Chen et al [25]

An initial configuration and a deformed configuration of a quarter model of the CBM are schematically shown in Fig 4 The beam is clamped at one end and the other end is modeled by a roller to represent the symmetry boundary condition The apex height

of the beam is H A displacement A is applied on the roller In the deformed configuration, the apex height of the beam is H A Based on Eqs (1-4), the parametric B-spline curve of the beam with t ranging from 0 to 2 can be represented as

4 10 7 2 2

1 2 2 4

2 )

(

1 4

4 10 7 2 2

1 2 2 4

2

)

(

1 0 , 4

2

2 3 4

12 18 7 )

(

4 2

2 3 4

12 18 7 )

(

5 4

3 2

5 4

3 2

4 3

2

4 3

2

3 2

2 3

2

3 2

2 3

2

3 2

2

1

3 2

t Y t t t Y

t t Y

t t

g

Y

X t X t t t X

t t X

t t

f

X

t Y

t Y t t Y t t t

t

g

Y

X t X t t X t t t

t

f

X

B B

B B

B B

B B

B B B

B B

B

(8)

In this investigation, XY and xy represent the global and local Cartesian coordinate system, respectively t0 and t2 refer to the parameter value of the left end and the right end of the beam, respectively Segment 1, 2 and 3 of the beam are discretized into 16,

32 and 16 elements, respectively The parameter values of the left end and the right end

of the segment 2 can be solved by their known X coordinates as the design variable D2

is given The coordinates (X i,Y i) of the discretization points on the segments 1, 2 and 3 are obtained by the 16, 32 and 16 equal increments of the parameter value

The force and moment applied to the first, the i th, the (i1) th and the final

element of the beam are schematically shown in Fig 5(a-d), respectively, where q is the

number of elements As seen in the figure, X i Y i represents a global Cartesian

coordinate system for the i th element A local Cartesian coordinate system x iy i for each

ACCEPTED MANUSCRIPT

element is also shown in the figure As seen in Fig 5(b), P i and F i are the forces applied

at the right end of each element M i1 and M i are the moments applied at the left end and the right end of each element, respectively The axial displacement and transverse displacement at the right end of each element are represented by i and i, respectively The slope at the left end and the right end of each element are denoted by i and i, respectively L i is the span of the i th element i may be written as

i i

j j i

i i

, ,3,2

where i and i are the slope of the initial curve and the deflected curve of the beam,

respectively q is the number of elements The slope at the left end of each element in the

global coordinate system, i, can be written as

10tan

' 2

' 2 1

' 1

' 1 1

t t

f

t g

t t

f

t g

where K i and K i1 are the curvature at the left end and the right end of the i th element,

respectively K i can be expressed as

ACCEPTED MANUSCRIPT

2 2 ' 2 2

' 2

' 2 '' 2 '' 2 ' 2

2 2 ' 1 2 ' 1

' 1 '' 1 '' 1 ' 1

i t

g t f

t g t f t g t f

i t

g t f

t g t f t g t f

The expressions for f1(t), g1(t), f2(t) and g2(t) can be found by differentiation of Eq

(8) with respect to t In the development of the CBCM, the force, moment and

displacement of each element are normalized as

i

i i i

i i i

i i i i

i i i i

i i i

L L

EI

L M m EI

L F f EI

L P

2 2

(13) where I i is the second moment of the cross sectional area of the ith element The chained beam constraint model equations for curved beams can be expressed as [24]

1114001

140017001

152101

101564

6

612

2

i

i i i

i i

i

i i

i i i

i

p p

p m

63001114001

140017001

152101

101562

1122

12

2 2

i i i i i i

i i

i i

i

i i

yi i

i i i i i

p p

p

p t

2 1 2 1

15

.01

0cos

sin

0sin

cos

i i i i i

i i i

i i i

i i

i i

L L m

L L p

L L f

m p f

i

1

sin5

.0cos

1      (17)

ACCEPTED MANUSCRIPT

i i i i

i

1

cos5

.0sin

1      (18)

o q

to obtain the F-d curve of the CBM Due to geometry symmetry, a half model of the CBM

is considered in the finite element analyses Fig 6 shows the mesh of the half model The

displacement in the X direction at the symmetry plane is constrained to represent the

symmetry condition Fixed boundary conditions are applied to the anchors of the

mechanism A displacement is applied in the Y direction to the shuttle mass to obtain the

F-d curve

A thermoplastic of polyoxymethylene (POM) is assumed to be the material of the CBM During the deformation process of the mechanism, the material is assumed to be linear elastic and isotropic The Young’s modulus E and the Poisson’s ratio of the material are taken as 2.15 GPa and 0.38, respectively The element type of the finite element model

is a 4-node element CPE4R The number of elements in the model is 2392 A mesh sensitivity analysis is carried out to obtain converged solutions of the finite element analyses

A CBM with the dimensions listed in Table 1 (marked as designed dimensions) is considered for the verification of the CBCM Fig 7 shows the F-d curves of the CBM based on the CBCM and the finite element analyses (FEA) The curve computed by the

ACCEPTED MANUSCRIPT

CBCM is in close agreement with that based on the finite element analyses The CBCM provides an efficient and accurate means for design and analyses of the B-spline profiled compliant beams

2.3 Optimization

In this investigation, an optimization approach is adopted for design of the B-spline profiled CBM with specified switching forces and equilibrium positions The F-d curve of the CBM is calculated by the developed CBCM during the optimization process The values of v1, v and 2 v3 are specified as 1 mm, 10 mm and 1 mm, respectively, for all design

cases considered in this investigation The out-of-plane thickness of the mechanism T is

taken as 5.5 mm The domain size, D, is assigned as 110 mm The design variables are the coordinates of B2, B3, B4, H , and the Y coordinate of B5 The total number of the

design variables is 8 Table 1 also lists the specified values of D , T , v1, v , 2 v3, D m and

m

v A nondominated sorting genetic algorithm [27] is used to solve the three-objective optimization of the CBM Fig 8 shows a flowchart of the optimization process The number of generations and the population of each generation are taken as 50 and 100, respectively The evolutionary optimization procedure is programmed with the software MATLAB The genetic algorithm, the objective functions, the design constraints and the CBCM are written in a script file of MATLAB The values of Fmax, Fmin and S2 of the CBM are obtained from the F-d curve The profile of the B-spline curved beam is also created in a script file of MATLAB Parkinson et al [28] presented an optimization-based design of a CBM, where the F-d curve was calculated by a pseudo-rigid-body model They modeled thin flexible segments as pin joints, with torsional springs to represent bending

ACCEPTED MANUSCRIPT

stiffness Accuracy of their model depends on the estimation of the stiffness of the springs Wilcox and Howell [17] and Prasad and Diaz [30] designed their CBMs with finite element analyses linked to an optimization routine The computational cost of finite element analyses may be reduced by the developed CBCM for assessment of the F-d characteristics

of CBMs

Consider a design case with f1  f2 = 2000 mN and d = 11 mm Fig 9 displays

the population distribution of the 35th, 60th and 85th generation during the evolutionary process The three coordinates of the figure represent the values of the three objective functions of Eqs (5-7) As the evolution goes to the 85th generation, the individuals in the population converge to the dominant ones An optimal solution in indicated in the figure Table 1 lists the values of the design variables of the optimum solution Fig 10(a) shows the F-d curve of the optimum design The values of Fmax, Fmin and S2 are 2048 mN, -

2032 mN and 10.90 mm, respectively Fmax and Fmin are 2.4% higher and 1.6% lower, respectively, than the target value, 2000 mN S2 has a value nearly the same as the target value, 11 mm Fig 10(b) shows the Mises stress as a function of the displacement of the CBM The maximum Mises stress in the relevant displacement range of the CBM is 14.87 MPa, which is not higher than a typical value of the yield strength of POM material, 62 MPa The fact that the maximum Mises stress is well below the yield strength of the material justifies the assumption of a linear elastic material in this investigation

ACCEPTED MANUSCRIPT

3 Fabrication and testing

Prototypes of the optimal design were fabricated in order to verify the F-d characteristics of the mechanism with the designed dimensions as listed in Table 1 The prototypes were engraved with a 3-axis milling machine from the POM material Fig 11

is a photo of a fabricated CBM and the setup for the measurement of the F-d curve The mechanism was mounted on a translation stage A force gauge (DS2-5N, Zhiqu Precision Instruments Co., Ltd., China) fixed on a two-axis translation stage was used to measure the force applied to the CBM The rating and resolution of the force gauge are 5000 mN and

1 mN, respectively An aluminum probe with a pin inserted into a hole of the shuttle mass

of the CBM was utilized to apply a force to the CBM The displacement of the probe was recorded by a vernier micrometer of the translation stage

4 Results and discussions

Fig 12(a) and (b) shows F-d curves of the optimal design of the CBM based on experiments, the CBCM and the finite element analyses for the forward and backward motion, respectively The F-d curve obtained by CBCM for the CBM with designed

dimensions and E=2.15 GPa is a replica of the F-d curves shown in Figs 7 and 10 As

seen in Fig 7, the F-d curve based on the CBCM for the case with designed dimensions

and E=2.15 GPa agrees well with the finite element analyses The experiments were

repeated 5 times for both forward and backward motion The error bars shown in the figure indicate the range of the experimental values During the experiments, the tip of the force gauge was moved quasistaticaly to push the shuttle mass of the mechanism to obtain the F-d curves Initially, the mechanism rests in its first stable equilibrium position During the forward motion of the experiments, when the displacement reached the unstable

ACCEPTED MANUSCRIPT

equilibrium position of the mechanism, the shuttle mass lost contact with the probe tip and snapped to its second stable equilibrium position, S The event of the snap through 2

behavior is indicated by the discontinuous jump of the experimental results in Fig 12(a) Similarly, the snap through event was observed during the backward motion of the experiments as seen in Fig 12(b)

Table 2 lists the values of Fmax, Fmin and S based on the experiments, the CBCM 2

and the finite element analyses The experimental value of S , 10.90 mm, is equal to that 2

based on the CBCM, 10.90 mm The values of Fmax and Fmin of the experiments deviate significantly from those based on the CBCM with the designed dimensions The discrepancies can be attributed to machining error and uncertainty in material properties

An image dimension measurement system (IM-7020, Keyence Co., Japan) with a measurement precision of 5 micrometer was utilized to measure the dimensions of the fabricated prototype The IM-7020 is equipped with a low distortion lens designed to minimize distortion near the center and the outer of the view field and a multiple illumination unit to find optimal lighting to illuminate the entire target evenly Fig 13 shows the measured dimensions of the fabricated prototype The measured dimensions are also listed in Table 1 The coordinates of the control points of the B-spline curve based on the measured dimensions were approximated by curve fitting The width and span of the three segments and the apex height of the B-spline profiled beam have a certain range as listed in the table The developed CBCM cannot be used to find the F-d curve of the device with irregular profiles The F-d curve of the prototype with the measured dimensions was calculated by a finite element model (see the F-d curve marked by FEA (Measured dimension, E=2.15 GPa) in Fig 12) Note that 2.15 GPa is the value of the Young’s

ACCEPTED MANUSCRIPT

modulus of the POM material taken in the design stage The calculated values of Fmax,

min

F and S are also listed in Table 2 The results of 2 Fmax and Fmin based on the finite element model with measured dimensions and E=2.15 GPa still have a large difference from the experiments The Young’s modulus of the POM material may differ due to various processing conditions With an adjusted value of 1.875 GPa, the values of Fmax

and Fmin based on the finite element model with the measured dimensions are brought closer to the experiments (see Table 2) The relative errors in Fmax and Fmin values between the finite element model with the measured dimensions and E=1.875 GPa and the experiments are 4% and 3%, respectively It can be seen that the positon of S is less 2

sensitive to the change in the device dimensions and material properties compared to the values of Fmax and Fmin

Four design cases with various values of the targets f1, f2 and d were selected to demonstrate the design freedom of the switching force and location of the second stable equilibrium position of the B-splined profiled CBM For the four cases, the number of generations and the population of each generation of the optimization design were set to

300 and 100, respectively The POM material with Young’s modulus of 2.15 GPa was adopted Fig 14 shows the population distribution of various generations during evolutionary process for the four design cases The F-d curves of the four design cases are shown in Fig 15 Table 3 lists the values of the targets f1, f2 and d and the values of

max

F , Fmin and S2 of the optimized design The ratios of the f1/ f2 of the design case 1

to 4 are 2, 1, 2/3 and 1, and the values of d are 10, 10, 10 and 12, respectively The force

ratios are selected to demonstrate the design freedom of the B-spline profiled CBM and for

ACCEPTED MANUSCRIPT

comparison with a case with f1/ f2 =2/3 investigated by Huang et al [10], The largest discrepancy between the target value and the designed value occurs in the maximum force

of design case 3 with a relative error of 7.7% Compared to the work of Huang et al [10], the design case 3 with f1/ f2 =2/3 has a much smoother F-d curve as shown in Fig 15(c)

A kink near the point of Fmin in seen on their F-d curve Non-smooth force spectrum may cause chatter vibration during operation of the mechanism As shown in Fig 15, the F-d curves based on the CBCM are almost identical to those obtained by the finite element

analyses

Indeed, there have been several ways to achieve desired output force of bistale mechanisms Vangbo [31] compressed a beam with an axial load while the load has to be above the first critical load for buckling In order to put Vangbo’s design into practice, an elaborate control scheme may be devised to keep the axial load at the required level during device operation An array of pre-shaped curved beams [32] possesses various force outputs with respect to the forward and backward motion The array type design to reach the goal of force versatility may occupy too much of the area or volume of the available device space In this investigation, an optimization procedure with an analytical model to obtain F-d curves of B-spline profiled beams is adopted to design for force versatility of the CBM The F-d curves of the B-spline profiled beams prove to provide versatile force outputs to meet the design requirement The CBCM analytical model developed for the parametric curved beams could be efficient for design of beam-type compliant multistable mechanisms without resorting to the finite element method to obtain the nonlinear F-d curves of parametric curved beams due to their geometry complexity

ACCEPTED MANUSCRIPT

Chen et al [25] reported that it is inconvenient to implement CBCM with equal discretization for a beam of noncircular shape because the curvature changes along the

beam For a B-spline curve as explicit functions of an independent parameter t, it is

practical to discretize the beam into equal increments of the parameter value The parameter value of the left end and the right end of B-spline curved beam of the CBM are

0 and 2, respectively The parameter values of the left end and the right end of the segment

2 of the beam are solved given the known value of the design variable D2 In this investigation, the number of elements of segment 1, 2 and 3 of the beam are taken as 16,

32 and 16, respectively, for robustness of the optimization process against uncertainty in the value of the curvature of the B-spline curved beam Indeed, fewer elements could be enough for the proposed design The number of elements of segment 1, 2 and 3 of the beam can be taken as 6, 28 and 6, respectively, for the case shown in Fig 7, where the maximum value of the nondimensional curvature (defined in Eq (11)) of the elements is nearly 0.07 The beam constraint model developed by Awtar and Sen [24] requires that the nondimensional curvature to be less than 0.1

In elastic structures, load to cause loss of stability with their configuration to pass from one stable equilibrium state to another can be derived by employing the principle of the minimum total potential if a system is conservative [33] Based on the Lagrange-Dirichlet theorem, a stable equilibrium position is located where the potential energy has a local minimum value Therefore, the switching forces of CBMs are those that make the total potential assume a minimum value Li and Hao [14] synthesized a double-slider four-bar linkage with expected nonlinear force-displacement and energy-displacement behaviors They derived the conditions to synthesize a bistable mechanism with equal

ACCEPTED MANUSCRIPT

maximum potential energy in the forward and backward directions Indeed, equal potential energy can be a design target for the CBMs Equal switching force may be advantageous for bistable micro switches driven by comb-drives where the force developed by the comb-drives is in proportion to the applied electric potential which is limited by the conventional complementary metal-oxide-semiconductor (CMOS) technology Implementation of a bistable threshold acceleration sensing device requires equal switching force in both directions along the sensing direction It depends on the potential application to select a suitable criterion for design of the CBMs

5 Conclusions

Design for switching force of B-spline profiled CBM is demonstrated for various ratios of switching forces in forward and backward directions and the location of the second equilibrium position An analytical model to solve for the nonlinear force-displacement characteristics of the B-spline profiled beams is developed The model can be applied to various parametric curved beams, such as Bezier curves, to increase the design degree of freedom to achieve the design objectives Given the design flexibility of the B-spline profiled CBM, various ratios between the forward and backward switching forces are achieved and the resulting F-d curves are smooth Such a mechanism has potential applications where precise geometric activation and quantifiable load bearing capacity are required

ACCEPTED MANUSCRIPT

Acknowledgement

The computing facilities provided by the National Center for High-Performance Computing (NCHC) are greatly appreciated The machining facilities provided by the machine shop of National Chung Hsing University (NCHU) are greatly appreciated The authors are also thankful for the financial support from the Ministry of Science and Technology, R.O.C., under Grant No MOST 105-2221-E-005-060-

ACCEPTED MANUSCRIPT

[3] S Zhang, G Chen, Design of compliant bistable mechanisms for rear trunk lid of cars,

in Proceedings of the 4th international conference on Intelligent Robotics and Applications - Volume Part I, pp 291-299, December 6-8, 2011, Aachen, Germany [4] A Hinitt, A Conn, J Rossite, Non-back-drivable binary bistable DEA device, in Proceedings of the fifth International Conference on Electromechanically Active Polymer (EAP) transducers & artificial muscles, 45, EuroEAP, 2015

[5] E.J.J Kruglick, K.S.J Pister, Bistable MEMS relays and contact characterization, in Proc Solid-state Sensors and Actuators Workshop Cleveland, OH, 1998, pp 333-337 [6] J.G Fleming, A bistable membrane approach to micromachined displays, MRS Proceedings, 508, 219 doi:10.1557/PROC-508-219, 1998

[7] B Hälg, On a micro-electro-mechanical nonvolatile memory cell, IEEE Transactions