Hybrid flexure hinges

Hybrid flexure hingesRongzhou Lin, Xianmin Zhang, Xuejun Long, and Sergej Fatikow Citation: Rev. Sci. Instrum. 84, 085004 (2013); doi: 10.1063/1.4818522 View online: http://dx.doi.org/10.1063/1.4818522 View Table of Contents: http://rsi.aip.org/resource/1/RSINAK/v84/i8 Published by the AIP Publishing LLC. Additional information on Rev. Sci. Instrum.Journal Homepage: http://rsi.aip.org Journal Information: http://rsi.aip.org/about/about_the_journal Top downloads: http://rsi.aip.org/features/most_downloaded Information for Authors: http://rsi.aip.org/authors

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REVIEW OF SCIENTIFIC INSTRUMENTS 84, 085004 (2013)

Hybrid flexure hingesRongzhou Lin,1 Xianmin Zhang,1,a) Xuejun Long,1 and Sergej Fatikow2

1Guangdong Province Key Laboratory of Precision Equipment and Manufacturing Technology, School ofMechanical and Automotive Engineering, South China University of Technology, 510641 Guangzhou, China2Division for Micro-robotics and Control Engineering, University of Oldenburg, 26129 Oldenburg, Germany

(Received 21 April 2013; accepted 31 July 2013; published online 20 August 2013)

This paper designs and analyzes the hybrid flexure hinge composed of half a hyperbolic flexure hingeand half a corner-filleted flexure hinge. As it is transversely asymmetric, it has different performancewhen the fixed and free ends switch. Considering the diversion of rotation center from midpoint,closed-form equations are formulated to characterize both the active rotation and all other in-planeparasitic motion by the Castigliano’s second theorem. The maximum stress is evaluated as well. Theseequations are verified by the finite element analysis and experimentation. The compliance precisionratios are proposed to indicate flexure hinges’ ability of preserving the rotation center when theyhave the same displacement at the free end. The hybrid flexure hinges are compared with five kindsof common notch flexure hinges (circular, corner-filleted, elliptical, hyperbolic, and parabolic flexurehinges) quantitatively based on compliance, precision, compliance precision ratios, and the maximumstress. Conclusions are drawn regarding the performance of these six kinds of flexure hinges. © 2013AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4818522]

NOMENCLATURE

a half length of flexureb cross-sectional flexure widthc flexure thickness parameterd cutout deepnessf functionh flexure thicknessk stress concentration factorl distance to the free endt minimum thickness of flexurex, y, z reference axes, deflectionsA cross-sectional flexure areaB thickness of rigid barC complianceE Young’s modulusF forceI flexure cross-sectional moment of inertiaM bending momentU elastic strain energy

Greekβ, γ non-dimensional parametersθ rotation angle along z axisρ curvature radiusσ stress

SubscriptsFx, Fy force along the x, y axesM bending momenta axialb bending, beam flexurec circular flexurecf corner-filleted flexurecfh corner-filleted hyperbolic flexure

a)Author to whom correspondence should be addressed. Electronic mail:[email protected].

e elastic, elliptical flexureh hyperbolic flexurehcf hyperbolic corner-filleted flexuremax maximump parabolic flexurer rotation centerx, y reference axesθ rotation angle along z axis

Superscripts′ refers to the rotation center

I. INTRODUCTION

With advantages over traditional mechanisms like zerofriction and backlash, high resolution, light weight, and com-pact, compliant mechanisms with flexure hinges are widelyused in applications where high precision is required in a rel-atively small workspace such as translation micro-positioningstages, high-accuracy alignment devices for optical fibers, andmicro-electro-mechanical system.1–5

As the critical parts of compliant mechanisms, in-planeflexure hinges of various notch profiles have been designedand researched in previous works. Paros and Weisbord’swork6 established the milestone of theoretical method of an-alyzing notch flexure hinges. They first designed the circularflexure hinges, and deduced both exact and simplified equa-tions to calculate their compliance. Wu and Zhou7 deducedmore concise and exact equations than those of Paros andWeisbord for the circular flexure hinges.

Based on a modification of Paros and Weisbord’s work,Smith et al.8 presented closed-form equations for the ellipticalflexure hinges, which are limited by the circular flexure hingesand the simple beams in terms of compliance. The model ac-curacy was verified by both finite element analysis (FEA) andexperimentation.

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085004-2 Lin et al. Rev. Sci. Instrum. 84, 085004 (2013)

Tseytlin9 used the inverse conformal mapping of circu-lar approximating contour to deduce closed-form equationsof the circular flexure hinges and the elliptical flexure hinges.Predictions of the developed models are likely to be muchcloser to FEA and experimental data than other theoreticalequations.

Lobontiu et al.10 presented closed-form equations for thecorner-filleted flexure hinges. In comparison with the rightcircular flexure hinges, they are more bending-compliant andinduce lower stress but are less precise in rotation. The FEAand experimentation were carried out to confirm the model.

Xu and King11 made a comparison of right circular, el-liptical, and corner-filleted flexure hinges by FEA in termsof compliance, precision, and stress. The comparison revealsthat the elliptical profile has advantage in achieving long fa-tigue life while the corner-filleted design offers the highestflexibility.

Chen et al.12, 13 designed two kinds of hybrid flexurehinges, the right-circular corner-filleted flexure hinges and theright-circular elliptical flexure hinges. They are more flexi-ble than the right circular flexure hinges while all of themhave the same capability of constraining the shift of the rota-tion center. Chen et al.14, 15 proposed the elliptical-arc-filletedflexure hinges to bring elliptical arc, circular-arc-filleted,elliptical-filleted, elliptical, circular, circular-filleted, andright-circular flexure hinges together under one set of equa-tions. This solves the problem that theoretical equations areunique to each kind of flexure hinge. Chen et al.16 also pro-posed a generalized conic flexure hinge model that is suitablefor elliptical arc, parabolic and hyperbolic flexure hinges, andutilized the generalized equations for conic curves in polarcoordinates to deduce the compliance and precision matrices.

Lobontiu et al.17 designed parabolic and hyperbolicflexure hinges and used the Castigliano’s second theorem todeduce closed-formed equations. The comparisons betweenthem reveal that the parabolic flexure hinges are morerotation-compliant and induce less stress, while the hyper-bolic flexure hinges are less sensitive to parasitic effects.Lobontiu et al.18 developed closed-formed equations forconic-section flexure hinges including right circular, elliptic,parabolic, and hyperbolic flexure hinges. The non-dimensional comparisons of compliance and precisionbetween them reveal that elliptic, parabolic, and hyperbolicflexure hinges are more compliant than the right circularflexure hinge for large length-to-thickness ratio, while hyper-bolic flexure hinges have the best capability of constrainingthe shift of rotation center. Lobontiu et al.19, 20 proposeda general compliance model to analyze symmetric flexurehinges formed of serial segments of known compliance.

Zelenika et al.21 proposed a nonlinear parameter opti-mization of flexure hinge shapes including predefined andfreeform parametric shapes. The prismatic beam, circu-lar, elliptic, Grodzinski’s, Baud’s, Thum’s, and Bautz’s andfreeform optimized hinges are optimized and compared interms of compliance, strength, stress concentration factors,and parasitic shifts. These flexure hinges are both transverseand axial symmetry.

Tian et al.22 presented dimensionless empirical equationsand graph expressions of filleted V-shaped, cycloidal, and cir-

cular flexure hinges. The closed-form equations of these flex-ure hinges were developed according to the Castigliano’s sec-ond theorem and the numerical adaptive Simpson quadratureintegration. The influences of geometric parameters on theperformance of flexure hinges were investigated, as well asthe performance comparison among these three kinds of flex-ure hinges.

Most of these works are concentrated on researching thetransversely symmetric flexure hinges, and the other worksabout the transversely asymmetric flexure hinges consider therotation center is located at the midpoint. Thus, this paperresearches the hybrid flexure hinges which are transverselyasymmetric and considers the diversion of rotation centerfrom midpoint. The hybrid flexure hinge composed of halfa hyperbolic flexure hinge and half a corner-filleted flexurehinge is designed. The Castigliano’s second theorem is usedto deduce the closed-form equations and a general methodwhich does not assume the maximum stress happens at thethinnest part is proposed to evaluate the maximum stress.These equations are verified by the FEA and experimentation.The analytical model predictions are within 8% error marginscompared to FEA results and are within 13% error marginscompared to experimental results.

To our knowledge, no one has compared the same hybridflexure hinges with different fixed end, as well as with thefive kinds of common notch flexure hinges (circular, corner-filleted, elliptical, hyperbolic, and parabolic flexure hinges).The compliance precision ratios are proposed to indicate flex-ure hinges’ ability of preserving the rotation center when theyhave the same displacement at the free end. This paper makesa comprehensive comparison among these flexure hinges interms of compliance, precision, compliance precision ratio,and the maximum stress. All the analysis and comparisonsare within the range of small displacement.

II. DESIGN

Among the five kinds of common notch flexure hinges,the corner-filleted flexure hinges which have the largest rota-tion compliance and the hyperbolic flexure hinges which havethe highest precision are chosen to form the hybrid flexurehinges. As the hybrid flexure hinges are transversely asym-metric, they have different performance when the fixed endand free end switch. When the half hyperbolic flexure hingeis fixed, it is termed hyperbolic corner-filleted flexure hinge(HCF), which is shown in Fig. 1. When the half corner-filleted

y

xz

2a

b

t

c

o

h x( )

B

FIG. 1. Hyperbolic corner-filleted flexure hinge.



y

xz

h x( )2a

b

t

o

c

B

FIG. 2. Corner-filleted hyperbolic flexure hinge.

flexure hinge is fixed, it is termed corner-filleted hyperbolicflexure hinge (CFH), which is shown in Fig. 2.

The geometric parameters of the flexure hinges are illus-trated in Figs. 1 and 2. And two other geometric parametersand two non-dimensional parameters used in the followingsections are defined as

I (x) = bh(x)3

12, A(x) = bh(x), β = t

2c, γ = t

2a.

III. THEORETICAL ANALYSIS

The formulation in the paper is based on the followingsimplifying assumptions:17

(1) The flexure hinges consist of two notches that are longi-tudinally symmetric and transversely asymmetric.

(2) The flexure hinges discussed here are parts of plane de-vices where both the input and output are planar, whichallows conducting the research for only three degrees offreedom: two in-plane translations and one rotation, nor-mal to the translation plane.

(3) The flexure hinges are modeled and analyzed as small-displacement Euler-Bernoulli beams subjected to bend-ing produced by forces and moments and axial load,while shearing effects are neglected.

(4) The flexure hinges are considered to be fixed at one endand free at the other.

In the paper, for load parameters, the first subscript in-dicates the axis while the second one determines the pointalong the flexure longitudinal axis; for compliance and pre-cision factors, the first subscript represents the deformationdirection and the second one indicates the load that producesthe deformation.

A. Compliance

Sketch of a flexure hinge with loads is shown in Fig. 3.The hinge is modeled as a three-node cantilever beam,point 1 is the free end with loads, point 2 is the rotation center,and point 3 is the fixed end. The coordinate origin is locatedat the midpoint o.

The displacement-load relationship at the free end is ex-pressed as the form:18

Fy1

Fx1

Mz1Fy2

Fx2 x

y

x

aa

123lr

o

FIG. 3. Sketch of a flexure hinge with loads.

⎧⎪⎨⎪⎩

θ1

y1

x1

⎫⎪⎬⎪⎭ =

⎡⎢⎣

Cθ,M Cθ,Fy 0

Cy,M Cy,Fy 0

0 0 Cx,Fx

⎤⎥⎦

⎧⎪⎨⎪⎩

Mz1

Fy1

Fx1

⎫⎪⎬⎪⎭ , (1)

where Cy,M = Cθ ,Fy, according to the reciprocity principle.The displacement-load relationship can be formulated as

the following form by the Castigliano’s second theorem:

θ1 = ∂Ue

∂Mz1, y1 = ∂Ue

∂Fy1, x1 = ∂Ue

∂Fx1, (2)

where the elastic strain energy comprises bending and axialterms:

Ue = 1

2

(∫ a

−a

[Mz1 + Fy1(a − x)]2

EI (x)dx +

∫ a

−a

Fx12

EA(x)dx

).

(3)Equations (1)–(3) are used to obtain the following com-

pliance equations:

Cθ,M =∫ a

−a

1

EI (x)dx = 12

Ebf1, (4)

Cθ,Fy = Cy,M =∫ a

−a

a − x

EI (x)dx = 12

Eb(af1 − f2), (5)

Cy,Fy =∫ a

−a

(a − x)2

EI (x)dx = 12

Eb(a2f1 − 2af2 + f3), (6)

Cx,Fx =∫ a

−a

1

EA(x)dx = 1

Ebf4, (7)

where

f1 =∫ a

−a

1

h(x)3dx, (8)

f2 =∫ a

−a

x

h(x)3dx, (9)

f3 =∫ a

−a

x2

h(x)3dx, (10)

f4 =∫ a

−a

1

h(x)dx. (11)

1. Hyperbolic corner-filleted flexure hinges

For HCF, the variable thickness is

hhcf(x)

=

⎧⎪⎪⎨⎪⎪⎩

√t2 + (4c2 + 4ct)/a2x2 (−a ≤ x < 0)

t (0 ≤ x ≤ a − c)

2c + t − 2√

c2 − (x − a + c)2 (a − c < x ≤ a)

.

(12)



Substitute Eq. (12) into Eqs. (8)– (11), the followingequations are obtained:

f1,hcf = 1

2t2

{β

(β + 1) γ+ β − γ

βγ+ 1

2 (β + 2)2

×[

2β2 + 4β + 3

β + 1+ 6 (β + 1)√

β (β + 2)atan

√2

β+ 1

]},

(13)

f2,hcf

= 1

8tβγ

{β − γ

(β + 2)2

[2β2 + 4β + 3

β + 1+ 6 (β + 1)√

β (β + 2)

× atan

√2

β+ 1

]+ (β − γ )2

βγ+ 2βγ 2 + γ 2 − 2β3

γ (2β + 1) (β + 1)

},

(14)

f3,hcf = 1

8γ 2

{β(β − 2γ )(2β2 + 4β + 3) + 3(β + 1)2γ 2

2β2(β + 1)(β + 2)2

+ [3β(β − 2γ ) − (β + 1)2γ 2(2β2 + 4β − 3)]

× (β + 1) atan√

2/β + 1

β2.5 (β + 2)2.5 + γ

β (β + 1)

+ β3 − γ 3

3β3γ+ πγ 2

2

+ β3

γ (2β + 1)

[1

2√

2β + 1log

(β + 1 + √

2β + 1

β + 1 − √2β + 1

)

− 1

β + 1

]− β − γ + γ 2 (1 − β)

β2

}, (15)

f4,hcf = β

2γ√

2β + 1log

(β

β + 1 − √2β + 1

)+ β − γ

2βγ

+ β + 1√β (β + 2)

atan

√2

β+ 1 − π

4. (16)

2. Corner-filleted hyperbolic flexure hinges

For CFH, the variable thickness is

hcfh(x)

=

⎧⎪⎪⎨⎪⎪⎩

2c + t − 2√

c2 − (x + a − c)2 (−a ≤ x < c − a)

t (c − a ≤ x ≤ 0)√t2 + (4c2 + 4ct)/a2x2 (0 < x ≤ a)

.

(17)

As CFH is symmetric with HCF with respect to y-axis, sub-stitute Eq. (17) into Eqs. (8)–(11) and the following equationscan be obtained:

f1,cfh = f1,hcf, (18)

f2,cfh = −f2,hcf, (19)

f3,cfh = f3,hcf, (20)

f4,cfh = f4,hcf. (21)

B. Precision

The rotation center of longitudinally and transverselysymmetric flexure hinge is located at the midpoint, while itdiverges from the midpoint for the transversely asymmetricflexure hinge. Lipkin and Patterson23 define that a rotationabout the rotation-compliant axis only produces a paralleltranslation. The rotation center for the plane notch flexurehinges is the intersection of the rotation-compliance axis andthe plane, and its distance to the free end is

lr = Cy,M/Cθ,M = a − f2/f1. (22)

Ideally, the rotation center should not shift at all underany loads. But that is virtually impossible since the wholeflexure hinge deforms under the loads at the free end. Therelationship between displacement at the rotation center andloads at the free end is expressed as the form:18

⎧⎪⎨⎪⎩

0

y2

x2

⎫⎪⎬⎪⎭ =

⎡⎢⎣

0 0 0

C ′y,M C ′

y,Fy 0

0 0 C ′x,Fx

⎤⎥⎦

⎧⎪⎨⎪⎩

Mz1

Fy1

Fx1

⎫⎪⎬⎪⎭ . (23)

The displacements can be assessed by applying two fic-tional loads illustrated in Fig. 3, a horizontal one Fx2 anda vertical one Fy2, in addition to the real loads. The Cas-tigliano’s second theorem is used again to obtain the displace-ments of the rotation center in the form:

x2 = ∂Ue

∂Fx2, y2 = ∂Ue

∂Fy2, (24)

where the elastic strain energy comprises bending and axialterms:

Ue = 1

2

∫ a−lr

−a

{[Mz1 + Fy1(a − x) + Fy2(a − lr − x)]2

EI (x)

+ (Fx1 + Fx2)2

EA(x)

}dx. (25)

Equations (23)–(25) are used to obtain the followingclosed-form equations that characterize precision:

C ′y,M =

∫ a−lr

−a

a − lr − x

EI (x)dx = 12

Eb[(a − lr )f5 − f6], (26)

C′y,Fy =

∫ a−lr

−a

(a − x)(a − lr − x)

EI (x)dx

= 12

Eb[a(a − lr )f5 + (lr − 2a)f6 + f7], (27)

C ′x,Fx =

∫ a−lr

−a

1

EA(x)dx = 1

Ebf8, (28)

where

f5 =∫ a−lr

−a

1

h3(x)dx, (29)



f6 =∫ a−lr

−a

x

h3(x)dx, (30)

f7 =∫ a−lr

−a

x2

h3(x)dx, (31)

f8 =∫ a−lr

−a

1

h(x)dx. (32)

1. Hyperbolic corner-filleted flexure hinges

For HCF, the rotation center is located at the beam seg-ment and its distance to the free end is

lr,hcf = a − f2,hcf/f1,hcf. (33)

Substitute Eqs. (12) and (33) into Eqs. (29)–(32), the fol-lowing equations are obtained:

f5,hcf = β

2t2γ (β + 1)+ f2,hcf

t3f1,hcf, (34)

f6,hcf = − β2

4t (β + 1) (2β + 1) γ 2+ f 2

2,hcf

2t3f 21,hcf

, (35)

f7,hcf = β3

8 (2β + 1) γ 3

[1

2√

2β+1log

(β+1+√

2β+1

β+1−√2β+1

)

− 1

β + 1

]+ f 3

2,hcf

3t3f 31,hcf

, (36)

f8,hcf = β

2γ√

2β + 1log

(β

β + 1 − √2β + 1

)+ f2,hcf

f1,hcft.

(37)

2. Corner-filleted hyperbolic flexure hinges

For CFH, the rotation center is located at the beam seg-ment and its distance to the free end is

lr,cfh = a − f2,cfh/f1,cfh = a + f2,hcf/f1,hcf. (38)

Substitute Eqs. (13), (14), (17), and (38) to Eqs. (29)–(32), the following equations are obtained:

f5,cfh = 1

4t2 (β + 2)2

[2β2 + 4β + 3

β + 1+ 6 (β + 1)√

β (β + 2)

× atan

√2

β+ 1

]+ f2,cfh

t3f1,cfh+ β − γ

2t2βγ, (39)

f6,cfh = − 1

8tβ

{β − γ

(β + 2)2 γ

[6 (β + 1)√β (β + 2)

atan

√2

β+ 1

+ 2β2+4β+3

β+1

]+ 1

β+1

}+ f 2

2,cfh

2t3f 21,cfh

− (β−γ )2

8tβ2γ 2,

(40)

f7,cfh = 1

8γ 2

{(β−γ )2(2β2+4β+3)

2β2(β+1)(β+2)2+γ 2(2β2+4β+1)

2β(β+1)(β+2)

− [(2β2+4β−3)(β+1)2γ 2−3β(β−2γ )](β+1)

× atan√

2/β+1

β2.5 (β+2)2.5 + (β−γ ) γ

β2 (β+1)+πγ 2

2

}

+ f 32,cfh

3t3f 31,cfh

+ (β−γ )3

24β3γ 3, (41)

f8,cfh = β + 1√β (β + 2)

atan

√2

β+ 1 − π

4+ f2,cfh

f1,cfht+ β − γ

2βγ.

(42)

C. The maximum stress

The normal stress on cross-section is the result of bend-ing and axial loads when shearing effects, as previouslystated, are neglected. The maximum stress caused by the axialload is

σmax,Fx = ka

btFx1. (43)

The stress caused by bending on the upper fiber of posi-tion x is

σM+Fy = 6kb

bh2(x)[Mz1+Fy1(a − x)]. (44)

The maximum stress happens near the special positionssuch as the thinnest position, and the stress concentration in-fluences an area,10, 17 thus, for simplification, kb of each areanear the special position is treated as constant. The maximumstress happens near the poles or the discontinuities. For HCF,the maximum stress happens near:

xmax,hcf

=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

−a

Mz1

Fy1+ a−

√a2t2

4c (c+t)+

(Mz1

Fy1+a

)2

0

a − c

,−a ≤ x ≤ a.

(45)

For CFH, the maximum stress happens near:

xmax,cfh

=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

c − a

0

Mz1

Fy1+ a+

√a2t2

4c (c+t)+

(Mz1

Fy1+a

)2

a

,−a ≤ x ≤ a.

(46)

Substitute xmax and kb of the special positions intoEq. (44), respectively, and the largest one is the maximumstress caused by Mz1 and Fy1. For the hybrid flexure hinge, it



B

d

dc

c

2a

ρ

ρ

t

y

x

FIG. 4. Profile of a flexure hinge.

contains three kinds of stress concentration factors: kb,b for theconnection area of the rigid bar and the filleted, kb,cf for theconnection area of the beam and the filleted, and kb,h for thesegment near the thinnest position. Although the maximumstresses caused by bending and axial load, respectively, donot happen at the same position, the whole maximum stressis approximately equal to the sum of these two stresses. Themaximum stress can be evaluated with the known loads or thedisplacements which can be transformed into loads at the freeend.10

The stress concentration factors ka and kb can be found inthe textbooks like Masataka24 or Pilkey.25 Because the stresscaused by the axial force is relatively small when comparedwith the stress caused by the bending and the equations tocalculate ka are complicated, this kind of stress is neglected inthe paper. The equation to calculate kb is24

kb = 1 +[

B / t − 1

2 (4.27B / t − 4)

t

ρ

]0.83

. (47)

The relating geometric parameters are shown in Fig. 4.Equation (47) is suitable for the notch whose opening angle isapproximate 0 and the shape does not influence the result.

IV. VERIFICATION OF THE THEORETICALEQUATIONS

A. Finite element verification

This section is devoted to check the validity of the equa-tions for calculating compliance, precision, and the maximum

stress by FEA. ANSYS is used to conduct finite elementstatic-load simulations. The flexure hinge models are gener-ated by using 8-node, plane elements (PLANE 82) with twodegree-of-freedom on each node, which are translation in thenodal x- and y-axes.26 The modeled flexure hinges have a con-stant cross-sectional flexure width of 10 mm with a Young’smodulus of 110 GPa and Poisson ratio of 0.33 (titaniumalloy).

1. Verification of the closed-form equations

The analytical and FEA results of the compliance fac-tors are given in Tables I and II. Similar FEA is conducted inorder to evaluate the precision factors. The corresponding re-sults are given in III and IV. From Tables I–IV, the followingconclusions can be drawn:

(1) The analytical model predictions of the compliance fac-tors are confirmed by FEA within 5% error margins, andthat of the precision factors are within 8% error margins.

(2) The error becomes larger when either β closes to 1 or γ

closes to 0.05. It generates less error to use the equationswhen β < 1 and γ < 0.05.

(3) Because HCFs are symmetric with CFH about y-axis,they have the same geometric configuration and thesame rotation center, Cθ ,M,hcf is equal to Cθ ,M,cfh, Cx,Fx,hcf

is equal to Cx,Fx,cfh, and C ′y,M,hcf is equal to C′

y,M, cfh.

2. Verification of the maximum stress equations

This section chooses two random examples to verify themaximum stress equations. The key parameters are listedin Table V. The other geometric parameters are defined inSec. IV A. The loads are: Mz1 = 0.01 Nm, Fy1 = 1 N, andFx1 = 1 N.

The analytical and FEA results of the maximum stressare given in Table VI. From Table VI, it can be drawn thatthe analytical results and FEA results are conformity and themaximum stress caused by the axial force can be neglectedwhen compared with the maximum stress caused by bending.

B. Experimental verification

Experimentation is used to assess the validity of fourcompliance factors, Cθ ,M, Cθ ,Fy, Cy,M, and Cy,Fy. Cx,Fx is very

TABLE I. Comparison of compliance factors between analytical results (denoted by An) and FEA results for HCF.

Cy ,M,hcf = Cθ ,Fy ,hcf Cy ,Fy ,hcf Cx ,Fx ,hcf

Geometry Cθ ,M,hcf (N−1 m−1) (N−1 × 10−2) (N−1 m × 10−5) (N−1 m × 10−8)

a (mm) c (mm) t (mm) An FEA Err% An FEA Err% An FEA Err% An FEA Err%

4.0 0.2 0.2 7.14 7.20 0.83 2.00 2.01 0.50 7.59 7.68 1.17 2.93 2.92 − 0.344.0 0.4 0.2 6.21 6.26 0.80 1.60 1.60 0.00 5.33 5.38 0.93 2.61 2.60 − 0.388.0 0.2 0.2 14.41 14.38 − 0.21 8.00 8.02 0.25 60.71 61.03 0.52 5.88 5.87 − 0.178.0 0.4 0.2 12.76 12.74 − 0.16 6.40 6.41 0.16 42.67 42.82 0.35 5.27 5.27 0.004.0 0.2 0.4 1.01 1.04 2.88 0.32 0.32 0.00 1.37 1.42 3.52 1.59 1.58 − 0.634.0 0.4 0.4 0.88 0.90 2.22 0.25 0.25 0.00 0.95 0.98 3.06 1.45 1.44 − 0.698.0 0.2 0.4 2.03 2.06 1.46 1.27 1.28 0.78 10.94 11.12 1.62 3.19 3.18 − 0.318.0 0.4 0.4 1.79 1.80 0.56 1.00 1.01 0.99 7.59 7.69 1.30 2.93 2.92 − 0.3420.0 0.4 0.4 4.51 4.53 0.44 6.25 6.26 0.16 118.60 119.10 0.42 7.36 7.34 − 0.2740.0 0.4 0.4 9.06 9.04 − 0.22 25.00 24.99 − 0.04 948.60 949.00 0.04 14.74 14.72 − 0.14



TABLE II. Comparison of compliance factors between analytical results (denoted by An) and FEA results for CFH.

Cy ,M,cfh = Cθ ,Fy ,cfh Cy ,Fy ,cfh Cx ,Fx ,cfh

Geometry Cθ ,M,cfh (N−1 m−1) (N−1 × 10−2) (N−1 m × 10−5) (N−1 m × 10−8)

a (mm) c (mm) t (mm) An FEA Err% An FEA Err% An FEA Err% An FEA Err%

4.0 0.2 0.2 7.14 7.20 0.83 3.71 3.76 1.33 21.30 21.67 1.71 2.93 2.92 − 0.344.0 0.4 0.2 6.21 6.26 0.80 3.37 3.41 1.17 19.57 19.83 1.31 2.61 2.60 − 0.388.0 0.2 0.2 14.41 14.38 − 0.21 15.06 15.16 0.66 173.70 175.20 0.86 5.88 5.87 − 0.178.0 0.4 0.2 12.76 12.74 − 0.16 14.02 14.08 0.43 164.60 165.70 0.67 5.27 5.27 0.004.0 0.2 0.4 1.01 1.04 2.88 0.49 0.51 3.92 2.75 2.87 4.18 1.59 1.58 − 0.634.0 0.4 0.4 0.88 0.90 2.22 0.45 0.46 2.17 2.56 2.66 3.76 1.45 1.44 − 0.698.0 0.2 0.4 2.03 2.06 1.46 1.98 2.01 1.49 22.27 22.75 2.11 3.19 3.18 − 0.318.0 0.4 0.4 1.79 1.80 0.56 1.86 1.88 1.06 21.30 21.68 1.75 2.93 2.92 − 0.3420.0 0.4 0.4 4.51 4.52 0.22 11.80 11.85 0.42 340.60 342.70 0.61 7.36 7.34 − 0.2740.0 0.4 0.4 9.06 9.02 − 0.44 47.46 47.40 − 0.13 2746.00 2745.00 − 0.04 14.73 14.72 − 0.07

TABLE III. Comparison of precision factors between analytical results (denoted by An) and FEA results for HCF.

Geometry C′y,M,hcf (N−1 × 10−3) C′

y,Fy,hcf (N−1 m × 10−5) C′x,Fx,hcf (N−1 m × 10−8)

a (mm) c (mm) t (mm) An FEA Err% An FEA Err% An FEA Err%

4.0 0.2 0.2 4.98 5.07 1.78 2.49 2.56 2.73 1.68 1.68 0.004.0 0.4 0.2 3.69 3.74 1.34 1.61 1.65 2.42 1.50 1.50 0.008.0 0.2 0.2 20.28 20.23 − 0.25 20.20 20.26 0.30 3.38 3.36 − 0.608.0 0.4 0.2 15.51 15.46 − 0.32 13.45 13.47 0.15 3.06 3.05 − 0.334.0 0.2 0.4 0.81 0.85 4.71 0.45 0.49 8.16 0.89 0.88 − 1.144.0 0.4 0.4 0.60 0.63 4.76 0.30 0.32 6.25 0.83 0.83 0.008.0 0.2 0.4 3.26 3.35 2.69 3.67 3.80 3.42 1.78 1.78 0.008.0 0.4 0.4 2.49 2.55 2.35 2.49 2.57 3.11 1.68 1.68 0.0020.0 0.4 0.4 15.90 16.01 0.69 39.57 39.97 1.00 4.23 4.23 0.0040.0 0.4 0.4 64.02 63.87 − 0.23 318.36 318.28 − 0.03 8.48 8.46 − 0.24

TABLE IV. Comparison of precision factors between analytical results (denoted by An) and FEA results for CFH.

Geometry C′y,M,hcf (N−1 × 10−3) C′

y,Fy,hcf (N−1 m × 10−5) C′x,Fx,hcf (N−1 m × 10−8)

a (mm) c (mm) t (mm) An FEA Err% An FEA Err% An FEA Err%

4.0 0.2 0.2 4.98 5.11 2.54 3.49 3.60 3.06 1.25 1.24 − 0.814.0 0.4 0.2 3.69 3.76 1.86 2.58 2.64 2.27 1.10 1.10 0.008.0 0.2 0.2 20.28 20.77 2.36 28.56 29.30 2.53 2.50 2.51 0.408.0 0.4 0.2 15.51 15.82 1.96 21.98 22.45 2.09 2.21 2.22 0.454.0 0.2 0.4 0.81 0.85 4.71 0.56 0.60 6.67 0.71 0.70 − 1.434.0 0.4 0.4 0.60 0.63 4.76 0.42 0.44 4.55 0.63 0.61 − 3.288.0 0.2 0.4 3.26 3.35 2.69 4.53 4.68 3.21 1.42 1.40 − 1.438.0 0.4 0.4 2.49 2.55 2.35 3.49 3.59 2.79 1.25 1.24 − 0.8120.0 0.4 0.4 15.90 16.04 0.87 56.04 56.68 1.13 3.13 3.12 − 0.3240.0 0.4 0.4 64.02 64.30 0.44 452.41 454.99 0.57 6.25 6.25 0.00

TABLE V. The key parameters of two random examples.

a (mm) c (mm) t (mm) B (mm) ka kb,b kb,cf kb,h

G1 4.0 0.2 0.2 5.0 1 1.3 1.1682 1.0037G2 4.0 0.4 0.4 5.0 1 1.3 1.1686 1.0808



TABLE VI. Comparison of the maximum stresses between analytical re-sults (denoted by An) and FEA results.

Type HCF CFH

Geometry G1 G2 G1 G2

σFx ,max (Mpa) An 0.50 0.25 0.50 0.25FEA 0.50 0.25 0.50 0.25Err% 0.00 0.00 0.00 0.00

σMz+Fy ,max (Mpa) An 211.30 56.89 311.90 77.13FEA 211.00 52.99 294.30 75.71Err% 0.14 7.36 5.98 1.88

small and is neglected. Two hybrid flexure hinge samples aremachined by the wire cut electrical discharge machine withlow wire traveling speed (LSWEDM) on a 15 mm thick alu-minum alloy sheet. The Young’s modulus of the material istheoretically estimated to be E = 71 GPa. The other geomet-ric parameters are listed in Table VII.

The experimental equipment comprises the BOSE Elec-tro Force 3300 test instrument which has force sensors(0.1 N resolution) and LVDT displacement sensors (1 μmresolution), fixtures and hinge samples. The experimentationincludes two setups. In the first test (Fig. 5(a)), the applyingforce is parallel to the axis of the hinge. As the force will notpass through it, it will generate an axial force and a moment.The instrument can measure the applying force and the corre-sponding displacement at the applying point which is known,and the displacement originated from the axial compressionis neglected, so Cθ ,M, can be obtained from the test.

In the second test (Fig. 5(b)), the applying force is verti-cal to the axis of the hinge. As the applying force does not passthrough the end of the hinge, the displacement at the applyingpoint contained the effect of the four compliance factors. Thesame as the first test, the applying force and the correspond-ing displacement at the applying point which is known can bemeasured by the instrument, Cθ ,M is obtained from the firsttest, and Cθ ,Fy is equal to Cy,M, so two measurements at twodifferent applying points are needed to calculate Cθ ,Fy, Cy,M,and Cy,Fy.

The analytical and experimental results are given inTable VII. For the Cy,Fy,HCF of the second configuration, theerrors between the two results are as high as 35.23%, thisis because the γ is equal to 0.0668 and is higher than 0.05,the displacement caused by the shearing effect cannot be ne-glected. The other results are within 13% error margin com-pared to the analytical results. The discussion about the errororigination can be found in Ref. 8.

FIG. 5. The experimental setups: (a) parallel mode and (b) vertical mode.

V. NUMERICAL RESULTS

This section is devoted to compare the hybrid flexurehinges with the five kinds of common notch flexure hingesin terms of compliance, precision, compliance precision ratio,and the maximum stress. Although the closed-form equationsof different flexure hinges are formulated based on differentcoordinates that are parallel to each other, it will not influ-ence the absolute results. In order to make the comparison,the equations are changed based on the same coordinate andare given in the Appendix.

The constant-valued material and geometric parametersare: E = 110 GPa, b = 10 mm, and t = 0. 5 mm.

A. Comparison of compliance

Equations for compliance factors of the five kinds ofcommon notch flexures are given in Refs. 7, 8, 10, 17, and18. Figs. 6(a)–6(d) illustrate the comparison of compliancefactors Cθ ,M, Cθ ,Fy and Cy,M, Cy,Fy, and Cx,Fx, respectively.From Fig. 6, the following conclusions can be drawn:

(1) For all compliance factors, HCF and CFH are larger thanthe hyperbolic and smaller than the corner-filleted. Allcompliance factors increase non-linearly when either β

increases or γ decreases. And γ has a greater influencethan β on them. It indicates that a plays a more importantrole than c in determining compliance.

(2) For Cθ ,M and Cx,Fx, the descending order is: corner-filleted, HCF is equal to CFH, elliptical, circular,parabolic, hyperbolic.

(3) For Cθ ,Fy and Cy,M, the descending order is: corner-filleted, CFH, elliptical, circular, parabolic, hyperbolic.HCF intersects with the elliptical, and it is larger thanthe circular, smaller than CFH.

TABLE VII. Comparison of compliance factors between analytical results (denoted by An) and experimental results (denoted by Ex).

Geometry Cθ ,M (N−1 m−1×10−2) Cy ,M = Cθ ,Fy (N−1 × 10−4) Cy ,Fy (N−1 m × 10−6)

Type a (mm) c (mm) t (mm) An Ex Err% An Ex Err% An Ex Err%

HCF 15.00 2.91 1.54 5.06 5.11 0.98 5.12 6.06 0.50 6.53 7.49 12.82CFH 15.00 2.91 1.54 5.06 4.81 − 5.20 10.05 11.00 8.64 21.31 23.13 7.87HCF 14.97 3.78 2.00 2.23 2.08 − 7.21 2.31 2.48 6.85 2.96 4.57 35.23CFH 14.97 3.78 2.00 2.23 2.09 − 6.70 4.36 4.46 2.24 9.10 10.32 11.82



FIG. 6. Comparison of the compliance factors versus non-dimensional parameters β and γ : (a) comparison of Cθ ,M, (b) comparison of Cθ ,Fy and Cy ,M,,(c) comparison of Cy ,Fy, and (d) comparison of Cx ,Fx. (Pink square, CFH; green square, HCF; blue square, circular; yellow square, corner-filleted; red square,elliptical; black square, hyperbolic; and open square, parabolic.)

(4) For Cy,Fy, the descending order is: corner-filleted, CFH,elliptical, circular, parabolic, hyperbolic. HCF intersectswith the circular and the parabolic, and it is larger thanthe hyperbolic, smaller than the elliptical.

B. Comparison of precision

Equations for precision factors of the five kinds of com-mon notch flexure hinges are given in Refs. 10 and 18. Forthe transversely symmetric flexure hinges, the precision fac-

tors C ′x,Fx and C ′

y,Fy can be evaluated by

C ′x,Fx = Cx,Fx/2, C ′

y,Fy = aC ′y,M + Cy,Fy/2 − a2Cθ,M/2.

Figs. 6(a)–6(c) illustrate the comparison of precision fac-tors C ′

y,M, C ′y,Fy, and C ′

x,Fx, respectively. From Fig. 7, thefollowing conclusions can be drawn:

(1) For all precision factors, HCF and CFH are largerthan the hyperbolic and smaller than the corner-filleted.All precision factors increase non-linearly when either

FIG. 7. Comparison of the precision factors versus non-dimensional parameters β and γ : (a) comparison of C′y,M , (b) comparison of C′

y,Fy , and (c) comparisonof C′

x,Fx . (Pink square, CFH; green square, HCF; blue square, circular; yellow square, corner-filleted; red square, elliptical; black square, hyperbolic; and opensquare, parabolic.)



FIG. 8. Comparison of the compliance precision ratios versus non-dimensional parameters β and γ : (a) comparison of I1, (b) comparison of I2, (c) comparisonof I3, (d) comparison of I4, and (e) comparison of I5. (Pink square, CFH; green square, HCF; blue square, circular; yellow square, corner-filleted; red square,elliptical; black square, hyperbolic; and open square, parabolic.)

β increases or γ decreases. And γ has a greater influ-ence than β on them. It indicates that a plays a moreimportant role than c in determining precision.

(2) For C ′y,M, the descending order is: corner-filleted,

HCF is equal to CFH, elliptical, circular, parabolic,hyperbolic.

(3) For C ′y,Fy, the descending order is corner-filleted, CFH,

HCF, elliptical, circular, parabolic, hyperbolic.(4) For C ′

x,Fx, the descending order is corner-filleted, HCF,elliptical, circular, parabolic, hyperbolic. CFH intersectswith the elliptical, and it is larger than the circular,smaller than HCF.

C. Comparison of compliance precision ratio

Unlike the traditional rigid hinges, the whole flexurehinge deforms under the loads at the free end. The ideal flex-ure hinges have the large compliance at the free end and theirrotation centers do not shift. The compliance precision ratiosare proposed to evaluate the ability of preserving the rotationcenters when the hinges have the same displacement at thefree end. They are defined as

I1 = Cθ,M

C ′y,M

, I2 = Cθ,Fy

C ′y,Fy

, I3 = Cy,Fy

C ′y,Fy

,

I4 = Cy,M

C ′y,M

, I5 = Cx,Fx

C ′x,Fx

.

The larger compliance precision ratios indicate that the flex-ure hinges have the better ability.

Figs. 8(a)–8(e) illustrate the comparison of complianceprecision ratios I1, I2, I3, I4, and I5, respectively. From Fig. 8,the following conclusions can be drawn:

(1) For the six kinds of flexure hinges, I1 and I2 decreasenon-linearly when either β increases or γ decreases, andγ has a greater influence than β on them; I3 and I4 de-crease non-linearly when β increases, and γ has slightlyinfluence on them; I5 keeps constant basically.

(2) For I1, the descending order is: hyperbolic, parabolic,circular, elliptical, HCF is equal to CFH, corner-filleted.When the hinges rotate the same angle under z-axis mo-ment, the hyperbolic performs best at preserving the ro-tation center along y-axis. HCF and CFH have the betterability than the corner-filleted, and HCF has the sameability as CFH.

(3) For I2, the descending order is: hyperbolic, parabolic,circular, elliptical, CFH, HCF, corner-filleted. When thehinges rotate the same angle under y-axis force, the hy-perbolic performs best at preserving the rotation cen-ter along y-axis. CFH and HCF have better abilitythan the corner-filleted, and CFH has better ability thanHCF.

(4) For I3, the descending order is: hyperbolic, parabolic,circular, elliptical, corner-filleted. CFH intersects withthe parabolic and the circular, HCF intersects with the



FIG. 9. Comparison of the maximum stress versus non-dimensional parameters β and γ between the hybrid with: (a) circular, (b) corner-filleted, (c) elliptical,(d) hyperbolic, (e) parabolic, and (f) whole. (Pink square, CFH; green square, HCF; blue square, circular; yellow square, corner-filleted; red square, elliptical;black square, hyperbolic; and open square, parabolic.)

corner-filleted. When they translate the same y-axisdisplacement under y-axis force, the hyperbolic per-forms best at preserving the rotation center along y- axis.When β > 0.3, CFH has better ability than the other flex-ures expect the hyperbolic. In most of comparison range,the HCF has the worst ability.

(5) For I4, the descending order is: hyperbolic, parabolic,circular, elliptical, corner-filleted. CFH intersects withthe parabolic, the circular, and the elliptical. HCF in-tersects with the corner-filleted. When they translate thesame y-axis displacement under z-axis moment, the hy-perbolic performs best at preserving the rotation centeralong y-axis. When β > 0.4, CFH has better ability thanthe other flexures expect the hyperbolic. In most of com-parison range, HCF has the worst ability.

(6) For I5, the five kinds of common notch flexure hingesare 2. CFH has the best ability of preserving the rotationcenter along x-axis when they translate the same x-axisdisplacement under x-axis force, and HCF has the worstability.

D. Comparison of the maximum stress

The equations for calculating the maximum stress of thehybrid flexure hinges are given in Sec. III C, and the methodcan be used to evaluate the other hinges’ maximum stress.This paper only gives their numerical results. The maximumstress is determined by the geometric parameters which canbe expressed by the non-dimensional parameters β and γ andthe load parameters which can be changed into displacementparameters y1 and θ1.

Provided that the displacements are y1 = 0.2 mmand θ1 = 0.015, Figs. 9(a)–9(e) illustrate the comparisonof the maximum stress between the hybrid flexure hingeswith the circular, corner-filleted, elliptical, hyperbolic, andparabolic flexure hinges, respectively. They are combinedin Fig. 9(f). From Fig. 9, the following conclusions can bedrawn:

(1) The maximum stress decreases non-linearly when eitherβ increases or γ decreases.

(2) For the maximum stress, the descending order is: hyper-bolic, parabolic, circular, elliptical, corner-filleted. HCFand CFH intersect with these flexure hinges, and theirsize relations change when either β or γ changes.

Provided that the geometric parameters are a = 4.0 mmand c = 0.2 mm, Figs. 10(a)–10(e) illustrate the compari-son of the maximum stress between the hybrid flexure hingeswith the circular, corner-filleted, elliptical, hyperbolic, andparabolic flexure hinges, respectively. They are combined inFig. 10(f). From Fig. 10, the following conclusions can bedrawn:

(1) Relations between the maximum stress and the displace-ment parameters are not simple increase or decrease.

(2) For the maximum stress, the descending order is: hy-perbolic, parabolic, circular, corner-filleted, elliptical.HCF and CFH intersect with these flexure hinges,and their size relations change when either y1 or θ1

changes.(3) In most range of the displacement parameters, CFH has

the largest maximum stress, the next is HCF, and theirnumerical results are very close.



FIG. 10. Comparison of the maximum stress versus displacement parameters y1 and θ1 between the hybrid with: (a) circular, (b) corner-filleted, (c) elliptical,(d) hyperbolic, (e) parabolic, and (f) whole. (Pink square, CFH; green square, HCF; blue square, circular; yellow square, corner-filleted; red square, elliptical;black square, hyperbolic; and open square, parabolic.)

VI. CONCLUSION

This paper designs the hybrid flexure hinge composedof half a corner-filleted flexure hinge and half a hyperbolicflexure hinge. Equations are formulated to characterize com-pliance, precision, and the maximum stress, and checkedagainst FEA and experimentation. The analytical model pre-dictions are within 8% error margins compared to FEA re-sults and are within 13% error margins compared to exper-imental results when β < 1 and γ < 0.05. The six kindsof flexure hinges are compared based on compliance, preci-sion, compliance precision ratios, and the maximum stress.The comparisons are useful to choose different kinds offlexure hinges to redesign the hybrid flexure hinges. Thecomparison results change non-linearly versus β and γ .The hybrid flexure hinges have larger compliance than thehyperbolic flexure hinges and higher precision than thecorner-filleted flexure hinges. The corner-filleted hyperbolicflexure hinges are better than the hyperbolic corner-filletedflexure hinges in term of compliance and compliance pre-cision ratios. A general method is proposed to evaluate themaximum stress. The maximum stress changes non-linearlyversus non-dimensional parameters β, γ and displacementparameters y1, θ1.

ACKNOWLEDGMENTS

This research was supported by the National ScienceFoundation of China (Grant No. 91223201); the United Fundof Natural Science Foundation of China and Guangdongprovince (Grant No. U0934004); Project GDUPS (2010);

and the Fundamental Research Funds for the Central Uni-versities (Grant No. 2012ZP0004). This support is greatlyacknowledged.

APPENDIX: CLOSED-FORM EQUATIONSFOR THE FLEXURE HINGES

1. Circular flexure hinges

Cθ,M,c = 12

Ebt2s2f1,c, (A1)

Cy,M,c = Cθ,Fy,c = aCθ,M,c, (A2)

Cy,Fy,c = 12

Eb

[sin2 (2θ ) f1,c + f2,c

], (A3)

Cx,Fx,c = 1

Ebf3,c, (A4)

C ′y,M,c = 3

2Ebtf4,c, (A5)

C ′y,Fy,c = aC ′

y,M,c + Cy,Fy,c/2 − a2Cθ,M,c/2, (A6)

C ′x,Fx,c = Cx,Fx,c/2, (A7)



where

f1,c = 8s4(2s + 1)

(4s + 1)2

tan θ

[(4s + 1) tan2 θ + 1]2

+ 4s3(6s2 + 3s + 1)

(4s + 1)2

tan θ

1 + (4s + 1) tan2 θ

+ 12s4(2s + 1)

(4s + 1)2.5atan

(√4s + 1 tan θ

),

f2,c = −2s2(2s + 1)

(4s + 1)

tan θ

[(4s + 1) tan2 θ + 1]2

+ s(2s2 + 5s + 1)

(4s + 1)

tan θ

1 + (4s + 1) tan2 θ

+ (2s + 1)(2s2 − 4s − 1)

2(4s + 1)1.5atan

(√4s + 1 tan θ

),

f3,c = 2(2s + 1)√4s + 1

atan(√

4s + 1 tan θ)

− 2θ,

f4,c = 2s − 1 + 2s + 1 − 4s cos θ

(2s + 1 + 2s cos θ)2,

s = β2 + γ 2

4βγ 2,

θ = 1

2atan

(2βγ

β2 + γ 2

).

2. Corner-filleted flexure hinges

Cθ,M,cf = 6

Ebt2

[2

γ− 2

β+ 2β2 + 4β + 3

(β + 1) (β + 2)2

+ 6 (β + 1)√

β (β + 2)2.5atan

√2

β+ 1

], (A8)

Cy,M,cf = Cθ,Fy,cf = aCθ,M,cf, (A9)

Cy,Fy,cf

= 3

Eb

{(β − γ )(4β2 − 2βγ + γ 2)

3β3γ 3+ 2β2 + 4β + 3

(β + 1)(β + 2)2γ 2

+ 2β3 + 6β2 + 3β − 5 + 2β (1−β) /γ

2 [β (β + 2)]2 + (β + 1)

× 3 + 2β−7β2 − 8β3 − 2β4 + 6β2/γ 2−6β/γ

[β (β + 2)]2.5

× atan

√2

β+ 1 + π

2

}, (A10)

Cx,Fx,cf= 1

Eb

[β−γ

βγ+ 2 (β+1)√

β (β+2)atan

√2

β+1−π

2

]. (A11)

The equations for evaluating precision factors of corner-filleted flexure hinges are the same as these for CFH providedf2,cfh = 0.

3. Elliptical flexure hinges

Cθ,M,e = 6

Ebt2

β

(β + 2)2 γ

[2β2 + 4β + 3

β + 1

+ 6 (β + 1)√β (β + 2)

atan

√2

β+ 1

], (A12)

Cy,M,e = Cθ,Fy,e = aCθ,M,e, (A13)

Cy,Fy,e = 3

Eb

β

(β+2)2 γ 3

[3+6β+11β2+8β3+2β4

2 (β+1)+π

8

+ 3+5β−5β2−15β3−10β4−10β2

√β (β+2)

atan

√2

β+1

],

(A14)

Cx,Fx,e = 1

Eb

1

γ

[2 (β + 1)√

2/β + 1atan

√2

β+ 1 − π

2

], (A15)

C ′y,M,e = 3

2Ebt

β

(β + 1) γ 2, (A16)

C ′y,Fy,e = aC ′

y,M,e + Cy,Fy,e/2 − a2Cθ,M,e/2, (A17)

C ′x,Fx,e = Cx,Fx,e/2. (A18)

4. Hyperbolic flexure hinges

Cθ,M,h = 12

Ebt2

β

(β + 1) γ, (A19)

Cy,M,h = Cθ,Fy,h = aCθ,M,h, (A20)

Cy,Fy,h = 3

2Eb

β

(β+1) (2β+1)2 γ 3

{2[1+β(4+3β−2β2)]

+ β2 (β+1)√

2β+1log

(β+1+√

2β+1

β+1−√2β+1

)},

(A21)

Cx,Fx,h = 1

Eb

β√2β + 1γ

log

(β

β + 1 − √2β + 1

). (A22)

The equations for evaluating precision factors of hyper-bolic flexure hinges are the same as these for HCF providedf2,hcf = 0.



5. Parabolic flexure hinges

Cθ,M,p = 3

2Ebt2

β(5β + 3) + 3√

β(β + 1)2acot√

β

γ (β + 1)2,

(A23)

Cy,M,p = Cθ,Fy,p = aCθ,M,p, (A24)

Cy,Fy,p = 3

8Eb

1

(β + 1)2γ 3[β(3 + 6β − β2)

+√

β(3 + 7β + 5β2 + β3)atan√

1/β], (A25)

Cx,Fx,p = 1

Eb

√β

γacot

√β, (A26)

C ′y,M,p = 3

4Ebt

β(2β + 1)

(β + 1)2γ 2, (A27)

C ′y,Fy,p = aC ′

y,M,p + Cy,Fy,p/2 − a2Cθ,M,p/2, (A28)

C ′x,Fx,p = Cx,Fx,p/2. (A29)

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Hybrid flexure hinges