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Analysis of a novel flexure hinge with three degrees of freedom

Feng-Zone Hsiao and Tai-Wu Lin

Citation: Review of Scientific Instruments 72, 1565 (2001); doi: 10.1063/1.1340024 View online: http://dx.doi.org/10.1063/1.1340024

View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/72/2?ver=pdfcov Published by the AIP Publishing

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Republic of China

共Received 9 August 2000; accepted for publication 11 November 2000兲

The flexure hinge is widely used as the mechanism for a high precision positioning stage with a micrometer or nanometer resolution. In this article we propose a novel flexure hinge with three degrees of freedom in which the motions are restricted in the same plane. An analysis model is developed to analyze the flexure hinge. The results obtained are consistent with those of the finite element method. A characteristic study using the proposed model shows that this flexure hinge has a feature of linear load displacement. In addition, the most effective way to change the stiffness of this flexure hinge is to modify the notch’s radius. A prototype of this novel flexure hinge has been manufactured and the measured characteristics prove the advantages of this proposed model. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1340024兴

I. INTRODUCTION

The need for micromotion stage has found wide applica-tion in fields like the lithography process of the semiconduc-tor industry, optical device tuning, ultraprecision machining, analysis of material surface structure, and the direct manipu-lation of DNA in genetic engineering. Nowadays, trends of miniaturization push the resolution of stage toward to the scale of nanometer or even subnanometer. Stages with high precision using sliding and rolling bearings of various types are inherently subjected to stick-slip and backlash problems for small displacement. The flexure hinge is a way to avoid such problems. It can be used to smoothly transfer the mo-tion provided by actuators with sufficient resolumo-tion.

In fine stage design the flexure hinges with one or two axes of motion are usually used in which the fine motion is achieved either directly using the flexure hinge to transfer the small displacement from actuators like piezoelectric ones1,2 or using the flexure hinge magnification mechanism to obtain a larger stroke.3,4 In those applications the flexure hinge is usually the notch type which is formed through the machin-ing of the high precision numerical controlled cuttmachin-ing ma-chine. Because of the advancement of manufacturing tech-nology, the profiles of notch-type flexure hinges can have right-circular, corner-filleted, and elliptical forms.5 The notch-type flexure hinge usually has one axis of motion. Flexure hinges with two axes of motion can be formed either from the universal joint with coincident axes or by combin-ing two notch-type hcombin-inges with perpendicular rotation axes together. For the study of flexure hinges Paros and Weisbord6 presented an analysis formulation for evaluating the compliance of the notch-type flexure hinges with right-circular profiles and then investigated the characteristics of such kinds of flexure hinges. The limitation used in their

analysis is that the minimum hinge thickness is much smaller than both the cutting notch radius and the hinge height. Smith, Chetwynd, and Bowen7 showed an approximate for-mula, which was derived from the finite element analysis result, to evaluate the compliance of right-circular profile hinges for cases of larger minimum hinge thickness. The analysis formulations for the flexure hinges with elliptical profiles were presented by Smith et al.8Xu and King5 com-pared the performance of different flexure hinge profiles by the finite element method共FEM兲 and they concluded that the flexure hinge with elliptical profiles has a lower maximum stress and thus a long fatigue life while the corner-filleted profiles offer the highest flexibility.

In this article we propose a novel monolithic flexure hinge with three degrees of freedom, where motions in the same plane with two translation and one rotation are al-lowed. An analytical model is proposed for the analysis and design of the flexure hinge and the finite element method is used to compare the results obtained from this model. The characteristics of the flexure hinge are then studied. Based on this study, a prototype is manufactured and its characteristics are measured to verify the results from the analysis model.

II. CONFIGURATION

Figure 1 shows the geometric configuration of the novel flexure hinge, where two notches with radius ␳1 and␳2 are

located, respectively, at the left and top region and two con-centric arcs form the boundary of the corner region. The width of the horizontal and vertical straight section can be different and the thickness of the flexure hinge is uniform. The central points of the arcs at the left end notch generally have an offset with the edge of the adjacent straight section. A similar condition holds for the notch at the top end. The span angle of the outer arc共radius R0兲 in the corner region is

90° and the two ends of this arc are tangent to the edges of

a兲Electronic mail: fzh@srrc.gov.tw

1565

0034-6748/2001/72(2)/1565/9/$18.00 © 2001 American Institute of Physics

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their adjacent straight sections, respectively. If the left-hand side of the flexure hinge is clamped and the forces and mo-ment are applied at the top end, the flexure hinge will move in horizontal, vertical, and rotational directions. That is, the flexure hinge has three degrees of freedom in the same mo-tion plane. A monolithic stage using this type of flexure hinge is shown in Fig. 2, where four flexure hinges placed at the corners of the stage are oriented to form a symmetric structure. Due to the unique feature of the flexure hinge, this stage has 3 motional degrees of freedom.

III. ANALYSIS MODEL

Figure 3 shows the simplified model of the flexure hinge. In this model the offset is set to zero and the notch radius at the left and top ends is set to the same value. The central point of the two concentric arcs at the corner is set as the intersection of the two extension lines from the edges of the horizontal and vertical straight sections. That is, the span angle of the inner arc is 270°. Though we simplify the geo-metric configuration, the following analysis procedure can still be applied to the general configuration.

In Fig. 3 the flexure hinge is divided into three segments A, B, and C. Segments A and C are treated as beam and the one-dimensional beam theory model is used to evaluate the deformation of each section. The left end is clamped and

thus both the translation and rotation degrees of freedom are constrained. At the top end position the external force FX,FY, M are applied here and this position is also regarded

as the output point. The deflection curve for the neutral axis of the flexure hinge at segments A and C can be expressed by the differential equation

EI共x兲d

2y

dx2⫽M共x兲, 共1兲

where the notations x and y are defined in Fig. 4, E is the Young’s modulus, I is the cross sectional area moment of inertia of the neutral axis, and M (x) is the bending moment at the given point x, which can be expressed as follows:

M共x兲⫽M⫹FY共H⫺0.5u兲⫺FX共L⫺0.5u兲⫺FYx. 共2兲

The assumptions used in Eq. 共1兲 are small deflection, small rotation and d2y /dx2Ⰶ1.9Since the thickness of the flexure hinge is uniform, I(x) at the straight section is a constant value. At the notch region I(x) can be expressed as

I共x兲⫽2bh3共x兲/3, 共3兲

where b is the thickness of the flexure hinge and h is the half width of the notch region. The half width h at a given posi-tion x is expressed as

h共x兲⫽0.5u⫺

␳2⫺共␳⫺x兲2. 共4兲

Thus the lateral deflection of the beam can be obtained from the integration of Eq.共1兲 and we have the following expres-sions:

FIG. 1. General geometry of the flexure hinge.

FIG. 2. A stage using the new flexure hinge.

FIG. 3. Simplified model.

FIG. 4. Coordinate definition.

1566 Rev. Sci. Instrum., Vol. 72, No. 2, February 2001 F.-Z. Hsiao and T. W. Lin

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⭐x⭐H⫺u⫺Ri y

共x兲⫽c1 u3xc2 2u3x 2⫹A 3, 共6兲 y共x兲⫽ c1 2u3x 2 c2 6u3x 3⫹A 3x⫹A4.

At notch adjacent to the corner region: H-u-Ri⭐x⭐H⫺u

(0⭐␪⭐␲/2兲 y

共␪兲⫽共c3⫺c2Ri兲RiT关u,Ri,␪兴⫹c2Ri 2 P关u,Ri,␪兴 ⫹c4RiF关u,Ri,␪兴⫹A5, 共7兲 y共␪兲⫽共c3⫺c2Ri兲Ri 2Q关u,R i,␪兴⫹c2Ri 3S关u,R i,␪兴 ⫹c4Ri 2G关u,R

i,␪兴⫹A5共Ri⫺Ricos␪兲⫹A6.

Here the integral functions T•兴, P关•兴, F关•兴, Q关•兴, S关•兴, G关•兴, coefficients c1⬃c4and A1⬃A6are defined in the Appendix.

In deriving Eqs. 共5兲 and 共7兲 we use the coordinate transfor-mation in which the polar coordinate with origin at the center point of the arc boundary is used 共see Fig. 4兲. Each term in Eqs. 共5兲–共7兲 can be expressed explicitly, though they are lengthy expressions. Thus we have the deformation formula expressed explicitly as a function of x at a given geometry and loading condition.

The axial displacement can be obtained by using the stress–strain relations for a beam subjected to the uniaxial load F: ␴x⫽FA, ␧x⫽ ␴x E , 共8兲 ⌬x

xdx.

Thus from Eqs. 共7兲 and 共8兲 the deformation at the interface between segment A and B 共point p兲 can be obtained as fol-lows:

XpFX

Eb

W关u,2␳,␲兴⫺␳W关u,2␳,0兴⫹RiW

u,Ri,

␲ 2

⫺RiW关u,Ri,0兴⫹ H⫺u⫺Ri⫺2␳ u

, Yp⫽共c3⫺c2Ri兲Ri 2Q

u,R i, ␲ 2

⫹c2Ri 3S

u,R i, ␲ 2

⫹c4Ri 2 G

u,Ri, ␲ 2

⫹A5Ri⫹A6, 共9兲 ␪p⫽共c3⫺c2Ri兲RiT

u,Ri, ␲ 2

⫹c2Ri 2P

u,R i, ␲ 2

⫹c4RiF

u,Ri, ␲ 2

⫹A5.

In segment B, the energy method is used to obtain the displacement formula. Assume the plane section remains plane after loading and the contribution to strain energy from radial stress ␴r and out of plane stress ␴z are negligible

compared with the contribution of circumferential stress␴. The energy function U* is then expressed as

U*⫽

冕 冕

␴␾

2

2E

␶2

2G

rddA. 共10兲

Since segment B is bounded by two concentric arcs it is a curved beam. The stress for a curved beam under the loading shown in Fig. 5 is expressed as10

␴␾⫽ M eA

rn r ⫺1

N A , 共11兲 ␶⫽1.2 V␾ 2 A2,

where A is the cross sectional area and e and rn are defined as below rnA

dA ru⫺Ri ln共u兲⫺ln共Ri兲 , 共12兲 e⫽R⫺rn

rdA A ⫺rn.

From the free body diagram the force equilibrium condition shown in Fig. 5 can be expressed as

M⫽Mq⫺FxqR•sin␾⫺Fy qR共1⫺cos␾兲,

N⫽⫺Fxqsin␾⫹Fy qcos␾, 共13兲

V⫽Fxqcos␾⫹Fy qsin␾, 0⭐␾⭐ ␲

2 FIG. 5. Loading at Sec. B.

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where Mq, Fxq, and Fy qare the loadings at point q, which are related to the external loads by the following expressions:

Mq⫽⫺M⫹FX共L⫺u兲⫹12FYRi,

Fxq⫽⫺FX, 共14兲

Fy q⫽FY.

Based on the assumption of small displacement and linear elastic material, the deflection at point q can be expressed from Castigliano’s theorem

H⫽ ⳵U* ⳵Fxq ⫽ 1 EA

1⫺ R e

冊冉

Mq⫹ ␲R 4 FXR 2FY

1.2RGA

⫺␲4FX⫹ 1 2FY

, ⌬V⫽ ⳵U* ⳵Fy qMq EA

R e

1⫺ ␲ 2

⫺1

FX 2EA

R2 e ⫺R⫹ 1.2RE G

EAFY

R 2 e

3␲ 4 ⫺2

⫹R

2⫺ ␲ 4

⫹ 1.2␲ 4 RE G

, 共15兲 ⌬␪⫽⳵ U* ⳵Mq ⫽ 1 EAe

␲ 2 Mq⫹R⫻FX⫹R⫻FY

1⫺ ␲ 2

冊冊

EA1 共FX⫹FY兲.

In addition to the deformation caused by the loading at point q, the displacement at q actually includes the finite

rotation at point p as Fig. 6 shows. Neglecting the second order term and treating segment B as a rigid body one, the corresponding displacement⌬x, ⌬y from the finite rotation is expressed as below

⌬x⫽x⫺x0⫽x0共cos␪p⫺1兲⫺y0sin␪p,

共16兲 ⌬y⫽y⫺y0⫽x0sin␪p⫹y0共cos␪p⫺1兲.

Thus the displacement at point q can be obtained as follows:

Xq⫽Xp⫺⌬Hu⫹Ri 2 ⫻共cos␪p⫺sin␪p⫺1兲, Yq⫽Yp⫹⌬Vu⫹Ri 2 ⫻共sin␪p⫹cos␪p⫺1兲, 共17兲 ␪q⫽␪p⫺⌬␪.

Deformation of segment C can be obtained through the same procedure as that in segment A. Continuous conditions in both displacement and slope are applied at the junction of segments B and C. With the local coordinate transformation

⫽⫺x and⫽y, the expressions for the lateral deflection at segment C can be obtained as follows:

At the notch region adjacent to the corner region

共␪兲⫽c5RiT关u,Ri,␪兴⫹2c4Ri 2 P关u,Ri,␪兴 ⫹c2 2 RiF关u,Ri,␪兴⫹A7, 共18兲 ␩共␪兲⫽c5Ri 2 Q关u,Ri,␪兴⫹2c4Ri 3 S关u,Ri,␪兴 ⫹c2 2 Ri 2G关u,R

i,␪兴⫹A7Ricos␪⫹A8. FIG. 6. Finite rotation.

FIG. 7. Meshes and deformation.

TABLE I. Simulation results using both the analysis model and the finite element method.

Case Dx共␮m兲 Dy共␮m兲 ␪z共mrad兲

Geometry Loading FEM Analysis Error共%兲 FEM Analysis Error共%兲 FEM Analysis Error共%兲 A A 259.67 265.02 ⫺2.06 ⫺350.72 ⫺364.04 ⫺3.80 ⫺10.39 ⫺10.02 3.55 B A 286.54 292.42 ⫺2.05 ⫺398.81 ⫺412.42 ⫺3.41 ⫺10.98 ⫺10.94 0.42 A B 217.10 212.39 2.17 ⫺249.96 ⫺260.15 ⫺4.08 ⫺9.84 ⫺9.16 6.88 A C ⫺51.08 ⫺49.59 2.90 49.85 54.10 ⫺8.52 2.91 3.04 ⫺4.69 A D 425.79 427.42 ⫺0.38 ⫺550.83 ⫺570.47 ⫺3.57 ⫺17.08 ⫺16.14 5.50 Remark:

Geometry A:␳⫽5, Ri⫽5, u⫽12, H⫽45, L⫽30, b⫽16 mm. Geometry B:␳⫽4.8, Ri⫽5.2, u⫽11.6, H⫽46, L⫽30, b⫽15 mm.

Loading A: Fy⫽⫺60 N, Fx⫽M⫽0; Loading C: Fx⫽Fy⫽0, M⫽300 N mm. Loading B: Fx⫽60 N, Fy⫽M⫽0; Loading D: Fx⫽60, Fy⫽⫺60 N, M⫽300 N mm.

1568 Rev. Sci. Instrum., Vol. 72, No. 2, February 2001 F.-Z. Hsiao and T. W. Lin

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FIG. 8. Deformation at different FXand FY. FIG. 9. Deformation at different moment M.

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FIG. 10. Effect of geometry dimension on flexure compliance; FX⫽M⫽0, FY⫽20 N.

1570 Rev. Sci. Instrum., Vol. 72, No. 2, February 2001 F.-Z. Hsiao and T. W. Lin

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At the top notch region

共␪兲⫽共c8⫹2c4␳兲␳T关u,2␳,␪兴⫺2c4␳2P关u,2␳,␪兴⫹A11, 共20兲

␩共␪兲⫽共c8⫹2c4␳兲␳2Q关u,2␳,␪兴⫺2c4␳3S关u,2␳,␪兴 ⫹A11共␳⫺␳cos␪兲⫹A12.

The axial displacement can be obtained by using Eq. 共8兲. Thus the formulations for the displacement of the flexure hinge subjected to external loads at output point r can be obtained from Eqs.共8兲 and 共20兲 and listed as below

Xr⫽⫺共c8⫹2c4␳兲␳2Q关u,2␳,␲兴 ⫹2c4␳3S关u,2␳,␲兴⫺2␳A11⫺A12, Yr⫽YqFY Eb

RiW

u,Ri, ␲ 2

⫺RiW关u,Ri,0兴 ⫹␳W关u,2␳,␲兴⫺␳W关u,2␳,0兴⫹L⫺Ri⫺u⫺2

u

,

共21兲

r⫽共c8⫹2c4␳兲␳T关u,2␳,␲兴⫺2c4␳2P关u,2␳,␲兴⫹A11.

Each term in Eq. 共21兲 can be expanded in terms of either loading or geometric dimension and the explicit form is ex-pressed in the Appendix.

IV. FINITE ELEMENT METHODFEM

The motion of the flexure hinge is simulated by using the commercial FEM package ANSYS. Figure 7 shows the meshes used to analyze the flexure hinge. An eight-node el-ement SOLID73, with 6 motional degrees of freedom 共three translation and three rotation兲 on each node, is used to define the geometry and simulate the deformation. Since there are singularity problems at the left and top ends of the flexure hinge, the range of the geometric model at both ends is ex-tended to avoid the mesh singularity. The clamped boundary condition is applied at the left-extended region. To reduce the effect of large stress gradient around the loaded region 共Saint-Venant’s principle兲, the loading is applied at the end of the top extended region. However, because of the exten-sion at the top end, application of the horizontal force will give an additional moment at the notch end. A correction moment should be added in FEM analysis for cases of hori-zontal loading. Table I shows the simulation result obtained from both the analysis model and the finite element method, where the additional moment caused from horizontal load is compensated in FEM analysis. The enlarged scale of the typical deformation for the flexure hinge at a given loading is displayed in Fig. 7.

In Table I five different cases formed from two geometry and three external forces are analyzed. From different cases of comparison, both methods give a consistent result. The

⫺2␳)⬍7.67. The tolerance is within 16% if 1⬍ ␳/(u ⫺2␳) ⬍4.5.

V. CHARACTERISTICS

Using the analysis model we study the characteristic of the monolithic flexure hinge. Figure 8 shows the flexure hinge behavior for cases of different external horizontal load-ing FX and vertical loading FY. The effect of adding exter-nal moment M is shown in Fig. 9. From Figs. 8 and 9 a linear relation between loading and displacement exits in all x, y , andz directions. Furthermore, if one applies only

either the horizontal force, the vertical force, or the pure moment at the top end, the flexure hinge will have the hori-zontal motion, the vertical motion, and the rotating motion simultaneously. That is, the flexure hinge has a coupled mo-tion behavior. Since a linear relamo-tionship holds between the loading and the motion it is easy to compensate this coupled motion problem. From the linear relationship and coupled behavior the motion at the end point of the flexure hinge subject to external loading can be expressed as follows:

Dx⫽ZxxFX⫹ZxyFY⫹ZxM ,

Dy⫽Zy xFX⫹Zy yFY⫹ZyM , 共22兲

z⫽Z␪xFX⫹Z␪yFY⫹Z␪␪M .

Here ZIJ represents the I-direction compliance due to J-direction loading.

Case studies for the effect of geometry dimension on the flexure characteristics for the same loading condition are shown in Fig. 10, where the constraint u⬎Ri, u⬎2␳, H

⬎u⫹2⫹Ri, and L⬎u⫹2⫹Ri are applied. Only six

curves are shown here because the compliances ZIJ and ZJI

FIG. 11. Performance measurement of the prototype.

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have similar value. From Fig. 10 all the compliances show strong sensitivity to the change of the notch’s cutting radius

, flexure thickness b, and width u, where the most sensitive factor is␳. Changing the corner radius Rihas not as apparent

an effect as ␳ does. Changing the magnitude of H has a strong effect on the compliances Zy y, Zy x, and Z␪y; similar

conditions hold for the case of changing L except that the directions x and y are interchanged.

A prototype is fabricated to check the accuracy of the analysis model. The loading in either horizontal or vertical direction is applied to test the characteristics of this flexure prototype. Figure 11 shows the measurement setup of this prototype. A force gauge is used to measure the magnitude of the applied loads. Deformation of the prototype is mea-sured through the 0.1␮m accuracy linear variable differen-tial transformer probe. Table II shows the measured result for different loading conditions; the results estimated from the analysis model are also shown here for comparison. From Table II the analysis model provides a good estimation in the case of horizontal loading and a less accurate estimation in the case of vertical loading. The measured result also proves the linear load-displacement feature of the flexure hinge.

ACKNOWLEDGMENTS

This study was supported by the National Science Coun-cil, Republic of China, under Contract No. NSC88-2218-E-213-003.

APPENDIX

T关u,d,␪兴⫽

sin␪

共u⫺d⫻sin␪兲3d␪,

P关u,d,␪兴⫽

sin␪⫻cos␪ 共u⫺d⫻sin␪兲3d␪,

Q关u,d,␪兴⫽

T关u,d,␪兴⫻sin␪d␪, S关u,d,␪兴⫽

P关u,d,␪兴⫻sin␪d␪,

F关u,d,␪兴⫽

sin␪

共u⫺d⫻sin␪兲2d␪,

G关u,d,␪兴⫽

F关u,d,␪兴sin␪d␪,

W关u,d,␪兴⫽

sin␪ 共u⫺d⫻sin␪兲d␪, c1M⫹FY共H⫺u/2兲⫺FX共L⫺u/2兲 Eb/12 , c2⫽ FY Eb/12, c3⫽ M⫹FY共Ri⫹u/2兲⫺FXL Eb/12 , c4⫽ FX Eb/6, c5⫽ M⫺FX共L⫺u兲⫺FYu/2 Eb/12 , c6⫽ M⫺FX共L⫺u⫺RiEbu3/12 , c7⫽ FX Ebu3/12, c8⫽ M⫺2FX␳ Eb/12 , A1⫽⫺共c1⫺c2␳兲␳T关u,2␳,0兴⫺c2␳2P关u,2␳,0兴, A2⫽⫺共c1⫺c2␳兲␳2Q关u,2␳,0兴⫺c2␳3S关u,2␳,0兴, A3⫽A12c1␳⫺2c2␳ 2 u3 ⫹共c1⫺c2␳兲␳T关u,2␳,␲兴 ⫹c2␳2P关u,2␳,␲兴,

A4⫽A2⫹2共A1⫺A3兲␳⫺

2c1␳2 u3 ⫹ 4c2␳3 3u3 ⫹共c1⫺c2␳兲␳ 2Q关u,2,兴⫹c 2␳ 3S关u,2,兴, A5⫽A3⫹ c1 u3共H⫺u⫺Ri兲⫺ c2 2u3共H⫺u⫺Ri兲 2 ⫺共c3⫺c2Ri兲RiT关u,Ri,0兴⫺c2Ri 2 P关u,Ri,0兴 ⫺c4RiF关u,Ri,0兴,

TABLE II. Characteristics of the flexure prototype.

Dx Dy Dx Dy

Fx Measure Analysis Error Measure Analysis Error FY Measure Analysis Error Measure Analysis Error

共Kgw兲 共␮m兲 共␮m兲 共%兲 共␮m兲 共␮m兲 共%兲 共Kgw兲 共␮m兲 共␮m兲 共%兲 共␮m兲 共␮m兲 共%兲 1 38.6 34.7 ⫺10.1 ⫺42.6 ⫺42.5 ⫺0.2 1 ⫺42.4 ⫺43.4 2.4 72.5 59.3 ⫺18.2 2 78.3 69.3 ⫺11.5 ⫺85.2 ⫺85.1 ⫺0.1 2 ⫺82.5 ⫺86.9 5.3 142.1 118.5 ⫺16.6 3 118.6 104.0 ⫺12.3 ⫺126.9 ⫺127.6 0.6 3 ⫺118.5 ⫺130.4 10.0 208.5 177.7 ⫺14.8 4 158.8 138.6 ⫺12.7 ⫺168.3 ⫺170.2 1.1 4 ⫺153.4 ⫺173.8 13.3 275.3 236.9 ⫺13.9 5 200.6 173.2 ⫺13.7 ⫺210.9 ⫺212.7 0.9 5 ⫺188.3 ⫺217.4 15.5 339.3 296.1 ⫺12.7 6 240.7 207.8 ⫺13.7 ⫺254 ⫺255.3 0.5 6 ⫺223.7 ⫺260.9 16.6 401.7 355.2 ⫺11.6 ⫺1 ⫺37.2 ⫺34.7 ⫺6.7 38.5 42.5 10.4 ⫺1 35.4 43.4 22.6 ⫺57.4 ⫺59.3 3.3 ⫺2 ⫺74.1 ⫺69.4 ⫺6.3 79.7 85.0 6.6 ⫺2 73.1 86.8 18.7 ⫺115.2 ⫺118.6 3.0 ⫺3 ⫺111 ⫺104.1 ⫺6.2 122 127.5 4.5 ⫺3 111.6 130.1 16.6 ⫺179.9 ⫺177.9 ⫺1.1 ⫺4 ⫺146 ⫺138.8 ⫺4.9 161 170.0 5.6 ⫺4 149.5 173.5 16.1 ⫺238.1 ⫺237.3 ⫺0.3 ⫺5 ⫺177.4 ⫺173.5 ⫺2.2 195.3 212.4 8.8 ⫺5 190.2 216.8 14.0 ⫺298.5 ⫺296.6 ⫺0.6 ⫺6 ⫺204 ⫺208.2 2.1 222.8 254.9 14.4 ⫺6 223.6 260.1 16.3 ⫺360.8 ⫺356.0 ⫺1.3

1572 Rev. Sci. Instrum., Vol. 72, No. 2, February 2001 F.-Z. Hsiao and T. W. Lin

(10)

⫺共c3⫺c2Ri兲RiQ关u,Ri,0兴⫺c2RiS关u,Ri,0兴 ⫺c4Ri 2 G关u,Ri,0兴, A7⫽␪q⫺c5RiT

u,Ri, ␲ 2

⫺2c4Ri 2 P

u,Ri, ␲ 2

c2 2 RiF

u,Ri, ␲ 2

, A8⫽⫺Xq⫺c5Ri 2Q

u,R i, ␲ 2

⫺2c4Ri 3S

u,R i, ␲ 2

c22Ri 2 G

u,Ri, ␲ 2

, A9⫽A7⫹c5RiT关u,Ri,0兴⫹2c4Ri 2P关u,R i,0兴 ⫹c2 2 RiF关u,Ri,0兴,

A10⫽A8⫹A7Ri⫹c5Ri2Q关u,Ri,0兴⫹2c4Ri3S关u,Ri,0兴 ⫹c2

2 Ri

2

G关u,Ri,0兴,

A12⫽A10⫹A9共L⫺u⫺Ri⫺2␳兲⫹

c6 2 共L⫺u⫺Ri⫺2␳兲 2 ⫹c7 6 共L⫺u⫺Ri⫺2␳兲 3 ⫺共c8⫹2c4␳兲␳2Q关u,2␳,0兴⫹2c4␳3S关u,2␳,0兴.

1M. Taniguchi, M. Ikeda, A. Inagaki, and R. Funatsu, Int. J. Jpn. Soc.

Precis. Eng. 26, 35共1992兲.

2M. R. Howells, Opt. Eng.共Bellingham兲 34, 410 共1995兲. 3

F. E. Scire and E. C. Teague, Rev. Sci. Instrum. 49, 1735共1978兲.

4S. H. Chang and B. C. Du, Rev. Sci. Instrum. 69, 1785共1998兲. 5W. Xu and T. King, Precis. Eng. 19, 4共1997兲.

6J. M. Paros and L. Weisbord, Mach. Des. 37, 151共1965兲. 7

S. T. Smith, D. G. Chetwynd, and D. K. Bowen, J. Phys. E 20, 977

共1987兲.

8S. T. Smith, V. G. Badami, J. S. Dale, and Y. Xu, Rev. Sci. Instrum. 68,

1474共1997兲.

9J. M. Gere and S. P. Timoshenko, Mechanics of Materials共PWS

Engi-neering, Boston, 1984兲, p. 354.

10R. D. Cook and W. C. Young, Advanced Mechanics of Materials

共Mac-millan, New York, 1985兲, p. 416.

數據

FIG. 1. General geometry of the flexure hinge.
TABLE I. Simulation results using both the analysis model and the finite element method.
FIG. 8. Deformation at different F X and F Y . FIG. 9. Deformation at different moment M.
FIG. 10. Effect of geometry dimension on flexure compliance; F X ⫽M⫽0, F Y ⫽20 N.
+2

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