Characteristics of Beam-Based Flexure Modules...cal understanding of distributed-compliance ﬂexure mechanism behavior, and thus provide useful qualitative and quantitative de-sign

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Shorya Awtar1e-mail: [email protected]

Alexander H. Slocume-mail: [email protected]

Precision Engineering Research Group,Massachusetts Institute of Technology,

Cambridge, MA 01239

Edip SevincerOmega Advanced Solutions Inc.,

Troy, NY 12108e-mail: [email protected]

Characteristics of Beam-BasedFlexure ModulesThe beam flexure is an important constraint element in flexure mechanism design. Non-linearities arising from the force equilibrium conditions in a beam significantly affect itsproperties as a constraint element. Consequently, beam-based flexure mechanisms sufferfrom performance tradeoffs in terms of motion range, accuracy and stiffness, while ben-efiting from elastic averaging. This paper presents simple yet accurate approximationsthat capture the effects of load-stiffening and elastokinematic nonlinearities in beams. Ageneral analytical framework is developed that enables a designer to parametricallypredict the performance characteristics such as mobility, over-constraint, stiffness varia-tion, and error motions, of beam-based flexure mechanisms without resorting to tediousnumerical or computational methods. To illustrate their effectiveness, these approxima-tions and analysis approach are used in deriving the force–displacement relationships ofseveral important beam-based flexure constraint modules, and the results are validatedusing finite element analysis. Effects of variations in shape and geometry are also ana-lytically quantified. �DOI: 10.1115/1.2717231�

Keywords: beam flexure, parallelogram flexure, tilted-beam flexure, double parallelo-gram flexure, double tilted-beam flexure, nonlinear beam analysis, elastokinematic effect,flexure design tradeoffs

Introduction and BackgroundFrom the perspective of precision machine design �1–4�, flex-

res are essentially constraint elements that utilize material elas-icity to allow small yet frictionless motions. The objective of andeal constraint element is to provide infinite stiffness and zeroisplacements along its degrees of constraint �DOC�, and allownfinite motion and zero stiffness along its degrees of freedomDOF�. Clearly, flexures deviate from ideal constraints in severalays, the primary of which is limited motion along the DOF.iven a maximum allowable stress, this range of motion can be

mproved by choosing a distributed-compliance topology over itsumped-compliance counterpart, as illustrated in Fig. 1. However,istribution of compliance in a flexure mechanism gives rise toonlinear elastokinematic effects, which result in two very impor-ant attributes. First, the error motions and stiffness values alonghe DOC deteriorate with increasing range of motion along DOF,hich leads to fundamental performance tradeoffs in flexures �5�.econd, distributed compliance enables elastic averaging and al-

ows non-exact constraint designs that are otherwise unrealizable.or example, while the lumped-compliance multi-parallelogramexure in Fig. 1�a� is prone to over-constraint in the presence of

ypical manufacturing and assembly errors, its distributed-ompliance version of Fig. 1�b� is relatively more tolerant. Elasticveraging greatly opens up the design space for flexure mecha-isms by allowing special geometries and symmetric layouts thatffer performance benefits �5�. Thus, distributed compliance inexure mechanisms results in desirable as well as undesirablettributes, which coexist due to a common root cause. Because ofheir influence on constraint behavior, it is important to under-tand and characterize these attributes while pursuing systematiconstraint-based flexure mechanism design.

The uniform-thickness beam flexure is a classic example of a

1Corresponding author.Contributed by the Mechanisms and Robotics Committee of ASME for publica-

ion in the JOURNAL OF MECHANICAL DESIGN. Manuscript received December 29, 2005;nal manuscript received May 29, 2006. Review conducted by Larry L. Howell.aper presented at the ASME 2005 Design Engineering Technical Conferences andomputers and Information in Engineering Conference �DETC2005�, Long Beach,

A, USA, September 24–28, 2005.

ournal of Mechanical Design Copyright © 20

ded 10 Nov 2010 to 150.135.248.238. Redistribution subject to ASM

distributed-compliance topology. With increasing displacements,nonlinearities in force–displacement relationships of a beam flex-ure can arise from one of three sources—material constitutiveproperties, geometric compatibility, and force equilibrium condi-tions. While the beam material is typically linear-elastic, the geo-metric compatibility condition between the beam’s curvature anddisplacement is an important source of nonlinearity. In general,this nonlinearity becomes significant for transverse displacementsof the order of one-tenth the beam length, and has been thor-oughly analyzed in the literature using analytical and numericalmethods �6–8�. Simple and accurate parametric approximationsbased on the pseudo-rigid body method also capture this nonlin-earity and have proven to be important tools in the design andanalysis of mechanisms with large displacements �9�. However,the nonlinearity resulting from the force equilibrium conditionscan become significant for transverse displacements as small asthe beam thickness. Since this nonlinearity captures load-stiffening and elastokinematic effects, it is indispensable in deter-mining the influence of loads and displacements on the beam’sconstraint behavior. These effects truly reveal the design tradeoffsin beam-based flexure mechanisms and influence all the key per-formance characteristics such as mobility, over-constraint, stiff-ness variation, and error motions. Although both these nonlineareffects have been appropriately modeled in the prior literature, thepresented analyses are either unsuited for quick design calcula-tions �10�, case-specific �11�, or require numerical/graphical solu-tion methods �12,13�. While pseudo-rigid body models captureload-stiffening, their inherent lumped-compliance assumption pre-cludes elastokinematic effects.

Since we are interested in transverse displacements that are anorder of magnitude less than the beam length but generally greaterthan the beam nominal thickness, this third source of nonlinearityis the focus of discussion in this paper. We propose the use ofsimple polynomial approximations in place of transcendentalfunctions arising from the deformed-state force equilibrium con-dition in beams. These approximations are shown to yield veryaccurate closed-form force–displacement relationships of thebeam flexure and other beam-based flexure modules, and helpquantify the associated performance characteristics and tradeoffs.

Furthermore, these closed-form parametric results enable a physi-

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al understanding of distributed-compliance flexure mechanismehavior, and thus provide useful qualitative and quantitative de-ign insights.

The term flexure module is used in this paper to refer to aexure building-block that serves as a constraint element in aotentially larger and more complex flexure mechanism. Flexureodules are the simplest examples of flexure mechanisms.

Characteristics of Flexure ModulesThis section presents some key performance characteristics that

apture the constraint behavior of flexure modules, and will beeferred to in the rest of this paper. These include the concepts ofobility �DOF/DOC�, stiffness variations, error motions, and cen-

er of stiffness.For a flexure module, one may intuitively or analytically assign

tiff directions to be DOC and compliant directions to DOF. Forxample, in the beam flexure illustrated in Fig. 2, the transverseisplacements of the beam tip, y and �, obviously constitute thewo degrees of freedom, whereas axial direction x displacement isdegree of constraint. However, for flexure mechanisms compris-

ng of several distributed-compliance elements, this simplistic in-erpretation of mobility needs some careful refinement.

In a given module or mechanism, one may identify all possibleocations or nodes, finite in number, where forces may be allowed.or planer mechanisms, each node is associated with three gener-lized forces, or allowable forces, and three displacement coordi-ates. Under any given set of normalized allowable forces of unitrder magnitude, some of these displacement coordinates will as-ume relatively large values as compared to others. The maximumumber of displacement coordinates that can be made indepen-ently large using any combination of the allowable forces, eachf unit order magnitude, quantifies the degrees of freedom of theexure mechanism. The remaining displacement coordinates con-

ribute to the degrees of constraint. For obvious reasons, normal-zed stiffness or compliance not only plays an important role inhe determination of DOF and DOC, but also provides a measureor their quality. In general, it is always desirable to maximizetiffness along the DOC and compliance along the DOF. Sinceoad-stiffening and elastokinematic effects result in stiffness varia-

ig. 1 „a… Lumped-compliance and „b… distributed-complianceulti-parallelogram mechanisms

Fig. 2 Generalized beam flexure

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tions, these nonlinearities have to be included for an accurateprediction of the number and quality of DOF and DOC in adistributed-compliance flexure mechanism. A situation where thestiffness along a DOF increases significantly with increasing dis-placements represents a condition of over-constraint. These con-siderations are not always captured by the traditional Gruebler’scriterion.

A second measure of the quality of a DOF or DOC is errormotion, which affects the motion accuracy of a given flexure mod-ule or mechanism. While it is common to treat any undesireddisplacement in a flexure mechanism as a parasitic error, we pro-pose a more specific description of error motions. The desiredmotion, or primary motion, is one that occurs in the direction ofan applied generalized force along a DOF. Resulting motions inany other direction are deemed as undesired or error motions. In apurely linear elastic formulation, this has a straightforward impli-cation. If the compliance, or alternatively stiffness, matrix relatingthe allowable forces to the displacement coordinates has any off-diagonal terms, the corresponding forces will generate undesiredmotions. However, a nonlinear formulation reveals load-stiffening, kinematic, and elastokinematic effects in the force–displacement relationships, which can lead to additional undesiredmotions. Since these terms are revealed in a mechanism’s de-formed configuration, an accurate characterization of undesiredmotions should be performed by first applying unit order allow-able forces in order to nominally deform the mechanism. Fromthis deformed configuration, only the generalized force along thedirection of primary motion is varied while others are kept con-stant. The resulting changes in displacements along all other dis-placement coordinates provide a true measure of undesired mo-tions in a mechanism. The undesired motions along the otherdegrees of freedom are defined here as cross-axis error motion,while those along the degrees of constraint are referred to as para-sitic error motions. Each of these error motions can be explicitlyforce dependent, purely kinematic, elastokinematic, or any com-bination thereof. This shall be illustrated in the following sectionsby means of specific flexure module examples.

Such a characterization is important because it reveals the con-stituents of a given error motion. Any error component that isexplicitly force dependent, based on either elastic or load-stiffening effects, may be eliminated by an appropriate combina-tion of allowable forces. In some cases, this may be accomplishedby simply varying the location of an applied force to provide anadditional moment. This observation leads to the concept of cen-ter of stiffness �COS�. If the rotation of a certain stage in a flexuremechanism is undesired, then the particular location of an appliedforce that results in zero stage rotation is defined as the COS ofthe mechanism with respect to the given stage and applied force.Obviously, the COS may shift with loading and deformation. Ki-nematic terms that contribute to error motions are dependent onother displacements and, in general, may not be eliminated by anycombination of allowable forces without over-constraining themechanism. Optimizing the shape of the constituent distributed-compliance elements can only change the magnitude of these ki-nematic terms while a modification of the mechanism topology isrequired to entirely eliminate them. Elastokinematic contributionsto error motions, on the other hand, can be altered by either ofthese two schemes—by appropriately selecting the allowableforces, as well as by making geometric changes. This informationprovides insight regarding the kind of optimization and topologi-cal redesign that may be needed to improve the motion accuracyin a flexure mechanism.

3 Beam FlexureFigure 2 illustrates a varying-thickness beam with generalized

end forces and end displacements in a deformed configuration.Displacements and lengths are normalized by the overall beamlength L, forces by E�Izz /L

2, and moments by E�Izz /L. The two

equal end-segments have a uniform thickness t, and the middle

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ection is thick enough to be considered rigid. The symbol E� issed to denote Young’s modulus for a state of plane stress, andlate modulus for plane strain. All nondimensional quantities areepresented by lower case letters throughout this discussion.

The case of a simple beam �ao=1/2� is first considered. Forransverse displacements, y and �, of the order of 0.1 or smaller,eam curvature may be linearized by assuming small slopes.ince the force equilibrium condition is applied in the deformedonfiguration of the beam, the axial force p contributes to theending moments. Solving Euler’s equation for the simple beamields the following well-known results �3,10�, where the normal-zed tensile axial force p�k2

f =k3 sinh k

k sinh k − 2 cosh k + 2y +

k2�1 − cosh k�k sinh k − 2 cosh k + 2

�

�1�

m =k2�1 − cosh k�

k sinh k − 2 cosh k + 2y +

k2 cosh k − k sinh k

k sinh k − 2 cosh k + 2�

y = � k − tanh kk3

�f + � cosh k − 1k2 cosh k

�m�2�

� = � cosh k − 1k2 cosh k

�f + � tanh kk

�mx = xe + xk =

p

d− �y ��r11 r12

r21 r22��y

�� 3�

here

d = 12/t2

r11 =k2�cosh2 k + cosh k − 2� − 3k sinh k�cosh k − 1�

2�k sinh k − 2 cosh k + 2�2

r12 = r21 = −k2�cosh k − 1� + k sinh k�cosh k − 1� − 4�cosh k − 1�2

4�k sinh k − 2 cosh k + 2�2

r22 =− k3 + k2 sinh k�cosh k + 2� − 2k�2 cosh2 k − cosh k − 1�

4k�k sinh k − 2 cosh k + 2�2

+2 sinh k�cosh k − 1�

4k�k sinh k − 2 cosh k + 2�2

n the presence of a compressive axial force, expressions analo-ous to Eq. �1�–�3� may be obtained in terms of trigonometricunctions instead of hyperbolic functions. The axial displacementis comprised of two components—a purely elastic component xe

hat results due to the elastic stretching of the beam, and a kine-atic component xk that results from the conservation of beam

rc-length. The kinematic component of the axial displacementay be alternatively stated in terms of the transverse loads f and, instead of displacements y and �. However, it should be rec-

gnized that this component fundamentally arises from a condi-ion of geometric constraint that requires the beam arc-length totay constant as it takes a new shape.

Based on the approximations made until this stage, the aboveesults should be accurate to within a few percent of the trueehavior of an ideal beam. Although the dependence of transversetiffness on axial force, and axial stiffness on transverse displace-ent is evident in expressions �1�–�3�, their transcendental nature

ffers little parametric insight to a designer. We therefore proposeimplifications that are based on an observation that the transverseompliance terms may be accurately approximated by inverse lin-

ar or inverse quadratic expressions

ournal of Mechanical Design


C = �k − tanh k

k3� � cosh k − 1

k2 cosh k�

� cosh k − 1k2 cosh k

� � tanh kk

�

�

1

3�1 + 25

p�1

2�1 + 512

p�1

2�1 + 512

p��1 + 1

10p�

�1 + 1330

p +1

96p2� �4�

A comparison of the series expansions of the actual hyperbolicfunctions and their respective approximations, provided in the Ap-pendix, reveals that the series coefficients remain close even forhigher order terms. The actual and approximate functions for thethree compliance terms are plotted in Fig. 3 for a range of p=−2.5 to 10. Since p=−�2 /4�−2.47 corresponds to the firstfixed-free beam buckling mode �f=m=0�, all the complianceterms exhibit a singularity at this value of p. The approximatefunctions accurately capture this singularity, but only for the firstfixed–free beam buckling mode.

The fact that some compliance terms are well approximated byinverse linear functions of the axial force p indicates that thestiffness terms may be approximated simply by linear functions ofp. These linear approximations are easily obtained from the seriesexpansion of the hyperbolic functions in expression �1�, whichshow that the higher order terms are small enough to be neglectedfor values of p as large as ±10

K = k3 sinh k

k sinh k − 2 cosh k + 2



k2 cosh k − k sinh k

k sinh k − 2 cosh k + 2

� 12�1 +1

10p� − 6�1 + 1

60p�

− 6�1 + 1 p� 4�1 + 1 p� �5�

Fig. 3 Normalized compliance terms: actual „solid lines… andapproximate „dashed lines…

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imilarly, the following linear approximations can be made for theyperbolic functions in the geometric constraint relation �3�

xk � − �y ��3

5�1 − p

420� − 1

20�1 − p

70�

−1

20�1 − p

70� 1

15�1 − 11p

420� �y� � �6�

he maximum error in approximation in all of the above cases isess than 3% for p within ±10. This excellent mathematical matchetween the transcendental functions and their respective approxi-ations has far-reaching consequences in terms of revealing the

ey characteristics of a beam flexure. Although shown for theensile axial force case, all of the above approximations hold validor the compressive case as well. Summarizing the results so far ingeneral format

� fm� = �a c

c b��y

�� + p�e h

h g��y

�� 7�

x =1

dp + �y �� i k

k j��y

�� + p�y ��r q

q s��y

�� 8�

he coefficients a, b, c, e, g, h, i, j, k, q, r, and s are all nondi-ensional numbers that are characteristic of the beam shape, and

ssume the values for a simple beam with uniform thickness ashown in Table 1.

In general, these coefficients are functionals of the beam’s spa-ial thickness function t�X�. A quantification of these coefficientsn terms of the beam shape provides the basis for a sensitivitynalysis and shape optimization. The particular shape variationllustrated in Fig. 2 is discussed in further detail later in this sec-ion.

The maximum estimated error in the analysis so far is less than% for transverse displacements within ±0.1 and axial forceithin ±10. Unlike the original results �1� and �3�, expressions �7�

nd �8� express the role of the normalized axial force p in theorce–displacement relations of the beam in a simple matrix for-at. The two components of transverse stiffness, commonly re-

erred to as the elastic stiffness matrix and the geometric stiffnessatrix, are clearly quantified in expression �7�. This characterizes

he load-stiffening effect in a beam, and consequently the loss inhe quality of DOF in the presence of a tensile axial force. Of

able 1 Stiffness, kinematic, and elastokinematic coefficientsor a simple beam

12 e 1.2 i −0.6 r 1/7004 g 2/15 j −1/15 s 11/6300−6 h −0.1 k 1/20 q −1/1400

articular interest is the change in axial stiffness in the presence of

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a transverse displacement, as evident in expression �8�. It may beseen that the kinematic component defined earlier may be furtherseparated into a purely kinematic component, and an elastokine-matic component. The latter is named so because of its depen-dence on the axial force as well as the kinematic requirement ofconstant beam arc-length. This component essentially captures theeffect of the change in the beam’s deformed shape due to the axialforce, for given transverse displacements. Since this term contrib-utes additional compliance along the axial direction, it compro-mises the quality of the x DOC. For an applied force f, the dis-placement x and rotation � are undesired. Since these correspondto a DOC and DOF, respectively, any x displacement is a parasiticerror motion and � rotation is a cross-axis error motion. While thelatter is explicitly load dependent and can be eliminated by anappropriate combination of the transverse loads f and m, theformer has kinematic as well as elastokinematic components andtherefore cannot be entirely eliminated.

These observations are significant because they parametricallyillustrate the role of the force along a DOC on the quality of DOFdue to load-stiffening. Furthermore, the range of motion alongDOF is limited to ensure an acceptably small stiffness reductionand error motion along the DOC due to elastokinematic effects.These are the classic tradeoffs in flexure performance, and are notcaptured in a traditional linear analysis.

The accuracy of the above derivations may be verified usingknown cases of beam buckling, as long as the magnitude of thecorresponding normalized buckling loads is less than 10. Bucklinglimit corresponds to the compressive load at which the transversestiffness of a beam becomes zero. Using this definition and ex-pression �4� for a fixed–free beam, one can estimate the bucklingload to be pcrit=−2.5, which is less than 1.3% off from the clas-sical beam buckling prediction of pcrit=−�

2 /4. Similarly, thebuckling load for a beam with zero end slopes is predicted to be−10 using expression �5�, as compared to −�2 derived using theclassical theory. Many other nontrivial results may be easily de-rived from the proposed simplified expressions, using appropriateboundary conditions. For example, in the presence of an axial loadp, the ratio between m and f required to ensure zero beam-endrotation is given by −�1+ 160p� /2�1+

110p�, which determines the

COS of the beam with respect to the beam-end and force f. Anexample that illustrates a design tradeoff is that of a clamped–clamped beam of actual length 2L transversely loaded in the cen-ter. While symmetry ensures perfect straight-line motion along they DOF, and no error motions along x or �, it also results in anonlinear stiffening effect along the DOF given by f=2�a− �ied / �1+rdy2��y2y, which significantly limits the range of al-lowable motion. This nonlinear stiffness behavior is derived in afew steps from expressions �7� and �8�, whereas the conventionalmethods can be considerably more time consuming.

Next, we consider the specific generalization of beam geometryshown in Fig. 2. The nonlinear force–displacement relationshipsfor this beam geometry may be obtained mathematically by treat-ing the two end-segments as simple beams, and using the prior

results of this section

� fm� � 1

�3 − 6ao + 4ao2�ao

� 6 − 3− 3 3 − 3ao + 2ao

2 ��y� �+

p

5�3 − 6ao + 4ao2�2 3�15 − 50ao + 60ao

2 − 24ao3� − ao�15 − 60ao + 84ao

2 − 40ao3�

− ao�15 − 60ao + 84ao2 − 40ao

3� ao�15 − 60ao + 92ao2 − 60ao3 + 403 ao4� �y� � �9�



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x = 2ao� t212�p + �y ��10�3 − 6ao + 4ao2�2− 3�15 − 50ao + 60ao

2 − 24ao3� ao�15 − 60ao + 84ao

2 − 40ao3�

ao�15 − 60ao + 84ao2 − 40ao

3� − ao�15 − 60ao + 92ao2 − 60ao3 + 403 ao4� �y� �+ p

ao3�y ��

175�3 − 6ao + 4ao2�3 2�105 − 630ao + 1440ao

2 − 1480ao3 + 576ao

4� − �105 − 630ao + 1440ao2 − 1480ao

3 + 576ao4�

− �105 − 630ao + 1440ao2 − 1480ao

3 + 576ao4� �105 − 630ao + 1560ao2 − 2000ao3 + 1408ao4 − 560ao5 + 11209 ao6� �y� �

�10�

t is significant to note that these force–displacement relations aref the same matrix equation format as expressions �7� and �8�.bviously, the transverse direction elastic and geometric stiffness

oefficients are now functions of a0, a nondimensional number. Asxpected, for a fixed beam thickness, reducing the length of thend-segments increases the elastic stiffness, while reducing theeometric stiffness. In the limiting case of a0→0, which corre-ponds to a lumped-compliance topology, the elastic stiffness be-omes infinitely large, and the first geometric stiffness coefficienteduces to 1 while the others vanish. Similarly, the axial directionlastic stiffness as well as the kinematic and elastokinematic co-fficients are also functions of beam segment length a0. In the

Fig. 4 Transverse elastic stiffness coefficients

Fig. 5 Transverse geometric stiffness coefficients



limiting case of a0→0, the elastic stiffness becomes infinitelylarge, the first kinematic coefficient approaches 0.5, and the re-maining kinematic coefficients along with all the elastokinematiccoefficients vanish. This simply reaffirms the prior observationthat elastokinematic effects are a consequence of distributed com-pliance. For the other extreme of a0→0.5, which corresponds to asimple beam, it may be verified that the above transverse elasticand geometric stiffness coefficients, and the axial elastic stiffness,kinematic, and elastokinematic coefficients take the numericalvalues listed earlier in Table 1.

Observations on these two limiting cases agree with the com-mon knowledge and physical understanding of distributed-compliance and lumped-compliance topologies. Significantly, ex-

Fig. 6 Axial kinematic coefficients

Fig. 7 Axial elastokinematic coefficients

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ressions �9� and �10� provide an analytical comparison of thesewo limiting case topologies, and those inbetween. To graphicallyllustrate the effect of distribution of compliance, the stiffness andonstraint coefficients are plotted as functions of length parameter0 in Figs. 4–8. Assuming a certain beam thickness limited by theaximum allowable axial stress, one topology extreme, a0=0.5,

rovides the lowest elastic stiffness along the y DOF �Fig. 4�, andherefore is best suited to maximize the primary motion. However,his is prone to moderately higher load-stiffening behavior, as in-icated by the geometric stiffness coefficients in Fig. 5, and para-itic errors along the x DOC, as indicated by the kinematic coef-cients in Fig. 6. Most importantly, this topology results in theighest elastokinematic coefficients plotted in Fig. 7, which com-romise the stiffness and error motions along the x DOC, but athe same time make this topology best suited for approximate-onstraint design. Approaching the other topology extreme, a0

0, there are gains on several fronts. Elastic stiffness along the xOC improves �Fig. 8�, load-stiffening effects decrease �Fig. 5�,inematic effects diminish or vanish �Fig. 6�, and elastokinematicffects are entirely eliminated �Fig. 7�. However, these advantagesre achieved at the expense of an increased stiffness along theransverse direction, which limits the primary motion for a given

aximum allowable bending stress. It may be observed that of allhe beam characteristic coefficients, only the normalized axialtiffness is dependent on the beam thickness. A value of t=1/50 isssumed in Fig. 8. Thus, once again we are faced with perfor-ance tradeoffs in flexure geometry design. The significance of

he closed-form results Eqs. �9� and �10� is that for given stiffnessnd error motion requirements in an application, an optimal beamopology, somewhere between the two extremes, may be easilyelected. For example, a0=0.2 provides a beam topology that has50% higher axial stiffness, 78% lower elastokinematic effects,% lower stress-stiffening and kinematic effects, at the expense of27% increase in the primary direction stiffness.It is important to note that as the parameter a0 becomes small,

ernoulli’s assumptions are no longer accurate and correctionsased on Timoshenko’s beam theory may be readily incorporatedn the above analysis. However, the strength of this formulation ishe illustration that irrespective of the beam’s shape, its nonlinearorce–displacement relationships, which eventually determine itserformance characteristics as a flexure constraint, can always beaptured in a consistent matrix based format with variable nondi-ensional coefficients. Beams with even further generalized ge-

metries such as continuously varying thickness may be similarlyodeled. As mentioned earlier, this provides an ideal basis for a

Fig. 8 Axial elastic stiffness coefficient

hape sensitivity and optimization study. In all the subsequent

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flexure modules considered in this paper, although the figuresshow simple beam flexure elements, the presented analysis holdstrue for any generalized beam shape.

4 Parallelogram Flexure and VariationsThe parallelogram flexure, shown in Fig. 9, provides a con-

straint arrangement that allows approximate straight-line motion.The y displacement represents a DOF, while x and � are DOC.The two beams are treated as perfectly parallel and identical, atleast initially, and the stage connecting these two is assumed rigid.Loads and displacements can be normalized with respect to theproperties of either beam. Linear analysis of the parallelogramflexure module, along with the kinematic requirement of constantbeam arc-length, yields the following standard results �3�:

� fm� = 2�a c

c w2d + b��y

�� 2�a c

c w2d��y

��

x �p

2d+ iy2 �11�

The above approximations are based on the fact that elastic axialstiffness d is at least four orders of magnitude larger than elastictransverse stiffness a, b, and c, for typical dimensions and beamshapes. Based on this linear analysis, stage rotation � can beshown to be several orders of magnitude smaller than the y dis-placement. However, our objective here is to determine the morerepresentative nonlinear force–displacement relations for the par-allelogram flexure. Conditions of geometric compatibility yield

x =�x1

e + x2e�

2+

�x1k + x2

k�2

� xe + xk

w� =�x2

e − x1e�

2+

�x2k − x1

k�2

y1 = y −w�2

2; y2 = y +

w�2

2⇒ y1 � y2 � y �12�

Force equilibrium conditions are derived from the free body dia-gram of the stage in Fig. 9. While force equilibrium is applied ina deformed configuration to capture nonlinear effects, the contri-bution of � is negligible

p1 + p2 = p; f1 + f2 = f; m1 + m2 + �p2 − p1�w = m �13�These geometric compatibility and force equilibrium conditions,along with force–displacement results, Eqs. �7� and �8�, applied toeach beam, yield

f = f1 + f2 = �2a + pe�y + �2c + ph��

Fig. 9 Parallelogram flexure and free body diagram



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m1 + m2 = �2c + ph�y + �2b + pg��

xe =1

2d�p1 + p2�;

xk = �y �� i kk j

��y�� + �p1 + p2�

2�y ��r q

q s��y

��

� = �1d

+ �y ��r qq s

��y��m − �2c + ph�y − �2b + pg��

2w2�

�14�nlike in the linear analysis, it is important to recognize thateither f1 and f2, nor m1 and m2, are equal despite the fact that theransverse displacements, y and �, for the two beams are con-trained to be the same. This is due to the different values of axialorces p1 and p2, which result in unequal transverse stiffnesshanges in the two beams. Shifting attention to the expression forabove, the first term represents the consequence of elastic con-

raction and stretching of the top and bottom beams, respectively.he second term, which is rarely accounted for in the literature, is

he consequence of the elastokinematic effect explained in therevious section. Since the axial forces on the two beams arenequal, apart from resulting in different elastic deflections of thewo beams, they also cause slightly different beam shapes andherefore different elastokinematic axial deflections. Because ofts linear dependence on the axial load and quadratic dependencen the transverse displacement, this elastokinematic effect con-ributes a nonlinear component to the stage rotation. The purelyinematic part of the axial deflection of the beams is independentf the axial force, and therefore does not contribute to �. How-ver, it does contribute to the stage axial displacement x, whichlso comprises a purely elastic component and an elastokinematicomponent. Using Eqs. �12�–�14�, one can now solve the force–isplacement relationships of the parallelogram flexure. For sim-lification, higher order terms of � are dropped wherever appro-riate

=�4a2 + 4pea + p2e2 + f2 dr��2ma − 2fc + p�me − fh��

2w2d�2a + ep�3

=1

2w2�1

d+ y2r��m − y�2c + ph�� 15�

y =f − �2c + ph��

�2a + pe�

=f

�2a + pe�−

�2c + ph�2w2�2a + pe��1d + f2r�2a + pe�2��m − f �2c + ph��2a + pe��

�f

�2a + pe��16�

�p

2d+ y2i +

p

2y2r �17�

ssuming uniform thickness beams, with nominal dimensions of=1/50 and w=3/10, parallelogram flexure force–displacementelations based on the linear and nonlinear closed-form analysesCFA�, and nonlinear finite element analysis �FEA� are plotted inigs. 10–13. The inadequacy of the traditional linear analysis isvident in Fig. 10, which plots the stage rotation using expression15�. The nonlinear component of stage rotation derived from theelative elastokinematic axial deflections of the two beams in-reases with a compressive axial load p. Since the stage rotation isndesirable in response to the transverse force f, it is a parasiticrror motion comprising elastic and elastokinematic components,nd may therefore be eliminated by an appropriate combination of

ransverse loads. In fact, the COS location for the parallelogram



module is simply given by the ratio between m and f that makesthe stage rotation zero. This ratio is easily calculated from Eq.�15� to be �2c+ph� / �2a+pe�, and is equal to −0.5 in the absenceof an axial load, which agrees with common knowledge.

Expression �16� describes the transverse force displacement be-havior of the parallelogram flexure, which is plotted in Figs. 11and 12. Since � is several orders smaller than y, its contribution inthis expression is generally negligible and may be dropped. Thiseffectively results in a decoupling between the transverse momentm and displacement y. However, it should be recognized thatsince � is dependent on f, reciprocity does require y to be depen-dent on m, which becomes important only in specific cases, forexample, when the transverse end load is a pure moment. Thus,the relation between f and y is predominantly linear. As shown inFig. 12, the transverse stiffness has a linear dependence on theaxial force, and approaches zero for p=−20, which physicallycorresponds to the condition for buckling.

The axial force–displacement behavior is given by expression�17�, which also quantifies the dependence of axial compliance ontransverse displacements. Axial stiffness drops quadratically withy and the rate of this drop depends on the coefficient r, which is

Fig. 10 Parallelogram flexure stage rotation: CFA „lines…, FEA„circles…

Fig. 11 Parallelogram flexure transverse displacement: CFA

„lines…, FEA „circles…

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1aytmabmetbsmsiD

esa

Ft

Fs

6

Downloa

/700 for a simple beam. For a typical case when t=1/50, thexial stiffness reduces by about 30% for a transverse displacementof 0.1, as shown in Fig. 13. Based on the error motion charac-

erization in Sec. 2, any x displacement will be a parasitic errorotion, which in this case comprises of elastic, purely kinematic

s well as elastokinematic components. The kinematic component,eing a few orders of magnitude higher than the others and deter-ined by i=−0.6, dominates the error motion. The stiffness and

rror motion along the X direction represent the quality of DOC ofhe parallelogram module, and influence its suitability as a flexureuilding-block. This module may be mirrored about the motiontage, so that the resulting symmetry eliminates any x or � errorotions in response to a primary y motion, and improves the

tiffness along these DOC directions. However, this attempt tomprove the quality of DOC results in a nonlinear stiffening of theOF direction, leading to over-constraint.Next, a sensitivity analysis may be performed to determine the

ffect of differences between the two beams in terms of material,hape, thickness, length, or separation. For the sake of illustration,parallelogram flexure with beams of unequal lengths, L1 and L2,

ig. 12 Parallelogram and double parallelogram flexuresransverse stiffness: CFA „lines…, FEA „circles…

ig. 13 Parallelogram and double parallelogram flexures axial

tiffness: CFA „lines…, FEA „circles…

32 / Vol. 129, JUNE 2007


is considered. L1 is used as the characteristic length in the mecha-nism and error metric � is defined to be �1−L2 /L1�. Force–displacement relationships for Beam 1 remain the same as earlierEqs. �7� and �8�, whereas those for Beam 2 change as follows, forsmall �

� f2m2

� = ��1 + 3��a �1 + 2��c�1 + 2��c �1 + ��b ��y� �+ p2��1 + ��e hh �1 − ��g ��y� � �18�

x2e =

�1 − ��d

p2

�19�

x2k = �y ��1 + ��i k

k �1 − ��j ��y� �+ p2�y �� 1 − ��r �1 − 2��q�1 − 2��q �1 − 3��s ��y� �

Conditions of geometric compatibility Eq. �12� and force equilib-rium Eq. �13� remain the same. These equations may be solvedsimultaneously, which results in the following stage rotation forthe specific case of m=p=0

� =x2 − x1

2w� −

c�1 + ��2 − ��2w2

� yd

+ ry3� + iy22w

� �20�

Setting �=0 obviously reduces this to the stage rotation for theideal case Eq. �15�. It may be noticed that unequal beam lengthsresult in an additional term in the stage rotation, which has aquadratic dependence on the transverse displacement y. Thisarises due to the purely kinematic axial displacement componentsof the two beams that cancelled each other in the case of identicalbeams. The prediction of expression �20� is plotted in Fig. 14assuming simple beams, t=1/50, w=3/10, and two cases of �,along with FEA results.

Any other differences in the two parallel beams in terms ofthickness, shape, or material may be similarly modeled and theireffect on the characteristics of the parallelogram flexure accu-rately predicted. An important observation in this particular case isthat an understanding of the influence of � on the stage rotationmay be used to an advantage. If in a certain application the center

Fig. 14 “Nonidentical beam” parallelogram flexure stage rota-

tion: CFA „lines…, FEA „circles…



otloowt

tpmtiaf

�

Tr̄w

c̄

To=taytima

J

Downloa

f stiffness of the parallelogram flexure is inaccessible, a prede-ermined discrepancy in the parallelogram geometry may be de-iberately introduced to considerably reduce the stage rotation, asbserved in Fig. 14. For a given range of y motion, � may beptimized to place a limit on maximum possible stage rotation,ithout having to move the transverse force application point to

he COS.Another important variation of the parallelogram flexure is the

ilted-beam flexure, in which case the two beams are not perfectlyarallel, as shown in Fig. 15. This may result either due to pooranufacturing and assembly tolerances, or because of an inten-

ional design to achieve a remote center of rotation at C1. Assum-ng a symmetric geometry about the X-axis, and repeating annalysis similar to the previous two cases, the following nonlinearorce–displacement results are obtained for this flexure module

�1

2w2 cos ��m − �2c̄ + h̄ p

cos �� y

cos ��1

d+ � y

cos ��2r̄� − y�

w

�21�

y ��f cos � − m �

w�

�2ā + pcos �

ē� �22�

x �p

2d cos2 �+

y2

cos3 �� ī + p

2 cos �r̄� �23�

he newly introduced dimensionless coefficients ā, c̄, ē, h̄, ī, andare related to the original beam characteristic coefficients alongith the geometric parameter �, as follows

ā � �a − 2cw

� +a

2�2 +

b

w2�2� ; ē � �e − 2h

w� +

e

2�2 +

g

w2�2�

� �c − aw� − b �w

+ c�2� ; h̄ � �h − ew� − g �w

+ h�2� �24�ī � �i − 2k �

w+ j

�2

w2� ; r̄ � �r − 2q �

w+ s

�2

w2�

hese derivations are made assuming small values of � on therder of 0.1, and readily reduce to expressions �15�–�17� for �0. The most important observation based on expression �21� is

hat unlike the ideal parallelogram flexure, the stage rotation � hasn additional linear kinematic dependence on the primary motion, irrespective of the loads. This kinematic dependence dominateshe stage rotation for typical values of � on the order of 0.1. Thismplies that, as a consequence of the tilted beam configuration, the

otion stage has an approximate virtual center of rotation located

Fig. 15 Tilte

t C1. It should be noted that the virtual center of rotation and



center of stiffness are fundamentally distinct concepts. The formerrepresents a point in space about which a certain stage in a mecha-nism rotates upon the application of a certain allowable force,whereas the latter represents that particular location on this stagewhere the application of the said allowable force produces norotation. In general, the location of the virtual center of rotationdepends on the geometry of the mechanism and the location of theallowable force.

The magnitude of � in this case is only a single order less thanthat of y, as opposed to the several orders in the ideal parallelo-gram. Consequently � has not been neglected in the derivation ofthe above results. If transverse displacement y is the only desiredmotion then stage rotation � is a parasitic error motion, compris-ing an elastic, elastokinematic, and a dominant kinematic compo-nent.

The primary motion elastic stiffness in the tilted-beam flexure isa function of �, and the primary motion y itself has a dependenceon the end-moment m, as indicated by expression �22�, unlike theparallelogram flexure. This coupling, which is a consequence ofthe tilted-beam configuration, may be beneficial if utilized appro-priately, as shall be illustrated in the subsequent discussion on thedouble tilted-beam flexure module. For a typical geometry of uni-form thickness beam t=1/50 and w=3/10, and a transverse loadf=3, the transverse elastic stiffness and the y-� relation are plottedin Figs. 16 and 17, respectively.

The axial displacement in this case, given by expression �23�, isvery similar in nature to the axial displacement of the parallelo-

eam flexure

Fig. 16 Tilted-beam flexure transverse elastic stiffness: CFA

d-b

„lines…, FEA „circles…

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F„circles…

F„

F„

634 / Vol. 129, JUNE 2007

Downloaded 10 Nov 2010 to 150.135.248.238. Redistribution subject to ASM

gram flexure. As earlier, x displacement represents a parasitic er-ror motion comprising elastic, kinematic, and elastokinematicterms. Since the stage rotation is not negligible, it influences allthe axial direction kinematic and elastokinematic terms. Figures18 and 19 show the increase in the kinematic component of theaxial displacement and decrease in axial stiffness, respectively,with an increasing beam tilt angle. All these factors marginallycompromise the quality of the x DOC. The tilted-beam flexure isclearly not a good replacement of the parallelogram flexure ifstraight-line motion is desired, but is important as a frictionlessvirtual pivot mechanism. Quantitative results Eqs. �20� and �21�,regarding the stage rotation, are in perfect agreement with theempirical observations in the prior literature regarding the geo-metric errors in the parallelogram flexure �14�.

5 Double Parallelogram Flexure and VariationsThe results of the previous section are easily extended to a

double parallelogram flexure, illustrated in Fig. 20. The two rigidstages are referred to as the primary and secondary stages, asindicated. Loads f, m, and p are applied at the primary stage. Thetwo parallelograms are identical, except for the beam spacing, w1and w2. A linear analysis, with appropriate approximations, yieldsthe following force–displacement relationships

� fm� = a − a/2− a/2 2w12w22d

w12 + w2

2 �y� ��25�

x =p

d

To derive the nonlinear force–displacement relations for this mod-ule, results Eqs. �15�–�17� obtained in the previous section areapplied to each of the constituent parallelograms. Geometric com-patibility and force equilibrium conditions are easily obtainedfrom Fig. 20. Solving these simultaneously while neglectinghigher order terms in � and �1, leads to the following results

y1 �f

�2a − pe�; y − y1 �

f

�2a + pe��26�

⇒y =4af

�2a�2 − �ep�2

�1 =1

2w12�1d + y12r��m − y1�1 − p�2a + ep� + �2c − ph��

� − �1 =1

2w22�1d + �y − y1�2r��m − �y − y1��2c + ph�� 27�

⇒� =1

2w12�1d + f2�2a − pe�2r��m − f�2a − pe��1 − p�2a + ep�

+ �2c − ph�� + 12w2

2�1d + f2�2a + pe�2r��m − f�2a + pe� �2c + ph��

x1 = −p

2d+ y1

2�i − pr2� ; x + x1 = p2d + �y − y1�2�i + pr2 �

⇒ x =p

d+ py2

r��2a�2 + �ep�2� − 8aei�4a�2

�28�

Similar to the parallelogram flexure, the y displacement representsa DOF while x and � displacements represent DOC. Expression

ig. 17 Tilted-beam flexure stage rotation: CFA „lines…, FEA

ig. 18 Tilted-beam flexure kinematic axial displacement: CFAlines…, FEA „circles…

�26� describes the DOF direction force–displacement relation and
ig. 19 Tilted-beam flexure axial stiffness: CFA „lines…, FEAcircles…


dgdcstdbtl

lrcrag=rftona

mp

FC

J

Downloa

oes not include the weak dependence on m. Unlike the parallelo-ram flexure, it may be seen that the primary transverse stiffnessecreases quadratically with an axial load. This dependence is aonsequence of the fact that one parallelogram is always in ten-ion while the other is in compression, irrespective of the direc-ion of the axial load. A comparison between parallelogram andouble parallelogram flexure modules with uniform thicknesseams is shown in Fig. 12. Clearly, the transverse stiffness varia-ion, particularly for small values of axial force p, is significantlyess in the latter.

Expression �27� for primary stage rotation � in a double paral-elogram flexure is similar in nature to the parallelogram stageotation, but its nonlinear dependence on the axial load p is moreomplex. In the absence of a moment load, the primary stageotation � is plotted against the transverse force f, for differentxial loads, in Fig. 21. These results are obtained for a typicaleometry of uniform beam thickness t=1/50, w1=0.3, and w20.2. The change in the �− f relationship with axial forces is

elatively less as compared to the parallelogram flexure becauseor a given axial load, one of the constituent parallelograms is inension and the other is in compression. The increase in stiffnessf the former is somewhat compensated by the reduction in stiff-ess of the latter, thereby reducing the overall dependence onxial force.

Since the parasitic error motion � has elastic and elastokine-atic components only, it may be entirely eliminated by appro-

riately relocating the transverse force f, for a given p. Despite the

Fig. 20 Double parallelogram flexure

ig. 21 Double parallelogram flexure primary stage rotation:

FA „lines…, FEA „circles…



nonlinear dependence of � on transverse forces, the m / f ratiorequired to keep the primary stage rotation zero for p=0, is givenby

−w2

2�a + c� − w12c

�w22 + w1

2�a

For uniform thickness constituent beams, this ratio predictablyreduces to −0.5, which then changes in the presence of an axialforce.

The nondimensional axial displacement expression �28� revealsa purely elastic term as well as an elastokinematic term, but nopurely kinematic term. The purely kinematic term gets absorbedby the secondary stage due to geometric reversal. While the purelyelastic term is as expected, the elastokinematic term is signifi-cantly different from the parallelogram flexure, and is not imme-diately obvious. The axial compliance may be further simplifiedas follows

�x

�p�

1

d+

y2

2� r

2−

ei

a� �29�

Of the two factors that contribute elastokinematic terms, r /2 andei /a, the latter, being two orders larger than the former, dictatesthe axial compliance. Recalling expression �17�, this ei /a contri-bution does not exist in the parallelogram flexure, and is a conse-quence of the double parallelogram geometry. When a y displace-ment is imposed on the primary stage, the transverse stiffnessvalues for the two parallelograms are equal if the there is no axialload, and therefore y is equally distributed between the two. Re-ferring to Fig. 20, as a tensile axial load is applied, the transversestiffness of parallelogram 1 decreases and that of parallelogram 2increases by the same amount, which results in a proportionateredistribution of y between the two as given by Eq. �26�. Since thekinematic axial displacement of each parallelogram has a qua-dratic dependence on its respective transverse displacement, theaxial displacement of parallelogram 1 exceeds that of parallelo-gram 2. This difference results in the unexpectedly large elastoki-nematic component in the axial displacement and compliance. Ifthe axial force is compressive in nature, the scenario remains thesame, except that the two parallelograms switch roles.

For the geometry considered earlier, the axial stiffness of adouble parallelogram flexure is plotted against its transverse dis-placement y in Fig. 13, which shows that the axial stiffness dropsby 90% for a transverse displacement of 0.1. This is a seriouslimitation in the constraint characteristics of the double parallelo-gram flexure module. In the transition from a parallelogram flex-ure to a double parallelogram flexure, while geometric reversalimproves the range of motion of the DOF and eliminates thepurely kinematic component of the axial displacement, it provesto be detrimental to stiffness and elastokinematic parasitic erroralong the X direction DOC.

Expressions similar to �29� have been derived previously usingenergy methods �11�. It has also been shown that the maximumaxial stiffness can be achieved at any desired y location by tiltingthe beams of the double parallelogram flexure �15�. However, therate at which stiffness drops with transverse displacements doesnot improve because even though beams of one module are tiltedwith respect to the beams of the other, they remain parallel withineach module. Prior literature �16� recommends the use of thedouble tilted-beam flexure, shown in Fig. 22, over the doubleparallelogram flexure to avoid a loss in axial compliance resultingfrom a nonrigid secondary stage, but does not discuss the above-mentioned elastokinematic effect. To really eliminate this effect,one needs to identify and address its basic source, which is thefact that the secondary stage is free to move transversely when anaxial load is applied on the primary stage. Eliminating the trans-lational DOF of the secondary stage should therefore resolve thecurrent problem. In fact, it has been empirically suggested that an

external geometric constraint be imposed on the secondary stage,

JUNE 2007, Vol. 129 / 635


fewai

dtai

�

�

y

Itt�t

�

Swit

6

Downloa

or example by means of a lever arm, which requires it to havexactly half the transverse displacement of the primary stage,hen using a double parallelogram flexure �14�. However, this

pproach may lead to design complexity in terms of practicalmplementation.

It is shown here that imposing a primary stage rotation in theouble tilted-beam flexure, shown in Fig. 22, can help constrainhe transverse motion of the secondary stage. Results �21�–�23�pplied to the two tilted-beam flexure modules lead to the follow-ng force–displacement relations

1 �1

2w12 cos �

�m1 − �2c̄1 − h̄1 pcos �� y1cos ��1d + � y1cos ��2r̄1�−

y1�

w1

+ �1 � −1

2w2*2 cos �

�m1 + �2c̄2 + h̄2 pcos �� y1 − y�cos � ��1

d+ � y1 − y

cos ��2r̄2� + �y1 − y��

w2* �30�

1 ��f cos � − m1 �w1��2ā1 − pcos � ē1�

; y1 − y � −�f cos � + m1 �

w2*�

�2ā2 + pcos � ē2��31�

n the following analysis, we assume that the primary stage rota-ion � is constrained to zero, by some means. By eliminating m1,he internal moment at the secondary stage, between Eqs. �30� and31� the following relation between the primary and secondaryransverse displacements may be derived

�2c̄1 − h̄1p/cos ��2w1

2d cos2 �+

�

w1�y1

+ �− �2c̄2 + h̄2p/cos ��2w2

*2d cos2 �+

�

w2*��y1 − y�

� −1

�·

1

2w1w2*d cos2 �

�w12 + w2*2w1 + w2

* ��2ā1 − pcos � ē1�y1 + �2ā2 + pcos � ē2��y1 − y�� 32�

etting �=0 eliminates the left-hand side in the above relation,hich expectedly degenerates into expression �26�. However, with

ncreasing values of ��1/d� the left-hand side starts to dominate

Fig. 22 Double

he right-hand side, and eventually Eq. �32� is reduced to the

36 / Vol. 129, JUNE 2007


following approximate yet accurate relation between the trans-verse displacement of the two tilted-beam modules

y1w1

+y1 − y

w2* � 0

Thus, the purely kinematic dependence of � on y resulting fromthe tilted-beam configuration suppresses the redistribution of theoverall transverse displacement between the two modules due toelastokinematic effects in the presence of an axial load, as seen inthe double parallelogram flexure. This has been made possible dueto the coupling between the end-moments and transverse displace-ments revealed in expressions �31�. The transverse displacements,thus determined, are then used in estimating the axial directionforce–displacement relationships and constraint behavior

x1 � −p

2d cos2 �+

y12

cos3 ��i − 2k �

w1+ j

�2

w12�

−p

2 cos ��r − 2q �

w1+ s

�2

w12��

x + x1 �p

2d cos2 �+

�y1 − y�2

cos3 ��i + 2k �

w2* + j

�2

w2*2�

+p

2 cos ��r + 2q �

w2* + s

�2

w2*2�� 33�

⇒x �p

d cos2 �+

1

cos3 ��− ī1y12 + ī2�y1 − y�2�

+p

2 cos ��r1y1

2 + r2�y1 − y�2��In the above expressions, ā1, c̄1, ē1, h̄1, ī1, and r̄1, are as defined in

Eq. �24� with w=w1 and beam tilt angle �, while ā2, c̄2 ē2 h̄2, ī2,and r̄2 are defined with w=w2

* and a beam tilt angle of −�.For uniform thickness beams with a typical geometry of t

=1/50, w1=0.5, w2*=0.28, and a range of �, the axial stiffness

predicted by expression �33� is plotted in Fig. 23. It is seen thatwith increasing beam tilt angle �, there is a remarkable improve-ment in the quality of x DOC in terms of stiffness, especially incomparison to a double parallelogram flexure module. It is alsointeresting to note that this flexure configuration does not entirelycancel the purely kinematic component of the axial displacement.Thus, an improvement in the stiffness along a DOC is achieved atthe expense of larger parasitic error motion. Nevertheless, thisflexure module presents a good compromise between the desirable

d-beam flexure
tilte
performance measures.



btsprtsqdddCoilomtm

tptroepucOtscd

6

pmstdt

FF

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Downloa

The mathematical results presented above are amply supportedy physical arguments. For a given y displacement, when a rota-ion constraint is imposed on the primary stage, the secondarytage has a virtual center of rotation approximately located atoint C1, owing to module 1, and a different virtual center ofotation approximately located at point C2 due to module 2. Sincehese two centers of rotation for the rigid secondary stage arepaced apart, the secondary stage is better constrained. Conse-uently, the additional elastokinematic term encountered in theouble parallelogram flexure is attenuated because the depen-ence of the transverse displacements on the axial load p is re-uced. In the limiting case of � approaching zero, points C1 and2 move out to infinity, and no longer pose conflicting constraintsn the secondary stage, which therefore becomes free to translaten the transverse direction. This is the case of the double paral-elogram flexure, for which it is not possible to constrain the sec-ndary stage by imposing displacements or moments on the pri-ary stage because the transverse displacement and rotation of

he constituent parallelograms are not kinematically related andoments do not significantly affect translation.Of course, the effectiveness of the above strategy for improving

he axial stiffness depends upon the rotational constraint on therimary stage of the double tilted-beam flexure. Any mechanismhat utilizes this module should be carefully designed to meet thisequirement. Mirroring the design about the Y-axis, in Fig. 22,ffers limited success because the resulting configuration does notntirely constrain the primary stage rotation. A hybrid design com-rising a double parallelogram flexure and a double-tilt beam flex-re may work better because the former can provide the rotationonstraint necessary for the latter to preserve the axial stiffness.f course, this comes with a loss of symmetry, which influences

he axial parasitic error motion, thermal, and manufacturing sen-itivity, among other performance measures. Once again, this dis-ussion highlights the performance tradeoffs in flexure mechanismesign.

ConclusionWe have presented a nondimensional analytical framework to

redict the performance characteristics of beam-based flexureodules and mechanisms. It has been shown that the load-

tiffening and elastokinematic effects, which are not captured in araditional linear analysis, strongly influence the force–isplacement relations, and therefore the constraint properties, of

ig. 23 Double tilted-beam flexure axial stiffness: CFA „lines…,EA „circles…

he beam flexure. These nonlinear effects are modeled in a simple



yet accurate format using mathematical approximations for tran-scendental functions. The proposed formulation quantifies keymetrics including dimensionless elastic stiffness, geometric stiff-ness, kinematic, and elastokinematic coefficients, and relates themto the performance characteristics of the beam flexure. Signifi-cantly, it is shown using a specific beam-shape generalization thatonly these characteristic dimensionless coefficients vary withchanging beam shapes, without affecting the nature of the force–displacement relations. This provides a continuous comparisonbetween lumped-compliance and increasingly distributed-compliance flexure topologies, and an accurate analytical meansfor modeling elastic averaging. Furthermore, these coefficientsprovide the objective functions for beam shape optimization usingstandard techniques.

The results for a generalized beam are employed to analyzeseveral important beam-based flexure modules such as the paral-lelogram and double parallelogram flexures and their respectivevariations. The closed-form parametric results provide a qualita-tive and quantitative understanding of the modules’ force–displacement relations and constraint properties. The effects ofgeometric variations, reversal, and symmetry are also mathemati-cally addressed. This provides a basis for a geometric sensitivityanalysis to predict the consequences of manufacturing and assem-bly tolerances, and offers a systematic means for introducing pre-determined geometric imperfections in a flexure design to achievespecific desired attributes such as lower error motions or im-proved DOC stiffness.

An important theme that is repeatedly highlighted in this paperis the existence of performance tradeoffs in flexure design. Per-formance characteristics of beam-based flexure modules havebeen characterized and it is shown by means of illustrative ex-amples that the quality requirements for DOF and DOC, in termsof range of motion, error motions, and stiffness, are often contra-dictory. An attempt to improve one performance characteristic in-evitably undermines the others. However, a design that offers asuitable compromise for a given application may be objectivelyachieved using the analytical results presented in this paper.

Based on a estimate of modeling errors at each step in theanalysis, the closed-form predictions presented here are expectedto be accurate within 5–10%, depending on the flexure module.This is corroborated by a thorough nonlinear FEA performed inANSYS using BEAM4 elements, with the large displacementanalysis option turn on and shear coefficients set to zero. Althoughshear effects, which become increasingly important in shortbeams, have not been included, these can be readily incorporatedwithin the presented framework. While the proposed analysis doesnot match the generality of computational methods, it allowsquick calculations and parametric insights into flexure mechanismbehavior, and therefore is potentially helpful in flexure design.

The nonlinear load-stiffening and elastokinematic effects maybe used to accurately model the dynamic characteristics of flexuremechanisms, which are often employed in precision motion con-trol and vibration isolation. Furthermore, the thermal sensitivity offlexure modules may also be modeled by including the materialthermal behavior. Beam shape generalizations and module geom-etry variations beyond what are presented here may be investi-gated. A systematic treatment of the concepts of mobility, over-constraint, and elastic averaging in flexure mechanisms iscurrently being developed.

AppendixThis Appendix provides a comparison between the Taylor series

expansions of the actual transcendental functions and their alge-braic approximations for the compliance, stiffness and constraintterms discussed in Sec. 2. As mentioned earlier, normalized axial

2
force p�k .
JUNE 2007, Vol. 129 / 637


c

c

c

k

k

k

r

r¯

r

R

6

Downloa

11

Exact �k−tanh kk3 �= 13 − 215p+ 17315p2− 622835p3+¯Approximate

1

3�1+ 25p�=

1

3−

2

15p+

4

75p2−

8

375p3+¯

12

Exact �cosh k−1k2 cosh k �= 12 − 524p+ 61720p2− 2778064p3¯Approximate

1

2�1+ 512p�=

1

2−

2

24p+

25

288p2−

125

3456p3+¯

22

Exact � tanh kk �=1− 13p2+ 215p2− 17315p3+¯Approximate

�1+ 110p��1+ 1330p+

196p

2� =1−1

3p2+

193

1440p2−

2359

43,200p3+¯

11

Exact k3 sinh k

k sinh k−2 cosh k+2=12+

6

5p−

1

700p2+

1

63,000p3

Approximate 12+6

5p

12

Exact k2�1−cosh k�

k sinh k−2 cosh k+2=−6−

1

10p+

1

1400p2−

1

126,000p3

Approximate −6−1

10p

22

Exact k2 cosh k−k sinh k

k sinh k−2 cosh k+2=4+

2

15p−

11

6300p2+

1

27,000p3

Approximate 4+2

15p

11

Exact k2�cosh2 k+cosh k−2�−3k sinh k�cosh k−1�

2�k sinh k−2 cosh k+2�2=

3

5−

1

700p+

1

42,000p2−

37

97,020,000p3+¯

Approximate3

5−

1

700p

12

Exact −k2�cosh k−1�+k sinh k�cosh k−1�−4�cosh k−1�2

4�k sinh k−2 cosh k+2�2=−

1

20+

1

1400p−

1

84,000p2+

37

194,040,000p3+

Approximate −1

20+

1

1400p

22

Exact

−k3+k2 sinh k�cosh k+2�−2k�2 cosh2 k−cosh k−1�

4k�k sinh k−2 cosh k+2�2+

2 sinh k�cosh k−1�

4k�k sinh k−2 cosh k+2�2

=1

15−

11

6300p+

1

18,000p2−

509

291,060,000p3+¯

Approximate1

15−

11

6300p

eferences�1� Slocum, A. H., 1992, Precision Machine Design, Society of Manufacturing

Engineers, Dearborn, MI.�2� Blanding, D. K., 1999, Exact Constraint: Machine Design Using Kinematic

Principles, ASME, New York.�3� Smith, S. T., 2000, Flexures: Elements of Elastic Mechanisms, Gordon and

Breach Science, New York.�4� Ejik, J. V., 1985, “On the Design of Plate Spring Mechanism,” Ph.D. thesis,

Delft University of Technology, Delft, The Netherlands.

38 / Vol. 129, JUNE 2007


�5� Awtar, S., 2004, “Analysis and Synthesis of Planer Kinematic XY Mecha-nisms,” Sc.D. thesis, Massachusetts Institute of Technology, Cambridge MA�http://web.mit.edu/shorya/www�.

�6� Bisshopp, K. E., and Drucker, D. C., 1945, “Large Deflection of CantileverBeams,” Q. Appl. Math., 3�3�, pp. 272–275.

�7� Frisch-Fay, R., 1963, Flexible Bars, Butterworth, Washington, D.C.�8� Mattiasson, K., 1981, “Numerical Results From Large Deflection Beam and

Frame Problems Analyzed by Means of Elliptic Integrals,” Int. J. Numer.Methods Eng., 17, pp. 145–153.

�9� Howell, L. L., 2001, Compliant Mechanisms, Wiley, New York.



J

Downloa

�10� Plainevaux, J. E., 1956, “Etude des Deformations d’une Lame de SuspensionElastique,” Nuovo Cimento, 4, pp. 922–928.

�11� Legtenberg, R., Groeneveld, A. W., and Elwenspoek, M., 1996, “Comb-driveActuators for Large Displacements,” J. Micromech. Microeng., 6, pp. 320–329.

�12� Haringx, J. A., 1949, “The Cross-Spring Pivot as a Constructional Element,”Appl. Sci. Res., Sect. A, A1�4�, pp. 313–332.

�13� Zelenika, S., and DeBona, F., 2002, “Analytical and Experimental Character-ization of High Precision Flexural Pivots Subjected to Lateral Loads,” Precis.Eng., 26, pp. 381–388.



�14� Jones, R. V., 1988, Instruments and Experiences: Papers on Measurement andInstrument Design, Wiley, New York.

�15� Zhou, G., and Dowd, P., 2003, “Tilted Folded-Beam Suspension for Extendingthe Stage Travel Range of Comb-Drive Actuators,” J. Micromech. Microeng.,13, pp. 178–183.

�16� Saggere, L., Kota, S., and Crary, S. B., 1994, “A New Design for Suspensionof Linear Microactuators,” ASME J. Dyn. Syst., Meas., Control, 2, pp. 671–675.

JUNE 2007, Vol. 129 / 639


Characteristics of Beam-Based Flexure Modules...cal understanding of distributed-compliance ﬂexure mechanism behavior, and thus provide useful qualitative and quantitative de-sign

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