Characteristics of Beam-Based Flexure Modules

Transcription

1 Shorya Awtar Alexander H. Slocum Precision Engineering Research Grou, Massachusetts Institute of Technology, Cambridge, MA 039 Edi Sevincer Omega Advanced Solutions Inc., Troy, NY 08 Characteristics of Beam-Based Flexure Modules The beam flexure is an imortant constraint element in flexure mechanism design. Nonlinearities arising from the force equilibrium conditions in a beam significantly affect its roerties as a constraint element. Consequently, beam-based flexure mechanisms suffer from erformance tradeoffs in terms of motion range, accuracy and stiffness, while benefiting from elastic averaging. This aer resents simle yet accurate aroximations that cature the effects of load-stiffening and elastokinematic nonlinearities in beams. A general analytical framework is develoed that enables a designer to arametrically redict the erformance characteristics such as mobility, over-constraint, stiffness variation, and error motions, of beam-based flexure mechanisms without resorting to tedious numerical or comutational methods. To illustrate their effectiveness, these aroximations and analysis aroach are used in deriving the force dislacement relationshis of several imortant beam-based flexure constraint modules, and the results are validated using finite element analysis. Effects of variations in shae and geometry are also analytically quantified. DOI: 0.5/.773 Keywords: beam flexure, arallelogram flexure, tilted-beam flexure, double arallelogram flexure, double tilted-beam flexure, nonlinear beam analysis, elastokinematic effect, flexure design tradeoffs Introduction and Background From the ersective of recision machine design 4, flexures are essentially constraint elements that utilize material elasticity to allow small yet frictionless motions. The objective of an ideal constraint element is to rovide infinite stiffness and zero dislacements along its degrees of constraint DOC, and allow infinite motion and zero stiffness along its degrees of freedom DOF. Clearly, flexures deviate from ideal constraints in several ways, the rimary of which is limited motion along the DOF. Given a maximum allowable stress, this range of motion can be imroved by choosing a distributed-comliance toology over its lumed-comliance counterart, as illustrated in Fig.. However, distribution of comliance in a flexure mechanism gives rise to nonlinear elastokinematic effects, which result in two very imortant attributes. First, the error motions and stiffness values along the DOC deteriorate with increasing range of motion along DOF, which leads to fundamental erformance tradeoffs in flexures 5. Second, distributed comliance enables elastic averaging and allows non-exact constraint designs that are otherwise unrealizable. For examle, while the lumed-comliance multi-arallelogram flexure in Fig. a is rone to over-constraint in the resence of tyical manufacturing and assembly errors, its distributedcomliance version of Fig. b is relatively more tolerant. Elastic averaging greatly oens u the design sace for flexure mechanisms by allowing secial geometries and symmetric layouts that offer erformance benefits 5. Thus, distributed comliance in flexure mechanisms results in desirable as well as undesirable attributes, which coexist due to a common root cause. Because of their influence on constraint behavior, it is imortant to understand and characterize these attributes while ursuing systematic constraint-based flexure mechanism design. The uniform-thickness beam flexure is a classic examle of a Corresonding author. Contributed by the Mechanisms and Robotics Committee of ASME for ublicationinthejournal OF MECHANICAL DESIGN. Manuscrit received December 9, 005; final manuscrit received May 9, 006. Review conducted by Larry L. Howell. Paer resented at the ASME 005 Design Engineering Technical Conferences and Comuters and Information in Engineering Conference DETC005, Long Beach, CA, USA, Setember 4 8, 005. distributed-comliance toology. With increasing dislacements, nonlinearities in force dislacement relationshis of a beam flexure can arise from one of three sources material constitutive roerties, geometric comatibility, and force equilibrium conditions. While the beam material is tyically linear-elastic, the geometric comatibility condition between the beam s curvature and dislacement is an imortant source of nonlinearity. In general, this nonlinearity becomes significant for transverse dislacements of the order of one-tenth the beam length, and has been thoroughly analyzed in the literature using analytical and numerical methods 6 8. Simle and accurate arametric aroximations based on the seudo-rigid body method also cature this nonlinearity and have roven to be imortant tools in the design and analysis of mechanisms with large dislacements 9. However, the nonlinearity resulting from the force equilibrium conditions can become significant for transverse dislacements as small as the beam thickness. Since this nonlinearity catures loadstiffening and elastokinematic effects, it is indisensable in determining the influence of loads and dislacements on the beam s constraint behavior. These effects truly reveal the design tradeoffs in beam-based flexure mechanisms and influence all the key erformance characteristics such as mobility, over-constraint, stiffness variation, and error motions. Although both these nonlinear effects have been aroriately modeled in the rior literature, the resented analyses are either unsuited for quick design calculations 0, case-secific, or require numerical/grahical solution methods,3. While seudo-rigid body models cature load-stiffening, their inherent lumed-comliance assumtion recludes elastokinematic effects. Since we are interested in transverse dislacements that are an order of magnitude less than the beam length but generally greater than the beam nominal thickness, this third source of nonlinearity is the focus of discussion in this aer. We roose the use of simle olynomial aroximations in lace of transcendental functions arising from the deformed-state force equilibrium condition in beams. These aroximations are shown to yield very accurate closed-form force dislacement relationshis of the beam flexure and other beam-based flexure modules, and hel quantify the associated erformance characteristics and tradeoffs. Furthermore, these closed-form arametric results enable a hysi- Journal of Mechanical Design Coyright 007 by ASME JUNE 007, Vol. 9 / 65

2 Fig. a Lumed-comliance and b distributed-comliance multi-arallelogram mechanisms cal understanding of distributed-comliance flexure mechanism behavior, and thus rovide useful qualitative and quantitative design insights. The term flexure module is used in this aer to refer to a flexure building-block that serves as a constraint element in a otentially larger and more comlex flexure mechanism. Flexure modules are the simlest examles of flexure mechanisms. Characteristics of Flexure Modules This section resents some key erformance characteristics that cature the constraint behavior of flexure modules, and will be referred to in the rest of this aer. These include the concets of mobility DOF/DOC, stiffness variations, error motions, and center of stiffness. For a flexure module, one may intuitively or analytically assign stiff directions to be DOC and comliant directions to DOF. For examle, in the beam flexure illustrated in Fig., the transverse dislacements of the beam ti, y and, obviously constitute the two degrees of freedom, whereas axial direction x dislacement is a degree of constraint. However, for flexure mechanisms comrising of several distributed-comliance elements, this simlistic interretation of mobility needs some careful refinement. In a given module or mechanism, one may identify all ossible locations or nodes, finite in number, where forces may be allowed. For laner mechanisms, each node is associated with three generalized forces, or allowable forces, and three dislacement coordinates. Under any given set of normalized allowable forces of unit order magnitude, some of these dislacement coordinates will assume relatively large values as comared to others. The maximum number of dislacement coordinates that can be made indeendently large using any combination of the allowable forces, each of unit order magnitude, quantifies the degrees of freedom of the flexure mechanism. The remaining dislacement coordinates contribute to the degrees of constraint. For obvious reasons, normalized stiffness or comliance not only lays an imortant role in the determination of DOF and DOC, but also rovides a measure for their quality. In general, it is always desirable to maximize stiffness along the DOC and comliance along the DOF. Since load-stiffening and elastokinematic effects result in stiffness variations, these nonlinearities have to be included for an accurate rediction of the number and quality of DOF and DOC in a distributed-comliance flexure mechanism. A situation where the stiffness along a DOF increases significantly with increasing dislacements reresents a condition of over-constraint. These considerations are not always catured by the traditional Gruebler s criterion. A second measure of the quality of a DOF or DOC is error motion, which affects the motion accuracy of a given flexure module or mechanism. While it is common to treat any undesired dislacement in a flexure mechanism as a arasitic error, we roose a more secific descrition of error motions. The desired motion, or rimary motion, is one that occurs in the direction of an alied generalized force along a DOF. Resulting motions in any other direction are deemed as undesired or error motions. In a urely linear elastic formulation, this has a straightforward imlication. If the comliance, or alternatively stiffness, matrix relating the allowable forces to the dislacement coordinates has any offdiagonal terms, the corresonding forces will generate undesired motions. However, a nonlinear formulation reveals loadstiffening, kinematic, and elastokinematic effects in the force dislacement relationshis, which can lead to additional undesired motions. Since these terms are revealed in a mechanism s deformed configuration, an accurate characterization of undesired motions should be erformed by first alying unit order allowable forces in order to nominally deform the mechanism. From this deformed configuration, only the generalized force along the direction of rimary motion is varied while others are ket constant. The resulting changes in dislacements along all other dislacement coordinates rovide a true measure of undesired motions in a mechanism. The undesired motions along the other degrees of freedom are defined here as cross-axis error motion, while those along the degrees of constraint are referred to as arasitic error motions. Each of these error motions can be exlicitly force deendent, urely kinematic, elastokinematic, or any combination thereof. This shall be illustrated in the following sections by means of secific flexure module examles. Such a characterization is imortant because it reveals the constituents of a given error motion. Any error comonent that is exlicitly force deendent, based on either elastic or loadstiffening effects, may be eliminated by an aroriate combination of allowable forces. In some cases, this may be accomlished by simly varying the location of an alied force to rovide an additional moment. This observation leads to the concet of center of stiffness COS. If the rotation of a certain stage in a flexure mechanism is undesired, then the articular location of an alied force that results in zero stage rotation is defined as the COS of the mechanism with resect to the given stage and alied force. Obviously, the COS may shift with loading and deformation. Kinematic terms that contribute to error motions are deendent on other dislacements and, in general, may not be eliminated by any combination of allowable forces without over-constraining the mechanism. Otimizing the shae of the constituent distributedcomliance elements can only change the magnitude of these kinematic terms while a modification of the mechanism toology is required to entirely eliminate them. Elastokinematic contributions to error motions, on the other hand, can be altered by either of these two schemes by aroriately selecting the allowable forces, as well as by making geometric changes. This information rovides insight regarding the kind of otimization and toological redesign that may be needed to imrove the motion accuracy in a flexure mechanism. Fig. Generalized beam flexure 3 Beam Flexure Figure illustrates a varying-thickness beam with generalized end forces and end dislacements in a deformed configuration. Dislacements and lengths are normalized by the overall beam length L, forces by EI zz /L, and moments by EI zz /L. The two equal end-segments have a uniform thickness t, and the middle 66 / Vol. 9, JUNE 007 Transactions of the ASME

3 section is thick enough to be considered rigid. The symbol E is used to denote Young s modulus for a state of lane stress, and late modulus for lane strain. All nondimensional quantities are reresented by lower case letters throughout this discussion. The case of a simle beam a o =/ is first considered. For transverse dislacements, y and, of the order of 0. or smaller, beam curvature may be linearized by assuming small sloes. Since the force equilibrium condition is alied in the deformed configuration of the beam, the axial force contributes to the bending moments. Solving Euler s equation for the simle beam yields the following well-known results 3,0, where the normalized tensile axial force k where f = m = k 3 sinh k k sinh k cosh k + y + k cosh k k sinh k cosh k + k cosh k k sinh k cosh k + y + k cosh k k sinh k k sinh k cosh k + y = k tanh k k 3 cosh k = cosh k f + k m cosh k k f + tanh k cosh k k m x = x e + x k = d y r r r r y 3 d = /t r = k cosh k + cosh k 3k sinh kcosh k k sinh k cosh k + r = r = k cosh k + k sinh kcosh k 4cosh k 4k sinh k cosh k + r = k3 + k sinh kcosh k + k cosh k cosh k 4kk sinh k cosh k + + sinh kcosh k 4kk sinh k cosh k + In the resence of a comressive axial force, exressions analogous to Eq. 3 may be obtained in terms of trigonometric functions instead of hyerbolic functions. The axial dislacement x is comrised of two comonents a urely elastic comonent x e that results due to the elastic stretching of the beam, and a kinematic comonent x k that results from the conservation of beam arc-length. The kinematic comonent of the axial dislacement may be alternatively stated in terms of the transverse loads f and m, instead of dislacements y and. However, it should be recognized that this comonent fundamentally arises from a condition of geometric constraint that requires the beam arc-length to stay constant as it takes a new shae. Based on the aroximations made until this stage, the above results should be accurate to within a few ercent of the true behavior of an ideal beam. Although the deendence of transverse stiffness on axial force, and axial stiffness on transverse dislacement is evident in exressions 3, their transcendental nature offers little arametric insight to a designer. We therefore roose simlifications that are based on an observation that the transverse comliance terms may be accurately aroximated by inverse linear or inverse quadratic exressions Fig. 3 Normalized comliance terms: actual solid lines and aroximate dashed lines C = k tanh k k 3 cosh k cosh k k cosh k k cosh k tanh k k A comarison of the series exansions of the actual hyerbolic functions and their resective aroximations, rovided in the Aendix, reveals that the series coefficients remain close even for higher order terms. The actual and aroximate functions for the three comliance terms are lotted in Fig. 3 for a range of =.5 to 0. Since = /4.47 corresonds to the first fixed-free beam buckling mode f=m=0, all the comliance terms exhibit a singularity at this value of. The aroximate functions accurately cature this singularity, but only for the first fixed free beam buckling mode. The fact that some comliance terms are well aroximated by inverse linear functions of the axial force indicates that the stiffness terms may be aroximated simly by linear functions of. These linear aroximations are easily obtained from the series exansion of the hyerbolic functions in exression, which show that the higher order terms are small enough to be neglected for values of as large as ±0 = k 3 sinh k k cosh k k sinh k cosh k + k sinh k cosh k K k cosh k k cosh k k sinh k k sinh k cosh k + k sinh k cosh k Journal of Mechanical Design JUNE 007, Vol. 9 / 67

4 Table Stiffness, kinematic, and elastokinematic coefficients for a simle beam a e. i 0.6 r /700 b 4 g / 5 j / 5 s / 6300 c 6 h 0. k / 0 q / 400 Similarly, the following linear aroximations can be made for the hyerbolic functions in the geometric constraint relation x k y y 6 The maximum error in aroximation in all of the above cases is less than 3% for within ±0. This excellent mathematical match between the transcendental functions and their resective aroximations has far-reaching consequences in terms of revealing the key characteristics of a beam flexure. Although shown for the tensile axial force case, all of the above aroximations hold valid for the comressive case as well. Summarizing the results so far in a general format m = f a x = i k + y d c c b y + e h h g y 7 k j y + y r q q s y 8 The coefficients a, b, c, e, g, h, i, j, k, q, r, and s are all nondimensional numbers that are characteristic of the beam shae, and assume the values for a simle beam with uniform thickness as shown in Table. In general, these coefficients are functionals of the beam s satial thickness function tx. A quantification of these coefficients in terms of the beam shae rovides the basis for a sensitivity analysis and shae otimization. The articular shae variation illustrated in Fig. is discussed in further detail later in this section. The maximum estimated error in the analysis so far is less than 5% for transverse dislacements within ±0. and axial force within ±0. Unlike the original results and 3, exressions 7 and 8 exress the role of the normalized axial force in the force dislacement relations of the beam in a simle matrix format. The two comonents of transverse stiffness, commonly referred to as the elastic stiffness matrix and the geometric stiffness matrix, are clearly quantified in exression 7. This characterizes the load-stiffening effect in a beam, and consequently the loss in the quality of DOF in the resence of a tensile axial force. Of articular interest is the change in axial stiffness in the resence of a transverse dislacement, as evident in exression 8. Itmaybe seen that the kinematic comonent defined earlier may be further searated into a urely kinematic comonent, and an elastokinematic comonent. The latter is named so because of its deendence on the axial force as well as the kinematic requirement of constant beam arc-length. This comonent essentially catures the effect of the change in the beam s deformed shae due to the axial force, for given transverse dislacements. Since this term contributes additional comliance along the axial direction, it comromises the quality of the x DOC. For an alied force f, the dislacement x and rotation are undesired. Since these corresond to a DOC and DOF, resectively, any x dislacement is a arasitic error motion and rotation is a cross-axis error motion. While the latter is exlicitly load deendent and can be eliminated by an aroriate combination of the transverse loads f and m, the former has kinematic as well as elastokinematic comonents and therefore cannot be entirely eliminated. These observations are significant because they arametrically illustrate the role of the force along a DOC on the quality of DOF due to load-stiffening. Furthermore, the range of motion along DOF is limited to ensure an accetably small stiffness reduction and error motion along the DOC due to elastokinematic effects. These are the classic tradeoffs in flexure erformance, and are not catured in a traditional linear analysis. The accuracy of the above derivations may be verified using known cases of beam buckling, as long as the magnitude of the corresonding normalized buckling loads is less than 0. Buckling limit corresonds to the comressive load at which the transverse stiffness of a beam becomes zero. Using this definition and exression 4 for a fixed free beam, one can estimate the buckling load to be crit =.5, which is less than.3% off from the classical beam buckling rediction of crit = /4. Similarly, the buckling load for a beam with zero end sloes is redicted to be 0 using exression 5, as comared to derived using the classical theory. Many other nontrivial results may be easily derived from the roosed simlified exressions, using aroriate boundary conditions. For examle, in the resence of an axial load, the ratio between m and f required to ensure zero beam-end rotation is given by + 60 / + 0, which determines the COS of the beam with resect to the beam-end and force f. An examle that illustrates a design tradeoff is that of a clamed clamed beam of actual length L transversely loaded in the center. While symmetry ensures erfect straight-line motion along the y DOF, and no error motions along x or, it also results in a nonlinear stiffening effect along the DOF given by f=a ied/+rdy y y, which significantly limits the range of allowable motion. This nonlinear stiffness behavior is derived in a few stes from exressions 7 and 8, whereas the conventional methods can be considerably more time consuming. Next, we consider the secific generalization of beam geometry shown in Fig.. The nonlinear force dislacement relationshis for this beam geometry may be obtained mathematically by treating the two end-segments as simle beams, and using the rior results of this section m f 3 6a o +4a o a o a o +a o y 35 50ao +60ao 4a 3 o a o 5 60a o +84a o 40a 3 o a o +4a o a o 5 60a o +84a o 40a 3 o a o5 60a o +9a o 60a 3 o a o 4 y 9 68 / Vol. 9, JUNE 007 Transactions of the ASME

5 x =a o t y 35 50ao +60ao 4a 3 o a o 5 60a o +84a o 40a 3 o a o +4a o a o 5 60a o +84a o 40a 3 o a o5 60a o +9a o 60a 3 o a o 4 y 3 a 3 o y ao + 440ao 480a 3 o + 576a 4 o a o + 440a o 480a 3 o + 576a 4 o a o +4a o a o + 440a o 480a 3 o + 576a 4 o a o + 560a o 000a 3 o + 408a 4 o 560a 5 o a o 6 y 0 It is significant to note that these force dislacement relations are of the same matrix equation format as exressions 7 and 8. Obviously, the transverse direction elastic and geometric stiffness coefficients are now functions of a 0, a nondimensional number. As exected, for a fixed beam thickness, reducing the length of the end-segments increases the elastic stiffness, while reducing the geometric stiffness. In the limiting case of a 0 0, which corresonds to a lumed-comliance toology, the elastic stiffness becomes infinitely large, and the first geometric stiffness coefficient reduces to while the others vanish. Similarly, the axial direction elastic stiffness as well as the kinematic and elastokinematic coefficients are also functions of beam segment length a 0.Inthe limiting case of a 0 0, the elastic stiffness becomes infinitely large, the first kinematic coefficient aroaches 0.5, and the remaining kinematic coefficients along with all the elastokinematic coefficients vanish. This simly reaffirms the rior observation that elastokinematic effects are a consequence of distributed comliance. For the other extreme of a 0 0.5, which corresonds to a simle beam, it may be verified that the above transverse elastic and geometric stiffness coefficients, and the axial elastic stiffness, kinematic, and elastokinematic coefficients take the numerical values listed earlier in Table. Observations on these two limiting cases agree with the common knowledge and hysical understanding of distributedcomliance and lumed-comliance toologies. Significantly, ex- Fig. 4 Transverse elastic stiffness coefficients Fig. 6 Axial kinematic coefficients Fig. 5 Transverse geometric stiffness coefficients Fig. 7 Axial elastokinematic coefficients Journal of Mechanical Design JUNE 007, Vol. 9 / 69

6 flexure modules considered in this aer, although the figures show simle beam flexure elements, the resented analysis holds true for any generalized beam shae. Fig. 8 Axial elastic stiffness coefficient ressions 9 and 0 rovide an analytical comarison of these two limiting case toologies, and those inbetween. To grahically illustrate the effect of distribution of comliance, the stiffness and constraint coefficients are lotted as functions of length arameter a 0 in Figs Assuming a certain beam thickness limited by the maximum allowable axial stress, one toology extreme, a 0 =0.5, rovides the lowest elastic stiffness along the y DOF Fig. 4, and therefore is best suited to maximize the rimary motion. However, this is rone to moderately higher load-stiffening behavior, as indicated by the geometric stiffness coefficients in Fig. 5, and arasitic errors along the x DOC, as indicated by the kinematic coefficients in Fig. 6. Most imortantly, this toology results in the highest elastokinematic coefficients lotted in Fig. 7, which comromise the stiffness and error motions along the x DOC, but at the same time make this toology best suited for aroximateconstraint design. Aroaching the other toology extreme, a 0 0, there are gains on several fronts. Elastic stiffness along the x DOC imroves Fig. 8, load-stiffening effects decrease Fig. 5, kinematic effects diminish or vanish Fig. 6, and elastokinematic effects are entirely eliminated Fig. 7. However, these advantages are achieved at the exense of an increased stiffness along the transverse direction, which limits the rimary motion for a given maximum allowable bending stress. It may be observed that of all the beam characteristic coefficients, only the normalized axial stiffness is deendent on the beam thickness. A value of t=/50 is assumed in Fig. 8. Thus, once again we are faced with erformance tradeoffs in flexure geometry design. The significance of the closed-form results Eqs. 9 and 0 is that for given stiffness and error motion requirements in an alication, an otimal beam toology, somewhere between the two extremes, may be easily selected. For examle, a 0 =0. rovides a beam toology that has 50% higher axial stiffness, 78% lower elastokinematic effects, 6% lower stress-stiffening and kinematic effects, at the exense of a 7% increase in the rimary direction stiffness. It is imortant to note that as the arameter a 0 becomes small, Bernoulli s assumtions are no longer accurate and corrections based on Timoshenko s beam theory may be readily incororated in the above analysis. However, the strength of this formulation is the illustration that irresective of the beam s shae, its nonlinear force dislacement relationshis, which eventually determine its erformance characteristics as a flexure constraint, can always be catured in a consistent matrix based format with variable nondimensional coefficients. Beams with even further generalized geometries such as continuously varying thickness may be similarly modeled. As mentioned earlier, this rovides an ideal basis for a shae sensitivity and otimization study. In all the subsequent 4 Parallelogram Flexure and Variations The arallelogram flexure, shown in Fig. 9, rovides a constraint arrangement that allows aroximate straight-line motion. The y dislacement reresents a DOF, while x and are DOC. The two beams are treated as erfectly arallel and identical, at least initially, and the stage connecting these two is assumed rigid. Loads and dislacements can be normalized with resect to the roerties of either beam. Linear analysis of the arallelogram flexure module, along with the kinematic requirement of constant beam arc-length, yields the following standard results 3: m = f a c c w d + b y a c c w d y x d + iy The above aroximations are based on the fact that elastic axial stiffness d is at least four orders of magnitude larger than elastic transverse stiffness a, b, and c, for tyical dimensions and beam shaes. Based on this linear analysis, stage rotation can be shown to be several orders of magnitude smaller than the y dislacement. However, our objective here is to determine the more reresentative nonlinear force dislacement relations for the arallelogram flexure. Conditions of geometric comatibility yield x = x e + x e w = x e x e + x k + x k x e + x k + x k x k y = y w ; y = y + w y y y Force equilibrium conditions are derived from the free body diagram of the stage in Fig. 9. While force equilibrium is alied in a deformed configuration to cature nonlinear effects, the contribution of is negligible + = ; f + f = f; m + m + w = m 3 These geometric comatibility and force equilibrium conditions, along with force dislacement results, Eqs. 7 and 8, alied to each beam, yield f = f + f = a + ey + c + h Fig. 9 Parallelogram flexure and free body diagram 630 / Vol. 9, JUNE 007 Transactions of the ASME

7 m + m = c + hy + b + g = d x k = y i k k + y r q q x e = d + ; j y + + s y y r q q s y m c + hy b + g w 4 Unlike in the linear analysis, it is imortant to recognize that neither f and f, nor m and m, are equal desite the fact that the transverse dislacements, y and, for the two beams are constrained to be the same. This is due to the different values of axial forces and, which result in unequal transverse stiffness changes in the two beams. Shifting attention to the exression for above, the first term reresents the consequence of elastic contraction and stretching of the to and bottom beams, resectively. The second term, which is rarely accounted for in the literature, is the consequence of the elastokinematic effect exlained in the revious section. Since the axial forces on the two beams are unequal, aart from resulting in different elastic deflections of the two beams, they also cause slightly different beam shaes and therefore different elastokinematic axial deflections. Because of its linear deendence on the axial load and quadratic deendence on the transverse dislacement, this elastokinematic effect contributes a nonlinear comonent to the stage rotation. The urely kinematic art of the axial deflection of the beams is indeendent of the axial force, and therefore does not contribute to. However, it does contribute to the stage axial dislacement x, which also comrises a urely elastic comonent and an elastokinematic comonent. Using Eqs. 4, one can now solve the force dislacement relationshis of the arallelogram flexure. For simlification, higher order terms of are droed wherever aroriate = 4a +4ea + e + f drma fc + me fh w da + e 3 = w d + y rm yc + h 5 f c + h y = a + e f = a + e c + h w a + e d + f r a + e m c + h f a + e f a + e 6 x d + y i + y r 7 Assuming uniform thickness beams, with nominal dimensions of t=/ 50 and w=3/ 0, arallelogram flexure force dislacement relations based on the linear and nonlinear closed-form analyses CFA, and nonlinear finite element analysis FEA are lotted in Figs The inadequacy of the traditional linear analysis is evident in Fig. 0, which lots the stage rotation using exression 5. The nonlinear comonent of stage rotation derived from the relative elastokinematic axial deflections of the two beams increases with a comressive axial load. Since the stage rotation is undesirable in resonse to the transverse force f, it is a arasitic error motion comrising elastic and elastokinematic comonents, and may therefore be eliminated by an aroriate combination of transverse loads. In fact, the COS location for the arallelogram Fig. 0 Parallelogram flexure stage rotation: CFA lines, FEA circles module is simly given by the ratio between m and f that makes the stage rotation zero. This ratio is easily calculated from Eq. 5 to be c+h/a+e, and is equal to 0.5 in the absence of an axial load, which agrees with common knowledge. Exression 6 describes the transverse force dislacement behavior of the arallelogram flexure, which is lotted in Figs. and. Since is several orders smaller than y, its contribution in this exression is generally negligible and may be droed. This effectively results in a decouling between the transverse moment m and dislacement y. However, it should be recognized that since is deendent on f, recirocity does require y to be deendent on m, which becomes imortant only in secific cases, for examle, when the transverse end load is a ure moment. Thus, the relation between f and y is redominantly linear. As shown in Fig., the transverse stiffness has a linear deendence on the axial force, and aroaches zero for = 0, which hysically corresonds to the condition for buckling. The axial force dislacement behavior is given by exression 7, which also quantifies the deendence of axial comliance on transverse dislacements. Axial stiffness dros quadratically with y and the rate of this dro deends on the coefficient r, which is Fig. Parallelogram flexure transverse dislacement: CFA lines, FEA circles Journal of Mechanical Design JUNE 007, Vol. 9 / 63

8 Fig. Parallelogram and double arallelogram flexures transverse stiffness: CFA lines, FEA circles /700 for a simle beam. For a tyical case when t=/50, the axial stiffness reduces by about 30% for a transverse dislacement y of 0., as shown in Fig. 3. Based on the error motion characterization in Sec., any x dislacement will be a arasitic error motion, which in this case comrises of elastic, urely kinematic as well as elastokinematic comonents. The kinematic comonent, being a few orders of magnitude higher than the others and determined by i= 0.6, dominates the error motion. The stiffness and error motion along the X direction reresent the quality of DOC of the arallelogram module, and influence its suitability as a flexure building-block. This module may be mirrored about the motion stage, so that the resulting symmetry eliminates any x or error motions in resonse to a rimary y motion, and imroves the stiffness along these DOC directions. However, this attemt to imrove the quality of DOC results in a nonlinear stiffening of the DOF direction, leading to over-constraint. Next, a sensitivity analysis may be erformed to determine the effect of differences between the two beams in terms of material, shae, thickness, length, or searation. For the sake of illustration, a arallelogram flexure with beams of unequal lengths, L and L, is considered. L is used as the characteristic length in the mechanism and error metric is defined to be L /L. Force dislacement relationshis for Beam remain the same as earlier Eqs. 7 and 8, whereas those for Beam change as follows, for small f +c y +c +b m = +3a + +e h h g y 8 x e = d x k = y +i k k + y j y r q q 3s y 9 Conditions of geometric comatibility Eq. and force equilibrium Eq. 3 remain the same. These equations may be solved simultaneously, which results in the following stage rotation for the secific case of m==0 = x x w c+ w y d + ry3 + iy w 0 Setting =0 obviously reduces this to the stage rotation for the ideal case Eq. 5. It may be noticed that unequal beam lengths result in an additional term in the stage rotation, which has a quadratic deendence on the transverse dislacement y. This arises due to the urely kinematic axial dislacement comonents of the two beams that cancelled each other in the case of identical beams. The rediction of exression 0 is lotted in Fig. 4 assuming simle beams, t=/50, w=3/0, and two cases of, along with FEA results. Any other differences in the two arallel beams in terms of thickness, shae, or material may be similarly modeled and their effect on the characteristics of the arallelogram flexure accurately redicted. An imortant observation in this articular case is that an understanding of the influence of on the stage rotation may be used to an advantage. If in a certain alication the center Fig. 3 Parallelogram and double arallelogram flexures axial stiffness: CFA lines, FEA circles Fig. 4 Nonidentical beam arallelogram flexure stage rotation: CFA lines, FEA circles 63 / Vol. 9, JUNE 007 Transactions of the ASME

9 Fig. 5 Tilted-beam flexure of stiffness of the arallelogram flexure is inaccessible, a redetermined discreancy in the arallelogram geometry may be deliberately introduced to considerably reduce the stage rotation, as observed in Fig. 4. For a given range of y motion, may be otimized to lace a limit on maximum ossible stage rotation, without having to move the transverse force alication oint to the COS. Another imortant variation of the arallelogram flexure is the tilted-beam flexure, in which case the two beams are not erfectly arallel, as shown in Fig. 5. This may result either due to oor manufacturing and assembly tolerances, or because of an intentional design to achieve a remote center of rotation at C. Assuming a symmetric geometry about the X-axis, and reeating an analysis similar to the revious two cases, the following nonlinear force dislacement results are obtained for this flexure module w c + h cos m cos y cos + y d cos r y w x y f cos m w ē ā + cos r d cos + y cos ī cos The newly introduced dimensionless coefficients ā, c, ē, h, ī, and r are related to the original beam characteristic coefficients along with the geometric arameter, as follows ā a c w + a + b w ; ē e h w + e + g w c c aw b w + c; h h ew g w + h 4 ī i k w + j w ; r r q w + s w These derivations are made assuming small values of on the order of 0., and readily reduce to exressions 5 7 for =0. The most imortant observation based on exression is that unlike the ideal arallelogram flexure, the stage rotation has an additional linear kinematic deendence on the rimary motion y, irresective of the loads. This kinematic deendence dominates the stage rotation for tyical values of on the order of 0.. This imlies that, as a consequence of the tilted beam configuration, the motion stage has an aroximate virtual center of rotation located at C. It should be noted that the virtual center of rotation and center of stiffness are fundamentally distinct concets. The former reresents a oint in sace about which a certain stage in a mechanism rotates uon the alication of a certain allowable force, whereas the latter reresents that articular location on this stage where the alication of the said allowable force roduces no rotation. In general, the location of the virtual center of rotation deends on the geometry of the mechanism and the location of the allowable force. The magnitude of in this case is only a single order less than that of y, as oosed to the several orders in the ideal arallelogram. Consequently has not been neglected in the derivation of the above results. If transverse dislacement y is the only desired motion then stage rotation is a arasitic error motion, comrising an elastic, elastokinematic, and a dominant kinematic comonent. The rimary motion elastic stiffness in the tilted-beam flexure is a function of, and the rimary motion y itself has a deendence on the end-moment m, as indicated by exression, unlike the arallelogram flexure. This couling, which is a consequence of the tilted-beam configuration, may be beneficial if utilized aroriately, as shall be illustrated in the subsequent discussion on the double tilted-beam flexure module. For a tyical geometry of uniform thickness beam t=/ 50 and w=3/ 0, and a transverse load f=3, the transverse elastic stiffness and the y- relation are lotted in Figs. 6 and 7, resectively. The axial dislacement in this case, given by exression 3, is very similar in nature to the axial dislacement of the arallelo- Fig. 6 Tilted-beam flexure transverse elastic stiffness: CFA lines, FEA circles Journal of Mechanical Design JUNE 007, Vol. 9 / 633

10 gram flexure. As earlier, x dislacement reresents a arasitic error motion comrising elastic, kinematic, and elastokinematic terms. Since the stage rotation is not negligible, it influences all the axial direction kinematic and elastokinematic terms. Figures 8 and 9 show the increase in the kinematic comonent of the axial dislacement and decrease in axial stiffness, resectively, with an increasing beam tilt angle. All these factors marginally comromise the quality of the x DOC. The tilted-beam flexure is clearly not a good relacement of the arallelogram flexure if straight-line motion is desired, but is imortant as a frictionless virtual ivot mechanism. Quantitative results Eqs. 0 and, regarding the stage rotation, are in erfect agreement with the emirical observations in the rior literature regarding the geometric errors in the arallelogram flexure 4. Fig. 7 Tilted-beam flexure stage rotation: CFA lines, FEA circles 5 Double Parallelogram Flexure and Variations The results of the revious section are easily extended to a double arallelogram flexure, illustrated in Fig. 0. The two rigid stages are referred to as the rimary and secondary stages, as indicated. Loads f, m, and are alied at the rimary stage. The two arallelograms are identical, excet for the beam sacing, w and w. A linear analysis, with aroriate aroximations, yields the following force dislacement relationshis m = f a a/ a/ w w d w + w y 5 x = d Fig. 8 Tilted-beam flexure kinematic axial dislacement: CFA lines, FEA circles To derive the nonlinear force dislacement relations for this module, results Eqs. 5 7 obtained in the revious section are alied to each of the constituent arallelograms. Geometric comatibility and force equilibrium conditions are easily obtained from Fig. 0. Solving these simultaneously while neglecting higher order terms in and, leads to the following results y f a e ; y y f a + e 4af y = a e = w d + y rm y + c h a + e 6 = w d + y y rm y y c + h 7 Fig. 9 Tilted-beam flexure axial stiffness: CFA lines, FEA circles d + f a e r m + c h + w d + f a + e r f m a + e c + h = w f a e a + e x = d + y i r ; x + x = d + y y i + r x = d + + e 8aei yra 4a 8 Similar to the arallelogram flexure, the y dislacement reresents a DOF while x and dislacements reresent DOC. Exression 6 describes the DOF direction force dislacement relation and 634 / Vol. 9, JUNE 007 Transactions of the ASME

11 nonlinear deendence of on transverse forces, the m/f ratio required to kee the rimary stage rotation zero for =0, is given by Fig. 0 Double arallelogram flexure does not include the weak deendence on m. Unlike the arallelogram flexure, it may be seen that the rimary transverse stiffness decreases quadratically with an axial load. This deendence is a consequence of the fact that one arallelogram is always in tension while the other is in comression, irresective of the direction of the axial load. A comarison between arallelogram and double arallelogram flexure modules with uniform thickness beams is shown in Fig.. Clearly, the transverse stiffness variation, articularly for small values of axial force, is significantly less in the latter. Exression 7 for rimary stage rotation in a double arallelogram flexure is similar in nature to the arallelogram stage rotation, but its nonlinear deendence on the axial load is more comlex. In the absence of a moment load, the rimary stage rotation is lotted against the transverse force f, for different axial loads, in Fig.. These results are obtained for a tyical geometry of uniform beam thickness t=/50, w =0.3, and w =0.. The change in the f relationshi with axial forces is relatively less as comared to the arallelogram flexure because for a given axial load, one of the constituent arallelograms is in tension and the other is in comression. The increase in stiffness of the former is somewhat comensated by the reduction in stiffness of the latter, thereby reducing the overall deendence on axial force. Since the arasitic error motion has elastic and elastokinematic comonents only, it may be entirely eliminated by aroriately relocating the transverse force f, for a given. Desite the Fig. Double arallelogram flexure rimary stage rotation: CFA lines, FEA circles w a + c w c w + w a For uniform thickness constituent beams, this ratio redictably reduces to 0.5, which then changes in the resence of an axial force. The nondimensional axial dislacement exression 8 reveals a urely elastic term as well as an elastokinematic term, but no urely kinematic term. The urely kinematic term gets absorbed by the secondary stage due to geometric reversal. While the urely elastic term is as exected, the elastokinematic term is significantly different from the arallelogram flexure, and is not immediately obvious. The axial comliance may be further simlified as follows x d + y r a ei 9 Of the two factors that contribute elastokinematic terms, r/ and ei/a, the latter, being two orders larger than the former, dictates the axial comliance. Recalling exression 7, this ei/a contribution does not exist in the arallelogram flexure, and is a consequence of the double arallelogram geometry. When a y dislacement is imosed on the rimary stage, the transverse stiffness values for the two arallelograms are equal if the there is no axial load, and therefore y is equally distributed between the two. Referring to Fig. 0, as a tensile axial load is alied, the transverse stiffness of arallelogram decreases and that of arallelogram increases by the same amount, which results in a roortionate redistribution of y between the two as given by Eq. 6. Since the kinematic axial dislacement of each arallelogram has a quadratic deendence on its resective transverse dislacement, the axial dislacement of arallelogram exceeds that of arallelogram. This difference results in the unexectedly large elastokinematic comonent in the axial dislacement and comliance. If the axial force is comressive in nature, the scenario remains the same, excet that the two arallelograms switch roles. For the geometry considered earlier, the axial stiffness of a double arallelogram flexure is lotted against its transverse dislacement y in Fig. 3, which shows that the axial stiffness dros by 90% for a transverse dislacement of 0.. This is a serious limitation in the constraint characteristics of the double arallelogram flexure module. In the transition from a arallelogram flexure to a double arallelogram flexure, while geometric reversal imroves the range of motion of the DOF and eliminates the urely kinematic comonent of the axial dislacement, it roves to be detrimental to stiffness and elastokinematic arasitic error along the X direction DOC. Exressions similar to 9 have been derived reviously using energy methods. It has also been shown that the maximum axial stiffness can be achieved at any desired y location by tilting the beams of the double arallelogram flexure 5. However, the rate at which stiffness dros with transverse dislacements does not imrove because even though beams of one module are tilted with resect to the beams of the other, they remain arallel within each module. Prior literature 6 recommends the use of the double tilted-beam flexure, shown in Fig., over the double arallelogram flexure to avoid a loss in axial comliance resulting from a nonrigid secondary stage, but does not discuss the abovementioned elastokinematic effect. To really eliminate this effect, one needs to identify and address its basic source, which is the fact that the secondary stage is free to move transversely when an axial load is alied on the rimary stage. Eliminating the translational DOF of the secondary stage should therefore resolve the current roblem. In fact, it has been emirically suggested that an external geometric constraint be imosed on the secondary stage, Journal of Mechanical Design JUNE 007, Vol. 9 / 635

12 Fig. Double tilted-beam flexure for examle by means of a lever arm, which requires it to have exactly half the transverse dislacement of the rimary stage, when using a double arallelogram flexure 4. However, this aroach may lead to design comlexity in terms of ractical imlementation. It is shown here that imosing a rimary stage rotation in the double tilted-beam flexure, shown in Fig., can hel constrain the transverse motion of the secondary stage. Results 3 alied to the two tilted-beam flexure modules lead to the following force dislacement relations r w cos m c h y w cos cos y cos d + y + w * +c + h cos m cos y y cos + y y d cos r + y y * w f cos m y ā ; cos ē w f cos + m y y ā + cos ē w * 30 3 In the following analysis, we assume that the rimary stage rotation is constrained to zero, by some means. By eliminating m, the internal moment at the secondary stage, between Eqs. 30 and 3 the following relation between the rimary and secondary transverse dislacements may be derived c h /cos w d cos + w y + c + h /cos w * d cos + w *y y w w * d cos w * + w w + w * ā cos ē y +ā + cos ē y y 3 Setting =0 eliminates the left-hand side in the above relation, which exectedly degenerates into exression 6. However, with increasing values of /d the left-hand side starts to dominate the right-hand side, and eventually Eq. 3 is reduced to the following aroximate yet accurate relation between the transverse dislacement of the two tilted-beam modules y + y y * 0 w w Thus, the urely kinematic deendence of on y resulting from the tilted-beam configuration suresses the redistribution of the overall transverse dislacement between the two modules due to elastokinematic effects in the resence of an axial load, as seen in the double arallelogram flexure. This has been made ossible due to the couling between the end-moments and transverse dislacements revealed in exressions 3. The transverse dislacements, thus determined, are then used in estimating the axial direction force dislacement relationshis and constraint behavior x d cos + y cos 3 k i + j w x + x x cos r q w + s w w d cos + y y cos 3 i +k * w + j w * + cos r +q w * + s w * d cos + cos 3 ī y + ī y y + cos r y + r y y 33 In the above exressions, ā, c, ē, h, ī, and r, are as defined in Eq. 4 with w=w and beam tilt angle, while ā, c ē h, ī, and r are defined with w=w * and a beam tilt angle of. For uniform thickness beams with a tyical geometry of t =/50, w =0.5, w * =0.8, and a range of, the axial stiffness redicted by exression 33 is lotted in Fig. 3. It is seen that with increasing beam tilt angle, there is a remarkable imrovement in the quality of x DOC in terms of stiffness, esecially in comarison to a double arallelogram flexure module. It is also interesting to note that this flexure configuration does not entirely cancel the urely kinematic comonent of the axial dislacement. Thus, an imrovement in the stiffness along a DOC is achieved at the exense of larger arasitic error motion. Nevertheless, this flexure module resents a good comromise between the desirable erformance measures. 636 / Vol. 9, JUNE 007 Transactions of the ASME