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A common feature to most of them-machines andm-devices is that their structures are getting moreand more complex and are often three-dimensional (3-D) whiletheir sizes are becoming smaller, which imposes a criticalchallenge to their manufacturing issues. The existing MEMSandLIGAtechnologieshavebeenwidelyusedfor2-Dand2.5-Dm-manufacturing applications, however, they do not provide acapability for 3-Dm-manufacturing 2. Therefore, an importantand challenging research topic has been to designm-machines orm-devices that are capable of 3-Dm-manufacturing at thenanometric accuracy level.This study proposes a novel rotary flexural bearing that iscapable of achieving rotational/oscillational motions of highaccuracy and a design methodology for such a bearing. Thebearing is targeted for use inm-manufacturing and precisionmetrology, such asm-EDM 3,m-ECM 4, ultrasonicm-machining5, laserm-machining 6, and coordinate measuring machines.The design of the bearing is based on the principle of flexuralmechanisms that realizes rotational/oscillational motions of onecomplete revolution through the elastic deformation of the elasticflexures.2. The proposed rotary flexural bearingFig. 1 shows a schematic view of the rotary flexural bearingwhich has three bearing sections and is configured as am-spindleunit. The bearing consists of inner and outer bearing cages, abearing shaft, and am-coupling that is connected to am-servomotor (an external power source). The rotational/oscilla-tional motions of the bearing shaft are guided by the bearing,which is expected to be of extremely high accuracy. The wholedesign is compact in size without any redundancy. The use of am-coupling can minimize the erroneous torque transmission causedby the possible misalignment between the bearing shaft andthe servomotor shaft, as well as vibrations and/or error motions oftheservomotor. Inthisway, therotational/oscillationalaccuracyofthe bearing can be maintained.2.1. General design considerationsThe bearing must satisfy the following requirements:?It should be able to rotate/oscillate in one complete revolution. Itshould have sufficient strength and fatigue life for a sustainedperiod of time.?It should possess rotational motion accuracy at the nanometerlevel or better.?It should be compact to fit into the limited spaces in variousmicro-machines and devices.In the proposed design, the inner and outer bearing cages arenested and connected at one end (left end in Fig. 1).Although the bearing can be designed as a monolithic structurewithout any joints, the proposed two-piece design is purely basedCIRP Annals - Manufacturing Technology 57 (2008) 179182A R T I C L EI N F OKeywords:SpindleFinite element methodRotary flexural bearingA B S T R A C TThis study proposes a design methodology for a novel rotary flexural bearing that is based on the motionprinciples of elastic flexures. The bearing is capable of providing rotational oscillations of one completerevolution and is characterized by potentially high repeatability, smooth motions, no mechanical wearand no lubrication requirements, no gaps or interfaces, zero maintenance, in addition to its compactness.From the structural characteristics and the basic working principles of the flexural bearings, the studyprovides a design analysis on the various aspects of the bearing, including material selection, stressanalysis and calculations (such as nonlinear finite element analysis, static and fatigue strength designs),motion error analysis and error reduction strategy, parametric design, etc.? 2008 CIRP.* Corresponding author.Contents lists available at ScienceDirectCIRP Annals - Manufacturing Technologyjournal homepage: http:/ see front matter ? 2008 CIRP.doi:10.1016/j.cirp.2008.03.033on the fabrication considerations since the monolithic designwould be extremely difficult to fabricate. By axis-symmetricallyarranging the elastic flexures in the inner and outer cages, thebearing is flexible in the circumferential direction, but stiff in theother directions. The rotational oscillation of 3608 (one completerevolution)orlargercanbeobtained.Ifalargerangulardisplacement (e.g., 3608) is desirable, more bearing sectionscan be added to the design although to do so makes the bearinglonger and less stiff. Otherwise, the bearing can have a compactand relatively stiff design.It should be pointed out that theoretically, the bearing shouldbe free of motion errors. Practically, it would have motion errorsbecause of various errors involved in the bearing fabrication andassembly processes. Motion errors can also be induced due tomaterial defects in the bearing. The bearing is designed under thefollowing considerations.2.1.1. Use of straight flexuresCompared to the other types of flexures, straight flexures havecertain advantages, such as distributed compliance over the wholeflexure length rather than lumped compliance localized at certainpoints under stress conditions. Straight flexures can effectivelysuppress the stress concentration 7, which in turn provides morecompliance and longer fatigue life at the material endurance limit.Furthermore, the straight flexures can be small in thickness butlarge in other dimensions for high compliance in the rotationaldirection and high stiffness in the other directions.2.1.2. Use of axis symmetrySymmetryisapowerfuldesigntoolinminimizingoreliminating bearing errors. In this design, the identical elasticflexures are axis-symmetrically arranged and uniformly distrib-uted over the circumference of the bearing, which is expected tohelp suppress motion errors in the radial, axial and tilt directions.Meanwhile, such a bearing is insensitive to temperature rise in theworking environment because errors due to thermal expansiontend to cancel each other. Additionally, the axis-symmetric designcan largely simplify the bearing fabrication. It also facilitates aneasy compensation for errors due to geometric inaccuraciesstemming from the fabrication process, which helps improvethe overall performance of the bearing.2.1.3. Even number of elastic flexuresPerfectaxissymmetryofelasticflexuresisimpracticalduetothefact that there exist geometric errors in the flexural bearing duringthefabricationandassemblyprocesses.Anyerrorsinthesymmetricdistributionoftheelasticflexuresmayresult inerrormotionsofthebearing. To minimize the geometric errors in the fabrication andassembly processes, a good strategy is to use an even number ofelastic flexures in the bearing design. When the wire electricdischargemachining(WEDM)methodisusedtomachinetheelasticflexures, for example, two opposing flexures can be simultaneouslycut. The simultaneous machining of the two opposing flexures notonly minimizes the geometric difference between the two flexures,but also relaxes the machining tolerance of the entire bearing.2.1.4. Multiple bearing sections in seriesFor a complete revolution of rotation, the bearing needs tohave at least 3608 angular displacement. It is impossible for asingle-section bearing to achieve such a large deflection. This isbecause too large deflection in a single bearing section couldover stress the elastic flexures, resulting in permanent (plastic)deformation or even fracture. Over deflection could also causethe so-called necking and cross interference phenomenon,as demonstrated in Fig. 2. To obtain a large oscillation rangewithout such a problem, the bearing is designed using multiplesections in series.2.1.5. Nested design of bearing cagesThe bearing utilizes bending deflections in the circumfer-ential direction to realize its rotational motion. A bearing sectionwill have a decrease in its length if subjected to torsion. Thedecrease in length can directly contribute to the axial errormotion of the bearing. To minimize or eliminate such an errormotion, nested design of bearing cages is proposed. In thisdesign, an inner bearing cage is inserted into an outer cage of thesimilar length and is further connected to the outer bearing cageat one end. If the other end of the outer cage is fixed, the free-end (the right side end in Fig. 3) of the inner bearing cage willhave very little or even no axial motion errordrwhen it issubjected to an external torque. This is because the axial errormotion of the inner bearing cage is effectively canceled by thatof the outer cage.The nested and axis-symmetric design can effectively cancelthe error motions due to thermal expansion of the bearingmaterial. This is because the inner and outer bearing cages wouldhave a uniform expansion in both radial and axial directions if thebearing should be exposed to a temperature field. Moreover, thenested design not only effectively increases the oscillation rangeof the bearing, but also reduces its overall dimensions forcompactness.Fig. 1. Schematic view of the bearing configured as am-spindle.Fig. 2. Necking phenomenon of a bearing section.Fig. 3. Nested design of outer and inner cages effectively reduces bearing axial errordr.H.P. Luo et al./CIRP Annals - Manufacturing Technology 57 (2008) 1791821802.1.6. Corner filletThe corner fillet at the connections of the elastic flexures in theindividualbearingsectionshouldbeproperlydesignedtominimize the stress concentration so as to increase the fatiguelife of the bearing.In addition to the above considerations, the design of thebearing also includes material selection, strength analysis andcalculations (with both static and fatigue considerations), analysisand suppression of error motions in the radial, axial and tiltdirections, stiffness analysis and calculations, etc.2.2. Material selectionSince the bearing realizes its rotational/oscillational motionsbased on the elastic deflections of the circumferentially arrangedflexures, it is subjected to cyclic stress conditions. In selecting amaterial for the bearing, fatiguestrength and flexibilityareofprimeconsideration. The bearingmust be compact in size to minimize theeffect of the gravitational force and to meet the applicationrequirements form-machines andm-devices. The following con-siderations have been given for the material selection:?High static strength. To achieve the maximum possible deflectionof the elastic flexures in the bearing, the bearing material shouldhavearatiooftheyieldstrengthtoelasticmodulus(ss/E)aslargeas possible 8. This is considered the most important materialrequirement.?Lowmaterialdensityr.Thedensityof thebearingmaterial shouldbe as low as possible to minimize the deflection to thegravitational force which could bend the bearing axis and thuscause error motions.?High elastic modulus. To have a good dynamic performance, thebearing must have high elastic modulus combined with lowmass density 9.?Good machinability. The bearing material must be easy tomachine. The machined bearing should possess good surfacefinish, surface integrity, and dimensional accuracies.?High fatigue strength. High fatigue strength allows the bearing tohave a long lifetime under the cyclic loading conditions.?Long-term stability. The material should have a long-termstability under various environmental conditions, includingthe corrosive and elevated temperature environments. It shouldnot have aging and creeping problems.In the above considerations for material selection as well as thestatic and dynamic performances of the material, a comprehensiveparameter is introduced for material selection 9,McA1Er? ?a1?A2ssE?a2?(1)where a1and a2are dynamic and static performance indicators formaterialselection;A1andA2areweightingfactorsoftherespectivedynamic and static performance indicators; E andrare Youngsmodulus and mass density of the material, respectively. UsingEq. (1), the comprehensive parameter Mcis calculated as 2400 Pa/(kg/m3) for titanium alloy Ti6Al4V which is compared to 1199for beryllium copper, and 370 for spring steel.Among the selected bearing materials, titanium alloy is best touse in terms of its comprehensive parameter and endurance limit(700 MPa for titanium alloy as compared to 321 and 490 forberyllium copper and spring steel, respectively). Moreover, thismaterial can achieve a high surface finish and dimensionalaccuracy when machined with the wire electro-discharge machin-ing method. In addition, titanium alloy has excellent corrosionresistance which is even better than that of stainless steels. Basedon the above considerations, titanium alloy has been thereforeselected for the bearing.It should be pointed out that although titanium alloy is amaterial of comprehensive performance, it is sensitive to surfacedefects and stress concentration (fatigue notch sensitivity or stressconcentration sensitivity). For this reason, in the process of thebearing fabrication, the elastic flexures should be machined with asurface roughness less than Ra 2.5mm and with smooth edges, butno sharp notches or pits.2.3. Design calculationsStrengthisthe priorityforthe bearing.Thebearingmust notfailduring its cyclic rotations/oscillations. Stress analysis needs to beperformedandthedetailedstressestobecalculatedforthebearingstructure.Inthedesigncalculations,finiteelementmethod(FEM)wasusedonasinglebearingsectionfortherespectiveinnerandouterbearingcages which were formed by the serial connection of the individualbearing sections. In this way, the amount of work in the FEMcalculationswassignificantlyreducedasopposedtothecalculationsof the entire bearing. Fig. 4 shows an individual bearing sectionsubjectedto both clockwise and counter-clockwiserotations. Stressdistributions of and the maximum stress points within the bearingsectionwerethusobtainedinthiswayintheFEMprocess.Whenthebearing section is deformed by a torque, the elastic flexures of thebearing are deformed due to the combined bending and twistingeffects. Because the elastic flexures are radially confined at theconnection portion and distributed in the circumferential directionof the bearing cage, they cannot have end rotation or warping. Theycan be subjected to tension, twisting, and bending, and are thus inthe three-directional stress state.2.4. Nonlinear finite element analysisSince the flexural bearing undergoes large deformations duringa working process, the problem becomes geometrically nonlineareven though the actual strain may be small and is well within theelastic limit. In this study, ANSYS9.0 was adopted in the FEMcalculations of the bearing. Displacement (angular in this study)loading method was used in the calculations. In the nonlineardeformation problems, the displacement loading method usuallyspeeds up the calculations.2.5. Analysis and minimization of axial errorThe axial error of the flexural bearing comes from two differentsources. The first and also major axial error source is due to theelastic motions of the bearing. When a bearing section is given anangular displacement, it has a reduction in its length. As the entirebearing is given an angular displacement, both inner and outercages have reductions in their respective lengths. Although thelength reductions of the two cages cancel each other because of thecoupling effect of the bearing cages, if the cancellation does notcome to zero, an axial error motion occurs. Fortunately, such anaxial error motion can be minimized or even eliminated bycarefully designing the inner and outer bearing cages so that bothcages have the same length reduction under external loadconditions.The second and also minor axial error source results from thetilterrormotion.Anytilterrormotion,ifprojectedtotheaxisofthebearing, causes the bearing to have axial error motion, althoughsuch an effect is secondary and negligible. Fig. 5 shows an FEMresult of axial error motion of nested single inner/outer bearingFig. 4. Individual bearing section subjected to clockwise and counterclockwiserotations.H.P. Luo et al./CIRP Annals - Manufacturing Technology 57 (2008) 179182181sections subjected to the external torque conditions. Due to thegeometrically nonlinear phenomenon, the axial error motion isnonlinear in terms of the applied torque.2.6. Fatigue analysis and designBecause the bearing is subjected to cyclic stress conditions, thefatigue problem must be taken into consideration at the designstage in order for the bearing to have a long lifetime. Whentitanium alloy is used for the bearing, its SN curve (or Wo hlercurve) has an endurance limit below which the material never failsunder cyclic loading conditions. The design used the endurancelimit of the material with the fatigue safety factor larger than theallowable fatigue safety factor.The stress level of the bearing is proportional to its angulardisplacement. The highest stress level is expected when thebearing reaches its maximum angular displacement. The elasticflexures in the bearing are subjected to asymmetrically cyclic andtri-axial complex stress conditions. Under the conditions of uni-axial, constant amplitude and asymmetrical cyclic stress, fatiguesafety factor is represented as 10nss?1ke=esbsacssm(2)wherecsis called average stress influence factor and is related tocyclic stress, material properties, stress concentration factor andheat treatment method of the material. It can also be obtained inthe following equation based on material pulsating cyclic fatiguelimits0,cs2s?1?s0s0(3)wheres?1iscyclicfatiguestrengthofperfectsymmetry.Itmustbepointed out that fatigue strength of the bearing can be affected bymany factors, such as surface integrity and dimensional accuraciesof the bearing cages, material defects and heat treatmentconditions, environmental and loading conditions 8. The bearingis supposed to have a long lifetime when its fatigue safety factor nsis equal to or larger than the allowable fatigue safety factor ns.It is worth noting that using the endurance limit of the bearingmaterial can theoretically allow designing a bearing of unlimitedlifetime. Practically, the bearing lifetime may be limited due to anumber of reasons. Examples include that the fatigue strength ofthe bearing material may not solely be determined by the cyclicstresses, other factors, such as the state of stress, bearingmachining and post-processing conditions and bearing applicationenvironment can bring uncertainties to the bearing lifetime. Inaddition, the endurance limit of the bearing material is normallyobtained from the SN test, which is typically performed under theuni-axial loading conditions. For the tri-axial loading conditions,the endurance limit on the SN curve should be different. In thisconsideration, a better way to determining the bearing lifetime isto actually test the bearing under the practical loading conditions.In contrast to the design for unlimited lifetime, the bearing mayalso be designed to have a limited lifetime for applications wherethe bearingdoesnot necessarilyhavefrequentcyclicoscillations.Ahigher endurance limit can be used in the design, which can lead toa more compact design and better accuracies.3. PrototypingWire EDM was used for the fabrication of the bearing. In thefabrication process, it was found that the bearing flexures tend tohave deflections due to machining stress, heat generation, andspark-induced vibrations. To minimize the flexure deflectionsduring the machining process, a special fixture was designed andfab