Stiffness Reduction Strategies for Additively Manufactured Compliant Mechanisms

284 Pages·2017·7.72 MB·English

BBrriigghhaamm YYoouunngg UUnniivveerrssiittyy
BBYYUU SScchhoollaarrssAArrcchhiivvee
Theses and Dissertations
2016-04-01
SSttiiffffnneessss RReedduuccttiioonn SSttrraatteeggiieess ffoorr AAddddiittiivveellyy MMaannuuffaaccttuurreedd
CCoommpplliiaanntt MMeecchhaanniissmmss
Ezekiel G. Merriam
Brigham Young University - Provo
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BBYYUU SScchhoollaarrssAArrcchhiivvee CCiittaattiioonn
Merriam, Ezekiel G., "Stiffness Reduction Strategies for Additively Manufactured Compliant Mechanisms"
(2016). Theses and Dissertations. 5873.
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StiffnessReductionStrategiesforAdditively
ManufacturedCompliantMechanisms
EzekielG.Merriam
Adissertationsubmittedtothefacultyof
BrighamYoungUniversity
inpartialfulfillmentoftherequirementsforthedegreeof
DoctorofPhilosophy
LarryL.Howell,Chair
SpencerP.Magleby
PaulW.Richards
EricR.Homer
MarkB.Colton
DepartmentofMechanicalEngineering
BrighamYoungUniversity
April2016
Copyright©2016EzekielG.Merriam
AllRightsReserved
ABSTRACT
StiffnessReductionStrategiesforAdditively
ManufacturedCompliantMechanisms
EzekielG.Merriam
DepartmentofMechanicalEngineering,BYU
DoctorofPhilosophy
Thisworkdevelopsandexaminesdesignstrategiesforreducingthestiffnessof3D-printed
compliant mechanisms. The three aspects of a flexure that determine its stiffness are well known:
material, boundary conditions, and geometry. In a highly constrained design space however, flex-
ure stiffness may remain unacceptably high even while arriving at the limits of design constraints.
In this work, changes to geometry and boundary conditions are examined that lead to drastically
reduced stiffness behavior without changing flexure thickness, width, or length. Changes to ge-
ometry can result in very complex mechanisms. However, 3D-printing enables almost arbitrarily
complexgeometries. Thisdissertationpresentsthreedesignstrategiesforstiffnessreduction: static
balancing,latticeflexures,andcompoundjoints.
Static balancing refers to changes in the boundary conditions that result in a near-zero net
change in potential energy storage over the useful deflection of a flexure. In this work, I present a
method for static balancing that utilizes non-dimensional parameters to quickly synthesize a joint
design with stiffness reduced by nearly 90%. This method is not only simple and straightforward,
it is applicable to a wide range of flexure topologies. The only requirements on the joint to be
balanced are that it must be approximated as a pin joint and torsion spring, and it must have a
well-understoodstiffnesswhensubjectedtoacompressiveload.
Latticeflexuresresultfrommodificationstogeometrythatreducecross-sectionalareawith-
out changing width. However, the reduction in stiffness is greater than the reduction in cross sec-
tional area. This can occur because the bending load is now carried by beams partially in torsion.
Twolatticegeometriesareproposedandanalyzedindetailusinganalyticandnumerictechniques.
It is shown that the off-axis stiffness behavior of lattice flexures can be better than that of conven-
tionalbladeflexureswhilebendingstiffnessisreduced>60%.
Compound joints are those that consist of arrays of flexures arranged co-axially. This
arrangement provides increased range of motion, generally decreased stiffness, and improved sta-
bility. Additionally,amethodishereinpresentedtoreducetheparasiticcentershiftofacompound
jointtonearlyzeroataspecifieddeflection.
The penultimate chapter demonstrates how all three strategies can be used together, and
includesnewresultstofacilitatetheircombination.
Keywords: compliantmechanisms,staticbalancing,latticeflexures,compoundjoints,load-dependent
stiffness,3Dprinting
ACKNOWLEDGMENTS
Icouldnothavecompletedthisdissertationwithoutthehelpofmanypeople. LarryHowell
has been more than an advisor to me over the years; he has truly been a mentor and a source of
wisdom in many of the large and small things that make a man worth admiring. Kevin Cole, Ken
Forster, Nick Hawkins, Miriam Busch, Judy Stoudt, and the other members of the department
staff have been wonderful support in fabricating prototypes, building DAQs, and handling the
paperworkrequiredtocompletemystudies.
My peers have also been an invaluable source of support. In particular, Jason Lund, Shan-
non Zirbel, Jared Bruton, and Todd Nelson have helped solve many issues, from de-bugging code
to securing funding to technical discussion and simple comedy relief. Abraham Lee’s particle
swarm optimiztion algorithm proved invaluable in many of my design efforts. Patrick Walton’s
helpconstructingseveralofmyprototypesisgratefullyacknowledged.
This work would have been impossible without the financial support of NASA’s Office
of the Chief Technologist; its support is gratefully acknowledged, along with the mentorship of
Jonathan Jones and Douglas Hofmann. Additionally, Jonathan Jones and Kenneth Cooper of
NASA Marshall Space Flight Center are profusely thanked for the fabrication of the titanium pro-
totypes. AdditionalfundingwasprovidedbyNASAGrantNo.NNX13AF52G.
Finally, my wife Kayla deserves the highest praise for her unflinching and unfailing sup-
port. Shehasbeenmywellspringofencouragementandmotivation. Itistoherthatthisdissertation
isdedicated.
TABLEOFCONTENTS
LISTOFTABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LISTOFFIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Chapter1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 ProblemStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 ImprovingRangeofMotion . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 ReducingActuationEffort . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 AdditiveManufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.4 CombiningAdditiveManufacturingwithCompliantMechanisms . . . . . 6
1.3 ResearchObjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 ReducingActuationEffort . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 ImprovingRangeofMotion . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 ResearchMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 RelevancetoAdditiveManufacturing . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Collaboration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter2 Designof3DPrintedTitaniumCompliantMechanisms . . . . . . . . . . 13
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 MaterialConsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 PorosityofEBM-ProducedParts . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 ThicknessCorrectionFactor . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 AllowableStress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 GeometryConstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 FeatureGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 ThermalStressesandPartWarping . . . . . . . . . . . . . . . . . . . . . 17
2.4.3 ManufacturingClearances . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.4 PowderandSupportRemoval . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter3 Non-DimensionalApproachforStaticBalancingofRotationalFlexures . 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 BalancingofLoad-IndependentHinge-SpringSystem . . . . . . . . . . . . . . . . 24
3.4 ExtendingtoLoad-DependentJoints . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 ExampleDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 ExperimentalResultsandDiscussion . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
iv
Chapter4 AMethodforDeterminingLoad-DependentStiffnessofFlexures . . . . . 35
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 LoadDependentStiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5.1 Cross-AxisFlexuralPivot(CAFP) . . . . . . . . . . . . . . . . . . . . . . 40
4.5.2 TriangleFlexure(TF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5.3 CartwheelHinge(CH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.5.4 Small-LengthFlexuralPivot(SLFP) . . . . . . . . . . . . . . . . . . . . . 42
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.6.1 Cross-AxisFlexuralPivot(CAFP) . . . . . . . . . . . . . . . . . . . . . . 43
4.6.2 TriangleFlexureandCartwheelHinge . . . . . . . . . . . . . . . . . . . . 44
4.6.3 Small-LengthFlexuralPivot(SLFP) . . . . . . . . . . . . . . . . . . . . . 45
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chapter5 LatticeFlexures: GeometriesforStiffnessReductionOfBladeFlexures . 47
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2.1 StiffnessofLatticeFlexures . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2.2 Off-AxisStiffnessConsiderations . . . . . . . . . . . . . . . . . . . . . . 54
5.3 PrototypeTestingandPerformance . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3.1 BendingStiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3.2 TorsionalStiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter6 CompoundJoints: BehaviorandBenefitsofFlexureArrays . . . . . . . . 67
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2.1 RangeofMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2.2 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2.3 CenterShift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2.4 Off-AxisStiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.3 ExperimentalSetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3.1 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3.2 CenterShift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.4.1 StiffnessandOff-AxisStiffness . . . . . . . . . . . . . . . . . . . . . . . 76
6.4.2 CenterShift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.5 ExampleMechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.6.1 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.6.2 CenterShift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
v
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Chapter7 Integration of Advanced Stiffness-Reduction Techniques Demonstrated
ina3D-PrintableJoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 LatticeFlexures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.2.2 Load-DependentStiffnessBehavior . . . . . . . . . . . . . . . . . . . . . 89
7.3 StaticBalancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.4 PrintableBalancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.5 Flexure-BalancerIntegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.6 ExperimentalValidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.7 DiscussionandConclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter8 ConclusionsandRecommendations . . . . . . . . . . . . . . . . . . . . . . 105
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
AppendixA CodesandScriptsusedinStaticBalancing . . . . . . . . . . . . . . . . . 119
A.1 FindingtheΠ-GroupRelationship . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.2 BalancedSpringDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.3 FEAConfirmation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
AppendixB CodesandScriptsusedintheAnalysisofLatticeFlexures . . . . . . . . . 139
B.1 ANSYSScripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.1.1 LatticeMacros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
B.1.2 Off-AxisStiffnessAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 147
B.1.3 AnalysisofConventionalBladeFlexures . . . . . . . . . . . . . . . . . . 158
B.2 MatLabScripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
AppendixC CodesandScriptsusedintheAnalysisofCompoundJoints . . . . . . . . 171
C.1 CenterShiftScripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
C.1.1 ANSYSScripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
C.1.2 MatLabScripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
C.2 Off-AxisStiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
C.2.1 ANSYSScripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
C.2.2 MatLabScripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
C.3 ImageProcessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
AppendixD CodesandScriptsusedinStaticBalancingofaCompoundLattice-Flexured
CAFP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
vi
D.1 DeterminingLoad-DependentStiffness . . . . . . . . . . . . . . . . . . . . . . . 213
D.2 SystemDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
D.3 BalancerDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
D.4 ConfirmationofFinalDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
AppendixE SetupandUseoftheTorque-RotationMeasuringApparatus . . . . . . . 259
E.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
E.2 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
E.2.1 DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
E.2.2 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
E.2.3 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
E.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
E.3.1 Spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
E.3.2 LabviewDAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
E.4 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
E.4.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
E.4.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
E.4.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
vii
LISTOFTABLES
2.1 Summaryoftitaniumstrengthdata. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 TabulatedvaluesofΠ andΠ ,withtheexpectedreductioninjointstiffness. . . . 27
1 2
3.2 DesignparametersforbalancedCAFP. . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 DesignparametersforhypotheticalLoad-IndependentCAFP. . . . . . . . . . . . . 31
4.1 Meshdensityusedindifferentjointtypes. . . . . . . . . . . . . . . . . . . . . . . 37
4.2 PublisheddatausedforFEAvalidation. . . . . . . . . . . . . . . . . . . . . . . . 40
5.1 Comparisonoflatticeflexureoff-axisstiffnesstoconventionalbladeflexure. . . . . 59
5.2 Geometricparametersfortestspecimens. . . . . . . . . . . . . . . . . . . . . . . 60
5.3 SummaryofmeasuredCAFPstiffnessandlatticeflexurestiffnessreduction. . . . . 62
5.4 SummaryofmeasuredtitaniumCAFPstiffnessandlatticeflexurestiffnessreduction. 62
6.1 Coefficientsforsurfacefitsfornon-dimensionalstiffnessofcompoundjoints. . . . 82
6.2 ComparisonofFEmodelandliterature. . . . . . . . . . . . . . . . . . . . . . . . 83
7.1 Valuesoftheβ coefficientstodescribethestiffnessoflattice-flexuredCAFPs. . . 91
i
7.2 Parametersforthebalancedlatticeflexuresystem. . . . . . . . . . . . . . . . . . . 92
7.3 Variablestobeoptimizedinthedesignofthebalancerspring. . . . . . . . . . . . 96
7.4 Balancerparametersachievedwithprintablebalancer. . . . . . . . . . . . . . . . . 97
7.5 Actuationenergyreductionmeasuredinatitaniumprototype. . . . . . . . . . . . . 102
E.1 Partnumbersofpurchasedhardware. . . . . . . . . . . . . . . . . . . . . . . . . . 260
E.2 Wiringconnectionofthetorquetransducer. . . . . . . . . . . . . . . . . . . . . . 266
viii
LISTOFFIGURES
1.1 Afoldedbeamsuspension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Anillustrationofstaticbalancing. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 AcomplianttitaniumhingeproducedwithEBM. . . . . . . . . . . . . . . . . . . 5
1.4 Atwo-degree-of-freedompointingmechanismproducedwithEBMintitanium. . . 6
1.5 Examplesofadditivelymanufacturedcompliantmechanisms. . . . . . . . . . . . . 7
1.6 Anearlylattice-flexuredCAFP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Compoundjointexamples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 AcomplianttitaniumhingeproducedwithEBM. . . . . . . . . . . . . . . . . . . 14
2.2 Titaniumprototypesusedtovalidatemodels. . . . . . . . . . . . . . . . . . . . . 16
2.3 Twodesignmodificationsthatincreaselikelihoodofbuildsuccess. . . . . . . . . . 17
2.4 Buildfailureduetowarpingofslenderflexures. . . . . . . . . . . . . . . . . . . . 18
2.5 Supportmaterialremoval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Anidealizedload-independentsystem. . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Across-axisflexuralpivotwithassociatedvariables. . . . . . . . . . . . . . . . . 24
3.3 TherelationshipbetweenΠ andΠ . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1 2
3.4 NormalizedbalancedstiffnessasafunctionofΠ . . . . . . . . . . . . . . . . . . 28
1
3.5 ComparisonofunbalancedandbalancedstiffnessofaCAFP. . . . . . . . . . . . . 31
3.6 Aprototypejointdesignedusingthemethoddetailedinthischapter. . . . . . . . . 32
3.7 Thestiffnessofthebalancedandunbalancedjointsplottedforcomparison. . . . . 33
3.8 Thepercentreductioninstiffnessofthebalancedjoint. . . . . . . . . . . . . . . . 34
4.1 Jointarrangementandloading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 ComparisonofFEAresultswithanalyticsolutionforasymmetricCAFP. . . . . . 39
4.3 Schematicsforeachflexureanalyzed. . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 Finiteelementmodelsforeachflexureanalyzed. . . . . . . . . . . . . . . . . . . 42
4.5 Stiffnessbehaviorofthetriangleflexure. . . . . . . . . . . . . . . . . . . . . . . . 43
4.6 Stiffnessbehaviorofthecartwheelhinge. . . . . . . . . . . . . . . . . . . . . . . 44
4.7 Stiffnessbehaviorofthesmall-lengthflexuralpivot. . . . . . . . . . . . . . . . . . 45
5.1 Examplesofdifferentflexuretypes. . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Anearlylattice-flexuredCAFP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 NomenclaturefortheX-typeflexure. . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4 NomenclaturefortheV-typeflexure. . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.5 Plotofstiffnessreductionforthetwolatticetypes. . . . . . . . . . . . . . . . . . 53
5.6 Off-axisstiffnessbehavioroftheX-typelatticewithsquarelatticeelements. . . . . 56
5.7 Off-axisstiffnessbehavioroftheV-typelatticewithsquarelatticeelements. . . . . 57
5.8 Off-axisstiffnessbehaviorcomparedtoabladeflexure. . . . . . . . . . . . . . . . 58
5.9 CAFPprototypesusedvalidatestiffnessmodels. . . . . . . . . . . . . . . . . . . . 60
5.10 Titaniumprototypesusedinthemeasurementoflatticeflexurestiffness. . . . . . . 61
5.11 Experimentalsetupformeasuringbendingstiffness. . . . . . . . . . . . . . . . . . 61
5.12 Experimentalsetupfordeterminingtorsionalstiffness. . . . . . . . . . . . . . . . 63
5.13 Off-axis(torsional)stiffnessbehaviorofbothlatticetypes. . . . . . . . . . . . . . 63
x