Frameworks, Tensegrities, and Symmetry

299 Pages·2022·6.627 MB·English

by  Robert Connelly, Simon D. Guest

FRAMEWORKS,TENSEGRITIES,ANDSYMMETRY
This introduction to the theory of rigid structures explains how to analyze the
performanceofbuiltandnaturalstructuresunderloads,payingspecialattentionto
theroleofgeometry.Thebookunifiestheengineeringandmathematicalliteratures
by exploring different notions of rigidity – local, global, and universal – and how
they are interrelated. Important results are stated formally, but also clarified with
awiderangeofrevealingexamples.Animportantgeneralizationistotensegrities,
wherefixeddistancesarereplacedwith“cables”notallowedtoincreaseinlength
and “struts” not allowed to decrease in length. A special feature is the analysis of
symmetric tensegrities, where the symmetry of the structure is used to simplify
mattersanditallowsthetheoryofgrouprepresentationstobeapplied.Writtenfor
researchers and graduate students in structural engineering and mathematics, this
workisalsoofinteresttocomputerscientistsandphysicists.
robert connelly is Professor of Mathematics at Cornell University and a
pioneer in the study of tensegrities. His research focuses on discrete geometry,
computational geometry, and the rigidity of discrete structures and its relations
to flexible surfaces, asteroid shapes, opening rulers, granular materials, and
tensegrities.In2012hewaselectedafellowoftheAmericanMathematicalSociety.
simon d. guest isProfessorofStructuralMechanicsintheStructuresGroupof
the Department of Engineering at the University of Cambridge. His research
straddles the border between traditional structural mechanics and the study of
mechanisms,andincludesworkon“morphing”and“deployable”structures.

FRAMEWORKS, TENSEGRITIES,
AND SYMMETRY
ROBERT CONNELLY
CornellUniversity,NewYork
SIMON D. GUEST
UniversityofCambridge
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Informationonthistitle:www.cambridge.org/9780521879101
DOI:10.1017/9780511843297
©RobertConnellyandSimonD.Guest2022
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Firstpublished2022
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Names:Connelly,Robert(Mathematician),author.|
Guest,S.D.(SimonD.),author.
Title:Frameworks,tensegrities,andsymmetry/RobertConnelly,
SimonD.Guest.
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2022.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2021029104(print)|LCCN2021029105(ebook)|
ISBN9780521879101(hardback)|ISBN9780511843297(epub)
Subjects:LCSH:Structuralanalysis(Engineering)|Rigidity(Geometry)|
Engineeringmathematics.|BISAC:MATHEMATICS/DiscreteMathematics
Classification:LCCTA645.C6552021(print)|LCCTA645(ebook)|
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Contents
Preface pagexi
1 Introduction 1
1.1 Prerequisites 6
1.2 Notation 6
PartI TheGeneralCase 7
2 FrameworksandRigidity 9
2.1 Introduction 9
2.2 DefinitionofaFramework 9
2.3 DefinitionofaFlex 10
2.4 DefinitionsofRigidity 11
2.5 Exercises 16
3 First-OrderAnalysisofFrameworks 17
3.1 Introduction 17
3.2 Kinematics 17
3.3 Statics 27
3.4 Static/KinematicDuality 35
3.5 GraphicalStatics 37
3.6 First-OrderStiffness 41
3.7 Example:StructuralAnalysisofaPin-JointedCantilever 42
3.8 TheBasicRigidityTheorem 47
3.9 AnotherExampleofInfinitesimalRigidity 49
3.10 ProjectiveTransformations 55
3.11 Exercises 59
vii
viii Contents
4 Tensegrities 62
4.1 Introduction 62
4.2 RigidityQuestions 64
4.3 InfinitesimalRigidity 64
4.4 StaticRigidity 66
4.5 ElementaryForces 67
4.6 FarkasAlternative 68
4.7 EquivalenceofStaticandInfinitesimalRigidity 69
4.8 Roth–WhiteleyCriterionforInfinitesimalRigidity 71
4.9 First-OrderStiffness 72
4.10 ApplicationtoCirclePackings 73
4.11 Exercises 76
5 EnergyFunctionsandtheStressMatrix 78
5.1 Introduction 78
5.2 EnergyFunctionsandRigidity 78
5.3 QuadraticEnergyFunction 81
5.4 Equilibrium 81
5.5 ThePrincipleofLeastEnergy 84
5.6 TheStressMatrix 86
5.7 TheConfigurationMatrix 89
5.8 UniversalConfigurationsExist 90
5.9 ProjectiveInvariance 92
5.10 UnyieldingandGloballyRigidExamples 94
5.11 UniversalTensegrities 95
5.12 SmallUnyieldingTensegrities 95
5.13 AffineMotionsRevisited 98
5.14 TheFundamentalTheoremofTensegrityStructures 101
5.15 Exercises 107
6 PrestressStability 110
6.1 Introduction 110
6.2 AGeneralEnergyFunction 110
6.3 QuadraticForms 119
6.4 ReducingtheCalculation 121
6.5 Second-OrderRigidity 121
6.6 CalculatingPrestressabilityandSecond-OrderRigidity 125
6.7 Second-OrderDuality 127
6.8 TriangulatedSpheres 130
6.9 Exercises 132
Contents ix
7 GenericFrameworks 135
7.1 Introduction 135
7.2 DefinitionofGeneric 135
7.3 InfinitesimalRigidityisaGenericProperty 136
7.4 NecessaryConditionsforBeingGenericallyRigid 139
7.5 GenericRigidityinthePlane 140
7.6 PebbleGame 143
7.7 VertexSplitting 146
7.8 GenericGlobalRigidity 147
7.9 Applications 153
7.10 Exercises 153
8 FiniteMechanisms 155
8.1 Introduction 155
8.2 FiniteMechanismsUsingtheRigidityMap 156
8.3 FiniteMechanismsUsingSymmetry 157
8.4 AlgebraicMethodsforCreatingFiniteMechanisms 161
8.5 Crinkles 162
8.6 ATriangulatedSurfacethatisaFiniteMechanism 165
8.7 Carpenter’sRuleProblem 170
8.8 AlgebraicSetsandSemi-AlgebraicSets 176
8.9 Exercises 181
PartII SymmetricStructures 185
9 GroupsandRepresentationTheory 187
9.1 Introduction 187
9.2 WhatisSymmetry? 187
9.3 WhatisaGroup? 188
9.4 HomomorphismsandIsomorphismsofGroups 193
9.5 Representations 195
10 First-OrderSymmetryAnalysis 205
10.1 InternalandExternalVectorSpaces 205
10.2 DecompositionofInternalandExternalVectorSpaces 206
10.3 InternalandExternalVectorSpacesasRG-Modules 209
10.4 Symmetry Operations, Equilibrium and Compatibility –
RG-Homomorphisms 218
10.5 DecompositionofInternalandExternalRG-Modules 222
10.6 IrreducibleSubmodules 229