Tensegrity Structures: Form, Stability, and Symmetry

307 Pages·2015·8.875 MB·English

by  Jing Yao Zhang, Makoto Ohsaki (auth.)

Mathematics for Industry 6
Jing Yao Zhang
Makoto Ohsaki
Tensegrity
Structures
Form, Stability, and Symmetry
Mathematics for Industry
Volume 6
Editor-in-Chief
MasatoWakayama(KyushuUniversity,Japan)
ScientificBoardMembers
RobertS.Anderssen(CommonwealthScientificandIndustrialResearchOrganisation,Australia)
HeinzH.Bauschke(TheUniversityofBritishColumbia,Canada)
PhilipBroadbridge(LaTrobeUniversity,Australia)
JinCheng(FudanUniversity,China)
MoniqueChyba(UniversityofHawaiiatMānoa,USA)
Georges-HenriCottet(JosephFourierUniversity,France)
JoséAlbertoCuminato(UniversityofSãoPaulo,Brazil)
Shin-ichiroEi(HokkaidoUniversity,Japan)
YasuhideFukumoto(KyushuUniversity,Japan)
JonathanR.M.Hosking(IBMT.J.WatsonResearchCenter,USA)
AlejandroJofré(UniversityofChile,Chile)
KerryLandman(TheUniversityofMelbourne,Australia)
RobertMcKibbin(MasseyUniversity,NewZealand)
GeoffMercer(AustralianNationalUniversity,Australia)(Deceased,2014)
AndreaParmeggiani(UniversityofMontpellier2,France)
JillPipher(BrownUniversity,USA)
KonradPolthier(FreeUniversityofBerlin,Germany)
OsamuSaeki(KyushuUniversity,Japan)
WilSchilders(EindhovenUniversityofTechnology,TheNetherlands)
ZuoweiShen(NationalUniversityofSingapore,Singapore)
Kim-ChuanToh(NationalUniversityofSingapore,Singapore)
EvgenyVerbitskiy(LeidenUniversity,TheNetherlands)
NakahiroYoshida(TheUniversityofTokyo,Japan)
Aims&Scope
The meaning of “Mathematics for Industry” (sometimes abbreviated as MI or MfI) is different
fromthatof“MathematicsinIndustry”(orof“IndustrialMathematics”).Thelatterisrestrictive:it
tendstobeidentifiedwiththeactualmathematicsthatspecificallyarisesinthedailymanagement
andoperationofmanufacturing.Theformer,however,denotesanewresearchfieldinmathematics
thatmayserveasafoundationforcreatingfuturetechnologies.Thisconceptwasbornfromthe
integrationandreorganizationofpureandappliedmathematicsinthepresentdayintoafluidand
versatileformcapableofstimulatingawarenessoftheimportanceofmathematicsinindustry,as
well as responding to the needs of industrial technologies. The history of this integration and
reorganizationindicatesthatthisbasicideawillsomedayfindincreasingutility.Mathematicscan
beakeytechnologyinmodernsociety.
Theseriesaimstopromotethistrendby(1)providingcomprehensivecontentonapplications
of mathematics, especially to industry technologies via various types of scientific research, (2)
introducingbasic,useful,necessaryandcrucialknowledgeforseveralapplicationsthroughcon-
crete subjects, and (3) introducing new research results and developments for applications of
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orientedtechnologicalworldandevennewresearchfieldsofmathematics.
More information about this series at http://www.springer.com/series/13254
Jing Yao Zhang Makoto Ohsaki
(cid:129)
Tensegrity Structures
Form, Stability, and Symmetry
123
JingYao Zhang MakotoOhsaki
NagoyaCityUniversity HiroshimaUniversity
Nagoya Higashi-Hiroshima
Japan Japan
ISSN 2198-350X ISSN 2198-3518 (electronic)
Mathematics forIndustry
ISBN 978-4-431-54812-6 ISBN 978-4-431-54813-3 (eBook)
DOI 10.1007/978-4-431-54813-3
LibraryofCongressControlNumber:2015932232
SpringerTokyoHeidelbergNewYorkDordrechtLondon
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Preface
Aims and Scope
Tensegritystructuresarenowmorethan60-yearsold,sincetheirbirthasartworks.
However,theyarenot“old”noroutoffashion!Onthecontrary,theyarebecoming
more and more present in many different fields, including but not limited to
engineering, biomedicine, and mathematics. These applications make use of the
unique mechanical as well as mathematical properties of tensegrity structures in
contrast to conventional structural forms such as trusses and frames.
Ourprimaryobjectiveinwritingthisbookistoprovideatextbookforself-study
whichiseasilyaccessiblenotonlytoengineersandscientists,butalsotoupper-level
undergraduate and graduate students. Both students and professionals will find
materialofinteresttotheminthebook.Withthisobjectiveinmind,thepresentation
ofthisbookisdetailedwithmanyexamples,andmoreover,itisself-contained.
There are already several existing books on tensegrity structures; most of them
present approaches to realization and practical applications of those structures. By
contrast,thisbookisdevotedtohelpingthereadersachieveadeeperunderstanding
offundamentalmechanical andmathematical propertiesof tensegritystructures.In
particular, emphasis is placed on the two key problems in preliminary design of
tensegrity structures—self-equilibrium and (super-)stability, by extensively utiliz-
ing the concept offorce density and high level of symmetry of the structures.
Subjects and Contents
Tensegrity structures are similar in appearance to conventional bar-joint structures
(trusses), however, their members carry forces (prestresses) evenwhen no external
load is applied. This means that their nodes and members have to be balanced by
v
vi Preface
the prestresses so as to maintain their equilibrium. Furthermore, most tensegrity
structures are intrinsically unstable in the absence of prestresses, and it is the
introduction of prestresses that makes them stable. For these reasons, finding the
self-equilibrated configuration and investigation of stability are the two key
problems in the preliminary design of tensegrity structures.
Finding the configuration associated with prestresses, in the state of self-equi-
librium,iscalledform-findingorshape-finding.Itisacommondesignproblemfor
tension structures, including tensile membrane structures and cable-nets. The
problemisdifficultbecausetheconfigurationandprestressescannotbedetermined
separatelyasaresultofthehighinterdependencybetweenthem.Furtherdifficulties
arise from the fact that tensegrity structures maintain their stability without any
support.
A structure is stable if and only if it has the locally minimum total potential
energy, or strain energy in the absence of external loads. Stability investigation of
tensegritystructuresisnecessarybecausetheirstabilitycannotbeguaranteedascan
that of cable-nets or membrane structures carrying tension only in their structural
elements. This comes from the fact that tensegrity structures are composed of
(continuous) tensile members and (discontinuous) compressive members. More-
over, it is possible for tensegrity structures to be super-stable, which is a more
robust stability criterion, if proper prestresses are associated with the proper con-
nectivity pattern.
In this book, basic concepts and applications of tensegrity structures are intro-
duced in Chap. 1. Chapter 2 formulates the matrices and vectors necessary for the
study of self-equilibrium and stability. The analytical conditions for self-equilib-
rium of several highly symmetric tensegrity structures with simple geometries are
given in Chap. 3. Chapter 4 defines the three stability criteria—stability, prestress-
stability, and super-stability—and derives the necessary conditions and sufficient
conditions for super-stability. The force density method, which guarantees super-
stability, is presented in Chap. 5 for numerical form-finding of relatively complex
tensegrity structures. Utilizing the analytical formulations for highly symmetric
structures given in Appendix D, the self-equilibrium and super-stability conditions
arederivedfortheprismatictensegritystructuresinChap.6andthoseforthestar-
shaped structures in Chap. 7; both these classes of structures are of dihedral
symmetry. Additionally, Chap. 8 presents the self-equilibrium and super-stability
conditions for structures with tetrahedral symmetry.
At the end of the preface, we have to give our deepest thanks to our families,
friends, and former and current students for their supports. Part of the work on
symmetry has been conducted in close collaboration with Dr. Simon D. Guest
of the University of Cambridge and Professor Robert Connelly of Cornell Uni-
versity; they showed us a new way to study tensegrity structures. Mr. Masaki
Preface vii
OkanoofNagoyaCityUniversityreadthefirsthalfofthebookcarefullyandfound
manymistakes,whichwethenwereabletocorrect.Wealsoappreciatetheproposal
of writing this book by Dr. Yuko Sumino of Springer Japan; she has always been
helpful during the preparation and publication of the book.
Nagoya, December 2014 Jing Yao Zhang
Higashi-Hiroshima Makoto Ohsaki
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Applications in Architecture. . . . . . . . . . . . . . . . . . . 3
1.2.2 Applications in Mechanical Engineering . . . . . . . . . . 4
1.2.3 Applications in Biomedical Engineering . . . . . . . . . . 5
1.2.4 Applications in Mathematics . . . . . . . . . . . . . . . . . . 5
1.3 Form-Finding and Stability . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Existing Form-finding Methods . . . . . . . . . . . . . . . . 7
1.3.3 Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Definition of Configuration . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Basic Mechanical Assumptions. . . . . . . . . . . . . . . . . 16
2.1.2 Connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 Geometry Realization . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Equilibrium Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Equilibrium Equations by Balance of Forces . . . . . . . 23
2.2.2 Equilibrium Equations by the Principle
of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Static and Kinematic Determinacy. . . . . . . . . . . . . . . . . . . . . 30
2.3.1 Maxwell’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.2 Modified Maxwell’s Rule . . . . . . . . . . . . . . . . . . . . 35
2.3.3 Static Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.4 Kinematic Determinacy. . . . . . . . . . . . . . . . . . . . . . 38
2.3.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
ix
x Contents
2.4 Force Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.1 Definition of Force Density Matrix. . . . . . . . . . . . . . 43
2.4.2 Direct Definition of Force Density Matrix . . . . . . . . . 45
2.4.3 Self-equilibrium of the Structures with Supports. . . . . 46
2.5 Non-degeneracy Condition for Free-standing Structures. . . . . . 48
2.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Self-equilibrium Analysis by Symmetry . . . . . . . . . . . . . . . . . . . . 55
3.1 Symmetry-based Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Symmetric X-cross Structure . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3 Symmetric Prismatic Structures. . . . . . . . . . . . . . . . . . . . . . . 62
3.3.1 Dihedral Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.2 Connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.3 Self-equilibrium Analysis. . . . . . . . . . . . . . . . . . . . . 69
3.4 Symmetric Star-shaped Structures. . . . . . . . . . . . . . . . . . . . . 75
3.4.1 Connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4.2 Self-equilibrium Analysis. . . . . . . . . . . . . . . . . . . . . 78
3.5 Regular Truncated Tetrahedral Structures. . . . . . . . . . . . . . . . 83
3.5.1 Tetrahedral Symmetry. . . . . . . . . . . . . . . . . . . . . . . 85
3.5.2 Self-equilibrium Analysis. . . . . . . . . . . . . . . . . . . . . 88
3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.1 Stability and Potential Energy. . . . . . . . . . . . . . . . . . . . . . . . 97
4.1.1 Equilibrium and Stability of a Ball Under Gravity. . . . 97
4.1.2 Total Potential Energy. . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Equilibrium and Stiffness. . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.1 Equilibrium Equations. . . . . . . . . . . . . . . . . . . . . . . 102
4.2.2 Stiffness Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3 Stability Criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3.2 Prestress-stability . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.3.3 Super-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.4 Necessary and Sufficient Conditions for Super-stability. . . . . . 128
4.4.1 Geometry Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.4.2 Sufficient Conditions. . . . . . . . . . . . . . . . . . . . . . . . 132
4.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135