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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 mathematicsintherealworld.Thesepointsmayprovidethebasisforopeninganewmathematics- 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 ©SpringerJapan2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerJapanKKispartofSpringerScience+BusinessMedia (www.springer.com) 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

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