a literature review of tensegrity by brigitte servatius
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A Literature Review of Tensegrity
Brigitte Servatius
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1. 41 Papers on Tensegrity
1 Tensegrity Frameworks, Roth, Whiteley, 19812 On generic rigidity in the plane., Lovsz, Yemini, 1982
3 Survey of research work on structures., Minke, 19824 Cones, infinity and 1-story buildings, Whiteley, 19835 Statics of frameworks, Whiteley, 19846 Prismic tensigrids, Hinrichs, 19847 Scene analysis and motions of frameworks, Whiteley, 19848 1-story buildings as tensegrity frameworks, Recski, 19869 A Fuller explanation. Edmondson, 1987.10 Immobile kinematic chains, Kuznetsov, 1989
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11 Rigidity and polarity. II, Whiteley, 198912 Duality between plane trusses and grillages, Tarnai, 198913 1st infinitesimal mechanisms, Calladine, Pellegrino, 199114 1-story buildings as tensegrity frameworks II, Recski 1991
15 1-story buildings as tensegrity frameworks III Recski 199216 Combinatorics to statics, 2nd survey, Recski, 199217 Rigidity, Connellly, 199318 Globally rigid symmetric tensegrities,
Connelly, Terrell, 199519 Graphs, digraphs, and the rigidity of grids, Servatius, 199520 Second-order rigidity and prestress stability,
Connelly, Whiteley 1996
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21 Controllable tensegrity, Skelton and Sultan 199722 Tensegrities and rotating rings of tetrahedra, Guest, 2000
23 Dynamics of the shell class of tensegrity structures,Skelton 2001
24 Cyclic frustum tensegrity modules,Nishimura, Murakami, 2001
25 Modelling and control of class N SP tensegrity structures,
Kanchanasaratool, Williamson, 200226 Two-distance preserving functions,Khamsemanan, Connelly, 2002
27 Tensegrity and the viscoelasticity of the cytoskeleton,Caladas, et.al. 2002
28 Motion control of a tensegrity platform,Kanchanasaratool, Williamson, 2002
29 Stability conjecture in the theory of tensegrity structures,Volokh, 2003
30 Equilibrium conditions of a tensegrity structure,Williamson, Skelton, Han, 2003
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31 Tensegrities, Heunen, van Leijenhorst, Dick, 2004
32 Algebraic tensegrity form-finding,Masic, Skelton, Gill, 2005
33 Towards understanding tensegrity in cells,Shen, Wolynes, 2005
34 Structures in hyperbolic space, Connelly, 2006
35 dynamic analysis of a planar 2-DOFtensegrity mechanism, Arsenault, Gosselin, 200636 Modeling virus self-assembly pathways,
Sitharam, Agbandje-Mckenna, 200637 Improving the DISPGB algorithm using
the discriminant ideal, Manubens, Montes, 200638 From graphs to tensegrity structures: geometric and
symbolic approaches, de Guzman, Orden, 200639 Stability conditions for tensegrity structures,
Zhang, Ohsaki, 2007
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Tensegrity frameworks
Roth, B.; Whiteley, W.
Trans. Amer. Math. Soc. 265 (1981), no. 2, 419446.
A tensegrity framework G(p) is an abstract graph G witheach edge called a bar, cable, or strut, together with an as-signment of a point pi in Euclidean space Rn for each vertexof G. Let p = ( , pi, ) Rnv, where v is the (finite)number of vertices ofG. A flexing ofG(p) is a continuous (orequivalently, analytic) path of the vertices x(t), 0 t 1,such that x(0) = p, bars stay the same length, cables do not
increase in length, and struts do not decrease in length.
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If all flexings ofG(p) are restrictions of rigid motions ofRn,then G(p) is said to be rigid, or not a mechanism. For atensegrity framework an infinitesimal motion of G(p) is an
assignment of vectors i to each vertex of G such that, foreach bar, cable, or strut, (pi pj)(i j) is = 0, 0, or 0, respectively. If = ( , i, ) is the restriction ofthe derivative of a rigid motion ofRn, is said to be trivial. IfG(p) has only trivial infinitesimal motions then G(p) is said
to be infinitesimally rigid.
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A stress for G(p) is an assignment of a scalar {i,j} for each
edge ofG such that for each vertex i,
j {i,j}(pi pj) = 0,where the sum is over the edges adjacent to i. A stress isalso required to be nonpositive on cables and nonnegative onstruts. The stress is said to be proper if the stresses on allthe cables and struts are nonzero.
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The first four sections of the paper are basically a review of
notation and a description of equivalent notions of rigidityand infinitesimal rigidity in terms of tensegrity structures.The basic results here are essentially a review in these termsof the papers by H. Gluck
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The basic result of the paper under review is Theorem 5.2,which says that a tensegrity framework G(p) is infinitesi-mally rigid if and only if G(p) has a proper stress and G(p)is infinitesimally rigid, where G is the graph obtained bychanging all the cables, struts, and bars to bars. This re-
sult is used to prove analogues for tensegrity frameworks ofGlucks result about bar frameworks. Here, for instance, itcan only be said that the set {p Rnv: G(p) is infinitesi-mally rigid}is open in Rnv and not that it is dense (Theorem5.4). Another important result is Theorem 5.10, which saysroughly that if G(p) is infinitesimally rigid, if each pi is re-placed by Lpi = qi, where L is a projective map of Rn andif certain cables and struts are changed to get G, then thenew G(q) is infinitesimally rigid.
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Section 6 specializes to tensegrity frameworks in the plane.
Applying their stress criterion for infinitesimal rigidity, theauthors provide another proof of the infinitesimal rigidity ofCauchy polygons described in the reviewers paper, as wellas many other related frameworks.
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Section 7 deals with analogous results in R3, where manyof the tensegrity frameworks of the type considered by R.Buckminster Fuller are shown to be infinitesimally rigid.
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On generic rigidity in the plane.
Lovsz, L.; Yemini, Y.SIAM J. Algebraic Discrete Methods 3 (1982), no. 1, 9198.
A finite graph whose vertices are points in Euclidean spaceis viewed as a collection of rigid bars and universal joints (a
tensegrity structure, a la Buckminster Fuller). The prob-lem of assigning a degree of floppiness to a graph in theplane in such a way that degree 0 means rigid is dealt with(a triangle is rigid, a square is not); and the authors use ma-troid theory to prove G. Lamans theorem showing how to
compute this number.
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The matroid methods are also used to show that 6-connectedgraphs are rigid in the plane but 5-connected graphs need notbe. The techniques are more algebraic than those used by
other authors in dealing with similar problems, and hopeis expressed in the present paper that higher-dimensionalanalogues to Lamans theorem will now be easier to find.Finally the authors conjecture that any n(n + 1)-connectedgraph will be rigid in n-dimensional Euclidean space.
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Survey of research work on structures
Minke, GernotStructural Topology 1982, no. 6, 2132.
This report concerns the work of the Research Laboratory forExperimental Building at the University of Kassel, in WestGermany. We describe the design and use of equipment forsoap film simulation of minimal length-sum networks con-necting configurations of given points in the plane.This research has application to city planning, in the designof roadway and pedestrian networks. Subsequent sectionsdeal with the research on tensegrity structures, with space
grid cable structures, with minimum-cost fabric-coveredframe structures, and with grid shell and pneumatic struc-tures.
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Cones, infinity and 1-story buildings.
Whiteley, WalterStructural Topology 1983, no. 8, 5370.
Consider a 1-story building constructed with a series of ver-tical columns (bars and joints possibly of different lengths),a roof framework connecting the tops of the columns, plusa minimum of three additional wall braces going to the topsof some columns. We examine the static rigidity (or equiv-alently, infinitesimal rigidity) of this framework by viewingit as a cone from the roof to the point at infinity at thecommon end of all the vertical columns. We conclude thatthe framework is statically rigid if and only if the orthogonalprojection of the roof onto a horizontal plane is staticallyrigid in the plane, and the three wall braces do not meet a
common vertical line.
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This analysis is extended to tensegrity structures (with ca-bles), to structures with extra wall braces (and fewer roofbraces) and to multi-story buildings with vertical columns.
In all cases the static rigidity of the structure is tested by thestatic rigidity of the appropriate single plane projection.We conclude with some mathematical recreation. We in-troduce the spherical model for statics as a truly projectivegeometry study.
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Infinitesimally rigid polyhedra. I. Statics of
frameworks.
Whiteley, Walter
Trans. Amer. Math. Soc. 285 (1984), no. 2, 431465.The reviewer can hardly do better than use the authorsabstract to provide a sample of the good things in store forthe reader. In this paper the concept of static rigidity forframeworks is used to describe the behavior of bar and jointframeworks built around convex (and other) polyhedra. Forexample, the author proves that a triangulated polyhedroncan be rigid even when it has vertices interior to its naturalfaces, extending Aleksandrovs extension of Cauchys rigiditytheorem. He also studies the static rigidity of tensegrityframeworks (with cables and struts in place of bars) (and)
detailed analogues of Aleksandrovs theorem for convex 4-polytopes.All this and more is presented with attention paid to theshape and history of this growing part of mathematics andwith a care which brings accuracy to an area often plagued
by theorems which are only almost true.Back to Main List
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Prismic tensigrids.
Hinrichs, Lowell A.Structural Topology 1984, no. 9, 314.
In this paper the author studies symmetric rigid structuresin space built from struts and cables and so arranged that
just one strut meets each vertex. It is this last propertywhich makes these objects interesting sculpturesthe viewerwonders why they dont collapse.The regularity imposed is that the group G of symmetries acttransitively on the vertices of the structure. In consequence,G acts transitively on the struts. The author uses a theoremon the rigidity of general tensegrities due to B. Roth and
W. Whiteley to find all structures in which the cables arepartitioned into two G-orbits.As is traditional for this journal, the illustrations are clear,and invite the reader to build models.
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A correspondence between scene analysis and
motions of frameworks.
Whiteley, WalterDiscrete Appl. Math. 9 (1984), no. 3, 269295.
It is known that projections of a spherical polyhedron into aplane correspond to a static stress on the projection. Herea different correspondence is established between a planepicture of a scene and a plate and bar framework in theplane. This provides an equivalence between these two areasof study. This also allows a correspondence between scenes
with occlusion and tensegrity frameworks where the linearsystems involve inequalities.
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One-story buildings as tensegrity frameworks.
Chakravarty, N.; Holman, G.; McGuinness, S.; Recski, A.Structural Topology No. 12 (1986), 1118.In this short, accessible paper the authors re-prove theoremsof the reviewer and H. Crapo on the bracing of plane gridsof squares and of one-story buildings.They then generalize to allow cables and struts as bracing el-ements. Thus, for example, a plane grid of squares is rigid if
and only if the directed bipartite graph describing the place-ment of the struts and cables is strongly connected. Thisresult may also be found in a book by J. A. Baglivo and J.E. Graver.The proofs rely on the direct analysis of some simple systems
of equations rather than on the machinery (or language) ofmatroid theory.
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A Fuller explanation.
The synergetic geometry of R. Buckminster FullerEdmondson, Amy C.Design Science Collection. A Pro Scientia Viva Title.Birkhauser Boston, Inc., Boston, MA, 1987.
This book is a disciples affectionate tribute to R. Buckmin-ster Fuller (18951983), inventor, architect, engineer, andphilosopher.It begins with a portrait and a view of his most famous build-
ing: the great geodesic dome in Montreal which was builtin 1967 as the U.S. Pavilion. He was deeply interested inPlatonic and Archimedean polyhedra, especially the cuboc-tahedron, which he quaintly renamed vector equilibrium.The figures on page 115 illustrate the cuboctahedral numbers
1 +
n
f=1(10f2
+ 2) as clusters of congruent balls
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Chapter 15 describes Fullers concept of tensegrity, the
most remarkable instance of which is a skeletal object con-sisting of 30 rigid struts firmly held together by 90 threads.It is symmetrical by the icosahedral group of rotations. The60 ends of the 30 struts are the vertices of one of the two chi-ral Archimedean solids, the snub dodecahedron. The threadsrun along 90 of the 150 edges, namely the 60 sides of the 12pentagonal faces and 30 shared by pairs of triangles. Nearlyparallel to those 30, and nearly three times as long, are 30diagonals which are likewise reversed by the 30 half-turnsbelonging to the icosahedral group. These 30 diagonals arethe positions for the 30 struts. The same chapter provides
a fuller explanation for the structure of geodesic domes,based on tessellations of a sphere by 20T nearly congruent,nearly equilateral triangles, where T = b2 + bc + c2, b and cbeing nonnegative integers
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Page 265 shows the Dymaxion map, in which the wholeworld is projected onto the faces of a regular icosahedronwhich is then cut along certain edges and spread out flat.
He conceived this idea in 1943, simultaneously with Profes-sor Irving Fisher of Yale. Fisher placed the North and Southpoles at two opposite vertices, with the unhappy result thatsome countries (such as Algeria) are ripped apart by anedge separating two faces. It took Fuller two years of ex-perimenting to find an orientation in which all twelve icosa-hedral vertices land in the oceanan essential requirementif land masses are not to be ripped apart.Reviewed by H. S. M. Coxeter
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On immobile kinematic chains and a fallacious
matrix analysis.
Kuznetsov, E. N.Trans. ASME J. Appl. Mech. 56 (1989), no. 1, 222224.
T. Tarnai asked the following questions about pin jointed barframeworks: (1) What criterion determines whether self-
stress stiffens an assembly which is both statically and kine-matically indeterminate? and (2) How can matrix methodsbe used to decide whether kinematical indeterminacy takesthe form of an infinitesimal or a finite mechanism?S. Pellegrino and C. R. Calladine attempted to answer these
questions.
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In the present paper the author points out, among otherthings, that the analysis by Pellegrino and Calladine is atthe least inadequate because it fails to provide informationabout whether a certain quadratic form is positive definite, inorder to detect when a certain potential function (associatedto the stress-strain characteristics on the members) is at alocal minimum. This is one very natural test to detect thestability of a structure in a special position.
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In the reviewers opinion, part of the confusion comes fromnot making the distinction between pre-stress stability,where the second derivative test works for the potential func-
tion involved, and second-order rigidity, where second-order terms are introduced and are compatible with the geo-metric distance constraints. If a structure is pre-stress stable,then it is second-order rigid, but not conversely. See a forth-coming paper by the reviewer and W. Whiteley.
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Rigidity and polarity. II. Weaving lines and
tensegrity frameworks.
Whiteley, WalterGeom. Dedicata 30 (1989), no. 3, 255279.
A weaving is a finite collection of straight lines in the plane,where certain pairs of the lines are designated so that one
passes over or under the other at the crossing. A liftingof a weaving is a corresponding collection of straight linesin 3-space that projects orthogonally onto the given weavinglines such that the crossings have the corresponding weav-ing patterns. One kind of problem is to determine when a
weaving has a lifting.
The author explores the question of what happens when a
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p q ppconfiguration of weaving lines is polarized in the plane. Linesof the weaving correspond to points, and crossings of theweaving correspond to lines between corresponding points.
Surprisingly, the proper structure to consider for the polarconfiguration is a tensegrity framework, which is a collec-tion of points in the plane with certain pairs of those pointsdesignated as either a cable or a strut. A strict in-finitesimal flex of a tensegrity framework is the assignmentof a vector pi to each point pi of the tensegrity frameworksuch that (pi pj)(pi pj) < 0 for (i, j) a cable, and(pipj)(pipj) > 0 for (i, j) a strut. This is a rather strongform of nonrigidity for the tensegrity framework, where thecables shorten and the struts lengthen. As a consequence ofthe polarity, the author shows that a weaving has a strict
lifting, where none of the lifted lines intersect, if and onlyif the corresponding polar tensegrity framework has a strictinfinitesimal flex.The author also explores similar problems with grillages,which are configurations of lines in 3-space, where certain
pairs are forced to intersect.Back to Main List
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Duality between plane trusses and grillages.
Tarnai, T.Internat. J. Solids Structures 25 (1989), no. 12, 13951409.
In the paper, projective plane duality, that is, a point-to-line,line-to-point, incidence-to-incidence correspondence betweenplane trusses and grillages of simple connection is treated.By means of linear algebra it is proved that the rank of the
equilibrium matrix of plane trusses and grillages does notchange under projective transformations and polarities; con-sequently the number of infinitesimal inextensional mecha-nisms and the number of independent states of self-stress arepreserved under these transformations. The results obtainedare also applied to structures with unilateral constraints, andby using several examples it is shown that plane tensegritytrusses have projective dual counterparts among grillageswhich can be physically modelled with popsicle sticks byweaving.
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First-order infinitesimal mechanisms.
Calladine, C. R., Pellegrino, S.Internat. J. Solids Structures 27 (1991), no. 4, 505515.
This paper discusses the analytical conditions under which apin-jointed assembly, which has s independent states of self-stress and m independent mechanisms, tightens up when itsmechanisms are excited. A matrix algorithm is set up to dis-tinguish between first-order infinitesimal mechanisms (whichare associated with second-order changes of bar length) andhigher-order infinitesimal or finite mechanisms.
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It is shown that, in general, this analysis requires the com-putation of s quadratic forms in m variables, which can beeasily computed from the states of self-stress and mechanisms
of the assembly. If any linear combination of these quadraticforms is sign definite, then the mechanisms are first-order in-finitesimal. An efficient and general algorithm to investigatethese quadratic forms is given. The calculations required areillustrated for some simple examples. Many assemblies ofpractical relevance admit a single state of self-stress (s = 1),and in this case the algorithm proposed is straightforwardto implement. This work is relevant to the analysis and de-sign of pre-stressed mechanisms, such as cable systems andtensegrity frameworks.
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O t b ildi t it f k II
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One-story buildings as tensegrity frameworks II
Recski, AndrasStructural Topology No. 17 (1991), 4352.
In a paper with the same title as this one, N. Chakravartiet al. showed that the minimal number of diagonal cablesneeded to rigidify a k l one-story building is k + l 1(when k, l 2 and k + l 5). In this paper the authorshows which sets of that size do the job in two particular
interesting cases.As usual, regard the cables as edges in the bipartite graphK(A, B), where |A| = k and |B| = l. When the subgraphcorresponding to k + l 1 bracing cables is not a tree thecables rigidify the building if and only if k l = 1 and thesubgraph is a directed circuit with 2 min(k, l) vertices.When the cables are parallel they rigidify the grid if andonly if, for every proper subset X ofA, |N(X)| > (k/l)|X|,where N(X) denotes the set of those vertices in B adjacentto at least one vertex in X. The proofs are straightforward.The paper contains interesting examples.
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One-story buildings as tensegrity frameworks.
III.
Recski, Andras; Schwaerzler, WernerDiscrete Appl. Math. 39 (1992), no. 2, 137146.
The authors of Parts I and II showed that the minimal num-ber of diagonal cables needed to rigidify a k l one-storybuilding is k + l 1 (when k, l 2 and k + l 5), andshowed which sets of that size do the job in two particularinteresting cases. In this paper the authors characterize the
minimum systems in the general case.
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As usual, regard the cables as forming a directed subgraph Gof the bipartite graph K(A, B), where |A| = k and |B| = l.G has an edge from a A to b B just when there isa cable between the northeast and southwest corners of the
corresponding room in the building. The edge goes the otherway when the cable joins the other two corners.Suppose G is not connected. Then G is rigid if andonly if it has two asymmetric components each of which isstrongly connected. (If the components of G have vertexsets A1, A2
A and B1, B2
B then these components are
asymmetric when |Ai| l = |Bi| k, i = 1, 2.)
S G i t d A AB th i di t d th
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Suppose G is connected. An AB-path is a directed pathstarting at some x A and ending at some y B. LetN(X) be the set of endpoints ofAB-paths starting in X A. Similarly, define a BA-path and N(Y). Then G is rigidif and only if|N(X)| k > |X| l for all proper subsets XofA or |N(Y)| l > |Y| k for all proper subsets Y ofB.The proof uses a generalization of a generalization of theKoenig-Hall theorem on the existence of perfect matchingsin a bipartite graph.
This paper should mark the end of the sequence of papersby Recski et al. on this subject.
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A li ti f bi t i t t ti
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Applications of combinatorics to staticsa sec-
ond survey.
Recski, Andras
Discrete Math. 108 (1992), no. 1-3, 183188.
Some recent results are presented concerning the algorithmicaspects of 2-dimensional generic rigidity and 1-story build-ings as tensegrity frameworks. Most of these results wereobtained after the completion of the first survey.
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Ri idit
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Rigidity.
Connelly, RobertHandbook of convex geometry, Vol. A, B, 223271, North-
Holland, Amsterdam, 1993.
This chapter surveys the intersection between rigidity andconvex geometry. The central result is Cauchys rigidity the-orem (1813), which says: Two convex polyhedra comprisedof the same number of equal similarly placed faces are su-perposable or symmetric. There are two main categories
for generalizations; one is in the category of polyhedra andsimilar discrete objects while the other is in the category ofappropriate smooth surfaces. The author gives a very niceand up-to-date survey of the former with an extensive list ofliterature. A nice feature is that a comparison between the
two categories of generalizations is also given.
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The author begins by discussing Dehns (1916) infinitesimalversion of Cauchys theorem and Aleksandrovs (1958) exten-
sion. Dehns theorem says: Given a convex compact poly-tope p in three-space with all faces triangles, if one formsthe associated bar-framework G(p) by taking the edges asbars and vertices as joints, the G(p) is infinitesimally rigid.When not all faces of p are triangles and one triangulates p
by adding new vertices only on the edges of p, Aleksandrovproved that the associated bar framework of the triangula-tion remains infinitesimally rigid in three-space. The authordescribes in great detail the various generalizations of thesetwo theorems by present-day authors:In another direction, Dehns theorem and Aleksandrovs the-orem have also been generalized to higher dimensions. Thisled to the proof of Barnettes lower bound theorem for tri-angulated convex polytopes and a new lower bound theoremfor nontriangulated convex polytopes.
The relatively new concepts of second order rigidity and pre
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The relatively new concepts of second-order rigidity and pre-stress stability and their connections with convex surfaces arealso discussed extensively in the final section. Other areascovered include Gruenbaum and Shephards conjectures con-cerning rigidity of certain tensegrity frameworks in the planeas well as Maxwell-Cremona theory and spider webs.There are many different concepts of rigidity. In the pastthis has caused considerable confusion. The author givesvery clear definitions of various types of rigidity and their
relationships.
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Globally rigid symmetric tensegrities
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Globally rigid symmetric tensegrities
Connelly, R.; Terrell, M.Structural Topology No. 21 (1995), 5978.
L. A. Hinrichs defined a class of symmetric tensegrity frame-works Pn(j,k), n = 3, 4, , and j, k = 1, 2, , n 1.He called them prismic tensegrids. The authors remarkthat these frameworks are never infinitesimally rigid. How-
ever, they prove that Pn(j,k) is globally rigid iff k = 1 ork = n 1. Since global rigidity implies rigidity, this alsoshows that Pn(j, 1) and Pn(j,n 1) are rigid. This is themain result of the paper.The authors also raise the question of the rigidity ofPn(j,k)for other values j and k. Since the method used in this
paper does not apply, the authors also propose some possiblelines of attack. The proof of the main theorem uses somegeneral methods developed by Connelly and Connelly andW. Whiteley
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Graphs digraphs and the rigidity of grids
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Graphs, digraphs, and the rigidity of grids.
Servatius, BrigitteUMAP J. 16 (1995), no. 1, 4369.
Here is pretty play with problems about the bracing of planegrids of squares for undergraduates who know enough linearalgebra to be comfortable with the rank of a (simple) systemof linear equations.After an algebraic prologue, graph theory assumes center
stage. No prerequisites are needed. The author gently intro-duces bipartite graphs and shows how their connectedness isequivalent to the rigidity of the braced grids to which theycorrespond. Trees and cycles occur naturally when the plotturns to minimal and minimally redundant bracings. Cables,
tensegrity and directed graphs follow in act three.This 23-page paper contains 21 useful exercises together withtheir solutions, notes for the instructor and suggestions forfurther reading, all without seeming rushed. Its an exposi-tory tour de force.
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Second-order rigidity and prestress stability for
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Second order rigidity and prestress stability for
tensegrity frameworks.
Connelly, Robert; Whiteley, Walter
SIAM J. Discrete Math. 9 (1996), no. 3, 453491.
A tensegrity framework is one with cables, bars and struts.The authors define two concepts of rigidity for such aframeworkprestress stability and second-order rigidity.
The concept of prestress stability is borrowed from struc-tural engineering. The authors also exhibit a hierarchy ofrigidityfirst-order rigidity implies prestress stability impliessecond-order rigidity implies rigidity for any framework. Aseries of examples also show that none of these implications
can be reversed.
It is well known that first-order rigidity has a duality ex-
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It is well known that first order rigidity has a duality exhibited in the equivalence of the so-called static rigidity andinfinitesimal rigidity. In this paper a duality is also developedfor second-order rigidity. Using this duality, the authors de-velop a second-order stress test which states: A second-orderflex exists if and only if for every proper self-stress of theframework the quadratic form it defines is nonpositive whenevaluated at the given first-order flex.A convex b-c polygon is a framework in the plane with points
the vertices of a convex polygon, bars on the edges and cablesinside connecting certain pairs of vertices. Roth conjecturedthat a b-c polygon is rigid if and only if it is first-order rigid.Using the second-order stress test, the authors prove the con-trapositive form of Roths conjecture in its full generality.
These results are the main contributions of the paper.
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Controllable tensegrity, a new class of smart
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Controllable tensegrity, a new class of smart
structures
Skelton and Sultan
Smart structures and materials 1997: mathematics and con-trol in smart structures (San Diego, CA), 166177, Proc.SPIE, 3039, SPIE, Bellingham, WA, 1997.
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Tensegrities and rotating rings of tetrahedra: a
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g g g
symmetry viewpoint of structural mechanics.
Guest, S. D.
Science into the next millennium: young scientists give theirvisions of the future, Part II.R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci.358 (2000), no. 1765, 229243.
Symmetry is a common attribute of both natural and en-gineering structures. Despite this, the application of sym-metry arguments to some of the basic concepts of struc-tural mechanics is still a novelty. This paper shows some ofthe insights into structural mechanics that can be obtained
through careful symmetry arguments, and demonstrates howsuch arguments can provide a key to understanding the para-doxical behaviour of some symmetric structures.
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Dynamics of the shell class of tensegrity struc-
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y g y
tures.
Skelton, Pinaud, and Mingori,
Dynamics and control of structural and mechanical systems.J. Franklin Inst. 338 (2001), no. 2-3, 255320.
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Initial shape-finding and modal analyses of cyclic
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frustum tensegrity modules.
Nishimura, Murakami,
Comput. Methods Appl. Mech. Engrg. 190 (2001), no.43-44, 57955818.Initial equilibrium and modal analyses of Kenneth Snelsonscyclic frustum tensegrity modules with an arbitrary numberof stages are presented.
There are m( 3) bars at each stage. The Maxwell numberof the modules is 6 2m and is independent of the numberof stages in the axial direction.Calladines relations reveal that there are 25m infinitesimalmechanism modes. For multi-stage modules the necessaryconditions for axial assembly of one-stage modules with the
same internal element-forces are investigated.One-stage modules with geometrically similar frustum mod-ules satisfy the necessary conditions. For pre-stressed config-urations, modal analyses were conducted to investigate themode shapes of infinitesimal mechanism modes.
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Modelling and control of class NSP tensegrity
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structures.
Kanchanasaratool, Williamson,
Internat. J. Control 75 (2002), no. 2, 123139.
The method of constrained particle dynamics is used to de-
velop a dynamic model of order 12N for a general class oftensegrity structures consisting of N compression members(i.e. bars) and tensile members (i.e. cables).This model is then used as the basis for the design of afeedback control system which adjusts the lengths of the barsto regulate the shape of the structure with respect to a given
equilibrium shape. A detailed design is provided for a 3-barstructure.
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Two-distance preserving functions.
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Khamsemanan, Connelly,Beitraege Algebra Geom. 43 (2002), no. 2, 557564.
Let c, s be positive real numbers such that c/s