Author | Title | Year | Journal/Proceedings | Reftype | DOI/URL |
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Arsenault, M. | Développement et analyse de mécanismes de tenségrité [BibTeX] |
2006 | School: Université Laval |
phdthesis | |

BibTeX:
@phdthesis{arsenaultphd2006, author = {Arsenault, M.}, title = {Développement et analyse de mécanismes de tenségrité}, school = {Université Laval}, year = {2006} } |
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Arsenault, M. & Gosselin, C. | Kinematic, Static, and Dynamic Analysis of a Planar One-Degree-of-Freedom Tensegrity Mechanism | 2005 | Journal of Mechanical Design Vol. 127(6), pp. 1152-1160 |
article | DOI |

Abstract: The use of tensegrity systems as structures has been extensively studied. However, their development for use as mechanisms is quite recent even though they present such advantages as reduced mass and a deployment capability. The object of this paper is to apply analysis methods usually reserved for conventional mechanisms to a planar one-degree-of-freedom tensegrity mechanism. This mechanism is obtained from a three-degree-of-freedom tensegrity system by adding actuation to the latter as well as by making some assumptions of symmetry. Analytical solutions are thus developed for the mechanism's direct and inverse static problems. Furthermore, the working curve, singularities, and stiffness of the mechanism are detailed. Finally, a dynamic model of the mechanism is developed and a preliminary control scheme is proposed. |
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BibTeX:
@article{arsenault2005, author = {Arsenault, M. and Gosselin, C.}, title = {Kinematic, Static, and Dynamic Analysis of a Planar One-Degree-of-Freedom Tensegrity Mechanism}, journal = {Journal of Mechanical Design}, year = {2005}, volume = {127}, number = {6}, pages = {1152-1160}, doi = {http://dx.doi.org/10.1115/1.1913705} } |
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Arsenault, M. & Gosselin, C.M. | Kinematic and static analysis of a 3-PUPS spatial tensegrity mechanism | 2009 | Mechanism and Machine Theory Vol. 44(1), pp. 162-179 |
article | DOI |

Abstract: The development of tensegrity mechanisms is motivated by their reduced inertia which is made possible by an extensive use of cables and springs. In this paper, a new spatial tensegrity mechanism is introduced. The direct and inverse static problems of the mechanism are solved by minimizing its potential energy. For a simplified case where external and gravitational loads are neglected, analytical solutions to these problems are found and are then used to compute the boundaries of the mechanism’s actuator and Cartesian workspaces.e |
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BibTeX:
@article{arsenault2008, author = {M. Arsenault and C. M. Gosselin}, title = {Kinematic and static analysis of a 3-PUPS spatial tensegrity mechanism}, journal = {Mechanism and Machine Theory}, year = {2009}, volume = {44}, number = {1}, pages = {162-179}, doi = {http://dx.doi.org/10.1016/j.mechmachtheory.2008.02.005} } |
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Azadi, M., Behzadipour, S. & Faulkner, G. | A Variable Spring using a Tensegrity Prism | 2009 | Proceedings of the 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2009 | conference | |

Abstract: Prestressable pin jointed structures are statically indeterminate but kinematically they can be either overdeterminate or indeterminate. Tensegrities are special cases of the latter class. Although, all prestressable structures have variable stiffness that can be controlled by prestress, it is only a subgroup of them such as tensegrities that make effective variable stiffness springs. An N-gon tensegrity prism is investigated as a basis for making a variable stiffness spring called Tensegrity Prism spring. The appropriate deflection direction is found and force-displacement and stiffness equations of a translational tensegrity prism spring with n bars are found. In addition, the main characteristics of a variable tensegrity prism spring and the methods that improve them are discussed. |
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BibTeX:
@conference{azadi2009, author = {M. Azadi and S. Behzadipour and G. Faulkner}, title = {A Variable Spring using a Tensegrity Prism}, booktitle = {Proceedings of the 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2009}, year = {2009} } |
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Baker, A. & Crane III, C. | Analysis of three degree of freedom 6x6 tensegrity platform | 2006 | Proceedings of DETC/CIE 2006, ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 10-13, 2006, Philadelphia, Pennsylvania, USA | conference | |

Abstract: The mechanism studied in this paper is a three degree of freedom 6x6 tensegrity structure. A tensegrity structure is one that balances internal (pre-stressed) forces of tension and compression. These structures have the unique property of stabilizing themselves if subjected to certain types of disturbances. The structure analyzed in this paper consists of two rigid bodies (platforms) connected by a total of six members. Three of the members are noncompliant constant-length struts and the other three members consist of springs. For typical parallel mechanisms, if the bottom platform is connected to the ground and the top platform is connected to the base by six compliant leg connectors, the top platform will have six degrees of freedom relative to the bottom platform. However, because three of the six members connecting the two platforms are noncompliant constant-length struts, the top platform has only three degrees of freedom.The primary contribution of this paper is the analysis of the three degree of freedom tensegrity platform. Specifically, given the location of the connector points on the base and top platforms, the lengths of the three noncompliant constant-length struts, and the desired location of a point embedded in the top platform measured with respect to a coordinate system attached to the base, all possible orientations of the top platform are determined. |
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BibTeX:
@conference{baker2006, author = {Baker, A. and Crane III, C.D.}, title = {Analysis of three degree of freedom 6x6 tensegrity platform}, booktitle = {Proceedings of DETC/CIE 2006, ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 10-13, 2006, Philadelphia, Pennsylvania, USA}, year = {2006} } |
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Bayat, J. & Crane III, C. | Closed-Form Equilibrium Analysis of Planar Tensegrity Mechanisms | 2006 | Proceedings of Florida Conference on Recent Advances in Robotics, Miami, Florida, May 25-26, 2006 | conference | URL |

Abstract: This paper presents a closed-form analysis of a series of planar tensegrity mechanisms to determine all possible equilibrium configurations for the device when no external forces or moments are applied. The equilibrium position is determined by identifying the configurations at which the potential energy stored in the two springs is a minimum. For a two-spring system, a 28th degree polynomial expressed in terms of the length of one of the springs is developed where this polynomial identifies the cases where the change in potential energy with respect to a change in the spring length is zero. Three spring systems are also analyzed. This more complex systems was solved using the Continuation Method. Numerical examples are presented. |
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BibTeX:
@conference{bayat2006, author = {Bayat, J. and Crane III, C.D.}, title = {Closed-Form Equilibrium Analysis of Planar Tensegrity Mechanisms}, booktitle = {Proceedings of Florida Conference on Recent Advances in Robotics, Miami, Florida, May 25-26, 2006}, year = {2006}, url = {http://www.eng.fiu.edu/mme/robotics/fcrar2006/sessions.htm} } |
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Buckminster Fuller, R. | Tensile-Integrity Structures [BibTeX] |
1962 | US Patent Number 3,063,521 | misc | |

BibTeX:
@misc{fuller1962, author = {Buckminster Fuller, R}, title = {Tensile-Integrity Structures}, year = {1962} } |
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Calladine, C. & Pellegrino, S. | Authors' closure [BibTeX] |
1991 | International Journal of Solids and Structures Vol. 27(4), pp. 521-522 |
article | DOI |

BibTeX:
@article{calladine1991-b, author = {Calladine, C.R. and Pellegrino, S.}, title = {Authors' closure}, journal = {International Journal of Solids and Structures}, year = {1991}, volume = {27}, number = {4}, pages = {521--522}, doi = {http://dx.doi.org/10.1016/0020-7683(91)90139-7} } |
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Calladine, C.R. | Buckminster Fuller's `tensegrity' structures and Clerk Maxwell's rules for the construction of stiff frames | 1978 | International Journal of Solids and Structures Vol. 14(2), pp. 161-172 |
article | DOI |

Abstract: Maxwell has shown that b bars assembled into a frame having j joints would, in general, be simply stiff if b = 3j-6. Some of Buckminster Fuller's "Tensegrity" structures have fewer bars than are necessary to satisfy Maxwell's rule, and yet are not "mechanisms" as one might expect, but are actually stiff structures. Maxwell anticipates special cases of this sort, and states that their stiffness will "be of a low order". In fact, the conditions under which Maxwell's exceptional cases occur also permit at least one state of "self-stress" in the frame.Linear algebra enables us to find the number of "incipient" modes of low-order stiffness of the frame in terms of the numbers of bars, joints and independent states of self-stress. Self-stress in the frame has the effect of imparting first-order stiffness to the frame, and it seems from experiments that a single state of self-stress can stiffen a large number of modes. It is this factor which Fuller exploits to make satisfactory structures. |
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BibTeX:
@article{calladine1978, author = {Calladine, C. R.}, title = {Buckminster Fuller's `tensegrity' structures and Clerk Maxwell's rules for the construction of stiff frames}, journal = {International Journal of Solids and Structures}, year = {1978}, volume = {14}, number = {2}, pages = {161--172}, doi = {http://dx.doi.org/10.1016/0020-7683(78)90052-5} } |
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Calladine, C.R. & Pellegrino, S. | Further remarks on first-order infinitesimal mechanisms | 1992 | International Journal of Solids and Structures Vol. 29(17), pp. 2119-2122 |
article | DOI |

Abstract: This paper analyses a particular structural assembly that has two independent inextensional mechanisms and two independent states of self-stress. It is shown that, for certain values of a variable bar length, this assembly is a first-order infinitesimal mechanism; and yet it cannot be stiffened by the imposition of a state of prestress. |
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BibTeX:
@article{calladine1992, author = {Calladine, C. R. and Pellegrino, S.}, title = {Further remarks on first-order infinitesimal mechanisms}, journal = {International Journal of Solids and Structures}, year = {1992}, volume = {29}, number = {17}, pages = {2119--2122}, doi = {http://dx.doi.org/10.1016/0020-7683(92)90060-7} } |
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Calladine, C.R. & Pellegrino, S. | First-order infinitesimal mechanisms | 1991 | International Journal of Solids and Structures Vol. 27(4), pp. 505-515 |
article | DOI |

Abstract: This paper discusses the analytical conditions under which a pin-jointed assembly, which has s independent states of self-stress and m independent mechanisms, tightens up when its mechanisms are excited. A matrix algorithm is set up to distinguish between first-order infinitesimal mechanisms (which are associated with second-order changes of bar length) and higher-order infinitesimal or finite mechanisms. It is shown that, in general, this analysis requires the computation of s quadratic forms in m variables, which can be easily computed from the states of self-stress and mechanisms of the assembly. If any linear combination of these quadratic forms is sign definite, then the mechanisms are first-order infinitesimal. An efficient and general algorithm to investigate these quadratic forms is given. The calculations required are illustrated for some simple examples. Many assemblies of practical relevance admit a single state of self-stress (s = 1), and in this case the algorithm proposed is straightforward to implement. This work is relevant to the analysis and design of pre-stressed mechanisms, such as cable systems and tensegrity frameworks. |
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BibTeX:
@article{calladine1991, author = {Calladine, C. R. and Pellegrino, S.}, title = {First-order infinitesimal mechanisms}, journal = {International Journal of Solids and Structures}, year = {1991}, volume = {27}, number = {4}, pages = {505--515}, doi = {http://dx.doi.org/10.1016/0020-7683(91)90137-5} } |
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Carwardine, G. | Improvements in equipoising mechanism. [BibTeX] |
1935 | UK Pat. 433 617 (filed 10 February and 7 March 1934, granted 12 August 1935) | misc | |

BibTeX:
@misc{carwardine1935, author = {Carwardine, G}, title = {Improvements in equipoising mechanism.}, year = {1935} } |
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Connelly, R. | Tensegrity structures: Why are they stable? [BibTeX] |
1999 | Rigidity Theory and Applications, pp. 47-54 | incollection | URL |

BibTeX:
@incollection{connelly1999, author = {Connelly, R}, title = {Tensegrity structures: Why are they stable?}, booktitle = {Rigidity Theory and Applications}, publisher = {Kluwer Academic/Plenum Publishers}, year = {1999}, pages = {47--54}, url = {http://www.math.cornell.edu/~connelly/stability.pdf} } |
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Connelly, R. & Back, W. | Mathematics and Tensegrity | 1998 | American Scientist Vol. 86(2), pp. 142-151 |
article | DOI |

Abstract: Tensegrity, a coined word describing a structure that retains its integrity under tension, is a concept developed by the American sculptor, Kenneth Snelson. The wonder and beauty of Snelson's sculptures surely lie in their three-dimensional nature. But these assemblies also pose interesting and difficult questions for mathematicians. Mathematically, what is a tensgrity? Why is it stable? Can tensegrities be classified or listed? The authors' recent work has aimed to find a proper three-dimensional generalization for tensegrities. Using the mathematical tools of group theory and representation theory, coupled with the powerful graphic and computational capabilities of computers, they have drawn up a complete catalogue of tensegrities with certain prescribed types of stability and symmetry, including some that have never been seen before. |
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BibTeX:
@article{connelly1998, author = {Connelly, R. and Back, W.}, title = {Mathematics and Tensegrity}, journal = {American Scientist}, year = {1998}, volume = {86}, number = {2}, pages = {142--151}, doi = {http://dx.doi.org/10.1511/1998.2.142} } |
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Connelly, R. & Terrell, M. | Globally Rigid Symmetric Tensegrities | 1995 | Structural Topology Vol. 21, pp. 59-77 |
article | URL |

Abstract: Si on construit une structure de tenségrité avec des câbles et des étais, quand sera-t-elle globalement rigide, au sens où il n’existe aucune autre configuration non congruente des points satisfaisant les contraintes des câbles et des étais? Nous étudions ici une farnille de ce type de structures qui possèdent une symétrie dièdre et nous caractérisons complètement celles qui sont globalement rigides. Nous utilisons certaines techques de contrainte-énergie qui requièrent la démonstration qu’une certaine matrice symétrique est positive semidéfinie avec le bon rang. |
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BibTeX:
@article{connelly1995, author = {Connelly, R. and Terrell, M.}, title = {Globally Rigid Symmetric Tensegrities}, journal = {Structural Topology}, year = {1995}, volume = {21}, pages = {59--77}, url = {https://e-revistes.upc.edu/handle/2099/1099} } |
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Connelly, R. & Whiteley, W. | Second-order rigidity and prestress stability for tensegrity frameworks | 1996 | SIAM Journal of Discrete Mathematics Vol. 7(3), pp. 453-491 |
article | DOI |

Abstract: This paper defines two concepts of rigidity for tensegrity frameworks (frameworks with cables, bars, and struts): prestress stability and second-order rigidity. We demonstrate a hierarchy of rigidity -- first-order rigidity implies prestress stability implies second-order rigidity implies rigidity -- for any framework. Examples show that none of these implications are reversible, even for bar frameworks. Other examples illustrate how these results can be used to create rigid tensegrity frameworks.This paper also develops a duality for second-order rigidity, leading to a test which combines information on the self stresses and the first-order flexes of a framework to detect second-order rigidity. Using this test, the following conjecture of Roth is proven: a plane tensegrity framework, in which the vertices and bars form a strictly convex polygon with additional cables across the interior, is rigid if and only if it is first-order rigid. |
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BibTeX:
@article{connelly1996, author = {Connelly, R. and Whiteley, W.}, title = {Second-order rigidity and prestress stability for tensegrity frameworks}, journal = {SIAM Journal of Discrete Mathematics}, year = {1996}, volume = {7}, number = {3}, pages = {453--491}, doi = {http://dx.doi.org/10.1137/S0895480192229236} } |
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Connelly, R. & Whiteley, W. | The Stability of Tensegrity Frameworks [BibTeX] |
1992 | International Journal of Space Structures Vol. 7(2), pp. 153-163 |
article | |

BibTeX:
@article{connelly1992, author = {Connelly, R. and Whiteley, W.}, title = {The Stability of Tensegrity Frameworks}, journal = {International Journal of Space Structures}, year = {1992}, volume = {7}, number = {2}, pages = {153--163} } |
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Crane III, C.D., Bayat, J., Vikas, V. & Roberts, R. | Kinematic Analysis of a Planar Tensegrity Mechanism with Pre-Stressed Springs [BibTeX] |
2008 | Advances in Robot Kinematics: Analysis and Design, pp. 419-427 | inbook | DOI |

BibTeX:
@inbook{crane2008, author = {Crane III, Carl D. and Bayat, Jahan and Vikas, Vishesh and Roberts, Rodney}, title = {Kinematic Analysis of a Planar Tensegrity Mechanism with Pre-Stressed Springs}, booktitle = {Advances in Robot Kinematics: Analysis and Design}, publisher = {Springer Netherlands}, year = {2008}, pages = {419-427}, doi = {http://dx.doi.org/10.1007/978-1-4020-8600-7_44} } |
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Crane III, C.D., Duffy, J. & Correa, J. | Static Analysis of Tensegrity Structures | 2005 | Journal of Mechanical Design Vol. 127(2), pp. 257-268 |
article | DOI |

Abstract: Two problems are addressed in this paper. First, the mathematical model to perform the static analysis of an antiprism tensegrity structure subjected to a wide variety of external loads is presented. The virtual work approach is used to deduce the equilibrium equations and a method based on Newton's Third Law is used to verify the numerical results. Two numerical examples are provided to demonstrate the use of the mathematical model, as well as the verification method. The second problem deals with the development of a mathematical model to perform the static analysis of a prestressed antiprism tensegrity structure subjected to an arbitrary length reduction of its connecting ties. Again, a virtual work approach is used to deduce the equilibrium equations and the numerical results are verified using a Newtonian approach. One example is provided to illustrate the mathematical model. |
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BibTeX:
@article{crane2005, author = {Crane III, C. D. and Duffy, J. and Correa, J.}, title = {Static Analysis of Tensegrity Structures}, journal = {Journal of Mechanical Design}, year = {2005}, volume = {127}, number = {2}, pages = {257--268}, doi = {http://dx.doi.org/10.1115/1.1804194} } |
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Deng, H. & Kwan, A.S.K. | Unified classification of stability of pin-jointed bar assemblies | 2005 | International Journal of Solids and Structures Vol. 42(15), pp. 4393-4413 |
article | DOI |

Abstract: Based on the energy criterion and geometrical nonlinearity theory, this paper broadens conventional concepts of structural stability to explain some non-generic stability phenomena of pin-jointed bar assemblies in a unified and coherent way. A novel classification for stability conditions of such kind of structures is put forward, using analysis of the constitution of the tangential stiffness matrix. Some classical issues, including geometrical stability and stability of mechanisms, are re-investigated under this new concept as part of the formal theoretical development. Effects of bars stiffness are introduced into the necessary and sufficient conditions of intrinsic stability (stability of structure devoid of internal forces). The stability conditions for mechanisms, whether they acquire stiffness from self-stressing or external loading, are also probed. The stability of infinitesimal mechanism is expounded through consideration of high-order variations of the potential energy. Some discussions are provided at the end to build up an integrated understanding of stability of pin-jointed bar assemblies. |
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BibTeX:
@article{deng2005, author = {Deng, H. and Kwan, A. S. K.}, title = {Unified classification of stability of pin-jointed bar assemblies}, journal = {International Journal of Solids and Structures}, year = {2005}, volume = {42}, number = {15}, pages = {4393--4413}, doi = {http://dx.doi.org/10.1016/j.ijsolstr.2005.01.009} } |
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Estrada, G.G., Bungartz, H.-J. & Mohrdieck, C. | Numerical form-finding of tensegrity structures | 2006 | International Journal of Solids and Structures Vol. 43(22-23), pp. 6855-6868 |
article | DOI |

Abstract: A novel and versatile numerical form-finding procedure that requires only a minimal knowledge of the structure is presented. The procedure only needs the type of each member, i.e. either compression or tension, and the connectivity of the nodes to be known. Both equilibrium geometry and force densities are iteratively calculated. A condition of a maximal rank of the force density matrix and minimal member length, were included in the form-finding procedure to guide the search of a state of self-stress with minimal elastic potential energy. It is indeed able to calculate novel configurations, with no assumptions on cable lengths or cable-to-strut ratios. Moreover, the proposed approach compares favourably with all the leading techniques in the field. This is clearly exemplified through a series of examples. |
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BibTeX:
@article{gomezestrada2006, author = {G. Gomez Estrada and H.-J. Bungartz and C. Mohrdieck}, title = {Numerical form-finding of tensegrity structures}, journal = {International Journal of Solids and Structures}, year = {2006}, volume = {43}, number = {22-23}, pages = {6855-6868}, doi = {http://dx.doi.org/10.1016/j.ijsolstr.2006.02.012} } |
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Fest, E., Shea, K. & Smith, I.F.C. | Active Tensegrity Structure | 2004 | Journal of Structural Engineering Vol. 130(10), pp. 1454-1465 |
article | DOI |

Abstract: Most active structures involve direct control of single parameters when there is a closed form relationship between the response required and the control parameter. Building on a previous study of an adjustable structure, this paper describes geometric active control of a reusable tensegrity structure that has been enlarged to five modules with improved connections and is equipped with actuators. Closely coupled strut and cable elements behave nonlinearly (geometrically) even for small movement of the 10 telescopic struts. The control criterion for maintaining the upper surface slope has no closed form relationship with strut movement. The behavior of the structure is studied under 25 load cases. A newly developed stochastic search algorithm successfully identifies good control commands following computation times of up to 1 h. Sequential application of the commands through sets of partial commands helps to avoid exceeding limits during intermediate stages and adds robustness to the system. Reuse of a previously calculated command reduces the response time to less than 1 min. Feasible storage and reuse of such commands confirm the potential for improving performance during service. |
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BibTeX:
@article{fest2004, author = {E. Fest and K. Shea and I. F. C. Smith}, title = {Active Tensegrity Structure}, journal = {Journal of Structural Engineering}, year = {2004}, volume = {130}, number = {10}, pages = {1454-1465}, doi = {http://dx.doi.org/10.1061/(ASCE)0733-9445(2004)130:10(1454)} } |
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Fowler, P.W. & Guest, S.D. | A symmetry extension of Maxwell's rule for rigidity of frames | 2000 | International Journal of Solids and Structures Vol. 37(12), pp. 1793-1804 |
article | DOI |

Abstract: A symmetry extension of Maxwell's rule for rigidity of frames is found. This rule subsumes and strengthens Maxwell's 1864 rule by requiring that the internal and external degrees of freedom of a pin-jointed structure are not only numerically equal, but are also equisymmetric. A number of special cases of Maxwell's original rule are studied to show the improved insight that the symmetry-adapted version can give. |
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BibTeX:
@article{fowler2000, author = {Fowler, P. W. and Guest, S. D.}, title = {A symmetry extension of Maxwell's rule for rigidity of frames}, journal = {International Journal of Solids and Structures}, year = {2000}, volume = {37}, number = {12}, pages = {1793--1804}, doi = {http://dx.doi.org/10.1016/S0020-7683(98)00326-6} } |
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French, M.J. & Widden, M.B. | The spring-and-lever balancing mechanism, George Carwardine and the Anglepoise lamp | 2000 | Proceedings of the Institution of Mechanical Engineers part C -- Journal of Mechanical Engineering Science Vol. 214(3), pp. 501-508 |
article | DOI |

Abstract: Two short and direct methods of exact analysis of the spring-and-lever balancing mechanism are presented. It is shown that perfect balance can be achieved by the use of a close-coiled spring whose free length is effectively zero and whose stiffness is chosen appropriately. The two-degree-of-freedom balancing mechanism, commonly seen in desk lamps but useful in many other situations, is then analysed. The treatment is extended to allow for the weight of the components of the mechanism itself. The originator of these devices was George Carwardine, whose patents over the period 1931-35 show the evolution of his ideas on this subject. A short biographical note is included as an appendix. |
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BibTeX:
@article{french2000, author = {French, M J and Widden, M B}, title = {The spring-and-lever balancing mechanism, George Carwardine and the Anglepoise lamp}, journal = {Proceedings of the Institution of Mechanical Engineers part C -- Journal of Mechanical Engineering Science}, year = {2000}, volume = {214}, number = {3}, pages = {501--508}, doi = {http://dx.doi.org/10.1243/0954406001523137} } |
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Guest, S.D. | The stiffness of prestressed frameworks: A unifying approach | 2006 | International Journal of Solids and Structures Vol. 43(3-4), pp. 842-854 |
article | DOI |

Abstract: A simple derivation of the tangent stiffness matrix for a prestressed pin-jointed structure is given, and is used to compare the diverse formulations that can be found in the literature for finding the structural response of prestressed structures. |
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BibTeX:
@article{guest2006, author = {Guest, S D}, title = {The stiffness of prestressed frameworks: A unifying approach}, journal = {International Journal of Solids and Structures}, year = {2006}, volume = {43}, number = {3--4}, pages = {842--854}, doi = {http://dx.doi.org/10.1016/j.ijsolstr.2005.03.008} } |
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Guest, S.D. | Tensegrities and rotating rings of tetrahedra: a symmetry viewpoint of structural mechanics | 2000 | Philosophical Transactions of the Royal Society of London Vol. 358(1765), pp. 229-243 |
article | DOI |

Abstract: Symmetry is a common attribute of both natural and engineering structures. Despite this, the application of symmetry arguments to some of the basic concepts of structural mechanics is still a novelty. This paper shows some of the insights into structural mechanics that can be obtained through careful symmetry arguments, and will show how these can provide a key to understanding the paradoxical behaviour of some symmetric structures. |
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BibTeX:
@article{guest2000, author = {Guest, S. D.}, title = {Tensegrities and rotating rings of tetrahedra: a symmetry viewpoint of structural mechanics}, journal = {Philosophical Transactions of the Royal Society of London}, year = {2000}, volume = {358}, number = {1765}, pages = {229--243}, doi = {http://dx.doi.org/10.1098/rsta.2000.0529} } |
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Hanaor, A. | Prestressed pin-jointed structures -- flexibility analysis and prestress design | 1988 | Computers & Structures Vol. 28(6), pp. 757-769 |
article | DOI |

Abstract: Two classes of prestressable structures exist. Class I are geometrically rigid and statically indeterminate structures. Class II are statically and kinematically indeterminate structures with infinitesimal mechanisms. In structures of class I, prestress by means of imposed lack of fit can substantially enhance the design, particularly when failure is governed by compression member buckling. Structures of class II depend on prestress for their geometric integrity.A unified algorithm for the analysis and for the design of prestress of pin-jointed structures of both classes, consisting of any combination of bars and cables, is presented. The algorithm is based on the flexibility method of structural analysis, and assumes small displacements. It is capable of identifying states of prestress and mechanisms and optimizing the prestress in compression controlled structures. |
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BibTeX:
@article{hanaor1988, author = {Hanaor, A.}, title = {Prestressed pin-jointed structures -- flexibility analysis and prestress design}, journal = {Computers & Structures}, year = {1988}, volume = {28}, number = {6}, pages = {757--769}, doi = {http://dx.doi.org/10.1016/0045-7949(88)90416-6} } |
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Herder, J.L. | Energy-Free Systems. Theory, conception and design of statically balanced spring mechanisms [BibTeX] |
2001 | School: Delft University of Technology, The Netherlands |
phdthesis | URL |

BibTeX:
@phdthesis{herderphd2001, author = {Herder, J L}, title = {Energy-Free Systems. Theory, conception and design of statically balanced spring mechanisms}, school = {Delft University of Technology, The Netherlands}, year = {2001}, url = {http://mms.tudelft.nl/staff/herder/statbal.htm} } |
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Hinrichs, L.A. | Prismic Tensigrids [BibTeX] |
1984 | Structural Topology Vol. 9, pp. 3-14 |
article | URL |

BibTeX:
@article{hinrichs1984, author = {Hinrichs, L. A.}, title = {Prismic Tensigrids}, journal = {Structural Topology}, year = {1984}, volume = {9}, pages = {3--14}, url = {https://e-revistes.upc.edu/handle/2099/1009} } |
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Ingber, D.E. | The Architecture of Life [BibTeX] |
1998 | Scientific American Vol. 6(1), pp. 48-57 |
article | URL |

BibTeX:
@article{ingber1998, author = {Ingber, D E}, title = {The Architecture of Life}, journal = {Scientific American}, year = {1998}, volume = {6}, number = {1}, pages = {48--57}, url = {http://www.sciamdigital.com/index.cfm?fa=Products.ViewIssuePreview&ARTICLEID_CHAR=E0933DE8-00FC-49CA-ABA0-E4EE1F88D0E} } |
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Juan, S.H. & Tur, J.M.M. | Tensegrity frameworks: Static analysis review | 2008 | Mechanism and Machine Theory Vol. 43(7), pp. 859-881 |
article | DOI |

Abstract: This paper hands in a review of the basic issues about the statics of tensegrity structures. Definitions and notation for the most important concepts, borrowed from the vast existing literature, are summarized. All of these concepts and definitions provide a complete mathematical framework to analyze the rigidity and stability properties of tensegrity structures from three different, but related, points of view: motions, forces and energy approaches. Several rigidity and stability definitions are presented in this paper and hierarchically ordered, from the strongest condition of infinitesimal rigidity to the more wide concept of simple rigidity, so extending some previous classifications already available. Important theorems regarding the relationship between these definitions are also put together to complete the static overview of tensegrity structures. Examples of different tensegrity structures belonging to each of the rigidity and stability categories presented are described and analyzed. Concluding the static analysis of tensegrity structures, a review of existing form-finding methods is presented. |
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BibTeX:
@article{juan2008, author = {S. H. Juan and J. M. Mirats Tur}, title = {Tensegrity frameworks: Static analysis review}, journal = {Mechanism and Machine Theory}, year = {2008}, volume = {43}, number = {7}, pages = {859-881}, doi = {http://dx.doi.org/10.1016/j.mechmachtheory.2007.06.010} } |
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Kangwai, R.D. & Guest, S.D. | Symmetry-adapted equilibrium matrices | 2000 | International Journal of Solids and Structures Vol. 37(11), pp. 1525-1548 |
article | DOI |

Abstract: A number of authors have previously used group representation theory to block-diagonalise the stiffness matrix of a symmetric structure. This paper describes how similar techniques can be used to block-diagonalise the equilibrium matrix of a symmetric structure. This is shown to provide useful insight into the static and kinematic response of such systems. In particular, it simplifies finding and classifying states of self-stress and mechanisms, as well as reducing the computational effort required for a Force Method analysis. |
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BibTeX:
@article{kangwai2000, author = {Kangwai, R D and Guest, S D}, title = {Symmetry-adapted equilibrium matrices}, journal = {International Journal of Solids and Structures}, year = {2000}, volume = {37}, number = {11}, pages = {1525-1548}, doi = {http://dx.doi.org/10.1016/S0020-7683(98)00318-7} } |
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Kangwai, R.D. & Guest, S.D. | Detection of finite mechanisms in symmetric structures | 1999 | International Journal of Solids and Structures Vol. 36(36), pp. 5507-5527 |
article | DOI |

Abstract: Using group representation theory, it is possible to block-diagonalise the equilibrium matrix of a symmetric structure. This analysis can identify the symmetry properties of any states of self-stress or mechanisms present in the structure. This paper will show that in some cases, this linear analysis, combined with symmetry arguments, can show that particular mechanisms of a symmetric structure must be finite. |
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BibTeX:
@article{kangwai1999, author = {Kangwai, R D and Guest, S D}, title = {Detection of finite mechanisms in symmetric structures}, journal = {International Journal of Solids and Structures}, year = {1999}, volume = {36}, number = {36}, pages = {5507--5527}, doi = {http://dx.doi.org/10.1016/S0020-7683(98)00234-0} } |
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Kebiche, K., Kazi-Aoual, M.N. & Motro, R. | Geometrical non-linear analysis of tensegrity systems | 1999 | Engineering Structures Vol. 21(9), pp. 864-876 |
article | DOI |

Abstract: A calculation method for structures with large deformations and displacements, based on works by Bathe [7, 8, 9], is developed so as to determine the tangent stiffness matrix and the internal stress vector. The formulation is established for a `bar' element. Application of this method for tensegrity systems allowed the study of behaviour for a simple self-stressed system, the four-strut tensegrity system, in case of `traction, compression, flexion and torsion' loading. Except for compression, the structure is rigidified when the loads increase. Influence of the self-stress level on this behaviour is also evaluated. Secondly a structure, generated by assembly of several four-strut tensegrity systems, has been calculated only under traction, concentrated and distributed. The behaviour under uniformly distributed load can be related to the `isolate' cell behaviour, even if some difficulties appear, mainly because of boundary conditions and possible choices for common cables between two adjacent cells. |
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BibTeX:
@article{kebiche1999, author = {Kebiche, K. and Kazi-Aoual, M. N. and Motro, R.}, title = {Geometrical non-linear analysis of tensegrity systems}, journal = {Engineering Structures}, year = {1999}, volume = {21}, number = {9}, pages = {864--876}, doi = {http://dx.doi.org/10.1016/S0141-0296(98)00014-5} } |
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Masic, M. | Design, optimization and control of tensegrity structures [BibTeX] |
2004 | School: University of California, San Diego, United States |
phdthesis | |

BibTeX:
@phdthesis{masicphd2004, author = {Masic, M.}, title = {Design, optimization and control of tensegrity structures}, school = {University of California, San Diego, United States}, year = {2004} } |
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Masic, M., Skelton, R.E. & Gill, P.E. | Optimization of tensegrity structures | 2006 | International Journal of Solids and Structures Vol. 43(16), pp. 4687-4703 |
article | DOI |

Abstract: This paper concerns the design of tensegrity structures with optimal mass-to-stiffness ratio. Starting from an initial layout that defines the largest set of allowed element connections, the procedure seeks the topology, geometry and prestress of the structure that yields optimal designs for different loading scenarios. The design constraints include strength constraints for all elements of the structure, buckling constraints for bars, and shape constraints. The problem formulation accommodates different symmetry constraints for structure parameters and shape. The static response of the structure is computed by using the nonlinear large displacement model. The problem is cast in the form of a nonlinear program. Examples show layouts of 2D and 3D asymmetric and symmetric structures. The influence of the material parameters on the optimal shape of the structure is investigated. |
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BibTeX:
@article{masic2006, author = {Masic, M. and Skelton, R. E. and Gill, P. E.}, title = {Optimization of tensegrity structures}, journal = {International Journal of Solids and Structures}, year = {2006}, volume = {43}, number = {16}, pages = {4687--4703}, doi = {http://dx.doi.org/10.1016/j.ijsolstr.2005.07.046} } |
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Masic, M., Skelton, R.E. & Gill, P.E. | Algebraic tensegrity form-finding | 2005 | International Journal of Solids and Structures Vol. 42(16-17), pp. 4833-4858 |
article | DOI |

Abstract: This paper concerns the form-finding problem for general and symmetric tensegrity structures with shape constraints. A number of different geometries are treated and several fundamental properties of tensegrity structures are identified that simplify the form-finding problem. The concept of a tensegrity invariance (similarity) transformation is defined and it is shown that tensegrity equilibrium is preserved under affine node position transformations. This result provides the basis for a new tensegrity form-finding tool. The generality of the problem formulation makes it suitable for the automated generation of the equations and their derivatives. State-of-the-art numerical algorithms are applied to solve several example problems. Examples are given for tensegrity plates, shell-class symmetric tensegrity structures and structures generated by applying similarity transformation. |
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BibTeX:
@article{masic2005, author = {Masic, M. and Skelton, R. E. and Gill, P. E.}, title = {Algebraic tensegrity form-finding}, journal = {International Journal of Solids and Structures}, year = {2005}, volume = {42}, number = {16--17}, pages = {4833--4858}, doi = {http://dx.doi.org/10.1016/j.ijsolstr.2005.01.014} } |
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Maxwell, J.C. | On the Calculation of the Equilibrium and Stiffness of Frames [BibTeX] |
1864 | Philosophical Magazine Vol. XXVII, pp. 294 |
article | |

BibTeX:
@article{maxwell1864, author = {J. C. Maxwell}, title = {On the Calculation of the Equilibrium and Stiffness of Frames}, journal = {Philosophical Magazine}, year = {1864}, volume = {XXVII}, pages = {294} } |
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Micheletti, A. & Williams, W. | A marching procedure for form-finding for tensegrity structures | 2007 | Journal of Mechanics of Materials and Structures Vol. 2(5), pp. 857-883 |
article | URL |

Abstract: We give an algorithm for solving the form-finding problem, that is, for finding stable placements of a given tensegrity structure. The method starts with a known stable placement and alters edge lengths in a way that preserves the equilibrium equations. We then characterize the manifold to which classical tensegrity systems belong, which gives insight into the form-finding process. After describing several special cases, we show the results of a successful test of our algorithm on a large system. |
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BibTeX:
@article{micheletti2007, author = {Micheletti, A. and Williams, W.O.}, title = {A marching procedure for form-finding for tensegrity structures}, journal = {Journal of Mechanics of Materials and Structures}, year = {2007}, volume = {2}, number = {5}, pages = {857--883}, url = {http://pjm.math.berkeley.edu/jomms/2007/2-5/p03.xhtml} } |
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Moored, K. & Bart-Smith, H. | Investigation of clustered actuation in tensegrity structures | 2009 | International Journal of Solids and Structures Vol. 46(17), pp. 3272-3281 |
article | DOI |

Abstract: As tensegrity research is moving away from static structures toward active structures it is becoming critical that new actuation strategies and comprehensive active structures theories are developed to fully exploit the properties of tensegrity structures. In this paper a new general tensegrity paradigm is presented that incorporates a concept referred to as clustered actuation. Clustered actuation exploits the existence of cable elements in a tensegrity structure by allowing cables to be run over frictionless pulleys or through frictionless loops at the nodes. This actuation strategy is a scalable solution that can be utilized for active structures that incorporate many active elements and can reduce the number of actuators necessary for complex shape changes. Clustered actuation also has secondary benefits, specifically reducing the force requirements of actuators in dynamic structures, reducing the number of pre-stress modes to potentially one global mode and relieving element size limitations that occur with embedded actuation. Newly formulated clustered equilibrium equations are developed using energy methods and are shown to be a generalization of the classic tensegrity governing equations. Pre-stress analysis, mechanism analysis and stability of clustered structures are discussed. Lastly, examples compare the mechanics of a clustered structure to an equivalent classic structure and the utility of clustering is highlighted by allowing for actuation throughout a class 1 (no bar-to-bar connections) tensegrity while not embedding the actuators into the structure. |
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BibTeX:
@article{moored2009, author = {K.W. Moored and H. Bart-Smith}, title = {Investigation of clustered actuation in tensegrity structures}, journal = {International Journal of Solids and Structures}, year = {2009}, volume = {46}, number = {17}, pages = {3272-3281}, doi = {http://dx.doi.org/10.1016/j.ijsolstr.2009.04.026} } |
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Motro, R. | Tensegrity : structural systems for the future [BibTeX] |
2003 | book | ||

BibTeX:
@book{motro2003, author = {Motro, R}, title = {Tensegrity : structural systems for the future}, publisher = {London : Kogan Page}, year = {2003} } |
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Motro, R. | Tensegrity Systems: The State of the Art [BibTeX] |
1992 | International Journal of Space Structures Vol. 7(2), pp. 75-83 |
article | |

BibTeX:
@article{motro1992, author = {Motro, R}, title = {Tensegrity Systems: The State of the Art}, journal = {International Journal of Space Structures}, year = {1992}, volume = {7}, number = {2}, pages = {75--83} } |
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Murakami, H. | Static and dynamic analyses of tensegrity structures. Part I. Nonlinear equations of motion | 2001 | International Journal of Solids and Structures Vol. 38(20), pp. 3599-3613 |
article | DOI |

Abstract: In order to present basic equations for static and dynamic analyses of a class of truss structures called tensegrity structures, large-deformation kinematics and kinetics were presented in both Eulerian and Lagrangian formulations. The two sets of equations of motion yield the same values even if different stress and strain measures were employed for their computation. The Eulerian formulation was implemented in an updated Lagrangian finite element code using Newton's method with consistently linearized equations of motion. By utilizing the linearized Lagrangian equations of motion at pre-stressed initial configurations, harmonic modal analyses of a three-bar tensegrity module and a six-stage tensegrity beam were conducted. In the second part of the paper, linearized equations were utilized to investigate the equilibrium configurations of basic tensegrity modules and the stiffness of pre-stressed tensegirty structures. |
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BibTeX:
@article{murakami2001a, author = {Murakami, H.}, title = {Static and dynamic analyses of tensegrity structures. Part I. Nonlinear equations of motion}, journal = {International Journal of Solids and Structures}, year = {2001}, volume = {38}, number = {20}, pages = {3599--3613}, doi = {http://dx.doi.org/10.1016/S0020-7683(00)00232-8} } |
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Murakami, H. | Static and dynamic analyses of tensegrity structures. Part II. Quasi-static analysis | 2001 | International Journal of Solids and Structures Vol. 38(20), pp. 3615-3629 |
article | DOI |

Abstract: Linearized Lagrangian equations developed in the first part of the paper were employed for static analyses of cyclic cylindrical tensegrity modules. Linearized equilibrium equations at natural configurations were used to investigate initial shape, static and kinematic indeterminancy, pre-stress and infinitesimal mechanism modes, and the sensitivity analysis of initial geometry. Linearized equilibrium equations at pre-stressed initial configurations were utilized to investigate pre-stress stiffening and to distinguish first-order mechanisms from higher-order mechanisms. To estimate critical loads for bar buckling and cable slacking, nonlinear equilibrium equations were employed to compute element forces. Further, the equivalence between the twist angle theorem obtained from a geometrical consideration and the equilibrium analysis was established for cyclic cylindrical tensegrity modules. It is concluded that infinitesimal mechanism modes and pre-stresses characterize the static and dynamic response of tensegrity structures. |
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BibTeX:
@article{murakami2001b, author = {Murakami, H.}, title = {Static and dynamic analyses of tensegrity structures. Part II. Quasi-static analysis}, journal = {International Journal of Solids and Structures}, year = {2001}, volume = {38}, number = {20}, pages = {3615--3629}, doi = {http://dx.doi.org/10.1016/S0020-7683(00)00233-X} } |
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Murakami, H. & Nishimura, Y. | Static and dynamic characterization of regular truncated icosahedral and dodecahedral tensegrity modules | 2001 | International Journal of Solids and Structures Vol. 38(50-51), pp. 9359-9381 |
article | DOI |

Abstract: Static and dynamic properties of a pair of dual spherical tensegrity modules invented by Buckminster Fuller are investigated. They are regular truncated icosahedral and dodecahedral tensegrity modules. The computation of the Maxwell number and the use of Calladine's relation reveal that regular truncated icosahedral and dodecahedral tensegrity modules possess 55 infinitesimal mechanism modes. A reduced equilibrium matrix is presented for the initial shape finding to economically impose the existence of a pre-stress mode. Both the initial shape and the corresponding pre-stress mode are analytically obtained by using graphs of the icosahedral group and the reduced equilibrium matrix. For both icosahedral and dodecahedral modules the maximum values of the cable tension is always less than the absolute value of bar compression. In order to classify a large number of infinitesimal mechanism modes, modal analyses are conducted. Infinitesimal mechanism modes have the stiffness due to pre-stress and are associated with lowest natural frequencies. Their natural frequencies increase proportionally to the square root of the amplitude of pre-stress. It is found that there are only 15 distinct natural frequencies associated with the infinitesimal mechanism modes. |
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BibTeX:
@article{murakami2001c, author = {Hidenori Murakami and Yoshitaka Nishimura}, title = {Static and dynamic characterization of regular truncated icosahedral and dodecahedral tensegrity modules}, journal = {International Journal of Solids and Structures}, year = {2001}, volume = {38}, number = {50-51}, pages = {9359-9381}, doi = {http://dx.doi.org/10.1016/S0020-7683(01)00030-0} } |
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Nishimura, Y. & Murakami, H. | Initial shape-finding and modal analyses of cyclic frustum tensegrity modules | 2001 | Computer Methods in Applied Mechanics and Engineering Vol. 190(43-44), pp. 5795-5818 |
article | DOI |

Abstract: Initial equilibrium and modal analyses of Kenneth Snelson's cyclic frustum tensegrity modules with an arbitrary number of stages are presented. There are m (>=3) bars at each stage. The Maxwell number of the modules is 6−2m and is independent of the number of stages in the axial direction. Calladine's relations reveals that there are 2−5m infinitesimal mechanism modes. For multi-stage modules the necessary conditions for axial assembly of one-stage modules with the same internal element-forces are investigated. One-stage modules with geometrically similar frustum modules satisfy the necessary conditions. For pre-stressed configurations, modal analyses were conducted to investigate the mode shapes of infinitesimal mechanism modes. |
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BibTeX:
@article{nishimura2001, author = {Y. Nishimura and H. Murakami}, title = {Initial shape-finding and modal analyses of cyclic frustum tensegrity modules}, journal = {Computer Methods in Applied Mechanics and Engineering}, year = {2001}, volume = {190}, number = {43-44}, pages = {5795-5818}, doi = {http://dx.doi.org/10.1016/S0045-7825(01)00198-0} } |
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Oppenheim, I. & Williams, W. | Vibration and damping in three-bar tensegrity structure | 2001 | Journal of Aerospace Engineering Vol. 14(3), pp. 85-91 |
article | DOI |

Abstract: The most interesting examples of tensegrity structures are underconstrained and display an infinitesimal flex. In the direction of that flex the force-displacement relationship is highly nonlinear, resulting from geometric stiffening and influenced by the effect of prestress at equilibrium. A tensegrity structure would therefore display nonlinear vibrations when excited in the direction of the infinitesimal flex, the "frequency" decreasing with amplitude. Movement in the direction of the flex occurs with only infinitesimal change in member length, and therefore under conventional models of material damping in members the motion would not vanish as rapidly as it would for a conventional oscillator. We study one particular tensegrity geometry for which we present the force-displacement relationship in analytical form and then examine the nonlinear vibrations. We observe the role of damping and we discuss those implications for the design of tensegrity structures in space applications. |
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BibTeX:
@article{oppenheim2001, author = {Oppenheim, I.J. and Williams, W.O.}, title = {Vibration and damping in three-bar tensegrity structure}, journal = {Journal of Aerospace Engineering}, year = {2001}, volume = {14}, number = {3}, pages = {85--91}, doi = {http://dx.doi.org/10.1061/(ASCE)0893-1321(2001)14:3(85)} } |
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Oppenheim, I.J. & Williams, W.O. | Geometric Effects in an Elastic Tensegrity Structure | 2000 | Journal of Elasticity Vol. 59(1-3), pp. 51-65 |
article | DOI |

Abstract: Tensegrity structures are under-constrained, 3-dimensional, self-stressing structural systems. They demonstrate an infinitesimal flex and when loaded they display a nonlinear geometric stiffening. In earlier work many examples of the resulting force?displacement relationship have been demonstrated numerically, and some aspects of the force?displacement relationship have been derived analytically. In this article an energy formulation is presented for the case of a simple but representative tensegrity structure, yielding an exact solution for the force?displacement relationship. The solution makes understandable the different appearance of the force?displacement relationship when comparing a system at zero prestress to one at high prestress, or when comparing a system with almost-inextensible members to one with highly extensible members. The exact solution also is offered as a benchmark against which numerical solutions should be tested. Furthermore, the formulation and the solution reveal conditions of asymmetry of response that have not been noted previously. |
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BibTeX:
@article{oppenheim2000, author = {Oppenheim, I. J. and Williams, W. O.}, title = {Geometric Effects in an Elastic Tensegrity Structure}, journal = {Journal of Elasticity}, year = {2000}, volume = {59}, number = {1--3}, pages = {51--65}, doi = {http://dx.doi.org/10.1023/A:1011092811824} } |
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Pandia Raj, R. | Novel Symmetric Tensegrity Structures [BibTeX] |
2009 | School: University of Cambridge |
phdthesis | |

BibTeX:
@phdthesis{pandiaraj2009_thesis, author = {Pandia Raj, R.}, title = {Novel Symmetric Tensegrity Structures}, school = {University of Cambridge}, year = {2009} } |
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Pandia Raj, R. & Guest, S.D. | A Tensegrity Catalogue using Point Group Theory | 2008 | 6th International Conference on Computation of Shell and Spatial Structures, IASS-IACM 2008 | conference | |

Abstract: We show how applied point group theory can be used to generate the tensegrity catalogue that was originally produced by Connelly and Back, and suggest some simple extensions of the catalogue. |
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BibTeX:
@conference{pandiaraj2008, author = {Pandia Raj, R. and Guest, S. D.}, title = {A Tensegrity Catalogue using Point Group Theory}, booktitle = {6th International Conference on Computation of Shell and Spatial Structures, IASS-IACM 2008}, year = {2008} } |
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Pandia Raj, R. & Guest, S.D. | Form-finding of Repetitive Tensegrity Structures | 2007 | Proceedings of IASS-2007 | conference | |

Abstract: This paper will show that symmetry techniques can help in form-finding for repetitive tensegrity structures. Structures consisting of a repetition of identical units abound in structural engineering; we will consider the case where these structures are prestressed in their initial configuration, which we refer to as repetitive tensegrity structures. By considering translation as a symmetry operation, we can gain insight intothe form-finding process, as well as the structural mechanics of these structures. |
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BibTeX:
@conference{pandiaraj2007, author = {Pandia Raj, R. and Guest, S. D.}, title = {Form-finding of Repetitive Tensegrity Structures}, booktitle = {Proceedings of IASS-2007}, year = {2007} } |
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Pandia Raj, R. & Guest, S.D. | Using Symmetry for Tensegrity Formfinding | 2006 | Journal of the International Association for Shell and Spatial Structures Vol. 47(3), pp. 245-252 |
article | |

Abstract: Symmetry can simplify the form-finding process for tensegrity structures; and this paper will describe one simplification technique. Our method is based on the commonly used force density method, but the calculations are done using a symmetry-adapted coordinate system. The standard force-density method assumes a known connectivity for the structure. A tension coefficient (tension divided by length) must then be found for every member so that an equilibrium solution is possible. Finding the nodal coordinates is straightforward once a suitable set of tension coefficient is found; but finding suitable tension coefficients may be non-trivial. In this paper we simplify the correct choice of tension coefficients by the use of symmetry - in addition to the connectivity of the structure, we assume that the structure has certain symmetry properties, greatly reducing the difficulty of finding possible configurations. The paper will show simple examples of the method where a simple analytical solution gives all possible symmetric tensegrities with a given connectivity. |
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BibTeX:
@article{pandiaraj2006, author = {Pandia Raj, R. and Guest, S. D.}, title = {Using Symmetry for Tensegrity Formfinding}, journal = {Journal of the International Association for Shell and Spatial Structures}, year = {2006}, volume = {47}, number = {3}, pages = {245--252} } |
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Pellegrino, S. | Structural computations with the singular value decomposition of the equilibrium matrix | 1993 | International Journal of Solids and Structures Vol. 30(21), pp. 3025-3035 |
article | DOI |

Abstract: The Singular Value Decomposition of the equilibrium matrix makes it possible to answer any question of a static, kinematic, or static/kinematic nature for any structural assembly, within a unified computational framework. The paper introduces this decomposition, relates it to the physical properties of a given assembly, and derives formulae for stresses, displacements, etc. A simple example illustrates the calculations. |
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BibTeX:
@article{pellegrino1993, author = {Pellegrino, S}, title = {Structural computations with the singular value decomposition of the equilibrium matrix}, journal = {International Journal of Solids and Structures}, year = {1993}, volume = {30}, number = {21}, pages = {3025--3035}, doi = {http://dx.doi.org/10.1016/0020-7683(93)90210-X} } |
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Pellegrino, S. | Analysis of Prestressed Mechanisms | 1990 | International Journal of Solids and Structures Vol. 26(12), pp. 1329-1350 |
article | DOI |

Abstract: A new theory is presented for the matrix analysis of prestressed structural mechanisms made from pin-jointed bars. The response of a prestressed mechanism to any external action is decomposed into two almost separate parts, which correspond to extensional and inextensional modes. A matrix algorithm which treats these two modes separately is developed and tested. It is shown that the equilibrium requirements for the assembly, in its initial configuration as well as in deformed configurations which are obtained through infinitesimal inextensional displacements, can be fully described by a square equilibrium matrix. It is also shown that any set of extensional nodal displacements has to satisfy some equilibrium conditions as well as standard compatibility equations, and that the resulting system of linear equations defines a square kinematic matrix. Theoretical as well as experimental evidence supporting this approach is given in the paper ; two simple experiments which were of crucial importance in arriving at the equilibrium conditions on the extensional displacements are described.The interaction between the two modes of action of a prestressed mechanism is discussed, together with a rapidly converging iterative procedure to handle it. A study of the non-linear effect by which the self-stress level in a statically indeterminate assembly rapidly increases if an inextensional mode is excited, supported by further experimental results, concludes the paper. This work is relevant to the analysis of most cable systems, pneumatic domes, fabric roofs, and "Tensegrity" frameworks. |
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BibTeX:
@article{pellegrino1990, author = {Pellegrino, S}, title = {Analysis of Prestressed Mechanisms}, journal = {International Journal of Solids and Structures}, year = {1990}, volume = {26}, number = {12}, pages = {1329--1350}, doi = {http://dx.doi.org/10.1016/0020-7683(90)90082-7} } |
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Pellegrino, S. & Calladine, C.R. | Matrix analysis of statically and kinematically indeterminate frameworks | 1986 | International Journal of Solids and Structures Vol. 22(4), pp. 409-428 |
article | DOI |

Abstract: The paper is concerned with the structural mechanics of assemblies of bars and pinjoints, particularly where they are simultaneously statically and kinematically indeterminate. The physical significance of the four linear-algebraic vector subspaces of the equilibrium matrix is examined, and an algorithm is set up which determines the rank of the matrix and the bases for the four subspaces. In particular, this algorithm gives full details of any states of self-stress and modes of inextensional deformation which an assembly may possess. A scheme is devised for the segregation of inextensional modes into rigid-body modes (up to six of these may be allowed by the foundation constraints) and "internal" mechanisms. In some circumstances a state of self-stress may impart first-order stiffness to an inextensional mode. A matrix method for detecting this effect is devised, and it is shown that if there is no state of self-stress which imparts first-order stiffness to a given mode, then that mode can undergo rather large distortion which involves either zero change in length of the bars or, possibly, changes in length of third or higher order in the displacements. The significance of negative stiffness, as indicated by the matrix method, is discussed. The paper contains simple examples which illustrate all of the main points of the work. |
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BibTeX:
@article{pellegrino1986, author = {Pellegrino, S and Calladine, C R}, title = {Matrix analysis of statically and kinematically indeterminate frameworks}, journal = {International Journal of Solids and Structures}, year = {1986}, volume = {22}, number = {4}, pages = {409--428}, doi = {http://dx.doi.org/10.1016/0020-7683(86)90014-4} } |
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Pirentis, A. & Lazopoulos, K. | On the elastica solution of a T3 tensegrity structure | 2006 | Archives of Applied Mechanics Vol. 76(7-8), pp. 481-496 |
article | DOI |

Abstract: Stability studies of a T3 tensegrity structure are performed. This structure is composed of three slender struts interconnected by six nonlinear elastic tendons and is prestressed. The struts are governed by linear constitutive laws and are allowed to buckle. Since tensegrity is used for modeling structures with quite large deformations, for example the cytoskeleton, and bifurcation theory—valid for small solutions of the nonlinear equations—does not directly apply, a general procedure for studying the stability behavior of the particular tensegrity model based upon the elastica theory is presented. The reference placement is defined by the prestress, and the equilibrium placements are defined by the applied force and moment. |
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BibTeX:
@article{pirentis2006, author = {Pirentis, A.P. and Lazopoulos, K.A.}, title = {On the elastica solution of a T3 tensegrity structure}, journal = {Archives of Applied Mechanics}, year = {2006}, volume = {76}, number = {7--8}, pages = {481--496}, doi = {http://dx.doi.org/10.1007/s00419-006-0051-7} } |
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Pugh, A. | Introduction to tensegrity [BibTeX] |
1976 | book | ||

BibTeX:
@book{pugh1976, author = {Pugh, A}, title = {Introduction to tensegrity}, publisher = {Berkeley University of California Press}, year = {1976} } |
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Schek, H. | The force density method for form finding and computation of general networks | 1974 | Computer Methods in Applied Mechanics and Engineering Vol. 3(1), pp. 115-134 |
article | DOI |

Abstract: A new method for network analysis, the “force density method”, is presented. This concept is based upon the force-length ratios or force densities which are defined for each branch of the net structure. It is shown that the force densities are very suitable parameters for the description of the equilibrium state of any general network: the associated node coordinates of the structure are obtained by solving one single system of linear equations. A Gaussian transformation of the topological branch-node matrix yields the equation matrix.A method based on force densities renders a simple linear “analytical form finding” possible. If the free choice of the force densities is restricted by further conditions, the linear method is extended to a nonlinear one. A damped least squares approach is then introduced. In this version the method may be applied to the detailed computation of networks. Moreover, an interesting cross-connection between geometrical minimum way nets and prestressed unloaded structures is derived using the force densities. |
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BibTeX:
@article{schek1974, author = {Schek, H.J.}, title = {The force density method for form finding and computation of general networks}, journal = {Computer Methods in Applied Mechanics and Engineering}, year = {1974}, volume = {3}, number = {1}, pages = {115--134}, doi = {http://dx.doi.org/10.1016/0045-7825(74)90045-0} } |
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Schenk, M. | Theory and Design of Statically Balanced Tensegrity Mechanisms [BibTeX] |
2006 | School: Delft University of Technology, The Netherlands |
mastersthesis | URL |

BibTeX:
@mastersthesis{schenkmsc2006, author = {Schenk, M.}, title = {Theory and Design of Statically Balanced Tensegrity Mechanisms}, school = {Delft University of Technology, The Netherlands}, year = {2006}, url = {http://www.markschenk.com/tensegrity/} } |
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Schenk, M., Guest, S.D. & Herder, J.L. | Zero Stiffness Tensegrity Structures | 2007 | International Journal of Solids and Structures Vol. 44(20), pp. 6569-6583 |
article | DOI |

Abstract: Tension members with a zero rest length allow the construction of tensegrity structures that are in equilibrium over a continuous range of positions and thus exhibit mechanism-like properties; equivalently, they have zero stiffness. The zero-stiffness modes are not internal mechanisms, as they involve first-order changes in member length, but are a direct result of the use of the special tension members. These modes correspond to an infinitesimal affine transformation of the structure that preserves the length of conventional members, they hold over finite displacements and are present if and only if the directional vectors of those members lie on a projective conic. This geometric interpretation provides several interesting observations regarding zero stiffness tensegrity structures. |
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BibTeX:
@article{schenk2007, author = {Schenk, M. and Guest, S. D. and Herder, J. L.}, title = {Zero Stiffness Tensegrity Structures}, journal = {International Journal of Solids and Structures}, year = {2007}, volume = {44}, number = {20}, pages = {6569--6583}, doi = {http://dx.doi.org/10.1016/j.ijsolstr.2007.02.041} } |
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Schenk, M., Herder, J.L. & Guest, S.D. | Design of a Statically Balanced Tensegrity Mechanism | 2006 | Proceedings of DETC/CIE 2006, ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 10-13, 2006, Philadelphia, Pennsylvania, USA | conference | URL |

Abstract: The combination of static balancing and tensegrity structures has resulted in a new class of mechanisms: Statically Balanced Tensegrity Mechanisms. These are prestressed structures that are in equilibrium in a wide range of positions, and thus exhibit mechanism-like properties. This paper describes the design of a prototype model of a statically balanced tensegrity mechanism based on a classic tensegrity structure. |
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BibTeX:
@conference{schenk2006, author = {Schenk, M. and Herder, J. L. and Guest, S. D.}, title = {Design of a Statically Balanced Tensegrity Mechanism}, booktitle = {Proceedings of DETC/CIE 2006, ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 10-13, 2006, Philadelphia, Pennsylvania, USA}, year = {2006}, url = {http://store.asme.org/product.asp?catalog%5Fname=Conference+Papers&category%5Fname=Mechanism+Synthesis%2C+Components+and+Applications%5FDETC2006TA%2D2&product%5Fid=DETC2006%2D99727} } |
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Skelton, R.E., Helton, J.W., Adhikari, R., Pinaud, J.P. & Chang, W. | An Introduction to the Mechanics of Tensegrity Structures [BibTeX] |
2002 | The Mechanical systems design handbook : modeling, measurement, and control. | incollection | URL |

BibTeX:
@incollection{skelton2002, author = {Skelton, R. E. and Helton, J. W. and Adhikari, R. and Pinaud, J. P. and Chang, W.}, title = {An Introduction to the Mechanics of Tensegrity Structures}, booktitle = {The Mechanical systems design handbook : modeling, measurement, and control.}, publisher = {Boca Raton : CRC Press}, year = {2002}, url = {http://www.engnetbase.com/ejournals/books/book_summary/summary.asp?id=590} } |
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Skelton, R.E. & Sultan, C. | Controllable tensegrity: a new class of smart structures | 1997 | Proceedings of SPIE -- Volume 3039, Smart Structures and Materials 1997: Mathematics and Control in Smart Structures, Vasundara V. Varadan, Jagdish Chandra, Editors, June 1997, pp. 166-177 | conference | DOI |

Abstract: Tensegrity structures represent a special class of tendon space structures, whose members may simultaneously perform the functions of strength, sensing, actuating and feedback control. Thus, these structures ideally match the definition of smart structures. This paper introduces the concept of controllable tensegrity as a new class of smart structures capable of large displacement. The kinematics and nonlinear dynamics of one element of this class is derived and analyzed. Pre-stressability conditions are given and a particular equilibrium identified. The equations of motion are then linearized about this equilibrium and linear parametric models generated. These are next used for controller design. For control system design some of the tendons are chosen as actuators and some as sensors and a family of dynamic controllers designed such that the control energy is minimized while requiring output variance constraints to be satisfied. Another family of controllers is designed such that the output variance is minimized while requiring input variance constraints to be satisfied. The performances of these controllers are evaluated. |
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BibTeX:
@conference{skelton1997, author = {Skelton, R. E. and Sultan, C.}, title = {Controllable tensegrity: a new class of smart structures}, booktitle = {Proceedings of SPIE -- Volume 3039, Smart Structures and Materials 1997: Mathematics and Control in Smart Structures, Vasundara V. Varadan, Jagdish Chandra, Editors, June 1997, pp. 166-177}, year = {1997}, doi = {http://dx.doi.org/10.1117/12.276535} } |
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Snelson, K.D. | Personal website [BibTeX] |
2005 | misc | URL | |

BibTeX:
@misc{snelson2005, author = {Snelson, K D}, title = {Personal website}, year = {2005}, url = {http://www.kennethsnelson.net} } |
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Snelson, K.D. | Personal communication between K. Snelson and R. Motro. [BibTeX] |
1990 | misc | URL | |

BibTeX:
@misc{snelson1990, author = {Snelson, K D}, title = {Personal communication between K. Snelson and R. Motro.}, year = {1990}, url = {http://www.grunch.net/snelson/rmoto.html} } |
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Snelson, K.D. | Continuous tension, discontinuous compression structures [BibTeX] |
1965 | US Patent Number 3,169,611 | misc | |

BibTeX:
@misc{snelson1965, author = {Snelson, K D}, title = {Continuous tension, discontinuous compression structures}, year = {1965} } |
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Sultan, C. | Modeling, design and control of tensegrity structures with applications [BibTeX] |
1999 | School: Purdue University |
phdthesis | |

BibTeX:
@phdthesis{sultanphd1999, author = {Sultan, C.}, title = {Modeling, design and control of tensegrity structures with applications}, school = {Purdue University}, year = {1999} } |
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Sultan, C., Corless, M. & Skelton, R.E. | Linear dynamics of tensegrity structures | 2002 | Engineering Structures Vol. 24(6), pp. 671-685 |
article | DOI |

Abstract: The linearized equations of motion for tensegrity structures around arbitrary equilibrium configurations are derived. For certain tensegrity structures which yield particular equilibrium configurations of practical interest, the linearized models of their dynamics around these configurations are presented. Evidence which indicates that these equilibria are stable is given and some stiffness and dynamic properties of these structures are investigated. |
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BibTeX:
@article{sultan2002, author = {Sultan, C. and Corless, M. and Skelton, R. E.}, title = {Linear dynamics of tensegrity structures}, journal = {Engineering Structures}, year = {2002}, volume = {24}, number = {6}, pages = {671--685}, doi = {http://dx.doi.org/10.1016/S0141-0296(01)00130-4} } |
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Sultan, C., Corless, M. & Skelton, R.E. | The prestressability problem of tensegrity structures: some analytical solutions | 2001 | International Journal of Solids and Structures Vol. 38(30-31), pp. 5223-5252 |
article | DOI |

Abstract: In this paper we formulate the general prestressability conditions for tensegrity structures. These conditions are expressed as a set of nonlinear equations and inequalities on the tendon tensions. Several examples of tensegrity structures for which the prestressability conditions can be analytically solved are then presented. |
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BibTeX:
@article{sultan2001, author = {Sultan, C. and Corless, M. and Skelton, R. E.}, title = {The prestressability problem of tensegrity structures: some analytical solutions}, journal = {International Journal of Solids and Structures}, year = {2001}, volume = {38}, number = {30-31}, pages = {5223--5252}, doi = {http://dx.doi.org/10.1016/S0020-7683(00)00401-7} } |
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Sultan, C. & Skelton, R. | Tendon control deployment of tensegrity structures | 1998 | Proceedings of SPIE -- Volume 3323, Smart Structures and Materials 1998: Mathematics and Control in Smart Structures, Vasundara V. Varadan, Editor, July 1998, pp. 455-466 | conference | DOI |

Abstract: In this paper we consider the problem of deployment of tensegrity structures. Our idea is to make use of a certain set of equilibria to which the undeployed and deployed configurations belong. In the state space this set is represented by a connected equilibrium manifold and can be completely characterized analytically. The deployment is conducted such that the deployment trajectory is close to the equilibrium manifold and the deployment time is minimized. |
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BibTeX:
@conference{sultan1998, author = {Sultan, C. and Skelton, R.E.}, title = {Tendon control deployment of tensegrity structures}, booktitle = {Proceedings of SPIE -- Volume 3323, Smart Structures and Materials 1998: Mathematics and Control in Smart Structures, Vasundara V. Varadan, Editor, July 1998, pp. 455-466}, year = {1998}, doi = {http://dx.doi.org/10.1117/12.316325} } |
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Tarnai, T. | Zero stiffness elastic structures | 2003 | International Journal of Mechanical Sciences Vol. 45(3), pp. 425-431 |
article | DOI |

Abstract: In the paper, examples of neutral states of equilibrium, rare in the theory of elastic stability, are presented where the magnitude of the load maintaining equilibrium of the structure does not change even in the case of large displacements. The structures are able to change their shape without any additional external load. They behave like mechanisms. The first example is related to bifurcation of equilibrium paths, and the second is related to the snap-through type loss of stability of unloaded structures having a state of self-stress. |
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BibTeX:
@article{tarnai2003, author = {Tarnai, T}, title = {Zero stiffness elastic structures}, journal = {International Journal of Mechanical Sciences}, year = {2003}, volume = {45}, number = {3}, pages = {425--431}, doi = {http://dx.doi.org/10.1016/S0020-7403(03)00063-8} } |
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Tarnai, T. | Duality between plane trusses and grillages | 1989 | International Journal of Solids and Structures Vol. 25(12), pp. 1395-1409 |
article | DOI |

Abstract: In the paper, projective plane duality, that is, a point-to-line, line-to-point, incidence-to-incidence correspondence between plane 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 not change under projective transformations and polarities: consequently the number of infinitesimal inextensional mechanisms and the number of independent states of self-stress are preserved under these transformations. The results obtained are also applied to structures with unilateral constraints, and by using several examples it is shown that plane tensegrity trusses have projective dual counterparts among grillages which can be physically modelled with popsicle sticks by weaving. |
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BibTeX:
@article{tarnai1989, author = {Tarnai, T.}, title = {Duality between plane trusses and grillages}, journal = {International Journal of Solids and Structures}, year = {1989}, volume = {25}, number = {12}, pages = {1395--1409}, doi = {http://dx.doi.org/10.1016/0020-7683(89)90108-X} } |
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Tarnai, T. | Simultaneous static and kinematic indeterminacy of space trusses with cyclic symmetry | 1980 | International Journal of Solids and Structures Vol. 16(4), pp. 347-359 |
article | DOI |

Abstract: This paper examines the structural rigidity of pin-jointed space trusses with cyclic symmetry. Although the necessary condition of static and kinematic determinacy, the so-called Maxwell's rule, is satisfied, the structures are both statically and kinematically indeterminate. The physical meaning of the static and kinematic indeterminacy is presented. In the case of both statically and kinematically indeterminate cylindrical trusses having planes of mirror symmetry, it is proved that although the cylindrical truss consists of individually both unstable and redundant "rings", the degree of static and kinematic indeterminacy is independent of the number of the "rings". |
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BibTeX:
@article{tarnai1980, author = {Tarnai, T.}, title = {Simultaneous static and kinematic indeterminacy of space trusses with cyclic symmetry}, journal = {International Journal of Solids and Structures}, year = {1980}, volume = {16}, number = {4}, pages = {347-359}, doi = {http://dx.doi.org/10.1016/0020-7683(80)90087-6} } |
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Tibert, A.G. & Pellegrino, S. | Review of Form-Finding Methods for Tensegrity Structures | 2003 | International Journal of Space Structures Vol. 18(4), pp. 209-223 |
article | DOI |

Abstract: Seven form-finding methods for tensegrity structures are reviewed and classified. The three kinematical methods include an analytical approach, a non-linear optimisation, and a pseudo-dynamic iteration. The four statical methods include an analytical method, the formulation of linear equations of equilibrium in terms of force densities, an energy minimisation, and a search for the equilibrium configurations of the struts of the structure connected by cables whose lengths are to be determined, using a reduced set of equilibrium equations. It is concluded that the kinematical methods are best suited to obtaining only configuration details of structures that are already essentially known. The force density method is best suited to searching for new configurations, but affords no control over the lengths of the elements of the structure. The reduced coordinates method offers a greater control on elements lengths, but requires more extensive symbolic manipulations. |
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BibTeX:
@article{tibert2003, author = {Tibert, A. G. and Pellegrino, S.}, title = {Review of Form-Finding Methods for Tensegrity Structures}, journal = {International Journal of Space Structures}, year = {2003}, volume = {18}, number = {4}, pages = {209--223}, doi = {http://dx.doi.org/10.1260/026635103322987940} } |
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Tibert, G. | Deployable Tensegrity Structures for Space Application [BibTeX] |
2002 | School: Royal Institute of Technology, Stockholm, Sweden |
phdthesis | URL |

BibTeX:
@phdthesis{tibertphd2002, author = {Tibert, G}, title = {Deployable Tensegrity Structures for Space Application}, school = {Royal Institute of Technology, Stockholm, Sweden}, year = {2002}, url = {www.diva-portal.org/diva/getDocument?urn_nbn_se_kth_diva-3317-2__fulltext.pdf} } |
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Vassart, N., Laporte, R. & Motro, R. | Determination of mechanisms' order for kinematically and statically indeterminate systems | 2000 | International Journal of Solids and Structures Vol. 37(28), pp. 3807-3839 |
article | DOI |

Abstract: This paper deals with the mechanisms in kinematically and statically indeterminate reticulated systems. Knowledge of length variation amplitude for members in association with an assigned mechanism allows determination of mechanism order. This is a fundamental characteristic of these systems, mainly for stability considerations. We submit on one part, simple tests allowing distinction between "order one mechanisms" and mechanisms of higher order, and on the other part an algorithm giving access to order exact value for all mechanisms associated with a given reticulated system. With this algorithm, order one mechanisms and higher order mechanisms are identified. Simple examples are given in the text and illustrate these aspects. In conclusion, we submit a stop criterion for the algorithm which gives access to the finite mechanisms for most of constructive reticulated systems. |
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BibTeX:
@article{vassart2000, author = {Vassart, N. and Laporte, R. and Motro, R.}, title = {Determination of mechanisms' order for kinematically and statically indeterminate systems}, journal = {International Journal of Solids and Structures}, year = {2000}, volume = {37}, number = {28}, pages = {3807--3839}, doi = {http://dx.doi.org/10.1016/S0020-7683(99)00178-X} } |
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Vassart, N. & Motro, R. | Multiparametered Formfinding Method: Application to Tensegrity Systems | 1999 | International Journal of Space Structures Vol. 14(2), pp. 147-154 |
article | |

Abstract: A method allowing a multiparametered formfinding for prestressed and selfstressed reticulated systems with tensile and compressive members is presented. Known methods, based on geometric analysis and dynamic (dynamic relaxation) considerations have been developed for these systems but they allow generally the evolution of only one parameter. But, in case of shape finding of non-regular new forms or when the sought-after form is subject to a set of geometrical constraints, it becomes obligatory to elaborate a multiparametered form-finding process.The proposed numerical method, which is described in this paper, exploits the force density method, already used for form finding of pure tensile structures. However, equilibrium matrix of pure tensile structures as cable nets systems, admits always an inverse, which might be false when tensile and compressive members coexist in the system. In this paper, different processes allowing to define prestressed (or selfstressed) equilibrium geometry are described. Except for the relational structure which is considered as known at the beginning of the process, two sets of form-finding parameters can be identified for this method : prestress (or selfstress) coefficients of members and coordinates or redundant nodes. The proposed method does not yield a unique geometry but it is very convenient for a multiparametered formfinding, and has produced very interesting results, especially for Tensegrity Systems. Application of this method of multiparametered formfinding to Tensegrity Systems, provides the designer with an efficient way to achieve interesting new selfstressed geometries, such as the generation of double-layer grids by agglomeration of Tensegrity modules. |
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BibTeX:
@article{vassart1999, author = {Vassart, N. and Motro, R.}, title = {Multiparametered Formfinding Method: Application to Tensegrity Systems}, journal = {International Journal of Space Structures}, year = {1999}, volume = {14}, number = {2}, pages = {147--154} } |
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Volokh, K. & Vilnay, O. | New classes of reticulated underconstrained structures | 1997 | International Journal of Solids and Structures Vol. 34(9), pp. 1093-1104 |
article | DOI |

Abstract: The problem of synthesis of underconstrained structures is discussed. Two new classes of reticulated space underconstrained structures are presented. It is shown that they can be stiffened by prestressing. A version of the procedure of underconstrained structures initial stability checking is described |
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BibTeX:
@article{volokh1997, author = {K. Volokh and O. Vilnay}, title = {New classes of reticulated underconstrained structures}, journal = {International Journal of Solids and Structures}, year = {1997}, volume = {34}, number = {9}, pages = {1093-1104}, doi = {http://dx.doi.org/10.1016/S0020-7683(96)00080-7} } |
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Wang, B.-B. | Definitions and Feasiblity Studies of Tensegrity Systems [BibTeX] |
1998 | International Journal of Space Structures Vol. 13(1), pp. 41-47 |
article | |

BibTeX:
@article{wang1998, author = {Wang, B-B.}, title = {Definitions and Feasiblity Studies of Tensegrity Systems}, journal = {International Journal of Space Structures}, year = {1998}, volume = {13}, number = {1}, pages = {41--47} } |
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Weisstein, E.W. | Affine Transformation [BibTeX] |
1999 | MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/AffineTransformation.html | misc | URL |

BibTeX:
@misc{wfaffine, author = {Weisstein, E. W.}, title = {Affine Transformation}, year = {1999}, url = {http://mathworld.wolfram.com/AffineTransformation.html} } |
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Weisstein, E.W. | Conic Section [BibTeX] |
1999 | misc | URL | |

BibTeX:
@misc{wfconic, author = {Weisstein, E. W.}, title = {Conic Section}, year = {1999}, note = {From Mathworld -- a Wolfram Web Resource. http://mathworld.wolfram.com/ConicSection.html}, url = {http://mathworld.wolfram.com/ConicSection.html} } |
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Williamson, D., Skelton, R.E. & Han, J. | Equilibrium conditions of a tensegrity structure | 2003 | International Journal of Solids and Structures Vol. 40(23), pp. 6347-6367 |
article | DOI |

Abstract: This paper characterizes the necessary and sufficient conditions for tensegrity equilibria. Static models of tensegrity structures are reduced to linear algebra problems, after first characterizing the problem in a vector space where direction cosines are not needed. This is possible by describing the components of all member vectors. While our approach enlarges (by a factor of 3) the vector space required to describe the problem, the advantage of enlarging the vector space makes the mathematical structure of the problem amenable to linear algebra treatment. Using the linear algebraic techniques, many variables are eliminated from the final existence equations. |
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BibTeX:
@article{williamson2003, author = {Williamson, D. and Skelton, R. E. and Han, J.}, title = {Equilibrium conditions of a tensegrity structure}, journal = {International Journal of Solids and Structures}, year = {2003}, volume = {40}, number = {23}, pages = {6347--6367}, doi = {http://dx.doi.org/10.1016/S0020-7683(03)00400-1} } |
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Zhang, J., Guest, S. & Ohsaki, M. | Symmetric prismatic tensegrity structures. Part II: Symmetry-adapted formulations | 2009 | International Journal of Solids and Structures Vol. 46(1), pp. 15-30 |
article | DOI |

Abstract: This paper presents analytical formulations for the symmetry-adapted equilibrium, force density and geometrical stiffness matrices for prismatic tensegrity structures with dihedral symmetries. An analytical expression for the infinitesimal mechanisms is also derived. |
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BibTeX:
@article{zhang2009, author = {J.Y. Zhang and S.D. Guest and M. Ohsaki}, title = {Symmetric prismatic tensegrity structures. Part II: Symmetry-adapted formulations}, journal = {International Journal of Solids and Structures}, year = {2009}, volume = {46}, number = {1}, pages = {15-30}, doi = {http://dx.doi.org/10.1016/j.ijsolstr.2008.07.035} } |
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Zhang, J., Guest, S. & Ohsaki, M. | Symmetric Prismatic Tensegrity Structures: Part I. Configuration and Stability | 2008 | International Journal Solids and Structures Vol. 46(1), pp. 1-14 |
article | DOI URL |

Abstract: This paper presents a simple and efficient method to determine the self-equilibrated configurations of prismatic tensegrity structures, nodes and members of which have dihedral symmetry. It is demonstrated that stability of this class of structures is not only directly related to the connectivity of members, but is also sensitive to their geometry (height/radius ratio), and is also dependent on the level of self-stress and stiffness of members. A catalogue of the structures with relatively small number of members is presented based on the stability investigations. |
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BibTeX:
@article{zhang2008, author = {Zhang, J.Y. and Guest, S.D. and Ohsaki, M.}, title = {Symmetric Prismatic Tensegrity Structures: Part I. Configuration and Stability}, journal = {International Journal Solids and Structures}, year = {2008}, volume = {46}, number = {1}, pages = {1-14}, url = {http://tensegrity.aistructure.com/prismatic/}, doi = {http://dx.doi.org/10.1016/j.ijsolstr.2008.08.032} } |
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Zhang, J.Y. & Ohsaki, M. | Stability Conditions for Tensegrity Structures [BibTeX] |
2007 | International Journal of Solids and Structures | article | DOI |

BibTeX:
@article{zhang2007a, author = {Zhang, J. Y. and Ohsaki, M.}, title = {Stability Conditions for Tensegrity Structures}, journal = {International Journal of Solids and Structures}, year = {2007}, doi = {http://dx.doi.org/10.1016/j.ijsolstr.2006.10.027} } |
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Zhang, L., Maurin, B. & Motro, R. | Form-Finding of Nonregular Tensegrity Systems | 2006 | Journal of Structural Engineering Vol. 132(9), pp. 1435-1440 |
article | DOI |

Abstract: The potential applications of tensegrity structures are not only increasing in civil engineering but also in fields like biomechanics. The key step in designing tensegrity, the form-finding problem, has been investigated by many researchers but until now they have tended to focus on methods for regular shapes. Since there is an increasing need for design tools devoted to more various and complex systems, the objective of this paper is to present the form-finding of nonregular tensegrity structures with a numerical approach. It is based on the dynamic relaxation method with kinetic damping, and new tensegrity configurations in more intricate and creative forms can be obtained this way. During the form-finding process, either the force or length of some elements can be fixed by an appropriate choice of related stiffnesses. The application of the process is illustrated by several numerical examples. It can be concluded that an improvement in tensegrity form-finding has been achieved extending research from regular shapes toward "freer" shapes. |
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BibTeX:
@article{zhang2006, author = {L. Zhang and B. Maurin and R. Motro}, title = {Form-Finding of Nonregular Tensegrity Systems}, journal = {Journal of Structural Engineering}, year = {2006}, volume = {132}, number = {9}, pages = {1435-1440}, doi = {http://dx.doi.org/10.1061/(ASCE)0733-9445(2006)132:9(1435)} } |

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