On the model flexibility of Siamese dipyramids

Gorkavyy, Vasilii Alekseevich; Fesenko, Ivan

doi:10.1007/s00022-018-0462-y

Cited by 8 publications

(22 citation statements)

References 10 publications

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“…where E denotes the modulus of elasticity 4 , L i j is the deformed length of the bar and L i j its original one. The formula (1) is based on the so-called Cauchy/Engineering (CE) strain, which can also be extended to triangular elements playing a central role in the plane stress 5 analysis within the finite element method (e.g. see [7,Chapter 6]).…”

Section: Elastic Gl Strain Energy Of Bars and Triangular Platesmentioning

confidence: 99%

“…Fig. 1) with a pinned base given by k 1 = (0, 0) T , k 2 = (9, 0) T , k 3 = (7, 4) T which is equipped with the intrinsic metric (L 14 , L 25 , L 36 , L 45 , L 46 , L 56 ) = (11,10,5,8,5,5). The two undeformed realizations G(k) and G(k) snap into each other over G(k ) which was computed based on a framework consisting of (a) six bars using GL/CE strain (cyan/magenta dotted), (b) three bars and one triangular plate using GL strain (blue dashed).…”

Section: Singularity-distancementioning

confidence: 99%

“…One can apply the proposed approach to almost all known examples 2 of snapping spatial frameworks (cf. [10]), as for example the Siamese dipyramids [4,5], the fourhorn [18] or Wunderlich's snapping octahedra, icosahedra and dodecahedra, which are reviewed in [15]. But the presented method is not limited to the listed triangular platehinge structures -also known as panel-hinge frameworks 3 -as it can also handle structures including bars, as for example the 3-legged planar parallel manipulator or an spatial hexapod of octahedral structure, which are both of practical importance.…”

Section: Introductionmentioning

confidence: 99%

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On the Snappability and Singularity-Distance of Frameworks with Bars and Triangular Plates

Nawratil

2021

Springer Proceedings in Advanced Robotics

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In a recent article the author presented a method to measure the snapping capability -shortly called snappability -of bar-joint frameworks based on the total elastic strain energy by computing the deformation of all bars using Hooke's law and the definition of Cauchy/Engineering strain. Within the paper at hand, we extend this approach to isostatic frameworks composed of bars and triangular plates by using the physical concept of Green-Lagrange strain. An intrinsic pseudometric based on the resulting total elastic strain energy density cannot only be used for evaluating the snappability but also for measuring the distance to the closest singular configuration. The presented methods are demonstrated on the basis of the 3-legged planar parallel manipulator.

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Section: Elastic Gl Strain Energy Of Bars and Triangular Platesmentioning

confidence: 99%

Section: Singularity-distancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Snappability and Singularity-Distance of Frameworks with Bars and Triangular Plates

Nawratil

2021

Springer Proceedings in Advanced Robotics

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“…Next we consider a polyhedron that is provably rigid (as a polyhedron), but such that physical models of it "feel" flexible: they can be deformed by some finite amount, without any noticeable stretching or bending of the faces. This family of examples, called Siamese dipyramids, was introduced to the mathematics literature by Goldberg [17], and partially analyzed recently by Gorkavyy and Fesenko [18]. The authors built our own physical model of a Siamese dipyramid using cardboard for the faces and Scotch tape for the hinges, and it did not feel at all rigid; we could deform it significantly without noticeably bending or creasing the cardboard or ripping the hinges.…”

Section: Siamese Dipyramidmentioning

confidence: 99%

“…In example (j), one disc has been flattened and one has been thickened. See [18] for a description of how to construct these examples.…”

Section: Siamese Dipyramidmentioning

confidence: 99%

Almost‐Rigidity of Frameworks

Holmes-Cerfon

Theran

Gortler

2020

Comm Pure Appl Math

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We extend the mathematical theory of rigidity of frameworks (graphs embedded in d‐dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call “almost‐rigidity”; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost‐rigidity. We briefly discuss how the theory may be applied to tensegrities. © 2020 Wiley Periodicals LLC.

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Evaluating the Snappability of Bar-Joint Frameworks

Nawratil

2020

Advances in Robot Kinematics 2020

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On the model flexibility of Siamese dipyramids

Cited by 8 publications

References 10 publications

On the Snappability and Singularity-Distance of Frameworks with Bars and Triangular Plates

On the Snappability and Singularity-Distance of Frameworks with Bars and Triangular Plates

Almost‐Rigidity of Frameworks

Evaluating the Snappability of Bar-Joint Frameworks

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