self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a...
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![Page 1: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/1.jpg)
Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical
solution
by Li-Yuan Zhang, Yue Li, Yan-Ping Cao, Xi-Qiao Feng, and Huajian Gao
Proceedings AVolume 468(2147):3323-3347
November 8, 2012
©2012 by The Royal Society
![Page 2: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/2.jpg)
Relationships of different states of tensegrity structures.
Li-Yuan Zhang et al. Proc. R. Soc. A 2012;468:3323-3347
©2012 by The Royal Society
![Page 3: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/3.jpg)
Regular and truncated platonic solids: (a) regular polyhedra, (b) truncated regular polyhedra, (c) critical truncated polyhedra and (d) hyper-truncated polyhedra.
Li-Yuan Zhang et al. Proc. R. Soc. A 2012;468:3323-3347
©2012 by The Royal Society
![Page 4: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/4.jpg)
Truncated regular tetrahedral tensegrity: (a) the edges and vertices of a truncated regular tetrahedron, and (b) the strings, bars and nodes of the corresponding truncated regular
tetrahedral tensegrity.
Li-Yuan Zhang et al. Proc. R. Soc. A 2012;468:3323-3347
©2012 by The Royal Society
![Page 5: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/5.jpg)
Self-equilibrium solutions with the minimum eigenvalue of force density matrix being negative, hence violating the positive semi-definite condition of super-stability for truncated regular (a)
cubic, (b) octahedral, (c) dodecahedral and (d) icosahedral tens...
Li-Yuan Zhang et al. Proc. R. Soc. A 2012;468:3323-3347
©2012 by The Royal Society
![Page 6: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/6.jpg)
Un-truncated tetrahedron: (a) its geometry and (b) the corresponding tensegrity.
Li-Yuan Zhang et al. Proc. R. Soc. A 2012;468:3323-3347
©2012 by The Royal Society
![Page 7: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/7.jpg)
Self-equilibrium and possibly super-stable solutions for truncated regular (a) tetrahedral, (b) cubic, (c) octahedral, (d) dodecahedral and (e) icosahedral tensegrities.
Li-Yuan Zhang et al. Proc. R. Soc. A 2012;468:3323-3347
©2012 by The Royal Society
![Page 8: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/8.jpg)
Critical truncated tensegrity structures associated with (a) tetrahedron, (b) cube and octahedron, and (c) dodecahedron and icosahedron.
Li-Yuan Zhang et al. Proc. R. Soc. A 2012;468:3323-3347
©2012 by The Royal Society
![Page 9: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/9.jpg)
A special self-equilibrated state of truncated tetrahedral tensegrity with zero force densities in the remaining-strings.
Li-Yuan Zhang et al. Proc. R. Soc. A 2012;468:3323-3347
©2012 by The Royal Society
![Page 10: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/10.jpg)
Hyper-truncated tetrahedron: (a) its geometry and (b) the corresponding tensegrity.
Li-Yuan Zhang et al. Proc. R. Soc. A 2012;468:3323-3347
©2012 by The Royal Society
![Page 11: Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: a unified analytical solution by Li-Yuan Zhang, Yue Li, Yan-Ping](https://reader036.vdocuments.us/reader036/viewer/2022062518/56649f325503460f94c4ee56/html5/thumbnails/11.jpg)
Example self-equilibrated configurations of truncated regular polyhedral tensegrities which are (a) super-stable and (b) not super-stable.
Li-Yuan Zhang et al. Proc. R. Soc. A 2012;468:3323-3347
©2012 by The Royal Society