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Research Papers

Analytical Form-Finding for Highly Symmetric and Super-Stable Configurations of Rhombic Truncated Regular Polyhedral Tensegrities

[+] Author and Article Information
Li-Yuan Zhang

School of Mechanical Engineering,
University of Science and Technology Beijing,
Beijing 100083, China
e-mail: zhangly@ustb.edu.cn

Shi-Xin Zhu

School of Mechanical Engineering,
University of Science and Technology Beijing,
Beijing 100083, China

Xiao-Fei Chen

Beijing Institute of Astronautical
Systems Engineering,
Beijing 100076, China

Guang-Kui Xu

International Center for Applied Mechanics,
State Key Laboratory for Strength and
Vibration of Mechanical Structures,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: guangkuixu@mail.xjtu.edu.cn

1Corresponding author.

Manuscript received September 21, 2018; final manuscript received December 4, 2018; published online January 8, 2019. Assoc. Editor: Pedro Reis.

J. Appl. Mech 86(3), 031006 (Jan 08, 2019) (11 pages) Paper No: JAM-18-1541; doi: 10.1115/1.4042216 History: Received September 21, 2018; Revised December 04, 2018

Tensegrities have exhibited great importance and numerous applications in many mechanical, aerospace, and biological systems, for which symmetric configurations are preferred as the tensegrity prototypes. Besides the well-known prismatic tensegrities, another ingenious group of tensegrities with high symmetry is the truncated regular polyhedral (TRP) tensegrities, including Z-based and rhombic types. Although Z-based TRP tensegrities have been widely studied in the form-finding and application issues, rhombic TRP tensegrities have been much less reported due to the lack of explicit solutions that can produce their symmetric configurations. Our former work presented a unified solution for the rhombic TRP tensegrities by involving the force-density method which yet cannot control structural geometric sizes and may produce irregular shapes. Here, using the structural equilibrium matrix-based form-finding method, we establish some analytical equations, in terms of structural geometric parameters and force-densities in elements, to directly construct the self-equilibrated, symmetric configurations of rhombic TRP tensegrities, i.e., tetrahedral, cubic/octahedral, and dodecahedral/icosahedral configurations. Moreover, it is proved, both theoretically and numerically, that all of our obtained rhombic TRP tensegrities are super-stable and thus can be stable for any level of the force-densities without causing element material failure, which is beneficial to their actual construction. This study helps to readily design rhombic tensegrities with high symmetry and develop novel biomechanical models, mechanical metamaterials, and advanced mechanical devices.

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Figures

Grahic Jump Location
Fig. 1

Truncated polyhedral tensegrities: (a) Z-based class and (b) rhombic class

Grahic Jump Location
Fig. 2

Truncated regular tetrahedron and rhombic truncated regular tetrahedral tensegrity: (a) polyhedron, (b) polyhedral topology, (c) tensegrity topology, and (d) tensegrity. In figure (a), bT denotes the length of the edges of tetrahedron without truncation, while aT denotes the length of the truncating edge of truncated tetrahedron. Thus, the length of the remaining edge is calculated by bT−2aT. In figure (d), qTs1, qTs2, and qTb denote the force-densities of the type-1 string, type-2 string, and bar of tetrahedral tensegrity, respectively.

Grahic Jump Location
Fig. 3

Self-equilibrated and super-stable tetrahedral tensegrity: (a) variations of structural geometric size (solid line) and force-densities in elements (dashed lines) with respect to the twist angle and (b) structural configurations with twist angles (1) −5π/18, (2) −2π/9, (3) −π/6, (4) −π/18, and (5) π/18

Grahic Jump Location
Fig. 4

Truncated regular tetrahedral tensegrities and their mechanical responses: (a) tetrahedral tensegrities of (1) rhombic class and (2) Z-based class and (b) variations of constraining force and structural stiffness with respect to applied displacement

Grahic Jump Location
Fig. 5

Truncated regular octahedron and rhombic truncated regular octahedral tensegrity: (a) polyhedron, (b) polyhedral topology, (c) tensegrity topology, and (d) tensegrity. In figure (a), bO denotes the length of the edges of octahedron without truncation, while aO denotes the length of the truncating edge of truncated octahedron. Thus, the length of the remaining edge is calculated by bO−2aO. In figure (d), qOs1, qOs2, and qOb denote the force-densities of the type-1 string, type-2 string, and bar of octahedral tensegrity, respectively.

Grahic Jump Location
Fig. 6

Self-equilibrated and super-stable octahedral tensegrity: (a) variations of structural geometric size (solid line) and force-densities in elements (dashed lines) with respect to the twist angle and (b) structural configurations with twist angles (1) −7π/36, (2) −5π/36, (3) −π/12, (4) π/36, and (5) 5π/36

Grahic Jump Location
Fig. 7

Truncated regular icosahedron and rhombic truncated regular icosahedral tensegrity: (a) polyhedron, (b) polyhedral topology, (c) tensegrity topology, and (d) tensegrity. In figure (a), bI denotes the length of the edges of icosahedron without truncation, while aI denotes the length of the truncating edge of truncated icosahedron. Thus, the length of the remaining edge is calculated by bI−2aI. In figure (d), qIs1, qIs2, and qIb denote the force-densities of the type-1 string, type-2 string, and bar of icosahedral tensegrity, respectively.

Grahic Jump Location
Fig. 8

Self-equilibrated and super-stable icosahedral tensegrity: (a) variations of structural geometric size (solid line) and force-densities in elements (dashed lines) with respect to twist angle and (b) structural configurations with twist angles (1) −13π/90, (2) −4π/45, (3) −π/30, (4) 7π/90, and (5) 17π/90

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