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A Unified Approach for Dynamic Analysis of Tensegrity Structures with Arbitrary Rigid Bodies and Rigid Bars

Jiahui Luo, Xiaoming Xu, Zhigang Wu, Shunan Wu

Abstract

This paper proposes a unified approach for dynamic modeling and simulations of general tensegrity structures with rigid bars and rigid bodies of arbitrary shapes. The natural coordinates are adopted as a non-minimal description in terms of different combinations of basic points and base vectors to resolve the heterogeneity between rigid bodies and rigid bars in three-dimensional space. This leads to a set of differential-algebraic equations with a constant mass matrix and free from trigonometric functions. Formulations for linearized dynamics are derived to enable modal analysis around static equilibrium. For numerical analysis of nonlinear dynamics, we derive a modified symplectic integration scheme which yields realistic results for long-time simulations, and accommodates non-conservative forces as well as boundary conditions. Numerical examples demonstrate the efficacy of the proposed approach for dynamic simulations of Class-1-to-$k$ general tensegrity structures under complex situations, including dynamic external loads, cable-based deployments, and moving boundaries. The novel tensegrity structures also exemplify new ways to create multi-functional structures.

A Unified Approach for Dynamic Analysis of Tensegrity Structures with Arbitrary Rigid Bodies and Rigid Bars

Abstract

This paper proposes a unified approach for dynamic modeling and simulations of general tensegrity structures with rigid bars and rigid bodies of arbitrary shapes. The natural coordinates are adopted as a non-minimal description in terms of different combinations of basic points and base vectors to resolve the heterogeneity between rigid bodies and rigid bars in three-dimensional space. This leads to a set of differential-algebraic equations with a constant mass matrix and free from trigonometric functions. Formulations for linearized dynamics are derived to enable modal analysis around static equilibrium. For numerical analysis of nonlinear dynamics, we derive a modified symplectic integration scheme which yields realistic results for long-time simulations, and accommodates non-conservative forces as well as boundary conditions. Numerical examples demonstrate the efficacy of the proposed approach for dynamic simulations of Class-1-to- general tensegrity structures under complex situations, including dynamic external loads, cable-based deployments, and moving boundaries. The novel tensegrity structures also exemplify new ways to create multi-functional structures.
Paper Structure (21 sections, 44 equations, 19 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 44 equations, 19 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Unifying rigid bodies and rigid bars using natural coordinates
  3. Rigid bodies of arbitrary shapes
  4. 3D rigid bodies
  5. 3D rigid bars
  6. Unified formulations and mass matrices
  7. Modeling tensegrity structures
  8. Ball joints
  9. Boundary conditions
  10. Generalized forces
  11. Tensile cables
  12. Dynamic analysis formulations
  13. Dynamic equation
  14. Linearized dynamics around static equilibrium
  15. Modified symplectic integration scheme for nonlinear dynamics
  16. ...and 6 more sections

Figures (19)

  • Figure 1: A 3D rigid body described by four types of natural coordinates. Rigid bodies are drawn by red lines. Basic points and base vectors are colored in green.
  • Figure 2: A 3D rigid bar described by two types of natural coordinates.
  • Figure 3: The basic point $\bar{\bm{r}}_{I,i}$, the base vectors $\bar{\bm{u}}_{I}$, $\bar{\bm{v}}_{I}$ and $\bar{\bm{w}}_{I}$, the mass center $\bar{\bm{r}}_{I,g}$, and a generic point $\bar{\bm{r}}_{I}$ in the local Cartesian frame of (a) a 3D rigid body or (b) a 3D rigid bar.
  • Figure 4: (a) Two 3D rigid bodies or (b) a 3D rigid body and a 3D rigid bar connected by a ball joint, which is represented by a circle filled with light blue.
  • Figure 5: Two 3D rigid bodies subjected to gravity, a concentrated force, and tension forces of a cable. The points of action are colored in blue.
  • ...and 14 more figures