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

In this paper, we develop a unified framework to study the dynamics of tensegrity systems that can include any arbitrary rigid bodies and rigid bars. The natural coordinates are adopted as a completely non-minimal modeling approach to describe both rigid bodies and rigid bars in terms of different combinations of basic points and base vectors. Various types of coordinates combinations are then unified into polymorphic expressions which succinctly encompass Class-1-to-$k$ tensegrities. Then, we employ the Lagrange-d'Alembert principle to derive the dynamic equation, which has a constant mass matrix and is free from trigonometric functions as well as centrifugal and Coriolis terms, owing to the non-minimal formulations. For numerical analysis of nonlinear dynamics, a modified symplectic integration scheme is derived, accommodating non-conservative forces and prescribed boundary conditions. Additionally, formulations for statics and linearized dynamics around static equilibrium states are derived to help determine cable actuations and calculate natural frequencies and mode shapes, which are commonly needed for structural analyses. Numerical examples are given to demonstrate the proposed approach's abilities in the modeling of tensegrity structures composed of Class-1-to-$k$ modules and conducting dynamic simulations with complex conditions, including slack cables, gravity loads, seismic grounds, and cable-based deployments. Finally, two novel designs of tensegrity structures exemplify new ways to create multi-functional composite structures.