Multiple equilibrium states of a curved-sided hexagram: Part I-Stability of states

TL;DR

This paper investigates the stability of multiple equilibrium configurations of a curved-sided hexagram ring made of uniform rods, analyzing conditions for stability and demonstrating various states experimentally.

Contribution

It introduces a comprehensive analysis of equilibrium states and stability conditions for curved hexagram rings, including experimental validation and stability criteria based on rod geometry and curvature.

Findings

Identified four stable equilibrium states of the hexagram ring.

Determined stability limits related to natural curvature and cross-section geometry.

Validated stability conditions through experimental demonstrations.

Abstract

The stability of the multiple equilibrium states of a hexagram ring with six curved sides is investigated. Each of the six segments is a rod having the same length and uniform natural curvature. These rods are bent uniformly in the plane of the hexagram into equal arcs of 120deg or 240deg and joined at a cusp where their ends meet to form a 1-loop planar ring. The 1-loop rings formed from 120deg or 240deg arcs are inversions of one another and they, in turn, can be folded into a 3-loop straight line configuration or a 3-loop ring with each loop in an "8" shape. Each of these four equilibrium states has a uniform bending moment. Two additional intriguing planar shapes, 6-circle hexagrams, with equilibrium states that are also uniform bending, are identified and analyzed for stability. Stability is lost when the natural curvature falls outside the upper and lower limits in the form of a bifurcation mode involving coupled out-of-plane deflection and torsion of the rod segments. Conditions for stability, or lack thereof, depend on the geometry of the rod cross-section as well as its natural curvature. Rods with circular and rectangular cross-sections will be analyzed using a specialized form of Kirchhoff rod theory, and properties will be detailed such that all four of the states of interest are mutually stable. Experimental demonstrations of the various states and their stability are presented. Part II presents numerical simulations of transitions between states using both rod theory and a three-dimensional finite element formulation, includes confirmation of the stability limits established in Part I, and presents additional experimental demonstrations and verifications.