Stability analysis through folds: An end-loaded elastica with a lever arm

TL;DR

This work tackles stability analysis of parameter-dependent variational problems with fixed-free boundaries by extending distinguished bifurcation diagrams to such settings and introducing a stability index based on the sign of the second variation $\delta^{2}J$. It applies the framework to a planar elastica end-loaded through a rigid lever arm, using pseudo-arclength continuation to generate solution branches and folds that signal stability changes without full eigenvalue computations. The main findings reveal multi-stability and snap-back instabilities across varied parameters including load magnitude $P$, direction $\alpha$, arm length $\epsilon$, lever angle $\psi$, and fixed-end rotation $\theta_{o}$, with folds encoding the direction of stability exchange. The results have practical implications for soft robotics and actuator design, and point toward extensions to three-dimensional Kirchhoff rod theory and more complex boundary constraints.

Abstract

Many physical systems can be modelled as parameter-dependent variational problems. In numerous cases, multiple equilibria co-exist, requiring the evaluation of their stability, and the monitoring of transitions between them. Generally, the stability characteristics of the equilibria change near folds in the parameter space. The direction of stability changes is embedded in a specific projection of the solutions, known as distinguished bifurcation diagrams. In this article, we identify such projections for variational problems characterized by fixed-free ends - a class of problems frequently encountered in mechanics. Using these diagrams, we study an Elastica subject to an end load applied through a rigid lever arm. Several instances of snap-back instability are reported, along with their dependence on system parameters through numerical examples. These findings have potential applications in the design of soft robot arms and other actuator designs.

Figures (12)

• Figure 1: The direction of index change near simple folds in the distinguished bifurcation diagram for problems with fixed-free ends: a) Stability transition when the parameter in the fixed end $s=0$ is varied. b) Stability transition when the parameter in the free end $s=l$ is varied.
• Figure 2: Schematic of an elastica with a load arm attached at the free end.
• Figure 3: Plot illustrating the solutions of $\cot{\sqrt{P}} - \sqrt{P}\epsilon=0$ for different values of $\epsilon$, along with the local bifurcation characteristics associated with $\epsilon=+/-0.25$ shown at top. The family of configurations branching out at bifurcations are indicated.
• Figure 4: The distinguished bifurcation diagram for arm $\epsilon = 0.25$ rotating for loads (a)$P=\frac{\pi^{2}}{4}$ and (b) $P=\frac{9\pi^{2}}{4}$. The intermediate configurations along the family of stable equilibrium are labeled and depicted at the top. The stable configurations before and after the snap-back instability are also displayed. In (b), all equilibria are unstable. Although the folds are present no information on snapping behavior can be inferred, as no stable equilibrium exists. The tip trace in the foldless region is shown (in green).
• Figure 5: The distinguished bifurcation diagram for an arm $\epsilon = 0.5$ rotating for loads (a) $P=\frac{\pi^{2}}{4}$ and (b)$P=\frac{9\pi^{2}}{4}$. Intermediate configurations along this family are labeled and depicted at the top. Additionally, the stable configurations before and after the snap-back instability are displayed. The trajectory of the arm's tip preceding the snap-back instability is indicated in green.