Stability Analysis of Cantilever-like Structures with Applications to Soft Robotic Arms

The application of variational structure for analyzing problems in the physical sciences is widespread. Cantilever-like problems, where one end is subjected to a fixed value and the other end is free, have been less studied, especially in terms of their stability despite their abundance. In this article, we develop the stability conditions for these problems by examining the second variation of the energy functional using the generalized Jacobi condition, which includes computing conjugate points. These conjugate points are determined by solving a set of initial value problems from the resulting linearized equilibrium equations. We apply these conditions to investigate the nonlinear stability of intrinsically curved elastic cantilevers subject to a tip load. Kirchhoff rod theory is employed to model the elastic rod deformations. The role of intrinsic curvature in inducing complex nonlinear phenomena, such as snap-back instability, is particularly emphasized. This snap-back instability is demonstrated using various examples, highlighting its dependence on various system parameters. The presented examples illustrate the potential applications in the design of flexible soft robotic arms and mechanisms.