Stability of Cantilever-like Structures with Applications to Soft Robot Arms

The application of variational principles for analyzing problems in the physical sciences is widespread. Cantilever-like problems, where one end is fixed and the other end is free, have received less attention in terms of their stability despite their prevalence. In this article, we establish stability conditions for these problems by examining the second variation of the energy functional through the generalized Jacobi condition. This requires computing conjugate points determined by solving a set of initial value problems from the linearized equilibrium equations. We apply these conditions to investigate the nonlinear stability of intrinsically curved elastic cantilevers subject to an end load. The rod deformations are modelled using Kirchhoff rod theory. The role of intrinsic curvature in inducing complex nonlinear phenomena, such as snap-back instability, is particularly emphasized. The numerical examples highlight its dependence on the system parameters. These examples illustrate potential applications in the design of flexible soft robot arms and innovative mechanisms.