Configurational forces and nonlinear structural dynamics

Abstract

Configurational, or Eshelby-like, forces are shown to strongly influence the nonlinear dynamics of an elastic rod constrained with a frictionless sliding sleeve at one end and with an attached mass at the other end. The configurational force, generated at the sliding sleeve constraint and proportional to the square of the bending moment realized there, has been so far investigated only under quasi-static setting and is now confirmed (through a variational argument) to be present within a dynamic framework. The deep influence of configurational forces on the dynamics is shown both theoretically (through the development of a dynamic nonlinear model in which the rod is treated as a nonlinear spring, obeying the Euler elastica, with negligible inertia) and experimentally (through a specifically designed experimental setup). During the nonlinear dynamics, the elastic rod may slip alternatively in and out from the sliding sleeve, becoming a sort of nonlinear oscillator displaying a motion eventually ending with the rod completely injected into or completely ejected from the sleeve. The present results may find applications in the dynamics of compliant and extensible devices, for instance, to guide the movement of a retractable and flexible robot arm.

A Sliding-Rod Variable-Strain Model for Concentric Tube Robots

“…Equation (15) says that the distributed constraint force is equal and opposite between two consecutive tubes and that it takes the form:…”

“…Here we relax this assumption, although still in a static setting, which allows calculation of the actuation insertion forces required for the equilibrium for the first time. The spaghetti problem appears in the planar models of animal locomotion [14], structural stability [15], and finite element analysis [16]. Thus, to the authors' knowledge, it is also the first time the spaghetti problem is tackled in three dimensions and with an additional relative rotation.…”

A Sliding-rod Variable-strain Model for Concentric Tube Robots

“…1). Equation (15) says that the distributed constraint force is equal and opposite between two consecutive tubes and that it takes the form:…”

“…To the authors' knowledge, it is the first time that such a model is proposed for CTRs. The spaghetti problem appears in the planar models of animal locomotion [14], structural stability [15], and finite element analysis [16]. Thus, to the authors' knowledge, it is also the first time the spaghetti problem is tackled in three dimensions and with an additional relative rotation.…”

On contact point motion in the vibration analysis of elastic rods

“…The goal of this paper is to clarify the third approach, use it to explain when the first is applicable, and highlight why the second is an unnecessary complication if only linear vibrations are considered. We do all this in the context of the foregoing contact problem, but the technique we outline can be readily applied to a number of other situations involving variable-length rods, such as a roller support or sleeve constraint [18,19]. Over the course of our analysis, which we present in considerable detail, we explain several counter-intuitive results from the literature.…”