Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Static and Linearized Dynamic Examples

In the present paper, a three-dimensional shear deformable beam finite element is presented, which is based on the absolute nodal coordinate formulation (ANCF). The orientation of the beam’s cross section is parameterized by means of slope vectors. Both a structural mechanics based formulation of the elastic forces based on Reissner’s nonlinear rod theory, as well as a continuum mechanics based formulation for a St. Venant Kirchhoff material are presented in this paper. The performance of the proposed finite beam element is investigated by the analysis of several static and linearized dynamic problems. A comparison to results provided in the literature, to analytical solutions, and to the solution found by commercial finite element software shows high accuracy and high order of convergence, and therefore the present element has high potential for geometrically nonlinear problems.

On the Kinematics of the Octopus’s Arm

The kinematics of the octopus’s arm is studied from the point of view of robotics. A continuum three-dimensional kinematic model of the arm, based on a nonlinear rod theory, is proposed. The model enables the calculation of the strains in various muscle fibers that are required in order to produce a given configuration of the arm—a solution to the inverse kinematics problem. The analysis of the forward kinematics problem shows that the strains in the muscle fibers at two distinct points belonging to a cross section of the arm determine the curvature and the twist of the arm at that cross section. The octopus’s arm lacks a rigid skeleton and the role of material incompressibility in enabling the configuration control is studied.

Planar Vibration of an Elastica Arch: Theory and Experiment

This investigation examines the planar, linear vibration of a deep arch that is described by a simply supported elastica. The arch is formed from an elastic rod that buckles nonlinearly under the action of a large, steady end-load. A theoretical model is proposed that governs the planar response of the rod about a generally curved, pre-stressed equilibrium. The model utilizes a geometrically nonlinear rod theory to describe the planar bending and extension of the rod centerline. The equations of motion are linearized about an elastica equilibrium and numerical solutions for free vibration are determined using a variational formulation of the associated eigenvalue problem. Natural frequencies and mode shapes are computed over a large range of centrally and eccentrically applied end-loads. Results from an experimental modal test provide support for the model.