Lie Group Variational Integrator for the Geometrically Exact Rod with Circular Cross-Section Incorporating Cross-Sectional Deformation

Abstract

In this paper, we derive the continuous space-time equations of motion of a three-dimensional geometrically exact rod, or the Cosserat rod, incorporating planar cross-sectional deformation. We then adopt the Lie group variational integrator technique to obtain a discrete model of the rod incorporating both rotational motion and cross-sectional deformation as well. The resulting discrete model possesses several desirable features: it ensures volume conservation of the discrete elements by considering cross-sectional deformation through a local dilatation factor, it demonstrates the beneficial properties associated with the variational integrator technique, such as the preservation of the rotational configuration, and energy conservation with a bounded error. An exhaustive set of numerical results under various initial conditions of the rod demonstrates the efficacy of the model in replicating the physics of the system.

Figures (10)

• Figure 1: Deformation of the three-dimensional geometrically exact rod.
• Figure 2: The cross-sectional deformation of the rod due to external force.
• Figure 3: Flying beam: (a) Initial configuration of the beam. (b) The time-dependent function $g(t)$ is used to define external force and moments at the base of the beam.
• Figure 4: Flying beam: Deflection of the base of the beam for the standard (blue) and the modified (red) models. The component of the position of the 3D beam along $e_i$ is denoted by $x_i$ for $i = 1,2,3$.
• Figure 5: Flying beam: Time evolution of the 3D beam including cross-sectional deformation, base trajectory (green) and tip trajectory (red).

Theorems & Definitions (3)

• Remark 5
• Remark 6
• Remark 7