A Simple Energy-conserving Torsion-free Beam Element for Multibody Applications

Abstract

Slender flexible beams are very common in many applications in different fields related to multibody dynamics: robotics, aerospace mechanisms, computer graphics, etc. Typically, these components experience small strains but large displacements and rotations. What is more, ofter their torsional strains are very small or null, due to their slenderness or the specific type of kinematical pairs that connect them with other parts of the mechanism. The numerical simulation of the dynamical behavior of such systems demand the development of beam elements capable of representing these nonlinear effects, while providing accurate results when used with a time-marching scheme. Many different approaches have been proposed in the literature for dealing with these types of problems. The main ones are segmentation or lumped methods [3, 4, 5], floating frame methods, and nonlinear finite element formulations. Naturally, each approach have different advantages and drawbacks for different applications, and can be used with different time integration schemes. Among these schemes, the so-called geometric integration methods have gained popularity in the last two decades. Specifically, energy-consistent (both energy-conserving or energy-decaying) methods present great appeal for the robust and physically-accurate integration of the dynamics of such systems. Energy-consistent formulations for beams with the nonlinear finite element approach have been reported in the literature but, to the best of the author's knowledge, there are no such formulations available for the segmentation or lumped approach. We propose in this work the energy-conserving extension of the simple torsion-free beam element proposed in [2], which belongs to this category. In this model, the beam is a collection of articulated non- linear trusses (large displacements and strains) with the proper tensile stiffness derived from a nonlinear potential, parametrized with the cartesian coordinates of the extremes. The bending stiffness is represented by another potential, such that a single element is composed by two segments (three nodes) that overlap with the neighbors. Since both tensile and bending forces are defined through discrete potentials, it is possible to develop a simple energy-conserving formulation, following [1]. With this formulation, total mechanical energy is exactly preserved in the discrete setting when no dissipative effects are present in the model. The main benefit of this approach is the enhancement of the stability of the numerical scheme. It is remarkable too that the formulation does not employ rotational degrees of freedom and can be applied to two and three-dimensional problems. The basic formulation of this idea will be presented along with some validation tests that will assess the accuracy of the model, and some exploration on potential applications.