Positional FEM formulation for flexible multi-body dynamic analysis

This paper presents a simple formulation to deal with flexible multi-body dynamic systems by the finite element method. The proposed methodology is based on the minimum potential energy theorem written regarding nodal positions. Velocity, acceleration and strain are achieved directly from positions, not displacements. A non-dimensional space is created and the relative curvature and fibers length are calculated for both reference and deformed configurations and used to calculate the strain energy at general points. The classical Newmark equations are used to integrate time. Damping is introduced into the mechanical system by a rheonomic energy functional. The final formulation has the advantage of being simple and easy to teach, when compared to classical counterparts. The behavior of a bench-mark problem (spin-up maneuver) is studied regarding the influence of mass representation on its overall transient and steady-state behavior. Three other examples are presented to show the applicability of the technique, namely, a flexible slider–crank mechanism, a flexible beam flight and a Peaucellier-type mechanism. The results are compared with other authors’ numerical solutions.