Fast and simple dynamic analysis of flexible multi-body systems

Abstract

A simple method to study the dynamics of beam-like multi-body systems is presented. The case of large elastic displacements and small strain is considered. Elastic displacements may be so large that the initial configuration of the system is completely changed. This may happen in compliant mechanisms, for instance, which have become widespread in the last few years. Classical large-displacement beam theory leads to an incremental time-consuming procedure that uses translations and rotations as nodal coordinates. Rotations (in fact, the Euler-Rodrigues parameters), axial strain, and shear angles are proposed as nodal coordinates. This approach gives the exact equations, written for actual, deformed system configuration. Movement equations are derived for both the Euler-Bernoulli and Timoshenko models. Motion equations are expressed in the actual total nodal generalized coordinates, not in their increments, therefore resulting in a non-incremental approach. The mass and stiffness matrices are polynomials in nodal coordinates, which explains the rapid convergence. In static analysis, even for highly distorted systems, the load may be applied in only one load step and the results are obtained in few iterations, giving high accuracy. Newmark method is used to integrate the non-linear motion equations and, usually, no more than two iterations for each time step are required. A floating reference-frame formulation and absolute nodal-coordinate formulation are presented in the work. Finally, several numerical examples are presented.