Graph-Theoretic Approach for the Dynamic Simulation of Flexible Multibody Systems

M J Richard,M Bouazara

doi:10.1155/2012/530132

Abstract

This paper provides a general description of a variational graph-theoretic formulation for simulation of flexible multibody systems (FMSs) which includes a brief review of linear graph principles required to formulate this algorithm. The system is represented by a linear graph, in which nodes represent reference frames on flexible bodies, and edges represent components that connect these frames. The method is based on a simplistic topological approach which casts the dynamic equations of motion into a symmetrical format. To generate the equations of motion with elastic deformations, the flexible bodies are discretized using two types of finite elements. The first is a 2 node 3D beam element based on Mindlin kinematics with quadratic rotation. This element is used to discretize unidirectional bodies such as links of flexible systems. The second consists of a triangular thin shell element based on the discrete Kirchhoff criterion and can be used to discretize bidirectional bodies such as high-speed lightweight manipulators, large high precision deployable space structures, and micro/nano-electromechanical systems (MEMSs). Two flexible systems are analyzed to illustrate the performance of this new variational graph-theoretic formulation and its ability to generate directly a set of motion equations for FMS without additional user input.

Highlights

Stringent tolerances on mechanical components have created increasingly severe demands on the quality of new mechanical designs
Since the equations governing the motion of flexible multibody systems (FMSs) are highly nonlinear and dynamically coupled, one must exploit some kind of linear graph principles [1,2,3,4] to properly define the interconnection between the bodies
By combining the mechanical system topology with the variational virtual work constitutive equations, a new systematic graph-theoretic formulation has been introduced and used to describe the time-varying configuration of spatial FMS. This method assembles automatically the governing equations of motion in a symmetrical format where the structure and organization of the mass matrix parallels that of structural finite element mass and stiffness matrices, which are derived using variational methods

Summary

Introduction

Stringent tolerances on mechanical components have created increasingly severe demands on the quality of new mechanical designs. By combining linear graph theory [5,6,7,8] with the principle of virtual [9,10,11] work and finite elements, a dynamic formulation is obtained that extends graphtheoretic (GT) modelling methods to the analysis of 3D beams and shell surfaces of FMS. Advances in Mechanical Engineering the principle of virtual work, a new variational Lagrangian formulation is obtained that extends graph-theoretic modelling methods to the analysis of FMS. It shows how a Lagrangian multiplier technique can be incorporated into a flexible finite element algorithm. In order to create a system graph that results in correct kinematic and flexible dynamic equations for any choice of spanning tree, it is necessary to introduce a dependent virtual work element

System Representation by GT Modelling

Virtual Work Terminal Equations

Examples

Conclusion