On the use of the finite element method and classical approximation techniques in the non-linear dynamics of multibody systems

Abstract

The dynamic formulations for deformable bodies based on the finite element method and classical approximation techniques are compared. The finite element method and classical approximation techniques such as Rayleigh-Ritz methods are used to develop a set of generalized Newton-Euler equations for deformable bodies that undergo large translational and rotational displacements. In the finite element formulation, a stationary (total) Lagrangian approach is used to formulate the generalized Newton-Euler equations for each finite element in terms of a set of invariants that depend on the assumed displacement field. The deformable body invariants are obtained by assembling the invariants of the finite elements using a standard finite element Boolean matrix approach. This leads to the non-linear generalized Newton-Euler equations for the deformable bodies. These equations are presented in a simple closed form which is useful in developing recursive formulations for multibody systems consisting of interconnected deformable bodies. Both lumped and consistent mass formulations are discussed. Numerical results are presented for comparison of the finite element method and classical approximation methods in the non-linear dynamics of multibody systems.