A Finite Element Geometrically Nonlinear Dynamic Formulation of Flexible Multibody Systems Using a New Displacements Representation

Abstract

In previous work (Mayo, 1993), the authors developed two geometrically nonlinear formulations of beams inflexible multibody systems. One, like most related methods, includes geometric elastic nonlinearity in the motion equations via the stiffness terms (Mayo and Domínguez, 1995), but preserving terms, in the expression for the strain energy, of a higher-order than most available formulations. The other formulation relies on distinguishing the contribution of the foreshortening effect from that of strain in modelling the displacement of a point. While including exactly the same nonlinear terms in the expression for the strain energy, the stiffness terms in the motion equations generated by this formulation are exclusively limited to the constant stiffness matrix for the linear analysis because the terms arising from geometric elastic nonlinearity are moved from elastic forces to inertial, reactive and external forces, which are originally nonlinear. This formulation was reported in a previous paper (Mayo et al, 1995) and used in conjunction with the assumed-modes method. The aim of the present work is to implement this second formulation on the basis of the finite-element method. If, in addition, the component mode synthesis method is applied to reduce the number of degrees of freedom, the proposed formulation takes account of the effect of geometric elastic nonlinearity on the transverse displacements occurring during bending without the need to include any axial vibration modes. This makes the formulation particularly efficient in computational terms and numerically more stable than alternative geometrically nonlinear formulations based on lower-order terms.