Robust analysis of flexible multibody systems and joint clearances in an energy conserving framework

Abstract

Flexible multibody systems (MBS) appear in a number of mechanical applications, in which the model must consider the deformation of some or all of the bodies. A classical method for considering flexibility has been the floating frame technique (25), generally limited to small strains. A more general approach based on inertial coordinates may be formulated by nonlinear finite element methods (27; 28; 11), which are versatile and computationally efficient. Furthermore, employing energy-momentum time integration algorithms they prove to be extremely stable for nonlinear stiff behaviours which frequently arise in such systems. We present briefly the overall dynamic formulation, but we focus mainly on the formulation of constraints for joints with or without clearances. Integration algorithms that conserve both momentum and energy have been proposed in (29; 26; 16; 15), attracting considerable attention in the last few years. One of the main benefits of their robustness is their ability to perform stable long-term simulations in nonlinear systems. The approach followed here, in contrast to other energy-momentum formulations (29; 16; 8; 5; 9; 23), differs in two key aspects: 1) a rotation-free parametrisation for rigid bodies, based on inertial cartesian coordinates of body points, forming a dependent set which is subject to constraints; 2) a penalty formulation for constraints. As will be explained below, this allows for a simple, efficient and robust numerical implementation. Penalty methods are associated to a non-exact fulfilment of constraints, and in order to ensure sufficient numerical accuracy, large enough penalty parameters must be employed. These may lead to a stiff behaviour and further difficulties in the numerical solution of the problem. Their effect on the system is analogous to introducing very stiff elements between the constrained degrees of freedom. As a consequence, a penalty approach introduces high-frequency components in addition to the already existing ones in flexible multibody systems, due to wave propagation in the deformable bodies. This causes severe numerical difficulties for most time integration algorithms (8), being the main drawback of penalty formulations. However,