Energy-Momentum Conserving Time Integration of Modally Reduced Flexible Multibody Systems

Many conventional time integration schemes frequently adopted in flexible multibody dynamics fail to retain the fundamental conservation laws of energy and momentum of the continuous time domain. Lack of conservation, however, in particular of angular momentum, may give rise to unexpected, unphysical results. To avoid such problems, a scheme for the consistent integration of modally reduced multibody systems subjected to holonomic constraints is developed in the present paper. As opposed to the conventional approach, in which the floating frame of reference formulation is combined with component mode synthesis for approximating the flexible deformation, an alternative, recently proposed formulation based on absolute coordinates is adopted in the analysis. Owing to the linear relationship between the generalized coordinates and the absolute displacement, the inertia terms in the equations of motion attain a very simple structure. The mass matrix remains independent of the current state of deformation and the velocity dependent term known from the floating frame approach vanishes due to the absence of relative coordinates. These advantageous properties facilitate the construction of an energy and momentum consistent integration scheme. By the mid-point rule, algorithmic conservation of both linear and angular momentum is achieved. In order to consistently integrate the total energy of the system, the discrete derivative needs to be adopted when evaluating the strain energy gradient and the derivative of the algebraic constraint equations.