Modeling Flexible Bodies in Multibody Systems in Joint-Coordinates Formulation Using Spatial Algebra

Abstract

This paper presents an efficient approach of using spatial algebra operator to formulate the kinematic and dynamic equations for developing capabilities to model flexible bodies within general purpose multibody dynamics solver. The proposed approach utilizes the joint coordinates and modal elastic coordinates as the system generalized coordinates. The recursive nonlinear equations of motion are initially formulated using the Cartesian body coordinates (CBC) and the joint coordinates (JC) to form an augmented set of differential algebraic equations. The system connectivity matrix is derived from system topological relations and is used to project the Cartesian quantities into the joint subspace leading to minimum set of differential equations. The modal transformation matrix is used to describe the finite element kinematics in terms of a small set of generalized modal coordinates. Although the resulting stiffness matrix is constant, the mass matrix depends on the generalized elastic modal coordinates and needs to be updated at each time step. To reduce the computational efforts, a set of precomputed inertia shape invariants (ISI) can be identified and used to update the flexible body mass matrix. In this proposed joint-coordinates formulation, the transformation operations required for the flexible body inertia matrix are different from those in case of CBC formulation. The necessary ISI and the algorithm to reconstruct the modal mass matrix will be presented in this paper.