A high-precision co-rotational formulation of 3D beam elements for dynamic analysis of flexible multibody systems

Abstract

Evaluating inertia forces is a central and complicated task for dynamic analysis of flexible multibody systems (FMS) involving large displacements and rotations. In a number of co-rotational approaches, cubic interpolations are adopted to formulate both inertia and internal forces, where the inertia term may be complicated in derivation and several Gauss points are needed in the numerical integration. The paper presents a high-precision co-rotational formulation of 3D beam elements for dynamic analysis of FMS. The method of switching the rotation vector and its complement is adopted to avoid the singularity problem in spatial finite rotations. In contrast to the traditional co-rotational formulation, the governing equations in the paper are formulated based on the principle of virtual power, without requiring the variation of rotation matrix. In the framework of the proposed co-rotational formulation, the stiffness matrix of small-deformation beam element can be used directly, including the Euler–Bernoulli and Timoshenko–Reissner beam elements. More essentially, the inertia terms of the beam element are formulated by discretizing the beam element into three lumped masses at the left and right end nodes, and an auxiliary node at the middle point of the beam element, resulting in a lumped mass matrix. The equivalence ensures that the total mass is completely accurate and the moment of inertia is high-precision if the beam element undergoes small elastic deformation in the local coordinate system. In the local coordinate system of co-rotational formulations, small elastic deformation can be guaranteed so that the inertia forces are formulated analytically without needing the Gauss integration. Finally, four numerical examples are considered to evaluate the accuracy of the formulation against to the geometrically exact beam theory and the previous co-rotational beam formulations. The proposed method can give high-precision numerical results from the comparison, even if using a small number of elements.