Efficient numerical integration techniques for multibody systems based on incremental forms of finite rotation

Abstract

In analyses of multibody systems, a numerical treatment of finite rotation is of key importance due to its mathematical complexity and numerical difficulties. Methods describing the finite rotation are naturally classified into vectorial and non-vectorial parameterizations. In particular, this study addresses a method incorporating a discretization with respected to time into the vectorial parameterizations. It introduces an approximation form for the rotation kinematic compatibility equation which expresses a relation between the angular velocities and the time derivatives of the rotation parameters. The model used herein is based on the fact that the resulting equations are eventually solved by numerical integration techniques. The resulting set of equations of motion for the rotation can be expressed in terms of the incremental angular velocity only. On the other hand, this study employs a scaling method based on a preconditioning technique, since an iteration matrix in the Newton-Raphson method includes higher order terms for the time step size. This remedy is also effective in order to avoid severe ill-condition in numerical integrations for constrained systems with configuration level constraints. In addition, the present models are enhanced by incorporating with the Euler parameters which are often used as singularity free expression for the finite rotation. Then, this study applies the present expressions for the finite rotation to numerical integration methods based on the energy-momentum preserving scheme and the generalized-α scheme. The present numerical models are verified through comparisons with the conventional models based on the standard Newton-Euler equations with the Euler angles in a typical benchmark problem for a simple multibody system. Then, we discuss its performance as the numerical models, such as convergency and computation time.