Describing Rigid-Flexible Multibody Systems Using Natural and Absolute Nodal Coordinates

Abstract

This paper deals with the dynamic description of interconnected rigid and flexible bodies. The absolute nodal coordinate formulation is used to describe the motion of flexible bodies and natural coordinates are used to describe the motion of the rigid bodies. The absolute nodal coordinate formulation is a non-incremental finite element procedure specially suitable for the dynamic analysis of flexible bodies exhibiting rigid body motion and large deformations. Nodal coordinates, that include global position vectors and global slopes, are all defined in a global inertial coordinate system. The advantages of using the absolute nodal coordinate formulation include constancy in the mass matrix, and the need for only a minimal set of non-linear constraint equations when connecting different flexible bodies with kinematic joints. When bodies within the system can be considered rigid, the above-mentioned advantages of the equations of motion can be preserved provided natural coordinates are used. In the natural coordinates method, the coordinates used to describe rigid bodies include global position vectors of basic points and global unit vectors. As in the absolute nodal coordinate formulation, rotational coordinates are avoided and the mass matrix is also constant. This paper provides computer implementation of this formulation that only uses absolute coordinates for general two-dimensional multibody systems. The constraint equations needed to define kinematic joints between different bodies can be linear or non-linear. The linear constraint equations, that include those needed to define rigid connections and revolute joints, are used to define constant connectivity matrices that reduce the size of the system coordinates. These constant connectivity matrices are also used to obtain the system mass matrix and the system generalized forces. However, the non-linear constraint equations that account for sliding joints, require the use of the Lagrange multipliers technique. Numerical examples are provided and compared to the results of other existing formulations.