A Virtual Body and Joint for Constrained Flexible Multibody Dynamics

Abstract

This research developes a relative coordinate formulation for the multibody flexible dynamics. The velocity transformation method is notationally compact, because the Cartesian generalized velocities are simultaneously transformed to the relative generalized velocities in a matrix form. However, inherent computational efficiency in the recursive kinematics between two adjacent bodies has not been exploited. This research presents a recursive formulation which is both notationally compact and computationally efficient. The velocity transformation method is used to derive the equations of motion and their derivatives. Matrix operations associated with the velocity transformation matrix in the resulting equations of motion and their derivatives are classified into several categories. A joint library of the generalized recursive formulas is developed for each category. When one category is encountered in implementing the equations of motion and their derivatives, the corresponding recursive formulas in the category are invoked. When a new force or joint module is added to a general purpose program in the relative coordinate formulation, the modules for the rigid body are not reusable for the flexible body. Since the flexible body dynamics handles additional generalized coordinates associated with deformation, implementation of the flexible dynamics is generally complicated and prone to coding mistakes. A virtual rigid body is introduced at every joint and force reference frames. A virtual flexible body joint is introduced between two body reference frames of the virtual and original bodies. This makes a flexible body subjected to only the kinematic admissibility condition for the virtual flexible body joint. As a result, only extra work to handle the flexible bodies is to add the virtual flexible body joint modules in all recursive formulas. Since computation time in a relative coordinate formulation is approximately proportional to the number of the relative coordinates, computational overhead due to the additional virtual bodies and joints is minor. Meanwhile, implementation convenience is dramatically improved.