The goal of this study is to present a method for deriving equations of motion capable of modeling the controlled motion of an open loop multibody structure comprised of an arbitrary number of rigid bodies and slender beams. The procedure presented here for deriving equations of motion for flexible multibody systems is carried out by means of the Principle of Virtual Work (often referred to in the dynamics literature as d'Alembert's Principle).

We first consider the motion of a general flexible body relative to the inertial space, and then derive specific formulas for both rigid bodies and slender beams. Next, we make a small motions assumption, with the end result being equations for a Rayleigh beam, which include terms which account for the axial motion, due to bending, of points on the beam central axis. This process includes a novel application of the exponential form of an orthogonal matrix, which is ideally suited for truncation. Then, the generalized coordinates and quasi-velocities used in the mathematical model, including those needed in the spatial discretization process of the beam equations are discussed. Furthermore, we develop a new set of recursive relations used to compute the inertial motion of a body in terms of the generalized coordinates and quasi-velocities.

This research was motivated by the desire to model the controlled motion of a flexible space robot, and consequently, we use the multibody dynamics equations to simulate the motion of such a structure, providing a demonstration of the computer program. For this particular example we make use of a new sequence of shape functions, first used by Meirovitch and Stemple to model a two dimensional building frame subjected to earthquake excitations.