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Dynamic analysis of multibody systems has gained tremendous popularity in the past two decades. The research has been the subject of numerous publications []–[]. While most of the significant theory has been developed, some computational issues remain to be resolved. The latter is a result of the strategic development of the kinematics involved and the formalism used in for the development of the equations of motion. In the recursive formulation developed in reference []–[], relative coordinates were used in conjunction with Boolean matrices and the tensor notation, which made the notation somehow difficult to follow. While the tensor notation adopted kept the equations in compact form, it hid some vital information from interested readers. This chapter focuses on developing an all-purpose algorithm for the dynamic simulation of flexible treelike systems, making use of matrix representation at all levels. The equations developed are applicable to high-speed systems undergoing large amounts of rotation, and a strict finite-element formulation is presented. The inertial moment of the elements, the damping forces of the structure, and geometrical stiffening forces are all included in an explicit form to be able to monitor their contribution and how they influence the dynamics of multibody systems.

The dynamic simulation of flexible multibody systems has been a real challenge to dynamists and researchers for the past two decades. Areas of interest include robotics, aircraft dynamics, vehicle dynamics and rotorcraft systems. Over the years, numerous papers have been published, tackling a broad range of theoretical problems. More precisely, the inclusion of geometric nonlinearities in the dynamics has proven to be a major contributing factor in predicting accurately the dynamics of flexible (elastic) beams and plates and optimization, which is a recurrence in variational design.