An efficient method for the kinematic and dynamic analysis of spatial multibody systems, which consist of interconnected rigid and flexible bodies, is presented. To derive the equations of motion in terms of relative and elastic coordinates, the velocity transformation technique applicable to flexible multibody systems is developed. The position transformation equations that relate the relative and elastic coordinates to the Cartesian coordinates for the two contiguous flexible bodies are derived. The velocity transformation matrix is derived systematically corresponding to the types of kinematic joints connecting the bodies and system path matrix. The equations of motion are then formulated in terms of relative and elastic coordinates by the velocity transformation matrix. The Euler parameters are used as the rotational coordinates to represent the orientation of body reference frame for the convenience of algebraic manipulation. To reduce the number of elastic coordinates, deformation of the flexible body is approximated by a linear combination of normal vibration modes obtained from the finite element analysis.
A computer program has been developed based on the proposed algorithm and several examples are taken to test the method developed here. The dynamic response of a simply supported beam due to the self-weight, for which an analytical solution is available, is analyzed. The results show good agreement with the analytical solution. Spatial motion of a flexible manipulator with angle drivers is simulated. Tip deviations of the flexible model from the rigid body kinematic solution are shown. As an application of the method for the analysis of a more complex system, a launcher system is simulated. The results are compared with the absolute coordinate formulations. Through these examples, the algorithm is shown to be general and efficient.
