Systems of rigid and flexible bodies undergoing large rigid body motions but small elastic deformations are investigated. In order to get the correct equations of motion one has to consider geometric nonlinearities even in the elastic coordinates. Different possibilities of independently choosing these coordinates are presented. The flexible bodies are discretized using a Ritz-Galerkin approximation. This discretization leads to ordinary differential equations for the description of the clastic vibrations of the flexible bodies as well as for the description of the rigid body motions.