Due to the highly complex structure of the equations of motion, there exists a basic demand for procedures with minimum effort. This is achieved by the projection method applied to systems consisting of rigid and elastic bodies which undergo fast rigid body motions with superimposed small elastic deflections. The outlined method leads to different left and right Jacobians for the partial differential equations along with simple operators for determination of corresponding boundary conditions. When a Ritz series expansion is used for approximate solution, the left and right Jacobians become identical. The procedure is demonstrated for plate vibrations without rigid body motion and then applied to a single moving beam and finally augmented to multi beam systems. Special attention is hereby given to the effects of dynamical coupling which influence bending stiffness and connect bending with torsion. This review article has 55 references.