In the linear of elastodynamics, it is assumed that the elastic deformation does not have a significant effect on the rigid body motion of mechanical systems. In many recent investigations, the significance of the coupling between the rigid body and elastic displacements is demonstrated. In this investigation, a simple model, which clearly demonstrates that there are applications in which the reference motion can have strong dependence on the elastic deformation, is used to examine the coupling between the rigid body and elastic modes. A closed form solution is obtained and used to examine the effect of the in-plane and out-of-plane bending vibrations on the reference displacements. It is shown that instability of elastic modes can significantly influence the reference displacements. More significantly, as demonstrated in this paper, a linear formulation that maintains the stability of rotating beams at relatively large values of the angular velocity can be developed, despite the fact that the amplitude of vibration becomes large. The effect of the longitundinal displacement on the coupling between the rigid body and the deformation modes is also discussed using the simple beam model presented in this paper. Stability regions are identified as function of the angular velocity of the beam and the effect of the deformation modes instability on the reference motions is examined. It is demonstrated that stability limits depend on the difference between the axial and bending stiffness coefficients. (C) 1996 Academic Press Limited
