On the stability and instability regions of nonconservative continuous system under partially follower forces

The transition between stability and instability of a uniform cantilever beam subjected to a partially follower load is discussed by means of the finite element method. The stability diagram in terms of the load versus the non-conservativeness loading parameter is obtained both in the hypothesis of linearized (small displacements) and non-linearized (large displacements) analysis. It is found that the regions of divergence and flutter instability of the finite element model of continuous system are quite different from the corresponding ones obtained for the classical 2 degrees of freedom Ziegler's model (i.e. the inverted double pendulum), both quantitatively and qualitatively. The transition from the 2 degrees of freedom model to the continuous model is investigated through the study of some multi degrees of freedom approximation models (i.e. the 3, 4, 5 and 10 degrees of freedom models) of the continuous column. Finally, the effect of damping in the stability diagram of the finite element model of continuous system is discussed and the destabilizing effect of small damping is emphasized

Finite element solution of the stability problem for nonlinear undamped and damped systems under nonconservative loading

A nonlinear finite element analysis of elastic structures which can be studied by a 3D beam theory, subjected to conservative as well as to nonconservative forces, is presented. The stability behaviour of the system is studied by means of an eigenvalue analysis. The stiffness matrix of the eigenvalue problem is asymmetric (i.e., non-self-adjoint system). The flutter and divergence modes of instability, as well as the values of the critical load, are identified for a number of numerical examples belonging to the benchmark tests proposed by NAFEMS (1990). The results demonstrate the reliability of this finite element formulation. In particular the effect of damping on the stability behaviour of such structures is investigated and the destabilizing effect of small damping is underlined. Finally, the need to define a number of benchmark tests for nonlinear-nonconservative analyses in presence of damping is included