Journal of Applied Mathematics and Physics ( Z A M P ) 0044-2275/87/006874-09 $ 3.30/0 Vol. 38, November 1987 Birkhfiuser Verlag Basel, 1987 Dynamic instability of a non-uniform beam modeled in a two-degree-of-freedom system By David Levanony and Menahem Baruch, Dept. of Aeronautical Engineering Technion - Israel Institute of Technology Haifa 3 20 00, Israel Introduction This paper can be considered as a continuation of the previous works by Lee and Reissner [1] and Neer and Baruch [2], where the instability of a non-uniform beam under a parallel conservative compressive force or a non-conservative "follower" force was analyzed. While in Ref. 1 the static instability of the system was investigated, in Ref. 2 the static and dynamic instability was examined by modeling the beam in a one-degree-of-freedom dynamic system. It must be stated here that one of the reviewers of Ref. 2 stated that the "simple dynamic model may be oversimplified and that a better problem would be to consider a rotational inertia term in addition to the translation term". In fact, the analysis of the results obtained by applying a one-degree-of-freedom dynamic system more than indicated that the two-degree-of-freedom system would reveal a clearer picture of the behaviour of the structure. The non-uniform beam is presented here by a two-degree-of-freedom dy- namic model obtained by adding to the translational mass a rotational inertia term. For the tangential force the one-degree-of-freedom instability obtained in Ref. 2 and characterized by an infinite frequency is replaced here by a classical flutter instability. For the parallel force, as obtained previously [1, 2], only the static instability has a physical significance. Problem description The structure considered here is the same as in [1] and [2], namely, a non-uniform beam of span 2a (Fig. 1) with both ends clamped. The force P is applied at midspan (x = 0) towards - a. The support of x = + a has axial freedom, so that the entire force P is taken by the segment - a < x < 0. The stiffnesses are D 1 and D2, respectively. No mass had to be considered in [1] for the analysis of the static instability. In [2], in order to allow a simple dynamic approach, a translational concentrated mass was added at the midpoint (x = 0)