Some considerations on dynamic stability

Abstract eng:
Within the second order theory, this work considers some classic stability problems, whose critical load corresponds to a dynamic instability. Attention is primarily focused on plane systems with just one lumped mass. This idealization, together with neglecting the axial strain, reduces the system Lagrangian coordinates to only one. Thus, static methods can be applied to derive the analytical expression of the stiffness coefficient and study the dynamic stability, starting from a well-known example, that is, a cantilever beam with a lumped mass at its free end subject to a follower concentrated load (Panovko & Gubanova 1967). In this benchmark the first asymptote of the stiffness coefficient corresponds to the critical load, due to divergence at infinity. Such critical load is equal to the buckling load by divergence of an auxiliary structure, differing from the original one in that the lumped mass is replaced by a constraint blocking the corresponding Lagrangian coordinate: a possible explanation for this coincidence is presented. This finding is corroborated by the numerical study of a cantilever beam with a lumped mass at its free end subject to a uniform distributed follower load, whose critical load is compared with the one of the corresponding auxiliary structure, presented by (Leipholz 1970). Studying the stiffness coefficient of a parametrized example, which contains the above mentioned one (Panovko & Gubanova 1967) as a particular case, a paradox is met, since it is found that, for a specific range of values of the parameter, apparently no instability is detected. It follows that the only possible dynamic instability could be flutter, which needs a more complex mass distribution. Finally, two examples are presented, where divergence instability occurs when the applied load causes traction.

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