Abstract

The Nicolai problem concerning the stability of a quasisymmetric cantilever beam embedded in a three-dimensional space, under a compressive dead load and a follower torque, is addressed. The effect of external and internal damping on stability is investigated. The partial differential equations of motion, accounting for the pretwist contribution, are recast in weak form via the Galerkin method, and a linear algebraic problem, governing the stability of the rectilinear configuration of the beam, is derived. Perturbation methods are used to analytically compute the eigenvalues, starting with an unperturbed, undamped, symmetric, untwisted beam, axially loaded, in both the subcritical and critical regimes. Accordingly, an asymmetry parameter, the torque, the damping, and the load increment are taken as perturbation parameters. Maclaurin series are used for semisimple eigenvalues occurring in subcritical states, and Puiseux series for the quadruple-zero eigenvalue existing at the Euler point. Based on the eigenvalue behavior described by the asymptotic expansions, the stability domains are constructed in the two or three-dimensional space of the bifurcation parameters. It is found that dynamic bifurcations occur in the subcritical regime, and dynamic or static bifurcations in the critical regime. It is shown that stability is governed mostly by the bifurcation of the lowest eigenvalue. In all cases the Nicolai paradox is recovered, and the beneficial effects of asymmetry and damping are highlighted.

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Mathematical Subject Classification 2010

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