Vol. 3, No. 1, 2015

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ISSN: 2325-3444 (e-only)
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Effects of damping on the stability of the compressed Nicolai beam

Angelo Luongo, Manuel Ferretti and Alexander P. Seyranian

Vol. 3 (2015), No. 1, 1–26
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.

Keywords
stability of beams, nonconservative system, Nicolai paradox, perturbation methods, semisimple eigenvalues, quadruple-zero eigenvalue
Mathematical Subject Classification 2010
Primary: 74H10, 74H55, 74H60
Milestones
Received: 7 March 2013
Revised: 13 November 2013
Accepted: 31 December 2013
Published: 13 February 2015

Communicated by Antonio Carcaterra
Authors
Angelo Luongo
Dipartimento di Ingegneria Civile, Edile-Architettura e Ambientale
Università degli studi dell’Aquila
Via Giovanni Gronchi 18
Zona industriale di Pile
67100 L’Aquila
Italy
Manuel Ferretti
Dipartimento di Ingegneria Civile, Edile-Architettura e Ambientale
Università degli studi dell’Aquila
Via Giovanni Gronchi 18
Zona industriale di Pile
67100 L’Aquila
Italy
Alexander P. Seyranian
Institute of Mechanics
Moscow State Lomonosov University
Michurinsky pr. 1
Moscow
119192
Russia