Effects of damping on the stability of the compressed Nicolai beam

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.

Paradoxes in dynamic stability of mechanical systems: investigating the causes and detecting the nonlinear behaviors

“… 2014 ; Luongo et al. 2014 ; Seyranian and Glavardanov 2014 ), see Fig. 1 c, thus entailing instability; the presence of a small asymmetry, that we label with a parameter , is able to shift the critical load of a small amount only (Seyranian and Mailybaev 2011 ; Seyranian et al.…”

“…( 1 ), the linear operator and the trilinear forms associated to the nonlinearities of the discretized system, in nondimensional form, read: in which are (time-dependent) amplitudes of the trial function adopted for the Galerkin projection, and are the nondimensional mass, inertia moments with respect to the two principal inertia axes and intensity of the follower torque, respectively, defined as: where E is Young modulus of the elastic material, and are the mass per unit length and the inertia moment, respectively, taken as the characteristics of an ideal symmetric system, from which the actual system can be generated via a perturbation (Luongo et al. 2014 ). The other quantities appearing in Eq.…”

“…( 2014 ), Luongo et al. ( 2014 ) and Seyranian and Glavardanov ( 2014 ). An extension of the problem to second-order perturbations was also performed in Luongo and Ferretti ( 2014 ).…”

A discrete formulation of Kirchhoff rods in large-motion dynamics

“…The nonlinear nature of the analyzed mechanical system implies that many features that are difficult to be tackled (see, e.g., in the context of nonlinear vibrations Battisti et al (2017); Chróścielewski et al (2019); Deü et al (2008); Lazarus et al (2012); Thomas et al (2016), nonlinear motions Boyer et al (2002); Boyer and Primault (2004) as well as in dynamic stability of mechanical systems D'Annibale (2013, 2017); Luongo et al (2016Luongo et al ( , 2015; Spagnuolo and Andreaus (2019)). Therefore, having a formulation which is as simple as possible is always an opportunity to pursue.…”

Selected local stability problems of channel section flanges made of aluminium alloys

“…The years of research have not been sufficient to thoroughly solve the problems of stability loss, which is of a sudden nature and difficult to contain. The stability loss problems are subject to ongoing research [18,31,32,52,54]. Experimental research is still being conducted to investigate the phenomenon of stability loss, to verify the proposed analytical or numerical solutions and to improve the direct strength method (DSM) [7,25,41,45,62].…”