Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2691963, 10 pages

https://doi.org/10.1155/2017/2691963

## Flexural-Torsional Flutter and Buckling of Braced Foil Beams under a Follower Force

^{1}International Research Center on Mathematics and Mechanics of Complex Systems, University of L’Aquila, 67100 L’Aquila, Italy^{2}Department of Civil, Construction-Architectural and Environmental Engineering, University of L’Aquila, 67100 L’Aquila, Italy

Correspondence should be addressed to Angelo Luongo

Received 30 May 2017; Revised 13 August 2017; Accepted 24 August 2017; Published 12 October 2017

Academic Editor: Salvatore Caddemi

Copyright © 2017 Manuel Ferretti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The flutter and buckling behavior of a cantilever foil beam, loaded at the tip by a follower force, are addressed in this paper. The beam is internally and externally damped and braced at the tip by a linear spring-damper device, which is located in an eccentric position with respect to beam axis, thus coupling the flexural and torsional behaviors. An exact linear stability analysis is carried out, and the linear stability diagram of the trivial rectilinear configuration is built up in the space of the follower load and spring’s stiffness parameters. The effects of the flexural-torsional coupling, as well as of the damping, on the flutter and buckling critical loads are discussed.

#### 1. Introduction

Dynamic stability of elastic systems loaded by nonconservative and* configuration-dependent* loads, such as follower forces [1, 2], has been thoroughly investigated by many researchers in the last century [3–10]. Some experimental evidences, proving the existence of such a kind of actions in the real world applications, are available, for example, in [10–12] and in the critical review [13], notwithstanding the fact that the engineering world is still suspicious of their existence and physical meaning. However, the effects on dynamic stability due to the presence of follower forces are very important in several engineering branches, such as in aerospace [10, 14, 15], in flexible pipes conveying fluid [16–18], and in vehicle brakes [19, 20].

Researchers have devoted great attention in the last years to the so-called Beck’s beam (see, e.g., [4, 5, 13]), namely, a cantilever beam loaded at the tip by a follower force (i.e., a force which keeps its direction tangential to the centerline), and, eventually, in the presence of conservative loads, and/or of distributed (internal and external) as well as lumped forms of damping. This structure indeed represents a paradigmatic system for the comprehension of stability issues in one-dimensional nonconservative systems; in fact, the loss of stability may happen either by divergence, in the presence of conservative loads or lumped springs, or by flutter, also said to be Hopf bifurcation in Dynamical System Theory, depending on the mechanical properties of the structure [21–24]. Moreover Beck’s beam is also able to show one of the most amazing phenomena, occurring in the dynamical behavior of elastic systems loaded by follower forces, namely, the* destabilizing effect of damping*, or the “Ziegler Paradox”; see, for example, [3–6, 25–27]. It occurs when a vanishingly small and positive-definite damping is added to such a system, entailing a finite reduction of the flutter critical load with respect to that of the undamped system. Several contributions can be found in the literature, which are devoted to giving an explanation of this occurrence and to present different case studies, pointing out the phenomenon; among the others, the reader can refer to [8, 9, 24, 28–33].

In most of the previously cited papers a* planar* Beck’s beam, under in-plane loads, is considered, whose trivial rectilinear configuration loses its stability in the same plane. However, when considering spatial beams, the loss of stability can occur out of that plane due to a flexural-torsional mechanical coupling. This phenomenon is well known in buckling analysis of spatial structures, such as thin-walled members, which, indeed, can exhibit a flexural-torsional Eulerian bifurcation, when subjected to conservative forces [34, 35].

The flexural-torsional coupling may become important when issues relevant to dynamic stability are addressed. In this framework classical examples can be found mainly in aerospace engineering, for example, when the flutter behavior of a wing, immersed in a gas flow, namely, under nonconservative and* velocity-dependent* loads, is considered [5, 36]. Other examples, when* configuration-dependent* loads act, can be found in [37], where the flutter instability of a cantilever beam containing a tip mass, subjected to a transverse follower force at the tip, and in the presence of airflow, is addressed; in [38], where the lateral-torsional stability of deep cantilever beams loaded by a transverse follower force at the tip, is studied; in [39], where the lateral stability of a slender beam, under a transverse follower force is addressed; in [21], where the flexural-torsional bifurcations of a cantilever beam under the simultaneous action of a nonconservative follower force and a conservative couple at the free end, have been analyzed; and finally in [40], where the bending-torsional flutter analysis of a cantilever, containing an arbitrarily placed mass, under a follower force and airflow, is analyzed. Remarkably, in the greatest part of the previous papers, a* foil beam*, namely, a beam for which one of the two inertia moments is much larger than the other, is considered as the mathematical model of aircraft’s wing.

This paper is framed in the scenario illustrated above. Indeed, to the best of author’s knowledge, there are no contributions in the literature addressing the flutter and buckling analyses of a* spatial* Beck’s column, so that the present work is a first step toward the study of the problem. To this end, reference will be made to the simplest model as possible, namely, a clamped-free foil beam, loaded at the tip by a tangential follower force, internally and externally damped. Moreover, in order to couple the flexural and torsional behavior even in the linear range, it is assumed that the beam is braced at the tip by a linear spring-damper device, which is orthogonal to the axis line, and eccentric with respect to it.

The paper is organized as follows. In Section 2 the equations of motion of the model are presented. In Section 3 the eigenvalue problem is addressed and an exact linear stability analysis is carried out, both in the presence and in the absence of damping. In Section 4 a numerical analysis is developed, and the linear stability diagrams are built up in the space of the follower load and spring’s stiffness parameters and for different damping coefficients. Finally, in Section 5 some conclusions are drawn.

#### 2. Model

The foil beam is modeled as one-dimensional, inextensible and twistable, polar continuum (see, e.g., [41]), embedded in a three-dimensional space spanned by the unit vectors , , and (Figure 1). It is assumed that is the* strong* and the* weak* axis of the cross-section, that is, the inertia moments are ; consistently, the -plane is here referred to as the strong plane, while -plane is the weak one. If forces in the strong plane are smaller or, at most, comparable with those acting in the weak plane, then the beam can be considered unflexurable in the strong plane, and the relevant bending moment is a reactive stress. Concerning torsional stiffness, it is of the same order of magnitude of the weak bending stiffness, as it happens for compact cross-sections. Therefore, if torsional moments are smaller than the bending moment , torsion is also negligible, so that the foil beam behaves as a planar beam in bending. Throughout the paper it is assumed that the torsional moment is comparable with the bending moment, so that the foil beam behaves as a flexural-torsional beam.