Influence of symmetries and imperfections on the non-linear vibration modes of archetypal structural systems Diego Orlando a , Paulo B. Gonc - alves a,n , Giuseppe Rega b , Stefano Lenci c a Department of Civil Engineering, Pontifical Catholic University of Rio de Janeiro, PUC-Rio, Rio de Janeiro, 22451-900, Brazil b Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome 00197, Italy c Department of Civil and Building Engineering, and Architecture, Polytechnic University of Marche, Ancona 60131, Italy article info Article history: Received 17 June 2011 Received in revised form 15 September 2012 Accepted 7 October 2012 Available online 16 October 2012 Keywords: Symmetry and symmetry breaking Geometric imperfections Inverted pendulum Non-linear normal modes Multimodes Modal coupling and interaction Mode bifurcation and instability abstract The objective of the present work is to investigate how symmetries, initial geometric imperfections and energy level influence the number and stability of the non-linear normal modes and the existence of multimode solutions in structural systems liable to unstable buckling. To this aim, two archetypal models exhibiting interactive buckling phenomena, which lead to several unstable post-buckling paths and high imperfection sensitivity, are considered. As many structural elements, these models have several planes of symmetry. The inherent symmetries and post-buckling solutions have a marked influence on the underlying potential function and, consequently, on the non-linear dynamics of the system, whose stable solutions are limited by the energy level associated with the saddles lying on the boundary of the pre-buckling well. A detailed non-linear modal analysis is accomplished for increasing energy levels, by first considering the nominally perfect systems. Then, the symmetry breaking effect of initial geometric imperfections on the number and stability of the non-linear normal modes is investigated. Finally, some examples illustrate the influence of the superabundance of modes on the resonant forced behavior of the system. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Nominally perfect structural models usually present some inherent symmetries, varying from a finite number (e.g. a column with a rectangular cross-section) to infinite symmetries (e.g. a spherical shell). These symmetries influence the governing differ- ential equations and some characteristics of these structures such as the eigenvalues and eigenmodes associated with buckling and vibration, and possible bifurcations under static and dynamic loads. Symmetries in the system’s stiffness may lead to coincident buckling loads, and coupling of the associated buckling modes [1,2]. Also coincident vibration frequencies may occur, leading to internal resonances. For example, 1:1 resonance is inherent to several structural elements such as rings and shells of revolution [3–6]. Small initial geometric imperfections break some of the symmetries, causing the unfolding of structurally unstable bifur- cations, with consequences on the buckling and vibration char- acteristics and may be viewed in this context as a source of small detuning parameters [7]. Thus, closeness of eigenfrequencies is expected in many real symmetric structures with imperfections. In fact, several studies have shown that, when a system is invariant under the action of a group of symmetry, this can have tremendous consequences on its bifurcations, producing a rich variety of solutions and unexpected phenomena, as shown by Golubitsky and co-workers [8–11]. The importance of symmetries in bifurcations is treated, for example, in [7] and [12] who discuss their correlation with group theory. In recent years an increasing number of papers has dealt with the non-linear normal modes (NNMs) of structural and mechan- ical systems. NNMs can be regarded as a generalization of linear normal modes (LNMs). The initial concept was introduced by Rosenberg [13,14], who defined a NNM of a discrete, conservative, non-linear system as a synchronous periodic oscillation where all material points of the system reach their extreme values or pass through zero simultaneously. His works laid the foundations for future research in this field, e.g., those of Rand [15,16] and Vakakis and their co-workers [17,18]. In 1991, Shaw and Pierre [19] introduced a more general concept of NNMs. They define NNMs as motions on invariant manifolds which are tangent to, and of the same dimension as, the linear eigenspaces in the system phase space. This definition contains the previous ones as special cases. Later Boivin et al. [20] introduced the concept of multimode invariant manifolds, which can be understood as an extension of the NNMs when two or more non-linear modes interact. Non-linear multimodes may be observed in structural Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/nlm International Journal of Non-Linear Mechanics 0020-7462/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2012.10.004 n Corresponding author. Tel.: þ55 21 3527 1188. E-mail addresses: dgorlando@gmail.com (D. Orlando), paulo@puc-rio.br (P.B. Gonc - alves), giuseppe.rega@uniroma1.it (G. Rega), lenci@univpm.it (S. Lenci). International Journal of Non-Linear Mechanics 49 (2013) 175–195