Asymptotic construction of nonlinear normal modes for continuous systems

Abstract

This paper provides a review of some asymptotic methods for construction of nonlinear normal modes for continuous system (NNMCS). Asymptotic methods of solving problems relating to NNMCS have been developed by many authors. The main features of this paper are that (i) it is devoted to the basic principles of asymptotic approaches for constructing of NNMCS; (ii) it deals with both traditional approaches and, less widely used, new approaches; and (iii) it pays a lot of attention to the analysis of widely used simplified mechanical models for the analysis of NNMCS. The author has paid special attention to examples and discussion of results. Keywords Nonlinear oscillations . Normal mode . Asymptotic approach . Continuous system 1 Definition of nonlinear normal modes for continuous system First definition of nonlinear normal modes for continuous system (NNMCS) is based on exact separation of spatial and time variables [1]. Exact separation of spatial and time variables is possible for some simplified models of continuous systems (Kirchhoff model of nonlinear beam [2], Berger model of nonlinear plate [3,4], Berger-like model of shallow shell [5-7]) for simply supported ends. Naturally, solutions of this type were constructed earlier [2], but Wah was the first who pointed to links between these solutions and those proposed by R.M. Rosenberg for discrete system concepts of NNM [8][9][10]. Naturally, area of applicability of Wah definition is very restricted.Shaw and Pierre proposed the following definition of NNMCS [11]: "If the displacement u 0 (t) = u(s 0 , t) and velocity v 0 (t) = v(s 0 , t) of a single point, s = s 0 , are known, then the entire displacement and velocity field can be determined by the dynamics of that single point u(s, t) = U (u 0 (t), v 0 (t), s, s 0 ), v(s, t) = V (u 0 (t), v 0 (t), s, s 0 )."

Nonlinears Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

“…The basic idea of this approach is a construction of certain particular selfsimilar solutions of a nonlinear problem, which can be considered as asymptotics of a wide class of other solutions. A realization of this approach for nonlinear dynamics of elastic systems (beams, plates, shallow shells) is presented by Andrianov and his coauthors [168][169][170][171][172]. The paper [172] is a survey of publications on the NNMs and similar solutions in nonlinear continuous elastic systems.…”

Exact augmented perpetual manifolds as a tool for the determination of similar rigid body modes: Corollary for determining backbone curves of autonomous systems

“…Until 1991, the approach of determining the NNMs of a mechanical system was based on 'direct attack' to the equations of motion of the underlying conservative system by neglecting the dissipative forces, and their determination leads to the backbone curves of the system. Then by adding small damping (as it is true in structural mechanical systems) and periodic excitation, the FRFs in similar fashion can be determined with resonance points those that are very close to the backbone curves of the system; a review of these approaches in Adrianov (2008); Kerschen et al (2009); Mikhlin and Avramov (2010); Nayfeh (1995) can be found. A research relevant to NNMs that is straightforward related with the development of the theory in this article is by Vakakis PhD thesis of Vakakis (1991).…”

Reduced-order models for large-amplitude vibrations of shells including in-plane inertia

“…They generally rely on an asymptotic development for approximating the invariant manifold [6][7][8][9][10][11]. More recently, purely numerical methods have been proposed [12][13][14], but they are often restricted either by complexity (i.e.…”