Nonlinear Normal Modes, Part I: A Useful Framework for the Structural Dynamicist

The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companion, Part II. Because there is virtually no application of the NNMs to largescale engineering structures, these papers are an attempt to highlight several aspects that might drive their development in the future. Specifically, we support that (i) numerical methods for the continuation of periodic solutions pave the way for an effective and practical computation of NNMs, and (ii) time-frequency analysis is particularly suitable for the analysis of the resulting dynamics. Another objective of the present paper is to describe, in simple terms, and to illustrate the fundamental properties of NNMs. This is achieved to convince the structural dynamicist not necessarily acquainted with them that they are a useful framework for the analysis of nonlinear vibrating structures. 1 Nonlinear Normal Modes: A Brief Historical Perspective and Current State-of-the-Art The concept of a normal mode is central in the theory of linear vibrating systems. Besides their obvious physical interpretation, the linear normal modes (LNMs) have interesting mathematical properties. They can be used to decouple the governing equations of motion; i.e., a linear system vibrates as if it were made of independent oscillators governed by the eigensolutions. Two important properties that directly result from this decoupling are: 1. Invariance: if the motion is initiated on one specific LNM, the remaining LNMs remain quiescent for all time. 2. Modal superposition: free and forced oscillations can conveniently be expressed as linear combinations of individual LNM motions. In addition, LNMs are relevant dynamical features that can be exploited for various purposes including model reduction (e.g., substructuring techniques [1]), experimental modal analysis [2], finite element model updating [3] and structural health monitoring [4]. Clearly, though, linearity is an idealization, an exception to the rule; nonlinearity is a frequent occurrence in real-life applications [5]. For instance, in an aircraft, besides nonlinear fluid-structure interaction, typical nonlinearities include backlash and friction in control surfaces, hardening nonlinearities in engine-to-pylon connections, saturation effects in hydraulic actuators, plus any underlying distributed nonlinearity in the structure. Furthermore, the next generations of aircraft are using materials such as glass-fiber or carbon-fiber composites to a greater extent for structural weight reduction. These materials entail new challenges for performance prediction, because they exhibit a structural behavior deviating significantly from linearity. Their increased use also creates more interfaces between different materials, which are further sources of nonlinear behavior.