Nonlinear normal modes and their application in structural dynamics

Abstract

Recent progress in the area of nonlinear modal analysis for structural systems is reported. Systematic methods are developed for generating minimally sized reduced-order models that accurately describe the vibrations of large-scale nonlinear engineering structures. The general approach makes use of nonlinear normal modes that are defined in terms of invariant manifolds in the phase space of the system model. An efficient Galerkin projection method is developed, which allows for the construction of nonlinear modes that are accurate out to large amplitudes of vibration. This approach is successfully extended to the generation of nonlinear modes for systems that are internally resonant and for systems subject to external excitation. The effectiveness of the Galerkin-based construction of the nonlinear normal modes is also demonstrated for a realistic model of a rotating beam.

Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction

“…This SSM is already unique among class C 6 invariant manifolds tangent to the slow spectral subbundle of a small-amplitude, periodic NNM, which is guaranteed to exist by Theorem 2. To compute this NNM and its slow SSM, we again use a linear change of coordinates (37) to obtain the equations of motion in the forṁ…”

Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells

“…Since then, extensive studies have been performed both on their applications and also their methods of computation (Vakakis 1997, Pierre et al 2006). The invariant manifold (IM) approach, proposed by Shaw and Pierre (1993) is one of these methods that defines the NNM as a twodimensional invariant manifold in phase space.…”

An Explanation for Why Natural Frequencies Shifting in Structures with Membrane Stresses, Using Backbone Curve Models

“…Many authors have studied undamped, unforced systems including beams, cables, membranes, plates and shells, see for example [11,12,13]. The free response of nonlinear systems has been studied using several different analytical approaches: nonlinear normal modes (NNMs) [14,15,16] and backbone curves [17,18,19].…”