Identifying the significance of nonlinear normal modes

Abstract

Nonlinear normal modes (NNMs) are widely used as a tool for understanding the forced responses of nonlinear systems. However, the contemporary definition of an NNM also encompasses a large number of dynamic behaviours which are not observed when a system is forced and damped. As such, only a few NNMs are required to understand the forced dynamics. This paper firstly demonstrates the complexity that may arise from the NNMs of a simple nonlinear system—highlighting the need for a method for identifying the significance of NNMs. An analytical investigation is used, alongside energy arguments, to develop an understanding of the mechanisms that relate the NNMs to the forced responses. This provides insight into which NNMs are pertinent to understanding the forced dynamics, and which may be disregarded. The NNMs are compared with simulated forced responses to verify these findings.

Explicit backbone curves from spectral submanifolds of forced-damped nonlinear mechanical systems

“…The non-resonance conditions (9) and (24) are satisfied for l = 1 and hence the two-dimensional timevarying SSM exists. We assume forcing at the first mass (f 1 = ε cos(Ωt) and f j = 0 j = 2, ..., N ).…”

On the geometrically exact low-order modelling of a flexible beam: formulation and numerical tests

“…Note that there is some slight difference between the two results in the 9.5-10 Hz range; however, due to the speed of decay it is unlikely that either method closely captures the localized backbone structure in this region. For the precise tracking of this backbone curve, one could employ more sophisticated methods based upon shooting or pseudo-arclength continuation [42][43][44], however, this is not pursued here.…”

ε 2-Order Normal Form Analysis for a Two-Degree-of-Freedom Nonlinear Coupled Oscillator

“…In this paper, we will show that the ǫ 2 terms are required in the direct normal form method of Neild & Wagg [9] to give the correct solutions. Typically the direct normal form method, [9], is applied to systems where the nonlinear, damping and forcing terms are assumed to be of order ε 1 small (or higher orders of ε) when compared to the linear terms [10][11][12][13][14]. The linear terms are the natural frequencies, taken to be of order ε 0 , meaning that the ε 1 nonlinear terms are typically an order smaller than the natural frequencies.…”