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.
Highlights
Nonlinear normal modes represent an analogue of the established theory of linear normal modes [1] for nonlinear systems
This paper has considered the dynamic behaviour of a simple nonlinear model of a beam
It has been shown that this system may exhibit a large number of Nonlinear normal modes (NNMs) branches, i.e. the loci of responses of the underlying conservative system
Summary
Nonlinear normal modes represent an analogue of the established theory of linear normal modes [1] for nonlinear systems. Due to the very low damping, a weak energy transfer mechanism between the modes may be sufficient for an NNM to be attractive, and the phase-unlocked NNMs may be able to provide this via the harmonics This is confirmed using numerical continuation results, which show that a section of the forced response follows a phaseunlocked branch. While this demonstrates that phase-unlocked branches may represent dynamic behaviours that underpin the forced responses, it is concluded that they are only significant for structures with extremely low levels of damping
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