Abstract
Nonlinear normal modes (NNMs) are a key tool for investigating the behaviour of nonlinear dynamic systems. Previous work has shown that branches of NNMs can be isolated from other NNM responses, in a similar manner to isolated, or detached, resonance curves in the forced responses. Their isolated nature poses a significant challenge for the prediction and measurement of these NNM branches. This paper illustrates how isolated NNMs may exist in two-degree-of-freedom systems, with cubic nonlinearities, that exhibit a 1:3 resonance. This is first introduced using a general two-mass oscillator, before considering a two-mode reduced-order model of a continuous cross-beam structure that exhibits a coupling between its primary bending and torsional modes. In both cases, a combination of analytical and numerical techniques is used to show how the isolated NNM branch may evolve from a set of bifurcating NNM branches. A nonlinear force appropriation technique is used to experimentally measure the NNMs of the cross-beam structure. By comparing these measurements to the numerical studies, it is shown that some of these NNMs are on the isolated branch, representing the first experimental confirmation of isolated NNM branches.
Highlights
When considering the responses of harmonically-forced nonlinear systems, it is well-known that multiple solutions can be observed at specific frequencies [1]
This approach requires that the Nonlinear Normal Modes (NNMs) branches of the system are known before the energy-based method is used to predict where the isolas cross the NNM branches
Evolution of an isolated nonlinear normal mode from a bifurcation In Ref. [15] it is shown that a two-mass oscillator may exhibit an isolated NNM branch with a 1:1 resonance between the modes
Summary
When considering the responses of harmonically-forced nonlinear systems, it is well-known that multiple solutions can be observed at specific frequencies [1]. [15], Section 2 of the current paper begins by showing that isolated NNM branches may exist in a simple two-mode system, representative of an asymmetric two-mass oscillator This system exhibits a 1:3 resonance between the two modes (i.e. the second mode responds at three times the frequency of the first) and differs from the oscillator considered in Ref. This model exhibits an isolated NNM branch, which is shown to evolve from a bifurcated set of branches as the symmetry of the structure is varied. Are NNMs on an isolated branch measured experimentally, but it is shown that, for certain forcing amplitudes, responses on the isolated branch are inevitable and represent significant features of the response
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