Abstract
In this paper the backbone curves of a two-degree-of-freedom nonlinear oscillator are used to interpret its behaviour when subjected to external forcing. The backbone curves describe the loci of dynamic responses of a system when unforced and undamped, and are represented in the frequency–amplitude projection. In this study we provide an analytical method for relating the backbone curves, found using the second-order normal form technique, to the forced responses. This is achieved using an energy-based analysis to predict the resonant crossing points between the forced responses and the backbone curves. This approach is applied to an example system subjected to two different forcing cases: one in which the forcing is applied directly to an underlying linear mode and the other subjected to forcing in both linear modes. Additionally, a method for assessing the accuracy of the prediction of the resonant crossing points is then introduced, and these predictions are then compared to responses found using numerical continuation.
Highlights
Predicting the dynamic responses of structures with nonlinear characteristics is a research field that has the ultimate aim of improving the design and efficiency of future structures
The backbone curves were found using a normal form approach, which allows for the calculation of harmonics and higher orders of accuracy; for the example considered here, inclusion of these was not needed to achieve accurate results
Whilst the example system responds with a frequency ratio of 1:1, the techniques adopted here are not limited to such cases, and can more generally be applied to n:m resonant systems
Summary
Predicting the dynamic responses of structures with nonlinear characteristics is a research field that has the ultimate aim of improving the design and efficiency of future structures. This paper follows the approach of [18], in which the unforced, undamped responses of the system are found analytically using the second-order normal form technique, first developed in [15]. These responses are referred to as backbone curves – for an early use of these see [19]. This is followed by a discussion of the dynamic response of a two degree-of-freedom oscillator with cubic nonlinearities. The second-order normal form technique is not detailed in the main text, but summarised in Appendix A
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