a b s t r a c tIn 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.

A fundamental drawback of vibration-based energy harvesters is that they typically feature a resonant mass/spring mechanical system to amplify the small source vibrations; the limited bandwidth of the mechanical amplifier restricts the effectiveness of the energy harvester considerably. By extending the range of input frequencies over which a vibration energy harvester can generate useful power, e.g. through adaptive tuning, it is not only possible to open up a wider range of applications, such as those where the source frequency changes over time, but also possible to relax the requirements for precision manufacture or the need for mechanical adjustment in situ. In this paper, a vibration-based energy harvester connected to a generalized electrical load (containing both real and reactive impedance) is presented. It is demonstrated that the reactive component of the electrical load can be used to tune the harvester system to significantly increase the output power away from the resonant peak of the device. An analytical model of the system is developed, which includes non-ideal components arising from the physical implementation, and the results are confirmed by experiment. The −3 dB (half-power) bandwidth of the prototype energy harvester is shown to be over three times greater when presented with an optimized load impedance compared to that for the same harvester presented with an optimized resistive only load.

A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations.

This paper considers the dynamic response of coupled, forced and lightly damped nonlinear oscillators with two degree-of-freedom. For these systems, backbone curves define the resonant peaks in the frequency-displacement plane and give valuable information on the prediction of the frequency response of the system. Previously, it has been shown that bifurcations can occur in the backbone curves. In this paper we present an analytical method enabling the identification of the conditions under which such bifurcations occur. The method, based on second order nonlinear normal forms, is also able to provide information on the nature of the bifurcations and how they affect the characteristics of the response.This approach is applied to a two-degree-of-freedom mass, spring, damper system with cubic hardening springs. We use the second-order normal form method to transform the system coordinates and identify which parameter values will lead to resonant interactions and bifurcations of the backbone curves. Furthermore, the relationship between the backbone curves and the complex dynamics of the forced system is shown.

SUMMARY Much work has been conducted on vibration absorbers, such as tuned mass dampers (TMD), where significant energy is extracted from a structure. Traditionally, this energy is dissipated through the devices as heat. In this paper, the concept of recovering some of this energy electrically and reuse it for structural control or health monitoring is investigated. The energy‐dissipating damper of a TMD is replaced with an electromagnetic device in order to transform mechanical vibration into electrical energy. That gives the possibility of controlled damping force whilst generating useful electrical energy. Both analytical and experimental results from an adaptive and a semi‐active tuned mass damper/harvester are presented. The obtained results suggest that sufficient energy might be harvested for the device to tune itself to optimise vibration suppression. Copyright © 2014 John Wiley & Sons, Ltd.