Systematic experimental exploration of bifurcations with noninvasive control

Copyright © 2013 American Physical SocietyWe present a general method for systematically investigating the dynamics and bifurcations of a physical nonlinear experiment. In particular, we show how the odd-number limitation inherent in popular noninvasive control schemes, such as (Pyragas) time-delayed or washout-filtered feedback control, can be overcome for tracking equilibria or forced periodic orbits in experiments. To demonstrate the use of our noninvasive control, we trace out experimentally the resonance surface of a periodically forced mechanical nonlinear oscillator near the onset of instability, around two saddle-node bifurcations (folds) and a cusp bifurcation.Engineering and Physical Sciences Research Council (EPSRC

The Significance of Nonlinear Normal Modes for Forced Responses

Nonlinear normal modes (NNMs) describe the unforced and undamped periodic responses of nonlinear systems. NNMs have proven to be a valuable tool, and are widely used, for understanding the underlying behaviour of nonlinear systems. They provide insight into the types of behaviour that may be observed when a system is subjected to forcing and damping, which is ultimately of primary concern in many engineering applications. The definition of an NNM has seen a number of evolutions, and the contemporary definition encompasses all periodic responses of a conservative system. Such a broad definition is essential, as it allows for the wide variety of responses that nonlinear systems may exhibit. However, it may also lead to misleading results, as some of the NNMs of a system may represent behaviour that will only be observed under very specific forcing conditions, which may not be realisable in any practical scenario. In this paper, we investigate how the significance of NNMs may differ and how this significance may be quantified. This is achieved using an energy-based method, and is validated using numerical simulations

Equation-Free Analysis of Macroscopic Behavior in Traffic and Pedestrian Flow

Equation-free methods make possible an analysis of the evolution of a few coarse-grained or macroscopic quantities for a detailed and realistic model with a large number of fine-grained or microscopic variables, even though no equations are explicitly given on the macroscopic level. This will facilitate a study of how the model behavior depends on parameter values including an understanding of transitions between different types of qualitative behavior. These methods are introduced and explained for traffic jam formation and emergence of oscillatory pedestrian counter flow in a corridor with a narrow door

Modelling and experimental characterization of an energy harvester with bi-stable compliance characteristics

This paper presents a novel design for a vibrational energy harvester. The design uses high permeability magnetic materials which brings about two key advantages. First, it gives strong coupling between the mechanical and electrical domains, thus enabling effective energy conversion. Second, it gives the device a bi-stable compliance characteristic, which gives the harvester a broad-band frequency response. An explicit analytical model is developed using a combination of experimental data and finite element modelling in order to accurately incorporate the magnetic forces. The model is then validated using dynamic tests of the experimental rig. The main features of the dynamic response of the bi-stable oscillator are highlighted and benefits discussed in the context of energy harvesting. Finally, comments are made on the relationship between the complicated behaviour resulting from the bi-stable compliance characteristic and the benefits of increased electrical coupling

Identifying the significance of nonlinear normal modes

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

Tuning a resonant energy harvester using a generalized electrical load

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