Research paperVibrational energy harvesting by exploring structural benefits and nonlinear characteristics
Introduction
Over the past years, there has been a growing attention in energy harvesting techniques for employing ambient vibrations in surrounding or machines. Traditional solutions for energy harvesting are usually based on linear resonant mechanical structures composed of springs and mass with the transducer through which mechanical energy can be converted to electrical energy based on electromagnetic [1], [2], [3], piezoelectric [4], [5], [6], [7], [8] and electrostatic [9], [10], [11] mechanisms. However, these energy harvesting systems are often effective only if the ambient vibration frequency is close to their resonant frequency. Due to time-varying and broadband properties of ambient vibrations, only a small fraction of mechanical energy can be converted to electrical energy with linear energy harvesting systems. To improve energy harvesting performance, various solutions have been proposed in order that broader range of operating frequencies of energy harvesters can be achieved. Nonlinear energy harvesting techniques, as an alternative method, have been proved to be capable of providing better performance in terms of the amount of energy extracted from a wider range of ambient vibrations [12], [13], [14], [15], [16], [17].
Recently, it has been demonstrated that bistable structure configurations can be used to overcome the bandwidth issues [12], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. In particular, Erturk et al. [21], [22] introduced a piezomagnetic structure for substantial enhancement of piezoelectric power generation in vibration energy harvesting. The bistability could arise because of appropriate placement of the magnets which create a bistable system switching between two stable states. Based on this, Zhou et al. [20], [33], [34] enhanced the broadband frequency responses by tuning the angular orientation of the permanent magnets. Different from this approach, Ando et al. [12] proposed a bistable clamped-clamped polyethylene terephthalate beam and two piezoelectric transducer, and the bistability of their system is created through beam switching due to the environmental vibrations. Arrieta et al. [18], [19] proposed a bistable composite plate with bonded piezoelectric patches for broadband nonlinear energy harvesting. Both analytical investigation and measurement show that the bistable energy harvesting system can provide better energy harvesting performance with board frequency responses. However, because of the dynamic properties of bistable systems, bifurcations and chaos may be generated and leading to unstable risk.
Several other researchers have attempted to apply nonlinear stiffness or nonlinear damping to enhance the energy harvesting performance [14], [35], [36]. Ramlan et al. [37] proposed a system composed of two types of nonlinear springs connected to a mass together with a linear viscous damper. A bigger amount of power harvested using the developed nonlinear device compared to the linear one. Nonlinear cubic damping has been investigated by Tehrani et al. [38] to increase the dynamic range of an energy harvester, and it is noticed that the nonlinear damping also leads to nonlinear stiffness properties.
To optimise energy harvesting performance, some other researchers such as Roundy [39] and Leland [40] attempted to analyse and optimise the effect of design parameters on power output efficiency. For this purpose, Wu et al. [32] developed a tuneable resonant frequency power harvesting device with cantilever beams, from which the resonant frequency can be shifted to match that of the external vibrations. Challa et al. [41] presented an approach which uses the magnetic force to alter the overall stiffness of the energy harvesting device in order to further change the resonant frequency of the device to lower frequencies for better energy harvesting. While successful in enabling energy harvesting over a broader range of frequencies, the amplitude of the vibration may be reduced which may affect the overall efficiency and power output of the energy harvesting device.
This paper present a novel nonlinear energy harvesting system composed by levers and X-shape supporting structures, in which the X-shape structure has been recently proposed for vibration isolation [42], [43], [44], [45]. The equivalent stiffness and damping of the system are nonlinear because of the geometry relationship between the base and platform motions within the structure. The objective is to explore the displacement-dependent nonlinear damping characteristics and the tuneable stiffness characteristics of the X-shape structure for advantageous energy harvesting performance.
In [49], it is shown that the tuneable stiffness of the X-shape structure can be employed for ultra-low frequency energy harvesting and several X-shape structures can be used in parallel to achieve a coupled effect for energy harvesting in a wider frequency band. However, although a single X-shape structure is much simpler to implement in practice, a lower energy harvesting peak frequency by tuning the structure parameters may contradict with higher harvesting peak value and thus only a discounted energy harvesting performance is achieved. To solve this, an easy-to-design lever-type system is applied to the X-shape structure, which can be used for tuning the dynamic responses of the system without compromising the advantageous nonlinear damping effect of the system. Due to the flexibility of the system configuration, the equivalent stiffness and damping can be freely tuned by adjusting several structure parameters for much better energy harvesting performance. The nonlinear energy harvesting system without levers and the corresponding linear ones are investigated for comparisons in order to validate the advantageous performance of the proposed hybrid energy harvesting system.
It is shown that (a) the nonlinear stiffness can be designed easily by structural parameters to achieve low resonance frequency and it is a fairly weak nonlinearity without chaos and bifurcation; This is a special feature of this proposed energy harvesting system, compared with most other nonlinear energy harvesting systems; (b) the displacement-dependent nonlinear damping effect of the X-shape structure is very beneficial and tuneable for vibration based energy harvesting; this is not well discussed in the literature; This presents a unique feature which is potentially more beneficial and practical to vibration energy harvesting; (c) the lever system is very helpful for tuning the dynamic response of the X-shape structure including the resonance peak and frequency (achieving low resonant frequency but high harvesting peak); this is not reported in the literature and provides another special feature of this energy harvesting system.
Section snippets
The novel nonlinear energy harvesting system
The energy harvesting system is composed by an n-layer x-shape structure, (two) levers with attached masses, and a linear spring and an energy-harvesting transducer functioning as a damper, placed horizontally, which are shown in Fig. 1. Each layer of the structure has two rods combined with one joint, and the length of one rod is 2l. The assembly angle of the rod with respect to horizontal direction is θ. Two levers are assembled on the centre joints of the X-shape structure. One of the lever
Dynamic properties of the system
Based on the dynamic equation of the system, the nonlinear dynamic stiffness of the system can be determined, and the relationship between the restoring force and vertical displacement can be written as where the equivalent linear stiffness coefficient β1 is a dominant factor to represent the dynamic property of the system, β2 β3 β4 are the equivalent nonlinear stiffness coefficient.
With this restoring force, the equivalent potential energy of the system is given by
Power dissipation prediction
In order to predict the power dissipation in the frequency domain, the Harmonic Balance Method (HBM) [48] is applied to analyse nonlinear systems. To deal with the nonlinear problem, Li et al. [49] used the Takagi-Sugeno fuzzy model-based approach to linearize the nonlinear network control system. Using this approach, the nonlinear system was divided into a series of linear sub-systems, and then all the sub-systems with membership functions were blended to approximate the original nonlinear
Conclusions
A systematic investigation on a novel nonlinear energy harvesting system by employing structure benefits and geometric nonlinearities is presented in this study. The dynamic property in terms of equivalent dynamic stiffness has been analyzed. For small amplitude dynamic responses, the system performs weekly nonlinear properties, which has no bifurcation and chaotic issues. The advantageous energy harvesting performance of the proposed system is analyzed with the HBM based on the dynamic
Acknowledgements
The authors gratefully acknowledge the support from a GRF Project of HK RGC (No 15206514), a NSFC project (No 61374041) of China, and a grant from the Innovation and Technology Commission of the HKSAR Government to the Hong Kong Branch of National Rail Transit Electrification and Automation Engineering Technology Research Center.
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