2.7 Mathematical Modeling and Solution Strategies
2.7.2 Explored resonance conditions
2.7.1.4 Flow induced excitation
The kinetic energy available in the fluid is another source of energy that has been utilized for energy harvesting purposes. Fluid phenomena such as galloping (Barrero- Gil et al. [84], Sirohi and Mahadik [24]), vortex induced vibration (VIV), wake galloping [104–106, 150] and flutter [101] have been explored in developing PEH systems.
2.7.1.5 Simultaneous excitation
Simultaneous excitation, i.e., direct with parametric excitation or direct with gal- loping [222], provides more opportunities to the PEH system for ambient energy harvesting. Several recent studies involve the analysis of PEH system under such simultaneous excitations. A combination of parametric and direct excitation is stud- ied by Xia et al. [223], where direct excitation helps in overcoming the necessity of threshold amplitude of excitation to parametric excitation. In another work, rich dynamic responses are observed (Yan and Abdelkefi [122], [6]) when concurrent gal- loping and base excitations provide input energy to the harvester. In the next section resonance conditions used to harvest energy in PEH systems have been discussed.
periodic excitation. These can be utilized to harness more power. In the following subsection literature related to the primary and secondary resonance conditions has been reviewed.
2.7.2.1 Primary and secondary resonance
Primary resonance occurs when the frequency of external excitation coincides with one of the natural frequencies of the system. Away from the resonance condition, the response amplitude, and hence the output voltage reduces significantly. Hence Cheng et al. [224] used self tuning and Niri and Salamone [225] used passive tuning method to enhance the output voltage of the PEH. Available narrow bandwidth also limits the harvester functionality. Tadesse et al. [226] has proposed to separately excite the multiple modes of linear harvester for enhanced performance. Qi et al.
[227] proposed that with an array of beams one can harvest energy with multiple resonance conditions simultaneously. Generally the natural frequencies are separated with a large bandwidth, but by utilizing a flexible attachment to the harvester (Seo et al. [228]) or folded configuration (Gong et al. [229]), one may bring the natural frequencies closer within a practical range. By using a magnetic forces, several authors (Leland and Wright [230], Challa et al. [231], Al-Ashtari et al. [232]) proposed a frequency tunable technique in PEH systems.
Nonlinear harvesting systems are capable of harvesting not only under primary res- onance but also under secondary resonance conditions. For example in Leadenham and Erturk [178] the superharmonic resonance condition is explored. The limited literature suggests that these secondary resonances are lesser explored for harvesting applications. In the next subsection literature related to internal resonance has been covered.
2.7.2.2 Internal resonance
Nonlinearity brings an interesting dynamic phenomenon i.e., internal resonance (IR) in multimode systems. While in the case of linear systems the different modes can be excited separately by adjusting the external excitation but in nonlinear systems coupling among different modes is possible [26, 233–235]. Internal resonance arises
when the modal frequencies are in integer relationship with each other. For example if the second modal frequency is near twice the 1st mode frequency, one may observe 1:2 internal resonance condition. In other words, these commensurable or nearly commensurable frequency ratios [25] causes mode coupling phenomenon [71] and energy transfer takes place between participating modes of the system. This leads to periodic, quasi-periodic, and chaotic responses [12, 26, 27, 233, 236]. Such responses attracted scientific community from time to time and inspired them to apply internal resonance conditions in multi-dimensional interconnected systems for a large number of application fields.
Recent developments in nonlinear energy harvesting systems show that the mecha- nism of internal resonance can be utilized to improve the energy harvesting capa- bilities of harvester systems. Perhaps the consideration of internal resonance is less explored in order to harvest more energy over wide bandwidth in the field of en- ergy harvesting from vibration. The commensurate modal frequencies can be used to broaden the steady state bandwidth Chen et al. [237] due to nonlinear energy transfer which occurs among the commensurate modes. In order to seek better performance in the field of PEH, few recently developed systems [179, 237–239] incorporated the internal resonance condition of 1:2. Similarly, Yan and Hajj [208] investigated an autoparametric vibration based PEH for enhanced frequency bandwidth. Where the frequency ratio of beam to base structure is taken as one half that causes the beam motion. A hybrid PEH system with internal resonance (1:2) is explored by Yang and Towfighian [68], where an axially movable magnet based bistable configuration is analysed. It is observed that large magnetization moment and closely placed mag- nets lead to wide bandwidth. Xiong et al. [240] achieved 1:2 internal resonance by incorporating an auxiliary oscillator to the main nonlinear harvester and they have shown experimentally that 130% improvement in bandwidth can be achieved. In all these internal resonance based PEH systems an auxiliary system is attached to the main system in such a way that the combined system exhibits internal resonance.
To enhance the steady state bandwidth limit, an axially loaded energy harvester with 1:2 internal resonance is studied by Jiang et al. [238]. Chen et al. [237] considered the 1:2 internal resonance with L-shaped energy harvester to improve the bandwidth.
An autoparametric vibration based energy harvester is investigated by Yan and Hajj [208], where nonlinear response enhances the frequency bandwidth. Inspired from
the trees which exploit nonlinear mechanisms such as multimodal frequency distri- butions and nonlinearities to damp out the vibrations caused by aerodynamic forces, Harne et al. [179] studied L-shaped vibration energy harvesting system under pure harmonic, mixed harmonic and stochastic excitations which exploits the 1:2 inter- nal resonance. The robustness and practicality of utilizing nonlinear mechanism to enhance the performance of energy harvesting systems are investigated analyti- cally, numerically, and experimentally. Xu and Tang [239] proposed a piezoelectric cantilever beam attached with a pendulum for multi-directional energy harvesting purposes. System dynamics is analysed for 1:2 internal resonance which is due to nonlinear coupling between cantilever beam and the pendulum. L-shaped beam- mass structure with quadratic nonlinearity tuned to two-to-one internal resonance is studied by Cao et al. [241] for bandwidth enhancement. Alazmi et al. [242] ex- amined the vibration and voltage response of an electromagnetic ferrofluid-based vibratory energy harvester with primary and a 1:2 internal resonance conditions.
Rocha et al. [243] presented a two degree of freedom frame structure for energy har- vesting with nonlinear piezoelectric material which exhibits 1:2 internal resonance between symmetric and the sway modes. Chen and Jiang [244] proposed a new pos- sibility to enhance energy harvesting by tuning harvesters at the internal resonance of 1:2. Recent experimental work of Shaw et al. [245] provides valuable insight into frequency responses of nonlinear systems consists of a cantilever beam with a non- linear spring at the tip with deliberately induced 1:3 internal resonance. Complex dynamics of periodic and quasi periodic response have been observed. The Effi- ciency of PEH systems is increased by the application of internal resonance Chunbo et al. [220]. Mode coupling leads to interaction and energy transfer between higher and lower modes. Here a cantilever beam is considered under vertical stochastic excitation.