Post-grazing dynamics of a vibro-impacting energy generator
Introduction
Energy Harvesting (EH) from ambient vibrations was proposed almost two decades ago as an attractive alternative to power supplies or as renewable sources of energy for rechargeable batteries. Since then the gaps in the linear theory of EH have been filled with different methods of energy conversion, based on single-degree-of freedom, multi-degree-of freedom and/or continuous (rods and beams) linear systems on the nano [1], micro [2] and macro scales [3], [4], [5]. The excitement regarding the potential of linear EH systems has significantly decreased since then, due to low energy densities of the linear devices, narrow bandwidth and high natural frequency in nano- and micro-scale systems, which are difficult to match in many practical applications. These and other adverse factors lead to insufficient output necessary to power or recharge a battery. The deficiencies in the development of linear EH devices has slowed the proliferation of wireless sensors, particularly critical in the Internet of Things paradigm.
The above limitations in the linear theory of EH have motivated wide-spread efforts on parametrically excited [6], [7], [8], nonlinear [9], [10], [11] and non-smooth systems. The idea behind parametrically excited systems is the use of large system responses near instabilities, e.g. see [12], [13], [14], [15], among others. Within the huge range of nonlinear EH systems [16], [17], [18], there are some particular themes of note; natural single-potential nonlinearities (classical continuous nonlinear systems like the Van-der-Pol oscillator, Lingala et al. [19], Duffing oscillator, Ghouli et al. [20], Zhu [21], Sebald et al. [22], the pendulum, etc.), natural or imposed geometrical nonlinearities (systems with double [23], [24], [25], triple [26], [27], [28] or multiple stable equilibriums [29], [30]), systems with a nonlinear interaction such as flow-induced vibration systems (see [31], [32], [33], [34], [35], [36] and references therein), and systems with strongly nonlinear or discontinuous nonlinearities like dry friction, piecewise discontinuity or vibroimpacts [37], [38]. It has been shown that the nonlinear mechanisms for EH are far more beneficial than linear ones. This observation follows from the typical structure of the response amplitude vs. forcing frequency or backbone curve, showing a wider bandwidth with higher response amplitude away from a main resonance frequency. However, the design and optimization of a nonlinear energy harvester is far more complex, with limited explicit analytical results, thus requiring extensive complementary experiments or numerics. The available approximation techniques can estimate the response within only a narrow range of parameters imposed by the mathematical assumptions necessary for the applied averaging procedure, typically based on a weakly nonlinear model with small forcing.
Vibro-impact systems have rich phenomenological behaviors, manifesting a variety of routes to nonlinear phenomena like bifurcations, grazing and chaos [39], [40], [41], [42]. These effects have been studied in deterministic [43], [44], [45], [46] and stochastic vibro-impact systems [47], [48], [49]. The models of vibro-impact systems include piecewise linear stiffness [50], [51] as well as rigid barriers for instantaneous impacts leading to a velocity jump for inelastic impacts. EH devices that utilize vibro-impact dynamics as a main energy absorption mechanism were developed and studied in a number of publications [52], [53], [54], [55]. Other interesting applications of vibro-impact systems include propulsion mechanisms, where the internal impacts are designed to drive the entire system forward or backward. Such systems can be used for autonomous robots and medical devices [56]. While often the study of such systems is limited to computational results only, certain settings allow an analytical or semi-analytical treatment when the motion is composed of a sequence of trajectories described (semi-)analytically. Such an approach translates the piece-wise continuous behavior into a sequence of maps, amenable to analytical treatment [57] that provides explicit parametric expressions for a simple periodic motion. This methodology has certain benefits since it allows bifurcation and stability analyses of various periodic regimes that may occur in the system. Of course, for more complex motions a larger series of maps is necessary, making this particular derivation more tedious and cumbersome.
Recently, Yurchenko et al. [58] proposed a novel vibro-impact energy harvesting (VI-EH) device utilizing dielectric elastomeric (DE) membranes. There it was shown that the performance of such VI-EH depends strongly on the relationship between the excitation and device parameters, leading to various vibro-impact regimes with a low or high power output. The device consists of a forced cylinder with a ball moving freely inside of it, impacting DE membranes covering both ends of the cylinder. Each membrane is composed of the DE material sandwiched between two compliant electrodes, acting as a variable capacitance capacitor. The impacts of the DE membrane by the ball influence its motion while deforming the membrane, leading to energy harvesting via the properties of variable capacitance. The analytical results of [59] gave parametric conditions for a simple periodic motion, consisting of two alternating bottom and top impacts per forcing period. Building on the method of [57], these results considered the asymmetric case of the inclined VI-EH device, providing explicit expressions for impact velocity, phase shift of impact relative to the oscillatory forcing, and time between impacts, in terms of parameters such as the length of the cylinder, excitation parameters, and incline angle. Furthermore, the linear stability of this periodic motion demonstrates the range of parameters where it influences the corresponding VI-EH power output. However, this study did not consider adjacent parameter regimes where more complex periodic motions, period doubling bifurcations, and chaotic motion were observed numerically.
Here we consider a broader class of periodic motions of the VI-EH, referred to here as :/ where and are the number of impacts on the bottom and top membrane, respectively, per period of the excitation, and is the ratio of the period of the motion of the VI-EH to . Then, the motion studied in Serdukova et al. [59] is 1:1/1 motion. Throughout this paper we mainly focus on motions where as these types of solutions appear over significant parameter ranges when the cylinder is inclined. For convenience of notation, throughout we do not include when and include only if .
The goal of this paper is to address a number of problems that have received limited attention for impacting systems of this type. We develop a generalized semi-analytical approach for analysis of :1 periodic behavior, applied explicitly to the 2:1 case. This approach is particularly valuable in cases when the transition to :1 motion follows more complex solutions appearing from period doubling or chaotic behavior. By using the maps to develop a series of expressions for a single impact velocity within the periodic solution, it is straightforward to generalize to other types of periodic solutions.This result moves beyond previous results in Luo and Guo [57]and [59], as the generalized approach avoids the cumbersome calculations used to get explicit expressions in those studies. These analytical results provide the basis for our stability analyses of 2:1 solutions and for the comparison of the energy output for 1:1 and 2:1 motions. The comparisons with computations reveal additional unexplored phenomena, not previously documented in the dynamics of such a system: bi-stability of 2:1 motion and 3:1 motion, with two different types of transitions between these behaviors. While we postpone a full analytical treatment of this bi-stability to future work, the results of this paper illustrate the importance of different types of bifurcations on the energy efficiency of the VI-EH. Thus our analysis here provides a necessary foundation for parametric comparisons between these different types of bifurcations, and for their impact on the energy output.
The paper is organized as follows. A mechanical model and equations of motion of the VI-EH are described in Section 2, together with a review of results from [59] for 1:1 behavior, presented within the larger context of new results from this paper. In Section 3 we outline a semi-analytical method for obtaining parametric conditions for general :1 periodic solutions, illustrating this method for 2:1 periodic motion. Specifically the results are derived through three nonlinear maps, corresponding to the three impacts per period, combined with the impact conditions. A linear stability analysis for this motion is given in Section 4, and contrasted with additional routes to grazing behavior. The voltage output of the 2:1 periodic motion is shown in Section 5 and compared with that for the 1:1 periodic motion, together with comparisons of different metrics for the average energy available for harvesting. Finally, conclusions are drawn together with recommendations for the device design.
Section snippets
Dynamical model of the vibro-impacting energy harvester
The focus of this paper is a nonlinear vibro-impact energy harvesting device comprised of an externally forced capsule with a freely moving ball inside. Each end of the capsule is closed by a membrane of DE material with compliant electrodes. The friction between the ball and the capsule is neglected, so that the motion of the ball is driven purely through impacts with one of the DE membranes and by gravity. The impact of the ball with the DE membrane not only excites the ball but also causes
Analytical expressions for periodic 2:1 motion
In this section we obtain analytical expressions for the parametric dependence of the 2:1 motion, using the maps and for the sequence of impacts over the intervals for .
Note that this calculation is a particular application of the general approach for deriving :1 periodic solutions for (6)and (7). An :1 periodic solution is composed of applications of followed by and . The unknowns needed to define the motion are values of the impact velocity
Linear stability analysis
The critical points as shown in Fig. 5 are obtained from a linear stability analysis around the quadruples corresponding to the asymmetric period- solutions. A complete review of this method can be found in Shaw and Holmes [50], Luo and Guo [57], Luo [60].
Considering a small perturbation to the fixed point we obtain the equation for linearized about with
Energy output
Here we investigate the output voltage of the 2:1 behavior and compare these results with the 1:1 motion published in Serdukova et al. [59]. Three variables corresponding to output voltage are shown, output voltage at the impact, average output per impact and averaged output per unit of time . The derivation of is summarized in Yurchenko et al. [58] and are defined aswhere is the sample size of impacts and
Conclusions
In this paper we determine semi-analytical solutions and stability conditions for the 2:1 motion of an inclined vibro-impacting energy harvester. These results also provide insight into the VI-EH’s energy harvesting potential. The device is composed of a ball moving in a cylinder with dielectric elastomer material at the cylinder ends. It is driven by a harmonic forcing with period and positioned with an incline angle . Energy is generated through impacts of the ball with the DE material
CRediT authorship contribution statement
Larissa Serdukova: Methodology, Software, Writing - original draft, Writing - review & editing. Rachel Kuske: Conceptualization, Methodology, Validation, Writing - review & editing. Daniil Yurchenko: Conceptualization, Methodology, Validation, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors gratefully acknowledge partial funding for this work from NSF Division CMMI 2009270 and EPSRC EP/V034391/1.
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