actuators

Article Piezoelectric Vibration-Based Energy Harvesting Enhancement Exploiting Nonsmoothness

Rodrigo Ai 1, Luciana L. S. Monteiro 1, Paulo Cesar. C. Monteiro Jr. 2, Pedro M. C. L. Pacheco 1 and Marcelo A. Savi 3,*

1 CEFET/RJ, Department of Mechanical Engineering, 20.271.110 Rio de Janeiro, Brazil; [email protected] (R.A.); [email protected] (L.L.S.M.); [email protected] (P.M.C.L.P.) 2 Universidade Federal do Rio de Janeiro, COPPE - Ocean Engineering Program, 21.945.970 Rio de Janeiro, Brazil; [email protected] 3 Universidade Federal do Rio de Janeiro, COPPE - Department of Mechanical Engineering, Center for Nonlinear Mechanics, 21.941.972 Rio de Janeiro, Brazil * Correspondence: [email protected]



Received: 7 February 2019; Accepted: 6 March 2019; Published: 10 March 2019


Abstract: Piezoelectric vibration-based energy harvesting systems have been used as an interesting alternative power source for actuators and portable devices. These systems have an inherent disadvantage when operating in linear conditions, presenting a maximum power output by matching their resonance frequencies with the ambient source frequencies. Based on that, there is a significant reduction of the output power due to small frequency deviations, resulting in a narrowband harvester system. Nonlinearities have been shown to play an important role in enhancing the harvesting capacity. This work deals with the use of nonsmooth nonlinearities to obtain a broadband harvesting system. A numerical investigation is undertaken considering a single-degree-of-freedom model with a mechanical end-stop. The results show that impacts can strongly modify the system dynamics, resulting in an increased broadband output power harvesting performance and introducing nonlinear effects as dynamical jumps. Nonsmoothness can increase the bandwidth of the harvesting system but, on the other hand, limits the energy capacity due to displacement constraints. A parametric analysis is carried out monitoring the energy capacity, and two main end-stop characteristics are explored: end-stop stiffness and gap. Dynamical analysis using proper nonlinear tools such as Poincaré maps, bifurcation diagrams, and phase spaces is performed together with the analysis of the device output power and efficiency. This offers a deep comprehension of the energy harvesting system, evaluating different possibilities related to complex behaviors such as dynamical jumps, bifurcations, and chaos.

Keywords: energy harvesting; piezoelectricity; nonsmooth systems; nonlinear dynamics; impact; vibration

1. Introduction Vibration-based energy harvesting is becoming a remarkable technology since it allows the use of alternative sources of vibration to supply small devices, eliminating the need for frequent battery replacements or power cables. This is especially interesting for applications that include oil drilling or production and for aerospace structures [1–3]. An archetypal energy harvesting system model is a mechanical oscillator connected to an electronic circuit by a piezoelectric element. A typical experimental device is built of a cantilever beam with a tip mass and piezoelectric patches excited in the transverse direction by its base harmonic or random movement that is representative of ambient vibration.

Actuators 2019, 8, 25; doi:10.3390/act8010025 www.mdpi.com/journal/actuators Actuators 2019, 8, 25 2 of 15

In general, ambient vibration provides the energy harvesting system excitation and, once the excitation frequency is close to the natural system frequency, a maximum output power is captured. This linear analysis defines a narrowband harvester system since the ambient vibrations are usually varying in frequency or totally random with energy distributed over a wide frequency range [4]. Several researchers are investigating alternatives that can enhance energy harvesting capacity by introducing nonlinearities into the system [5,6]. A usual approach is to explore bistable structures with double-well potential and, depending on vibration conditions, it is possible to achieve high-orbit motion visiting the two potentials. This is essentially a Duffing-type oscillator that can be experimentally built using magnetic forces to modify the effective stiffness of the harvester [7–18]. Besides this kind of nonlinearity, constitutive nonlinear effects have also been exploited with the same objective. The majority of the literature addresses linear electro-mechanical conversion approaches [19–23]. Crawley and Anderson [24] explored nonlinear aspects related to piezoelectric coupling, showing that there is a significant dependence of strains. Triplett and Quinn [25] investigated nonlinear piezoelectric coupling behavior and some aspects related to the mechanical nonlinearities. Stanton et al. [17] proposed a quadratic dependence of the piezoelectric coupling coefficient on the induced strain. Experimental tests showed a good agreement with the numerical results. Silva et al. [26] investigated the hysteretic behavior of piezoelectric coupling, comparing the results with linear models. The results suggested that there is an optimum hysteretic behavior that can increase the harvested power output of energy harvesting systems. Silva et al. [27] showed a comparison among experimental data and numerical simulations performed with distinct nonlinear piezoelectric coupling models. The conclusions showed that the inclusion of nonlinear terms reduces the discrepancies predicted by linear models. Moreover, nonlinear aspects such as dynamical jumps are associated with dramatic changes in system responses. Another interesting nonlinear approach to enhance energy harvesting system capacity is the synergistic use of smart materials. The inclusion of shape memory alloy (SMA) elements can enhance system performance by using the SMA’s unique properties related to solid-phase transformations that allow us to exploit either stiffness change or energy dissipation. Avirovik et al. [28] developed a hybrid device coupling piezoelectric elements with SMAs for dual functionality, both as an actuator and an energy harvester. Silva et al. [29] employed a numerical analysis of an SMA–piezoelectric energy harvesting system and indicated that the inclusion of the SMA element can be used to extend the operational range of the system. The tunability of an energy harvesting system is a special procedure to be employed, and different alternatives can be implemented to alter the vibration behavior of the harvester: the use of mechanical preload [30]; the inclusion of asymmetric tip mass [31]; or the alteration of the structural energy harvester geometry [32,33] Lesieutre and Davis [34] showed that compressive axial preloads can increase the effective coupling coefficient of an electrically driven piezoelectric bimorph element. Leland and Wright [35] developed an energy harvester by applying an axial compressive load to tune the resonance by changing its effective stiffness. Betts et al. [36] presented a nonlinear device through an arrangement of bistable composites combined with piezoelectric elements for broadband energy harvesting of ambient vibrations. The results showed that it is possible to improve the power harvest over that by conventional devices. Bai et al. [31] showed that an asymmetric tip mass can induce nonlinear and hysteretic behavior to a piezoelectric energy harvester with a free-standing thick-film bimorph structure. Friswell et al. [33] presented a nonlinear piezoelectric energy harvester using an inverted elastic beam–mass system with nonlinear electro-mechanical coupling. The results showed that the system nonlinearity has two potential wells for large tip masses that are responsible for rich system behavior including chaos. The authors showed that the bandwidth of harvested power can be increased compared with that for a linear harvester. The use of nonsmooth nonlinearity is another approach to increase the bandwidth of energy harvesting systems. The basic idea is to confine energy harvester displacements using end-stops. Actuators 2019, 8, 25 3 of 15

Nevertheless, this approach has the disadvantage that the output power saturation at high excitation levels reduces the amount of generated power for high displacements. Despite this disadvantage, transforming an energy harvester into a broadband system can be achieved by using appropriate parameters to enhance the power generated at different frequencies [30,37–42]. Nonsmooth dynamical systems present rich behavior with unusual, complex mathematical descriptions. Savi et al. [43] and Divenyi et al. [44,45] presented numerical and experimental efforts to investigate discontinuous oscillators. Soliman et al. [37] investigated a harvester with a stopper where the results showed that the performance is highly influenced by the stiffness ratio of the rigid support, the energy harvester element, and the velocity of the cantilevered beam at the impact point. Rysak et al. [42] developed an energy harvesting system through an aluminum beam with a piezoceramic patch subjected to harmonic excitation and impacts. Hardening effect characteristics and a broader frequency range increasing the efficiency of the energy harvesting process were observed. Jacquelin et al. [46] developed a model for a piezoelectric impact energy harvester system consisting of two piezoelectric beams and a seismic mass. They analyzed the influence of different parameters (gap, seismic mass, and beam length, among others) on the system performance. Vijayan et al. [40,47] considered a vibro-impact system with the capacity to convert low-frequency responses to high frequencies using nonlinear impacts. Kaur and Halvorsen [48] developed an experimental setup and a lumped model of an electrostatic energy harvester with end-stop impacts. The results showed that the harvester device response is highly sensitive to the end-stop position at fixed bias and small acceleration amplitudes. Much of these research efforts have been devoted to exploring the effect of the main parameters of nonsmooth energy harvester systems using experimental or/and numerical approaches but, in general, the power output improvement analyses are not coupled with dynamics system analysis, which is herein performed in order to allow us to better comprehend nonsmooth energy harvesting devices. This work develops a numerical investigation exploiting nonsmoothness in piezoelectric vibration-based energy harvesting systems. A piezoelectric energy harvesting single-degree-of-freedom model with a one-side mechanical end-stop is investigated. Numerical simulations are carried out presenting a parametric analysis that shows the main parameter’s influence on the system dynamical behavior. Dynamical analysis using proper nonlinear tools such as Poincaré maps, bifurcation diagrams, and phase spaces is performed together with analysis of the device output power and efficiency. This offers deep comprehension of the energy harvesting system and the evaluation of different possibilities related to complex behaviors such as dynamical jumps, bifurcations, and chaos. The results show that impacts can strongly modify the system dynamics, resulting in broadband output power harvesting performance. This shows that deep nonlinear dynamical analysis needs to be performed for design purposes.

2. The Vibration-Based Energy Harvesting System A nonsmooth vibration-based energy harvesting system is analyzed by considering a single-degree-of-freedom system with discontinuous support, shown in Figure1. The system is composed of a mechanical oscillator with mass m connected to a linear spring with stiffness k and linear damping with coefficient c. The oscillator movement is restricted by a massless support, separated by a gap g. The support is represented by a linear spring with stiffness ks and linear damping with coefficient cs. This oscillator is subjected to a base excitation u = u(t), and the mass displacement is represented by y; z is the mass displacement relative to the base. The mechanical system is connected to an electric circuit by a piezoelectric element that converts mechanical energy into electrical energy. The electrical circuit is represented by an electrical resistance, R; V denotes the voltage across the piezoelectric element, Cp is the capacitance term, and the electro-mechanical coupling is provided by Actuators 2019, 8, x 4 of 15

The electrical differential equation for both situations is given by 1 (3) 𝜃𝑧 +𝐶 𝑉 + 𝑉=0. 𝑅 The instantaneous energy harvesting electrical power is defined by 𝑃=𝑉⁄𝑅, and the output and input average powers are given by 1 Actuators 2019, 8, 25 𝑃 = (𝑉 ⁄)𝑅 𝑑𝑡 (4) 4 of 15 𝑡 the piezoelectric element represented by1Θ. The system dynamics has two modes, with contact and without contact, which are described𝑃 = by the following(𝑚𝑢 )𝑧 𝑑𝑡 equations: (5) 𝑡 .. . .. mz + cz + kz − θV = −mu if z < g (without contact) (1) where the harmonic acceleration is given by 𝑢 = 𝛿sin (𝜔𝑡). The system performance can be evaluated .. . .. by considering mthez + conversion(c + cs)z + efficiency,kz + ks(z − 𝜂g, )which− θV = establishes−mu if z ≥ a grelation(with contact between). the electrical (2) (output) and mechanical (input) powers, 𝜂=𝑃⁄𝑃.

Figure 1. 1.Archetypal Archetypal model model of a vibration-based of a vibration-base energy harvestingd energy system harvesting with discontinuous system support. with discontinuous support. The electrical differential equation for both situations is given by 3. Numerical Simulations . . 1 θz + C V + V = 0. (3) Numerical simulations were carried out usingp theR system parameters presented in Table 1, based on the work of Kim et al. [23]. In general, the parametric analysis considers different values of support The instantaneous energy harvesting electrical power is defined by P = V2/R, and the output stiffness, 𝑘, and gap, 𝑔. and input average powers are given by s Z t 1 2 2 Pout = (V /R) dt (4) t 0 s Z t 1  ..
. 2 Pin = mu z dt (5) t 0 .. where the harmonic acceleration is given by u = δ sin(ωt). The system performance can be evaluated by considering the conversion efficiency, η, which establishes a relation between the electrical (output) and mechanical (input) powers, η = Pout/Pin. Actuators 2019, 8, 25 5 of 15

3. Numerical Simulations Numerical simulations were carried out using the system parameters presented in Table1, based onActuators the work 2019,of 8, x Kim et al. [23]. In general, the parametric analysis considers different values of support5 of 15 stiffness, ks, and gap, g. Table 1. System parameters [23]. Table 1. System parameters [23]. −1 −1 𝟐 −1 𝒎 (kg) 𝒌(N·m ) 𝒄=𝒄𝒔 (N·s·m ) 𝜹 (𝐦/𝐬 ) 𝑪𝒑 (𝐅) 𝑹 (𝛀) 𝜣 N·V( ) −1 −1 2 −1 0.00878m (kg) 4150k (N ·m ) c=0.219cs (N ·s·m ) δ (m/s2.5 ) 4.194Cp (F) × 10 R100(Ω) × 10 Θ (N−·0.004688V ) 0.00878 4150 0.219 2.5 4.194 × 10−8 100 × 103 −0.004688 The equations of motion were integrated using the fourth-order Runge–Kutta method. Time steps less than 10−4 s were assumed after a convergence analysis. Numerical simulations were carried out to The equations of motion were integrated using the fourth-order Runge–Kutta method. Time steps provide a parametric analysis evaluating the energy harvesting capacity. Displacement, power, and less than 10−4 s were assumed after a convergence analysis. Numerical simulations were carried out efficiency were monitored as a function of forcing frequency and different values of support stiffness to provide a parametric analysis evaluating the energy harvesting capacity. Displacement, power, (𝑘 ) and gap (𝑔). In this regard, it is interesting to define the nondimensional parameter 𝛽 that and efficiency were monitored as a function of forcing frequency and different values of support establishes a ratio between support and oscillator stiffness: 𝛽=𝑘/𝑘. stiffness (k ) and gap (g). In this regard, it is interesting to define the nondimensional parameter β that Figures 2 presents the system frequency response resulting from numerical simulations showing establishes a ratio between support and oscillator stiffness: β = k /k. the maximum displacement under a slow quasi-static variations of forcing frequency. Different 𝛽 Figure2 presents the system frequency response resulting from numerical simulations showing values were considered to show the influence of the end-stop on system dynamics. For 𝑔 = 70 μm, the the maximum displacement under a slow quasi-static variation of forcing frequency. Different β values vibrating mass does not touch the support at maximum displacement, named WI (without impact), were considered to show the influence of the end-stop on system dynamics. For g = 70 µm, the vibrating resulting in a non-contact behavior and a narrowband harvester system. As 𝑔 decreases to 50 μm, the mass does not touch the support at maximum displacement, named WI (without impact), resulting in a impacts lead to a typical nonsmooth characteristic with dynamical jumps. This behavior is more non-contact behavior and a narrowband harvester system. As g decreases to 50 µm, the impacts lead to evident as the support stiffness increases, when the resonance peaks become less sharp and broader. a typical nonsmooth characteristic with dynamical jumps. This behavior is more evident as the support Impact causes the excitation of higher frequencies, increasing the bandwidth. Nevertheless, the stiffness increases, when the resonance peaks become less sharp and broader. Impact causes the increase in the support stiffness also restricts the displacement. It is noticeable that the displacement excitation of higher frequencies, increasing the bandwidth. Nevertheless, the increase in the support response increases monotonically until the vibrating mass reaches the support and the slope of the stiffness also restricts the displacement. It is noticeable that the displacement response increases frequency–response curve drops abruptly. This behavior is displayed over a large range of monotonically until the vibrating mass reaches the support and the slope of the frequency–response frequencies, especially when the support stiffness increases. After a dynamical jump, the response curve drops abruptly. This behavior is displayed over a large range of frequencies, especially when the becomes identical to that associated with the linear system. support stiffness increases. After a dynamical jump, the response becomes identical to that associated with the linear system.

Figure 2. Maximum displacement versus forcing frequency: comparison between linear model without Figure 2. Maximum displacement versus forcing frequency: comparison between linear impacts (g = 70 µm) and a system incorporating impacts using g = 50 µm at different values of β. model without impacts (𝑔 = 70 μm) and a system incorporating impacts using 𝑔 = 50 μm at different values of 𝛽.

Based on this conclusion, the support stiffness and gap can be used to establish a proper system design. Figure 3 shows situations for the same ratio between support and oscillator stiffness (𝛽) using different gaps: 𝑔=30 μm and 10 μm. Note that the gap reduction causes a decrease in the maximum displacement due to the contact constraint, which implies that the effectiveness of the harvester system is reduced but, on the other hand, there is an extension of the higher-frequency range. In order Actuators 2019, 8, 25 6 of 15

Based on this conclusion, the support stiffness and gap can be used to establish a proper system Actuators 2019, 8, x 6 of 15 design. Figure3 shows situations for the same ratio between support and oscillator stiffness (β) using = µ µ differentActuatorsto clarify 2019 gaps: this, 8, xbehavior,g 30 m Figureand 10 4 showsm. Note the that maximu the gapm displacement reduction causes response a decrease for different in the maximum values6 of 15of displacement𝑔 considering due a fixed to the contactvalue of constraint, 𝛽 = 200. whichA significant impliesthat reduction the effectiveness of the maximum of the harvester displacement system istoassociated reducedclarify this but,with behavior, onan theamplification other Figure hand, 4 showsof therethe bandwidththe is anmaximu extension ism noticeable. displacement of the higher-frequency response for range.different In values order toof clarify𝑔 considering this behavior, a fixed Figure value4 showsof 𝛽 the = 200 maximum. A significant displacement reduction response of the formaximum different displacement values of g consideringassociated with a fixed an valueamplification of β = 200 of .the A significantbandwidth reduction is noticeable. of the maximum displacement associated with an amplification of the bandwidth is noticeable.

(a) (b) Figure 3. Maximum displacement versus forcing frequency: comparison between linear (a) (b) model without impacts (𝑔 = 70 μm) and a system incorporating impacts with different FigureFigurevalues 3. of3.Maximum 𝛽Maximum: (a) 𝑔 = displacement 30 displacement μm; (b) 𝑔 versus = versus forcing10 μm. frequency:forcing frequency: comparison comparison between linear between model without linear impactsmodel without (g = 70 µ m)impacts and a system(𝑔 = 70 incorporating μm) and a system impacts withincorporating different values impacts of β :(witha) g different= 30 µm; (valuesb) g = of10 𝛽µ:m. (a) 𝑔 = 30 μm; (b) 𝑔 = 10 μm.

Figure 4. Maximum displacement versus forcing frequency β = 200 with different gap values, g. Figure 4. Maximum displacement versus forcing frequency 𝛽 = 200 with different gap

Thevalues, previous 𝑔. results are now presented in the form of phase spaces and Poincaré sections (FigureFigure5). It is4. noticeableMaximum thatdisplacement the phase spaceversus is forcing split into frequency two regions 𝛽 associated = 200 with different with contact gap and non-contactvalues,The previous 𝑔. behaviors. results The are systemnow presented without in contact the form has of a phase symmetric spaces orbitand Poincaré since the sections contact (Figure mode response5). It is noticeable is not reached. that the In phase this regard, space theis split decrease into two of gapregionsg tends associated to present with more contact considerable and non- The previous results are now presented in the form of phase spaces and Poincaré sections (Figure changescontact behaviors. in the phase The spaces system since without it increases contact the has contact a symmetric region. Allorbit presented since the responses contact mode are related response to 5). It is noticeable that the phase space is split into two regions associated with contact and non- period-1is not reached. behavior, In this characterized regard, the bydecrease a Poincar of gapé section g tends with to present a single more point. co Asnsiderable expected, changes the effect in the is contact behaviors. The system without contact has a symmetric orbit since the contact mode response morephase pronounced spaces since when it increases the support the contact stiffness region. increases. All presented responses are related to period-1 isbehavior, not reached. characterized In this regard, by a thePoincaré decrease section of gap with g tends a single to present point. more As expected, considerable the changeseffect is inmore the phasepronounced spaces when since theit increases support stiffnessthe contact increases. region. All presented responses are related to period-1 behavior, characterized by a Poincaré section with a single point. As expected, the effect is more pronounced when the support stiffness increases. Actuators 2019, 8, x 7 of 15 Actuators 2019, 8, 25x 7 of 15

(a) (b)

(c) Figure 5. Phase spaces and Poincaré sections at resonance frequencies. A comparison Figure 5.5.Phase Phase spaces spaces and Poincarand Poincaréé sections sections at resonance at resonance frequencies. frequencies. A comparison A between comparison a linear 𝑔 μ betweenmodel without a linear impacts model (WI, without using g =impacts 70 µm) and(WI, a using system 𝑔 incorporating == 7070 μm) and impacts a system with incorporating different values impactsimpactsof β:(a) g withwith= 10 µdifferentdifferentm, (b) g = valuesvalues 30 µm, ofof and 𝛽:: (((ca))g 𝑔= = 50 10µ m.μμm, (b) 𝑔 = 30μm, and (c) 𝑔 = 50 μμm..

The influenceinfluence ofof impacts impacts is is now now analyzed analyzed from from the energythe energy harvesting harvesting perspective. perspective. Figure Figure6 shows 6 showsthe average the average power power curves curves for all cases for all discussed. cases discusse Thesed. curvesThese curves show that show the that same the conclusions same conclusions related relatedto displacement to displacement and broadband and broadband apply to apply harvested to harvested energy. energy.

(a) (b)

Figure 6. Cont. Actuators 2019, 8, x 8 of 15 Actuators 2019, 8, x 8 of 15 Actuators 2019, 8, 25 8 of 15

(c) (c) Figure 6. Average power versus forcing frequency using 𝑔 = 70 μm (without impact) and incorporatingFigureFigure 6. 6.Average Average impacts power power versuswith versus different forcing forcing frequency values frequency usingof 𝛽: g(a=) using 70𝑔 =µ m50 𝑔 (without μ =m, 70 ( bμ)m impact) 𝑔 (without= 30 andμm, incorporating impact)and (c) 𝑔and = 10impactsincorporating μm. with different impacts values with of βdifferent:(a) g = 50 valuesµm, (b )ofg 𝛽=: 30(aµ) m,𝑔 = and 50 (μcm,) g =(b 10) 𝑔µ m.= 30 μm, and (c) 𝑔 = 10 μm. Dynamical jumps are are associated with dramatic ch changesanges in in system system response response that that can reduce the systemDynamical amplitude amplitude jumps and and the theare harvested harvestedassociated energy. energy. with dramatic Figure Figure 7 7ch showsshowsanges thethein system averageaverage response powerpower ininthat thethe can steadysteady reduce statestate the forforsystem the the up-sweep up-sweep amplitude and and and down-sweep down-sweep the harvested frequency frequency energy. cases. cases. Figure The The 7 showsgap gap increase increase the average tends tends to topower increase increase in the the the steady frequency frequency state wherefor the the up-sweep jump occurs and down-sweep and also the jumpfrequency magnitude. cases. The gap increase tends to increase the frequency where the jump occurs and also the jump magnitude.

(a) (b) (a) (b)

(c) (c) Figure 7.7.Dynamical Dynamical jumps jumps of averageof average power power for frequency for frequency up-sweep up-sweep and down-sweep and down-sweep assuming assumingβFigure= 100: (7.a )𝛽 Dynamicalg = =50 100µ:m, (a ()b 𝑔jumps) g= =50 30 μofµm,m, average ( andb) 𝑔( c=) 30gpower= μ 10m,µ and m.for (frequencyc) 𝑔 = 10 μ m.up-sweep and down-sweep assuming 𝛽 = 100: (a) 𝑔 = 50 μm, (b) 𝑔 = 30 μm, and (c) 𝑔 = 10 μm. Actuators 2019, 8, 25 9 of 15 Actuators 2019, 8, x 9 of 15

Another way way to to evaluate evaluate energy energy harvesting harvesting capa capacitycity is is in in terms terms of of system system efficiency efficiency (𝜂) (η, ),which which establishes a power output/input output/input ratio. ratio. Figure Figure 88 sh showsows the the system system efficiency efficiency under under slow slow quasi-static quasi-static variation of the forcing frequency for for different different 𝛽β values.values. Once Once again, again, a asystem system without without impact impact is is compared with systems incorporating incorporating impacts: impacts: 𝑔g = 50 50 μµm,m, 𝑔g == 30 30 μµm,m, and and 𝑔g == 10 10 μm.µm. The The results results show show more or less the same same trend, trend, but but it it is is possible possible to to identify identify a a variation variation in in the the impact impact region. region.

(a) (b)

(c)

Figure 8.8. EfficiencyEfficiency versus versus forcing forcing frequency: frequency: comparison comparison between between a linear a linear model model without without impacts impacts(g = 70 µ(𝑔m) = and70 μm) systems and systems incorporating incorporating impacts withimpacts different with valuesdifferent of values support of stiffness, supportk s: stiffness,(a) g = 50 µ𝑘m,: ( (ab)) 𝑔g = 50 30 μµm, and(b) 𝑔 (c =) g30= μ 10m,µ m.and (c) 𝑔 = 10 μm.

Numerical simulationssimulations showshow that that the the use use of nonsmoothnessof nonsmoothness can becan useful be useful to spread to thespread frequency the frequencyband of an band energy of harvestingan energy harvesting system, generating system, age broadbandnerating a system.broadband On system. the other On hand, the other displacement hand, displacementlimits can restrict limits the can amount restrict of generatedthe amount power. of generated Hence, there power. is a Hence, competition there betweenis a competition amplitude betweenand distribution amplitude along and the distribution frequency. along Basically, the freque the gapncy. reduction Basically, decreases the gap reduction the maximum decreases value the but maximumenlarges the value frequency but enlarges band. This the conclusion frequency illustrates band. This the necessityconclusion to investigateillustrates thethe most necessity interesting to investigateconditions forthe amost specific interesting application. conditions for a specific application. In this this regard, regard, variations variations of of support support stiffness stiffness and and gap gap are are the the essential essential points points to tobe be investigated. investigated. The forthcoming analysis analysis considers considers average average values values for for simulations simulations for for different different frequencies frequencies within within the range range of of 500 500 to to 1400 1400 rad/s. rad/s. Theref Therefore,ore, average average values values of output of output power power 𝑃P, out𝑃, P, andout, andefficiency efficiency 𝜂, 𝜂η̅,, areη, are evaluated. evaluated. We We note note that that this this approach approach allo allowsws one one to toevaluate evaluate the the average average energy energy harvesting harvesting capacity through through a a frequency frequency ra range.nge. Figure Figure 99 showsshows the the average average harvested harvested power power output output (𝑃 (Pout) for) for different valuesvalues ofof stiffness stiffness ratio ratio represented represented by by parameter parameterβ. Differentβ. Different gaps gaps are investigated:are investigated:g = 70 𝑔 µ=m (without70 μm (without contact), contact),g = 50 𝑔µ m,= 50g =μm, 30 𝑔µ m,= 30 and μm,g and= 10 𝑔µ =m. 10 Impacts μm. Impacts tend totend maximize to maximize the power the power output, output,and an increaseand an inincreaseβ causes in aβ monotoniccauses a monotonic increase of increase power upof topower an asymptotic up to an value.asymptotic Nevertheless, value. Nevertheless,the energy harvesting the energy capacity harvesting tends tocapacity decrease tend fors small to decrease values of for gap small due tovalues excessive of gap displacement due to excessive displacement restrictions. Note that the case with 𝑔 = 10 μm is associated with power Actuators 2019, 8, x 10 of 15

output smaller than that for the linear, non-impact case. Therefore, there is a limit case where the use of impact is interesting for energy harvesting purposes. Figure 10 considers the average efficiency (𝜂̅) values for the same situations. Efficiency follows the same trend observed for the power output. Nevertheless, this analysis allows one to define the transition where the nonsmoothness is interesting. The case for 𝑔 = 50 μm is more efficient than all the cases, including the non-impact system (𝑔 = 70 μm) and the cases with 𝑔 = 30 μm and 𝑔 = 10 μm for all β values. It should be pointed out that the case with 𝑔 = 30μm is the transition case, presenting Actuators 2019, 8, 25 10 of 15 a greater efficiency compared with the non-impact system case only for β = 10. In order to establish a comparison, we consider the case with 𝛽 = 200, and all cases are compared with a reference value defined for the non-impact system (𝑔 = 70 μm): when 𝑔 = 50 μm, the system with impacts is 23.1% restrictions. Note that the case with g = 10 µm is associated with power output smaller than that for more efficient, generating 56.7 μW; when 𝑔 = 30 μm, the system is 4.2% less efficient, generating 48.8 the linear,μW; non-impact when 𝑔 = 10μ case.m, the Therefore, system is 47.4% there less is efficient, a limit casegenerating where 21.9 the μW. use Therefore, of impact power is interesting output for energy harvestingandActuators efficiency 2019, purposes.8, xcan be analyzed together in order to define energy harvesting capacity. 10 of 15

output smaller than that for the linear, non-impact case. Therefore, there is a limit case where the use of impact is interesting for energy harvesting purposes. Figure 10 considers the average efficiency (𝜂̅) values for the same situations. Efficiency follows the same trend observed for the power output. Nevertheless, this analysis allows one to define the transition where the nonsmoothness is interesting. The case for 𝑔 = 50 μm is more efficient than all the cases, including the non-impact system (𝑔 = 70 μm) and the cases with 𝑔 = 30 μm and 𝑔 = 10 μm for all β values. It should be pointed out that the case with 𝑔 = 30μm is the transition case, presenting a greater efficiency compared with the non-impact system case only for β = 10. In order to establish a comparison, we consider the case with 𝛽 = 200, and all cases are compared with a reference value defined for the non-impact system (𝑔 = 70 μm): when 𝑔 = 50 μm, the system with impacts is 23.1% more efficient, generating 56.7 μW; when 𝑔 = 30 μm, the system is 4.2% less efficient, generating 48.8 μW; when 𝑔 = 10μm, the system is 47.4% less efficient, generating 21.9 μW. Therefore, power output and efficiency can be analyzed together in order to define energy harvesting capacity.

Figure 9.FigureAverage 9. Average harvested harvested power power output output for different for different frequencies frequencies (interval (interval of 500 of to500 1400. to 1400. rad/s), for differentrad/s), values for of different gap, as avalues function of gap, of β as. a function of 𝛽.

Figure 10 considers the average efficiency (η) values for the same situations. Efficiency follows the same trend observed for the power output. Nevertheless, this analysis allows one to define the transition where the nonsmoothness is interesting. The case for g = 50 µm is more efficient than all the cases, including the non-impact system (g = 70 µm) and the cases with g = 30 µm and g = 10 µm for all β values. It should be pointed out that the case with g = 30 µm is the transition case, presenting a greater efficiency compared with the non-impact system case only for β = 10. In order to establish a comparison, we consider the case with β = 200, and all cases are compared with a reference value defined for the non-impact system (g = 70 µm): when g = 50 µm, the system with impacts is 23.1% more efficient, generating 56.7 µW; when g = 30 µm, the system is 4.2% less efficient, generating 48.8 µW; when g = 10 Figureµm, the 9. Average system harvested is 47.4% power less efficient, output for generating different frequencies 21.9 µW. (interval Therefore, of 500 power to 1400. output and efficiency canrad/s), be analyzed for different together values of in gap, order as a to function define of energy 𝛽. harvesting capacity.

Figure 10. Average efficiency for different frequencies (interval of 500 to 1400 rad/s), for different values of gap, as a function of β. Actuators 2019, 8, x 11 of 15

Figure 10. Average efficiency for different frequencies (interval of 500 to 1400 rad/s), for

Actuatorsdifferent2019, 8, 25values of gap, as a function of 𝛽. 11 of 15

In order to have a global comprehension of energy harvesting system dynamics, bifurcation diagramsIn order were to built have considering a global comprehension slow quasi-static of energy variation harvesting of the forcing system frequency dynamics, parameter bifurcation and diagramsplotting a werePoincaré built section considering of the slowresponse. quasi-static This is of variation special interest of the forcing since different frequency kinds parameter of response and plottingcan alter a Poincarthe energyé section harvesting of the response.capacity. Fi Thisgure is of11a special shows interest the bifurcation since different diagram kinds for of g response = 50 μm canconsidering alter the β energy = 200. The harvesting results show capacity. period-1 Figure motion 11a shows for all thefrequencies bifurcation and diagram a discontinuity for g = close 50 µm to consideringthe dynamicalβ = jump. 200. The The results details show of the period-1 system dynamics motion for are all depicted frequencies in Figure and a discontinuity11b and Figure close 11c tothrough the dynamical phase spaces jump. and The Poincaré details of sections the system for dynamicstwo different are depictedfrequencies, in Figure 708 and11b,c 936 through rad/s, phaserespectively. spaces andIt is Poincarnoticeableé sections that, although for two differentthe system frequencies, dynamics is 708 always and 936 related rad/s, to respectively.period-1 motion, It is noticeableit is associated that, althoughwith different the system solutions dynamics with di isstinct always amplitudes related to period-1and, therefore, motion, different it is associated energy withharvesting different capacities. solutions The with average distinct amplitudespower value and, for therefore,708 rad/s different is 180 𝜇W energy, andharvesting that for 936 capacities. rad/s is The3.91 average 𝜇W, showing power a value large for difference 708 rad/s between is 180 µ them.W, and that for 936 rad/s is 3.91 µW, showing a large difference between them.

(a)

(b) (c)

FigureFigure 11. 11.(a )(a Bifurcation) Bifurcation diagram diagram for g =for 50 µg m= and50μβm= and 200. β Phase = 200. space Phase and space Poincar andé sections Poincaré for (bsections) ω = 708 for rad/s (b) ω and = 708 (c) ω rad/s= 936 and rad/s. (c) ω = 936 rad/s.

When considering a gap g = 10 µm (Figure 12a), the system dynamics presents several bifurcations at low frequencies (500–700 rad/s) and in the region close to the dynamical jump. The phase spaces and Poincaré sections for three different frequencies offer another view confirming the dynamical system Actuators 2019, 8, x 12 of 15

When considering a gap g = 10 μm (Figure 12a), the system dynamics presents several Actuatorsbifurcations2019, 8 ,at 25 low frequencies (500–700 rad/s) and in the region close to the dynamical jump.12 ofThe 15 phase spaces and Poincaré sections for three different frequencies offer another view confirming the dynamical system complexity: see Figure 12b for ω = 536 rad/s, Figure 12c for ω = 684 rad/s, and complexity:Figure 12d seefor Figureω = 1228 12b rad/s. for ω =Different 536 rad/s, kinds Figure of response12c for ω =can 684 be rad/s, noticed and in Figure the detailed 12d for ωpictures:= 1228 rad/s.chaotic Different (536 rad/s), kinds period-2 of response (684 can rad/s), be noticed and period-1 in the detailed (1228 pictures:rad/s). It chaoticshould (536 be highlighted rad/s), period-2 that (684nonsmooth rad/s), andcharacteristics period-1 (1228 cause rad/s). dramatic, It should sudden be highlightedchanges in thatthe system nonsmooth dynamics. characteristics The efficiency cause dramatic,values are sudden more or changes less the in same the system between dynamics. 500 and The 1000 efficiency rad/s (see values Figure are 8c), more and or after less the that, same the betweenefficiency 500 increases and 1000 with rad/s the (seefrequency. Figure8 Thec), and efficien aftercy that, values the for efficiency the identifi increasesed simulations with the frequency. are 0.1057 The(536 efficiency rad/s), 0.09799 values (684 for therad/s), identified and 0.2627 simulations (1228 rad/s). are 0.1057 Note (536 that rad/s), period-1 0.09799 response (684 rad/s),has greater and 0.2627amplitudes (1228 and, rad/s). therefore, Note that generates period-1 more response energy has wi greaterth an amplitudesefficiency that and, is therefore, of interest. generates Hence, morealthough energy the with case anwith efficiency a small thatgap has is of lower interest. efficiency Hence, than although do the the other case investigated with a small cases, gap has its lowerricher efficiencynonlinear than dynamics do the can other be investigateduseful to enhance cases, its its energy richer nonlinearharvesting dynamics capacity. can be useful to enhance its energy harvesting capacity.

(a) (b)

(c) (d)

FigureFigure 12. 12.(a )(a Bifurcation) Bifurcation diagram diagram for g = for 10µ gm = and 10μβm= 200.and Phase β = 200. spaces Phase andPoincar spacesé sectionsand Poincaré for (b) ωsections= 536 rad/s, for (b (c)) ωω == 536 684 rad/s, and(c) ω (d =) ω684= 1228rad/s, rad/s. and (d) ω = 1228 rad/s. 4. Conclusions 4. Conclusions ThisThis paperpaper exploitsexploits nonsmoothnessnonsmoothness forfor piezoelectricpiezoelectric vibration-basedvibration-based energyenergy harvesting.harvesting. AA single-degree-of-freedomsingle-degree-of-freedom oscillatoroscillator withwith discontinuousdiscontinuous supportsupport connectedconnected toto anan electricelectric circuitcircuit byby aa piezoelectricpiezoelectric elementelement waswas investigated.investigated. AA parametricparametric analysisanalysis waswas carriedcarried outout forfor varyingvarying valuesvalues ofof supportsupport stiffness stiffness and and gap. gap Numerical. Numerical simulations simulations were were carried carried out comparingout comparing nonsmooth nonsmooth systems systems with awith linear a linear system system without without impact. impact. A frequency A frequency response response analysis analysis showed showed the sensitivity the sensitivity of the power of the generatedpower generated to the support to the support stiffness stiffness and gap, and which gap, compete which compete to define to the define energy the harvesting energy harvesting capacity. Largercapacity. gaps Larger reduce gaps the possibilityreduce the of possibility impacts, leading of impacts, the system leading to thethe linear system regime to the without linear impacts. regime On the other hand, small gaps are related to small amplitudes that considerably decrease the device efficiency but increase the frequency bandwidth. Concerning support stiffness, the results indicate Actuators 2019, 8, 25 13 of 15 that increased support stiffness makes it possible for a system to produce more power than a linear system using a gap with an appropriate value. Besides this, nonsmooth characteristics cause complex responses with dramatic and sudden changes to the system dynamics. Nonsmoothness can strongly modify the system dynamics, tending to increase the broadband output power harvesting performance. On the other hand, its constraints limit the energy harvesting capacity due to displacement restrictions. Dynamical investigation suggests that there are optimum values of support stiffness and gap that increase the power output of energy harvesters operating in broadband source vibration conditions. Besides this, richer nonlinear dynamics related to a high nonsmoothness level can be useful to increase energy harvesting capacity. In conclusion, the results indicate that the use of nonsmoothness is an interesting option to improve the operational bandwidth of energy harvesting systems. A proper dynamical analysis with a parametric study is an interesting methodology for the design of a nonsmooth energy harvesting system and points out situations where the energy harvesting capacity is enhanced.

Author Contributions: Investigation, R.A., L.L.S.M., P.C.C.M.J., P.M.C.L.P. and M.A.S.; Software, R.A, L.L.S.M. and P.C.C.M.J.; Supervision, L.L.S.M., P.C.C.M.J., P.M.C.L.P. and M.A.S.; Writing—original draft, L.L.S.M. and M.A.S..; Writing—review & editing, L.L.S.M. and M.A.S. Funding: The authors would like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES, and FAPERJ. The Air Force Office of Scientific Research (AFOSR) is also acknowledged. Conflicts of Interest: The authors declare no conflict of interest.

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