Dynamics of symmetric and asymmetric potential well-based piezoelectric harvesters: A comprehensive review

3.1.1. Dynamic behavior of the linear harvesters

As mentioned in above discussion, a harvester with linear restoring force and quadratic potential function is known as the linear mono-stable harvester. The quadratic potential function of the linear harvester can be represented by equation (6) in the dimensionless form. A dimensionless parameter q , where q > 0 , is introduced to facilitate the stiffening or relaxing of the linear stiffness and corresponding quadratic potential functions of the system are shown in Figure 5(a). Steady-state, non-dimensional response amplitude ( u ¯ ) of the linear harvester shows a peak response ( u ¯ max ) at resonance, that is when Ω = 1 (see Figure 5(b)). Also, a narrow frequency bandwidth ( d Ω ) is observed at half-power points (Rao, 2011), which are usually considered at an amplitude of u ¯ max / 2 . Thus, for the linear harvester, large amplitude oscillations that are required to produce sufficiently higher strains and electrical output from the harvester, exist over a very narrow bandwidth. Consequently, the linear harvester shows a satisfactory energy harvesting performance only when excited near resonant conditions.

U lin ( u ) = 1 2 q K 1 u 2

(6)

OPEN IN VIEWER

Figure 5.

(Color online) (a) Non-dimensional (ND) potential energy function U lin ( u ) of the linear harvester and the effect of tuning parameter q on it. The bold curve in orange color refers to the linear harvester without any tuning ( q = 1 ). The solid green circle at u = 0 represents the only stable equilibrium position. (b) The steady-state frequency response of the linear harvester ( q = 1 ), showing the peak amplitude u ¯ max and the frequency bandwidth d Ω . (c) The effect of the resonance tuning with the parameter q on the frequency response of the linear harvester. (d) The phase-portrait diagram in ( u , u · ) space showing the orbits of oscillations of the tuned harvesters excited with frequency Ω = 0 . 5 under similar forcing conditions and ζ = 0 . 04 .

In order to improve the off-resonance performance of the linear harvester, resonance tuning of the harvester can be exercised. This is achieved by modifying (stiffening or relaxing) the stiffness of the harvester, using the parameter q as mentioned above. For 0 < q < 1 , detuning of the resonance frequency is observed and for q > 1 resonance frequency increases as the stiffness of the system increases. The effect of such a tuning method under same forcing conditions on the frequency response of the linear harvester can be realized through Figure 5(c). As the stiffness of the system relaxes, the resonance peak shifts towards the lower frequencies ( Ω < 1 ) and the response amplitude increases. Conversely, increasing the stiffness of the system causes the resonance peak to shift toward the higher frequencies ( Ω > 1 ) and the response amplitude decreases. Thus, the natural frequency of the linear harvester can be tuned so as to match with the excitation frequency and improve its performance. The phase-portrait diagram of the tuned linear harvesters in ( u , u · ) space is presented in Figure 5(d), where all the cases of the tuned harvesters are excited at Ω = 0 . 5 under similar forcing conditions. It can be observed that the orbit of oscillations of the harvester under steady state conditions, which is also known as the limit cycle (Nayfeh and Mook, 2004), decreases as the stiffness of the system increases with the parameter q .

By observing the frequency responses and the phase-portrait diagram of the tuned harvesters depicted in Figure 5(c) and (d), respectively, it may seem that, relaxing the stiffness of the system is more suitable choice as it improves the response amplitude than increasing the stiffness. However, it should be noticed that the frequency bandwidth ( d Ω ) of such a harvester decreases as the resonance peak becomes more pronounced. Therefore, the resonance tuning approach may not be suitable in case of the excitations having a wide range of frequencies. In the following text, we present a brief discussion on the linear harvester models that are reported in the literature.

3.1.2. Models and studies related to linear harvester

A simple single degree-of-freedom (SDOF), lumped-mass dissipative model is shown in Figure 6. This model is applicable for piezoelectric, electromagnetic, and electrostatic transduction mechanisms (Williams and Yates, 1996). This model is described by equation (7), in which x ( t ) and x b ( t ) represent displacement of the mass and the base, respectively. The model assumes linear behavior for damping and stiffness of the system. The energy converted into electrical energy is equal to the power absorbed due to electrical damping and the maximum output power is obtained at resonant condition. This model predicts fairly close results for some electromagnetic harvesters, but needs to be modified for electrostatic and piezoelectric type harvesters. Also, in order to incorporate the interaction of the mechanical and electrical systems, the model needs improvements. Further variations of this model are presented along with the modifications to account for the electrical and few other parameters in White et al. (2001).

m x ·· + ( d m + d e ) x · + kx = − m x ·· b

(7)

OPEN IN VIEWER

Figure 6.

Schematic of a simple single degree-of-freedom dissipative model presented in Williams and Yates (1996).

Analysis and simulations of this SDOF dissipative model highlighted several key aspects. In order to lower the natural frequency of the system and bring it closer to the ambient excitation frequencies, it was needed to lower the stiffness and increase the mass of the system. Also, damping should be lowered and optimized to improve the quality factor of the system (Roundy et al., 2003). Considering these requirements the cantilevered structure with proof mass and piezoelectric patch turned out to be the most suitable configuration. The modeling of the cantilevered configuration involves combined modeling of the mechanical structure and the piezoelectric element. Based on the −31 or −33 mode of operation, the mechanical and electrical behaviors of the piezoelectric element are modeled using linear constitutive relations in 1-D form, as given by equations (1a) and (1b).

Considering the piezoelectric element represented by an equivalent circuit as shown in Figure 3 and a cantilever structure as a SDOF structure subjected to base excitation, coupled electromechanical equations are reported by Roundy et al. (2003). The linear model presented by them is similar to the one presented by Williams and Yates (1996) in many ways, except that it was described using three state-space variables due to presence of electrical equation. Using a similar approach, Roundy and Wright (2004) derived the electromechanical equations for resistive and capacitive loads considering the −31 mode of operation. In their lumped-mass model, the tip mass was considered as a point load, acting through the center of mass. The rotary inertia of the tip mass was also neglected primarily due to lower influence on the fundamental vibration mode. Using this model, the cantilevered design with PZT-5H was optimized, which was later validated through experiments. Another lumped-mass model was presented by Dutoit et al. (2005) considering −33 mode of operation. In this model, the mass of the piezoelectric element was also considered along with the tip mass. An analysis was presented for optimizing the power output using open circuit (OC) and closed circuit (CC) conditions for the piezoelectric element, the study showed existance of two optimal frequencies, the resonance and the anti-resonance frequency, for the system (Du Toit, 2005).

The linear SDOF models discussed above represented the cantilevered configuration as a lumped parameter spring-mass-damper model, which simplifies the modeling of electromechanical equations. The results of such SDOF models closely imitate the actual behavior of the harvesters, but only near to the fundamental vibration mode. Erturk and Inman (2008b) reported that the SDOF models with the harmonic base excitation may produce highly inaccurate results for both the transverse and the longitudinal vibrations of the cantilevered beam and bar structures. These inaccuracies are observed for certain values of the ratio of the tip mass to the beam/bar mass. In order to account for these inaccuracies, some correction factors are introduced to improve the performance of such models.

Further improvements over lumped parameter models were achieved through a distributed parameter model, which considers the effect of mass distribution over the entire length of piezoelectric patch and the cantilever beam. In this regard, a cantilevered structure bonded with a piezoelectric patch in the unimorph configuration and without any tip mass is shown in Figure 7. The piezoelectric patch covers the entire length of the cantilever beam and the electrical output from the patch is connected to a resistive load. This arrangement represents a distributed parameter system and was modeled using the Euler-Bernoulli beam theory (EBT) in Chen et al. (2006) and Eggborn (2003). The governing equation for the transverse deflection of a beam using the EBT is given by equation (8), as follows.

∂ 2 ∂ z 2 [ EI ( z ) ∂ 2 x ( z , t ) ∂ z 2 ] + m ( z ) ∂ 2 x ( z , t ) ∂ z 2 = f ~ ( z , t )

(8)

x ( z , t ) = ∑ i = 1 k ψ i ( z ) r i ( t ) = ψ ¯ ( z ) r ¯ ( t )

(9)

where, EI ( z ) represents flexural rigidity, m ( z ) represents mass per unit length, f ~ ( z , t ) represents external distributed load acting on the beam, and w ( z , t ) represents transverse displacement. Sodano et al. (2003, 2004b) presented an approximate distributed parameter model based on the Rayleigh-Ritz approximation given by equation (9), in which ψ ¯ ( z ) is the vector of assumed mode shape functions and r ¯ ( t ) is the temporal coordinate vector, with k terms. The model was derived using the variational principles and the response of the model for the first few modes was verified with the experimental response under the harmonic excitation. A similar modeling approach, without considering the tip mass, was followed by Du Toit (2005) that uses the EBT for representing the axial strain in the beam in terms of the displacement of the neutral axis and the distance from the neutral axis. This model was further validated through experiments in DuToit and Wardle (2007). Subsequently, the distributed parameter models that consider the tip mass effect were proposed. These models were based on the known mode shapes of the cantilever structure (Inman, 2017; Rao, 2011) and modal analysis (Du Toit, 2005; Oguamanam, 2003; Rama Bhat and Wagner, 1976; To, 1982). An approach which considers the fundamental vibration mode and the effect of the tip mass was presented by Lu et al. (2003) for MEMS applications.

OPEN IN VIEWER

Figure 7.

Schematic of a distributed parameter model.

An analytical method based on the EBT and considering the tip mass was presented by Erturk and Inman (2008a), in which the closed form solutions using the displacement influence functions for the transverse base displacement and small base rotation of the beam were reported. In their method, the internal strain rate damping and external air damping were represented separately to obtain better accuracy. Further, Elvin and Elvin (2009) have shown that the solution obtained by the Rayleigh-Ritz converges to the exact solution given in Erturk and Inman (2008b), when sufficient number of modes are considered. The closed form solutions using distributed parameter approach for piezoelectric bimorph cantilever in series and parallel arrangement were presented by Erturk and Inman (2009). Similar studies were reported which include the dynamic stiffness approach, the analytical modal mass method, influence of the beam shape and size, and influence of the physical size as well as position of the tip mass center (Bonello and Rafique, 2011; Goldschmidtboeing and Woias, 2008; Hu et al., 2007a; Kim et al., 2010; Lin et al., 2007; Wang and Meng, 2013).

A lumped mass model for a stack type piezoelectric energy harvesting circuit configurations with and without an inductor, as shown in Figure 8(a) and (b), respectively, was presented in Adhikari et al. (2009). The non-dimensional electromechanical equations are derived for the circuit configurations with and without an inductor. The base excitation x b ( t ) was assumed as Gaussian broadband white noise. Using linear random vibration theory (Papoulis and Pillai, 2002), the expressions for the mean power output E [ P ] for both the configurations were derived. It was observed from the expressions that the E [ P ] was directly proportional to the dimensionless time constant α , dimensionless coupling coefficient κ 2 , and inversely proportional to the damping ratio ζ . Additionally, for the case with an inductor, E [ P ~ ] was found to be directly proportional to the ratio of mechanical and electrical natural frequencies, denoted by β . Thus, the work presented in Adhikari et al. (2009) described the procedure to quantify the average power output from the harvester that is subjected to random Gaussian noise excitation and also provided a framework for simulating non-Gaussian random excitations. Several studies are reported on the performance of harvester subjected to deterministic (Dutoit et al., 2005; Erturk and Inman, 2008b; Sodano et al., 2004b) and random excitations (Adhikari et al., 2009; Soliman et al., 2008).

OPEN IN VIEWER

Figure 8.

Schematics of the stack type lumped mass PEH configuration (a) with an inductor connected in parallel with the load resistance and (b) the configuration without an inductor. Adapted from Adhikari et al. (2009).

In the above mentioned studies, the mechanical parameters such as the mass, natural frequency (or stiffness), damping, and coupling coefficient of the system were considered to be exactly known. Only the external excitation was considered to be a random process. However, in reality, the variations in these mechanical parameters are plausible and it can affect the performance of the harvester. The variability in these system parameters may exist due to error in estimation of natural frequency and damping characteristics, error due to modeling approximations (e.g., lumped mass model instead of distributed parameter model), variability introduced due to manufacturing and assembly processes, variability introduced due to operating conditions and prolonged use, and many more.

In regard to the above said notion, Ali et al. (2010) presented a semi-analytical approach as well as Monte Carlo Simulations (MCS), for studying performance of the linear harvesters by considering up to 20% uncertainty in the system parameters. A stack type lumped mass piezoelectric harvester model without an inductor was considered in their work. The mass of the system was assumed to be known exactly, whereas the variations in the stiffness, damping, and coupling coefficient were modeled as random variables. The electromechanical equations were presented in which the natural frequency ω n , and the damping coefficient ζ were considered as the random variables. The expression for the mean harvested power E [ P ] was derived, which involved double integral for the two random variables mentioned above. Similar expression considering additional uncertainty in the coupling coefficient would need a solution of a triple integral. Since closed form solutions of these expressions were difficult to obtain, the numerical results using MCS were presented. The results of the simulations under harmonic excitation by considering uncertainty only in ω n with Gaussian random process with unit mean and standard deviation of σ ω are shown in Figure 9(a). It was observed that the maximum mean power decreased by more than 75% as variability in the natural frequency was increased to 20%. When the uncertainties in both ω n and ζ were considered together, the variations in ζ didn’t affect the mean power considerably, as evident from Figure 9(b). Instead, a slight increase in the mean power was noticed as σ ζ was increased. Similarly, the variations in the coupling coefficient affected the standard deviation of power only for low values of σ ω , as observed from Figure 9(c). Thus, it was realized that the variability in the natural frequency has a pronounced effect compared to the variability in the damping and coupling coefficients.

OPEN IN VIEWER

Figure 9.

(a) Variation of the mean power as a function of standard deviation of the natural frequency ( σ ω ) and the normalized frequency ( Ω ). (b) Effect of variation of the standard deviation of the natural frequency ( σ ω ) and the damping coefficient ( σ ζ ) on the mean harvested power normalized by the maximum deterministic power. (c) Dependence of the standard deviation of the normalized harvested power on the standard deviation of the natural frequency ( σ ω ) and the standard deviation of the dimensionless coupling coefficient ( σ κ ). For all results, ω ¯ n = 670 . 5 rad s⁻¹, α = 0 . 8649 was considered. Reproduced from Ali et al. (2010) with permission. © IOP Publishing. All rights reserved.

Major limitations of these harvester designs are the lack of tunability, poor off-resonance and low frequency excitation response. For an exemplary energy harvester, it should overcome these limitations without compromising on the scale, simplicity of design, and harvested energy density. The most dominant drawback of the linear resonant harvester is the narrow frequency bandwidth over which the average power is harvested. Many researchers have presented various frequency tuning methods to improve the effective bandwidth of the linear PEH. Tuning methods may also introduce nonlinear stiffness into the harvester. The resonance frequency tuning can be achieved using mechanical methods (Eichhorn et al., 2008; Hu et al., 2007b; Leland and Wright, 2006; Loverich et al., 2008; Morris et al., 2008; Shahruz, 2006; Wu et al., 2008), magnetic methods (Challa et al., 2008; Reissman et al., 2009; Zhu et al., 2008), and piezoelectric methods (Peters et al., 2009; Roundy and Zhang, 2005; Wu et al., 2006). It was shown that, using the mechanical methods, up to 98% frequency tuning in the range of 50–230 (Hz) is achievable. Also, using magnetic and piezoelectric methods, a tunability of 37% and 30% was achieved, respectively. Soliman et al. (2009) developed a piecewise-linear oscillator for the electromagnetic harvester. Their architecture included a mechanical stopper due to which the stiffness of the system increases after certain amount of deflection. These and several other methods for resonance frequency tuning were reviewed in Ibrahim and Ali (2012), Tang et al. (2010), and Zhu et al. (2009). The effect of tuning methods on the potential well of the system can be visualized from Figure 10. Tuning of a linear harvester may introduce hardening or softening nonlinearities depending on the tuning approach used. Accordingly, a hardening or softening nonlinear mono-stable potential may govern the response of the harvester. Moreover, some tuning methods may also impart the bi-stability to the systems potential as shown in Figure 10.

OPEN IN VIEWER

Figure 10.

Effect of tuning methods on the potential function of the system. Frequency tuning of the harvester can change the linear potential function of the system into nonlinear mono-stable or bi-stable potential function.

3.3.1. Dynamic behavior of nonlinear mono-stable configurations

A nonlinear mono-stable potential can be expressed by equation (10) in the dimensionless form and corresponding nonlinear restoring force can be expressed as F nlm = K 1 u + K 3 u 3 . Here, subscript nlm stands for nonlinear mono-stable, K 1 = 1 represents the linear stiffness coefficient and K 3 represents the cubic nonlinearity coefficient. The nonlinearity introduced by K 3 is of a hardening nature for K 3 > 0 and of a softening nature for K 3 < 0 . This effect of the cubic nonlinearity coefficient K 3 on the potential well is illustrated in Figure 11(a). In either cases, the nonlinear potential has a single stable equilibrium position at u = 0 , marked by a solid green circle. However, for a softening case, two more unstable equilibrium positions, called as saddle points, appear on the either sides of the stable equilibrium position. These saddle points are marked by the hollow red circles in Figure 11(a).

U nlm ( u ) = 1 2 K 1 u 2 + 1 4 K 3 u 4

(10)

OPEN IN VIEWER

Figure 11.

(Color online) (a) Non-dimensional (ND) potential wells of the hardening and softening type of nonlinear mono-stable configurations along with the linear configuration. The solid green circle at u = 0 represents the only stable equilibrium position and the hollow red circles represent the saddle points of the softening type of potential well. (b) The phase-portrait diagram for a softening type ( K 3 = − 1 ) of nonlinear mono-stable potential well in ( u , u · ) space, showing the stable spiral, left, and right saddle points. The corresponding basins of attraction are shown in the green, blue and yellow colors.(c) The steady-state frequency response curves corresponding to the potential wells shown in Figure 10, depict the skew of the resonance peak and non-unique solution branches. The stable solution branches are drawn by the solid lines and the unstable solution branch is drawn by a dashed line.

The phase-portrait diagram, without external forcing and ζ = 0 . 04 , for a softening type of nonlinear mono-stable potential configuration ( K 3 = − 1 ), is shown in Figure 11(b). It can be verified that the stable equilibrium position, marked by a solid green circle at u = 0 , is a stable spiral and the two unstable equilibrium positions, marked by the hollow red circles, are saddle points (Strogatz, 1994). The stable (yellow and cyan colored arrowheads) and unstable (orange, green, and dark gray colored arrowheads) manifolds of both the saddle points are obtained using the backward and forward integration in time, respectively. The basins of attraction, which are shown in the green, blue, and yellow colors, correspond to the stable spiral, left saddle, and right saddle points, respectively. The trajectories whose initial conditions fall into the green colored basin have bounded solutions that will reach the stable fixed point. On the other hand, the trajectories whose initial conditions fall into the blue or yellow colored basins, that is the basins of the saddle points, have unbounded solutions that will go further away with in time. The phase-portrait diagrams of the linear ( K 3 = 0 ) and the hardening type ( K 3 > 0 ) nonlinear mono-stable harvesters shall depict only a stable spiral at u = 0 and all the trajectories will have bounded solutions that will reach the stable fixed point.

The frequency-response curves corresponding to the nonlinear mono-stable potential wells, shown in Figure 11(a), are presented in Figure 11(c). As an effect of hardening or softening nonlinearities, the resonance peak skews towards right or left, respectively, in comparison with the frequency-response of the linear harvester with K 3 = 0 . This skew is responsible for the widening of the frequency bandwidth and the presence of non-unique solution branches for certain range of excitation frequencies. As observed from Figure 11(c), a nonlinear mono-stable oscillator may have either one or three solution branches over the range of excitation frequencies. Out of the three solution branches, the largest and smallest amplitude solution branches are stable (represented by a solid line) and are known as the resonant and non-resonant branches, respectively. The middle solution branch is unstable (represented by a dashed line). The frequencies at which the unstable solution branch collides with the large and small amplitude stable solution branches are known as the jump-down and jump-up frequencies, respectively, and they also represent the saddle-node bifurcation points (Nayfeh and Balachandran, 2004). The response characteristic of the nonlinear mono-stable harvester are obtained for the excitations near primary resonance ( Ω ≈ 1 ). For a nonlinear potential with a cubic nonlinearity, the subharmonic ( Ω ≈ 3 ) and superharmonic ( Ω ≈ 1 / 3 ) resonances, which are also called as the secondary resonances, also exist (Kovacic and Brennan, 2011).

When an oscillator is subjected to quasi-statically increasing excitation frequency, its response follows the resonant solution branch up to the jump-down frequency (the higher saddle-node bifurcation point), at which the response amplitude jumps down to the non-resonant solution branch. Similarly, in case of the quasi-statically decreasing excitation frequency, the response amplitude follows the non-resonant branch up to the jump-up frequency (the lower saddle-node bifurcation point), at which the response amplitude jumps up to the resonant solution branch. This behavior is illustrated in Figure 11(c). The numerical responses of the forward and reverse frequency sweep simulations for a hardening ( K 3 = 5 ) and softening ( K 3 = − 1 ) type nonlinear mono-stable configurations are presented in Figure 12(a) and (b), respectively. For all these simulations, values of non-dimensional forcing f = 0 . 0459 , frequency sweep rate of 0.005 Hz/s, and damping ζ = 0 . 04 were considered. Also, analytical frequency-response showing stable and unstable branches are superimposed over the numerical response for comparison purpose in Figure 12(a) and (b). The analytical response of the linear harvester is shown to impart the sense of the decrease and increase in the amplitude of the skewed peak in the hardening and softening cases, respectively (Brennan et al., 2008). In the following text, we discuss several models and studies related to the nonlinear mono-stable configurations.

OPEN IN VIEWER

Figure 12.

(Color online) The numerical responses of the forward and reverse frequency sweep simulations (shown in blue and green colors, respectively) for (a) the hardening ( K 3 = 5 ) and (b) the softening ( K 3 = − 1 ) type of nonlinear mono-stable configurations with ( K 1 = 1 ). The stable and unstable analytical frequency response is superimposed over the numerical response. The frequency-response of the linear harvester ( K 3 = 0 ) is plotted for comparing the amplitudes of the skewed resonance peaks in the hardening and softening cases.

3.3.2. Models and studies based on nonlinear mono-stable configurations

A commonly studied method to widen the frequency bandwidth of the PEH is introducing nonlinear stiffness using the permanent magnets in various arrangements. The method is easy and does not require major reconfiguration of the linear PEH. The effect of such nonlinearity on the system’s performance was studied by several researchers (Daqaq et al., 2014; Mann and Sims, 2009; Quinn et al., 2011; Ramlan et al., 2010). Burrow and Clare (2007) designed an electromagnetic energy harvester that uses optimal magnetic material with varying reluctance and magnetic force over the displacement range. Thus, the varying reluctance forces combined with restoring forces of the cantilever beam results in nonlinear compliance characteristics (Burrow et al., 2008).

Quinn et al. (2007) presented numerical investigations about the capability of energy harvester with an attachment, based on essentially nonlinear elements. The arrangement, shown in Figure 13(a), comprised of a linear mechanical oscillator with an essential cubic nonlinear attachment (denoted by g ( z , z · ) ) is known as Nonlinear Energy Sink (NES) and it can passively engage in resonance with multiple modes of the host system for a wide range of excitation frequencies (Lee et al., 2005b; Vakakis et al., 2003). Numerical response of such a system was obtained under harmonic and band-limited excitation. Under direct excitation, as depicted in Figure 13(b), the performance with the essential nonlinearity exceeded that of the linear harvester by orders of magnitude for low frequency excitation. However, the response of the nonlinear harvester was observed to degrade beyond resonance as the forcing amplitude was increased. For random band-limited excitation, it was observed that the power generated by the nonlinear oscillator exceeded the linear harvester by several orders of magnitudes (refer Figure 7a in Quinn et al., 2007). When the attachment was excited at a tuned frequency that was lower than the natural frequency of the host system, the performance of the linear and nonlinear harvester was found comparable.

OPEN IN VIEWER

Figure 13.

(a) Schematic of an energy harvester with linear mechanical oscillator ( m s ) coupled with an attachment ( m a ), having electrical damping ( C e ) and nonlinear stiffness ( K a = K 3 ). (b) The average power harvested by linear (dashed line) and nonlinear attachment (solid line) under harmonic base excitation. Adapted from Quinn et al. (2007).

Mann and Sims (2009) designed a device in which the magnetic restoring forces were used to levitate an oscillating magnet at the center. A mathematical model was derived considering the harmonic base excitation that resembled to the Duffing type equation and the frequency response was obtained using the method of multiple scales (Nayfeh and Mook, 2004). The response of the system showed large oscillations caused by nonlinear restoring forces over a wide frequency range. Also, maximum power was generated at a frequency away from the linear resonance frequency. However, introduction of a hardening nonlinearity causes widening of bandwidth in one direction only. Stanton et al. (2009) presented a cantilevered harvester structure with a tip magnet mass and another fixed magnet was attached to the frame, with a possibility of bi-directional tuning of resonance. This device was suitable for harvesting energy from the excitation with slowly varying frequencies. In their device, it was possible to tune the nonlinear magnetic interaction around the tip mass so as to impart hardening or softening nonlinearity. It was observed that the increase in the frequency bandwidth for the hardening response was lesser than the softening response.

Barton et al. (2010) used electromagnetic transduction mechanism with the high magnetic permeability materials. This resulted in an improved coupling between mechanical and electrical domains leading to better energy extraction. The model was studied under pure harmonic and a narrow band Gaussian white noise excitation. The model simulations and experiments revealed the wide-band characteristics of the harvester under pure harmonic and a narrow band Gaussian white noise excitation. It was possible to further increase the bandwidth using the varying resistive load. Also, the presence of superharmonic resonances, another benefit of such a nonlinear system, was observed as shown in Figure 14.

OPEN IN VIEWER

Figure 14.

Occurrence of superharmonic resonances for electromagnetic energy harvester with varying electrical loads. Harvester response amplitude is shown at resonance peak in multiple of excitation amplitude above each peak. Reproduced from Barton et al. (2010), Copyright © 2010, with permission from AMSE.

Masana and Daqaq (2011a) proposed an axially loaded beam harvester subjected to transverse excitation as shown in Figure 15(a). An electromechanical model was developed using the EBT and incorporating nonlinear strain-displacement relations. The governing equations were derived using the Hamilton’s principle, considering the work done by non-conservative forces, namely, transverse external excitation f ( t ) , mechanical damping forces, external axial force P , and in dissipating electrical energy into the resistive load R . For simplification, terms such as longitudinal inertia, flexural rigidity, and coupling terms that have negligible influence compared with the axial rigidity, were neglected and single-mode approximation was used. Only pre-buckling configuration, i.e. without exceeding critical buckling load, was considered to avoid debonding and depolarization of the piezoelectric patch due higher stresses. Using the method of multiple scales, the analytical expressions for the steady-state response amplitude, the voltage drop across a resistive load, and the output power were obtained. The tunability of harvester with varying axial load is shown in Figure 15(b). It was concluded that the nonlinearity caused by the application of axial load improves the frequency bandwidth by enhancing the electromechanical coupling and also facilitates the tuning effects.

OPEN IN VIEWER

Figure 15.

(Color online) (a) Representation of an axially loaded beam harvester under transverse excitation. (b) Tuning of first natural frequency with varying axial load. Reproduced from Masana and Daqaq (2011a), Copyright © 2011, with permission from AMSE.

Further, considering the fact that the ambient vibrations can be multi-directional, Daqaq et al. (2009) presented a lumped parameter nonlinear model for parametrically excited cantilever beam with piezoelectric harvesting circuit. Using multiple scales method, the authors obtained approximate analytical expressions for the steady-state beam deflection, voltage drop, and output power. Also, the effects of the electromechanical coupling coefficient θ , load resistance R , and excitation amplitude F on the output power were studied. Unlike in the case of directly excited cantilever beam, where the beam oscillates even for the small excitation amplitudes, parametrically excited beam oscillates only when the forcing exceeds a critical forcing F cr . This is because the non-trivial solution for the response amplitude exists only for certain range of excitation amplitudes and frequencies. It was observed that the range of detuning parameter σ , which represents the nearness to the resonance frequency, decreases as the coupling coefficient θ increases. Thus, a critical value of coupling coefficient was identified.

Parametric excitation through external forcing, also known as heteroparametric excitation, required a threshold acceleration level to activate parametric resonance, thus making it inconvenient for practical harvesting applications. In this regard, Jia et al. (2012) have extensively investigated the ways to overcome the initiation threshold amplitude using various design approaches (Jia et al., 2013) and passive techniques (Jia et al., 2014). Further, a multi-DOF compound resonator exhibiting autoparametric resonance was explored by Jia and Seshia (2014) and Jia et al. (2018) for amplification of low amplitude excitations and was found more useful as it lowered the initiation threshold.

The nonlinear harvester configurations discussed above are based on the mono-stable Duffing oscillator equation with hardening or softening characteristics. Daqaq (2010) studied the response of such uni-modal Duffing type harvesters under white Gaussian and colored random excitations. An electromagnetic harvester with nonlinear restoring force F nl = x + β x 3 , with β > 0 as the stiffness nonlinearity coefficient, was considered. Analytical expressions were presented for the expected value of the mean square velocity, mean square current, and the expected value of power under Gaussian white excitations. The mean output power was unaffected by the variations in stiffness nonlinearity, but, it was affected by the electromechanical coupling coefficient, excitation spectral density, and effective damping of the system. Thus, to improve the mean output power, it was suggested to explore other types of nonlinearities such as nonlinearities that depend on velocity, displacement and velocity together (i.e. damping type nonlinearities), and inertia nonlinearities.

Subsequently, approximate expressions using Van Kampen expansion (Rodríguez and Kampen, 1976) were presented to obtain the response of the harvester under colored or narrow-band excitations. It was realized from Figure 16(a) that for large value of excitation bandwidth γ , as that of white noise excitations, when excitation’s center frequency was equal to the natural frequency of the harvester ( ω c = ω n ), the mean square velocity and hence the power output were not influenced by the variations in the nonlinearity (represented by β ). However, the output power decreased as nonlinearity was increased for smaller bandwidths. Thus, for the colored excitations, nonlinearity reduces the performance of the harvester for ω c = ω n and best performance was achieved by the linear case with β = 0 . Moreover, it can be noted from Figure 16(b), that the performance of the harvester increases with ω c for a constant value of the nonlinearity coefficient β . Tuning of the harvester above their natural frequencies with hardening type stiffness nonlinearities improved the performance. However, compared to the linear harvester with β = 0 and ω c = ω n , the maximum value of the power obtained for the nonlinear harvester with β = 0 . 5 and β = 1 at optimal ω c was found to be 5.5 and 7.5 times lower, respectively. Thus, it was concluded that, the stiffness type of nonlinearities reduce the performance of the harvester subjected to colored excitations with different bandwidths and excitation’s center frequencies.

OPEN IN VIEWER

Figure 16.

Effect of nonlinearity coefficient β on the mean square velocity for (a) different values of excitation bandwidth γ when ω c = ω n = 1 and for (b) different values of excitation center frequency ω c when γ = 1 . Reprinted from Daqaq (2010), Copyright © 2010, with permission from Elsevier.

As realized while discussing the dynamic behavior of the nonlinear mono-stable harvesters, there exist two stable and one unstable solution branches over certain range of excitation frequencies. Out of the two stable solutions, the large amplitude or resonant branch of solutions produces higher power output. However, in practical conditions the ambient excitations being purely random in nature seldom attain the large amplitude solution, unless the excitations have sufficiently high amplitude. Although, several studies regarding the response of nonlinear harvesters under random excitations have confirmed the enhancement in the frequency bandwidth, the mean power produced was relatively lower than that of the linear harvesters and nonlinear harvesters under harmonic excitation (Barton et al., 2010; Daqaq, 2010; Mann and Sims, 2009; Quinn et al., 2007; Stanton et al., 2009). The reason for such an inferior performance of the nonlinear harvester can be attributed to the inability of the device to attain and sustain the large amplitude solution.

Sebald et al. (2011b) suggested a method called Fast Burst Perturbation (FBP) to facilitate the displacement jump from the small amplitude solution to the large amplitude solution, using external adapted excitation comprised of the pulses of voltage or current. A dimensionless lumped parameter model was presented with three independent variables, namely, mechanical quality factor, electromechanical coupling, and electrical loading reduced time constant. Harmonic balance method (HBM) was used to solve the coupled electromechanical model and obtain the frequency response of the harvester. The frequency response curves for a nonlinear harvester under varying electrical loading are shown in Figure 17(a). It was observed that, except for open and close circuit conditions with extreme values of electrical loading, the jump occurs at low frequencies for non-extreme values of electrical loading due to high value of electromechanical coupling k 2 . For low values of the electromechanical coupling factor, the displacement amplitude b 0 was observed to be independent of the electrical loading. The maximum normalized output power P max _ norm , corresponding to an optimum value of electrical loading, was observed to be independent of the excitation amplitude a 0 , as can be seen in Figure 17(b). Similarly, the P max _ norm converged to a constant value for large values of the coupling factor k 2 , indicating the limit of maximum power that can be harvested. Thus, similar power output as that of the linear case, but with the increased frequency bandwidth was achieved in this study.

OPEN IN VIEWER

Figure 17.

(a) Frequency response showing displacement amplitude ( b 0 ) normalized with excitation amplitude ( a 0 ) with model parameters a 0  = 0.1, and k 2  = 20%. The results of various electrical loading cases ranging from γ  = 0.01 to 100 are represented by different curves. (b) The variation of the maximum normalized output power P max _ norm with frequency for excitation levels a 0 = 10 − 3 , 10 − 2 , 10 − 1 and k 2  = 5%. Reproduced from Sebald et al. (2011b) with permission. © IOP Publishing. All rights reserved.

The harmonic balance solution provided information about the two stable solutions at a given frequency, but how to attain these solution was unclear. Using the sweep this was addressed. When excited with a sine excitation in the tri-valued solution region, the FBP was applied to cause the steady-state solution jump from a lower solution state to a higher solution state. It was noted that there exists a limiting value of frequency above which no increase in the FBP amplitude produces the jump to a higher solution. Further, it was shown that the FBP can also be applied through electrical voltage on the piezoelectric element. The energy spent during the electrical voltage burst could be recovered back after few oscillations of higher state solution. It was shown that the FBP aids in attaining the large amplitude solution, but the jump phenomenon was not assured every time and the events of switching back to the lower solution state were also observed as the time progressed. Moreover, energy gain was lower for the broadband noise signal compared to the colored noise signal, suggesting that the FBP technique was most suited for the narrow-band or colored noise excitation with moderate amplitude of the FBP.

In Sebald et al. (2011a), authors presented an extended model considering the nonlinear losses along with experimental results. Experiments with the harmonic excitation with 1 g amplitude have shown that the linear harvester produced the maximum power of 2.2 mW with a half-power bandwidth of 1.4 % . Conversely, the nonlinear harvester registered a maximum power of 0.92 mW with a half-power bandwidth of 7.75 % . Further, the performance of linear harvester was better than the nonlinear harvester in the region where the application of the FBP was not possible. However, the nonlinear harvester was found more suitable due to its wide-band characteristics for the narrow-band noise as well as the harmonic excitation along with the FBP method.

A comparison among the modeling approaches, such as the nonlinear distributed parameter model using Galerkin discretization, classical mode shapes, lumped parameter model, and the experiments was presented by Abdelkefi and Barsallo (2014). Now, as we have gone thorough various studies and realized about several characteristics of the nonlinear mono-stable configuration, we discuss further about the nonlinear restoring forces causing the bi-stability.

3.4.1. Dynamic behavior of bi-stable potential well configuration

A bi-stable potential well configuration can be expressed in the dimensionless form by equation (11) with a nonlinear restoring force of the form F bi = − K 1 u + K 3 u 3 . Here, the negative linear stiffness coefficient ( K 1 < 0 ) imparts a softening behavior near u = 0 , which is countered by the cubic nonlinearity coefficient ( K 3 ) of a hardening nature for larger values of u . The resulting potential well configuration has two wells that are separated by a potential barrier B 0 = K 1 2 / 4 K 3 , as shown in Figure 18(a). Such a potential function has three equilibrium positions, an unstable equilibrium position (a saddle point) at u = u 0 = 0 along with the two stable equilibrium positions at u = u 1 , 2 = ± K 1 / K 3 , and therefore known as the bi-stable potential well configuration. As illustrated in Figure 18(a), for K 1 = − 1 , as the cubic nonlinearity coefficient K 3 increases, the potential barrier B 0 and the distance between two stable equilibria W 0 decreases, creating shallower potential wells.

U bi ( u ) = − 1 2 K 1 u 2 + 1 4 K 3 u 4 + C 0

(11)

OPEN IN VIEWER

Figure 18.

(Color Online) (a) The effect of the cubic nonlinearity coefficient K 3 on the potential barrier B 0 and the distance between two stable equilibria W 0 , for K 1 = − 1 is illustrated here. The stable and unstable equilibrium positions are marked by the solid green circles and a hollow red circle, respectively. (b) The phase-portrait diagram for a bi-stable potential case with K 1 = − 1 and K 3 = 1 in ( u , u · ) space, showing the two stable spirals at u = ± 1 and a saddle point at u = 0 . The basins of attraction of the two stable fixed points are shown in the green and peach colors.

The phase-portrait diagram, without external forcing and ζ = 0 . 04 , for a bi-stable potential configuration with K 1 = − 1 and K 3 = 1 , is shown in Figure 18(b). It can be verified that the stable equilibrium positions, marked by the solid green circles at u = u 1 , 2 = ± 1 , are the stable spirals and an unstable equilibrium position, marked by a hollow red circle at u = u 0 = 0 , is a saddle point. The stable (yellow and cyan colored arrowheads) and unstable (orange and green colored arrowheads) manifolds of the saddle point are obtained using the backward and forward integration in time, respectively. The basins of attraction, which are shown in the green and peach colors, correspond to the left and right stable spirals, respectively. Unlike the case of softening type of nonlinear mono-stable potential as shown in Figure 11(b), all the trajectories with an initial condition in ( u , u · ) space have bounded solutions that will reach either of the stable fixed points forward in time.

An oscillator with a bi-stable potential well (hereafter called as the bi-stable oscillator) may exhibit intrawell, or interwell (or cross-well), or a blend of both the oscillations, depending on the strength of the external excitation. Under the low strength excitation, the intrawell oscillations (IW) with a softening type of frequency response are observed due to the negative linear stiffness coefficient K 1 (Kovacic and Brennan, 2011; Stanton et al., 2012). As the excitation strength increases sufficiently to overcome the potential barrier B 0 , both the intrawell (IW) and interwell (CW) oscillations can be observed. Such a nonlinear oscillator with a bi-stable potential exhibits variety of phenomenons related to the nonlinear dynamics. Based on the frequency and amplitude of the external excitation, both the intrawell and interwell oscillations may follow periodic, chaotic or co-existing attractor solutions (Nayfeh and Balachandran, 2004). Such an enriched nonlinear dynamical behavior of the bi-stable oscillator can be visualized through Figure 19, in which the non-dimensional numerical frequency and amplitude sweep responses of the bi-stable oscillator with K 1 = − 1 and K 3 = 1 are presented.

OPEN IN VIEWER

Figure 19.

(Color Online) (a, b) The non-dimensional numerical forward and reverse frequency sweep responses of the bi-stable oscillator under the harmonic excitation with an amplitude level of A = 35  m/s ². Here, n P and CH implies the period-n and chaotic oscillations, respectively. The intrawell and interwell oscillations are denoted by IW and CW, respectively. A period doubling bifurcation point is denoted by “pd.” (c, d) The non-dimensional numerical forward and reverse amplitude sweep responses of the bi-stable oscillator under the harmonic excitation with a forcing frequency of ω = 6  Hz. In (c), the magnified inset view depicts the mixed periodic and chaotic intrawell oscillations (mixed IW). In (d), the magnified inset view depicts the period-2 intrawell (2P IW) oscillations in between the chaotic (CH) oscillations. In all figures, the orange and yellow colored dots belong to the stroboscopic bifurcation map of the numerical forward and reverse sweep responses, respectively.

The numerical forward and reverse frequency sweep responses of the bi-stable oscillator, under the harmonic excitation with an amplitude A = 35  m/s ², is shown in Figure 19(a) and (b), respectively. In the low excitation frequency region ( Ω ≈ 0 to 0 . 8 ), both the forward and reverse sweep responses exhibit chaotic (CH) and period-1 interwell (1P CW) oscillations. This is followed by a hardening type of large-orbit period-1 interwell oscillations (1P CW) up to Ω ≈ 1 . 3 . In the frequency region Ω ≈ 1 . 3 to 2 . 3 , in addition to the large-orbit period-1 interwell (1P CW) oscillations, the period-5 interwell (5P CW), chaotic (CH), and period-1 intrawell (1P IW) oscillations co-exist. In the frequency region Ω ≈ 2 . 3 to 3 . 25 , the period-3 interwell oscillations (3P CW) are observed during the forward frequency sweep simulation, whereas the period-2 intrawell (2P IW) oscillations are observed after a period doubling bifurcation (pd), during the reverse frequency sweep simulation.

The numerical forward and reverse amplitude sweep responses of the bi-stable oscillator, under the harmonic excitation with the forcing frequency of ω = 6  Hz, is shown in Figure 19(c) and (d), respectively. The forward amplitude sweep response shows the mixed periodic and chaotic intrawell oscillations (Mixed IW) and a period doubling (pd) bifurcation at A ≈ 31  m/s ². This response is followed by the chaotic (CH) oscillations in the region A ≈ 35 to 40  m/s ². Similarly, the reverse amplitude sweep response shows the chaotic (CH), period-2 intrawell (2P IW) and the mixed intrawell (Mixed IW) oscillations. Finally, it is to be noted that the numerical sweep responses are presented here so as to provide a glimpse of the vivid and enriched nonlinear dynamical behavior of the bi-stable oscillator under various excitation scenarios. Now, in the following text, several models and studies related to the bi-stable potential configuration are discussed.

3.4.2. Models and studies based on bi-stable potential well configuration

The class of nonlinear harvesters based on the bi-stable Duffing oscillator equation has been explored widely in the literature. We begin the discussion with a piezoelectric inverted pendulum model with a tip magnet (Cottone et al., 2009). The ground vibration force was imitated by a magnetic excitation produced by two small magnets attached near to the base of the pendulum. An external magnet, having opposite polarity as that of the tip magnet, was fixed on the frame just opposing the tip magnet with a certain gap distance Δ . It was shown that the dynamic behavior of the system was affected by the distance Δ . When the distance Δ was more than a critical distance Δ c (i.e. Δ > Δ c ), the system possessed a mono-stable potential with only one equilibrium point at trivial position. For Δ < Δ c , two new symmetric equilibrium positions were observed as shown in Figure 19, transforming the mono-stable potential into a bi-stable potential at Δ c . A pronounced barrier between the two symmetric wells of the bi-stable potential was observed for Δ values smaller than Δ c (i.e. Δ < < Δ c ). When the system was subjected to the Gaussian random excitation, the barrier confined the quasi-linear oscillations of the pendulum within one of the wells and the performance of the harvester was comparable to a linear harvester. For the values of Δ near to Δ c , but lower than Δ c , the RMS displacement X rms attained its maximum value as shown in Figure 20(b). In this condition, the pendulum showed highly nonlinear behavior with noise assisted interwell jumps, causing the large amplitude displacements. Thus, it was realized that the bi-stable potential can significantly improve energy harvesting performance of the oscillator.

U ~ ( x ) = − 1 2 a x 2 + 1 4 b x 4

(12a)

b M = a 2 4 D log τ p

(12b)

However, Gammaitoni et al. (2009) shown that the analysis presented in Cottone et al. (2009) is a special case of a more general behavior. The improvement in the performance of the nonlinear oscillators is not only peculiar to the bi-stable potential case, but also applicable for the case of the nonlinear mono-stable or other multi-stable potential configurations. In their model, a quartic bi-stable potential U ~ ( x ) was considered for analysis, as given by equation (12a), in which a and b are linear and nonlinear stiffness coefficients, respectively. The potential U ~ ( x ) becomes mono-stable and bi-stable for a ≤ 0 and a > 0 , respectively. The system was subjected to the random forcing with Gaussian distribution, having zero mean and unitary standard deviation. It was observed that the maximum value of the RMS voltage ( V rms ) was recorded in the bi-stable region with a > 0 , instead of the mono-stable region with a ≤ 0 . In the bi-stable region, the oscillator motion was comprised of the intrawell (IW) and interwell (CW) oscillations as reported in Cottone et al. (2009). The average rate of large amplitude random crossings during the interwell oscillations, which is also known as the Kramer’s rate (Kramers, 1940), is proportional to exp ( − Δ U / D ) . Here, D = σ 2 τ represents the noise intensity and τ is the time constant. The value of b for maximizing the V rms for a given noise intensity was given by equation (12b). Thus, with a fixed value of b = b M , the maximum of X rms and V rms were functions of a only with a > 0 . Also, in order to verify whether the bi-stability condition was the only necessary condition for such a performance improvement, a mono-stable potential function of the form U ~ ( x ) = a x 2 n was considered with a > 0 and n = 1 , 2 , . . . . It was observed that, for the values of a greater than a threshold value a th , the nonlinear mono-stable potential exhibited improved performance than the linear mono-stable case. Thus, it was concluded that the benefits of the nonlinearity are not only limited to the bi-stable systems, but also are applicable to the nonlinear mono-stable systems with carefully selected nonlinearity coefficients.

OPEN IN VIEWER

Figure 20.

(a) Variation of potential function U ( x ) of the inverted pendulum for different values of the gap distance Δ . (b) Variation of the RMS displacement X rms of the oscillator with the distance Δ for three different levels of the standard deviation of the externally applied noise excitation σ . Reprinted figures with permission from Cottone et al. (2009). Copyright © 2009 by the American Physical Society. DOI: 10.1103/PhysRevLett.102.080601.

The research discussed above and few other studies (Mann and Owens, 2010; Shahruz, 2008; Tang et al., 2012) have shown that, implementing nonlinear potential to improve the broadband characteristic of a harvester over frequency tuning methods is sensible and effective. Continuing further exploration in this regard, Erturk et al. (2009) used a well known magneto-elastic oscillator configuration, the Moon and Holmes beam (Moon and Holmes, 1979). Such a bi-stable configuration has a sixth order magneto-elastic potential and it exhibits strange attractor motion. A lumped parameter, Duffing type electromechanical equation similar to equation (4) was considered. The nonlinear restoring force corresponding to the bi-stable potential was given as F nl = − 1 2 u ( 1 − u 2 ) , which ensured a unit value for the linearized dimensionless natural frequency of the intrawell oscillations about the two stable equilibrium positions. The numerical voltage response showed the chaotic oscillations, which later transformed into the large amplitude periodic interwell oscillations as a result of the increment in the forcing amplitude or by an impulse imparted with a nonzero velocity initial condition. This observation revealed that a system under low strength external excitations and having chaotic motion can be pushed into the large amplitude periodic orbit by applying a small impulse condition.

Subsequently, the broadband frequency response of the piezo-elastic structure (without the base magnets), with a linear mono-stable potential, was compared with the response of the piezo-magneto-elastic structure with a bi-stable potential function. Both the configurations were studied under the similar base excitation amplitude of 0.35 g and the frequencies ranging from 4.5−8 Hz. It was observed that (see Figure 21), for the frequencies lower than 10.6 Hz (the post-buckled natural frequency of the bi-stable case), the RMS voltage output of the piezo-magneto-elastic configuration was up to three times higher than that of the piezo-elastic configuration. On the contrary, the piezo-elastic configuration showed higher RMS voltage output only near its resonance frequency of 7.4 Hz. Thus, an order of higher power output was harvested using the piezo-magneto-elastic structure over a wider frequency range. Additionally, in Erturk and Inman (2011) a brief discussion on other transduction mechanisms, namely, electrostatic, electromagnetic, and magnetostrictive, based on the bi-stable magneto-elastic structure was presented.

OPEN IN VIEWER

Figure 21.

(a) Input excitation in the form of RMS acceleration of 0.35  g applied at various frequency values. (b) RMS voltage output from the harvesters under open-circuit condition, for various excitation frequency values. Reproduced from Erturk et al. (2009), with the permission of AIP Publishing.

A distributed parameter model based on the EBT for a discontinuously laminated piezoelectric cantilever beam was presented by Stanton et al. (2010b). The cantilever beam was attached with a tip magnet mass, opposite to which a permanent magnet was fixed to the frame. As the distance s between the tip magnet and the fixed magnet was varied, the linear potential of the device was transformed into a bi-stable potential, as a result of the saddle-node bifurcation (Nayfeh and Balachandran, 2004; Strogatz, 1994). A discretization strategy that involved superpositioning of the beam mode shapes was used to model the cantilever beam whose partial length was laminated. The piezoelectric bimorph laminates covering the partial length of the beam were connected in series connection modes (Koplow et al., 2006; Stanton and Mann, 2010). The superpositioned mode shape was then substituted in the energy formulation to obtain the Lagrangian functional (Hagood et al., 1990; Wang and Cross, 1999), following which coupled electromechanical equations were derived using the Euler-Lagrange conditions (Lanczos, 2012), considering the dynamics of the first mode shape only.

Subsequently, using the forward and reverse frequency sweep (0−20 Hz) and amplitude sweep ( 0 − 30  m/s ²) simulations, as discussed in Section 3.4.1, various nonlinear dynamical characteristics of the bi-stable harvester such as periodic, multi-periodic, chaotic and co-existing attractor motions were illustrated thoroughly. The methods of mechanical and electrical impulses were suggested to achieve the transition from the chaotic or low amplitude periodic attractor to the high amplitude periodic attractor for improved power harvesting. For large number of periods, the average power over one forcing period was given by equation (13). Here, k denoted the number of periods and T = 2 π / Ω denoted an oscillation period. Equations (13) was solved for sufficient number of periods to achieve the convergence to the average value of the power. It was observed that the average power harvested in chaotic motion was more than that harvested in periodic motion, as shown in Figure 22(a).

P ⌢ = 1 k T ∫ 0 k T V ( t ) R L d t

(13)

OPEN IN VIEWER

Figure 22.

(a) Plot explaining the method of convergence to the average power value over the number of periods N . The flat line represents the convergence of the period-5 oscillation and the wavy line represents the convergence of the chaotic oscillations. (b) The occurrence of symmetry breaking bifurcation with distance s as the bifurcation parameter. Stable and unstable theoretical solutions are represented by solid and dashed lines, respectively. Experimental equilibrium positions are represented by hollow circles. Reprinted from Stanton et al. (2010b). Copyright © 2010, with permission from Elsevier.

The effect of varying separation distance s on the nonlinear response of harvester was also reported. When the fixed magnet was pushed towards the tip magnet, the voltage response was observed to be superior over a range of separation distances, compared to the case in which the fixed magnet was drawn away from the tip magnet. Further, symmetry breaking bifurcations were identified using the experiments, instead of the perfect pitchfork bifurcations as shown in Figure 22(b). The experimental imperfections were attributed for this deviation, which included small magnet misalignment, dipole model assumption, and other uncontrollable errors.

Next, Masana and Daqaq (2011b) presented a relative performance study on the mono-stable and bi-stable potential configurations of a axially loaded beam harvester (similar to the one used in Masana and Daqaq (2011a)), subjected to harmonic excitations. The system has a mono-stable potential in the pre-buckling configurations and a bi-stable potential in the post-buckling configurations. When both the configurations were tuned at the primary resonance frequency of f 0 = 65  Hz (i.e. relatively deeper potential wells as shown in Figure 23(b)), the bi-stable harvester performed better only when the excitations were sufficiently large to cause the interwell oscillations. Although, the bi-stable design exhibited large amplitude chaotic and periodic response for a certain range of excitation frequencies, in overall sense, the performance of the mono-stable harvester was found to be consistent with the periodic motion. However, when both the designs were tuned at f 0 = 45  Hz (i.e. shallow potential wells as shown in Figure 23(a)), the interwell oscillations were observed for the bi-stable harvester even with the lower excitation levels. Still, the performance of both the configurations were comparable due to the similarities in the shape of the potential functions. Due to the presence of secondary (sub/super harmonic) resonances, the bi-stable design performed better in the vicinity of these resonances. For higher excitation levels, the bi-stable design exhibited a combination of the chaotic, periodic, and multi-periodic responses in the presence of primary and secondary resonances, and hence outperformed the mono-stable harvester. Therefore, it was concluded that the mono-stable and bi-stable designs with shallow potential wells are suitable for the low strength excitation levels. On the other hand, the bi-stable design with deeper and widely separated wells is preferred under high excitation amplitude levels with lower frequencies.

OPEN IN VIEWER

Figure 23.

Plot of the potential energy function of the axially loaded beam harvester for the frequencies (a) f 0  = 45 Hz and (b) f 0  = 65 Hz. The bi-stable and mono-stable cases are represented by the solid (black) line and the dashed (orange) line, respectively. The solid (black) circles represent the stable equilibrium positions of the bi-stable potential well. Reprinted from Masana and Daqaq (2011b). Copyright © 2011, with permission from Elsevier.

McInnes et al. (2008) proposed a nonlinear harvester model based on the stochastic resonance phenomenon, in order to enhance the performance of a bi-stable harvester. The stochastic resonance is observed, when a weak periodic input signal is assisted by some noise signal, enhancing the output signal of the system. External excitation should have sufficient strength to overcome the potential barrier, so that the interwell oscillations take place to generate more energy. The probability of the crossings of the potential barrier for a randomly excited system is determined by the Kramer’s rate ( r k ) (Kramers, 1940). For a system subjected to a noise excitation, if the height of the potential barrier can be controlled using a weak periodic signal, such that the average time ( 1 / r k ) of the noise driven interwell transitions synchronizes with the half period of the periodic signal, the interwell transitions would occur frequently, leading to the stochastic resonance. Thus, the noise input actually enhances the weak periodic forcing and the output signal of the system is amplified. Of course, the net input provided to the system has increased, but there is a significant improvement in the harvested power output of the system. A thorough discussion on the stochastic resonance with several applications can be found in Gammaitoni et al. (1998).

The model presented by McInnes et al. (2008) consisted of a beam clamped at both the ends in the post-buckled conditions (see Figure 24(a)), resulting in a system with a bi-stable potential. The weak periodic axial forcing caused the compression and relaxation of the beam, leading to the controlled variation of the potential barrier height. This beam was then excited with a noise excitation Q ( t ) = f ( X ( t ) ) , causing the stochastic resonance. The time dependent bi-stable potential U ( u , t ) of the beam was given by equation (14), in which u represented the dimensionless position coordinate, μ was used as a measure of the compressive load, and η and ω represented the amplitude and the frequency of the weak periodic forcing, respectively. Figure 24(b) shows the plot of U ( u , t ) at various time instances. The noise excitation Q ( t ) emulated the noise signal from a single cylinder engine, which was composed of several harmonics summed along with a strong white noise component, having zero mean and unit variance.

U ( ξ , t ) = − 1 2 μ ( 1 − η cos ( ω t ) ) u 2 + 1 4 u 4

(14)

OPEN IN VIEWER

Figure 24.

(a) A clamped-clamped beam model in the post-buckled condition was represented by a single DOF beam model with a single lumped mass and the spring-damper system. The distance A − A ′ was controlled using the periodic frequency ω . (b) Variation of the bi-stable potential well of the system due to the weak periodic forcing at different time instances. The model parameters are μ = 1 and forcing amplitude η = 0 . 7 . Reprinted from McInnes et al. (2008). Copyright © 2008, with permission from Elsevier.

The Kramer’s rate can be used to determine the periodic forcing frequency which is required to instigate the stochastic resonance phenomenon. When subjected to a noise excitation, the response of such a tuned system (with ω =1.2, η =0.7, and μ =1) exhibiting the stochastic resonance is shown in Figure 25(a). The system exhibits poor performance in the absence of the periodic forcing ( η = 0 ) (and hence the stochastic resonance), as shown in Figure 25(b). Figure 25(c) shows the comparison of the net energy output of the system with and without the periodic forcing. Thus, the benefits of the stochastic resonance to enhance the performance of a bi-stable energy harvester were demonstrated with a simple conceptual model.

OPEN IN VIEWER

Figure 25.

Response of a tuned system ω = 1 . 2 , showing the stochastic resonance with the forcing amplitude of (a) η = 0 . 7 , (b) η = 0 . (c) Plots of the energy output from the system with η = 0 . 7 and η = 0 are indicated by the solid and dashed lines, respectively. Reprinted from McInnes et al. (2008). Copyright © 2008, with permission from Elsevier.

The cantilever type of the bi-stable harvesters discussed above make use of the base magnets and a tip magnet to introduce the nonlinear stiffness into the system. Sometimes, use of a tip magnet in the MEMS devices present challenges during the design of the devices. Ferrari et al. (2011) proposed a bi-stable harvester model as shown in Figure 26(a), which uses a ferromagnetic cantilever beam and a single base magnet, producing the magnetic interaction. The beam was screen printed with a low curing temperature PZT on both the surfaces, forming a bimorph configuration. Figure 26(b) shows FE simulation results for the dependence of the horizontal and the vertical components of the attractive force on the transverse tip displacement. It was shown that the transverse tip displacement was caused mainly due to the vertical component of the attractive force. Furthermore, the transition of the potential of the system from a mono-stable to a bi-stable state was realized as the distance d was varied. The simulations based on a sine sweep have revealed the existence of the nonlinear resonance peak that was skewed towards the lower frequencies, indicating the softening behavior. The system was also subjected to a white noise excitation, for which the frequency bandwidth was centered at a frequency lower than the natural frequency of the harvester. The results of sine sweep simulations and the experiments under white noise excitation indicated a drastic increase in the RMS voltage output under the bi-stable potential case.

OPEN IN VIEWER

Figure 26.

(a) Schematic representation of the ferromagnetic cantilever beam having the magnetic interaction with a fixed permanent magnet. The resultant attractive force at the cantilever tip has components F Av and F Ah along the transverse and axial direction of the beam, respectively. (b) Variation of the component forces F Av and F Ah with the transverse tip displacement x , when simulated using COMSOL Multiphysics software. The magnetic induction field B experienced at the tip of the cantilever is shown in the inset. Reprinted from Ferrari et al. (2011). Copyright © 2011, with permission from Elsevier.

The bi-stable piezo-magneto-elastic beam presented by Erturk et al. (2009) (see Figure 2) was further studied by Litak et al. (2010). They considered the Gaussian white noise excitation with a zero mean signal. During the simulations using Runge–Kutta–Maruyama method (Naess and Moe, 2000), the increase in the value of the standard deviation of the excitation σ f up to a certain value also increased the standard deviation of the displacement σ x . It was suggested that, since the harvester system under the random excitations shows a better performance for a certain range of the variance ( σ f 2 ) of the input excitation, the design of the system must be tailored so as to operate under the stochastic resonance conditions.

Next, a theoretical analysis of the piezo-magneto-elastic system under the random excitation was presented by Ali et al. (2011). The analysis of such a nonlinear system under random excitation for obtaining the probability density function (PDF) of the output power involves solving of the Fokker-Planck (FP) equation (Daqaq, 2011; Jazwinski, 2007). However, Ali et al. (2011) presented an equivalent linear model that was based on the stochastic linearization approach (Nigam, 1983; Roberts and Spanos, 2003). The results of the numerical analysis under white noise excitation showed that, there exists a cut-off value for the standard deviation of excitation σ f , below which the system portrays a poor performance. This cut-off value was observed to have an inverse relation with the dimensionless time constant. The numerical results of the equivalent linear model were verified by comparing with the numerical results of the nonlinear model presented in Litak et al. (2010).

Friswell et al. (2012) proposed a piezo-elastic inverted cantilever beam configuration with a tip mass as shown in Figure 27(a), to study the energy harvesting possibilities from the structures exhibiting large displacement oscillations and low natural frequency. The electromechanical equations were derived considering geometric nonlinearity, polar inertia of the tip mass for both the unimorph and bimorph configurations. Figure 27(b) depicts the transition of the single stable equilibrium position of a mono-stable potential into the three equilibrium positions (two stable and one unstable) of the bi-stable potential, as the tip mass was varied. Also, the variation of the linearized natural frequency of the system with the tip mass was studied as shown in Figure 27(c), from which the critical buckling mass was found to be 10 g. The nonlinear characteristics of the inverted beam-mass system in post-buckled (bi-stable) state were then exploited using the numerical sweep simulations with sweep parameters such as the excitation amplitude, frequency, tip mass, and the load resistance. The common features of a bi-stable case such as the periodic, multi-periodic, chaotic, and co-existing solutions were observed and verified with the experiments as well. Almost twice the improvement in the frequency bandwidth for the bi-stable state was observed. However, in post-buckled state, the improved power output was observed only under sufficiently high excitation parameters, promoting the interwell transitions.

OPEN IN VIEWER

Figure 27.

(a) Schematic representation of the inverted beam harvester with a tip mass M t . (b) Bifurcation diagram showing the variation of the equilibrium position(s) with the tip mass M t as a bifurcation parameter. The unstable equilibrium positions are represented by the dashed line. (c) The variation in the corresponding natural frequency of the stable equilibrium positions with the tip mass M t (Friswell et al., 2012).

A relative comparison between linear mono-stable (piezo-elastic) and bi-stable (piezo-magneto-elastic) harvester configurations (Erturk and Inman, 2011) under the stochastic random excitation was presented by Zhao and Erturk (2013). Both the configurations were excited with δ -correlated Gaussian white noise and the numerical response was obtained by using the Runge-Kutta scheme as discussed in Zhao and Erturk (2012). Under low excitation levels, both configurations exhibited linear random vibrations and the mono-stable harvester outperformed the bi-stable one in producing the optimal power output. Similarly, under the high excitation levels, the mono-stable configuration again generated higher power output than the bi-stable case, despite of the frequent interwell oscillations of the later. This behavior was attributed to the fact that the bi-stable case has confined and bounded response as compared to the dynamically flexible response of the linear mono-stable response. Under the moderate excitation levels that are just enough to ensure the crossing of bi-stable potential barrier, it was observed that the bi-stable system has larger displacement and voltage response than the mono-stable one as an effect of the stochastic resonance. This observation was consistent with the results reported in Adhikari et al. (2009), Ali et al. (2010), and Litak et al. (2010). Thus, it was concluded that a prior knowledge of the noise excitation characteristics is necessary for justifying the use of a bi-stable harvester, as it performs well only under certain excitation conditions. Otherwise, the mono-stable harvester shall be preferred for the case of random excitation with unknown characteristics.

Another comparative study was presented by Daqaq (2012), in which the response of the nonlinear mono-stable and bi-stable harvesters was compared with the linear harvester, under the Gaussian white noise excitation by solving the Fokker–Plank–Kolmogorov (FPK) equations (Jazwinski, 2007). The three cases of the harvesters considered in that study were modeled by the potential function U ( x ) = 1 2 ( 1 − r ) x 2 + 1 4 δ x 4 , as shown in Figure 28(a). Here, r and δ represented the tuning parameter for the linear stiffness coefficient and the coefficient of the cubic nonlinearity, respectively. The results of the study showed that (see Figure 28(b)), for a mono-stable harvester with both the open and closed circuit conditions, the mean square voltage varied inversely with the nonlinearity δ for any value of the tuning parameter r . The average power output from the harvester also followed an inverse relation with the nonlinearity δ (see Figure 28(c)). Moreover, the optimum value of the load resistance ( R opt ∝ 1 / α opt , α is time constant) corresponding to the maximum value of average power was found to be decreasing as the nonlinearity δ was increased.

OPEN IN VIEWER

Figure 28.

(a) Potential functions associated with the three cases of the harvesters considered in Daqaq (2012). δ and r are the cubic nonlinearity coefficient and the linear stiffness tuning parameter, respectively. Here, δ = 0 and r < 1 represents the linear mono-stable case, δ > 0 and r ≤ 1 represents the nonlinear mono-stable case and δ > 0 and r > 1 represents the bi-stable case. (b) The effect of variation of δ on the mean square voltage 〈 V 2 〉 , for various values of r and α = 0 . 05 representing the open-circuit condition. (c) The effect of variation of the time constant ratio α on the average power 〈 P 〉 for various values of δ and r = 0 . The optimum value of the time constant ratio α opt is marked by the hollow circles. (d) The effect of variation of δ on the mean square voltage 〈 V 2 〉 of a bi-stable case with r = 1 . 1 and α = 0 . 05 , for various values of the noise intensity σ . (e) A parametric plot showing the variation of the potential barrier height h b with the separation distance between the wells x 0 , for few discrete values of δ and 1 < r ≤ 2 . Reprinted by permission from Springer Nature, Nonlinear Dynamics (Daqaq, 2012), Copyright © 2012.

The simulation results (see Figure 28(d)) for the bi-stable case (open circuit condition) revealed that, there exists an optimal value of the nonlinearity δ opt corresponding to the maximum value of mean square velocity. The height of the potential barrier h b (see Figure 28(e)) and the separation distance between the wells x 0 decreases as the nonlinearity δ was increased. However, for large values of the time constant ratio α = 10 (i.e. the closed circuit condition), no significant influence of the nonlinearity was observed on the output voltage, for both the mono-stable and bi-stable harvesters. It was concluded that the nonlinear mono-stable harvester can never outperform the linear mono-stable case in terms of the output voltage produced, but the bi-stable harvester designed with the δ opt can perform better than the linear harvester. However, since a optimal bi-stable potential with δ opt was found to be sensitive to the noise intensity, the performance of the bi-stable harvester may deteriorate under the noise excitation with unknown and variable characteristics. This observation was similar to the one mentioned in Zhao and Erturk (2013) regarding the bi-stable harvester.

Kumar et al. (2014) derived the FP equations for the bi-stable Duffing oscillator (piezo-magneto-elastic configuration) subjected to the Gaussian white noise excitation. These equations were solved to obtain the joint and marginal PDFs of the response and voltage, using the FE method that involves the Galerkin Projection approach. The effect of the white noise intensity σ on the mean square response 〈 x 2 〉 as well as the mean square voltage 〈 V 2 〉 was studied and a direct relation was observed between them. Also, a steep increase was observed in the 〈 V 2 〉 beyond a threshold value of the σ , which was attributed to the merging of the peaks in the displacement PDFs, indicating the increase in the probability of the interwell transitions. The coupling coefficient was observed to have a direct effect on the 〈 V 2 〉 . The optimum values of the linear and nonlinear stiffness coefficients were recorded corresponding to the maximum value of 〈 V 2 〉 . These optimum values vary with the noise intensity. Hence, it was concluded that, with the appropriate combination of the system parameters and the excitation level, effective energy harvesting can be ensured from the bi-stable harvesters.

Similar to the case of nonlinear mono-stable potential harvesters, as discussed in Section 3.3 of the manuscript, multi-potential harvesters also exhibit superharmonic and subharmonic solutions. If frequency of the response of an oscillator is an integer multiple (n) or integer sub-multiple (1/n) of the excitation frequency, then that response pertains to a superharmonic or subharmonic solution, respectively. For a multi-stable harvester, both intrawell (low orbit) and interwell (high orbit) superharmonic and subharmonic solutions may exist (Kovacic and Brennan 2011). In Figure 19, these solutions are represented by period-n (nP) solutions. Of these, interwell (high orbit) subharmonic solutions are particularly useful as they extend the broadband frequency response of the harvester.

In this regard, Huguet et al. (2019) investigated subharmonic solutions of a bi-stable harvester and reported stability robustness criterion for these solutions. The bi-stable harvester configuration consisted a buckled beam and a piezoelectric stack type converter. An analytical model based on the harmonic balance method (HBM) was presented to identify low and high orbit harmonic 1 (period-1 or primary resonances) as well as subharmonic-n (period-n) solutions. The subharmonic solutions were then characterized based on their stability against small disturbances as stable and unstable solutions. Stable high orbit subharmonic solutions were further scrutinized as robust, if they sustain an experimentally determined threshold magnitude of the large disturbance. Numerically obtained frequency spectra has revealed that the subharmonic high orbits of odd order (e.g. 1, 3, and 5), of the mass displacement exhibit only odd harmonics of fundamental frequency, while both even and odd harmonics of fundamental frequency were observed for the subharmonic high orbits of the even order.

In order to experimentally attain various subharmonic solutions predicted by HBM, square wave pulses of voltage were applied to the piezoelectric converter to disturb the current steady state and to jump on other steady states, including the subharmonic states. Using this, the analytical model was verified with experimental observations and was further used to propose optimization guidelines. It was further observed that increasing the mass predominantly increases the mean harvested power as compared to the increments in the stiffness and buckling level. Conversely, increasing the stiffness predominantly increases the broadband response (by approx. +25 Hz), especially for subharmonic-3 solution, as compared to the increments in the mass and buckling level. Interestingly, it was suggested to optimize the buckling level such that there are no co-existing harmonic-1 and subharmonic-3 high orbit solutions, so as to avoid jump to a lower amplitude solution. Thus, it was concluded that, by using the optimization guidelines, a bi-stable harvester can be improved and made more suitable for the excitation sources with fluctuating frequencies.

Further exploring the work of Friswell et al. (2012), the bi-stable harvester with an inverted cantilever beam and a tip mass, as shown in Figure 27(a), was investigated for the multiple solutions by Syta et al. (2016). Using numerical simulations, four types of attractor solutions, namely, period-1 intrawell, period-3 crosswell subharmonic, chaotic, and period-1 high orbit harmonic solution, were studied for their power output and basins of attractions. As expected, interwell harmonic, chaotic, and subharmonic solutions, in the descending order, produced better average harvested power than the period-1 interwell attractors. The period-1 harmonic attractor has largest basin of attraction, followed by the chaotic and period-3 subharmonic attractors. The Wada patterns were observed that are formed by complex mixtures of the basins of attractions of the three large orbit attractors. With the increments in the tip mass, basin of attraction for chaotic attractor increases by forming a fractal border with the basin of period-1 harmonic attractor. Another study exploiting the usefulness of the superharmonic solutions of order 2 to improve the energy harvesting performance from low frequency excitations was presented by Masana and Daqaq (2012). In their investigation, an axially loaded bi-stable harvester exhibiting large orbit periodic, chaotic, and coexisting solutions close to half of primary resonance frequency was demonstrated.

Use of cantilever beam arrangement with a tip magnet mass for harvesting energy from rotational or angular motion of an automobile tyre was demonstrated by Zhang et al. (2018). In such an arrangement (as shown in Figure 29), the periodic forcing was achieved by the tangential component of the gravitational force acting on the tip mass. The mono-stable or bi-stable nature of the potential well for this arrangement was determined by the position of a radially movable permanent magnet mass, which was mounted opposite to the tip magnet. The initial bi-stable potential well transforms into a mono-stable potential well as the external magnet radially moves away from the tip magnet under the influence of the centrifugal force, which increases with the rotational frequency of the tyre. The harmonic balance solutions for both these potential well configurations were exercised to find the low and high orbit stable as well as unstable solutions as a function of rotational frequency. It was shown that the high orbit solution for the bi-stable case sustained up to 40 rad/s rotational frequency (or up to the driving speed of approx. 40 km/h) for a specific optimized offset distance 3.6 cm of the movable magnet mass. Also, for the same set of parameters, the transformation of bi-stable potential well into mono-stable potential well occurred at the rotational frequency of 31.6 rad/s. As the rotational frequency increases the initial high orbit oscillations of the bi-stable system stabilized onto the high orbit solution of the transformed mono-stable system over a longer frequency bandwidth. Here, the optimized offset distance ensured a stronger initial bi-stability, which in turn imparted a higher amplitude of the periodic excitation force, causing the system to hop onto the high orbit solution. Thus, a marked improvement in the harvesting efficiency was achieved with the assistance of the centrifugal force in sustaining and stabilizing the high orbit oscillations over a larger rotational frequency bandwidth.

OPEN IN VIEWER

Figure 29.

A schematic of the piezoelectric cantilever beam mounted on an automobile tyre for harvesting energy from rotational motion. Reproduced from Zhang et al. (2018), with the permission of AIP Publishing.

On the other hand, an inverted cantilever configuration with a proof mass directed radially inwards and utilizing centripetal acceleration induced gravity to impart bi-stability was proposed by Horne et al. (2018). Time dependent forcing caused by centripetal acceleration modulated the bi-stable potential well, allowing hopping between the wells. With the higher rotational speeds, stronger bi-stability with deeper and wider potential wells emerges, resulting in higher response amplitudes. However, after a threshold value of centripetal acceleration, response amplitude declined as the barrier modulation was not sufficient to cause the hopping between the wells.

The research on the bi-stable harvesters discussed above and few other studies (Firoozy et al., 2017; Panyam et al., 2014) describe about the nonlinear characteristics (Panyam et al., 2014), various modeling and solution approaches (Panyam et al., 2014; Stanton et al., 2012), and the response under periodic and random excitations. Several researchers have tried to further enhance the performance of the bi-stable harvesters by various means. Some of these studies are briefly discussed in the following sub-subsection.

3.4.3. Improving the performance of the bi-stable harvesters

In terms of the power output, although a bi-stable configuration performs well only under some specific excitation scenarios, the broadband frequency response is something which is highly assured in case of the bi-stable harvesters. In this regard, Zhou et al. (2013) studied the effect of the orientation of the base magnets, on the broadband frequency characteristics of a distributed mass bi-stable harvester model, using rotatable base magnets. The model simulations and experimental results with the magnet angles ( α , magnet rotation angle with horizontal in this study), α = 0 ° , 30 ° , 60 ° and 90 ° were presented. In these simulations, the nonlinear magnetic force was modeled by fitting a polynomial correlation on the measured data of the magnetic force at the tip of the cantilever beam, using a sensitive dynamometer. It was reported that the frequency bandwidth which was 4 − 15 . 5  Hz for α = 0 ° , increases to 4 − 22  Hz for the magnet angles up to α = 90 ° . However, a careful observation of the results reveals that, only the up-sweep jump frequency has shifted towards the higher frequency values, as a result of the increased hardening behavior as the magnet angle changes from α = 0 ° to α = 90 ° . The effective increase in the frequency bandwidth was imperceptible, perhaps negative, considering the frequency at which the voltage starts to build up during the up-sweep simulations. Also, the maximum amplitude of the voltage was observed to diminish consistently from 43.2 V for α = 0 ° to about 30 V for α = 90 ° . Thus, the rotatable magnet arrangement essentially exhibits a tuning approach for the harvester, resulting in the upward shift of the jump frequency without any additional space requirements for the operation.

The bimorph cantilever beam arrangement with rotatable magnets, as presented in Zhou et al. (2013), was further thoroughly investigated by Jung et al. (2015). In the work, electromechanical equations based on the EBT, Kelvin-Voigt damping model, and linear piezoelectric constitutive relations were derived using the variational principles. The magnetic forces and moments at the cantilever tip were explicitly modeled using the approach described in Tsui et al. (1972) and Zheng and Zeng (2004). The bifurcation analysis with the magnet angle as a bifurcation parameter, has showed that the system attains mono-stable, bi-stable, and even tri-stable states. The numerical frequency responses of all the three states were then obtained using the HBM. It was shown that a mono-stable potential can be transformed into a tri-stable or a bi-stable state by mere changing the magnet angle, undergoing the large amplitude interwell oscillations under the appropriate input excitation conditions.

As mentioned earlier, the bi-stable and tri-stable energy harvesters (BEH and TEH) exhibit the high and low amplitude co-existing solutions, which can be attained by different initial conditions with same intensity of excitation. However, the high amplitude solution is unattainable under a low intensity excitation, since the excitation strength is insufficient to overcome the potential barrier. In such cases, high amplitude solutions can be achieved by applying an initial condition with nonzero velocity (Erturk and Inman, 2011; Sebald et al., 2011a). Using this concept, Zhou et al. (2015) reported an impact based method, comprising a projectile mechanism as depicted in Figure 30(a), to attain a high amplitude solution. The simulations were performed at several frequencies to identify the initial velocity conditions that would achieve the high amplitude solution. It was observed that as the excitation frequency was increased, the number of such initial conditions decreased rapidly, realizing that the impact based method is most effective at low excitation frequencies. The simulations and experiments were performed on both the BEH and TEH configurations under excitation levels of 0.35 g and 0.27 g and it was confirmed that the impact based method significantly improves the frequency bandwidth over which the high amplitude oscillations prevail, resulting in the high voltage output.

OPEN IN VIEWER

Figure 30.

(a) Illustration of the impact based method that was used to attain the high amplitude solutions. Reproduced from Zhou et al. (2015), with the permission of AIP Publishing. (b) The adaptive bi-stable harvester configuration with an electromagnet acting as a base magnet. Adapted from Hosseinloo and Turitsyn (2015). (c) Schematic representation of the elastic support harvester. Reproduced from Gao et al. (2014) with permission. © IOP Publishing. All rights reserved. (d) The schematic diagram of the improved bi-stable energy harvester (IBEH) with an additional magnet. Reprinted from Lan and Qin (2017), Copyright © 2017, with permission from Elsevier.

The performance of a bi-stable design solely depends on the interwell oscillations, whose probability of occurrence is less for the low intensity excitations. In such cases, only the intrawell oscillations are observed. To enhance the performance of a bi-stable harvesters under the low intensity excitations, Gao et al. (2014) discussed a bi-stable configuration as shown in Figure 30(c), having the base or external magnets with an elastic support. The elastic support system possessed two conjugate excitation dependent potential functions, one for cantilever beam and another for the external magnet with the elastic support. Such an arrangement enhanced the probability of the interwell transition even for the low intensity excitations. Further, Leng et al. (2015) presented an improved version of this model, in which the nonlinear magnetic potential was derived using the vector differentiation approach. The response of the system under the filtered Gaussian noise and pink noise (Halley and Kunin, 1999) was numerically simulated, experimentally verified, and the improvements in the harvesting performance were discussed. The benefits of such an arrangement with the two coupled cantilever beams were also reported by Lin et al. (2010).

An interesting approach to enhance the performance of a bi-stable harvester, an adaptive bi-stability, was discussed by Hosseinloo and Turitsyn (2015). A lumped parameter model was presented in their work, which uses an adaptive potential function governed by the buy-low-sell-high (BLSH) strategy. The BLSH logic allows the interwell transitions (by controlling barrier height) only when the extreme excitation levels are identified, otherwise it ensures intrawell motions only. The barrier height of a bi-stable potential function was controlled with an electromagnet as shown in Figure 30(b). The numerical simulations were performed, considering equal maximum displacement of the linear mono-stable, conventional bi-stable, and adaptive bi-stable designs subjected to the harmonic and walking motion excitations. It was observed that the adaptive bi-stable design with BLSH logic exhibits better performance and robustness against the parameter variations under the low frequency harmonic and random excitations. Another study based on the adaptive bi-stable potential with the PVDF as a transduction material was reported by Shan et al. (2018). In their work, the adaptive bi-stability was achieved using a second cantilever beam with a tip magnet, which acts as a spring, hence the name spring assisted bi-stable energy harvester (SABEH) was suggested. Although the performance of the SABEH was realized to be better than the conventional bi-stable harvester with the optimal spring stiffness, there was no significant improvement in the frequency bandwidth of the SABEH.

Lan and Qin (2017) discussed an improved bi-stable energy harvester (IBEH) model as shown in Figure 30(d), to enhance the performance of a bi-stable design under random excitation. This study was motivated by the fact that, increasing the amplitude of the interwell oscillations by widening the distance between the two potential wells also increases the height of the potential barrier between the two wells. The increased barrier height reduces the probability of the interwell transitions. Therefore, a third base magnet was introduced in between the two fixed base magnets at the trivial position. This arrangement decreased the height of the potential barrier and also maintained a constant desired distance between the two potential wells. The potential energy function of the IBEH was derived with the dipole model assumption. The response of the IBEH model under the Gaussian white noise excitation was obtained by the Monte Carlo method and Runge-Kutta method. It was shown that the IBEH indeed performs better than the conventional BEH for low excitation intensities. Also, there existed an optimum size of the middle base magnet corresponding to an excitation level, which maximizes the output of the IBEH. Finally, it was proposed that the IBEH design can also perform better than the tri-stable energy harvester (TEH) under the low intensity random excitation.

Therefore, it can be realized that the bi-stable energy harvester configuration has been extensively explored by the researchers. A review of the bi-stable harvesters discussing some conceptual designs, modeling methods, and measurement of effectiveness was presented by Pellegrini et al. (2013). Another review describing the analytical and experimental studies, solution techniques, and the challenges in further development of the bi-stable harvesters was presented by Harne and Wang (2013). Now, in the following subsection, the tri-stable and the quad-stable harvester configurations are discussed in brief, highlighting their important characteristics, advantages and limitations.

3.5.1. Dynamic behavior of tri-stable potential well configuration

A tri-stable potential well configuration can be expressed in the dimensionless form by equation (15a) and corresponding nonlinear restoring force can be expressed as F tri = K 1 u − K 3 u 3 + K 5 u 5 . Here, K 5 denotes the quintic nonlinearity coefficient, in addition to the linear ( K 1 ) and cubic nonlinearity ( K 3 ) coefficients discussed earlier. Out of these, the linear ( K 1 > 0 ) and quintic nonlinearity ( K 5 > 0 ) coefficients impart a hardening behavior, whereas the cubic nonlinearity coefficient ( K 3 < 0 ) induces a softening behavior into the potential function U tri ( u ) . This potential function has three potential wells with five equilibrium positions provided the condition K 3 2 − 4 K 1 K 5 ≥ 0 is satisfied. Three out of the five equilibrium positions, namely, u 0 and u 1 , 2 as given in equations (15b) and (15c), are stable in nature, whereas the remaining two unstable equilibrium positions (saddle points) are given by u 3 , 4 in equation (15d). Therefore, such a potential function is known as the tri-stable potential well configuration and is presented in the Figure 31(a), showing the stable and unstable equilibrium positions along with the two potential barriers B 1 and B 2 . In Figure 31(a), the tri-stable potential function is constructed in such a way that its outermost stable equilibrium positions (i.e. u 1 , 2 ) are similar to the bi-stable potential case studied in Section 3.4.1. Also, the heights of the potential barriers are equal ( B 1 = B 2 ), which means the three potential wells have equal depths as well. Under these conditions, it can be realized that the potential barriers of the tri-stable case ( B 1 & B 2 ) are significantly smaller than the potential barrier of the bi-stable case ( B 0 ), which results in the increased probability of the interwell oscillations for the tri-stable potential case.

U tri ( u ) = 1 2 K 1 u 2 − 1 4 K 3 u 4 + 1 6 K 5 u 6 + C 0 .

(15a)

u 0 = 0 ,

(15b)

u 1 , 2 = ± − K 3 + K 3 2 − 4 K 1 K 5 2 K 5 ,

(15c)

u 3 , 4 = ± − K 3 − K 3 2 − 4 K 1 K 5 2 K 5 .

(15d)

OPEN IN VIEWER

Figure 31.

(Color Online) (a) A comparison between the tri-stable and bi-stable potential functions with equal outermost stable equilibrium positions (i.e u 1 , 2 = ± 1 ), showing the relative height of the potential barriers B 0 , B 1 and B 2 . The stable and unstable equilibrium positions are marked by the solid green circles and the hollow red circles, respectively. (b) The type 1 ( B 1 > B 2 ), type 2 ( B 1 = B 2 ), and type 3 ( B 1 < B 2 ) cases of the commonly studied tri-stable potential functions alongwith their stable and unstable equilibrium positions. (c) The phase-portrait diagram for a type 2 tri-stable potential function as shown in (a), depicting the three stable spirals at u 0 and u 1 , 2 along with the two unstable saddle points at u 3 , 4 . The basins of attraction of the three stable fixed points are shown in the green, blue and peach colors.

Figure 31(b) shows the three types of the tri-stable potential functions based on the relative height of the potential barriers B 1 and B 2 . For the type 1 case with B 1 > B 2 , the central potential well is deeper than the two outermost potential wells. Conversely, for the type 3 case with B 1 < B 2 , the two outermost potential wells are deeper than the central potential well. For the type 2 case with B 1 = B 2 , all the three potential wells have equal depth. The dynamic characteristics of these three types of tri-stable potential cases have been explored by several researchers and are discussed in the next subsection.

The phase-portrait diagram, without external forcing and ζ = 0 . 04 , for a tri-stable potential configuration shown in Figure 31(a) with K 1 = 1 , K 3 = − 4 and K 5 = 3 , is plotted in Figure 31(c). Once again, it can be verified that the stable equilibrium positions, marked by the solid green circles at u = u 0 and u = u 1 , 2 (see equations(15b) and (15c)), are the stable spirals and the unstable equilibrium positions, marked by the hollow red circles at u = u 3 , 4 (see equation (15d)), are the saddle points. The stable manifold (yellow, cyan, blue, and red colored arrowheads) and the unstable manifold (orange, magenta, green, and gray colored arrowheads) of the saddle points are obtained using the backward and forward integration in time, respectively. The basins of attraction, which are shown in the green, blue, and peach colors, correspond to the left, center, and right stable spirals, respectively. Similar to the case of a bi-stable potential as shown in Figure 18(b), all the trajectories with an initial condition in ( u , u · ) space have bounded solutions that will reach one of the three stable fixed points.

Similar to a bi-stable oscillator, an oscillator with a tri-stable potential well (hereafter called as the tri-stable oscillator) may exhibit intrawell, or interwell (or cross-well), or a blend of both the oscillations, depending on the strength of the external excitation. Both the intrawell and interwell oscillations may follow periodic, chaotic, or co-existing attractor solutions. However, a tri-stable oscillator offers more probability of the interwell oscillations even under the low strength excitation, when compared to the bi-stable one due to its smaller potential barriers. Figure 32 presents the comparison of the non-dimensional forward and reverse numerical frequency sweep responses of the bi-stable and tri-stable potential configurations as shown in Figure 31(a), under the harmonic excitation with an amplitude level of A = 15  m/s ². The forward and reverse sweep responses of the tri-stable oscillator confirmed the periodic, chaotic, and mixed interwell as well as intrawell oscillations as mentioned earlier. As observed from Figure 32, at low and moderate frequencies ( Ω ≈ 0 . 2 to 2 ), the tri-stable oscillator undergoes large amplitude periodic and chaotic oscillations, compared to the bi-stable one which mostly exhibits the intrawell oscillations. Hence, a tri-stable harvester is preferred in case of low frequency and low amplitude excitation over the bi-stable ones. Now, in the following text, several models and studies related to the tri-stable potential configuration are discussed.

OPEN IN VIEWER

Figure 32.

(Color Online) The non-dimensional numerical forward and reverse frequency sweep responses of (a) the bi-stable and (b) the tri-stable potential configurations shown in Figure 31(a), under the harmonic excitation with an amplitude level of A = 15  m/s². The tri-stable oscillator exhibits interwell oscillations at low as well as moderate frequencies ( Ω ≈ 0 . 2 to 2 ), whereas the bi-stable oscillator mostly performs the intrawell oscillations. Here, n P and CH implies the period-n and chaotic oscillations, respectively. The intrawell and interwell oscillations are denoted by IW and CW, respectively. The mixed periodic and chaotic intrawell/interwell oscillations are denoted by “Mixed IW/CW.” Superharmonic resonance is denoted by “SUPHR.” Orange and yellow colored dots belong to the stroboscopic bifurcation map of the numerical forward and reverse sweep responses, respectively.

3.5.2. Models and studies based on tri-stable and quad-stable configurations

An energy harvester having a triple well potential function is formally identified as the tri-stable energy harvester (TEH). The tri-stability (refer Figure 4) in the magnetic potential function arises as a result of the variation in horizontal spacing ( d h ) between the two fixed base magnets (Kumar et al., 2015), vertical gap ( d v ) between the tip magnet and base magnets (Kim and Seok, 2014), or magnet angle ( α ) with respect to the horizontal (Jung et al., 2015). Moon and Holmes (1979) reported the existence of such a tri-stable potential in case of magneto-elastic cantilever beam oscillator. A TEH configuration has three relatively shallower potential wells compared to a BEH configuration for the similar displacement limits (see Figure 31(a)), which promote the interwell transitions resulting in a higher energy output. Zhou et al. (2014a) presented a distributed parameter, single-mode approximation model with a tri-stable configuration, in which the nonlinear restoring force was measured by an electronic dynamometer and represented using a fifth order polynomial fit. A tri-stable and a bi-stable configurations having similar displacement limits were realized by appropriately choosing the values of d h , d v , and α . Numerical forward frequency sweep simulations and experimental tests were conducted for both the configurations with PZT-5A bimorph, for a frequency range of 1 − 20  Hz and the excitation amplitudes of 4 m/s² and 5.85 m/s². The results showed that under the low intensity excitation, unlike the BEH model, which was unable to cross the potential barrier, the TEH configuration exhibited the high amplitude interwell oscillations over a relatively larger frequency bandwidth. Also, when the BEH model was tested with the excitation that has sufficient strength to overcome the potential barrier, the voltage output and the frequency bandwidth for the TEH model was observed to be higher than the BEH model. The fixed frequency excitation simulations revealed that the TEH model possesses larger frequency bandwidth of interwell oscillations than the BEH model.

Subsequently, similar studies with numerical reverse frequency sweep simulations and experiments were presented by the same authors in Zhou et al. (2014b). In that study, the nonlinear restoring forces were modeled using a ninth order polynomial fit. The results of the numerical simulations and experiments with the increasing excitation amplitudes of revealed that unlike the mono-stable and the bi-stable harvesters, the amplitude of oscillations and the frequency bandwidth of the TEH increased only up to a certain level as the excitation amplitude was increased. This characteristic of the TEH prevents the harvester structure, including the piezoelectric element, from detrimental and excessive oscillation amplitudes.

Kim and Seok (2015) proposed a distributed parameter piezoelectric bimorph model of the TEH, similar to the one presented in Jung et al. (2015), and explored its dynamical characteristics. The nonlinear magnetic force and the torque exerted on the cantilever tip was modeled with the point-dipole approximation (Yung et al., 1998), using the position vectors of the external base magnets, which in turn depends on the geometrical parameters d h and d v (refer Figure 4). Here, in order to capture the tri-stable behavior accurately, the nonlinear part of the magnetic force and torque were calculated by subtracting the linear part from the total part, rather than using the truncated Taylor’s expansion. As discussed earlier, in case of the harvester configurations with a single external base magnet (Cottone et al., 2009; Gammaitoni et al., 2009; Stanton et al., 2010b), a bi-stable state materializes as the distance d v was decreased, following a supercritical pitchfork (PF) bifurcation. However, in case of the configuration with two external base magnets, in addition to the supercritical PF bifurcation (at point PF1 or DPF in Figure 33(a)), giving rise to a bi-stable state, a subcritical PF bifurcation occurs (at point PF2) as the distance d v was further reduced, resulting in the emergence of a tri-stable state. Corresponding variation of the potential energy function for different values of d v is shown in Figure 33(c). At degenerate point DPF in Figure 33(a), co-dimension two PF bifurcation and nucleation of the saddle-node (SN) solution branch was observed, as depicted in Figure 33(b) showing a bifurcation set diagram. Further, bi-stable, tri-stable configurations having equal distance between the outermost potential wells and a linear case were compared. Since the potential barriers of the tri-stable case are considerably shallower than the bi-stable case, the interwell motion in the tri-stable and bi-stable cases occurred under the excitation amplitude of 0.35 g and 0.6 g , respectively. Also, the frequency bandwidth of the interwell motion for the tri-stable case was noticeably larger than the bi-stable case. Therefore, the observations presented in Kim and Seok (2015) and Zhou et al. (2014a) support the discussion presented in Section 3.5.1 by confirming that, for the similar displacement limits, a tri-stable system exhibits better performance under the low strength excitation compared to a linear and bi-stable systems.

OPEN IN VIEWER

Figure 33.

(Color Online) (a) Bifurcation analysis showing stable (solid blue line) and unstable (dotted red line) equilibrium solutions for d h = 11  mm. (b) The bifurcation set diagram in ( d v , d h ) space, showing the regions of mono-stable (I), bi-stable (II) and tri-stable (III) solutions. (c) Non-dimensional plots of the potential functions for various values of d v , corresponding to the bifurcation diagram shown in (a). Reprinted from Kim and Seok (2015). Copyright © 2015, with permission from Elsevier.

Moreover, the results of the preliminary simulations highlighted the fact that, the low strength excitation performance of a tri-stable case having equal depth for all potential wells (i.e. type 2 case with B 1 = B 2 in Figure 31(b)) was realized to be better than the tri-stable systems having nonuniform depth of central and outermost potential wells (i.e. type 1 and type 3 cases in Figure 31(b)). This important observation was further extensively studied and verified by numerical and experimental simulations by Kim et al. (2016). They presented a dimensionless electromechanical model of a multi-stable energy harvester (MEH), which considers the inertia effects and geometric nonlinearity that influences the coupling behavior. Also, the magnetic restoring force was modeled by a fifth order polynomial fit and the point-dipole assumption. The non-dimensional restoring force expression for the MEH system was given by equation (16), in which γ 1 , γ 3 and γ 3 are nonlinear spring constants that depend on the distances d v and d h . The multi-stability conditions were realized as shown in Figure 34(a), by defining two bifurcation parameters as α = 1 − γ 1 and μ = γ 3 / 4 γ 5 .

F R = − x ( 1 − γ 1 + γ 3 x 2 + γ 5 x 4 )

(16)

OPEN IN VIEWER

Figure 34.

(a) Bifurcation set diagram with ( μ , α ) space showing mono-stable (R1), bi-stable (R2) and tri-stable (R3) regions. (b) The tri-stable cases of type 1, 2, and 3 are represented by plots F, E, and D, respectively. Plots A, B, and C represents the bi-stable cases for decreasing μ value. All the plots (A–F) have equal maximum potential well depth. (c) Response of the type 1–3 tri-stable cases and the bi-stable case having equal depth, under harmonic excitation. (d) The variation of the overall potential well width (in mm) and depth (in mJ) with μ . Reproduced from Kim et al. (2016), with the permission of AIP Publishing.

In the MEH model, the potential function U ~ ( x ) of a tri-stable system having equal depth for all the three potential wells, has zero value at all the stable centers ( U ~ ( x c ) = 0 ), which leads to the condition α = 3 4 μ 2 for μ < 0 . This condition is represented by a dotted red line T U in Figure 34(a), which belongs to the tri-stable cases of type 2 having equal depth for all the three potential wells. The tri-stable region to the left and right side of the T U line represents the tri-stable cases of type 1 ( B 1 > B 2 ) and type 3 ( B 1 > B 2 ), respectively, as shown in Figure 31(b) of Section 3.5.1. The potential energy functions for the tri-stable cases of type 1, 2, and 3 are depicted by plots F, E and D, respectively, in Figure 34(b). Also, the plots A, B, and C represent the bi-stable cases with decreasing μ value and having equal depth for both the potential wells. All the plots A to F lie on a isoline that represents a constant sum of the depths of all the potential wells.

The frequency response of the tri-stable cases of type 1–3 along with the bi-stable case having equal sum of potential well depths, under the constant excitation of 0.21 g amplitude was obtained. It showed the superior performance of the type 2 case (see Figure 34(c)) over all the other cases mentioned. As the value of μ was decreased along the T U line, the overall width (i.e the distance between the two stable centers of the outermost wells) and the depth of the potential well increased. This improvement in the overall width of the potential well enhances the power output as a result of the increased amplitude of the interwell oscillations. On the other hand, the increased depth of the potential wells would reduce the probability of the interwell transitions. Therefore, it was necessary to find out the most effective type 2 tri-stable case out of the many cases represented by T U line, under the low amplitude excitation. For this purpose, numerical simulations and experiments with the low intensity harmonic excitation were performed on the type 2 tri-stable cases represented by TU1, TU2, and TU3 in Figure 34(d), with decreasing μ value. Near the DPF point, the rate of increase of the potential well depth was realized to be lower than the rate of increase of the overall width of the well. Therefore, the type 2 tri-stable cases nearer to the point DPF (such as TU1), exhibited better performance than the other type 2 cases, under low amplitude excitation of 0.12 g . However, it should be noted that the observations and conclusions presented in this study were based on the assumption that, the system was already brought up to the high amplitude interwell solution by some initial perturbation.

Similar study to investigate the effect of potential well depths on the harvesting performance of a TEH was presented by Cao et al. (2015b). The three cases of the TEH discussed and shown earlier in Figure 34(b) were subjected to the harmonic excitation with 0.12 g , 0.25 g , and, 0.36 g amplitudes and frequency range of 0 . 01 − 25  Hz. The numerical and experimental simulation results confirmed that, indeed the type 2 tri-stable case with shallow and equal depths for all the potential wells can perform better under the low amplitude and low frequency excitation, producing higher output in comparison with the type 1 and 3 tri-stable cases. Also, by noticing the poor performance of the type 3 case under low strength excitation, the importance of mindful design of the harvester based on the awareness of the excitation characteristics was emphasized.

Haitao et al. (2015) studied the response of a TEH in comparison with a BEH having similar displacement limits, with numerical and experimental simulations under harmonic and stochastic excitations. An electromechanical distributed parameter model was presented, in which the magnetic restoring force was modeled following the approach discussed in Stanton et al. (2009). The tri-stable and bi-stable configurations of similar displacement limits were realized by adjusting the distances d h and d v . The results under the harmonic excitation, once again, showed the superior performance of the TEH over the BEH with similar conclusions as presented in Jung et al. (2015), Kim and Seok (2015), and Zhou et al. (2014a). In case of stochastic excitation with Gaussian white noise characteristics, the variation of signal to noise ratio ( σ x / σ f ) with the excitation intensity σ f exhibited the presence of marked peaks, as a result of the coherence resonance conditions (Gang et al., 1993), causing regular interwell transitions. The occurrence of the coherence resonance for the TEH and BEH were registered at the excitation intensities of σ f = 0 . 03 and σ f = 0 . 07 , respectively. Therefore, for the low intensity excitations of periodic and random nature, the tri-stable case performed better than the bi-stable case.

Another nonlinear electromechanical model used to characterize the multi-stable equilibrium states of the cantilever type piezo-magneto-elastic energy harvester was presented by Kumar et al. (2015). The magnetic potential of the system was modeled using the concept of magnetic field with cylindrical symmetry, as observed in the case of finite solenoid (Derby and Olbert, 2010; Ranz, 2006; Tam and Holmes, 2014), instead of the point-dipole approach which was commonly used for modeling the magnetic restoring force. This approach was motivated by noticing the irrational nature of the point-dipole assumption, considering the finite physical size of the external base and tip magnets. Following this approach, the resultant magnetic field as a function of the tip displacement was expressed by Taylor’s expansion with terms up to sixth order. The expansion coefficients were functions of the system parameters such as the field strength of the base ( B s ) and tip ( B t ) magnets, the sizes of the base and tip magnets, as well as the distances d h and d v . Finally, the magnetic potential of the system ( Π m ) was written as a sixth order polynomial, in which the expression for its coefficients were obtained using a symbolic computation software. Further, characterization of the multi-stable configurations, namely, the mono-stable (both softening and hardening characteristics) and the bi-stable, and the tri-stable case was presented and depicted in the bifurcation set diagram, as shown in Figure 35 with ( B t , d ) space. Here, d = 2 d h .

OPEN IN VIEWER

Figure 35.

The bifurcation set diagram in ( B t , d ) space showing different multi-stable configuration regions, where d = 2 d h . Reprinted by permission from Springer Nature, The European Physical Journal—Special Topics (Kumar et al., 2015), Copyright © 2015.

Using the similar approach for modeling of the magnetic potential, Kumar et al. (2017) further studied the nonlinear dynamical aspects of the magneto-elastic oscillator (Tam and Holmes, 2014). With the help of the bifurcation set diagrams in ( d v , d ) and ( B s , d ) space, multi-stable positions were characterized. The response of these configurations under the harmonic excitation with varying amplitude was discussed and the presence of periodic, chaotic, and unbounded solutions in the bi-stable and the tri-stable configurations were depicted thorough the poincaré map and bifurcation diagrams. With the help of phase portrait plots for the mono-stable softening, bi-stable, and tri-stable configurations, the basins of attraction of the stable equilibrium points, the heteroclinic tangles (causing unbounded solutions in the mono-stable softening case) and the homoclinic tangles (leading to the chaos in the bi-stable and tri-stable case) were illustrated (Nayfeh and Balachandran, 2004). The model results were then verified through the experiments. In summary, authors in Kumar et al. (2017) attempted to reveal the complex nature of the magnetic interaction with the sixth order magnetic potential and its effects on the dynamical characteristics of the magneto-elastic oscillator were demonstrated.

The effects of variations of the system parameters as well as the external parameters on the dynamical behavior such as amplitude and phase of the response displacement ( r , ϕ ) and the response voltage ( V , ψ ) of a TEH under harmonic excitation were presented by Zhou et al. (2016). Theoretical studies using the HBM solutions have shown the presence of single and multi-solution (both intrawell and interwell) regions as the excitation amplitude was varied. The threshold response amplitude to attain a higher solution increased as the excitation frequency was increased, in case of the intrawell solutions. The damping coefficient ξ was noticed to affect the response, both r and V , inversely. The response displacement r was noted to have direct and inverse variations with the capacitance C p and the coupling coefficient θ , respectively. However, it was realized that the maximum value of the response voltage V exists for an optimal choice of C p and θ . Thus, the choice of the optimal C p , θ , and ξ govern the chances of occurrence of the interwell solution under specific excitation conditions, leading to a higher output from the TEH system.

Further, the effect of change in equilibrium positions on the performance of a TEH was studied by modeling three TEH configurations having different x 1 and x 2 (the unstable and stable equilibrium position after the trivial one) values as shown in Figure 36(a). The results of the numerical sweep simulations for the TEH cases with the smallest (a type 2 case) and largest (a case with deepest potential wells) x 1 and x 2 values possessed the largest and smallest frequency bandwidths, respectively, over which the high amplitude interwell solution existed for any initial condition. Also, under the trivial initial condition and fixed excitation frequency simulations, the threshold amplitude at which the jump to the high amplitude solution occurs was found to be lowest and highest for the former and later cases, respectively, as shown in Figure 36(b). In another instance, a thorough study aimed at characterizing the effective energy harvesting bandwidth and to asses the effect of the system parameter variation on performance of the TEH was presented by Panyam and Daqaq (2017).

OPEN IN VIEWER

Figure 36.

(a) Three cases of the TEH potential functions with various values of the equilibrium positions. (b) Response voltage ( V ) variation with the excitation amplitude ( A ) at the fixed frequency ω = 2 π , showing the threshold amplitude required for the jump to reach the high amplitude interwell solution with trivial initial conditions. Reprinted from Zhou et al. (2016). Copyright © 2016, with permission from Elsevier.

Multiple attractor solutions of a type 2 tri-stable potential well harvester were studied and experimentally verified by Zhou et al. (2018a). In their investigation, numerical amplitude sweep simulations were used to identify various periodic and chaotic attractor solutions, and the excitation amplitudes over which these solutions exist. Thus, period-1 intrawell, period-1 crosswell, period-3 crosswell, period-2 crosswell, chaotic, and period-1 mixed interwell and crosswell response solutions were identified and some interesting observations were made using phase portraits, poincaré maps, and Fourier spectra. Indeed, the presence of 1 / 2 nd or 1 / 3 rd subharmonics in the Fourier spectra was responsible for the period-2 or period-3 solutions, respectively. Moreover, multi-harmonic components of the subharmonic-n as well as the presence of third and higher (super) harmonic components modulated the period-n response of the oscillator, which is evident from the oval shaped dots in the poincaré maps. The presence of only superharmonic components (third and higher) causes period-1 intrawell, crosswell, and mixed intrawell plus crosswell responses. On the other hand, the presence of both sub and super harmonic components lead toward period-n or multi-periodic responses based on the relative strength of the subharmonic and superharmonic components. Thus, subharmonic solutions and their multi-harmonic components form the basis for the chaotic, period-n, and multi-periodic responses of the oscillator.

In order to improve the broadband frequency response and harvesting efficiency, many researchers have proposed configurational modifications, directly affecting the multi-stable potential of the harvester. These designs were often analyzed for their performance by connecting with the purely resistive electrical loads. Unconventionally, Yan et al. (2018) investigated the performance of a tri-stable harvester having a resistor-inductor (RL) resonant circuit connected in series with the piezoelectric element. Thus, instead of a first order ODE, the electrical equation was modeled as a coupled second order ODE. Using numerical frequency sweep simulations, influence of the frequency ratio ( ω e / ω ) of the RL circuit ( ω e ) and mechanical excitation ( ω ) frequencies was demonstrated. For a given frequency ratio ( ω e / ω ), while the displacement of the TEH with the pure resistor circuit and RL circuit was similar, the voltage amplitude for the TEH with RL circuit was higher than the former one. The voltage amplitude of the TEH with RL circuit drastically improved (more than 20 times) when the frequency ratio ( ω e / ω ) is around and less than 1. Depending on the electrical damping and the frequency ratio, the TEH exhibited period-n, chaotic and multi-periodic responses. In summary, such a resonant RL circuit with the optimized ranges for the electrical damping and frequency ratio can provide promising outcomes without requiring any configurational modifications.

A multi-stable energy harvester (MEH) model capable of characterizing up to five stable equilibrium positions by appropriately adjusting the distances d v and d h was proposed by Kim and Seok (2014). In their MEH model, a soft magnetic material (HyMu80) was used at the tip of the cantilever bimorph, considering its low coercievity and hysteresis along with the high magnetic susceptibility, which enhances the magnetic interaction at the tip. The magnetization of the soft magnetic material was modeled using the linear or saturated magnetization model (Abbott et al., 2007), after which the resultant magnetic force and moment at the tip were modeled using a similar process as described in Jung et al. (2015). Figure 37(a) shows the bifurcation set diagram in ( d v , d h ) space, describing various stability regions and the bifurcation points. The two paths A and B in Figure 37(a) describe the sequential transitions from the bi-stable to tri-stable region followed by another transition into the quad-stable region (for path B only). Figure 37(b) depicts the transition of a mono-stable potential case into a tri-stable potential case, which is followed by another transition into a quad-stable potential case, as d v is varied for a fixed value of d h = 8 . 4  mm. Similarly, Figure 37(c) describes the formation of the penta-stable potential case, as d v is varied for a fixed value of d h = 21 . 5  mm.

OPEN IN VIEWER

Figure 37.

(a) The bifurcation set diagram in ( d v , d h ) space showing various multi-stable regions and bifurcation points. The formation of (b) the quad-stable potential case as the distance d v is varied, for d h = 8 . 4  mm, and (c) the penta-stable potential case as the distance d v is varied, for d h = 21 . 5  mm. Reprinted from Kim and Seok (2014), Copyright © 2014, with permission from Elsevier.

It was inferred that the shallow and wide (dish like) nature of the tri-stable case may enhance the interwell oscillations under the low intensity excitation, whereas, due to the presence of deep outermost potential wells, the quad-stable and penta-stable potential cases might require high intensity excitation to exhibit the interwell oscillations. These inferences were indeed found true when verified with the numerical simulations with harmonic frequency and amplitude sweep excitation. Also, the low amplitude excitation performance of the quad-stable case, which exhibited two different levels of the interwell solution, was found better than the bi-stable case. However, in terms of the response amplitude and frequency bandwidth, the tri-stable case under low excitation outperformed all other configurations. The penta-stable potential case has a small size (width ≈ 2 mm) of potential well at the trivial equilibrium point, which has negligible influence and hence its dynamical behavior resembles to that of a quad-stable case. However, the larger distance between two potential wells that are adjacent to the trivial position increases the amplitude of the local interwell oscillations, as compared to the quad-stable case.

Recently, Zhou et al. (2017) presented a quad-stable energy harvester (QEH) model as shown in Figure 38(a), in which three external base magnets were used to achieve the quad-stable potential function. The magnetic potential energy U mnD between n th external magnet and the tip magnet D was obtained using the approach described in Leng et al. (2015). The performance of this quad-stable model was compared against a bi-stable model described in Stanton et al. (2009). Several BEH and QEH models with equal gap distance between the tip magnet and the central external magnet (i.e. equal d v distance) were presented, to examine the relative shape of the potential wells. It was observed that the potential wells of the QEH models were shallower than the BEH models, based on which the authors predicted a better low excitation intensity performance of the QEH model than the BEH model. The phase-portrait diagrams with the Poincaré points were obtained through the numerical simulations of the QEH model under the harmonic excitation, which revealed the existence of the periodic and chaotic attractors. With experiments, it was shown that the response amplitude and the bandwidth of the QEH model was substantially greater than the BEH model under the similar excitation scenarios.

OPEN IN VIEWER

Figure 38.

(Color online) (a) The schematic of the QEH configuration showing three external base magnets. (b) The formation of various QEH potential configurations as distance d v was varied, for d h = 20  mm. The potential functions with d v = 28, 23, and 21 mm represent type 1 (cyan), type 2 (green) and type 3 (black) QEH models, respectively. Reprinted from Zhou et al. (2018b). Copyright © 2018, with permission from Elsevier.

Subsequently, in Zhou et al. (2018b), the relative performance of the BEH and QEH models was explored under the random Gaussian white noise excitation. The quad-stable potential configurations with innermost two wells having largest depth (type 1), all wells with equal depth (type 2), and outermost two wells with largest depth (type 3) were modeled as shown in Figure 38(b). Once again, it was realized that the quad-stable case having equal depth for all the potential wells (i.e type 2 case) exhibits better performance than the other two cases. Also, when compared with the BEH model, the QEH model attains the high amplitude solution at a lower excitation intensity and sustains it over a wider range of the excitation intensities. Thus, the benefits of the QEH model over the BEH model were demonstrated, using numerical and experimental studies under the similar external and system parameters.

In summary, the studies discussed in this section presented the models, numerical simulations, and experiments on the tri-stable and the quad-stable potential configurations in comparison with the bi-stable configurations. The simulation and experimental results confirmed the superior performance of the TEH system, in terms of its interwell response and the frequency bandwidth due to its shallower potential wells, compared to the BEH system. Also, among the three different types of the TEH configurations as discussed in Section 3.5.1, the performance of the type 2 TEH case having equal depth for all the three potential wells was found better than other two cases. Similar observations were also reported for the QEH model. These observations highlighted the importance of the thoughtful design of the potential wells of the multi-stable harvester systems.