# On the dynamics of a nonlinear energy harvester with multiple resonant zones

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## Abstract

The dynamics of a nonlinear vibration energy harvester for rotating systems is investigated analytically through harmonic balance, as well as by numerical analysis. The electromagnetic harvester is attached to a spinning shaft at constant speed. Magnetic levitation is used as the system nonlinear restoring force for broadening the resonant range of the oscillator. The system is modelled as a Duffing oscillator with linear frequency variation under static, as well as harmonic excitation. Behaviour charts and backbone curves are extracted for the fundamental harmonic response and validated against frequency response curves for selected cases, using direct numerical integration. It is found that variation in stiffness, together with asymmetric forcing, gives rise to a novel structure of multiple resonant zones, incorporating mono-stable and bi-stable dynamics. Contrary to previously considered bi-stable energy harvesters, cross-well oscillations are realized through a transition from single-well potential energy to double-well with forward frequency sweep. Furthermore, in-well_oscillations present a hardening behaviour, unlike the well-known softening in-well response of bi-stable Duffing oscillators. The analysis shows that the proposed system has multiple resonant responses to a frequency sweep, influenced by consecutive interacting backbone curves similar to a multi-modal system. This combined effect of the transition to bi-stable dynamics and the hardening in-well oscillations induces resonant response of the harvester over multiple distinct frequency ranges. Thus, the system exhibits a broadened frequency response, enhancing its energy harvesting potential.

## Keywords

Energy harvesting Backbone curve Duffing oscillator In-well oscillations Bi-stable dynamics## Nomenclature

*B*,*C*,*D*Functions of \(\varOmega ,x_0 ,x_1 \)

- \(c_\mathrm{m} \)
Mechanical damping coefficient (Ns/m)

- \(c_\mathrm{el} \)
Electrical damping coefficient (Ns/m)

- \(d_0 \)
Separation of levitating magnet from its static counterparts (m)

- \(f_1 ,f_2 \)
Functions of \(\varOmega ,\,x_0 ,\,x_1 \) linking the derivatives of the response components

- \(F_\mathrm{cp} \)
Centripetal force (N)

- \(F_{\mathrm{b,t}} \)
Bottom/top magnetic force (N)

- \(F_\mathrm{m} \)
Moving magnet force (N)

*g*Gravitational acceleration (9.81 m/s\(^{2})\)

*I*Current (A)

*k*Linear stiffness coefficient (N/m)

- \(k_3 \)
Nonlinear stiffness coefficient \((\hbox {N/m}^{3})\)

*L*Inductance (H)

*m*Mass of the moving magnet (kg)

- \(P_\mathrm{L} \)
Electrical power load (W)

- \(P_{\mathrm{L,av}} \)
Average power load (W)

- \(R_\mathrm{i} \)
Internal resistance (\(\Omega \))

- \(R_\mathrm{L} \)
Load resistance (\(\Omega \))

*r*Eccentric radius (m)

*u*Response perturbation (m)

*v*Variable used in the derivation of stability conditions

*x*Radial displacement of the levitating central magnet (m)

- \(x_0 \)
Static displacement (m)

- \(x_1 \)
Harmonic amplitude of displacement (m)

*V*Harmonic amplitude of velocity (m/s)

- \(\beta \)
Normalized nonlinear stiffness coefficient \((1/\hbox {m}^{2}\hbox {s}^{2})\)

- \(\varDelta \)
Determinant of a matrix

- \(\zeta \)
Damping ratio

- \(\varTheta \)
Electromagnetic coupling factor (Vs/m)

- \(\mu \)
Floquet exponent

- \(\phi \)
Phase of the harmonic response (rad)

- \(\varOmega \)
Angular velocity of the shaft (rad/s)

- \(\omega _n \)
Linear resonant frequency (rad/s)

## 1 Introduction

Vibration energy harvesting is a relatively recent concept. Mechanical systems generally experience vibrations due to manufacturing or assembly imperfections [1], transient and impulsive loading [2] and wear/degradation. Although these causes have negative effects on the system structural integrity and ideal operation, the potential for energy harvesting from the vibratory modes has attracted much research [3, 4, 5]. It is envisaged that the otherwise dissipated energy can be supplied to devices of low power demand such as wireless sensors [3, 4, 6]. Besides improving the energy efficiency of the host structure, the compact design of a harvester would allow its positioning in confined and often hostile environments, such as in structural health monitoring [7], which pose difficult accessibility through traditional means with a power supply.

Most vibration energy harvesting concepts deal with ambient vibrations acting as translational base excitations for the harvesting attachment [8, 9]. Despite an overwhelming diversity of proposed concepts and designs, one can identify two major classes of vibration harvesters; cantilever beams with a tip mass in bending mode [10, 11, 12] (usually coupled with piezoelectric elements), or a base-excited seismic magnetic mass, oscillating in the proximity of an electric coil. The former class may also include continuous systems such as plates and beams with their deflections inducing voltage in piezoelectric elements [13], which may extend beyond mere vibration harvesting (e.g. fluid–structure interactions [14]). Stephen [4] considered direct mass and base excitation of a seismic mass, first proposed in [3], comprising a mass suspended from a linear spring. This concept was also used by Beeby et al. [5] for the development of a micro-generator. A common conclusion from all the analytical and experimental work has shown optimal device performance when operating in the vicinity of resonance. This finding points to the requirement for tuning of the harvester to the expected frequency of oscillations. However, ambient vibrations are subject to unpredictable variations, both in short- and long-term durations, rendering the tuning of a device unsuitable when the dominant frequency of oscillations shifts away from a tuned frequency. This problem occurs with systems under transient conditions as highlighted in [1, 2]. Therefore, complex and often expensive control methods are required to re-tune the harvester for an instantaneous prevailing condition. An alternative would be the use of multi-modal devices [15]. However, this approach can become impractical, given the design constraints often imposed with regard to any added weight and package space (compactness). Passive tuning has also been proposed, often related to frequency up-conversion [16] through impulsive interactions of the device with the instantaneous vibration response.

Another approach utilizes the system nonlinearities which lead to a broader frequency response. Mann and Sims [17] proposed the use of magnetic repulsion in order to impose strongly nonlinear restoring forces on a seismic magnetic mass. This concept has received much attention due to the novelty of an intentionally introduced nonlinearity. Barton et al. [18] suggested coupling between a magnetic tip mass and a ferrous coil core as a means of achieving nonlinear stiffness. Remick et al. [19] provided an investigation of dynamic instability of a strongly nonlinear harvester with repetitive impulsive excitation. These conditions occur in many mechanical systems such as in gear rattle, caused by impact of loose (unengaged) gearing [20]. A number of studies [21, 22] have explored the advantages of nonlinear harvesting over the classical linear approach, especially those related to the frequency range over which high quantities of energy can be captured.

A widely used model for systems with nonlinear forcing is based on the Duffing oscillator (cubic nonlinearity) [23]. The major advantage of nonlinear energy harvesting can be ascertained from the hardening spring influence upon the frequency response curve (FRC), in which an increase occurs in the response bandwidth [25]. This effect allows for a broader range of frequencies over which power can be generated efficiently, compared with a linear system. Duffing oscillators have long been studied because of their numerous applications in nonlinear engineering problems [23, 24, 25, 26]. Kovacic et al. [27, 28] studied the presence of a bias term in the harmonic excitation of a purely nonlinear Duffing system. It was found that this term can induce the frequency response to coincide with the hardening–softening characteristics, which has also been observed in the parametrically excited Duffing oscillators, along with branch detachments [29]. Increased complexity of the response (up to chaotic excursions) can be induced by the presence of strongly nonlinear terms (fast-slow mixed-modes, multi-stability etc. [30]). The large variation in dynamic behaviour necessitates detailed analysis of Duffing oscillators when employed in different environments and applications.

Although harvesting energy from translational oscillations has received much attention, there is a dearth of investigation of torsional systems which abound in powertrain and rotor dynamic applications. One can conceive the design of a piezogenerator, based on torsional stressing of its active elements [31, 32]. Furthermore, the piezoelectric cantilever design has been used for either torsional vibratory modes with piezoelectric attachments or in bending modes, extending radially from a shaft subjected to torsional oscillations [33].

In this paper, the magnetic levitation concept [17] is used for a torsional vibration energy harvester rigidly mounted onto a spinning shaft. The nonlinear restoring force pertains to the principle of energy harvesting within a broadband frequency response. The system is investigated with a constant shaft speed in order to reduce the complexity of the problem, thus enabling an analytical solution. The harmonic balance method is used to derive approximate analytical expressions for the system frequency response, which is validated against numerical integration of the system equations. The harvester is found to exhibit a complex dynamic behaviour when compared with other single-degree-of-freedom (SDOF) oscillators, mainly concerning the apparent coexistence of multiple backbone curves. This is potentially a significant contribution to the energy harvesting capabilities of the proposed concept, since it combines mono-stable with bi-stable dynamics over distinct frequency ranges for which significant energy levels can be attained. An estimation of the induced load power is also provided.

## 2 The proposed energy harvester and the examined system

*x*is the displacement of the oscillating magnet.

It was also shown that the linear component of the stiffness can be tuned to a desired resonant frequency through changes in the initial separation of the magnets, \(d_0 \), without affecting the nonlinear stiffness, \(k_3 \) [17].

*r*is the distance of the magnet from the centre of the shaft whilst at rest, and \(\varOmega \) is the shaft’s angular velocity.

*L*is the coil’s inductance,

*I*is the current, \(R_\mathrm{i} \) is the coil’s resistance, and \(R_\mathrm{L} \) is the resistant load. The coil’s inductance is often much smaller than the circuit’s resistance, thus: \(L\ll \left( {R_\mathrm{i} +R_\mathrm{L} } \right) \), leading to:

*V*is the amplitude of the oscillating magnet’s velocity.

## 3 Dynamics of the harvester

### 3.1 Symmetric layout of the harvester

### 3.2 Asymmetric layout of the harvester

*r*below the critical value leads to the existence of two separated backbone curves. Since these curves diverge from one another, the bandwidth of significantly high response amplitudes remains sufficiently broad. On the other hand, the existence of a single backbone curve effectively reduces the frequency range, where the presence of nonlinearity would be beneficial.

### 3.3 Stability characteristics of the harvester

It is important to ascertain the dynamic stability of the derived solutions, especially as a multitude of these may exist. Previous studies have shown that changes in the stability of a solution branch are associated with saddle-node bifurcations [26]. Furthermore, it has been shown that bifurcation points coincide with the jump phenomenon in systems with a Duffing-type nonlinearity [26] (i.e. a sudden discontinuous increase or decrease in the amplitude of the response when a parameter is slowly varied). In fact, the jump frequencies are of special interest in nonlinear systems since their values limit the range of coexisting solutions, practically determining the frequency range of interest for nonlinear analysis.

## 4 Results and discussion

The FRCs of the proposed system are obtained based upon the derivations in the previous section for Eq. (). The approximate analytical results are compared with direct numerical integration of Eq. (4) or (8), where applicable, in order to examine the validity of the analysis. Numerical FRCs are constructed through sweep of excitation frequencies in incremental steps, whilst other system parameters remain unchanged. Every point of the FRC is computed based upon the time series of the numerical integration. The latter is performed within a small time step, \(\hbox {d}t=2\pi /\left( {100\varOmega } \right) \). Each time-domain solution is allowed to reach steady-state conditions prior to selection of the last few periods for the determination of the response characteristics. The values of the static displacement and harmonic amplitude are computed from the Fourier coefficients of the time series, whilst neglecting higher order harmonic contributions.

### 4.1 Behaviour charts

*r*values just above point D, the response quickly returns to a single solution following the narrow DL branch. However, the number of coexisting solutions increases almost instantly into three with the crossing branch LK.

The above structure indicates the rich dynamics of the examined system. Moving to higher frequencies, the system response solutions undergo a three-to-one saddle-node bifurcation on the branch EF, whereas multiple solutions exist on branches MK and FN. Below point F, a richer structure is evident. Branches KF and KJ induce the appearance of five coexisting solutions, which reduce to three in branch FG. Before that, a small area is found for the *r* parameter space enclosed by JGH, where seven solutions coexist, including the symmetric case (\(r=0)\). The final observed region occurs between GI and IH, defining a short range for five coexisting solutions.

With an increasing nonlinear stiffness coefficient, the same structure is generally maintained in the behaviour chart (Fig. 2b). Nevertheless, some notable changes are observed in branch ABC, now intersecting with branch DK, thus creating a region of five coexisting solutions in the low frequency range. Moreover, the region enclosed by branches JGH is expanded and as a result a wider area with seven coexisting solutions is present.

*r*give rise to distinct solutions in this frequency range, as it can be observed in Fig. 3b (for \(r=0.005)\). A finite value of

*r*breaks the symmetry of the system leading to a

*biased*response. This effect is further pronounced in Fig. 3c for \(r=0.013\) due to a higher centrifugal force. The response undergoes three bifurcations at branches DE, EF and FN (Fig. 2a), demonstrating the expected changes in the response. An interesting observation is that DE gives rise to a detached branch in the FRC, even though it only extends to a narrow frequency range (as it can be seen in the inset to Fig. 3c). The bifurcations observed in the system response have significant effect on the state of system equilibria. This can greatly affect the operation of the energy harvesting device, especially when the oscillation boundaries are of concern. However, the focus of this paper is on the harmonic response of the system.

### 4.2 Backbone curves

Considering the damped dynamics of the system, the FRCs have an unusual structure as it can be seen for the case \(\zeta =0.03\). This exhibits strongly nonlinear behaviour with an increasing frequency. The low-energy branch, past the first backbone curve, is destabilized by a saddle-node bifurcation. This gives rise to another branch corresponding to the second backbone curve, the structure of which also replicates a hardening-type response. The emergence of this branch has a significant contribution to the system’s ability for vibration energy harvesting. It would normally be expected that the harmonic response of a regular Duffing-type oscillator with a single backbone curve asymptotically vanishes after the jump-down event with increasing frequency. However, the transition from single-well to double-well dynamics gives rise to the second backbone curve corresponding to in-well oscillations around each of the stabilized equilibria. Effectively, this behaviour induces a second resonant zone, leading to a high amplitude branch through a second high-energy stable path. Importantly, this second resonant zone is of hardening nature, allowing the harvester to cover higher frequency ranges with high response amplitudes. The two backbone curves (nonlinear regions) overlap only with insignificant damping, indicating that they may be exploited independently for energy harvesting purposes across the frequency spectrum. Nonetheless, even if the choice of system parameters leads to overlapping frequency ranges of the backbone curves, any jump-down from the first curve would land on the high-energy branch of the second, thus maintaining the beneficial contribution. Therefore, this system may act as two separate vibration energy harvesters working in a synergistic manner.

With any system (i.e. \(r\ne 0\)), the backbone curves are computed utilizing the conditions in Eq.s () and (). Figure 6 shows the backbone curves along with the frequency response of the harmonic amplitude obtained for the same system parameters as in Fig. 4, and \(r=0.002\,\hbox {m}\). This induces a breakage of the FRC at the point where the low-energy branch of the symmetric layout loses its stability. Therefore, a third backbone curve appears in the intermediate frequency range of the previously observed resonant zones in Fig. 4. The nonlinear branch, residing in the high frequency range (corresponding to the second backbone curve of the symmetric case), is now detached from the rest of the FRC, even though an overlap exists. In Fig. 3 it was shown that eccentricity leads to asymmetric frequency response, separating the multiple solutions for the static displacement to two distinct regions (in the positive and in the negative space). This observation points to the possibility of the levitating magnet oscillating closer to the top or the bottom static magnets. This behaviour is also observed in the FRC of the harmonic amplitude. Starting from the lowest frequency in Fig. 6a, one can observe a joint FRC shaped around the first two backbone curves. This FRC extends over the whole frequency range (diminishing in an asymptotic manner for \(f>25\,\hbox {Hz})\), corresponding to the positive static displacement seen in Fig. 6b. Furthermore, the region between the backbone curves, where only a single solution exists, connects the two local maxima. Since two local peaks exist for a single curve, then a local minimum also exists in the intermediate region. In fact, the maxima stem from the first condition used in the computations, Eq. (22a), whilst the minima are derived from the second condition, Eq. (22b). This allows for separate manipulation of these points, which in the context of energy harvesting requires the value of the local minimum to be as close as possible to the maxima, so that the response amplitude would retain high values in the intermediate region. On the other hand, the second FRC arising at \(f=19\,\hbox {Hz}\) corresponds to the negative static displacement in Fig. 6b. Thus, in order to fully exploit the potential of the harvester, one would have to impose a jump in the response from the positive space to the negative when the excitation frequency exceeds \(25\,\hbox {Hz}\).

Overall, as for the symmetric case, the onset of an additional backbone curve is equivalent to an additional nonlinear harvester, tuned to the frequency range where this curve would reside. It appears that a small eccentricity would bring beneficial influence on the usage of such a design for vibration energy harvesting, especially when the added curve covers a frequency range of otherwise relatively low response amplitudes.

Figure 6 shows that the backbone curves asymptotically tend to the vertical, as \(x_1 \rightarrow 0\). This is the typical shape of the backbone curves of oscillators with cubic nonlinearities. The physical interpretation is that of a linear-like behaviour of these oscillators when the excitation amplitude is sufficiently low. Even so, the separate backbone curves define multiple peaks in the shape of the FRC, with clear advantages for energy harvesting compared with a linear harvester, for which only a single resonant peak exists. Recalling the analysis in the previous section, it was shown that an intersection point of the backbone curves can exist. If the introduced eccentricity exceeds a critical value given by Eq. (23), then a real, non-negative solution exists for which the backbone curves intersect. Substituting the system parameters used in Fig. 6, it is found that: \(r_{crit} =0.0035\,\hbox {m}\). Thus, selecting higher values (\(r=0.004\,\hbox {m}\) and \(r=0.006\,\hbox {m})\) and repeating the steps to calculate the backbone curves, Fig. 8 shows that above the critical value of eccentricity the backbone curves merge, no longer defining multiple peaks in the FRC for every excitation amplitude. Instead, the merged part of the backbone curve resembles the structure of a typical nonlinear oscillator. Therefore, for relatively weak excitations the examined nonlinear harvester shows no advantage over a linear counterpart. In this respect, \(r_\mathrm{crit} \) is an important design parameter that can enhance the harvester’s adaptability to low excitation amplitudes. It is also noted that this curve results from merging the system’s extrema in a way that the locus of the local minima is now transformed to the locus of the peak amplitudes.

### 4.3 Harvested energy

## 5 Conclusions

The harmonic balance method is employed to analyse the dynamics of a vibration energy harvester with nonlinear magnetic restoring force. Due to the layout of the rotary system, the oscillating magnet is excited by the combined effect of centripetal force and gravity, resulting in a constant and a harmonic excitation component, respectively. The linear frequency of the system varies with the excitation frequency due to the action of the centripetal force. The aim of the analysis is to investigate the structure of the system’s dynamics. The overlapping boundaries of saddle-node bifurcations lead to the appearance of frequency regions with up to seven coexisting solutions, even when a small amount of eccentricity is present in the harvester. Most of these bifurcations result in limited changes in the amplitudes of oscillation with the frequency response dominated by stable branches. Jump phenomena account for the amplitude variations within a narrow frequency range only. Nevertheless, the possibility of a part of the FRC detaching from the main branches exists.

The FRC structure of the studied system is of particular interest. When the excitation is purely symmetric, the frequency response is dominated by two distinct backbone curves, which induce the corresponding resonant zones. The first curve concentrates the nonlinear normal modes, corresponding to oscillations in a single-well potential for frequencies below the \(\omega _n \) threshold. This incorporates a transition to cross-well oscillations when the threshold, leading to a double-well potential, is reached. The second curve describes the nonlinear modes of in-well hardening oscillatory characteristics. The damped response of the system is then determined by the combined influence of these modes. In regular bi-stable oscillators, in-well oscillations exhibit softening characteristics, resulting in interactions between the cross-well and in-well oscillations over a confined frequency range. In the case demonstrated here, the transition from single-well to double-well dynamics occurs with cross-well oscillations as the continuation of the first backbone curve, where the in-well oscillations follow hardening characteristics. The result is two resonant zones, covering relatively distinct frequency ranges, thus establishing a coexisting frame of effectively broadening frequency range of high response amplitudes. One can consider the analogy of two nonlinear harvesters, the responses of which cover different frequency ranges. With the proposed concept, this is achieved through a SDOF oscillator only. In fact, once the frequency is increased beyond the first FRC peak, the low-energy branch is destabilized and the response follows the higher-energy path, leading to the second peak.

In addition, the introduction of eccentricity in the system layout (triggering asymmetric forcing of the oscillator) results in the appearance of an additional backbone curve. This is induced by the loss of symmetry, yielding non-identical equilibria of in-well oscillations, unlike the previous case. A new multi-peak FRC is created that can be conceived synergistically, as the symmetric system was. Nevertheless, the response can return to a single-peak form if higher energy dissipation is attained. The appearance of additional resonant zones in the dynamics of this nonlinear system significantly enhances the energy harvesting capabilities. This is because high velocity response of the oscillating magnet can be sustained through passage over separate hardening resonant zones, thus resulting in broader spectra of power load output.

## Notes

### Funding

The authors wish to express their gratitude to the Engineering and Physical Sciences Research Council (EPSRC) for the financial support extended to the “Targeted energy transfer in powertrains to reduce vibration-induced energy losses” Grant (EP/L019426/1), under which this research was carried out. Research data for this paper are available on request from Stephanos Theodossiades (S.Theodossiades@lboro.ac.uk).

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