## Abstract

By exchanging the internal energy between coupled vibration modes, internal-resonance-based energy harvesters may provide an effective solution to broadening and enhancing bandwidth and power performance in dealing with natural vibration sources. With the development of piezoelectric-based transducers, thickness and face shear coefficients in proper piezoelectric elements can also generate power output from shear deformation on the core vibrating elements. However, in most cantilever-based energy harvesters that focused on bending modes, the shear responses were neglected. In this paper, we present an internal-resonance-based piezoelectric energy harvester with three-dimensional coupled bending and torsional modes, for the first time. The fine-tuned system leverages a two-to-one internal resonance between its first torsion and second bending modes to enhance the power output with piezoelectric effects. The dynamic behavior implies the coexistence of in-plane and out-of-plane motions under a single excitation frequency, and the corresponding strain changes in the bending and shear directions are captured by bonded piezoelectric transducers. Dependence between excitation levels and the internal-resonance phenomenon is justified as a critical system parameter study; the results also indicate that an intriguing non-periodic region exists near the center frequency. The outcomes of this study feature a multi-directional and multi-modal energy harvester that displays rich dynamic behaviors. The operational bandwidth is promising for broadband energy harvesting, and the output voltage is enhanced by capturing both in-plane and out-of-plane motions at the same time.

## 1 Introduction

### 1.1 A Brief Review on Vibration Energy Harvesters.

As the trend of state-of-the-art electronic devices progressively becomes miniaturized and wireless, it is doubtful whether conventional electrochemical batteries are still promising in such low-power consumption devices. To achieve targeted applications such as powering up the Internet of Things (IoT), sensor networks [1,2], and medical implants [3], routine replacement and relatively high expense of conventional batteries limit the continuous and long-term operations of the devices. With the required power level of these devices reduced to sub-milliwatt, vibration-based energy harvesting techniques are feasible to provide continuous power supplies by converting environmental vibration sources into electrical energy. Nonlinear vibration-based energy harvesting has recently been shown to be an effective means of improving energy harvesting efficiency by broadening effective bandwidth and capturing a wide frequency range of vibration sources. The selected/designed nonlinear techniques, which should be effective for specific themes, are primarily determined by the characteristics of ambient vibration sources, including the excitation level and direction.

Multi-stable energy harvesting is a typical approach for broadening the operational bandwidth. Yan et al. [4] utilized magnetic restoring forces as externally induced nonlinearities to introduce bistability. By altering the system design parameters, such as the distance between tip magnet(s) on the vibrating structure and the stationary magnets, the system displayed softening and hardening frequency responses [5]. Wang et al. [6] investigated a tri-stable energy harvester that had enhanced performance with low-orbit vibration. Bistability can also be achieved by pre-loading approaches on beam-mass structures [7–12] or snap-through phenomena of laminated plate structures [13,14]. The motion types of multi-stable energy harvesters are highly dependent on acceleration levels. Periodic inter-well motion needs a sufficiently high excitation level to obtain enough energy for crossing the potential barrier between the two wells. Under insufficient excitation levels, monostable systems surpassed bistable systems for both operational bandwidth and peak power level [15–17]. In dealing with small ambient excitation levels, parametrically excited energy harvesters also need design techniques to reduce the initial threshold amplitudes [18–23]. In addition, piezoelectric materials with high power density compared with electromagnetic type transducers are commonly utilized as strain-charge converters in vibration-based energy harvesting devices [24,25]. On the other hand, with additional damping effects, large-amplitude oscillations are more challenging to be triggered when piezoelectric benders are bonded to the vibrating structure (i.e., a cantilever beam). To scavenge naturally varying vibration sources, multi-modal and multi-degrees-of-freedom (MDOF) devices utilize multiple adjacent natural frequencies to attain multiple peaks over a certain frequency range [26–28]. Energy harvesters with multiple adjacent natural frequencies can provide a broader operational bandwidth, and some prototypes are also efficient under multi-directional ambient environmental vibration sources. However, the main concerns are the complexity of design and the sacrifice of power density. For some of the linear multi-modal energy harvesters, when the natural frequencies are commensurable or nearly commensurable, a nonlinear phenomenon named internal resonance can be possessed, which can potentially be used for broadband energy harvesting.

### 1.2 Internal-Resonance Benefits for Energy Harvesting.

One of the typical phenomena is that the frequency response under harmonic excitations bends to increasing and decreasing frequency directions from the center frequency as a double-jump phenomenon, which is considered an alternative solution to broaden the operational bandwidth [29]. In the presence of internal resonances, such dynamic responses can have broader resonance regions when the excitation frequency is within the resonance regions of the internally coupled modes [30–32]. However, in most cases, a two-to-one (or three-to-one) internal resonance in fabricated simple cantilever beam structures can be challenging to achieve for the first few primary resonances. Thus, tuning techniques such as inducing external magnetic coupling effects were employed correspondingly [33–36]. The natural frequencies of the devices can be tuned into designed values by the restoring forces from magnetic interactions. Alternatively, a fine-tuned structure can be obtained by adding auxiliary structures to the main structure as an MDOF system; for instance, L-shaped beam and mass structures [37–42]. L-shaped structures seem more practical than simple clamped-free beam-mass structures; with tailored system parameters such as beam dimensions and additional tip/side masses, commensurable natural frequencies can be obtained accordingly.

In spite of in-depth investigations on L-shaped energy harvesting devices using internal-resonance phenomena, achieving internal resonances between the first two bending modes requires a sufficiently large ratio between the attached masses and the beam(s). In some cases, a large side mass block is essential to connect the two orthogonal beams because of the mismatched thicknesses and widths, which introduces more complexity to the system. On the other hand, for L-shaped energy harvesters with internal-resonance phenomena, the selections of natural frequencies are always between two bending modes (i.e., the first two transverse modes), and the harvested electrical energy is limited to in-plane motions. These devices have raised a practical concern, with the rapid development of piezoelectric transducers, the shear piezoelectric constants *d*_{15} can also provide efficient power output for energy harvesters on shear mode operations [43–45]. Consequently, an internal-resonance-based energy harvester with nonlinear modal couplings between bending and torsional modes has the potential to achieve enhanced power output with simultaneous bending-torsion responses.

### 1.3 Contributions of This Work to the Field.

To provide an insight into the *internal-resonance phenomenon between bending and torsional modes*, this paper, for the first time, proposes an internal-resonance-based energy harvester with coupled in-plane and out-of-plane motions (i.e., internally coupled torsional and bending modes). Specifically, we study the response of a multi-mode energy harvesting device (i.e., an L-shaped beam-mass structure with bonded piezoelectric elements as transducers) that exhibits *a two-to-one internal resonance between its first torsional and second bending modes* under external excitations. It is of interest to explore the feasibility of combining both in-plane and out-of-plane motions under one single excitation frequency, the possibility of enhancing the harvested voltage/power from piezoelectric transducers from both bending and torsional motions, and broadening the overall operational bandwidth by double-jump phenomena.

## 2 Experimental Setup and Design

The first torsional and second bending modes are shown in Figs. 1(a) and 1(b). A finite element method (FEM) via ansys (version 18.2) was adopted for simulation and pre-testing of the proposed device. The results display in Figs. 1(a) and 1(b) show that, without an internal-resonance phenomenon, the first and second bending modes exhibit in-plane motions (*Y*–*Z* plane), and out-of-plane motions (*X*–*Z* plane) occur at the first torsion mode. Although most L-shaped energy harvesters with internal-resonance focused on the first bending mode, tuning the first bending and first torsional modes required large mass blocks, which caused considerably large mass ratios between additional mass blocks and beam(s) that could possibly damage the structure due to rapid large-amplitude oscillations. In addition, the paper attempts to examine whether the internal resonance can still be achieved for broadband energy harvesting at higher modes. Thus, we look into its second bending and first torsional modes to examine the functionality of coupling them together to introduce an internal resonance. The first bending mode of the proposed device that occurs at 6.7 Hz could also be utilized for energy harvesting as a linear resonator. The out-of-plane (torsional) motion of the L-shaped device is predominantly determined by the inertia of the horizontal beam and the tip mass, the torsional deformation of the vertical beam in the first three natural frequencies can therefore be neglected. The preliminary tests demonstrate that the optimized direction of base excitation *W*_{b}(*t*) is in the *X*-direction and the *Y*-direction for the first torsional and second bending modes, respectively. A schematic of the proposed device that undergoes both bending and torsional motions is shown in Fig. 1(c). Because the strain changes of the horizontal beam contain both bending and torsional deformations, piezoelectric transducers are chosen to be bonded onto the horizontal beam for efficient strain-charge conversion.

Figure 1(d) presents the fabricated device and the main experimental setup. To tune an internal resonance between the first torsional and second bending modes, a 0.5-mm-thick, 15.2-mm-wide aluminum (*ρ*_{Al} = 2650 kg/m^{3}) beam is bent into an L-shaped (right-angled) structure, with a 79.5 mm (*l*_{h}) horizontal beam (clamped end) and a 74 mm (*l*_{v}) vertical beam (free end). A 9.5 gram tip mass is fixed onto the free end for tuning purposes. To obtain both bending and torsional motions for energy harvesting purposes, two MFC (macro fiber composite) 0714-P2 made of piezoelectric (PZT) material CTS3222HD are bonded onto the horizontal beam. The one with the fibers parallel to the horizontal beam length measures the bending motion and the other with the fibers perpendicular to the horizontal beam length measures the torsional motion. Two Wenglor CP24MHT80 laser sensors measure the bending displacement at the center line position along the horizontal beam length in the *Z*-direction and the torsional displacement in the *X*-direction by attaching an external plane that is placed parallel to the *Z*-direction (Fig. 1(d)). A shaking table APS 113, which is driven by an APS 115 amplifier, provides harmonic excitations *W*_{b}(*t*) = *F*cos(2*πft*), where *F* is the forcing amplitude and *f* is the excitation frequency in unit of Hz. The acceleration amplitude of the base excitation is recorded by an accelerometer Kistler 8774A50 and its coupler Kistler 5134B, and the generated voltage from each of the PZT layer is measured across a resistive load of *R*_{L} = 150 kΩ. All the measured signals are recorded to a data acquisition board NI USB-6281. The steady-state frequency response of the proposed device is obtained by employing both an upward and a downward sweep with 0.05-Hz frequency internal, 10 s of settling time, a 1-kHz sampling rate and constant base acceleration levels.

## 3 Results and Discussion

With preliminary testings and optimizations completed, the measured first three modes of the L-shaped device are *f*_{1} = 6.7Hz, *f*_{2} = 10.9Hz, and *f*_{3} = 22.1 Hz. The proposed structure leverages the relationship between *f*_{2} and *f*_{3} of a 1:2 frequency ratio to introduce a two-to-one internal resonance. The device is firstly excited under 0.4 g peak-to-peak (g = 9.81*N*/*s*^{2}) up (from 9 Hz to 14 Hz) and down (from 14 Hz to 9 Hz) base accelerations in *X*-direction as out-of-plane excitations at the first torsional mode (*f* ≅ *f*_{2}). Figure 2 depicts the voltage–frequency and displacement–frequency responses from the PZT and laser sensors, respectively. Two peaks are revealed at 10.15 Hz and 12.2 Hz in both bending and torsion motions, which implies there is an internal resonance between the torsional and bending modes and resonance energy is transferred to the second bending mode. The frequency responses show that the two branches are asymmetrical, and there exists a non-periodic region between the left peak and the center frequency region (Figs. 2(a) and 2(b)). The right branch has an upward jump at 11.65 Hz, but the peak on the left occurs at 10.95 Hz, which is within the non-periodic region.

The torsional response distinguishes the left branch and the non-periodic region in the frequency response compared with the bending response. It is particularly important to see that although the energy transfers between the two coupled modes in the presence of an internal-resonance phenomenon, no evident jump occurs between the upward and downward sweeps. For the system behaviors at the off-resonance regime, the bending motions (in the *Y*–*Z* plane) have an almost zero response while torsion motions (the *X*–*Z* plane) exhibit a non-zero response, which is due to the direction of the base excitation. When the direction of excitation is parallel to the torsional motion, the device is prone to oscillate in the *X*–*Z* plane because of the inertia of the tip mass block and horizontal beam. Figures 2(c) and 2(d) depict the harvested voltages from the bending and torsional PZT layers, respectively. Based on the level of displacement, the bending motion results in a larger peak voltage, which is consistent with the measured results from the laser sensors. The peak-to-peak voltages *V*_{p−p} of bending and torsion motions are up to 9.15 V and 6.52 V, respectively. It can be seen that the output voltage from the torsional motion resembles the bending motion. Although the torsional PZT is bonded perpendicular to the beam length, it still generates charges in the fiber perpendicular direction. Hence, the output voltages in Figs. 2(c) and 2(d) consist of both bending and torsional motions and display a similar trend. As Fig. 2(b) is the measured torsional displacement only, the curve is different from Figs. 2(a), 2(c), and 2(d). Because the designed width of the beam is relatively narrow, the current *d*_{31} type PZT cannot efficiently convert the small amplitude of the torsional motions in the off-resonance regime. Thus, the voltage–frequency response displays a very small amplitude in the off-resonance regime in Fig. 2(d).

The steady-state responses of the proposed device under 0.4 g in-plane (*Y*-axis) base excitations at the second bending mode are shown in Fig. 3. The experimental results present large-amplitude oscillation and double-jump phenomena, and the two branches are bent into opposite directions from the center frequency. The overall resonance region is from 21.1 Hz to 23.3 Hz. The results shown in Figs. 2 and 3 demonstrate that the dynamic responses provide key evidence that the device exhibits a two-to-one internal resonance between the first torsional and second bending modes. Between the coupled two modes, excitation frequencies at the lower mode can trigger large-amplitude oscillations at higher modes, and vice versa. With piezoelectric transducers in both bending and torsional directions, the power density of the device is greatly enhanced. As shown in Fig. 3(a), the response of the downward sweep case displays asymmetric branches that the left branch has a much higher peak than the right one, and the bandwidths are slightly broader than the upward case (0.55 Hz versus 0.45 Hz). The frequency responses of the bending motion shown in Fig. 3(b) depict a more symmetric resonance region in terms of bandwidth and amplitude. The peak-to-peak voltages from bending (Fig. 3(c)) and torsional (Fig. 3(d)) PZTs are 6.63 V and 4.69 V, respectively.

To understand the difference between the dynamic behaviors across the bending and torsional motions and to examine the internal-resonance phenomenon further, the time trace and frequency spectral diagrams of the two internally coupled modes under base excitations are displayed in Fig. 4. The peaks of the right branches for both cases are selected for comparison. As shown in Figs. 4(a)–4(d), the frequency components show coexisting responses at 11.5 Hz and 23 Hz for bending and torsional motions, and there are no contributions for the first bending mode *f*_{1} (6.7 Hz). The time trace plots from torsional PZTs in Figs. 4(b)–4(d) show the coexistent responses clearly. Thus, we can conclude that the two-to-one internal resonance from the experimental results falls between the first torsional and second bending modes. It is predictable that the frequency component of the torsional PZT have lower responses to the first torsional mode, because the selected transducer is a *d*_{31} type bender that will still generate charges in the fiber perpendicular directions. For future industrial applications, piezoelectric transducers work for detecting shear stress/strain and can further enhance the power performance for torsional vibration.

The experimental results presented in Fig. 5 are the voltage phase plane diagrams measured at the steady-state responses at the left peak, near center frequency and right peak positions for both first torsional and second bending modes. The coexistent responses for bending and torsional motions are agreed with the frequency spectrum shown in Fig. 4. Although the time series and frequency spectral diagrams of the bending PZTs may not obviously illustrate if the torsional mode is internally coupled, the phase plane diagrams identify the two coexistent responses. Given that there exist initial geometric nonlinearities in the system, the phase plane diagrams depict asymmetric motions. Likewise, the left and right branches in frequency responses (Figs. 2 and 3) are not perfectly symmetric. At the center frequency position of an internal-resonance phenomenon, conceptual systems generally display the lowest amplitude. However, as shown in Fig. 5(b), under base excitations at the torsional mode, the device exhibits large-amplitude oscillations at 11.5 Hz, which is much higher than the response at the center frequency of the second bending mode shown in Fig. 5(e). As the amplitude of the center frequency position remains an effective level of energy harvesting, the device displays a continually effective bandwidth when the excitation is near/at the first torsional mode.

As mentioned earlier, the region between two peaks with non-uniform amplitudes shown in Fig. 2 implies there are rich dynamic behaviors in the system, and the output voltage from the bonded PZTs is even higher than those of the peak positions of the left and right branches. In order to study the different dynamics in the system, the time series of the overall region and time trace of each frequency interval are shown in Figs. 6(a) and 6(b), respectively. Eight different motions are revealed in Fig. 6(b); the results suggest that, under harmonic sweep excitations, when the base excitation frequency is within the first torsional mode and the excitation direction is in the X-direction, the system response contains non-periodic motions between the two peaks, and the bandwidth and the location of this region remain the same in either up or down sweep case.

It is intriguing to investigate the effects of the forcing excitation amplitude to demonstrate if a certain threshold amplitude exists in the internal-resonance phenomenon. The bandwidth performance of the proposed device under small excitations is also examined in this parametric study. Figure 7 shows the frequency responses of the device under different excitation levels. As can be seen, the internal resonance exists even under a 0.2 g acceleration level. For excitations within the first torsional mode (Figs. 7(a) and 7(b)), the left branch has a narrow region that contains nonunique solutions between the upward and downward harmonic sweeps for all three excitation levels. The effective bandwidths in the bending motion from the 0.2-g to 0.4-g cases with a 0.5-mm reference level are 1.35 Hz, 1.75 Hz, and 2.05 Hz, respectively. With increasing levels of base excitation, the bandwidth increments become slightly smaller for the cases that are excited at the first torsional mode. For the second bending mode, the double-jump phenomena are presented in Figs. 7(c) and 7(d). The two branches are bent into opposite directions from the center frequency at 22.3 Hz, the left branch in bending motion exhibits a much broader nonunique solution region, while the torsional motion has more symmetrical bifurcations. Unlike the first torsional mode, the bandwidths for the 0.3 g case are closely similar to the 0.4 g case (2.15 Hz versus 2 Hz). However, under 0.2 g base excitations, the bandwidth has a substantial drop to 1.5 Hz. The comparison illustrates that, nonlinear modal couplings occur in all three cases, the amplitude of vibration still depends on the base excitation level, although with a further increased excitation level, its impacts on the bandwidth increment become smaller. The non-periodic region can still be captured at a 0.2 g base excitation, although the bandwidth becomes narrow and closer to the center frequency at 11.15 Hz, the left branch in bending motion (Fig. 7(a)) still has a lower peak amplitude than the peak amplitude in non-periodic region. In terms of the voltage level shown in Fig. 8, the power dissipated in the loading resistor can be obtained by *P* = *V*^{2}/*R*_{L}. For excitations at the first torsional mode, the peak power levels measure from bending and torsional PZTs are up to 0.56 mW and 0.28 mW, respectively. The peak power levels reduce to 0.29 mW and 0.17 mW for bending and torsional PZTs with relatively lower-amplitude oscillations, respectively, when the excitations are within the resonance region of the second bending mode.

## 4 Conclusions

The work attempts to enhance the power and bandwidth performance of a vibration-based energy harvester by leveraging an internal resonance between bending and torsional modes on a simple L-shaped beam-mass structure. An enhanced power density is achieved as both bending and torsional strain changes are utilized for strain-charge conversion; a broader overall bandwidth is obtained in the presence of the internal resonance.

In summary, the paper presents a two-to-one internal-resonance-based piezoelectric energy harvester with internally coupled bending and torsional modes in an L-shaped structure. With fine-tuned system parameters, a two-to-one internal-resonance phenomenon was revealed between the first torsional and second bending modes under a low ambient frequency range. The system with nonlinear modal couplings exhibited both in-plane and out-of-plane motions simultaneously under a single excitation frequency and featured rich nonlinear dynamic behaviors such as the double-jump phenomena. In the presence of the internal resonance, the frequency responses of the coupled two modes were bent into opposite directions from center positions, which yielded broader operational bandwidths for both modes. With energy exchanges between the two modes, base excitations within the resonance region of the lower mode enabled large-amplitude oscillations at the higher mode and vice versa, which resulted in higher power density than conventional multi-mode/MDOF devices. In order to optimize the harvested power level to a greater extent and utilize the piezoelectric transducers more efficiently, with bending-torsion nonlinear modal couplings, the torsional strain changes due to the out-of-plane motions were also captured by PZTs and converted into electrical form. The effects of excitation levels were investigated as a parametric study to justify the robustness of the device under small excitation levels. To further enhance the power output, fully covered piezoelectric materials (i.e., unimorph or bimorph) that can more efficiently convert shear strain changes are suggested in future applications. Numerical and analytical investigations are recommended for further studies to investigate the nonlinear response and modal interactions between the bending and torsional modes of the proposed energy harvester.

## Acknowledgment

This work has been supported by the Australian Government Research Training Program. The support is greatly appreciated.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.