A Graded Metamaterial for Broadband and High-capability Piezoelectric Energy Harvesting

This work studies a broadband graded metamaterial, which integrates the piezoelectric energy harvesting function targeting low-frequency structural vibrations, lying below 100 Hz. The device combines a graded metamaterial with beam-like resonators, piezoelectric patches and a self-powered piezoelectric interface circuit for energy harvesting. Based on the mechanical and electrical lumped parameters, an integrated model is proposed to investigate the power performance of the proposed design. Thorough numerical simulations were conducted to analyse the spatial frequency separation capacity and the slow-wave phenomenon of the graded metamaterial for broadband and high-capability piezoelectric energy harvesting. Experiments with realistic vibration sources show that the harvested power of the proposed design yields a five-fold increase with respect to conventional harvesting solutions based on single cantilever harvesters. Our results reveal the significant potential on exploitation of graded metamaterials for energy-efficient vibration-powered devices.


Introduction
Energy harvesting has received considerable attention over the last two decades mainly in the context of transitioning toward the Internet of Things (IoT) architectures with milliwatt-level sensor nodes [1]. The exploration of renewable energy sources not only relates to academic advances, but embraces significant social and economic values [2,3,4]. Since vibration energy harvesting (VEH) leverages one of the most ubiquitous and accessible en-tric transducers on a mechanical transformer under the excitation of an ambient vibration source. The generated AC (alternating current) voltage can then be regulated with a power conditioning interface circuit into a DC (direct current) voltage [1]. Irrespective of the specific mechanical design, transducer, and interface circuit chosen, the two primary design targets for PEH systems are: 1. The high-capability target: to increase the harvested power at resonance; 2. The broadband target: to increase the off-resonance harvested power, i.e., to broaden the harvesting bandwidth.
Many studies have investigated these two targets for PEH systems from the mechanical and the electrical standpoints. To achieve the high-capability target (1), mechanical solutions include an increase in the number of active materials, or a decrease of the equivalent mechanical stiffness [7,8,9,10]. All mechanical solutions yield an increased electromechanical coupling coefficient, which results in a stronger energy harvesting capability [11].
Without altering the mechanical structure, advanced interface circuits [12,13] can also enhance the energy harvesting capability thanks to the increased impedance matching ability from the electrical side [14]. For fulfilling the broadband design target, most research efforts stem from the mechanical engineering community with a variety of available options, namely combinations of multiple vibrators with different resonant frequencies [15,16,17] and introduction of nonlinear dynamics in the vibrator [18,19,20,21]. Nevertheless, in strong coupling systems, the interface circuits can also contribute to the broadband target by additional electrically induced stiffness with phase-variable (PV) interface circuits [22,23].

Engineered materials can be mainly categorized into
Bragg scattering phononic crystals [27] and locally resonant metamaterials [28]. When aiming to harvest mechanical energy from low frequencies to fit the scenarios of IoT devices, the relatively long wavelength of low-frequency ambient vibrations renders the use of Bragg scattering structures inefficient. It is well known that locally resonant metamaterials can surpass the size limitation of Bragg scattering systems, and generate sub-wavelength bandgaps leveraging local resonance mechanisms [29]. Sugino et al. [30,31] pointed out that the utility in using locally resonant metamaterials to enhance energy harvesting lies in their unique properties to slow down the propagation of elastic waves and focus the mechanical energy into the local resonators within or close to the frequency range of a bandgap. Therefore, the high-capability target of PEH can be naturally addressed using metamaterials. Li et al. [32] proposed a piezoelectric cantilever-based metamaterial for simultaneous vibration isolation and energy harvesting. Chen et al. [33] also realized the same dualfunction with a membrane-type metamaterial and further increased the output power with double-layer resonators.
Despite the high-capability PEH by locally resonant metamaterials, the harvested power is noticeable only close to the bandgap frequency [24], hence in a relatively narrow band. Therefore, research efforts have also investigated multifunctional metamaterials for tackling the broadband target [24]. Hwang and Arrieta [34] adopted nonlinear resonators and realized an input-independent metamaterial to broaden the energy harvesting bandwidth. Without resorting to nonlinear dynamics, De Ponti et al. [35,36,37] explored the rainbow trapping phenomena [38,39,40]

Theoretical Analysis
In order to investigate the performance of energy harvesting systems, research efforts have been dedicated to theoretical [41] and numerical methods [35] to determine the harvested power and the bandwidth. Unlike the wave propagation at high frequencies typical of acoustic metamaterials satisfying the assumptions of traveling waves in a long structure when compared to the wavelength [29], a quasi-standing wave dominates the dynamic response of the finite structure under low-frequency vibrations [30].
From the electrical side, without the assumption of pure resistive loads for energy harvesting [35], the equivalent impedance of the AC-DC interface circuit should be considered [41]. In addition, the nonlinear coupling of interface circuits with mechanical structures [42] often renders the numerical simulation difficult due to the intensive computation and memory requirements. For these reasons, a simplified integrated model combining mechanical and electrical lumped parameters is developed in this paper.
The flow chart of the integrated model used in this section is shown in Fig. 1

Graded Metamaterial and Circuit Topology
We base our design on previous work by De Ponti et al. [36,35], which exploited a graded metamaterial and slow waves to enhance the harvested power.  metamaterial-based energy harvesters [30], the effective energy harvesting bandwidth can also be increased due to the spatial frequency separation, quintessential in rainbow devices [38,40]. By combining the contributions of different pairs of parasitic beams located at increasing distances, the broadband ability can also be achieved.
In order to transform the mechanical energy confined inside the metamaterial into electricity, piezoelectric patches are mounted on the parasitic beams. Conventionally, the bulk of research on metamaterial energy harvesting is restricted to the use of resistors as loads of the piezoelectric patches to measure an AC output power [36,43]

Equivalent Lumped Parameters
With reference to the flow chart of the integrated model in Fig. 1, a modal analysis method is implemented in order to derive the lumped parameters of the proposed design from the mechanical side, as shown in the schematic in Fig. 2    w abs =ẅ b +ẅ. The equivalent mass M r , stiffness K mr , damping coefficient D mr , and the equivalent force-voltage coupling coefficient α r can be formulated as: where ψ r is the first order mode shape of the beam section without piezoelectric layer [48] in the r th pair of parasitic beams. ζ r , β r and ω r describe the damping ratio, the coupling related factor and the first order resonance frequency of the r th beams, respectively.
The first-order resonance frequency ω r can be determined by solving the characteristic equation of the r th pair of parasitic beams. Along with the boundary conditions, the mode shape ψ r and the coupling related factor β r are computed. It should be noted that the derived lumped parameters are effective only at the beam's tip. The reaction force at the root of the beam derived from the lumped parameters is slightly different from that derived from the analytical model [48]. However, by properly choosing a relatively large tip mass, these parameters are still valid for calculating the reaction force at the root of the beam [49].
Additionally, we include the electrical side considering the lumped parameters of the interface circuit as shown in  Based on these four steps, the equivalent impedance Z e of SP-SECE interface circuit can be derived with impedance analysis (detailed in Appendix A). The electrically induced damping D er and stiffness K er of each parasitic resonator in the r th pair can be derived based on the electromechanical analogy in this coupling system, namely: When the system operates at frequency ω, the real part R e and the imaginary part X e of Z e form functions of ϕ, whose relationship is illustrated in the two-dimensional impedance plane depicted in Fig. 3

Integrated Model
As shown in Fig. 1, by combining the lumped parameters determined mechanically and electrically, each piezoelectric parasitic beam can be represented by a SDOF parasitic resonator with the SP-SECE interface circuit.
Therefore, the governing equation for each parasitic resonator can be formulated as: where K r = K mr + K er and D r = D mr + D er contain the combined effect of the mechanical and electrical induced stiffness and damping, respectively. As stated previously, the electrically induced components are determined by the chosen interface circuit. For SP-SECE, D er is positive, which means it absorbs mechanical energy and converts it into electricity. In order to evaluate the energy harvesting performance of the proposed design, the displacement amplitude U r of the r th pair parasitic resonators shall be determined.
By adding the reaction forces of each pair of parasitic resonators onto the host beam, the governing equation of the host beam can be expressed as: where δ represents the Dirac function. The integrated model of the graded-harvester is then represented by Eq.

and Eq. 4.
U r can be solved by modal analysis method with the electrically induced components from the SP-SECE interface circuit (detailed in Appendix B). Furthermore, the magnitude of the piezoelectric current of the r th pair of parasitic resonators in series can be expressed as [41]: Taking into account the rectifier loss, switching loss, and freewheeling loss [50] of the SP-SECE interface circuit under the relatively large V oc assumption, the harvested power for each pair of parasitic resonators can be formulated as: where ∆E and E h represent the total extracted energy and the harvested energy in one cycle. The ratio between ∆E and E h is denoted as the harvesting efficiency η h : where γ,Ṽ s ,Ṽ D , andṼ F describe the flipping factor of the SP-SECE interface circuit, the V oc normalized V s , V D and V F , respectively. Consequently, the harvested power is a percentage of the extracted power. All the parameters in Eq. 6 are constant under the specific base excitation condition, except the harvesting efficiency η h . Strictly speaking, the harvested power also changes with the load condition, which influencesṼ s . However, this dependency can be significantly weakened after the conduction of the freewheeling diode D 2 [50].
As a case study, the theoretical harvested power, computed using the integrated model described above, for different pairs of parasitic beams is plotted as a gray line in

Numerical Analysis
The grading design of the metamaterial not only broadens the energy harvesting bandwidth. It also affects the dispersion relationship and the wave field propagation, which further influences the energy harvesting ability of the designed graded-harvester. This section investigates wave propagation within the graded metamaterial using numerical simulations (FEM) and demonstrates the broadband and high-capability energy harvesting characteristic of the graded-harvester.

Grading Profile
In order to evaluate the harvesting ability in the bandgap range, it is necessary to investigate the role of different grading profiles. As shown in Fig. 4, the spatialfrequency analyses of three grading profiles for the para-   Fig. 4 (d) enables more localized modes at the beginning of the bandgap range due to the increased number of longer parasitic beams [43]. This creates a stronger bandgap and suggests a higher energy density, ideal for boosting energy harvesting, and it is thus chosen for the proposed design.

Numerical Results
The dispersion relationship provides a general and fundamental description of the wave propagation characteristics of linear metamaterials. It leads to a better understanding of how the propagating wave could enable higher energy harvesting ability spanning the grading frequency range [40]. Based, therefore, on the cubic-root grading profile, we discuss wave propagation in the graded metamaterial by dispersion analysis. Due to the out-of-plane base 1 It should be noted that the grading discussed here concerns the lengths rather than the resonance frequencies of parasitic beams determined by Eq. 1. Therefore the final spatial-frequency separation curves indicated with dashed lines in Fig. 4 (a)   seen from the imaginary wave numbers. The imaginary wave numbers are more prominent in the graded range than in the non-graded case, which also explains the exponential decay of wave propagation with the graded design.
The slow waves enable longer interaction with the local resonators, which leads to the amplification of the wave fields.
We introduce the amplification factor of the wave field in local resonators as a metric of the energy transfer capability from the metamaterials to local resonators. We firstly define the relative velocity of a local resonator as the velocity difference between its tip and root positions of a parasitic beam:u r =u r −ẇ (x r ). Then the amplification factor can be expressed as the ratio of the relative velocity amplitude of a local resonator to the velocity amplitude at its root position on the host beam:U r /Ẇ (x r ). The efficiency of this step is η h in Eq. 7.
In order to determine the total energy harvesting efficiency η = η m η e η h , we firstly simulate η m of each pair of parasitic beams. From energy conservation, the work done by the external load is finally consumed by the damping effect of the graded-harvester. We use an isotropic mechan-  Fig. 7. The efficiency of the 8 th parasitic beam atṼ s = 1 reaches the peak value at its resonance frequency, after which the efficiency drops to zero due to the spatial frequency separation mentioned above.
Compared to the one-pair case indicated with the blue line, the η of the 8 th parasitic beam increased 160% due to the wave field amplification by the graded metamaterial. This kind of relay of different parasitic beams forms the envelope surface of the total efficiency η, which maintains a high level in the grading frequency range and serves the high-capability energy harvesting target.

Experimental Results
where f in andẇ b represent the harmonic input force and the base velocity. The experimental efficiencies for the 8 th pair of parasitic beams and its one-pair case are 26% and 14%, respectively, which shows the high-capability energy harvesting of the proposed design. They also agree with the numerical results shown in Fig. 7. Broadband energy harvesting can also be naturally realized by varying the different harvesting parasitic beam pairs. Fig. 11 (b) shows the experimental waveform of the piezoelectric voltage v p and the storage voltage V s under 1 MΩ load of the 8 th pair of parasitic beams and its one-pair case. It can be seen that the piezoelectric voltage of the 8 th pair parasitic beams is higher due to the wave field amplification from the graded metamaterial, and the storage voltage is also higher.

Discussion
Locally resonant metamaterials fit the scenario of utilizing low-frequency vibrations for self-powered IoT devices.
With reference to statistic studies [55], we discuss the PEH performance of the proposed graded-harvester with respect great potential to power sensors or microcontrollers and realize self-powered IoT devices.

Conclusion
Based on the idea of graded metamaterials, this paper proposes a graded metamaterial-based energy harvester for broadband and high-capability piezoelectric energy harvesting focusing on ambient vibrations (<100 Hz).
The broadband energy harvesting target has been inherently satisfied by the spatial-frequency separation of the graded metamaterial design with an efficient grading profile. Furthermore, the high capability energy harvesting target has been investigated by dispersion analysis of lowfrequency wave propagation to reveal the slow wave phenomena and the wave field amplification mechanism of the graded metamaterial. By combining the two advantages of the graded metamaterial with the two main goals for piezoelectric energy harvesting, the power performance and the energy harvesting efficiency of the graded-harvester are thoroughly discussed with theoretical, numerical, and experimental analyses. Finally, experiments were carried out to validate the performance of the graded-harvester. It is shown that coupling of the graded metamaterial with the self-powered interface circuit results in a five-fold increase of the harvested power with respect to the conventional harvesting solution commonly adopted in IoT devices. Therefore our proposed design opens up new potential for self-sustainable IoT devices.

SECE interface circuit
In phase P 3 as shown in Fig. 3, the L i -C p resonance should be introduced at the i eq crossing zero point by conducting the switching path with T 2 . Nevertheless, the cascaded connection of T 1 and T 2 renders the conduction of Assuming that each bias-flip action takes much less time than a vibration cycle, the piezoelectric voltage v p can be formulated by the following piece-wise equation: cos ϕ − cos(ωt), ϕ ≤ ωt < π + ϕ; − cos ϕ − cos(ωt), π + ϕ ≤ ωt < 2π + ϕ. φ n (x)η n (t), where φ n and η n represent the mode shape and modal coordinate of the n th mode, respectively. By substituting this latter into Eq. 4, applying the orthogonality relationships for φ m , and assuming harmonic excitation and solutions [30], the relative displacement amplitude of r th pair of parasitic resonators can be expressed as: H n φ n (x r ) (1 + K er /k mr ) ω 2 r + j2 (1 + D er /D mr ) ζ r ω r ω − ω 2 , (B.1) where W b and H n represent the amplitude of the harmonic base excitation and the n th mode modal amplitude, respectively.
Then, by plugging Eq. B.1 into the modal form of Eq.
4, the governing equation can be simplified as: