A frequency up-conversion rotational energy harvester with auxetic structures for high power output

Rotational energy harvesters (REHs) have been explored to substitute conventional batteries for small electronic sensors. However, there still exists a challenge for REHs to scavenge sufficient energy under low-rotational-frequency excitations. In this paper, we propose a plucking REH with auxetic structures, which utilizes the frequency up-conversion and auxetic structures to enhance the power output of the rotational energy harvesting under low-rotational-frequency excitations. Finite element simulation is performed to analyze the performance of the proposed REH. The simulation results match well with the experimental ones. When the magnet distance is set to be 2.5 mm, the maximum power output of the auxetic plucking REH (APREH) is found to be 1.43 mW at 1.1 Hz, which is increased by 686% compared with the conventional plain plucking REH. In addition, compared with typical plucking REHs, the proposed APREH achieves the highest power output under low-rotational-frequency excitations, which validates the advantage of adopting auxetic structures in plucking REHs.


Introduction
With the rapid development of Internet of Things, wireless sensing has been widely adopted for monitoring the conditions of human body, industry machines, and buildings; thus, the power supply technology for remote wireless sensors has * Authors to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. received extensive attentions. To overcome the drawbacks of conventional batteries, mechanical energy harvesters [1][2][3][4][5] have been extensively investigated to supply sustainable power for wireless sensors, which can reduce environmental pollution and maintenance costs. Rotational energy harvester (REH) plays an important role in mechanical energy harvesting due to the widespread rotational energy such as human motion, automotives, and wind turbines. There are four commonly used mechanisms to transform mechanical energy into electrical power including electrostatic [6], electromagnetic [7], piezoelectric [8][9][10], and triboelectric [11,12] mechanisms. Among them, the piezoelectric one is widely adopted for its ease of integrability and high power density [13][14][15]. The traditional small-scale cantilevered energy harvester can produce the maximum power near the high resonant frequency. However, with a slight frequency shift, the performance of the energy harvester would drop dramatically [16]; thus, the power output of the energy harvester is not desirable under low frequency excitations. Considering the widespread low frequency rotational excitations in the environment, it is crucial to improve the power output of REHs under low-frequency excitations.
To overcome the drawback of narrow working bandwidth in conventional REHs, several methods have been proposed, such as nonlinearity [17], multi-beam [18], centrifugal force [19], and frequency up-conversion [20]. Frequency up-conversion is a process in which the low frequency excitation can be converted to the high frequency vibration of the structure. Impact, mechanical plucking, and magnetic plucking are three commonly used frequency up-conversion mechanisms. Fang et al [21] proposed a rotational impact energy harvester with centrifugal softening effect, which can improve the performance of the REH at low frequencies. Rui et al [22] proposed a three-beam impact energy harvester consisting of an exciting beam, a harvesting beam, and a protecting beam. In the testing, the three-beam impact energy harvester can obtain 52.1 µW at 4.2 Hz excitation. A rotational piezoelectric energy harvester based on mechanical plucking method was explored to harvest the energy from the human knee during walking, where a piezoelectric beam is deflected by a plectrum to vibrate freely [23]. From the experimental results, the energy harvester can achieve maximum power output of 830 µW at 1 Hz. Fang et al [24] theoretically and experimentally studied the vibrational interference in mechanical plucking energy harvesting, which can provide an insight into the working mechanism of the vibrational interference over a wide rotational frequency range. Other than impact and mechanical plucking, another non-contact frequency up-conversion technology, magnetic plucking, has been proposed for reducing the fatigue of cantilever-like structures. A piezoelectric energy harvester was proposed to be installed on a wind turbine, which can convert the low frequency rotational motion of the turbine (0.08-0.5 Hz) into electric energy based on magnetic plucking [25]. The resulting energy of 1.2 J for 24 h operation proved that the proposed design can meet the energy requirement of autonomous monitoring of wind turbine. Xie et al [26] explored a bi-stable REH that consists of a piezoelectric buckled beam, a center magnet, and a driving magnet. Over a rotation frequency range from 1 to 14 Hz, the proposed design can achieve power output from 6.91 to 48.01 µW. Rashidi et al [27] designed a REH adopting multiple driving magnets and piezoelectric beams for enhancement of power output, which can operate at a wide range of rotational speeds.
In addition to the working bandwidth, several methods have been exploited to enhance the power output of REHs including mechanical structure designs, multi-stable systems [28], and circuit designs. Different from the regular rectangle beam, a trapezoid cantilever beam was utilized to enhance the power output of the rotational piezoelectric energy harvester [29]. The simulation results showed that the stress distribution of the trapezoid beam was higher than that of the rectangle beam, which can contribute to a 60% increment in power output. He et al [30] proposed a magnetic excitation REH that can improve the generated energy by changing the installation angle of the magnet. Fu and Yeatman explored a rotational harvester with bi-stability and frequency up-conversion [31]. When working in the periodic double-well mode, the energy harvester was capable of achieving a larger vibration amplitude, which resulted in a 200% increase in power output compared with the harvester without bi-stability. A tri-stable piezoelectric energy harvester was proposed, which was easier to cross the potential barrier than the bi-stable one, resulting in better performance under low excitation levels [32]. Other than bi-stable and tri-stable systems, a multi-stable piezoelectric REH was studied to improve the performance in lowfrequency rotational motion [33]. By adjusting positions of magnets, four stable equilibrium positions can be achieved, resulting in shallower potential wells than bi-stable and tristable systems. In the testing, the multi-stable energy harvester generated much higher voltage than the bi-stable and tri-stable ones under low-frequency excitations. Fu et al [34] utilized a parallel synchronized switch on the inductor circuit to enhance the power output of piezoelectric energy harvester.
The aforementioned methods can be utilized to improve the performances of REHs under low frequency excitations. Recently, several researchers have utilized auxetic structures to enhance the power output of vibration energy harvesters [35][36][37]. Unlike normal structures, auxetic structures can expand or contract simultaneously in the longitudinal and lateral directions under external load, which is defined as negative Poisson's ratio [38]. Due to this special property, auxetic structures show excellent performance including lightweight, increased shear stiffness, high energy absorption and fracture resistance [39,40]. For piezoelectric energy harvesting in bending mode, the power output is proportional to the average longitudinal and lateral stresses of the piezoelectric patches. Due to the negative Poisson's ratio, auxetic structures can expand or contract simultaneously in two directions, resulting in the same sign of average longitudinal and lateral stresses. Therefore, auxetic structures can superimpose the power generated by d 31 and d 32 modes of the piezoelectric patches for improving the power output. There are several benefits to adopt auxetic structures in piezoelectric energy harvesters. First, since the auxetic structure can be manufactured directly on the substrate of the energy harvester, the auxetic structure shows the advantage of compact structure compared with other methods of increasing power output that require external magnets or stoppers. In addition, the porous property of the auxetic structure can reduce the resonant frequency of the energy harvester, which is beneficial for low-frequency energy harvesting. Furthermore, the auxetic structures, with different shapes, sizes, and connectivity, provide abundant types of designs for energy harvesters.
In this paper, a plucking REH with auxetic structures is proposed, by utilizing the frequency up-conversion method and auxetic structures to improve the performance of the harvester in low-frequency excitation conditions. The remaining sections of this paper are organized as following: section 2 describes the design details and finite element simulation of the proposed REH. Section 3 shows the experimental validations including the open-circuit test, impedance matching, maximum power output measurement, and parametric studies. A comparison of the proposed REH with other typical plucking REHs is made to show the advantage of the proposed design of generating high power output at low rotational frequencies. The key conclusion is given in section 4.

Design concept
The schematic diagram of the proposed REH is plotted in figure 1. A cantilever beam is fixed on a holder, which is made of polylactic acid by 3D printing. The auxetic structure is located at the root of the cantilever beam, which is completely covered by a piezo buzzer. In addition, a tip magnet (NdFeB) is mounted on the free end of the cantilever beam. Another driving magnet is attached to a rotating plate. During each rotational cycle of the plate, the magnetic force between the two magnets can pluck the cantilever beam, resulting in the free vibration of the beam. Via the direct piezoelectric effect, the piezo buzzer can transfer the dynamic vibration of the structure to the electric voltage.
As shown in figure 2(a), to explore the effects of the auxetic structure on the performance of the rotational energy harvesting, two types of plucking REHs with the auxetic substrate (APREH) and conventional plain substrate (PPREH) are compared. It can be seen that the two types of substrates have the same length (l b ), width (w b ), and thickness (t b ). The detailed geometric parameters of the auxetic structure are introduced in figure 2(b), which is called re-entrant hexagonal unit cell. Under longitudinal compression (along 1-axis), the auxetic structure contracts laterally (along 2-axis) rather than expanding like the structure with normal positive Poisson's ratio. This property of the auxetic structure is called negative Poisson's ratio. Due to the simultaneous expansions or compressions of the auxetic structure in two directions, the longitudinal (σ 11 ) and lateral stresses (σ 22 ) of the auxetic structure show the same sign. This same sign of the stresses can improve the power output of the REH, which will be explained in the following section.

Finite element simulation
Finite element simulation in COMSOL is performed to analyze the performance of the proposed REH. As shown in figure 3(a), the magnetic field module is first adopted to calculate the interaction force between the tip and driving magnets. At the initial position, there exists a 3 mm gap between the two cuboidal magnets. Moreover, the two magnets are assumed to own uniform and constant magnetic polarizations with parallel orientations. A cuboid area (500 × 500 × 10 mm) is utilized to simulate the air domain with magnetic insulation boundary condition, and the rotational radius of the driving magnet is set to be 170 mm. The key dimensions and materials of the finite element model are listed in tables 1 and 2, respectively. As the driving magnet rotates, the magnetic flux density distributions are plotted in figures 3(b)-(d), where the rotational angle of the driving magnet is set to be 0.02, 0, and −0.02 rad, respectively. When the rotational angle is 0, in figure 3(c), it can be observed the direction of the magnetic field between the two magnets is mainly longitudinal (along 1-axis). Therefore, the corresponding longitudinal magnetic force reaches the maximum value, while the transverse magnetic force (along 3axis) is nearly zero. On the other hand, in figures 3(c) and (d), the magnetic field between the two magnets shows both longitudinal and transverse components, resulting in longitudinal and transverse magnetic forces.
Based on the above simulation, the relationship between the magnetic force and rotational angle can be obtained as shown in figure 4. It can be found that, at 0 rad, the longitudinal magnetic force reaches the peak value of 2.22 N, while the transverse one is almost zero. In addition, at 0.02 and −0.02 rad, the amplitudes of the transverse and longitudinal forces reach 1.49 and −1.58 N, 0.24 and 0.23 N, respectively. In fact, the longitudinal force curve tends to be an even function, while the transverse force curve approximates an odd one. The simulation results of the magnetic force match well with the previous analysis of the magnetic flux density distribution.
Taking the magnetic force as the driving force for the REH, an electromechanical coupling model can be built to derive the power output of the REH under plucking. The solid mechanics component is used to describe the mechanical properties of the energy harvester, where the Rayleigh damping is adopted to provide a source of energy dissipation to ensure the accuracy of the dynamic analysis [41,42]. The Rayleigh damping coefficients should be adjusted based on the resonant frequency and the corresponding quality factor of the system. Then, the AC/DC component can build the static electric model of the piezoelectric patch. Moreover, the electric circuit component is utilized to analyze the power output of the piezoelectric patch connected to the outer resistance. To achieve the electromechanical coupling effect, Multiphysics coupling solver would be adopted to solve the model, which could achieve data interaction between different components (the solid mechanics, the AC/DC, and the electric circuit).
The exploded configuration of the electromechanical coupling model is plotted in figure 5(a), the fixed boundary  condition is applied to one end of the cantilever beam, and the tip magnet is bonded to the free tip of the beam. The piezo patch and brass film form a piezo buzzer that is attached to the beam through a layer of epoxy. Since the epoxy completely covers the auxetic structure, the epoxy and auxetic structure show the same pattern. It should be mentioned that the cantilever beam is made of polylactic acid, which is an insulator; thus, there would not be current produced in the beam due to the magnetic flux variation as the driving magnet rotates. Moreover, the sweeping and triangular meshing are adopted in meshing as shown in figure 5(b).
When the distance of the tip and driving magnets is set to be 3 mm, under plucking, the displacement distributions of the substrates in APREH and PPREH are depicted in figure 6. It can be seen, in figures 6(a) and (c), that the longitudinal displacements of the two substrates (along 1-axis) are both positive, which means the substrates are all stretched. For PPREH in figure 6(d), the substrate contracts transversely (along 2-axis), showing the normal positive Poisson's ratio. On the other hand, for APREH in figure 6(b), the auxetic structure shows transverse expansion, indicating the negative Poisson's ratio. Since the piezo patches are attached to the substrates, the piezo patch in APREH expands or contracts simultaneously in longitudinal and transverse directions, resulting in the same sign of the average longitudinal (σ 11 ) and transverse stresses (σ 22 ) in the piezo patch. However, in PPREH, the deformation directions of the auxetic structure in two directions are opposite, leading to the opposite signs of the two average stresses in the piezo patch. The average stresses of the piezo patches in APREH and PPREH are summarized in table 3, which are consistent with the previous analysis. In addition, it can be found the average stresses in APREH are higher than those in PPREH, which can be explained by the stress distributions of the piezo patches. As shown in figures 6(e) and (f), due to the geometric corners and singularities in the auxetic structure, the stress distribution in APREH is higher than that in PPREH, which lead to higher average stress. The average stresses in the piezo patches can determine the power output of the REHs, which can be illustrated in the following derivation of equations. When the piezo patch transduces the mechanical energy into electric one through direct piezoelectric effect, the    constitutive equations of the piezoelectric material can be expressed as [43]: where S ij and T kl represent the mechanical strain and mechanical stress, respectively. E k and D i denote the electric field and electrical displacement. s E ijkl , d kij , and ε T ik denote the elastic, piezoelectric, and permittivity constants. When the piezo patch works in bending mode, the open-circuit voltage output of the piezo patch can be calculated as [44,45]: where d 31 and d 32 are the piezoelectric constant components (d 31 = d 32 ). t p and ε T 33 are the thickness and permittivity of the piezo patch, respectively. When the piezo patch is connected to an optimal load resistor (equation (4)), the maximum power output of the piezo patch can be expressed as equation (5): where C p and A p represent the capacitance of the piezo patch and the area of the electrode, respectively. Then, the relationship between the maximum power output of the piezo patch and its average stresses can be simplified as: Due to the negative Poisson's ratio of the auxetic structure in APREH, σ 11 and σ 22 show the same sign; thus, the electric power generated by these two average stresses can be superimposed. However, in PPREH, the generated power by the two average stresses would cancel each other due to the oppositive signs of the two average stresses. In addition, the enhanced average stresses in APREH can further improve the power output. Therefore, the above mentioned two characteristics of the auxetic structure can significantly improve the generated power of APREH. More finite element simulation results will be compared with the experimental ones in the following section.

Experimental setup
To validate the proposed REH, the prototypes of PPREH and APREH are fabricated and tested as shown in figure 7, which are made of polylactic acid by 3D printing. It can be observed the substrate of APREH contains an auxetic structure, while the substrate of PPREH is plain. The REH is clamped to a translation stage that can adjust the distance between the tip and driving magnets. The driving magnet is bonded to a rotating plate that is driven by a servo motor (ECMA-E21310RS, Delta). Moreover, a servo driver (ASDB2-1021-B, Delta) and a motion controller (DKC-Y110, YiXing) are utilized to control the motor. As the driving magnet rotates, the resulting magnetic force would pluck the REH to vibrate freely and generate electric power. The corresponding displacement and output voltage of the energy harvester can be measured by a laser sensor (Omron ZX2-LD100) and an oscilloscope (InfiniiVision 3000 X-series, Keysight), respectively.

Open-circuit test
The experiments were first conducted to test the performances of the REHs under open-circuit condition. The distance between the tip and driving magnets was set to be 3 mm. At a rotational frequency of 1 Hz, the output voltages and displacements of the energy harvesters in the time domain are plotted in figures 8(a)-(d). The corresponding fast Fourier transform of the displacement responses is depicted in figures 8(e) and (f). From the results of fast Fourier transform, it can be observed that the first resonant mode and higher modes are excited under plucking. However, the amplitudes of higher modes are negligible compared with the first resonant mode, that is, the free vibrations of the energy harvesters are dominated by the first resonant mode. The first resonant frequencies of APREH and PPREH are found to be 35.42 and 51.75 Hz, respectively, indicating that the introduced auxetic structure can reduce the resonant frequency of the REH. This characteristic of auxetic REHs has the potential to be applied under low frequency excitation conditions. Moreover, the free vibration of PPREH decays much faster than that of APREH, indicating that the damping of PPREH is larger than that of APREH. In addition, the maximum output voltage and displacement of APREH and PPREH are found to be 66.8 V and 11.99 mm, 29.18 V and 5.84 mm, respectively, which validates the voltage boosting ability of the auxetic structure in APREH. The high output voltage of APREH is beneficial to energy management circuits in practical applications.
Then, to find the optimal frequencies of the energy harvesters under open-circuit condition, the rotational frequency was set from 0.1 to 1.8 Hz to obtain the frequency responses of the energy harvesters. The root mean square (RMS) values of output voltage and displacement are depicted in figure 9. The peak RMS voltage and displacement of APREH reach 18.21 V and 3.26 mm at 1 Hz. On the other hand, for PPREH, the peak RMS voltage and displacement are found to be 6.19 V and 1.27 mm at 1 Hz. Therefore, the optimal frequencies of the energy harvesters are both identified to be 1 Hz under opencircuit condition.
In the simulation, the frequency response curve is obtained point by point (0.1 to 1.8 Hz with an increment of 0.05 Hz), which means a series of simulations are conducted to acquire the frequency response curve. In each simulation, the rotating frequency of the driving magnet is set to be a certain value, resulting in a magnetic force, between the driving and tip magnets, of the same frequency. This magnetic force can pluck the energy harvester to vibrate freely. During vibration, two probes in COMSOL are utilized to detect the dynamic responses of the voltage output and tip displacement of the energy harvester. Based on the dynamic responses, RMS values of output voltage and displacement at the certain rotating frequency can be calculated, which contributes to a point of the frequency response curve. When a series of simulations with different rotating frequencies are completed, the frequency response curve can be depicted. It can be observed the experimental results match well with the finite element simulation ones. For the existing slight frequency drifts and amplitude deviations, the main reason comes from manufacturing errors, positioning errors, and imperfect boundary conditions of the REHs in the experiment.

Impedance matching and maximum power output
With the obtained optimal rotational frequency of 1 Hz, the piezo patches are connected to different external resistance to find the optimal resistance of the energy harvesters. The relationship between the power output and the load resistance is shown in figure 10(a), where the optimal resistance of APREH and PPREH is found to be 140 and 110 kΩ, respectively. As equation (4) shows, the optimal resistance is inversely proportional to the capacity of the piezo patch and the resonant frequency of the energy harvester. Since the introduced auxetic structure reduces the resonant frequency of APREH, the optimal resistance of APREH is larger than that of PPREH. Then, connected to the optimal resistance, the maximum power output of the REHs can be found by measuring the generated power at different rotational frequencies. Since the testing rotational frequencies, varying around 1 Hz, are quite low, the optimal resistance for all the rotational frequencies can be  treated as constant for simplifying the experimental test [24]. As depicted in figure 10(b), the maximum power output of APREH is found to be 1.03 mW at 1.1 Hz, which is increased by 711% compared with that of PPREH (0.127 mW at 1 Hz). As mentioned before, the introduced auxetic structure can significantly increase the power output of APREH. Besides, the free vibration of APREH decays slower than that of PPREH, which contributes to a higher RMS output voltage and further improves the power output of APREH.
A series of simulations are conducted to obtain the simulation results in figure 10. In figure 10(a), the rotating frequency of the driving magnet is fixed to be the optimal frequency obtained from figure 9. Then, in the electric circuit component of COMSOL, the outer resistance, connected to the piezoelectric patch, is set from 20 to 290 kΩ with an increment of 5 kΩ to obtain the impedance matching curve of the energy harvester. In figure 10(b), the outer resistance is fixed to be the optimal value obtained from figure 10(a). As the rotating frequency of the driving magnet changes from 0.1 to 1.8 Hz with an increment of 0.05 Hz, the maximum power output of the energy harvester can be found.

Parametric study
The distance between the tip and driving magnets has a major effect on the plucking force that further influences the power output of the proposed REH. Therefore, a parametric study has been performed to investigate the working performance of the energy harvester with different magnetic distances. Based on the above impedance matching experiment, APREH is tested with optimal resistance of 140 kΩ at 1.1 Hz, while PPREH is tested with optimal resistance of 110 kΩ at 1 Hz. Since the distance between the two magnets cannot be so small that the generated plucking force would break the cantilever beam, the distance is set from 2.5 to 6 mm with a gap of 0.5 mm. The experimental results are summarized in figure 11. The power output of the energy harvesters decreases with the increasing of magnet distance. This is due to the reason that the larger magnet distance results in the smaller plucking force, therefore reducing the vibration amplitudes and power output of the energy harvesters. As the distance increases from 2.5 to 6 mm, the power output of APREH and PPREH decreases from 1.43 to 0.06 mW and 0.182 to 0.007 mW, respectively. When the magnet distance is set to be 2.5 mm, the power output of APREH and PPREH reaches the maximum values, and the power output of APREH is increased by 686% compared with that of PPREH.
To prove the superiority of the proposed design, it is compared with typical plucking REHs as shown in table 4, where the power output, size, corresponding working rotational frequency, and normalized power density of the energy harvesters are listed. Considering the requirements of both high power output and low working frequency of REHs, the normalized power density is defined by taking the size and the rotational frequency of the energy harvester into account as following [46]: where P is the power output of the energy harvester at the rotational frequency f r , and V is the size of the energy harvester. It can be observed the normalized power density of the proposed design is higher than all others. This validates the advantage of the proposed APREH of generating the high power output under low-rotational-frequency excitations, which benefits from the adoption of auxetic structures and frequency upconversion.

Conclusion
A new plucking REH with auxetic structures is proposed. The harvester consists of a piezoelectric cantilever beam with auxetic structures, a tip magnet, and a driving magnet. As the rotation of the driving magnet, the magnetic force between two magnets would pluck the piezoelectric beam to vibrate freely and generate electric power. Therefore, utilizing the frequency up-conversion and auxetic structures, the proposed design can increase the power output of rotational energy harvesting under low rotational frequency excitations. Finite element simulation is performed to analyze the performance of the proposed REH, which matches well with the experimental results.
In experiments, when the magnet distance is set to be 3 mm, the maximum open-circuit voltage and displacement of the proposed APREH are found to be 66.8 V and 11.99 mm, respectively. Then, connected with optimal resistance 140 kΩ, the maximum power output of APREH reaches 1.03 mW at 1.1 Hz, which is increased by 711% compared with that of PPREH (0.127 mW at 1 Hz). A parametric study is also conducted to explore the effect of the magnet distance on the power output of the plucking REHs. At magnet distance 2.5 mm, the maximum power output of APREH and PPREH is found to be 1.43 mW at 1.1 Hz and 0.182 mW at 1 Hz, respectively, indicating an increase of 686% with auxetic structures. In addition, compared with other typical plucking energy harvesters, the proposed APREH shows remarkable power output under low rotational frequency excitations.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.