Numerical and experimental investigation of an auxetic piezoelectric energy harvester with frequency self-tuning capability

The fabricated prototype of the proposed piezoelectric energy harvester is shown in figure 1(a). It consists of a doubly clamped substrate with auxetic structure, a piezoelectric film, and a sliding proof mass. To ensure the smooth movement of the sliding proof mass, the geometric corners of the auxetic substrate should be minimised. Therefore, a peanut-shaped auxetic pattern adapted from the literature [51] is evenly fabricated on the 301 stainless steel beam using laser cutting. Then, a commercial off-the-shelf piezoelectric film (LDT4-028K/L) is bonded to the auxetic substrate through the epoxy adhesive. The piezoelectric film has a total thickness of 157 μm, comprising a PVDF piezoelectric layer (28 μm), a laminated polyester sheet (77 μm), and a protective polyester coating (52 μm). The length, width, and thickness of the composite piezoelectric beam are 171 mm, 22 mm, and 0.8 mm, respectively (figures 1(b) and (c)). Note that although piezoelectric ceramics such as lead zirconate titanate (PZT) are commonly used for vibration energy harvesting due to their higher power density [30, 40, 45, 46], they are brittle and lack of robustness under large displacement. As a result, the flexible PVDF film is applied in this design.

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The auxetic structure can be regarded as an array of unit cells with peanut-shaped holes. The geometry of the peanut-shaped hole is determined by two small circles and two large circles tangent to them, as depicted in figure 1(c). The sliding proof mass comprises four pieces of metal, manufactured from 6063 aluminium alloy and engineering steel, and is assembled with bolts and nuts (figure 1(a)). The dimensions of the sliding proof mass are shown in figure 1(d). There is a gap at the centre of the proof mass with a gap size of 1 mm, which allows the proof mass to freely move along the piezoelectric beam. In order to avoid the sliding proof mass being stuck on the piezoelectric beam, the width of the gap (22.3 mm) is slightly wider than that of the piezoelectric beam.

Instead of using a conventional plain substrate, the proposed piezoelectric energy harvester utilises a peanut-shaped auxetic structure on the substrate to boost its power output. On the one hand, the auxetic structure has a negative Poisson's ratio, which is characterised by lateral expansion (along x axis) under a longitudinal tension force (along y axis), as shown in figure 2. In this case, the lateral and longitudinal strains in the auxetic substrate have the same sign, showing opposite behaviour compared to a normal plain substrate. For a piezoelectric energy harvester operating in bending mode, the maximum power output of the energy harvester is equally influenced by both average lateral strain $\overline {{\varepsilon _{xx}}}$ and average longitudinal strain $\overline {{\varepsilon _{yy}}}$ in the piezoelectric film (as expressed in equation (1) [47]). In an AEH, negative $\overline {{\varepsilon _{xx}}}$ can be reduced or even have the same sign as $\overline {{\varepsilon _{yy}}} \,$ due to the advantageous lateral force exerted by the auxetic structure.

$$\begin{align}{\text{Maximum}}\,{\text{Power}} \propto {\left( {\overline {{\varepsilon _{xx}}} + \overline {{\varepsilon _{yy}}} } \right)^2}\end{align} \tag{ 1 }$$

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On the other hand, compared to the plain substrate, the substrate with an auxetic structure can lead to higher stress in the piezoelectric film, thereby enhancing the power output. According to the constitutive equations of piezoelectric effect (equations (2) and (3)) [52], an increase in the stress $T$ in the piezoelectric material contributes to the rise of the electrical displacement $D$. Therefore, the power output of the piezoelectric energy harvester can be significantly enhanced by the introduced auxetic structure owing to the above-mentioned two advantages.

$$\begin{align}D & = dT + \varepsilon E\,\left( {{\text{Direct effect}}} \right)\end{align} \tag{ 2 }$$

$$\begin{align}X & = sT + dE\,\left( {{\text{Converse effect}}} \right)\end{align} \tag{ 3 }$$

where $D$ is the electrical displacement, $d$ is the piezoelectric coefficient, $T$ is the mechanical stress, $\varepsilon$ is the dielectric constant, $E$ is the electric field, $X$ is the mechanical strain, and $s$ is the mechanical compliance.

Since vibration frequency levels of common industrial machinery are relatively low (e.g. the dominant frequency of a water pump is 49.5 Hz [53]), in this case, only the first mode of vibration of the proposed doubly clamped beam-based energy harvester is suitable for vibration energy harvesting. The passive frequency self-tuning mechanism of the proposed energy harvester is depicted in figure 3. Initially, the proof mass is placed at the side of the piezoelectric beam. Then, an external constant excitation is applied to the energy harvester with a frequency of f₁. If the vibrating beam can provide sufficient inertia force to the proof mass and the excitation frequency f₁ is within the frequency self-tuning range, the proof mass will autonomously move towards the centre of the beam. During this process, the vibration amplitude of the beam will gradually increase, leading to the occurrence of the hardening nonlinearity in the doubly clamped beam. As a result, the proof mass will not stop at the position where the modulated natural frequency of the energy harvester (as a linear system) matches the excitation frequency. Instead, the proof mass will stop near the centre of the beam to attain a larger vibration amplitude due to the existence of geometric nonlinearity and friction force during the sliding process [54]. Overall, the passive frequency self-tuning behaviour can be attributed to the interaction between the vibrating beam and the sliding proof mass. Specifically, the vibrating beam drives the motion of the proof mass. Consequently, this motion results in the change in the mass distribution of the system, which in turn alters the dynamic response of the vibrating beam. To quantify the frequency self-tuning capability realised by a sliding proof mass, Kim et al [55] presented a comprehensive mathematical model of the piezoelectric energy harvester with a doubly clamped plain beam and a sliding proof mass based on extended Hamilton principle by considering the nonlinear dynamics of the doubly clamped beam. However, the motion of the proposed energy harvester cannot be accurately expressed by analytical calculations due to the introduction of the auxetic structure. The mode shape of the proposed energy harvester is more complicated than its plain counterpart, and the cross sections of the substrate with the auxetic structure are not uniform due to the presence of hollows [46].

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FEM is conducted using COMSOL Multiphysics 5.6, which couples electrostatic physics, electrical circuit and solid mechanics, to estimate the eigenfrequencies of the energy harvester with a fixed proof mass and investigate the effect of the auxetic structure. As shown in figure 4(a), the finite element model of the proposed energy harvester consists of an auxetic substrate with fixed boundary conditions at both ends, a PVDF piezoelectric layer sandwiched between two polyester sheets for power generation, an epoxy layer with the same auxetic pattern as the substrate to bond the piezoelectric film to the substrate, and a proof mass fixed on the piezoelectric beam. To simplify the model, the bolts and nuts on the proof mass are ignored. However, since the weight of the bolt and nuts has a significant influence on the natural frequency of the energy harvester, the density of the proof mass 1 is assumed to be the same as the aluminium alloy 6063, while the density of the proof mass 2 is calculated based on the weight of the proof mass used in the experiments. Besides, considering the imperfect bonding caused by the adhesive, substrate–adhesive–polyester interfaces are modelled as thin elastic layers with a bonding strength of 200 GN m⁻³ [56], and the damping ratio of the system is assumed to be a single isotropic loss factor with a value of 0.001 [40]. The computational mesh configuration of the proposed AEH is shown in figure 4(b). To characterise the geometry of the auxetic piezoelectric beam, a combination of free triangular mesh and swept meshing are employed to build the mesh of the auxetic piezoelectric beam, while the proof mass is meshed using free tetrahedral mesh due to its good adaptability. To achieve precise simulation results within a reasonable computational time, natural frequency of the energy harvester with a proof mass fixed at the centre of the piezoelectric beam is initially calculated to determine the maximum size of the mesh. As depicted in figure 4(c), the natural frequency of the energy harvester shows little sensitivity to the maximum mesh size when it is smaller than 1 mm. Consequently, the maximum mesh size is set at 1 mm. In this configuration, the proposed finite element model comprises 77 391 mesh vertices and a total of 168 384 elements. As a comparison, FEM is also performed to analyse the behaviour of a conventional piezoelectric energy harvester with a plain substrate. The geometrical parameters and material properties of AEH and plain energy harvester (PEH) for FEM are summarised in tables 1 and 2, respectively.

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Table 1. Geometrical parameters of the energy harvesters in the modelling.

Component	Parameter	Symbol	Value	Unit
Piezoelectric layer	Length	${l_{\text{p}}}$	171	mm
	Width	${w_{\text{p}}}$	22	mm
	Thickness	${t_{\text{p}}}$	0.028	mm
Laminated polyester sheet	Length	${l_{\text{1}}}$	171	mm
	Width	${w_1}$	22	mm
	Thickness	${t_1}$	0.077	mm
Protective polyester coating	Length	${l_2}$	171	mm
	Width	${w_2}$	22	mm
	Thickness	${t_2}$	0.052	mm
Substrate	Length	${l_{\text{s}}}$	178	mm
	Width	${w_{\text{s}}}$	22	mm
	Thickness	${t_{\text{s}}}$	0.47	mm
Auxetic structure	Radius of two small circle	$r$	1.53	mm
	Radius of two large circles	$R$	2.71	mm
	Centroid distance of two small circles	$D$	5.08	mm
	Centroid distance of two large circles	$S$	6.78	mm
	Distance between peanut-shaped holes	$m$	0.75	mm
	Side length of the unit cell	${l_{\text{a}}}$	11	mm
	Total length of the auxetic pattern	$l$	165	mm
Epoxy adhesive layer	Length	${l_{\text{e}}}$	171	mm
	Width	${w_{\text{e}}}$	22	mm
	Thickness	${t_{\text{e}}}$	0.1	mm
	Bonding strength	$k$	200	GN m−3
Proof mass (four pieces)	Length	${l_{\text{m}}}$	32	mm
	Width	${w_{\text{m}}}$	11	mm
	Thickness	${t_{\text{m}}}$	3	mm

Table 2. Material properties of the energy harvesters in the modelling.

Component	Parameter	Symbol	Value	Unit
Piezoelectric element (PVDF)	Density	${\rho _{\text{p}}}$	1780	kg m−3
	Compliance matrix	$s_{11}^{\text{E}}$	3.78 × 10−10	1 Pa
		$s_{12}^{\text{E}}$	−1.48 × 10−10	1 Pa
		$s_{13}^{\text{E}}$	−1.72 × 10−10	1 Pa
		$s_{22}^{\text{E}}$	3.78 × 10−10	1 Pa
		$s_{23}^{\text{E}}$	−1.72 × 10−10	1 Pa
		$s_{33}^{\text{E}}$	1.09 × 10−9	1 Pa
		$s_{44}^{\text{E}}$	1.11 × 10−9	1 Pa
		$s_{55}^{\text{E}}$	1.11 × 10−9	1 Pa
		$s_{66}^{\text{E}}$	1.43 × 10−9	1 Pa
	Coupling matrix	${d_{31}}$	2.3 × 10−11	C/N
		${d_{32}}$	1.48 × 10−12	C/N
		${d_{33}}$	−3.3 × 10−11	C/N
	Relative permittivity	${\varepsilon _{11}}$	7.4	—
		${\varepsilon _{22}}$	9.3	—
		${\varepsilon _{33}}$	12.0	—
Polyester sheet	Density	${\rho _1}$	1780	kg m−3
	Poisson's ratio	${\nu _1}$	0.2	—
	Young's modulus	${E_1}$	3.5	GPa
Substrate (301 stainless steel)	Density	${\rho _{\text{s}}}$	7930	kg m−3
	Poisson's ratio	${\nu _{\text{s}}}$	0.3	—
	Young's modulus	${E_{\text{s}}}$	195	GPa
Epoxy	Density	${\rho _{\text{a}}}$	1250	kg m−3
	Poisson's ratio	${\nu _{\text{a}}}$	0.35	—
	Young's modulus	${E_{\text{a}}}$	1.0	GPa
Proof mass 1 (aluminium alloy 6063)	Density	${\rho _{\text{m}}}$	2700	kg m−3
	Poisson's ratio	${\nu _{\text{m}}}$	0.33	—
	Young's modulus	${E_{\text{m}}}$	69	GPa
Proof mass 2 (engineering steel)	Density	${\rho _{\text{m}}}$	8200	kg m−3
	Poisson's ratio	${\nu _{\text{m}}}$	0.3	—
	Young's modulus	${E_{\text{m}}}$	200	GPa

To facilitate the investigation of the performance of the auxetic structure, the proof mass is fixed at the centre of the piezoelectric beam. The modal analysis is conducted to determine the eigenfrequencies of the energy harvester. A sinusoidal body load equivalent to a constant vibration of 0.3 g acceleration [40] at the first eigenfrequency is applied to the energy harvester along the z axis. Since the doubly clamped piezoelectric beam of the energy harvester has a thin thickness and exhibits large deflection when subject to an excitation acceleration of 0.3 g, the impact of geometric nonlinearity should be considered. Besides, in order to explore the effect of the position of the proof mass on the natural frequency of the AEH, the first eigenfrequency of the AEH is calculated by varying the position of the proof mass fixed on the piezoelectric beam.

To validate the design concept, an experimental setup is organised to investigate the performance of the proposed piezoelectric energy harvester. As shown in figure 5(a), the energy harvester is vertically mounted on a vibration shaker (APS 113 ELECTRO-SEIS) to be excited by harmonic excitations. The shaker is driven by a power amplifier (APS 125) whose input signal is sourced from a function generator. The vibration frequency and amplitude of the shaker can be manually adjusted to ensure that provided excitation is at the desired level. An accelerometer (Brüel & Kjær 4507-B) is attached to the U-shape fixture to measure the base acceleration generated by the shaker. Two data acquisition modules (DAQ NI-9229 and NI-9234) are used to measure the voltage signal generated by the energy harvester and the acceleration signal measured by the accelerometer, respectively. Simultaneously, these two data acquisition units are connected to the computer, where the measured voltage and acceleration signals can be displayed and recorded by the NI SignalExpress software (figure 5(b)). Note that the measured output voltage is slightly lower than the open-circuit voltage due to the inherent 1 MΩ resistance of the data acquisition module NI-9229. Besides, the piezoelectric film is connected to a resistance box to measure the power output of the energy harvester with different resistances.

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As shown in figure 6, the first four eigenfrequencies of the AEH with a proof mass fixed at the centre of the piezoelectric beam are simulated to be 18.93 Hz, 65.56 Hz, 120.82 Hz and 248.57 Hz, respectively. Among four mode shapes, only the first mode of vibration of the AEH can be utilised for vibration energy harvesting in the proposed energy harvester design because this mode enables the sliding proof mass to move smoothly along the vibrating beam for passive frequency self-tuning. Besides, the first eigenfrequency of the PEH is found to be 36.44 Hz, which is significantly higher than that of the proposed AEH. This decrease in the natural frequency of the AEH can be attributed to the lower stiffness of the auxetic substrate in comparison to the plain substrate. Therefore, the introduction of the auxetic can considerably reduce the natural frequency of the energy harvester. The variation of the natural frequency of the AEH with the position of fixed proof mass is shown in figure 7. Despite minor discrepancies, FEM results match well with experimental results. The experimental results indicate that the natural frequency of the AEH exponentially increases from 18.0 to 48.3 Hz, as the proof mass moves from the centre of the piezoelectric beam to the fixed end. This implies that the position of the proof mass significantly affects the dynamic behaviour of the energy harvester.

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To investigate the influence of auxetic structure on the deformation of the piezoelectric beam, finite element simulations were conducted to analyse the displacement distributions of the substrate covered by the piezoelectric layer, as well as the stress and strain distributions in the piezoelectric layer. To ensure a fair comparison, these simulations were performed for both PEH and AEH with a poof mass fixed at the centre of the beam under an excitation acceleration of 0.3 g at their respective first eigenfrequencies. Considering the structural symmetry of the two energy harvesters, the displacement and stress distributions of the piezoelectric beam are symmetrical with respect to the centre of the beam. Hence, only the displacement and stress distributions of the left half of the energy harvesters are depicted for comparison. As shown in figures 8(a) and (b), the longitudinal displacements of both substrates first increase and then decrease along the y-axis, which means that the regions near the fixed end undergo stretching, while the regions near the centre experience compression. The lateral displacement distribution of plain substrate (figure 8(c)) indicates that the plain substrate contracts laterally in the stretching region and expands laterally in the compression region, which aligns with the expected positive Poisson's ratio behaviour. In contrast, the auxetic substrate expands laterally in most regions (figure 8(d)). This behaviour arises because the longitudinal stress varies across the cross section of the substrate when it is in a bending mode, which is characterised by one surface in tension and the other in compression. The auxetic substrate consistently exhibits lateral expansion when both tensile and compressive stresses coexist within the auxetic substrate.

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The stress distributions of piezoelectric films for the two energy harvesters are depicted in figures 8(e) and (f). It can be observed that the stress in the piezoelectric film of the AEH is higher than that of the PEH. To further evaluate the performance of the auxetic structure, the average strain and maximum stress in piezoelectric layers for both energy harvesters are summarised in table 3. For the piezoelectric film in the PEH, the average strains along the x axis ($\overline {{\varepsilon _{xx}}}$) and along the y axis ($\overline {{\varepsilon _{yy}}}$) have opposite signs, which means that $\overline {{\varepsilon _{xx}}}$ will diminish the power output generated by $\overline {{\varepsilon _{yy}}}$. In contrast, $\overline {{\varepsilon _{xx}}}$ and $\overline {{\varepsilon _{yy}}}$ in the piezoelectric layer of the AEH have the same signs, indicating that both stains can contribute to the power generation. Besides, the maximum stress in the piezoelectric film is 0.375 MPa for the PEH and 1.645 MPa for the AEH.

Table 3. Average strain and maximum stress in piezoelectric films.

Energy harvester	$\overline {{\varepsilon _{xx}}}$ (10−5)	$\overline {{\varepsilon _{yy}}}$ (10−5)	Maximum stress (MPa)
PEH	−0.328	1.891	0.375
AEH	1.093	3.866	1.645

To quantify the effect of the auxetic structure on vibration energy harvesting, the frequency responses of the AEH and the PEH were experimentally tested under an excitation acceleration of 0.3 g. The proof masses of both energy harvesters were fixed at the centre of the piezoelectric beam. The frequency responses of the energy harvesters were obtained through manual frequency sweep [17]. The excitation frequency was increased and decreased manually in steps of 0.5 Hz within the frequency ranges of interest. Considering that the AEH has a lower natural frequency compared to the PEH, the frequency range of interest for the AEH was defined as 15 Hz to 30 Hz, while the range for PEH was set as 30 Hz to 45 Hz. As shown in figure 9, the AEH achieves a maximum root mean square (RMS) output voltage of 2.07 V at a frequency of 25 Hz. In contrast, the PEH acquires a lower maximum RMS output voltage of 1.42 V at a frequency of 40 Hz. These results highlight the ability of the auxetic structure to enhance the voltage output of the energy harvester. Furthermore, obvious jump phenomena can be observed in the upward frequency responses of both energy harvesters. There also exist noticeable differences between upward and downward frequency responses as the structural stiffness varies with the vibration amplitude. These observations indicate the presence of geometric nonlinearity in the energy harvesters, which can be attributed to the occurrence of stretching strain in the doubly clamped beam under large vibration amplitudes.

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For an AEH with a sliding proof mass, passive frequency self-tuning of the energy harvester can be achieved through the automatic movement of the sliding proof mass. To demonstrate the improved energy harvesting performance of the energy harvester, a series of experiments were conducted at different frequencies under a constant excitation acceleration of 0.3 g. Figures 10(a)–(d) depict the output voltage of the energy harvester over time at 26 Hz, 28 Hz, 30 Hz and 31 Hz, respectively. As shown in figure 10(a), when the energy harvester was subjected to an excitation frequency of 26 Hz, the initial output voltage of the energy harvester is relatively low owing to the small vibration amplitude of the piezoelectric beam. However, as the proof mass moves towards the beam centre, the output voltage gradually increases due to the amplified vibration amplitude. Ultimately, the output voltage of the energy harvester is significantly improved and remained constant, accompanied by the beam vibrating in a large amplitude and the proof mass anchored at a position near the beam centre. The transient responses of the energy harvester at 28 Hz (figure 10(b)) and 30 Hz (figure 10(c)) exhibit a similar trend as the case at 26 Hz. However, in the case of 31 Hz (figure 10(d)), the output voltage of the energy harvester increases dramatically at the beginning, followed by a sudden decrease. This behaviour may be attributed to the rapid movement of the proof mass, which may hinder the vibrating beam from effectively capturing the high vibration amplitude. Consequently, the nonlinear energy harvester transitions from the high-energy orbit to the low-energy orbit when the proof mass reaches the position near the centre. These results demonstrate that the sliding proof mass can automatically move from the side of the piezoelectric beam towards the beam centre to help the energy harvester capture high output voltage at different excitation frequencies within the frequency tuning range.

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To investigate the effect of the initial position of the proof mass on frequency self-tuning, experiments were conducted with the sliding proof mass initially placed at three different positions. During the tests, the energy harvester was subjected to an excitation acceleration of 0.3 g at a constant frequency of 26 Hz. As shown in figure 11(a), when the initial position of the proof mass was 66 mm from the beam centre, the output voltage of the energy harvester consistently remains at an extremely low level. This is because the proof mass is kept at its initial position and the excited frequency is deviated from the resonance frequency of the structure, causing the beam to vibrate with a small vibration amplitude. The similar phenomenon was also discovered when the proof mass was initially placed at the beam centre, as shown in figure 11(c). The voltage output of the energy harvester is relatively low and remains unchanged over time. However, when the proof mass was initially placed at 44 mm from the beam centre, the energy harvester could realise frequency self-tuning to obtain high voltage output (figure 11(b)). These results imply that frequency self-tuning is only achieved when the proof mass is initially placed at an appropriate position. Specifically, the initial vibration amplitude of the piezoelectric beam should be sufficiently large to initiate the movement of the proof mass. Subsequently, the proof mass can autonomously move towards the beam centre to track the large vibration amplitude. Note that for the PEH under an excitation acceleration of 0.3 g, the proof mass cannot automatically move towards the beam centre regardless of its initial position. This may be attributed to the significantly higher stiffness of the plain substrate with smaller vibration amplitude in comparison to the auxetic substrate.

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Figure 12 compares the RMS output voltage between the AEH with a fixed proof mass and the AEH with a sliding proof mass at different frequencies under a constant excitation acceleration of 0.3 g. In each test, the energy harvester remained in a stationary state until it was subjected to a specified excitation frequency. In the case of the AEH with a fixed proof mass, the proof mass was fixed at the beam centre to maximise output voltage. This configuration achieves a maximum RMS output voltage of 1.07 V at a frequency of 21 Hz. In contrast, the energy harvester with a sliding proof mass exhibits a maximum RMS output voltage of 3.37 V at a frequency of 30 Hz. Moreover, although the maximum RMS output voltage of the AEH with a fixed proof mass at the beam centre can be improved to 2.07 V at 25 Hz through the manual upward frequency sweep in steps of 0.5 Hz due to the occurrence of geometric nonlinearity, the AEH with a fixed proof mass at the beam centre exhibits a lower maximum operating frequency and RMS output voltage compared to the AEH with a sliding proof mass. Therefore, the introduction of the sliding proof mass significantly enhances the voltage output of AEH and broadens its working bandwidth by 9 Hz under an excitation acceleration of 0.3 g.

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The RMS output voltage of the AEH with a sliding proof mass was investigated under different excitation accelerations and frequencies, as shown in figure 13. The RMS output voltage and maximum working frequency of the energy harvester increase with increasing acceleration because of the improved vibration amplitude of the piezoelectric beam over time. The increased vibration amplitude of the beam can enhance the geometry nonlinearity of the energy harvester. Thus, the high energy orbit of the energy harvester can be easily captured and maintained as the proof mass moves towards the beam centre. Furthermore, figure 14 depicts the time histories of the output voltage of the energy harvester during the frequency self-tuning process at a constant frequency of 26 Hz under three different excitation accelerations. To ensure a fair comparison, the initial position of the sliding mass remained consistent across these three tests, with a distance of 44 mm from the beam centre. It can be observed that transient response of the energy harvester under an excitation acceleration of 0.2 g is consistently much lower than that of the energy harvester under an excitation acceleration of 0.3 g and 0.4 g. This indicates that the piezoelectric beam consistently exhibits larger vertical displacement when subjected to higher excitation accelerations, resulting in stronger geometry nonlinearity.

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To further evaluate the performance of the energy harvesters, the peak RMS output power for each energy harvester was examined with different load resistances under an excitation acceleration of 0.3 g, as shown in figure 15. Considering the energy harvester as a current source, the theoretical RMS power output generated by the energy harvester across the connected resistance can be calculated using equations (4)–(6) [57]. According to the experimental results, the PEH with a fixed proof mass has the lowest output power compared to the other two PEHs. It achieves a maximum RMS output power of 3.37 μW at a load resistance of 300 kΩ. The AEH with a fixed proof mass demonstrates a higher maximum RMS power output of 5.85 μW at a load resistance of 420 kΩ. The AEH with a sliding proof mass exhibits the highest output power, reaching a maximum RMS power output of 18.78 μW when connected to a load resistance of 320 kΩ. This value is 2.21 times higher than that of the AEH with a fixed proof mass and 4.57 times higher than that of the PEH with a fixed proof mass. In conclusion, the auxetic structure can improve the power output of the energy harvester, and the introduction of the sliding proof can further improve the working bandwidth and power output of the energy harvester due to its frequency self-tuning capability

$$\begin{align}P = \frac{{{U_R} \times {I_R}}}{2}\end{align} \tag{ 4 }$$

$$\begin{align}{U_R} = R \times {I_R}\end{align} \tag{ 5 }$$

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$$\begin{align}{I_R} = \frac{{{I_0}}}{{\sqrt {1 + {R^2}{C^2}{\omega ^2}} }}\end{align} \tag{ 6 }$$

where $P$, ${U_R}$, ${I_R}$ denotes the RMS power output, voltage amplitude, and current amplitude across the connected resistance load resistance $R$, respectively. ${I_0}$ denotes the short-circuit current generated by the energy harvester, which can be measured in the experiments. $C$ denotes the capacitance of the piezoelectric film, with a value of approximately 14.8 pF, $\omega$ denotes the angular frequency of the external excitation applied to the energy harvester.

Since the proof mass exhibits a significant impact on vibration energy harvesting, the influence of the weight and the gap size of the sliding proof mass on the performance of the energy harvester was experimentally investigated. As shown in figure 16, with the increase in the weight of the sliding proof mass, the working frequencies of the energy harvester shift from the high frequencies to low frequencies. At the same time, the RMS output voltage of the energy harvester increases as the weight of the proof mass increases. This is because a heavier proof mass results in a larger vibration amplitude of the piezoelectric beam when the sliding proof mass stabilises near the beam centre.

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In addition, the transient responses of the AEH with a sliding proof mass of various gap sizes during the frequency self-tuning process were investigated. The experiments were conducted under an excitation acceleration of 0.3 g at a frequency of 26 Hz with the sliding proof mass initially placed at 44 mm from the beam centre. As shown in figure 17, the energy harvester with a sliding proof mass of a smaller gap size exhibits a slightly lower output voltage, and it takes more time for the proof mass to move towards the beam centre during the frequency self-tuning process with the decrease of the gap size. This behaviour can be attributed to the increase in the friction force and impacts between the sliding proof mass and the vibrating beam, which slows down the movement of the sliding proof mass. Furthermore, figure 18 shows the RMS output voltage of the AEH with a sliding proof mass of different gap sizes as a function of excitation frequency under an excitation acceleration of 0.3 g. The gap size of the proof mass has little impact on the RMS output voltage of the energy harvester under the same excitation frequency within the frequency self-tuning range. However, a slight increase in the maximum working frequency of the energy harvester is observed as the gap size of the sliding proof mass decreases.

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