An omnidirectional acoustic energy harvester based on six Helmholtz resonators

Abstract

An omnidirectional acoustic energy harvester (AEH) based on six Helmholtz resonators is proposed in this work. Compared with the previous structure, the insufficiency of the directionality and conversion efficiency of energy collection can be effectively improved due to the coupling of six resonators. Based on the distributed parameter model, the relationship of the electrical output, the input frequency with the structure size is obtained. The simulation results show that the maximum output voltage is 70.95 mV at the resonant frequency of 35 kHz. When the external load resistance is 14 kΩ, the maximum output power is 0.45 μW. Moreover, the energy conversion efficiency of this omnidirectional AEH can reach 23%, which is improved greatly compared with the traditional structure. Therefore, this AEH will have a wide range of application prospects in medical implantation equipment and other fields.

Keywords

AEH Helmholtz resonators the output power conversion efficiency

1. Introduction

Energy harvester is a development trend that is expected to replace traditional batteries in the future. The energy conversion methods are divided into electrostatic [1], electromagnetic [2] and piezoelectric [3], etc., and external excitation includes vibration energy [4], solar energy [5], magnetic energy [6], and acoustic energy [7] etc. Due to the excellent performance of piezoelectric materials and the significant advantages of acoustic as the excitation source, miniature piezoelectric energy harvesters that use acoustic as the excitation source have gradually become a research hotspot [8–11]. In response to the problem of insufficient external excitation, researchers at home and abroad mainly amplify sound waves with the cavity-shaped structure, and then use the piezoelectric effect to achieve the conversion from sound energy to electrical energy. In 2006, Horowitz et al. proposed an AEH with Helmholtz cavity. After the magnification of the cavity structure, its output power is 6 pW under the sound pressure level of 149 dB [12]. In 2008, F. Liu et al. presented the development of an AEH using an electromechanical Helmholtz resonator, and experimental results indicated that approximately 30 mW of output power was harvested for an incident sound pressure level of 160 dB with a flyback converter [13]. Their research provides a reliable basic reference for subsequent researchers. However, the traditional AEH based on Helmholtz resonant cavity has some performance defects, including: single energy harvesting direction, narrow resonance frequency band, and low output power. In order to solve the above problems, scholars mainly use the following three methods to improve output performance.

By designing multiple resonant cavities to improve output performance. In 2013, X. Peng et al. designed an acoustic energy harvester using dual Helmholtz resonators. Compared with the traditional Helmholtz resonator, the voltage generated by this structure is increased by 400%, and the output power was 16 times of the original [14].

By designing a novel cavity structure to improve output performance. In 2016, F. Khan et al. proposed an AEH with a tapered Helmholtz resonator, and they improved the output performance by changing the cavity structure. It could receive an output power of 90.61 μW with an excitation of 130 dB [15].

By designing a novel coupling structure. In 2020, P. Eghbalia et al. designed an AEH with an axenic latticed resonator backed by an acoustic rectangular tube. It could arrive at a large magnification factor of around 10.5 for a 100 dB sound pressure level at resonance [16].

However, none of these structures could solve the problems of insufficient external excitation, single energy collection direction, and narrow resonance frequency band at the same time. In our previous study, a quarter spherical acoustic energy harvester was designed. It could harvest multi-directional acoustic energy, had a widened resonance frequency band, and had an improved energy conversion efficiency. Its output voltage was about 28 mV, and its output power was about 0.05 μW [17].

When it comes to practical application, the higher standards is needed for some key performance indicators such as the output bandwidth, collection direction and energy conversion efficiency. On the basis of the quarter-spherical structure, an omnidirectional AEH based on six Helmholtz resonators is proposed in this work, which can collect the sound energy in the environment from all directions. As a result, some key performance indicators are significantly improved. In Section 2, the theoretical model of this omnidirectional AEH is discussed, and the relationship of the output voltage with the input sound pressure is obtained. In Section 3, the output performance of this omnidirectional AEH, including sound pressure level, output voltage, and output power, are studied. Finally, some conclusions are drawn in Section 4.

2. Principle and theory

As shown in the Fig. 1, the omnidirectional acoustic energy harvester contains six independent resonators and an external piezoelectric layer, which can collect the sound energy in the environment from all directions. Taking the rectangular coordinate system as a reference, a single set of energy harvesters A, B, and C can collect energy in the opposite directions of x, y, and z, and the energy harvesters D, E, and F can be used to collect energy in the positive direction of x, y, and z. All energy harvesters have the same size.

When excited by incident acoustic wave, the air in the neck moves downward from the static equilibrium position into the cavity. After that, the air in the cavity is compressed. Then, a downward pressure is formed and acts on the piezoelectric layer. Similarly, when the air in neck area moves down to the extreme position, the internal pressure is greater than external incentives, and it will push the air back to the upward direction. Due to inertia, the air in neck area returns to the equilibrium position and continues to move upward, and the pressure in cavity is lowered. As a result, the film produces a reverse strain. In this periodic, an equal amount of positive and negative bound charges appears on the upper and lower surface of the piezoelectric film. Thus, when the external charge has accumulated enough, an effective output voltage is produced.

The shell theoretical model is more suitable for this structure. However, it is difficult to obtain an analytical solution for the shell theoretical model. Therefore, a simplified beam theoretical model is used to study the one-way coupling mechanism of the omnidirectional acoustic energy harvester based on six Helmholtz resonators in this work. Only the output performance of the electrical energy generated by vibration is studied, and the coupling mechanism is not studied. The structural parameters that affect the performance can be obtained according to the simplified beam theoretical model, and the finite element software Comsol is used to simulate its output performance.

Fig. 1.

(a) Overall viewing (Separate structure for easy observation). (b) Single group viewing of omnidirectional AEH.

The sound path equation in the cavity can be obtained by electric acoustic analogy [18,19]: $\begin{eqnarray}\displaystyle M_{a}{\displaystyle \frac{dU}{dt}}+R_{a}U+{\displaystyle \frac{1}{C_{a}}}\int Udt=p_{i} & & \displaystyle\end{eqnarray}$ (1) where M _a is the acoustic mass, R _a is the acoustic resistance, C _a is the acoustic capacitance, and U is the body velocity.

The highest sound pressure of the cavity is generated at the acoustic resonant frequency, which is determined by the cavity volume and neck size of the cavity. The acoustic resonant frequency f _a and the maximum sound pressure amplification coefficient G _max are expressed as [20–22]: $\begin{eqnarray}\displaystyle f_{a}\text{} & =\text{} & \displaystyle \frac{c}{2{\pi}}\sqrt{\frac{S_{n}}{V_{c}L_{\mathit{eff}}}}\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle G_{\text{max}}\text{} & =\text{} & \displaystyle 2{\pi}\sqrt{V_{c}\left(\frac{L_{\mathit{eff }}}{S_{n}}\right)^{3}}\end{eqnarray}$ (3) where S _n is the cross-sectional area of the neck, V _c is the volume of the cavity and L _eff is the effective length of neck.

The force F ₀ on the piezoelectric layer can be obtained as $\begin{eqnarray}\displaystyle F_{0}=Gp_{i}S_{p} & & \displaystyle\end{eqnarray}$ (4) where S _p is the surface area of the piezoelectric layer.

A distributed parameter model is used to analyze piezoelectric conversion, and its advantage lies on its ability to predict high-order modal vibration, considering the dynamic mode shape and the stress distribution on the cantilever beam during the vibration process. In the distributed parameter model, one end of the piezoelectric vibrator is clamped, and the other end is free. The upper and lower electrodes are connected to an external load. The external vertical excitation received by the piezoelectric vibrator is g(t), and the small rotation excitation at the clamping end is h(t). The damping of the system mainly comes from two aspects, which are aeroelastic damping 𝜉_a and the strain rate damping 𝜉_s. In the vibration process, both of them satisfy the proportional damping criterion. The Euler–Bernoulli equation can be used to obtain the dynamic equation of the piezoelectric vibrator [23]: $\begin{eqnarray}\displaystyle \frac{\partial ^{2}M(x,t)}{\partial x^{2}}+{\xi}_{s}I_{b}\frac{\partial ^{5}y(x,t)}{\partial x^{4}\partial t}+m\frac{\partial ^{2}y(x,t)}{\partial t^{2}}+{\xi}_{a}\frac{\partial y(x,t)}{\partial t}=0 & & \displaystyle\end{eqnarray}$ (5) where M(x, t) is the bending moment of the piezoelectric cantilever beam, I _b is the section moment of inertia of the piezoelectric cantilever beam, m is the mass per unit length of the piezoelectric cantilever beam, and y(x, t) is the vibration displacement of the piezoelectric cantilever beam in the vertical direction.

The model is solved by decomposing the vibration displacement. The absolute vibration displacement of the clamping end can be decomposed into a vertical vibration excitation and a rotational vibration excitation. With the mode superposition method, the relative vibration displacement can be expressed as the series of the mode function, which is composed of the standardized mode function and the generalized coordinates of the vibration mode [24,25]. $\begin{eqnarray}\displaystyle y_{base}(x,t)\text{} & =\text{} & \displaystyle g(t)+xh(t)\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle g(t)\text{} & =\text{} & \displaystyle G_{0}e^{j{\omega}t}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle h(t)\text{} & =\text{} & \displaystyle H_{0}e^{j{\omega}t}\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle y_{rel}(x,t)\text{} & =\text{} & \displaystyle \mathop{\sum }_{r=1}^{\infty }{\phi}_{r}(x){\varphi}_{r}(t).\end{eqnarray}$ (9)

Based on the boundary conditions, the solution of the mode can be calculated as: $\begin{eqnarray}\displaystyle {\phi}_{r}(x)=\frac{1}{\sqrt{mR}}\left[\cosh \frac{{\lambda}_{r}}{R}x-\cos \frac{{\lambda}_{r}}{R}x-{\sigma}_{r}\left(\sinh \frac{{\lambda}_{r}}{R}x-\sin \frac{{\lambda}_{r}}{R}x\right)\right] & & \displaystyle\end{eqnarray}$ (10) where R is the radius of sphere, 𝜆_r is the eigenvalue of the rth mode, σ_r in formula (10) can be expressed as: $\begin{eqnarray}\displaystyle {\sigma}_{r}=\frac{\sinh {\lambda}_{r}-\sin {\lambda}_{r}}{\cosh {\lambda}_{r}+\cos {\lambda}_{r}}. & & \displaystyle\end{eqnarray}$ (11) The natural angular frequency in the rth mode is: $\begin{eqnarray}\displaystyle {\omega}_{r}={\lambda}_{r}^{2}\sqrt{\frac{EI_{b}}{MR^{2}}} & & \displaystyle\end{eqnarray}$ (12) where E is the elastic modulus, I _b is the moment of inertia, M is the equivalent mass.

With the orthogonal conditions and boundary conditions, the modal solution can be calculated as [26]: $\begin{eqnarray}\displaystyle {\varphi}_{r}(t)=\frac{m{\omega}^{2}({\gamma}_{r}^{w}G_{0}+{\gamma}_{r}^{{\theta}}H_{0})-{\chi}_{r}V_{m}}{{\omega}_{r}^{2}-{\omega}^{2}+j2{\xi}_{r}{\omega}_{r}{\omega}}e^{j{\omega}t} & & \displaystyle\end{eqnarray}$ (13) where 𝜒_r is the modal coupling coefficient, 𝜉_r is the mechanical damping ratio, ω_r is the rth resonance frequency, and ${\gamma}_{w}^{r}$ and ${\gamma}_{{\theta}}^{r}$ are respectively obtained as: $\begin{eqnarray}\displaystyle {\gamma}_{r}^{w}\text{} & =\text{} & \displaystyle \int _{0}^{R}{\phi}_{r}(x)dx\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle {\gamma}_{r}^{{\theta}}\text{} & =\text{} & \displaystyle \int _{0}^{R}x{\phi}_{r}(x)dx.\end{eqnarray}$ (15) Finally, the output voltage can be obtained as $\begin{eqnarray}\displaystyle V_{m}=\displaystyle \frac{\displaystyle \mathop{\sum }_{r=1}^{\infty }\frac{jm{\omega}^{3}{\psi}_{r}({\gamma}_{r}^{w}G_{0}+{\gamma}_{r}^{{\theta}}H_{0})}{{\omega}_{r}^{2}-{\omega}^{2}+j2{\xi}_{r}{\omega}_{r}{\omega}}}{\displaystyle \mathop{\sum }_{r=1}^{\infty }\frac{jm{\chi}_{r}{\psi}_{r}}{{\omega}_{r}^{2}-{\omega}^{2}+j2{\xi}_{r}{\omega}_{r}{\omega}}+\frac{1+j{\omega}{\tau}_{c}}{{\tau}_{c}}} & & \displaystyle\end{eqnarray}$ (16) where τ is the time constant.

When the tiny rotational motion is not considered, that is, h(t) = 0; and when in the first resonance mode, that is, r = 1, the formula ((16)) can be simplified as: $\begin{eqnarray}\displaystyle V_{m}=\frac{j{\tau}_{c}m{\omega}^{3}{\psi}_{1}{\gamma}_{1}^{w}G_{0}}{j{\omega}{\tau}_{c}{\chi}_{1}{\psi}_{1}+(1+j{\omega}{\tau}_{c})({\omega}_{1}^{2}-{\omega}^{2}+j2{\xi}_{1}{\omega}_{1}{\omega})}. & & \displaystyle\end{eqnarray}$ (17)

3. Results and discussions

The resonator was made of copper, and the piezoelectric layer was made of PZT-5H. The highest output power can be obtained only when the mechanical resonance and the acoustic resonance are coupled. It is a simpler and more convenient method to change the acoustic resonant frequency parameters to make it close to the mechanical resonant frequency. According to the equation ((2)), the acoustic resonant frequency is related to the neck size and cavity volume. Therefore, by adjusting the size of the neck of the resonator, the resonant frequency of the sound pressure can be changed to achieve the coupling of the two resonance frequencies.

According to the results of the previous theoretical analysis, when the radius of the neck is 0.15 mm and the length of the neck is 0.115 mm, the mechanical resonant frequency is about 35.1 kHz, and the acoustic resonant frequency is about 35.109 kHz. The two are basically in resonance and are in the best working condition.

The material parameters and the structural dimensions of this omnidirectional AEH are shown in Tables 1 and 2, respectively.

Table 1
Material parameters

Young’s modulus (GPa) Relative dielectric constant Piezoelectric constant (C/m²) Mass density (Kg/m³)

Cu 110 8960

PZT-5H 56 1433.6 −6.62 7500

	Young’s modulus (GPa)	Relative dielectric constant	Piezoelectric constant (C/m²)	Mass density (Kg/m³)
Cu	110			8960
PZT-5H	56	1433.6	−6.62	7500

Table 2

Structural dimensions

Radius of cavity (mm)	Length of neck (mm)	Radius of neck (mm)	Thickness of bottom (mm)	Thickness of PZT (mm)	Thickness of baffle (mm)
R _c	l _n	r _n	t1	t2	t3
1	0.115	0.15	0.1	0.05	0.1

In the simulation, Comsol 5.3a was used for finite element simulation to analyze the model, including four physical fields: solid mechanics, static electricity, pressure acoustics and electric circuit. The relationship of the output characteristics with the incident sound frequency of this system is studied. The sound field space uses a three-dimensional hexahedron with a side length of 4 mm, which is filled with ideal medium (air). The front, bottom and left sides of the cube are set as the incident pressure field. The back, top and right sides are set as hard sound field boundaries. The sound source is set as a far-field plane wave with a sound pressure of 110 pa, and the transmission direction is e = (−1, −1, 1). The energy harvester is located in the middle of the cube, and the boundary conditions of piezoelectric film satisfies mechanical clamping and electrical short-circuit. The positive electrode and the negative electrode are respectively provided on the upper and lower surface of the piezoelectric film, which are connected with an external resistor. The energy harvester periodically strains through the amplified incident sound wave. Due to the piezoelectric effect, the strain is converted into a periodic voltage output, which is collected by an external resistor. Finally, the external air grid is set as normal, and the internal energy harvester grid is set as fine, with a minimum unit of 0.02 mm to obtain more accurate results. On this basis, inspections of different mesh thicknesses are also carried out, and the results are basically unchanged.

3.1. Output sound pressure level

When simulating, the input acoustic frequency is set as independent variable. According to the equation ((3)) and the structural size of the model, it can be calculated that the maximum multiple of the external sound pressure is about 51. According to the equation ((18)), the amplified sound pressure level is 34 dB [27]. $\begin{eqnarray}\displaystyle SPL=20\log \frac{P}{P_{ref}}. & & \displaystyle\end{eqnarray}$ (18)

When the external sound pressure of the system is 110 pa, the equivalent sound pressure level is 135 dB. Since the sound pressure is amplified by 51 times, the theoretically calculated sound pressure level inside the cavity is about 169 dB. As shown in Fig. 2, the maximum sound pressure level of the simulation result is 170 dB at 35 kHz, which is almost the same as the theoretical result. It reflects the sound pressure amplification of the resonant cavity structure. Therefore, according to the calculation and the simulation results, it can be found intuitively that the Helmholtz resonant cavity can amplify the input by 51 times compared with the cavityless structure, and the output voltage of the energy harvester will also differ by several tens of times. The advantage of the cavity structure meets the need of practical applications.

Fig. 2.

The simulation results of sound pressure level.

3.2. Output voltage

When the two resonant frequencies of the energy harvester are coupled, the system reaches the best resonance state. The theoretical calculation and the simulation results are compared. Since the vibration of the system is mainly based on the first-order mode, the theoretical result can be obtained according to the equation ((17)). The maximum output voltage of the energy harvester is 67.61 mV at 35.1 kHz. For simulation results, the incident sound frequency is set as the independent variable, which ranges from 30 kHz to 40 kHz, with a 200 Hz interval. The resonant frequency is 35 kHz, and the maximum output voltage result is 70.95 mV. As shown in Fig. 3, the frequency interval of the simulation and calculation is very small, and the two curves have similar changing trends, so the results of voltage are relatively accurate. Compared with the theoretical result, the curve on the right side of the simulation result drops faster. The reason is that the damping value in the simulation is larger than that in the equation ((17)), which causes the peak voltage to decrease faster. When it comes to the resonant frequency band, 0.707 times the peak voltage is also regarded as the effective working voltage. So that the effective bandwidth of the simulation result is about 700 Hz. The effective bandwidth of the theoretical result is about 950 Hz, which is also caused by the difference of the damping value. With the electromechanical acoustic analogy method, the acoustics, mechanics, and kinematics in this model are studied using Euler–Bernoulli equation. The aeroelastic damping and the strain rate damping are considered in the theoretical model, and the finite element analysis of this model is carried out to incorporate the aeroelastic damping and the strain rate damping. In addition, the isotropic loss factor of Cu is studied to show better prediction of the performance of the proposed method.

Fig. 3.

Comparison of theoretical and simulated results.

Fig. 4.

Structure of the quarter spherical AEH.

The quarter-spherical energy harvester has been shown in Fig. 4. Comparing the voltage results of this structure with that of a quarter-spherical energy harvester in the same size, since all the simulation conditions are the same, the improvement in output performance can be visually demonstrated. It can be seen that the voltage performance of this structure has been significantly improved, the amplitude is about 153.4% as shown in Fig. 5.

Fig. 5.

Voltage comparison results of omnidirectional and quarter spherical AEH.

According to the displacement results obtained by the simulation, the superiority of the omnidirectional structure can be demonstrated. As shown in Fig. 6, the vibration displacement of the energy harvester at the resonant frequency of 35 kHz reaches the maximum, which is about 23.1 nm. At 34 kHz and 36 kHz near the resonant frequency, the vibration displacement is relatively small, respectively 17.6 nm and 4.15 nm, which also reflects the role of the resonance peak. It can be clearly found that whether it is at or near the resonance frequency, the vibration displacement of the overall resonator is not much different among the six cavities, less than 10%. It is found that the vibration is more balanced. The reason is that the six Helmholtz resonators of this AEH correspond to all directions in the environment, and the utilization rate of acoustic energy is also higher. Because the external piezoelectric film is connected as a whole, six Helmholtz resonators can drive the piezoelectric film together, so it can also generate a higher output voltage.

Fig. 6.

Vibration displacement of omnidirectional energy harvester at (a) 34 kHz (b) 35 kHz (c) 36 kHz.

3.3. Output power

The output power of the energy harvester is related to the output voltage and the external resistance. According to formula (19), the expression of output power can be expressed as [28]. $\begin{eqnarray}\displaystyle P_{0}=\frac{V_{rms}^{2}}{R}=\frac{\left(\frac{V_{0}/2}{\sqrt{2}}\right)^{2}}{R}=\frac{V_{0}^{2}}{8R}. & & \displaystyle\end{eqnarray}$ (19) When the resistance value of the external resistance of the energy harvester is equal to the internal equivalent resistance, the maximum output power can be obtained. Therefore, an external resistor is connected with the electrodes on the upper and lower surfaces of the piezoelectric layer in the simulation. After that, the resistance value is used as an independent variable, varying from 0 to 100 kΩ, with a step of 200 Ω. The relationship of the output power with the resistance can be obtained at the resonant frequency. When the resistance reaches the best value, the output power at the resonant frequency is the maximum output power.

Fig. 7.

The relationship between output power and (a) external resistance and (b) sound frequency.

As shown in Fig. 7(a), at the resonant frequency, when the resistance of the external resistor is 14 kΩ, the maximum output power result is 0.45 μW, which also reflects that the internal equivalent impedance of the energy harvester is 14 kΩ. When putting it into practical application, it should also be used as a reference standard to design external circuits. At the same time, after determining the best external resistance, the acoustic frequency is used as an independent variable to verify the result of the output power. At this time, the external resistance is 14 kΩ. It can be found from Fig. 7(b) that the maximum output power at the resonant frequency of 35 kHz is also 0.45 μW, which is the same as the result in the Fig. 7(a).

The input power is produced by the incident sound wave, which can be expressed as: $\begin{eqnarray}\displaystyle P_{i}=\frac{p_{i}^{2}S_{n}}{{\rho}c} & & \displaystyle\end{eqnarray}$ (20) where p _i is the sound pressure, S _n is the opening area, 𝜌 is the air density, and c is the sound speed.

The energy conversion efficiency is: $\begin{eqnarray}\displaystyle {\eta}=\frac{P_{0}}{P_{i}}. & & \displaystyle\end{eqnarray}$ (21) After calculating, the input power is 1.95 μW. According to the equation ((21)), the energy conversion efficiency of the system is about 23%. Compared with the previous structure, it has been greatly improved as shown in Table 3.

Table 3

Comparison of acoustic energy harvesters based on resonator structures

Author (Reference number)	Energy source	Type	Material	Dimensions	Resonant Frequency (kHz)	Energy conversion efficiency
Horowitz et al. [11]	Acoustic pressure (149 dB)	Diaphragm (ring)	PZT	d1 = 2.4 mm d2 = 2.23 mm hp = 0.267 μm	13.6	0.012%
Ji et al. [16]	Acoustic pressure (135 dB)	Diaphragm	PZT	d = 1.1 mm hp = 0.05 mm	40.6	2.56%
Kimura et al. [29]	Acoustic pressure (100 dB)	Diaphragm	PZT	d = 1.2 mm hp = 1.0 μm	16.7	0.35%
Li et al. [30]	Acoustic pressure (100 dB)	Cantilever	PZT	L = 40 mm b = 20 mm hp = 0.48 mm	0.199	15.7%
This work	Acoustic pressure (135 dB)	Diaphragm	PZT	d =1.1 mm hp = 0.05 mm	35	23%

4. Conclusions

An omnidirectional Helmholtz resonant cavity acoustic energy harvester is proposed in this work. By setting six Helmholtz resonant cavities, the acoustic energy in all directions can be collected, thus greatly improving the energy conversion and output performance. When simulating, the input acoustic frequency is set as independent variable, the sound pressure level, output voltage and output power are set as dependent variables. Results show that the sound pressure amplification factor of the energy harvester is about 51, the maximum output voltage at the resonant frequency of 35 kHz is 70.95 mV, and the maximum output power is 0.45 μW. Moreover, the effective bandwidth is about 700 Hz, and the energy conversion efficiency reaches to 23%. The electrical output is more efficient and meets the need of practical application.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (61904085, 61904089), Province Natural Science Foundation of Jiangsu (BK20190731), the China Postdoctoral Science Foundation (2017M621692), and Jiangsu Postdoctoral Foundation (1701131B).

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An omnidirectional acoustic energy harvester based on six Helmholtz resonators

Abstract

Keywords

1. Introduction

2. Principle and theory

Table 1 Material parameters Young’s modulus (GPa) Relative dielectric constant Piezoelectric constant (C/m2) Mass density (Kg/m3) Cu 110 8960 PZT-5H 56 1433.6 −6.62 7500

Footnotes

Acknowledgements

References

Table 1
Material parameters

Young’s modulus (GPa) Relative dielectric constant Piezoelectric constant (C/m²) Mass density (Kg/m³)

Cu 110 8960

PZT-5H 56 1433.6 −6.62 7500