A Novel Circular Plate Acoustic Energy Harvester for Urban Railway Noise

Abstract

To harvest acoustic energy from urban railways, a novel and practical acoustic energy harvester is developed. The harvester consists of a piezoelectric circular plate and a Helmholtz resonator. Based on the field test data of urban railways, the resonance frequencies of the piezoelectric circular plate and the Helmholtz resonator are near 800 Hz. The Helmholtz resonator is designed to amplify the sound pressure. Thus, a lumped parameter model is established. The piezoelectric circular plate is used to convert mechanical energy into electrical energy. The simulation results show that the output power of the harvester is approximately 25 μW and the maximum voltage is 0.149 V under the excitation of urban railway noise. The experiment device is also developed. The maximum output power of the harvester is 8.452 μW, and the maximum voltage is 0.082 V. The experimental and the numerical results are in good agreement and demonstrate the effectiveness of the proposed acoustic energy harvester.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Harvesting energy from the environment can generate power to drive low-power electronic devices, such as wireless sensors and health-monitoring sensors [1, 2]. Many scientists have investigated methods to harvest energy from vibration energy [3], acoustic energy [4, 5], wind energy [6], solar energy [7], and ocean waves [8]. A substantial amount of literature has focused on harvesting the linear or nonlinear vibration energy or the radiated noise caused by the structure vibration [4, 5, 9–15]. Therefore, energy harvesting is a promising research field with huge potential.

At present, the high-speed transit of urban railway vehicles has caused serious noise pollution. Double-sided microcellular foam noise barriers have been developed [16] and installed in the retaining wall of station platforms. Urban railway noise can be reduced through sound absorption and prevention of sound propagation. Thus, a noise energy harvester must be developed to harvest wheel-rail noise and to drive some track monitoring equipment.

On the basis of the piezoelectric effect, Li et al. developed an acoustic energy harvesting mechanism to harvest traveling sound at low audible frequency. The harvester contains a quarter-wavelength straight tube resonator with piezoelectric cantilever plates placed inside the tube [17]. Gao et al. developed acoustic energy harvesting for low-frequency sound in a subwavelength system. This harvester contains eight multiple-cavity units. Moreover, it has rich compound Mie resonances and strong robustness [18]. In another study, Yuan et al. proposed a multifunctional acoustic metastructure to realize both effective low-frequency sound isolation and acoustic energy harvesting. The system exhibits excellent noise isolation in the low-frequency band, and the incident sound energy can be converted into electrical energy by the piezoelectric patch [19].

All the above mentioned studies focus on acoustic energy harvesting by tube structure. However, the Helmholtz resonator is an effective method to amplify the sound pressure. Thus, many researchers have developed acoustic energy harvesters with Helmholtz resonators. For example, an electromechanical Helmholtz resonator with a polyvinylidene fluoride (PVDF) piezoelectric composite cantilever for harvesting acoustic energy was proposed [20]. The results show a maximum power of 0.19 μW with sound waves of 15 Pa (118 dB) at 850 Hz. Noh et al. investigated piezoelectric (PVDF composite) cantilevers with Helmholtz resonators to harvest acoustic energy. The generated power was approximately 0.1 µW through piezoelectric materials [21]. In addition, Yang et al. proposed an acoustic energy harvester consisting of a Helmholtz resonator and dual piezoelectric cantilever beams. Each beam consists of a magnet at the free end [22]. Yuan et al. designed an acoustic energy harvester that contains a Helmholtz resonator and a PVDF film to harvest acoustic energy from high-speed railways. The results of the experiment show that a single unit of the harvester can generate a maximum output voltage of 74.6 mV at 110 dB (SPL) [23]. An acoustic energy harvester with a piezoelectric generator and a Helmholtz resonator is also developed to harvest high-speed trains’ noise in the passenger, cab, and between car sections [24]. Yuan et al. developed a helix structure for low-frequency acoustic energy harvesting. The acoustic resonance frequency is 175 Hz [25]. Atrah et al. developed and simulated an acoustic energy harvester with Helmholtz resonator. The resonance frequency is 3.5 kHz [26]. Khan also developed an acoustic energy harvester with a tapered Helmholtz resonator. A proof mass was attached to the piezoelectric plate [27].

To convert mechanical energy into electrical energy, the piezoelectric cantilever beam has been widely used [17, 20–23]. However, the circular piezoelectric plate is also an effective way to harvest energy. It has a high coupling coefficient [28], thus increasingly attracting researchers’ attention in both acoustic energy harvesting and vibration energy harvesting [25–31]. Xue et al. [32] analyzed nonlinear characteristics of a circular piezoelectric plate near the resonance on the basis of von Karman theory. In the previous work, these authors also developed a circular piezoelectric plate array to harvest vibration energy from railways [33]. Although there are a lot of studies about developing a circular piezoelectric plate harvest with a Helmholtz resonator, the design is not specific to the engineering application. Most studies focus on the mid-high frequency range; for example, the frequency range of [27] is from 1100 Hz to 3450 Hz. However, mid-low-frequency sound sources are prevalent within the industrial environment [25], so it is important to investigate an acoustic energy harvester in the mid-low-frequency range.

In this study, a novel circular plate energy harvester with a Helmholtz resonator is developed to harvest acoustic energy from the urban railway systems. The noise characteristics are analyzed on the basis of the field test data of urban railways. The Helmholtz resonator and the circular piezoelectric plate are designed, modeled, and simulated. The resonance frequency is 831 Hz, which is lower than those in [26, 27]. The simulation results show that the proposed harvester can effectively generate electrical energy from the urban railway noise. The experiment system is also established, and the results of the experiment are in good agreement with the theoretical results. Therefore, the present work demonstrates the innovation and feasibility of the circular plate energy harvester with the Helmholtz resonator.

The manuscript is organized as follows. Section 2 describes the measurements and analyses of the urban railway noise. Section 3 presents the theoretical model of the acoustic energy harvester, and Section 4 presents the simulations of the acoustic energy harvester. Then, Section 5 shows the experiment results of the acoustic energy harvester and compares the analytical solutions with the results of the experiment. Lastly, Section 6 ends the paper with the concluding remark.

2. The Characteristics of Urban Railway Noise

The noises induced by urban railway vehicles can mainly be divided into wheel-rail noise, traction system noise, braking noise, aerodynamic noise, and track structure noise. When the vehicles are moving, the speed would generally be at 50 km/h to 80 km/h, and the wheel-rail noise is the main source of noise. The contribution of the wheel-rail noise for the total noise is far more than those of other kinds. In this study, the wheel-rail noise is the main object of noise energy harvesting.

To clarify the characteristics of the noise of rail transit, the authors measured the noise of rail transit on the spot. The measurements were taken from Songjiang University Town Station to Songjiang Xincheng Station of Shanghai Metro line 9. The distance between the two stations is approximately 800 m, and the speed of the vehicles through the measuring point is approximately 70 km/h. In the measurement process, a microphone was set in the internal shield wall. Then, the noise data were acquired. The measurement scheme diagram is shown in Figure 1.

[figure omitted; refer to PDF]

The time domain of sound pressure is shown in Figure 2(a). The duration the train takes to pass through the station is approximately 7 s. The sampling frequency is 12 kHz. The train has six car formations, and the length of each train is 22.8 m. The sound pressure is mainly approximately 10 Pa, and its peak value can reach 25 Pa. The test vehicle has 24 wheel sets with 6 sections. Thus, 24 sound pressure peaks are represented in Figure 2(a).

[figures omitted; refer to PDF]

The analysis of urban railway noise is the basis of energy harvesting. The frequency characteristics of urban railway noise determine the main design parameters of the noise energy harvester. The frequency domain of sound pressure is shown in Figure 2(b) by fast Fourier transform (FFT). The peak frequency of the sound pressure of urban railway noise is mainly concentrated in 500–1200 Hz.

Next, a 1/3 octave spectrum was obtained from the analysis of filtered noise data in the frequency domain, which is shown in Figure 3. The figure shows that the peak frequency band of the sound pressure is concentrated between 400 Hz and 1600 Hz. The noise energy is mainly concentrated in the 1/3 octave band of 500–1250 Hz. The 800 Hz frequency band has the largest contribution rate of noise energy, and it accounts for approximately 28% of the total energy. Therefore, the noise within the 800 Hz frequency band is the main object to harvest the energy of urban railway noise.

[figure omitted; refer to PDF]

Based on Newton’s second law of motion, the equation for the mass spring system is$$\begin{array}{}\text{(1)}& m\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}=P_n-P_c{A}_n,\end{array}$$where m = ρA_nL_eff is the mass of the air in the neck, ρ is the density of air, and A_n is the cross-sectional area of the neck. L_eff = L_n + 1.7r_n is the effective length of neck. Moreover, P_n is the incident sound pressure amplitude, and P_c is the sound pressure amplitude in bottom of the cavity.

According to the equation of state for ideal gas (Ruan et al., 2006), the equation of the gas in the Helmholtz resonator can be written as$$\begin{array}{}\text{(2)}& \mathrm{P}{\mathrm{\nu }}^{\gamma }=\mathrm{const},\end{array}$$where ν = 1/ρ is the volume of the gas and γ is the adiabatic index.

By determining the differential to equation (2), we can obtain$$\begin{array}{}\text{(3)}& \mathrm{d}P=-P\gamma \frac{\mathrm{d}\nu }{\nu }=-P\gamma \frac{\mathrm{d}{V}_c}{V_c}.\end{array}$$

In an adiabatic ideal gas, the velocity of sound can be expressed as $c=\sqrt{\gamma P/\rho }$. Thus, equation (3) can be expressed as$$\begin{array}{}\text{(4)}& \mathrm{d}P=\rho c^2\frac{\mathrm{d}v}{v}.\end{array}$$

During a very short time, the change of P_n and P_c is small. Thus, we believe that ρ ρ₀, c ≈ c₀, and P_c ≈ dP.$$\begin{array}{}\text{(5)}& P_c=-{\rho }_0{c}_0^2\frac{\mathrm{d}{V}_c}{V_c}.\end{array}$$

Substituting dV_c = −A_nx with equation (5), we have$$\begin{array}{}\text{(6)}& P_c=\frac{{\rho }_0{c}_0^2{A}_n}{V_c}x.\end{array}$$

Substituting equation (6) with equation (1), we can obtain$$\begin{array}{}\text{(7)}& m\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+kx=P_n{A}_n,\end{array}$$where the elasticity coefficient of the equivalent spring is $k={\rho }_0{c}_0^2{A}_n^2/V_c$. The solution of equation (7) is$$\begin{array}{}\text{(8)}& x=\frac{P_n{A}_n}{k-{\omega }^{2}m}.\end{array}$$

When resonance occurs, we can obtain k−ω²m = 0. Thus, we have$$\begin{array}{}\text{(9)}& \frac{{\rho }_0{c}_0^2{A}_n^2}{V_c}-{\rho }_0{L}_{\mathrm{eff}}{A}_n{\omega }^2=0.\end{array}$$

The resonant frequency of the Helmholtz resonator can be written as$$\begin{array}{}\text{(10)}& f_r=\frac{c_0}{2\pi }\sqrt{\frac{A_n}{V_c{L}_{\mathrm{eff}}}},\end{array}$$where c₀ is the speed of sound in air. The sound pressure amplification factor of the Helmholtz resonator is expressed as$$\begin{array}{}\text{(11)}& G=\frac{P_c}{P_n}=\frac{c_0^3}{{2\pi }^2{f}_r^3{V}_c}=2\pi \sqrt{V_c{\frac{L_{\mathrm{eff}}}{A_n}}^3}.\end{array}$$

From equations (10) and (11), the resonant frequency increased with the increase of the cross-sectional area of the neck. Then it decreased with the increase of the effective length of the neck and the volume of the cavity. The sound pressure amplification factor of the Helmholtz resonator increased with the increase of the cross-sectional area of the neck, the effective length of the neck, and the volume of the cavity.

3.3. Model of Piezoelectric Circular Plate

The dynamic sound pressure p(t) is provided by the Helmholtz resonator. The dynamic bending of the plate creates tensile and compressive strains in the piezoelectric layers, which generate charges on the electrodes via piezoelectric effect. The electrodes are connected to a load resistance standing for an electric load.

The bending differential equation of the piezoelectric circular plate is$$\begin{array}{}\text{(12)}& \frac{d}{\mathrm{d}r}\frac{1}{r}\frac{d}{\mathrm{d}r}r\frac{\mathrm{d}w}{\mathrm{d}r}=\frac{Q_r}{D},\end{array}$$where $w$ is the deflection and D is the equivalent flexural rigidity of piezoelectric circular plate. Substituting the shear stress $Q_r=\pi r^{2}p/2\pi r=pr/2$ with equation (12), we can obtain$$\begin{array}{}\text{(13)}& \frac{d}{\mathrm{d}r}\frac{1}{r}\frac{d}{\mathrm{d}r}r\frac{\mathrm{d}w}{\mathrm{d}r}=\frac{pr}{2D}.\end{array}$$

Solving equation (13), we can obtain$$\begin{array}{}\text{(14)}& w=\frac{p{r}^4}{64D}+\frac{C_1{r}^2}{4}+C_2\ln r+C_3,\end{array}$$where C₁, C₂, and C₃ are integral constants. In the center of the piezoelectric circular plate, the angle of deflection surface and the deflection is a limited value. Thus, C₂ = 0. C₁ and C₃ are determined by boundary conditions.

The boundary conditions can be expressed as equation (15) due to the sides of the piezoelectric circular plate with simple supported constraints.$$\begin{array}{}\text{(15)}& \begin{array}{cc}r=R,& w=0,\\ r=R,& M_r=0.\end{array}\end{array}$$

Substituting equation (15) with equation (14), we can obtain$$\begin{array}{}\text{(16)}& w=\frac{pt}{64D}{R^2-r^2}^2+\frac{4R^2{R}^2-r^2}{1+\lambda },\end{array}$$where λ is the equivalent Poisson’s ratio of the piezoelectric circular plate. The curvature differential equation of the piezoelectric circular plate can be expressed as$$\begin{array}{}\text{(17)}& K=\frac{1}{\rho x}=\frac{w^{″}r,t}{{1+w^{\prime 2}r,t}^{3/2}}\approx w^{″}r,t=\frac{pt}{16D}3r^2-\frac{R^{2}3+\lambda }{1+\lambda }.\end{array}$$

The position of the neutral layer of the piezoelectric circular plate is$$\begin{array}{}\text{(18)}& h=\frac{R{E}_b{T}_b^2+R{E}_p{T}_p^2+2R{E}_p{T}_p{T}_b}{2R{E}_p{T}_p+2R{E}_b{T}_b}.\end{array}$$

The distribution of the axial strain of the piezoelectric circular plate in the thickness direction is linear [33]. The axial strain distribution function of the piezoelectric circular plate is$$\begin{array}{}\text{(19)}& Sr,t=T_{m}Kr,t=\begin{array}{c}\frac{pt{T}_m}{16D}\frac{R^{2}3+\lambda }{1+\lambda }-3r^{2}r\in 0,\frac{R\sqrt{3+\lambda }}{\sqrt{31+\lambda }},\\ \frac{pt{T}_m}{16D}3r^2-\frac{R^{2}3+\lambda }{1+\lambda }r\in \frac{R\sqrt{3+\lambda }}{\sqrt{31+\lambda }},R,\end{array}\end{array}$$where T_m = 0.5h_p + h_b − h is the axial distance from each point of the piezoelectric circular plate to the neutral layer.

The electric charge Q (t) on the electrode surface can be calculated by using the first kind of piezoelectric equation.$$\begin{array}{}\text{(20)}& Qt=\frac{\pi d_{31}pt{T}_m{R}^{4}5{\lambda }^2+6\lambda +9}{96s_{11}^{E}D{1+\lambda }^2}+\pi R^2{\overline{\epsilon }}_{33}{E}_{3}t,\end{array}$$where d₃₁ is the piezoelectric constant, E₃ is the electric field, ${\overline{\epsilon }}_{33}$ is the one-dimensional dielectric permittivity, and $s_{11}^E$ is the elastic compliance constant.

Output current i (t) can be expressed as$$\begin{array}{}\text{(21)}& it=\frac{\pi d_{31}{T}_m{R}^{4}5{\lambda }^2+6\lambda +9}{96s_{11}^{E}D{1+\lambda }^2}\frac{\mathrm{d}pt}{\mathrm{d}t}-\frac{\pi R^2{\overline{\epsilon }}_{33}}{T_p}\frac{\mathrm{d}ut}{\mathrm{d}t}.\end{array}$$

The sound pressure can be expressed as p (t) = psin (ωt). Based on the equation u (t) = i (t) R_L, the output voltage is$$\begin{array}{}\text{(22)}& ut=\frac{\pi d_{31}{T}_p{T}_m{R}^{4}5{\lambda }^2+6\lambda +9\omega p_i{R}_L}{96s_{11}^{E}D{1+\lambda }^2\sqrt{T_p^2+{\omega }^2{R}_L^2{\pi R^2{\overline{\epsilon }}_{33}}^2}}\sin \omega t,\end{array}$$where R_L is the load resistance. The output power can be expressed as$$\begin{array}{}\text{(23)}& Pt=\frac{u{t}^2}{R_L}.\end{array}$$

4. Simulation Results

4.1. Simulations of Helmholtz Resonator

The finite element analysis model of the Helmholtz resonator is established in the COMSOL Multiphysics®. The material of the Helmholtz resonator is polymethyl methacrylate. A cylinder is added at the top of the Helmholtz resonator. This cylinder represents the volume of air that enters the Helmholtz resonator. The boundary of the Helmholtz resonator is polymethyl methacrylate impedance boundary, and the boundary of the cylinder is the air impedance boundary. The plane wave acts on the surface of the cylinder as the excitation. The finite element analysis model is shown in Figure 6. In simulation, the density, elastic modulus, and Poisson’s ratio of the polymethyl methacrylate are 1160 kg/m³, 2.8 GPa, and 0.36, respectively. The damping factor of the brass plate is 0.002 and the dielectric loss of the piezoelectric film is 0.02. The dynamic viscosity coefficient of the air is 15.11 × 10⁻⁶ m²/s.

[figure omitted; refer to PDF]

Figure 7 shows the frequency response of the Helmholtz resonator. The incident sound pressure is 2 Pa. As can be seen from the figure, sound pressure also increases rapidly within the frequency band of 700–900 Hz. The resonant frequency of the Helmholtz resonator is 831 Hz.

[figure omitted; refer to PDF]

Figures 8(a) and 8(b) show the sound pressure response and the sound pressure isosurface of the circular plate acoustic energy harvester, respectively, when the circular plate acoustic energy harvester is resonant. The sound pressure reaches its maximum value of 68.473 Pa when it approaches the end of the Helmholtz resonator. In addition, the sound pressure amplification factor is 34.24.

[figures omitted; refer to PDF]

4.2. Simulations of Circular Piezoelectric Plate

The finite element analysis model of the circular piezoelectric plate is also established. The model is shown in Figure 9. The circular piezoelectric plate is made up of two parts: the brass substrate and the PZT-5H piezoelectric film. The parameters are listed in Table 1.

[figure omitted; refer to PDF]

Table 1

Parameters of the PZT-5H piezoelectric film.

Parameters	Unit	PZT-5H	Brass
Density	kg/m3	7600	8920
Elastic modulus	GPa	60.6	106
Poisson’s ratio	/	0.34	0.35
Piezoelectric constant	m/V	274e−12	/
Dielectric constant	F/m	3.363e−8	/
Elastic compliances constant	m2/N	16.5e−12	/

Figure 10 shows the first-order vibration modal result and frequency of the circular piezoelectric plate. The first-order natural frequency of the circular piezoelectric plate is 831.7 Hz. According to the theoretical model in Section 3.3, the theoretical natural frequency is 812 Hz, and the error is 2.98%. Figure 11 shows the deformation of the circular piezoelectric plate with the Helmholtz resonator. The frequency of the excitation sound is 831 Hz. The simulation result of the deformation is 1.4623 × 10⁻³. In addition, the theoretical result of the deformation is 1.3284 × 10⁻³ according to Section 3.3. The error is 10.08%. These results demonstrate the accuracy of the theoretical model.

[figure omitted; refer to PDF]

The radius of piezoelectric plate has a great influence on the resonance frequency, so it is set as a constant in the simulation. In addition, the radius of piezoelectric plate is related to its capacitance C_p. The larger the radius of the piezoelectric plate, the larger the capacitance value. The load resistance R_L, capacitance C_p, and the vibration frequency f meet the following relationship: R_L = 1/2πfC_p. So the radius of the piezoelectric plate should be large enough to reduce the load resistance. In the simulation, the radius of piezoelectric plate is the same as the brass substrate.

4.3. Simulations of Harvester under Urban Railway Noise

The traffic line of Shanghai Metro is totally closed. For safety reasons, the authors are not allowed to enter the line and test the performance of the harvester under the real excitation of the urban railway noise. So the authors use the data of the urban railway noise and study the harvester by simulation.

Figure 12 shows the time domain of the sound pressure with the Helmholtz resonator. The excitation sound is the urban railway noise in Section 2. The maximum sound pressure can reach 80 Pa, and the main frequency is 800 Hz.

[figure omitted; refer to PDF]

By using the Runge–Kutta method, the theoretical model of the harvester is solved. Figure 13(a) shows the curves of the resistance-voltage relationship. The maximum voltage increases with the resistance and tends to a constant for the resistance beyond 6 kΩ. Figure 13(b) shows the curves of the resistance-power relationship. The maximum power is not monotonous with the change of the resistance, and it reaches its peak value when the load resistance R_L approaches 1.4 kΩ.

[figures omitted; refer to PDF]

The load resistance R_L is set as 1.4 kΩ. Figure 14 shows the output voltage of the harvester. The max output voltage is 0.149 V. Figure 15 shows the output power of the harvester. The peak value of the output power is approximately 25 μW and the RMS value of the output power is 15.825 μW. These simulation results demonstrate that the circular plate energy harvester with the Helmholtz resonator can effectively harvest the urban railway noise.

[figure omitted; refer to PDF]

Figure 19 shows the open circuit output voltage of the proposed harvester. As shown in the figure, the blue solid line represents the simulation results, and the red line with the scatters signifies the results of the experiment. Although the voltage of the experiment is higher than that of the simulation results, the trend of the results of the experiment and the simulation results are consistent. The peak value is at 830 Hz, which is the resonance frequency of the harvester.

[figure omitted; refer to PDF]

Figure 20 shows that the output voltage and the power are shown with the variable load resistance from 5 Ω to 5 kΩ. The incident sound is 2 Pa (100 dB) at 830 Hz. The output voltage increases with the increasing load resistance and tends to a certain constant after the load resistance is larger than 3 kΩ. The power achieves its maximum when the resistance is 0.8 kΩ. The peak output power is 8.452 μW, and the corresponding voltage is 0.082 V. The simulation results and the measured results show good agreement. It demonstrates that the proposed models in Section 3 can describe the developed acoustic energy harvester of urban railways. The optimal output power is 8.452 μW, which is higher than the output power at 1.24 μW in [23]. The volume of the developed harvester is 56.16 cm³, and the output power density is 0.15 μW/cm³. The volume of the harvester in [23] is about 112.51 cm³, and the output power density is 0.011 μW/cm³. Therefore, the proposed harvester in the study performs better than that in literature.

[figures omitted; refer to PDF]

6. Conclusions

This investigation developed a novel circular plate energy harvester for urban railway noise. The harvester consists of a Helmholtz resonator and a circular piezoelectric plate. The circular plate energy harvester with the Helmholtz resonator is analytically, numerically, and experimentally investigated. The simulation results and the experiment results show good agreement. Moreover, the results demonstrate that the harvester with the Helmholtz resonator can effectively harvest the urban railway noise. The investigation yields the following conclusions:

(1) The urban railway noise includes various frequencies. The noise energy is mainly concentrated within the frequency band of 400–1600 Hz, and the maximum noise energy appears in the 800 Hz frequency band.

(2) The resonant frequency and amplification ratio of the developed Helmholtz resonator are 836 Hz and 34.31, respectively. The results of the experiment are in good agreement with the simulation results.

(3) Through simulation investigation, the maximum output power of the harvester is approximately 25 μW and the maximum voltage is 0.149 V under the excitation of the urban railway noise. The simulation results demonstrate that the harvester with the Helmholtz resonator can effectively harvest the urban railway noise.

(4) Under the excitation of 2 Pa (830 Hz) in the experiment, the maximum output power of the circular plate acoustic energy harvester is approximately 8.452 μW, and the maximum voltage is 0.082 V.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11802170, 51575334, and U1834201), 2018 Shanghai “Chenguang Project” (no. 18CG66), and Natural Science Foundation of Shanghai (no. 19ZR1421700).

References

[1] D. W. Wang, M. X. Liu, W. J. Qian, X. Wu, Z. Q. Wu, "Parametrical investigation of piezoelectric energy harvesting via friction-induced vibration," Shock and Vibration, vol. 2020,DOI: 10.1155/2020/6190215, 2020.

[2] S. Duan, W. Q. Song, E. Zio, C. Cattani, M. Li, "Product technical life prediction based on multi-modes and fractional Lévy stable motion," Mechanical Systems and Signal Processing, vol. 161 no. 12,DOI: 10.1016/j.ymssp.2021.107974, 2021.

[3] A. X. Luo, Y. L. Zhang, X. T. Dai, Y. F. Wang, W. H. Xu, Y. Lu, M. Wang, K. Q. Fan, F. Wang, "An inertial rotary energy harvester for vibrations at ultra-low frequency with high energy conversion efficiency," Applied Energy, vol. 279,DOI: 10.1016/j.apenergy.2020.115762, 2020.

[4] M. A. Pillai, E. Deenadayalan, "A review of acoustic energy harvesting," International Journal of Precision Engineering and Manufacturing, vol. 15 no. 5, pp. 949-965, DOI: 10.1007/s12541-014-0422-x, 2014.

[5] R. Ahmed, F. Mir, S. Banerjee, "A review on energy harvesting approaches for renewable energies from ambient vibrations and acoustic waves using piezoelectricity," Smart Materials and Structures, vol. 26 no. 8,DOI: 10.1088/1361-665x/aa7bfb, 2017.

[6] H. Lin, M. He, Q. Jing, W. Yang, S. Wang, Y. Liu, Y. Zhang, J. Li, N. Li, Y. Ma, L. Wang, Y. Xie, "Angle-shaped triboelectric nanogenerator for harvesting environmental wind energy," Nanomaterials and Energy, vol. 56, pp. 269-276, DOI: 10.1016/j.nanoen.2018.11.037, 2019.

[7] M. Bagmanci, M. Karaaslan, E. Unal, O. Akgol, M. Bakır, C. Sabah, "Solar energy harvesting with ultra-broad band metamaterial absorber," International Journal of Modern Physics B, vol. 33 no. 8,DOI: 10.1142/s0217979219500565, 2019.

[8] L. Xu, T. Jiang, P. Lin, J. J. Shao, C. He, W. Zhong, X. Y. Chen, Z. L. Wang, "Coupled triboelectric nanogenerator networks for efficient water wave energy harvesting," ACS Nano, vol. 12 no. 2, pp. 1849-1858, DOI: 10.1021/acsnano.7b08674, 2018.

[9] Z. Q. Lu, D. H. Gu, D. Hu, W. Lacarbonara, L. Q. Chen, "Nonlinear vibration isolation via a circular ring," Mechanical Systems and Signal Processing, vol. 136,DOI: 10.1016/j.ymssp.2019.106490, 2020.

[10] V. Pakrashi, G. Marano, P. Cahill, S. F. Ali, M. Magno, "Vibration energy harvesting for monitoring dynamical systems," Shock and Vibration, vol. 2018,DOI: 10.1155/2018/8396029, 2018.

[11] Y. L. Zhang, A. X. Luo, Y. F. Wang, X. T. Dai, Y. Lu, F. Wang, "Rotational electromagnetic energy harvester for human motion application at low frequency," Applied Physics Letters, vol. 116 no. 5,DOI: 10.1063/1.5142575, 2020.

[12] M. A. Halim, K. Tao, S. Towfighian, D. Zhu, "Vibration energy harvesting: linear, nonlinear, and rotational approaches," Shock and Vibration, vol. 2019,DOI: 10.1155/2019/5381756, 2019.

[13] X. G. Guo, Y. L. Zhang, K. Q. Fan, C. K. Lee, F. Wang, "A comprehensive study of non-linear air damping and “pull-in” effects on the electrostatic energy harvesters," Energy Conversion and Management, vol. 203,DOI: 10.1016/j.enconman.2019.112264, 2020.

[14] C. Y. Zhang, G. Hu, Y. Daniil, P. F. Lin, S. H. Gu, D. R. Song, H. Y. Peng, J. L. Wang, "Machine learning based prediction of piezoelectric energy harvesting from wake galloping," Mechanical Systems and Signal Processing, vol. 160,DOI: 10.1016/j.ymssp.2021.107876, 2021.

[15] Y. Peng, D. Zhang, J. Luo, S. R. Xie, H. Y. Pu, Z. J. Li, "Power density improvement based on investigation of initial relative position in an electromagnetic energy harvester with self-powered applications," Smart Materials and Structures, vol. 30 no. 6,DOI: 10.1088/1361-665x/abf69e, 2021.

[16] X. Lei, "Methods for predicting the ambient vibration and noise resulting from rail transit," Proceedings of the Institution of Mechanical Engineers - Part F: Journal of Rail and Rapid Transit, vol. 234 no. 9, pp. 1054-1067, DOI: 10.1177/0954409719881860, 2020.

[17] B. Li, J. H. You, Y. J. Kim, "Low frequency acoustic energy harvesting using PZT piezoelectric plates in a straight tube resonator," Smart Materials and Structures, vol. 22 no. 5,DOI: 10.1088/0964-1726/22/5/055013, 2013.

[18] W. T. Gao, J. P. Xia, H. X. Sun, S. Q. Yuan, G. Yong, X. J. Liu, "Acoustic energy harvesting for low-frequency airborne sound based on compound mie resonances," Applied Physics Express, vol. 12 no. 4,DOI: 10.7567/1882-0786/ab07e5, 2019.

[19] M. Yuan, Z. P. Cao, J. Luo, R. Ohayon, "Acoustic metastructure for effective low-frequency acoustic energy harvesting," Journal of Low Frequency Noise, Vibration and Active Control, vol. 37,DOI: 10.1177/1461348418794832, 2018.

[20] H. Y. Lee, B. Choi, "A multilayer PVDF composite cantilever in the Helmholtz resonator for energy harvesting from sound pressure," Smart Materials and Structures, vol. 22 no. 11,DOI: 10.1088/0964-1726/22/11/115025, 2013.

[21] S. Noh, H. Lee, B. Choi, "A study on the acoustic energy harvesting with Helmholtz resonator and piezoelectric cantilevers," International Journal of Precision Engineering and Manufacturing, vol. 14 no. 9, pp. 1629-1635, DOI: 10.1007/s12541-013-0220-x, 2013.

[22] A. C. Yang, P. Li, Y. M. Wen, "High-efficiency broadband acoustic energy harvesting using Helmholtz resonator and dual piezoelectric cantilever beams," Review of Scientific Instruments, vol. 85 no. 6, pp. 15-35, DOI: 10.1063/1.4882316, 2014.

[23] Y. Wang, X. Zhu, T. Zhang, S. Bano, H. Pan, L. Qi, Z. Zhang, Y. Yuan, "A renewable low-frequency acoustic energy harvesting noise barrier for high-speed railways using a Helmholtz resonator and a PVDF film," Applied Energy, vol. 230, pp. 52-61, DOI: 10.1016/j.apenergy.2018.08.080, 2018.

[24] H. M. Noh, "Acoustic energy harvesting using piezoelectric generator for railway environmental noise," Advances in Mechanical Engineering, vol. 10 no. 7,DOI: 10.1177/1687814018785058, 2018.

[25] M. Yuan, Z. Cao, J. Luo, Z. Pang, "Helix structure for low frequency acoustic energy harvesting," Review of Scientific Instruments, vol. 89,DOI: 10.1063/1.5021526, 2018.

[26] A. B. Atrah, H. Salleh, "Simulation of acoustic energy harvester using Helmholtz resonator with piezoelectric backplate," .

[27] F. Khan Izhar, "Piezoelectric type acoustic energy harvester with a tapered Helmholtz cavity for improved performance," Journal of Renewable and Sustainable Energy, vol. 8 no. 5, pp. 705-709, DOI: 10.1063/1.4962027, 2016.

[28] T.-C. Yuan, J. Yang, L.-Q. Chen, "Nonlinear vibration analysis of a circular composite plate harvester via harmonic balance," Acta Mechanica Sinica, vol. 35 no. 4, pp. 912-925, DOI: 10.1007/s10409-019-00863-0, 2019.

[29] X. Zhao, T. Q. Yang, Y. Dong, X. C. Wang, "Energy harvester array using piezoelectric circular diaphragm for broadband vibration," Applied Physics Letters, vol. 104 no. 22, 2014.

[30] J. Yuan, T. Xie, W. Chen, X. Shan, S. Jiang, "Performance of a drum transducer for scavenging vibration energy," Journal of Intelligent Material Systems and Structures, vol. 20 no. 14, pp. 1771-1777, DOI: 10.1177/1045389x09343477, 2009.

[31] T.-C. Yuan, J. Yang, L.-Q. Chen, "Nonlinear dynamics of a circular piezoelectric plate for vibratory energy harvesting," Communications in Nonlinear Science and Numerical Simulation, vol. 59, pp. 651-656, DOI: 10.1016/j.cnsns.2017.12.010, 2018.

[32] H. Hongping Hu, H. P. Hu, "Nonlinear characteristics of a circular plate piezoelectric harvester with relatively large deflection near resonance," IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 55 no. 9, pp. 2092-2096, DOI: 10.1109/tuffc.901, 2008.

[33] T. C. Yuan, J. Yang, X. W. Liu, "Vibration energy harvesting system for railroad safety based on running vehicles," Smart Materials and Structures, vol. 23, 2014.