# Comprehensive Numerical Analysis of a Porous Piezoelectric Ceramic for Axial Load Energy Harvesting

^{1}

^{2}

^{*}

## Abstract

**:**

_{31}) and cylinder (d

_{33}) piezoelectric patches with base excitation. The effects of various porosities, proof mass locations, and different applied acceleration are used to determine the output voltage and power generation. The maximum output voltage and power are obtained equal to 2.25 V and 5.1 µW, respectively.

## 1. Introduction

^{2}at 10

^{8}Ω was observed [16]. Biomass as value-added material was utilized in the fabrication of piezoelectric nanogenerators and piezosupercapacitors for energy-oriented applications. The coconut husk-immobilized PVDF films were developed as piezoelectric films for the piezoelectric nanogenerator (PNG). The PNG has achieved a power density of 1.316 mW/cm

^{2}and an energy density of 76.33 mJ/cm

^{2}[17].

_{31}) is deformed by the duralumin base plate, (ii) the piezoelectric cylinders (d

_{33}) are compressed by the L-shaped steel beam, and (iii) various proof mass position can be used for the different eigenvalues and can be located in the duralumin base plate due to the design flexibilities above with various mechanical vibration inputs. The piezoelectric energy harvester is capable to energy harvesting. In this paper, finite element modelling and experimental analyses of axial-type piezoelectric energy harvester are carried out and compared. The various porous piezoelectric materials are used for energy harvesting in the case of axial loading conditions and base excitation. The effect of various porosities in the energy harvesting system under harmonic base excitation is analysed. The effects of the porosities in bimorphs and cylinders are analysed individually with a combined effect onto the harvester by using numerical simulations. The output voltage and power are calculated for various piezoelectric porosities, electric resistances and proof mass positions. The effects of accelerations and porosities are also analysed.

## 2. Modelling of Porous Piezoelectric Based Energy Harvester

^{s}) and, (ii) parallel circuit (ε

^{P}). In the serial circuit, the phase boundary runs normally in the direction of the applied field and, in the parallel circuit, the boundaries between adjacent phases are parallel to the direction of the applied field [26,27]:

^{s}= ν′/ε′ + ν″/ε″

^{P}= ε′ν′ + ε″ν″

_{d}is used in ANSYS software).

## 3. Experimental Setup

_{31}) and cylinders (d

_{33}) are mounted on the duralumin base plate and the L-shaped steel plate, respectively. Due to vibration, alternating deformations tension-compression in plates (2), and compression-tension in cylinders (1) occurred due to the reaction of the fixed supports. The displacement of the duralumin base plate is measured by the optical linear displacement transducer. The output voltage of the piezoelectric bimorphs and cylinders is collected by the electric circuit. The displacement and voltage signals are transmitted to the CPU via an external ADC/DAC E14-440D module. Power graph software is used for signal processing and measuring the output signals. The technical configurations of the measuring devices are used:

- (i)
- the frequency range is 1–1000 Hz for force oscillations;
- (ii)
- a range from 0 to 5 mm is measurable for lateral displacements of the device;
- (iii)
- the input electric voltage is 0.1–10 V for the electromagnetic oscillations exciter;
- (iv)
- the sensitivity is not less than 5 µm of the optical displacement sensor.

_{33}) as well as bimorphs (d

_{31}). The combined use of these elements allows increasing the output power and conversion efficiency of the converter. When these elements are combined, the converter’s output power and conversion efficiency can be increased. The output voltage of the piezoelectric bimorphs and cylinders is called the total cumulative voltage output. All the piezoelectric elements are connected in series.

## 4. Finite Element Modelling of the Energy Harvesting Device

## 5. Analysis of Results and Discussion

^{2}for all structural units.

#### 5.1. Result Validation

#### 5.2. Effect of Porosity in the Energy Harvesting Device

_{31}piezoelectric power generator is more suitable for small forces and low vibration levels, at the same time, the d

_{33}-type is most suitable for high-force vibrations. In the device, two piezoelectric bimorphs (d

_{31}) and two cylinders (d

_{33}) are used for the analysis. The poling of the bimorphs and cylinders are d

_{31}and d

_{33}, respectively. The first eigenvalue of the device is increased when the porosities of the piezoelectric bimorphs and cylinders increase. The first eigenvalue is 283 Hz and 295 Hz in cases of 0% and 80% porosity, respectively, as shown in Figure 6. The output voltage of the piezoelectric bimorphs increases when the porosity increases. The output voltage decreases when the porosity of the piezoelectric cylinder increases as shown in Figure 7a. The total cumulative voltage of the energy harvesting device decreases, when the porosities of piezoelectric materials increase. The output voltages measured are equal to 2.25 V and 1.75 V for 0% and 80% porosities, respectively, see Figure 7b. The measured values of output current are equal to 2.53 µA and 1.74 µA in the cases of 0% and 80% porosity, respectively. The current intensity decreases with increasing the porosity as shown in Figure 7c.

^{2}/R, where R is the resistive load and, V is the voltage across the resistive load.

_{31}sensitivity of the piezoelectric patches but in the case of d

_{33}sensitivity, it is decreased. Porous piezoelectric ceramics can be used in a variety of applications such as underwater acoustics, non-destructive testing, medical ultrasonic devices, etc. due to their high sensitivity.

#### 5.3. Effect of Porosity with Various Position of Proof Mass

#### 5.4. Effect of Load Resistance

#### 5.5. Effect of the Acceleration in the Energy Harvesting Device

^{−2}to 10 m⋅s

^{−2}. When acceleration increases, the output voltage of the bimorphs and cylinders increases, too. The total cumulative voltage (see, Figure 12a) and power (see, Figure 12b) increase, when the applied acceleration increases. The maximum voltage and power are equal to 4.5 V and 21 µW, respectively, at 10 m⋅s

^{−2}acceleration.

## 6. Conclusions

^{−2}acceleration. The inner structure of the porous polymer does not change, and the mechanical strength of the porous piezoceramic is relatively low. Porosity increases the d

_{31}sensitivity of the piezoelectric patches.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Erturk, A.; Inman, D.J. Piezoelectric Energy Harvesting; John Wiley & Sons: Hoboken, NJ, USA, 2011; ISBN 9780470682548. [Google Scholar]
- Parinov, I.A.; Cherpakov, A. V Overview: State-of-the-Art in the Energy Harvesting Based on Piezoelectric Devices for Last Decade. Symmetry
**2022**, 14, 765. [Google Scholar] [CrossRef] - Beeby, S.P.; Tudor, M.J.; White, N.M. Energy harvesting vibration sources for microsystems applications. Meas. Sci. Technol.
**2006**, 17, R175–R195. [Google Scholar] [CrossRef][Green Version] - Yang, Z.; Zhou, S.; Zu, J.; Inman, D. High-Performance Piezoelectric Energy Harvesters and Their Applications. Joule
**2018**, 2, 642–697. [Google Scholar] [CrossRef][Green Version] - Lee, B.S.; Lin, S.C.; Wu, W.J.; Wang, X.Y.; Chang, P.Z.; Lee, C.K. Piezoelectric MEMS generators fabricated with an aerosol deposition PZT thin film. J. Micromechanics Microengineering
**2009**, 19, 065014. [Google Scholar] [CrossRef] - Gao, X.; Yang, J.; Wu, J.; Xin, X.; Li, Z.; Yuan, X.; Shen, X.; Dong, S. Piezoelectric Actuators and Motors: Materials, Designs, and Applications. Adv. Mater. Technol.
**2020**, 5, 1900716. [Google Scholar] [CrossRef] - Martínez-Ayuso, G.; Friswell, M.I.; Haddad Khodaparast, H.; Roscow, J.I.; Bowen, C.R. Electric field distribution in porous piezoelectric materials during polarization. Acta Mater.
**2019**, 173, 332–341. [Google Scholar] [CrossRef] - Sharma, S.; Kiran, R.; Azad, P.; Vaish, R. A review of piezoelectric energy harvesting tiles: Available designs and future perspective. Energy Convers. Manag.
**2022**, 254, 115272. [Google Scholar] [CrossRef] - Khalili, M.; Ahmed, S.; Papagiannakis, A.T. Piezoelectric energy harvesting for powering a novel weigh-in-motion system. Energy Convers. Manag. X
**2022**, 15, 100259. [Google Scholar] [CrossRef] - Maurya, D.; Yan, Y.; Priya, S. Piezoelectric Materials for Energy Harvesting; CRC Press: Boca Raton, FL, USA, 2015; pp. 143–178. [Google Scholar] [CrossRef]
- Ascione, A.; Gherlone, M.; Orifici, A.C. Nonlinear static analysis of composite beams with piezoelectric actuator patches using the Refined Zigzag Theory. Compos. Struct.
**2022**, 282, 115018. [Google Scholar] [CrossRef] - Chand, R.R.; Tyagi, A. Investigation of the Effects of the Piezoelectric Patch Thickness and Tapering on the Nonlinearity of a Parabolic Converging Width Vibration Energy Harvester. J. Vib. Eng. Technol.
**2022**, 10, 1–18. [Google Scholar] [CrossRef] - Haldkar, R.K.; Cherpakov, A.V.; Parinov, I.A. Modeling, analysis and design optimizations of an axial-type piezoelectric energy generator for optimal output. Smart Mater. Struct.
**2022**, 31, 65019. [Google Scholar] [CrossRef] - Soloviev, A.N.; Chebanenko, V.A.; Parinov, I.A. Mathematical modelling of piezoelectric generators on the base of the Kantorovich method. In Analysis and Modelling of Advanced Structures and Smart Systems; Springer: Singapore, 2018; Volume 81, ISBN 9789811068959. [Google Scholar]
- Parinov, I.A.; Chang, S.H.; Yun-Hae, K.; Nao-Aki, N. Physics and Mechanics of New Materials and Their Applications; Springer Nature Switzerland AG: Cham, Switzerland, 2021; ISBN 9781626185357. [Google Scholar]
- Panda, S.; Hajra, S.; Jeong, H.; Panigrahi, B.K.; Pakawanit, P.; Dubal, D.; Hong, S.; Kim, H.J. Biocompatible CaTiO3-PVDF composite-based piezoelectric nanogenerator for exercise evaluation and energy harvesting. Nano Energy
**2022**, 102, 107682. [Google Scholar] [CrossRef] - Sahu, M.; Hajra, S.; Jadhav, S.; Panigrahi, B.K.; Dubal, D.; Kim, H.J. Bio-waste composites for cost-effective self-powered breathing patterns monitoring: An insight into energy harvesting and storage properties. Sustain. Mater. Technol.
**2022**, 32, e00396. [Google Scholar] [CrossRef] - Khalatkar, A.; Gupta, V.K.; Haldkar, R. Modeling and simulation of cantilever beam for optimal placement of piezoelectric actuators for maximum energy harvesting. Smart Nano-Micro Mater. Devices
**2011**, 8204, 82042G. [Google Scholar] [CrossRef] - Haldkar, R.K.; Parinov, I.A.; Cherpakov, A.V.; Shilyaeva, O. V Modelling vibrations of axial piezoelectric generator with active base. J. Phys. Conf. Ser.
**2021**, 2131, 022018. [Google Scholar] [CrossRef] - Khalatkar, A.M.; Kumar, R.; Haldkar, R.; Jhodkar, D. Arduino-Based Tuned Electromagnetic Shaker Using Relay for MEMS Cantilever Beam. In Proceedings of the Smart Technologies for Energy, Environment and Sustainable Development, Nagpur, India, 28–29 July 2018; Kolhe, M.L., Labhasetwar, P.K., Suryawanshi, H.M., Eds.; Springer: Singapore, 2019; pp. 795–801. [Google Scholar]
- Haldkar, R.K.; Parinov, I.A. Wind Energy Harvesting from Artificial Grass by Using Micro Fibre Composite. In Proceedings of the Physics and Mechanics of New Materials and Their Applications, Kitakyushu, Japan, 26–29 March 2021; Parinov, I.A., Chang, S.-H., Kim, Y.-H., Noda, N.-A., Eds.; Springer International Publishing: Cham, Switzerland, 2021; pp. 511–518. [Google Scholar]
- Nasedkin, A.V. Modelling of Piezocomposites with Mechanical Interface Effects; Springer International Publishing: Cham, Switzerland, 2021; Volume 127, ISBN 9783030427078. [Google Scholar]
- Nasedkin, A.; Rybjanets, A.; Kushkuley, L.; Eshel, Y.; Tasker, R. Different approaches to finite element modeling of effective moduli of porous piezoceramics with 3-3 (3-0) connectivity. Proc. IEEE Ultrason. Symp.
**2005**, 3, 1648–1651. [Google Scholar] [CrossRef] - Nasedkin, A.V.; Nasedkina, A.A.; Rybyanets, A.N. Numerical analysis of effective properties of heterogeneously polarized porous piezoceramic materials with local alloying pore surfaces. Mater. Phys. Mech.
**2018**, 40, 12–21. [Google Scholar] [CrossRef] - Nasedkin, A.; Nassar, M.E. Comprehensive numerical characterization of a piezoelectric composite with hollow metallic inclusions using an adaptable random representative volume. Comput. Struct.
**2022**, 267, 106799. [Google Scholar] [CrossRef] - Rybyanets, A.N. Porous piezoceramics: Theory, technology, and properties. IEEE Trans. Ultrason. Ferroelectr. Freq. Control.
**2011**, 58, 1492–1507. [Google Scholar] [CrossRef] - Wersing, W.; Lubitz, K.; Mohaupt, J. Dielectric, elastic and piezoelectric properties of porous pzt ceramics. Ferroelectrics
**1986**, 68, 77–97. [Google Scholar] [CrossRef] - Getman, I.; Lopatin, S. Theoretical and experimental investigation of porous PZT ceramics. Ferroelectrics
**1996**, 186, 301–304. [Google Scholar] [CrossRef] - Nasedkin, A.V.; Nasedkina, A.A.; Nassar, M.E. Homogenization of Porous Piezocomposites with Extreme Properties at Pore Boundaries by Effective Moduli Method. Mech. Solids
**2020**, 55, 827–836. [Google Scholar] [CrossRef] - Rybianets, A.N.; Nasedkin, A.V. Complete characterization of porous piezoelectric ceramics properties including losses and dispersion. Ferroelectrics
**2007**, 360, 57–62. [Google Scholar] [CrossRef] - Cherpakov, A.V.; Parinov, I.A.; Haldkar, R.K. Parametric and Experimental Modeling of Axial-Type Piezoelectric Energy Generator with Active Base. Appl. Sci.
**2022**, 12, 1700. [Google Scholar] [CrossRef] - Kudimova, A.; Mikhayluts, I.; Nadolin, D.; Nasedkin, A.; Nasedkina, A.; Oganesyan, P.; Soloviev, A. Computer design of porous and ceramic piezocomposites in the finite element package ACELAN. Procedia Struct. Integr.
**2017**, 6, 301–308. [Google Scholar] [CrossRef] - Rybianets, A.; Nasedkin, A. Porous ferroelectric ceramics: Theory, experiment and applications. In Proceedings of the International Congress on Ultrasonics, Vienna, Austria, 9–13 April 2007; pp. 2–5. [Google Scholar] [CrossRef]
- Soloviev, A.; Parinov, I.; Cherpakov, A. Modeling PEG Cantilever Type Added Mass and Active Base Using Porous Ceramics with Effective Properties. In Proceedings of the International Conference on Physics and Mechanics of New Materials and Their Applications, Kitakyushu, Japan, 26–29 March 2021; Springer: Cham, Switzerland, 2021; pp. 333–344. [Google Scholar]

**Figure 1.**Design of piezoelectric energy harvester: (

**a**) schematic diagram and, (

**b**) various elements.

**Figure 3.**Experimental setup: (1) piezoelectric cylinders, (2) piezoelectric bimorphs, (3) duralumin substrate, (4) proof mass (5) L-shaped steel plate, (6) rigid punching, (7) optical linear displacement transducers, (8) electromagnetic vibrator.

**Figure 7.**Porosity variations effect on voltage and current: (

**a**) piezoelectric bimorphs and cylinders, (

**b**) total cumulative voltage vs first-eigen frequency at various porosity and, (

**c**) current intensities at various porosities.

No. | Name | Geometric Parameters, mm | |||||
---|---|---|---|---|---|---|---|

1 | Piezoelectric cylinder | d_{p} | 14 | h_{p} | 12.5 | ||

2 | Piezoelectric bimorph | l_{p} | 20 | w_{p} | 15 | t_{p} | 0.5 |

3 | Duralumin base plate | a_{l} | 250 | a_{w} | 15 | a_{t} | 2 |

4 | Proof mass | b_{m} | 8 | h_{m} | 8 | t_{m} | 30 |

5 | L- shaped plate | l_{f} | 74 | h_{f} | 25 | t_{f} | 2 |

b_{f} | 15 |

Element Name | Materials Used | Density (kg/m^{3}) | Young Modulus (GPa) | |
---|---|---|---|---|

Base Plate | duralumin | 2600 | 0.7 | 0.33 |

Proof mass | steel | 7700 | 2.1 | 0.33 |

L-clamping bar | steel | 7700 | 2.1 | 0.33 |

Fixing supports | duralumin | 2600 | 0.7 | 0.33 |

Active electric load | resistor | ${10}^{6}\mathsf{\Omega}$ | ||

Damping value | $\xi =0.036$ |

Porosity in (%) | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
---|---|---|---|---|---|---|---|---|---|

Density(ρ), kg/m^{3} | 7500 | 6750 | 6000 | 5250 | 4500 | 3750 | 3000 | 2250 | 1500 |

${C}_{11}^{E}$, 10^{10}, N/m^{2} | 13.9 | 11.56 | 9.25 | 6.85 | 5.05 | 3.34 | 2.07 | 1.26 | 0.68 |

${C}_{12}^{E}$, 10^{10}, N/m^{2} | 7.78 | 6.15 | 4.66 | 3.14 | 2.10 | 1.16 | 0.62 | 0.28 | 0.13 |

${C}_{13}^{E}$, 10^{10}, N/m^{2} | 7.43 | 5.82 | 4.25 | 2.82 | 1.87 | 1.06 | 0.52 | 0.24 | 0.1 |

${C}_{33}^{E}$, 10^{10}, /m^{2} | 11.5 | 9.53 | 7.23 | 5.42 | 3.91 | 2.72 | 1.63 | 0.91 | 0.47 |

${C}_{44}^{E}$, 10^{10}, N/m^{2} | 2.56 | 2.23 | 1.83 | 1.44 | 1.10 | 0.74 | 0.44 | 0.23 | 0.1 |

${e}_{31}$, C/m^{2} | −5.2 | −4.23 | −3.14 | −2.07 | −1.32 | −0.75 | −0.43 | −0.21 | −0.1 |

${e}_{33}$, C/m^{2} | 15.1 | 13.38 | 11.37 | 9.59 | 7.68 | 5.93 | 3.93 | 2.30 | 1.25 |

${e}_{15}$, C/m^{2} | 12.7 | 10.96 | 8.96 | 6.91 | 5.00 | 3.30 | 1.95 | 1.00 | 0.44 |

${K}_{11}^{S}$/Ɛ_{0} | 730 | 663 | 582 | 509 | 439 | 349 | 263 | 191 | 122 |

${K}_{33}^{S}$/Ɛ_{0} | 635 | 567 | 492 | 413 | 345 | 270 | 199 | 130 | 75 |

Mode | Experimental | Numerical Simulation | Error, % |
---|---|---|---|

First eigen-frequency (EF), Hz | 275 | 283 | 2.83 |

Second eigen-frequency (EF), Hz | 576 | 593 | 2.86 |

Amplitude at first EF, mm | 0.205 | 0.226 | 9.2 |

Amplitude at second EF, mm | 0.065 | 0.055 | 15.38 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Haldkar, R.K.; Cherpakov, A.V.; Parinov, I.A.; Yakovlev, V.E. Comprehensive Numerical Analysis of a Porous Piezoelectric Ceramic for Axial Load Energy Harvesting. *Appl. Sci.* **2022**, *12*, 10047.
https://doi.org/10.3390/app121910047

**AMA Style**

Haldkar RK, Cherpakov AV, Parinov IA, Yakovlev VE. Comprehensive Numerical Analysis of a Porous Piezoelectric Ceramic for Axial Load Energy Harvesting. *Applied Sciences*. 2022; 12(19):10047.
https://doi.org/10.3390/app121910047

**Chicago/Turabian Style**

Haldkar, Rakesh Kumar, Alexander V. Cherpakov, Ivan A. Parinov, and Vladislav E. Yakovlev. 2022. "Comprehensive Numerical Analysis of a Porous Piezoelectric Ceramic for Axial Load Energy Harvesting" *Applied Sciences* 12, no. 19: 10047.
https://doi.org/10.3390/app121910047