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This paper focuses on numerical and experimental investigations of a novel design piezoelectric energy harvester. Investigated harvester is based on polygon-shaped cantilever array and employs multifrequency operating principle. It consists of eight cantilevers with irregular design of cross-sectional area. Cantilevers are connected to each other by specific angle to form polygon-shaped structure. Moreover, seven seismic masses with additional lever arms are added in order to create additional rotation moment. Numerical investigation showed that piezoelectric polygon-shaped energy harvester has five natural frequencies in the frequency range from 10 Hz to 240 Hz, where the first and the second bending modes of the cantilevers are dominating. Maximum output voltage density and energy density equal to 50.03 mV/mm^{3} and 604 ^{3}, respectively, were obtained during numerical simulation. Prototype of piezoelectric harvester was made and experimental investigation was performed. Experimental measurements of the electrical characteristics showed that maximum output voltage density, energy density, and output power are 37.5 mV/mm^{3}, 815.16 ^{3}, and 65.24

Modern electronic and mechatronic systems have high demand on wireless sensors, low-power electronic devices, and wireless data transfer systems. In general, such devices are used to control numerous physical parameters to store and transfer data wirelessly [

Nowadays, energy harvesting technologies can be successfully used as an alternative solution to the electrochemical batteries. However, a proper energy harvesting technology must be chosen to obtain the highest efficiency of alternative power supply [

Wang et al. proposed magnetically coupled piezoelectric energy harvester with an elastic magnifier. Goal of the investigations was to overcome potential well barriers and to obtain much large-amplitude bistable motion. Numerical and experimental investigation confirmed that proposed energy harvesting system provides high output power, while mass/stiffness ratio of the harvester has the highest value. The maximum measured output power was 6.76 mW when resistance load of 100 kΩ was applied at excitation frequency of 17.6 Hz. It is 26 times higher output power value compared to conventional energy harvester [

Jiang et al. analyzed a multistep piezoelectric buckled beam energy harvester. Magnetic forces generated by permanent magnets were used for excitation. Multistep mechanism was employed to increase bandwidth and power density of the piezoelectric energy harvesting system. Authors showed that proposed buckled beam system is able to provide high-voltage output at broadband excitation frequencies; that is, open-circuit voltage generated by energy harvester was ±5.5 V, while the maximum voltage of ±9.3 V was achieved at 4.4 Hz. Moreover, output power of the harvester reached 5.0

Upadrashta and Yang introduced a new design of nonlinear piezomagnetoelastic energy harvester [

Piezoelectric energy harvesters based on multifrequency operation principle can be used for wide bandwidth energy harvesting as well. Usually multifrequency energy harvesters consist of several cantilevers that can operate at different resonant frequencies. This type of the harvesters has simple design and usually has linear-type vibrations.

Zhou et al. investigated energy harvester based on multimode dynamic magnifier [

Lee et al. designed and investigated a segmented-type harvester that employs multiple modes of harvester vibrations [

Qi et al. investigated a clamped-clamped type piezoelectric energy harvester with side mounted cantilevers [

Rezaeisaray et al. published results of investigation about piezoelectric micro energy harvester with multiple degrees of freedom [

Toyabur et al. investigated a multimodal vibration energy harvester with multiple piezoelectric elements [

Dhote et al. designed and investigated a multimode piezoelectric energy harvester based on a trileg compliant orthoplanar spring with multiple masses [

A new polygon-shaped multifrequency piezoelectric energy harvester is proposed in this paper. Special design of the harvester ensures modal strain summation and multifrequency energy harvesting. Irregular design of the cross-sectional areas of the cantilevers and special design of the seismic masses improve strain distribution characteristic along the cantilevers length. Strain summation and advanced design of the proposed piezoelectric harvester allow increasing electrical output power of the energy harvesting system significantly.

Design of polygon-shaped cantilever array consists of eight piezoelectric cantilevers with irregular cross-sectional design, seven seismic masses, and a clamping system (Figure

Isometric view of the energy harvester: ① clamping bolt; ② clamping frame; ③ M3 bolts for junction between the clamping frame and the harvester; ④ body of the polygon-shaped harvester; ⑤ design of cross-sectional area.

Clamping frame serves as the coupling system between the body of harvester and the host (Figure

Scheme of the clamping system:

Numerical investigation of the polygon-shaped energy harvester was divided into two parts. At the beginning, numerical investigation was performed to calculate optimal geometrical parameters of the cantilevers; after that mechanical and electrical characteristics of the harvester were investigated. Modal-frequency analysis and harmonic response analysis were studied. Finite element model (FEM) was built using Comsol 5.2 software. Boundary conditions were set as follows: ends of the supporting beams were fixed rigidly; motion of the host structure was modeled as acceleration of the harvester base in

Material properties.

Material properties | Beryllium bronze | Piezoceramic PIC255 |
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Density [kg/m^{3}] | 8360 | |

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Elastic modulus [kg/mm^{2}] | | — |

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Poisson’s ratio | 0.34 | — |

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Isotropic structural loss factor | 0.02 | |

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Relative permittivity | — | In the polarization direction |

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Perpendicular to polarity | ||

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Elastic stiffness coefficient ^{2}] | — | |

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Dielectric loss factor, ^{−3}] | — | |

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Coupling factor, | — | |

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Piezoelectric voltage coefficient, ^{−3} Vm/N] | — | − |

Principle scheme of the piezoelectric polygon-shaped energy harvester: ① base of harvester; ②

Two optimization problems were solved sequentially in order to obtain optimal design of the harvester. Goal of the first optimization problem was to find optimal length of the cantilevers when sum of the square differences of the neighboring resonant frequencies of the harvester is minimized. The resonant frequencies dominated by the first and the second out of plane vibration modes of any cantilever were used. Lengths of the eight cantilevers were chosen as the design variables and optimization problem was formulated as follows:

Goal of the second optimization problem was to obtain optimal mass values of the seismic masses in order to maximize tip displacement of the harvester in

Both optimization problems were solved by frequency domain studies using linear search method. Acceleration of the base was set to 0.5 m/s^{2}. Values of

Optimized parameters of the harvester.

Parameter | Value, mm | Parameter | Value, g |
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The second part of numerical investigation was dedicated to analyzing mechanical and electrical characteristics of the harvester. Geometrical parameters of the harvester were stated with respect to the values obtained during optimization study. Modal analysis of the harvester was performed in order to indicate natural frequencies and modal shapes of the harvester (Figure

Modal shapes of the harvester: (a) 14.41 Hz; (b) 25.926 Hz; (c) 73.981 Hz; (d) 199.53 Hz; (e) 214.56 Hz.

Mechanical and electrical characteristics of the harvester were investigated at the frequency range of 10–240 Hz. Base acceleration amplitude was set to 0.5 m/s^{2}. Amplitude graph of harvester tip acceleration in

Acceleration amplitude versus excitation frequency.

Analysis of the frequency response characteristic confirmed results of the modal analysis. Resonant frequencies have good coincidence with the natural frequencies obtained during modal analysis. Obtained characteristic showed that the first three resonant frequencies are located at the narrow frequency range and are close to each other. It shows that energy harvester will be able to operate efficiently at this frequency range. On the other hand, 4th and 5th resonant frequencies are located at slightly higher frequencies. However, these resonant frequencies are adequately close to each other and provide multifrequency energy harvesting at that frequency range. Moreover, high acceleration amplitude of the energy harvester tip at the resonance frequencies has positive influence on the electrical characteristics of harvester.

Investigation of strain distribution along each cantilever was performed as well. Results of the calculations are given in Figure

Strain characteristics at the cantilevers; (a) at

Electrode configuration of piezoceramic layers was based on the strain distribution along the length of the beam. Electrode configuration allowed proper separation of positive and negative charges obtained when cantilevers of the harvester vibrate at the second vibration mode. Proper electrode configuration influences electrical output characteristics of multifrequency energy harvester. Algorithm of electrode configuration was formulated as follows: partitioning of the electrodes was made in the places where the value of stain tensor component is close to zero. Therefore, five vibration nodes of the harvester were analyzed and partitioning of the electrodes was made at indicated nodes. As a result, thirteen different electrodes (PZT_{1}–PZT_{13}) were composed on the top surfaces of the piezoceramic layers of the harvester (Figure

Investigation of electrical characteristics was performed as well. Mechanical boundary conditions of the harvester were the same as in the previous numerical investigation. Electrodes were connected in parallel in order to increase the charge value during harvester operation. The aims of investigation were to obtain output voltage density and energy density characteristics in frequency domain. Voltage density and energy density were introduced in order to compare electrical characteristics of the harvester at the different resonant frequencies. Voltage density can be expressed as follows:

Energy density of the harvester was calculated as follows:

Numerical investigation of the voltage density was performed while applying open-circuit boundary conditions. The results of calculations are given in Figure ^{3}. Other values of voltage density are lower. However, it shows high electrical potential of the energy harvester. Ratio between the highest voltage density and the lowest voltage density is 6.78. It shows that voltage density is sensitive to the resonant frequency mode. However, several voltage density peaks at narrow frequency range give an opportunity to obtain multifrequency-type energy harvesting.

Electrical characteristics of the harvester.

Analysis of energy density characteristic showed that energy harvester is able to provide acceptable energy feed at the different resonant frequencies. The highest energy density was obtained at the 2nd resonant frequency as well. Maximum value of the energy density reached 604 ^{3}. In addition, it must be mentioned that energy density reached high value, 481 ^{3}, at the 1st resonant frequency as well. Therefore, it can be concluded that energy harvester is able to provide high energy feed with overcapacity at the two different resonant frequencies. On the other hand, energy feed at the other resonant frequencies is much lower. It can be concluded that the highest efficiency of energy harvester will be achieved at the lower frequency, while operation at the higher frequency is less effective, but energy feed is appreciable.

Comparison of voltage density and energy density values was made in order to assess electrical characteristics of the energy harvester at different resonant frequencies (Figure ^{3} to 50.03 mV/mm^{3} and from 120 ^{3} to 604 ^{3}, respectively. It is shown that values of voltage and energy densities are obtained at wide range, but harvester is not able to provide constant voltage and energy at the different excitation frequencies. On the other hand, when comparing minimum values of the conventional cantilever array with the proposed multifrequency harvester, it can be noticed that the higher output values were obtained. In addition, stability of electrical characteristics is significantly improved using the proposed piezoelectric harvester.

Summary of electrical characteristics of the harvester.

Experimental investigations of mechanical and electrical characteristics were performed in order to confirm results of numerical investigation. Prototype of the energy harvester was made with the strict respect to the geometrical and physical parameters used during FEM modeling (Figure

Prototype of piezoelectric energy harvester.

Firstly, experimental investigation of the frequency response characteristic was performed with the aim to validate results of numerical investigation. Polytec OFV 056 scanning vibrometer was used for acceleration measurements. Excitation of the energy harvester was performed by electromagnetic shaker Thermotron DSX-4000. A controller of Polytec vibrometer was employed to generate frequency sweep function for the shaker. Results of harvester tip acceleration measurement are shown in Figure

Frequency response characteristic of the energy harvester.

Results of the measurement show that the energy harvester has five peaks in frequency range of 10–240 Hz. Also, it can be seen that two additional peaks appear close to the 1st resonant frequency. This was caused by specific characteristic of the function of excitation signal. However, performed measurements confirmed results of numerical modal analysis. Differences between measured and calculated frequencies do not exceed 5.02%. Summary of the differences between measured and calculated frequencies was made to compare obtained results (Figure

Differences between measured and calculated resonant frequencies.

Experimental investigation of electrical characteristics was performed as well. Output voltage density and energy density were measured in frequency range of 10–240 Hz. Special experimental setup was built for this purpose (Figure

Experimental setup;

Experimental setup consisted of function generator Tektronix AFG1062 and an amplifier that was used to drive electromagnetic shaker. Keyence LK-G155 laser sensor was used to measure base displacement, while Fluke 289 data logging multimeter was used to measure amplitude of the output voltage. Laser sensor and multimeter were connected to computer in order to record and manage data.

Firstly, characteristics of unrectified open-circuit voltage were measured versus excitation frequency for each piezoceramic layer. Voltmeter was connected directly to each piezoceramic layer. Base acceleration of the harvester was set to 0.475 m/s^{2}. The goal of this investigation was to indicate characteristics of unrectified open-circuit voltage for each piezoceramic layer and to assess electrical performance of the harvester. Results of measurements are given in Figure

Measured output voltage of all piezoceramic layers versus frequency.

The results of measurements confirmed that all piezoceramic layers generate voltage, while harvester operates at the resonant frequencies. The highest output voltages were obtained at the low resonant frequencies. The highest output voltage was 10.61 V and it was generated by

Next step of experimental investigation was dedicated to the measurements of total rectified output voltage density. Electronic interface was designed and connected to the energy harvester for this purpose. It consisted of thirteen full diode rectifiers. The rectifiers were made from low loss Schottky diodes BAT46 and low ESR electrolytic capacitor. Voltage density at the frequency range of 10–240 Hz was measured. Electronic interface was switched to the open-circuit condition; that is, electrical load was approximately equal to 10 MΩ. Results of the measurement are shown in Figure

Measured voltage density in frequency domain.

Analysis of the voltage density graph showed that voltage density peaks are obtained at the same frequencies as in the previous measurements. It confirmed good agreement between numerical calculations and experimental measurements. Moreover, measurement shows that voltage density level is not falling below 5.25 mV/mm^{3}. It means that energy harvester is able to provide sufficient voltage output at nonresonant frequency as well. The highest voltage density was obtained at the 2nd resonant frequency and it reached 37.5 mV/mm^{3}. It is 7.14 times higher than lowest voltage density level. Also it can be noticed that voltage densities at the remaining four resonant frequencies are much lower than those at the 2nd resonant frequency.

Experimental investigation of energy density at the same frequency range was performed as well. The same experimental setup was used (Figure

Energy density versus frequency characteristic.

Also it can be noticed that the highest energy density was obtained at the 2nd resonant frequency. Energy density at this frequency reached 815.16 ^{3}. It shows that maximum output power of the proposed harvester is 65.24 ^{3}. Also, it can be seen that energy density values are much lower at the higher resonant frequencies. Energy density value reached 93.23 ^{3} at the 4th resonance frequency and 11.5 ^{3} at the 3rd resonant frequency. Comparison of both voltage and energy densities at the different resonant frequencies is given in Figure

Voltage density and energy density of the harvester at the different resonant frequencies.

A novel design of multifrequency polygon-shaped energy harvester was proposed. The modifications of cross-sectional area, additional seismic masses, and special clamping allow increasing strain of the cantilevers and improving strain distribution along the piezoceramic layers.

Numerical and experimental investigations were performed. Modal analysis showed that the system has five resonant frequencies in the range of 10–240 Hz. The first and the second out of plane bending modes of the cantilevers are dominating in aforementioned frequency range. Moreover, numerical investigation of the strain and strain distribution showed that strain function along the piezoceramic layers is almost constant. Such strain function was achieved by modification of cross-sectional areas of the cantilevers. Numerical investigation of electrical characteristics showed that maximum voltage and energy densities were obtained at the 2nd resonant frequency and reached 50.03 mV/mm^{3} and 604 ^{3}, respectively.

Experimental investigation confirmed results of the numerical modeling. The highest difference between resonant frequencies is 5.02%. Moreover, experimental investigation of electrical characteristics showed that the highest voltage density and energy density were obtained at the 2nd resonant frequency. Both voltage and energy densities reached 37.5 mV/mm^{3} and 815.16 ^{3}, respectively. Maximum output power of the harvester is 65.24

The authors declare that there are no conflicts of interest regarding the publication of this paper.