# A Wideband Piezoelectric Energy Harvester Design by Using Multiple Non-Uniform Bimorphs

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

^{jφt}is defined as the harmonic base motion excitation, where Y is the amplitude of the base displacement, φ is the angular frequency of the harmonic excitation, and j is the unit imaginary number. The dynamic governing equation of the lateral vibration under base motion for a piezoelectric coupled beam with non-uniform geometry and structural strain rate damping can be expressed by,

_{s}is the strain rate damping coefficient. Moreover, boundary conditions for the non-uniform cantilever beam are defined as,

_{i}(x) are mass normalized eigenfunctions and T

_{i}(t) are modal participation coefficients for the ith vibration mode. Since the system is proportionally damped, eigenfunctions represented by X

_{i}(x) are normal mode eigenfunctions of the corresponding undamped free vibration problem [21,22]. By inserting Equation (3) into the governing equation of the problem, Equation (1), normal modes can be calculated by solving,

_{i}

^{2}are the undamped natural frequencies of the structure.

_{x}

^{−}

^{1}is a fourfold integration with respect to x. Additionally, C

_{1}to C

_{4}are constants which should be evaluated from the system boundary conditions. As a result, all X

_{k}components are calculated.

_{i}(t) are the solution of the ordinary differential equation,

_{i}𝜔

_{i}= C

_{s}𝜔

_{i}

^{2}/E and ξ

_{i}is the structural strain rate damping ratio of ith vibration mode. Finally, the vibration response can be obtained by Duhamel integral,

_{di}is the ith mode damped natural frequency. However, the steady state solution of Equation (7) is expressed as,

_{3}is the electrical displacement, d

_{31}(Coulomb/N or m/V) is the piezoelectric coefficient with polarization in direction 3 due to the external stress in direction 1, 𝜎

_{1}is the axial stress along 1-axis, e

_{33}

^{𝜎}is the permittivity at constant stress, and E

_{3}is the electrical field along 3-axis. It should be noted that 1, 2, and 3 directions are aligned with x, y, and z axes, respectively.

_{1}with bending strain 𝜀

_{1}, changing the component permittivity at constant stress into constant strain [26] and knowing that due to assumed uniform electric field E

_{3}(t) = V(t)/h

_{p}, Equation (11) can be rewritten as,

_{p}is the piezoelectric elasticity modulus and V(t) is the electric voltage over the piezoelectric area. Furthermore, the longitudinal bending strain 𝜀

_{1}(x,t) in the structure can be calculated as,

_{c}is the distance between the center of the piezoelectric layer and the beam’s neutral axis. Consequently, the electric displacement becomes,

_{3}(x,t) is related to the output electric charge q(t) by integration over the electrode area as,

_{p}is the number of piezoelectric patches bonded to the surface of the beam and clearly N

_{p}L

_{p}= L. Considering each piezoelectric patch as an individual harvester, absolute charge values from each piezoelectric patch are added to build the total charge generated by the harvester.

_{p}. Additionally, the goal of this research is to estimate the open circuit voltage across the piezoelectric layers. Hence, the backward coupling in the relations is ignored. Finally, the generated voltage v(t) can be calculated as,

## 3. A Piezoelectric Cantilever Harvester with Non-Uniform Design

_{1}= ξ

_{2}= 0.01. Lastly, two steady-state frequency responses, voltage output and tip motion, are investigated. According to the same mathematical model and convergence study for vibration analysis of non-uniform beams provided by Ref. [10], 20 terms of mode shape function series (n = 20) are used to accurately present the final vibration mode shape function for the first two vibration modes.

_{0}e

^{−mx/L}is considered where g

_{0}is the initial value, m is the taper ratio and 0 ≤ x ≤ L. It is noted that taper/variation function is defined from fixed to the free end of the cantilever structure. In order to effectively compare the electromechanical outputs, all of the beams in this section have similar length, volume, and mass.

## 4. Wideband Piezoelectric Energy Harvester

#### 4.1. Wideband Energy Harvester with Decaying Tapered Design

_{min}–𝜔

_{max}] = (475–965) rad/s ≈ (75–155) Hz. The bottom plots show the individual output of each harvester whereas the top plot displays the superimposed overall voltage. As shown in Figure 4, high peak voltage outputs (>10 V) at resonance can be obtained in a wide frequency range from 500 rad/s to 950 rad/s by employing 20 nonlinearly tapered cantilever harvesters. Increasing the excitation frequency leads to higher peak voltage output (10 V to 22 V) due to the higher natural frequencies and peak voltage outputs of the bimorph cantilevers with greater taper ratios in the wideband harvesting system.

#### 4.2. Wideband Energy Harvester with Increasing Tapered Design

_{0}e

^{mx/L}where g

_{0}is the initial value, m is the taper ratio, and 0 ≤ x ≤ L. Again, the geometrical taper ratio increments are chosen as 0.05. By employing 10 non-uniform bimorph energy harvesters with the same length, volume, and mass, the voltage output of a wideband energy harvester is obtained and presented in Figure 5. In contrast to the decaying geometry, the increasing profile decreases the natural frequencies of the cantilever structure which is definitely beneficial for energy harvesting from environmental vibration signals with very low frequencies like HVAC systems. The overall voltage output of the wideband energy harvester within an input excitation range of [𝜔

_{min}–𝜔

_{max}] = (300–475) rad/s ≈ (45–75) Hz changes from 4.5 V to 8 V as shown in Figure 5. The bottom plots show the individual output of each harvester, whereas the top plot displays the superimposed voltage output.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**A nonlinearly tapered cantilever beam with a pair of piezoelectric patches symmetrically and perfectly bonded to the top and bottom surfaces of it, side and front views [11].

**Figure 2.**Electromechanical frequency responses for an exponentially tapered bimorph piezoelectric energy harvester; (

**a**) voltage output; (

**b**) tip displacement.

**Figure 3.**A schematic illustration of a wideband piezoelectric energy harvesting system. An array of nonlinearly tapered cantilever piezoelectric-coupled beams mounted on a transversely vibrating base.

**Figure 4.**Voltage output frequency response corresponding to the wideband energy harvesting system, 20 harvesters with various nonlinear decaying geometry profiles.

**Figure 5.**Voltage output frequency response corresponding to the wideband energy harvesting system, 10 harvesters with various nonlinear increasing geometry profiles.

**Figure 6.**Voltage output frequency response corresponding to the wideband energy harvesting system, 30 harvesters (−0.45 ≤ m ≤ 0.95) with various nonlinear geometry profiles.

**Table 1.**Geometry, material, and electromechanical properties of the piezoelectric energy harvester.

L (m) | b_{0}(m) | h_{0}(m) | h_{p}(m) | E (GPa) | 𝜌 (kg/m ^{3}) | E_{p}(GPa) | 𝜌_{p} (kg/m ^{3}) | d_{31}(pm/V) | e33^{𝜀}(nF/m) |
---|---|---|---|---|---|---|---|---|---|

0.2 | 0.02 | 0.005 | 0.004 | 100 | 7165 | 66 | 7800 | −190 | 15.93 |

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**MDPI and ACS Style**

Keshmiri, A.; Wu, N.
A Wideband Piezoelectric Energy Harvester Design by Using Multiple Non-Uniform Bimorphs. *Vibration* **2018**, *1*, 93-104.
https://doi.org/10.3390/vibration1010008

**AMA Style**

Keshmiri A, Wu N.
A Wideband Piezoelectric Energy Harvester Design by Using Multiple Non-Uniform Bimorphs. *Vibration*. 2018; 1(1):93-104.
https://doi.org/10.3390/vibration1010008

**Chicago/Turabian Style**

Keshmiri, Alireza, and Nan Wu.
2018. "A Wideband Piezoelectric Energy Harvester Design by Using Multiple Non-Uniform Bimorphs" *Vibration* 1, no. 1: 93-104.
https://doi.org/10.3390/vibration1010008