Performance improvement of low frequency piezoelectric energy harvester incorporating holes with an in-house experimental set-up

Abstract

The piezoelectric energy harvesters can be an integral part of self-powered sensor system due to its compatibility with semiconductor technology. In this paper, a modified structural geometry has been proposed in order to improve the performance of energy harvesters with classical cantilever topology by incorporating through holes. A detail mathematical analysis has been carried out to manifest the effect of through hole on resonant frequency of piezo harvester using Rayleigh–Ritz method. The theoretical analysis is validated through simulation studies as well as experimental results. Accordingly, an in-house low cost experimental set-up has been developed using Polyvinylidinefluoride (PVDF) based macro-level energy harvester and loud speaker based vibration set-up. It has been observed that the resonant frequency is reduced from 26.23 Hz to 21.94 Hz due to incorporation of hole to classical cantilever for a given dimension. With input acceleration of 3.7 g, the developed macro-harvester set-up resonates at 23 Hz and is capable of producing power of 59.93 $\mu$W with hole based structure whereas power of 14.60 $\mu$W is obtained in case of without hole based structure. The macro-scale energy harvester with modified geometry has been also designed and simulated in finite element analysis method to validate the mathematical analysis and it is found that lower the resonant frequency higher the power obtained with the hole based topology. It is verified from the experimental, simulation and theoretical studies, that there is a dominant effect of through hole in the performance improvement of piezoelectric energy harvester.

Appendix

In order to compute resonant frequency analytically, the segments of the cantilever harvester can be divided into three sections along the length direction. The first segment will be a distance from reference frame to origin of the hole i.e x=0 to x=$\alpha$L. Then, the second segment consists of the through hole section along the length direction which comprises two factors i.e. location factor $\alpha$ and length factor $\eta$, thus the integration limit lies between $\alpha$L and ($\alpha$ + $\eta$ )L  . Finally, the third segment is from end of through hole ($\alpha$ + $\eta$ )L   to ‘L’. Numerator (8) can be boiled down to to the following (9):

$$\begin{aligned}&Numerator =N= \int \limits _0^L {I{{\left( {\frac{{{d^2}X}}{{d{x^2}}}} \right) }^2}dx} \,\,\end{aligned}$$

(8)

$$\begin{aligned}&N= 144I\left[ {\frac{{{x^5}}}{{5{L^8}}} - \frac{{{x^4}}}{{{L^7}}}+ \frac{{2{x^3}}}{{{L^6}}}- \frac{{2{x^2}}}{{{L^5}}} + \frac{x}{{{L^4}}}} \right] _0^{\alpha L}\nonumber \\&\qquad +144\chi I\left[ {\frac{{{x^5}}}{{5{L^8}}} - \frac{{{x^4}}}{{{L^7}}} + \frac{{2{x^3}}}{{{L^6}}} - \frac{{2{x^2}}}{{{L^5}}} + \frac{x}{{{L^4}}}} \right] _{\alpha L}^{(\alpha + \eta )L} \nonumber \\&\qquad + 144I\left[ {\frac{{{x^5}}}{{5{L^8}}} - \frac{{{x^4}}}{{{L^7}}} + \frac{{2{x^3}}}{{{L^6}}} - \frac{{2{x^2}}}{{{L^5}}} + \frac{x}{{{L^4}}}} \right] _{(\alpha + \eta )L}^L.\end{aligned}$$

(9)

$$\begin{aligned}&Numerator = \,G + H + K\, \end{aligned}$$

(10)

where

$$\begin{aligned}&G= 144I\left[ {\frac{{{x^5}}}{{5{L^8}}} - \frac{{{x^4}}}{{{L^7}}} + \frac{{2{x^3}}}{{{L^6}}} - \frac{{2{x^2}}}{{{L^5}}} + \frac{x}{{{L^4}}}} \right] _0^{\alpha L}\nonumber \\&\quad = 144I\left[ {\frac{{{\alpha ^5}}}{{5{L^3}}} - \frac{{{\alpha ^4}}}{{{L^3}}} + \frac{{2{\alpha ^3}}}{{{L^3}}} - \frac{{2{\alpha ^2}}}{{{L^3}}} + \frac{\alpha }{{{L^3}}}} \right] \end{aligned}$$

(11)

$$\begin{aligned}&H = \,144\chi I\left[ {\frac{{{x^5}}}{{5{L^8}}} - \frac{{{x^4}}}{{{L^7}}} + \frac{{2{x^3}}}{{{L^6}}} - \frac{{2{x^2}}}{{{L^5}}} + \frac{x}{{{L^4}}}} \right] _{\alpha L}^{(\alpha + \eta )L} \nonumber \\&H = \frac{{144}}{{{L^3}}}\chi I\left[ {\frac{{{{(\alpha + \eta )}^5}}}{5} - \frac{{{\alpha ^5}}}{5} - {{(\alpha + \eta )}^4} + {\alpha ^4}} \right] \nonumber \\&\qquad + \,\frac{{144}}{{{L^3}}}\chi I\,\,\,\left[ {2{{(\alpha + \eta )}^3} - 2{\alpha ^3} - 2{{(\alpha + \eta )}^2} + 2{\alpha ^2} + \eta } \right] \, \end{aligned}$$

(12)

$$\begin{aligned}&K = 144I\left[ {\frac{{{x^5}}}{{5{L^8}}} - \frac{{{x^4}}}{{{L^7}}} + \frac{{2{x^3}}}{{{L^6}}} - \frac{{2{x^2}}}{{{L^5}}} + \frac{x}{{{L^4}}}} \right] _{(\alpha + \eta )L}^L \nonumber \\&K = \frac{{144I}}{{{L^3}}}\left[ {\frac{1}{5} - \frac{{{{(\alpha + \eta )}^5}}}{5} + {{(\alpha + \eta )}^4}} \right] \nonumber \\&\qquad + \frac{{144I}}{{{L^3}}}\left[ { - 2{{(\alpha + \eta )}^3} + 2{{(\alpha + \eta )}^2} - (\alpha + \eta )} \right] \end{aligned}$$

(13)

Now,

$$\begin{aligned}&N = 144I\left[ {\frac{{{\alpha ^5}}}{{5{L^3}}} - \frac{{{\alpha ^4}}}{{{L^3}}} + \frac{{2{\alpha ^3}}}{{{L^3}}} - \frac{{2{\alpha ^2}}}{{{L^3}}} + \frac{\alpha }{{{L^3}}}} \right] \nonumber \\&\qquad + \frac{{144}}{{{L^3}}}\chi I\left[ {\frac{{{{(\alpha + \eta )}^5}}}{5} - \frac{{{\alpha ^5}}}{5} - {{(\alpha + \eta )}^4} + {\alpha ^4} + 2{{(\alpha + \eta )}^3}} \right] \nonumber \\&\qquad + \frac{{144}}{{{L^3}}}\chi I\left[ { - 2{\alpha ^3} - 2{{(\alpha + \eta )}^2} + 2{\alpha ^2} + \eta } \right] \nonumber \\&\qquad + \frac{{144I}}{{{L^3}}}\left[ {\frac{1}{5} - \frac{{{{(\alpha + \eta )}^5}}}{5} + {{(\alpha + \eta )}^4}} \right] \nonumber \\&\qquad + \frac{{144I}}{{{L^3}}}\left[ { - 2{{(\alpha + \eta )}^3} + 2{{(\alpha + \eta )}^2} - (\alpha + \eta )} \right] \, \end{aligned}$$

(14)

$$\begin{aligned}&N = \frac{{144I}}{{5{L^3}}}\left[ { - \left\{ {{{(\alpha + \eta )}^5} - {\alpha ^5}} \right\} + 5\left\{ {{{(\alpha + \eta )}^4}} \right\} } \right] \nonumber \\&\qquad + \frac{{144I}}{{5{L^3}}}\left[ { - 5{\alpha ^4}} \right] \nonumber \\&\qquad + \frac{{144I}}{{5{L^3}}}\left[ { - 10\left\{ {{{(\alpha + \eta )}^3} - {\alpha ^3}} \right\} + \ldots - 5\eta + 1} \right] \nonumber \\&\qquad + \frac{{144I}}{{5{L^3}}}\left[ {\chi \left\{ {\left\{ {{{(\alpha + \eta )}^5} - {\alpha ^5}} \right\} - 5\left\{ {{{(\alpha + \eta )}^4} - {\alpha ^4}} \right\} } \right\} } \right] \nonumber \\&\qquad + \frac{{144I}}{{5{L^3}}}\left[ {\chi \left\{ {10\left\{ {{{(\alpha + \eta )}^3} - {\alpha ^3}} \right\} - 10\left\{ {{{(\alpha + \eta )}^2} - {\alpha ^2}} \right\} } \right\} } \right] \nonumber \\&\qquad + \frac{{144I}}{{5{L^3}}}\chi 5\eta \end{aligned}$$

(15)

Hence,

$$\begin{aligned}&Numerator = \frac{{144I}}{{5{L^3}}}\left[ { - Q \times 1 + 1 + \chi \times Q} \right] \nonumber \\&\quad = \frac{{144I}}{{5{L^3}}}\left[ {1 - Q(1 - \chi )} \right] \, \end{aligned}$$

(16)

$$\begin{aligned}&Denominator = \int \limits _0^L {A{X^2}dx} \end{aligned}$$

(17)

$$\begin{aligned}&\int {{X^2}dx = \frac{{{x^9}}}{{9{L^8}}}\,} - \frac{{{x^8}}}{{{L^7}}} + \frac{{4{x^7}}}{{{L^6}}} - \frac{{8{x^6}}}{{{L^5}}} + \frac{{36{x^5}}}{{5{L^4}}} \end{aligned}$$

(18)

$$\begin{aligned}&\int \limits _0^L {A{X^2}dx} \, = bh\left[ {\frac{{{x^9}}}{{9{L^8}}} - \frac{{{x^8}}}{{{L^7}}} + \frac{{4{x^7}}}{{{L^6}}} - \frac{{8{x^6}}}{{{L^5}}} + \frac{{36{x^5}}}{{5{L^4}}}} \right] _0^{\alpha L}\nonumber \\&\qquad + (1 - \beta )bh\left[ {\frac{{{x^9}}}{{9{L^8}}} - \frac{{{x^8}}}{{{L^7}}} + \frac{{4{x^7}}}{{{L^6}}} - \frac{{8{x^6}}}{{{L^5}}} + \frac{{36{x^5}}}{{5{L^4}}}} \right] _{\alpha L}^{(\alpha + \eta )L}\nonumber \\&\qquad + bh\left[ {\frac{{{x^9}}}{{9{L^8}}} - \frac{{{x^8}}}{{{L^7}}} + \frac{{4{x^7}}}{{{L^6}}} - \frac{{8{x^6}}}{{{L^5}}} + \frac{{36{x^5}}}{{5{L^4}}}} \right] _{(\alpha + \eta )L}^L \end{aligned}$$

(19)

$$\begin{aligned}&\int \limits _0^L {A{X^2}dx} \, = S + T + U \end{aligned}$$

(20)

Where,

$$\begin{aligned}&S = bhL\left[ {\frac{{{\alpha ^9}}}{9} - {\alpha ^8} + 4{\alpha ^7} - 8{\alpha ^6} + \frac{{36{\alpha ^5}}}{5}} \right] \end{aligned}$$

(21)

$$\begin{aligned}&T = (1 - \beta )bhL\left[ {\frac{{{{(\alpha + \eta )}^9}}}{9} - \frac{{{\alpha ^9}}}{9} - {{(\alpha + \eta )}^8}} \right] \nonumber \\&\qquad + (1 - \beta )bhL\left[ {{\alpha ^8} + 4{{(\alpha + \eta )}^7} - 4{\alpha ^7}} \right] \nonumber \\&\qquad + (1 - \beta )bhL\left[ { - 8{{(\alpha + \eta )}^6} + 8{\alpha ^6} + \frac{{36{{(\alpha + \eta )}^5}}}{5} - \frac{{36{\alpha ^5}}}{5}} \right] \end{aligned}$$

(22)

$$\begin{aligned}&U = bh\left[ {\frac{{{x^9}}}{{9{L^8}}} - \frac{{{x^8}}}{{{L^7}}} + \frac{{4{x^7}}}{{{L^6}}} - \frac{{8{x^6}}}{{{L^5}}} + \frac{{36{x^5}}}{{5{L^4}}}} \right] _{(\alpha + \eta )L}^L \nonumber \\&U = bhL\left[ {\frac{1}{9} - \frac{{{{(\alpha + \eta )}^9}}}{9} + {{(\alpha + \eta )}^8} - 4{{(\alpha + \eta )}^7} - 5} \right] \nonumber \\&\qquad + bhL\left[ {8{{(\alpha + \eta )}^6} + \frac{{36}}{5} - \frac{{36{{(\alpha + \eta )}^5}}}{5}} \right] \end{aligned}$$

(23)

$$\begin{aligned}&\int \limits _0^L {A{X^2}dx} \, = bhL\left[ {\frac{{{\alpha ^9}}}{9} - {\alpha ^8} + 4{\alpha ^7} - 8{\alpha ^6} + \frac{{36{\alpha ^5}}}{5}} \right] \nonumber \\&\qquad + (1 - \beta )bhL\left[ {\frac{{{{(\alpha + \eta )}^9}}}{9} - \frac{{{\alpha ^9}}}{9} - {{(\alpha + \eta )}^8}} \right] \nonumber \\&\qquad + (1 - \beta )bhL\left[ {{\alpha ^8} + 4{{(\alpha + \eta )}^7} - 4{\alpha ^7}} \right] \nonumber \\&\qquad + (1 - \beta )bhL\left[ { - 8{{(\alpha + \eta )}^6}+8{\alpha ^6} + \frac{{36{{(\alpha + \eta )}^5}}}{5} - \frac{{36{\alpha ^5}}}{5}} \right] \nonumber \\&\qquad + bhL\left[ {\frac{1}{9} - \frac{{{{(\alpha + \eta )}^9}}}{9}} \right] \nonumber \\&\qquad + bhL\left[ {{{(\alpha + \eta )}^8} - 4{{(\alpha + \eta )}^7} - 5} \right] \nonumber \\&\qquad + bhL\left[ {8{{(\alpha + \eta )}^6} + \frac{{36}}{5} - \frac{{36{{(\alpha + \eta )}^5}}}{5}} \right] \end{aligned}$$

(24)

$$\begin{aligned}&\int \limits _0^L {A{X^2}dx} \, = bhL\left[ {\frac{{ - 1}}{9}\left\{ {{{(\alpha + \eta )}^9} - {\alpha ^9}} \right\} } \right] \nonumber \\&\qquad + bhL\left[ {\left\{ {{{(\alpha + \eta )}^8} - {\alpha ^8}} \right\} } \right] \nonumber \\&\qquad + bhL\left[ { - 4\left\{ {{{(\alpha + \eta )}^7} - {\alpha ^7}} \right\} } \right] \nonumber \\&\qquad + bhL\left[ {8\left\{ {{{(\alpha + \eta )}^6} - {\alpha ^6}} \right\} + \frac{{104}}{5}} \right] \nonumber \\&\qquad + bhL\left[ { - \frac{{36}}{5}\left\{ {{{(\alpha + \eta )}^5} - {\alpha ^5}} \right\} } \right] \nonumber \\&\qquad + (1 - \beta )\left[ {\frac{1}{9}\left\{ {{{(\alpha + \eta )}^9} - {\alpha ^9}} \right\} - \left\{ {{{(\alpha + \eta )}^8} - {\alpha ^8}} \right\} } \right] \nonumber \\&\qquad + (1 - \beta )\left[ {4\left\{ {{{(\alpha + \eta )}^7} - {\alpha ^7}} \right\} } \right] \nonumber \\&\qquad + (1 - \beta )\left[ { - 8\left\{ {{{(\alpha + \eta )}^6} - {\alpha ^6}} \right\} + \frac{{36}}{5}\left\{ {{{(\alpha + \eta )}^5} - {\alpha ^5}} \right\} } \right] \end{aligned}$$

(25)

So,

$$\begin{aligned}&\int \limits _0^L {A{X^2}dx} \, = bhL\left[ {(1 - \beta ) \times P - P + \frac{{104}}{{45}}} \right] \nonumber \\&\quad = bhL\frac{{104}}{{45}}\left[ {(1 - \beta )P\frac{{45}}{{104}} - P\frac{{45}}{{104}} + 1} \right] \end{aligned}$$

(26)

$$\begin{aligned}&\int \limits _0^L {A{X^2}dx} \, = bhL\frac{{104}}{{45}}\left[ {1 - \beta P\frac{{45}}{{104}}} \right] \end{aligned}$$

(27)

$$\begin{aligned} where\,\,P= & {} \frac{1}{9}\left\{ {{{(\alpha + \eta )}^9} - {\alpha ^9}} \right\} - \left\{ {{{(\alpha + \eta )}^8} - {\alpha ^8}} \right\} \\&\qquad + 4\left\{ {{{(\alpha + \eta )}^7} - {\alpha ^7}} \right\} - 8\left\{ {{{(\alpha + \eta )}^6} - {\alpha ^6}} \right\} \\&\qquad + \frac{{36}}{5}\left\{ {{{(\alpha + \eta )}^5} - {\alpha ^5}} \right\} \end{aligned}$$

Now,

$$\begin{aligned} {\omega ^2}= & {} \frac{E}{\rho }\frac{{Numerator}}{{Denomin ator}} = \frac{E}{\rho }\frac{{\frac{{144I}}{{5{L^3}}}\left[ {1 - Q(1 - \chi )} \right] \,}}{{bhL\frac{{104}}{{45}}\left[ {1 - \beta P\frac{{45}}{{104}}} \right] }}\nonumber \\= & {} \frac{E}{\rho }\frac{{\frac{{144\left\{ \nicefrac {bh^{3}}{12}\right\} }}{{5{L^3}}}\left[ {1 - Q(1 - \chi )} \right] \,}}{{bhL\frac{{104}}{{45}}\left[ {1 - \beta P\frac{{45}}{{104}}} \right] }} \end{aligned}$$

(28)

Hence,

$$\begin{aligned}&{\omega ^2} = \frac{E}{\rho } \times \frac{{27}}{{26}} \times \frac{{{h^2}}}{{{L^4}}} \times \frac{{\left[ {1 - Q(1 - \chi )} \right] \,}}{{\left[ {1 - \beta P\frac{{45}}{{104}}} \right] }} \end{aligned}$$

(29)

$$\begin{aligned}&{f_{Hole}} = \frac{1}{{2\pi }} \times \frac{h}{{{L^2}}}\sqrt{\frac{{27E}}{{26\rho }}\frac{{\left[ {1 - Q(1 - \chi )} \right] \,}}{{\left[ {1 - \beta P\frac{{45}}{{104}}} \right] }}} \end{aligned}$$

(30)