Design, modeling, and experimental verification of reversed exponentially tapered multimodal piezoelectric energy harvester from harmonic vibrations for autonomous sensor systems

Abstract

Energy harvesting from multiple modes using piezoelectricity ensures the harvesting of energy from the varied ambient, wideband vibration sources for wireless autonomous sensor systems. In the reported studies, a piezoelectric energy harvester (PEH) with high strain concentration and multimodal characteristics plays an important role in enhancing the harvester's vibration amplitude, performance, and frequency bandwidth. This paper proposes a novel multimodal piezoelectric energy harvester by taking advantage of multimodal techniques consisting of a reversed exponentially tapered beam (Primary beam) and six branched beams (Secondary beam) attached to the primary beam’s free end with a proper flange. This design provides wideband with closely placed vibration modes while the reversed exponentially tapered beam attached to the secondary beams configuration provides higher strain distribution and hence improved harvested power. The harvester is subjected to continuous transverse vibrations due to vertical sinusoidal base excitation of varying frequencies and acceleration ranges. As a result, the primary beam with the piezoelectric patch continually deforms and generates electrical energy. The harvester’s theoretical model was developed and derived from the Euler–Bernoulli beam theory. The proposed harvester was fabricated, and its performance evaluated through experimentation at frequencies ranging from 8 to 30 Hz. Experimental results and numerical simulations using COMSOL Multiphysics confirmed the accuracy of the proposed theoretical model. As ambient vibrations were available in a band of frequencies, the proposed multimodal harvester had the potential to capture energy from wideband ambient vibration sources and hence was advantageous over conventional single-mode harvesters in sourcing low-power autonomous sensors. An energy management system designed after investigating the charging behavior of the capacitor with the harvester revealed that the proposed harvester was suitable for source wireless autonomous sensor systems.

Appendices

Appendix A

1.1 (A) Reversed exponentially tapered primary composite beam element (Beam Sect. 1)

Beam Sect. 1 is a composite consisting of a reserved exponentially tapered beam made of Al-7075 bonded to a piezoelectric patch. Bending stiffness $EI({\xi }_{p})$ and mass per unit length $m({\xi }_{p})$ in beam Sect. 1 is given by (Salmani et al. 2015)

$$EI({\xi }_{p}) =\left[\frac{B({\xi }_{p}){E}_{b}}{3}{{(h}_{b}}^{3}-{{h}_{a}}^{3})+\frac{{B}_{pe}{E}_{pe}}{3}{{(h}_{c}}^{3}-{{h}_{b}}^{3})\right]$$

(35)

$$m({\xi }_{p}) ={B({\xi }_{p})\rho }_{p}{T}_{p}+{B}_{pe}{\rho }_{pe}{T}_{pe}$$

(36)

where $I\left({\xi }_{p}\right)={I}_{0}{e}^{c{\xi }_{p}}$, ${{B\left({\xi }_{p}\right)=B}_{0}e}^{c{\xi }_{p}}$, and $m\left({\xi }_{p}\right)={m}_{0}{e}^{c{\xi }_{p}}$ were the area moment of inertia, width, and mass per unit length of the beam which varied exponentially along its length. Where ${I}_{0}=\frac{{{B}_{0}T}_{p}^{3}}{12}$, ${B}_{0}$ and ${m}_{0}=$ ${{B}_{0}\rho }_{p}{T}_{p}$ were the beam's moment of inertia, width, and mass per unit length of the beam at ${\xi }_{p}=0$. ${E}_{p},{\rho }_{p},{L}_{p}$ and ${T}_{p}$ are the modulus of elasticity, density, length, and thickness of the exponentially tapered primary beam (Beam Sect. 1) and ${E}_{pe},{{\rho }_{pe},{L}_{pe},B}_{pe},$ and ${T}_{pe}$ the modulus of elasticity, density, length, width and thickness of the piezoelectric material bonded to the beam.

Fig. 17

Cross-section of a reversed exponentially tapered beam bonded to a piezoelectric patch

As shown in Fig. 17, h_a is the distance from the neutral axis to the bottom of the host beam, h_b distance from the neutral axis to the top of the host beam, h_c distance from the neutral axis to the piezoelectric layer's top, h_pc distance between the neutral axis and the piezoelectric layer's center and nm the ratio of young’s modulus. These parameters are as follows: (Usharani et al. 2017)

$${h}_{a}=-\frac{nm{T}_{p}^{2}+2{T}_{p}{T}_{pe}+{T}_{pe}^{2}}{2nm{T}_{p}+2{T}_{pe}}$$

(37)

$${h}_{b}=\frac{nm{T}_{p}^{2}-{T}_{pe}^{2}}{2nm{T}_{p}+2{T}_{pe}}$$

(38)

$${h}_{c}=\frac{nm{T}_{p}^{2}+2nm{T}_{p}{T}_{pe}+{T}_{pe}^{2}}{2nm{T}_{p}+2{T}_{pe}}$$

(39)

$${h}_{pc}={h}_{b}+({T}_{pe}/2)$$

(40)

$$E_r=\frac{{E}_{p}B\left({\xi }_{p}\right)}{{E}_{pe}{B}_{pe}}$$

(41)

1.2 (B) Rectangular branched secondary beam elements (Beam Sects. 2, 3, 4, 5, 6, 7)

Bending stiffness $E{I}_{s}$ and mass per unit length ${m}_{s}$ for beam Sects. 2, 3, 4, 5, 6, 7 are expressed as (Salmani et al. 2015; Usharani et al. 2017)

$$E{I}_{s}={E}_{s}{I}_{s}, {m}_{s}={\rho }_{s}{B}_{s}{T}_{s}, {I}_{s}=\frac{{B}_{s}{{T}_{s}}^{3}}{12}$$

(42)

where ${{E}_{s}, {I}_{s}, {\rho }_{s}, B}_{s}$ and ${T}_{s}$ represent the modulus of elasticity, a moment of inertia, density, width, and thickness of the branched beams.

Appendix B

The motion of the base excited harvester in the transverse direction is represented by transverse displacement $g\left(t\right)$ with superimposed rotational displacement $R(t)$. Base displacement ${w}_{b}\left({\xi }_{p},t\right)$ can be written as (Erturk and Inman 2008; Li et al. 2019a)

$${w}_{b}\left({\xi }_{p},t\right)=g\left(t\right)+{\xi }_{p}R(t)$$

(43)

The vibration base does not rotate, and hence the superimposed rotational displacement was zero ($R(t)$ = 0).

Total transverse displacement of the primary beam element is written as (Erturk and Inman 2008; Li et al. 2019a)

$${w}_{T}\left({\xi }_{p},t\right)={w}_{p}\left({\xi }_{p},t\right)+{w}_{b}\left({\xi }_{p},t\right)$$

(44)

where ${w}_{p}\left({\xi }_{p},t\right)$ is the transverse displacement relative to the fixed end of the primary beam.

The movable base for the secondary beam elements was the free end of the primary beam element. Base displacement for the secondary beam elements was given by (Erturk and Inman 2008; Li et al. 2019c)

$${w}_{b}\left({\xi }_{s},t\right)={\left[{w}_{b}\left({\xi }_{s}+{\xi }_{p},t\right)+{w}_{p}\left({\xi }_{p},t\right)+{\xi }_{p}{{w}_{p}}^{\mathrm{^{\prime}}}\left({\xi }_{p},t\right)\right]}_{{\xi }_{p}=1}$$

(45)

Total transverse displacement in each secondary beam element is written as (Erturk and Inman 2008; Li et al. 2019c)

$${w}_{T}\left({\xi }_{s},t\right)={w}_{s}\left({\xi }_{s},t\right)+{w}_{b}\left({\xi }_{s},t\right), s=2, 3, 4, 5, 6, 7$$

(46)

Total kinetic energy of the harvester in terms of eigenfunction and a time-dependent coordinate is given by

$$T=\frac{1}{2}{\int }_{0}^{1}m\left({\xi }_{p}\right)\left[{{\varphi }_{p}\left({\xi }_{p}\right)}^{2}.{\dot{\eta }\left(t\right) }^{2}+{\dot{g}\left(t\right)}^{2}+2{\varphi }_{p}\left({\xi }_{p}\right).\dot{\eta }\left(t\right)\dot{g}\left(t\right)\right]d{\xi }_{p}+\frac{1}{2}{m}_{tp}\left[{{\varphi }_{p}\left({\xi }_{p}\right)}^{2}.{\dot{\eta }\left(t\right) }^{2}+{\dot{g}\left(t\right)}^{2}+2{\varphi }_{p}\left({\xi }_{p}\right).\dot{\eta }\left(t\right)\dot{g}\left(t\right)\right]+\frac{1}{2}{J}_{tp}{{{\varphi }_{p}}^{\mathrm{^{\prime}}}}^{2}\left({\xi }_{p}\right).{\dot{\eta }\left(t\right)}^{2}+\frac{1}{2}\sum_{s=2}^{7}{m}_{s}{\int }_{0}^{1}\left[{\left({\varphi }_{s}\left({\xi }_{s}\right).\dot{\eta }\left(t\right)\right)}^{2}+{\dot{g}\left(t\right)}^{2}+{\left({\varphi }_{p}\left({\xi }_{p}\right).\dot{\eta }\left(t\right)\right)}^{2}+{\left({{\xi }_{p}{\varphi }_{p}}^{\mathrm{^{\prime}}}\left({\xi }_{p}\right).\dot{\eta }\left(t\right)\right)}^{2}+2{\varphi }_{s}\left({\xi }_{s}\right).\dot{\eta }\left(t\right)\dot{g}\left(t\right)+2\dot{g}\left(t\right){\varphi }_{p}\left({\xi }_{p}\right).\dot{\eta }\left(t\right)+2{\varphi }_{s}\left({\xi }_{s}\right){\varphi }_{p}\left({\xi }_{p}\right).{\dot{\eta }\left(t\right)}^{2}+2\dot{g}\left(t\right){{\xi }_{p}{\varphi }_{p}}^{\mathrm{^{\prime}}}\left({\xi }_{p}\right).\dot{\eta }\left(t\right)+2{\varphi }_{s}\left({\xi }_{s}\right){{\xi }_{p}{\varphi }_{p}}^{\mathrm{^{\prime}}}\left({\xi }_{p}\right).{\dot{\eta }\left(t\right)}^{2}+2{\varphi }_{p}\left({\xi }_{p}\right){{\xi }_{p}{\varphi }_{p}}^{\mathrm{^{\prime}}}\left({\xi }_{p}\right).{\dot{\eta }\left(t\right)}^{2}\right]d{\xi }_{s}\mathrm{at }{\xi }_{p}=1+\frac{1}{2}\sum_{s=2}^{7}{m}_{ts}\left[{\left({\varphi }_{s}\left({\xi }_{s}\right).\dot{\eta }\left(t\right)\right)}^{2}+{\dot{g}\left(t\right)}^{2}+{\left({\varphi }_{p}\left({\xi }_{p}\right).\dot{\eta }\left(t\right)\right)}^{2}+{\left({{\xi }_{p}{\varphi }_{p}}^{\mathrm{^{\prime}}}\left({\xi }_{p}\right).\dot{\eta }\left(t\right)\right)}^{2}+2{\varphi }_{s}\left({\xi }_{s}\right).\dot{\eta }\left(t\right)\dot{g}\left(t\right)+2\dot{g}\left(t\right){\varphi }_{p}\left({\xi }_{p}\right).\dot{\eta }\left(t\right)+2{\varphi }_{s}\left({\xi }_{s}\right){\varphi }_{p}\left({\xi }_{p}\right).{\dot{\eta }\left(t\right)}^{2}+2\dot{g}\left(t\right){{\xi }_{p}{\varphi }_{p}}^{\mathrm{^{\prime}}}\left({\xi }_{p}\right).\dot{\eta }\left(t\right)+2{\varphi }_{s}\left({\xi }_{s}\right){{\xi }_{p}{\varphi }_{p}}^{\mathrm{^{\prime}}}\left({\xi }_{p}\right).{\dot{\eta }\left(t\right)}^{2}+2{\varphi }_{p}\left({\xi }_{p}\right){{\xi }_{p}{\varphi }_{p}}^{\mathrm{^{\prime}}}\left({\xi }_{p}\right).{\dot{\eta }\left(t\right)}^{2}\right]\mathrm{at }{\xi }_{p}=1, {\xi }_{s}=1+\frac{1}{2}{J}_{ts}{\left.{{\varphi }_{s}^{\mathrm{^{\prime}}}}^{2}({\xi }_{s})\right|}_{{\xi }_{s}=1}{{\dot{\eta }}_{i}(t)}^{2}$$

(47)

Total potential energy of the harvester in terms of eigenfunction and a time-dependent coordinate is given by

$$U=\frac{1}{2}\left({\int }_{0}^{1}{{{EI({\xi }_{p})\varphi }_{p}}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}\left({\xi }_{p}\right)}^{2}.{\eta (t)}^{2}d{\xi }_{p}\right)-{B}_{pe}{e}_{31}\frac{v\left(t\right)}{{T}_{pe}}\frac{\left({h}_{c}^{2}-{h}_{b}^{2}\right)}{2}{\left.{{\varphi }_{p}}^{\mathrm{^{\prime}}}\left({\xi }_{p}\right).\eta (t)\right|}_{{\xi }_{p}=1}+\frac{1}{2}{C}_{pe}{v\left(t\right)}^{2}+\sum_{s=2}^{7}\left(\frac{1}{2}{EI}_{s}{\int }_{0}^{1}{{{\varphi }_{s}}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}\left({\xi }_{s}\right)}^{2}.{\eta (t)}^{2}d{\xi }_{s}\right)$$

(48)

Appendix C

The forward coupling term is given by

$$ \vartheta _{r} = \frac{{B_{{pe}} e_{{31}} }}{{T_{{pe}} }}\frac{{\left( {h_{c}^{2} - h_{b}^{2} } \right)}}{2}\left. {\varphi _{{rp}} \left( {\xi _{p} } \right)} \right|_{{\xi _{p} }} = 1 $$

(49)

The driving force term ${G}_{r}$ due to base excitation of the harvester was obtained by

$${G}_{r}={G}_{p}+{G}_{s}+{G}_{st}$$

(50)

For the primary beam element, the driving force term due to base excitation ${G}_{p}$ was

$$ G_{p} = \left( {\smallint _{0}^{1} m\left( {\xi _{p} } \right)\varphi _{{rp}} \left( {\xi _{p} } \right)d\xi _{p} } \right) + \left. {m_{{tp}} \varphi _{{rp}} \left( {\xi _{p} } \right)} \right|_{{\xi _{p} }} = 1 $$

(51)

For the secondary beam element, the driving force term due to base excitation ${G}_{s}$ was

$$ G_{s} = \sum\limits_{{s = 2}}^{7} {m_{s} } \int\limits_{0}^{{L_{s} }} {\left[ {\varphi _{{rs}} \left( {\xi _{s} } \right) + \varphi _{{rp}} \left( {\xi _{p} } \right) + \xi _{p} \varphi _{{rp}} \prime \left( {\xi _{p} } \right)} \right]d\xi _{s} \;{\text{at}}\;\xi _{p} = 1} $$

(52)

Driving force term ${G}_{st}$ due to base excitation of the secondary beam element’s tip mass is given by

$${G}_{st}=\sum_{s=2}^{7}{m}_{ts}{\left[{\varphi }_{rs}\left({\xi }_{s}\right)+{\varphi }_{rp}\left({\xi }_{p}\right)+{{\xi }_{p}{\varphi }_{rp}}^{\mathrm{^{\prime}}}\left({\xi }_{p}\right)\right]}_{{{\xi }_{s,}=1, \xi }_{p}=1}$$

(53)

The internal capacitance of the piezoelectric layer

$${C}_{pe}={\varepsilon }_{33}^{s}{B}_{pe}{L}_{pe}/{T}_{pe}$$

(54)

The piezoelectric patch current is given by

$$i\left(t\right)={\delta }_{r} \cdot {\dot{\eta }}_{r}\left(t\right)$$

(55)

The backward coupling term is given by

$$ \delta _{r} = \left( {\frac{{B_{{pe}} e_{{31}} }}{{T_{{pe}} }}\frac{{\left( {h_{c}^{2} - h_{b}^{2} } \right)}}{2}\left. {\varphi _{{rp}} \left( {\xi _{p} } \right)} \right|_{{\xi _{p} = 1}} } \right) $$

(56)