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Use of Angularity on Piezoelectric Crystal to Create Frequency Phase Shift for a Wide-Band Energy Harvester

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Journal of Vibration Engineering & Technologies Aims and scope Submit manuscript

Abstract

Purpose

Piezoelectric materials, when positioned angularly with respect to their crystallographic axis, exhibit a fascinating phenomenon: the creation of a frequency phase shift (FPS) during resonance. Traditionally, FPS arises due to changes in shape and material properties, causing the resonance frequency point to shift. However, present research has revealed an unexplored avenue: when a piezoelectric crystal vibrates along an axis that deviates from the normal poling axis, it manifests FPS phenomena. This unique behavior can be harnessed for a wide-bandwidth energy harvester—a novel application that has not been investigated by other researchers.

Methods

Rather than delving into the theoretical nature of FPS, this study focuses on practical implementation of this phenomenon. Clear mathematical assumptions, coupled with experimental validation, confirm the significant impact of angular position on piezoelectric crystals. Overall, 3 methods of mathematical modeling (with explicit formulas in the appendix), experimental setup and finite element method (FEM) have been used in this study.

Results and Conclusions

The results from all the three methods for solving flexural motion demonstrate how power output depends on the angular position of the model and the dimensional variations of the beam and the piezoelectric material. The newly designed energy harvester produces 5 μWatt of power in both analytical and experimental approaches, under an acceleration of \(1 \, m{s}^{-2}\). Operating within a 20 Hz bandwidth for half-power level and utilizing two perpendicular beams, this design ensures that power output remains nonzero regardless of changes in the assembly’s direction of excitation.

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Author information

Authors and Affiliations

  1. Department of Multidisciplinary Engineering, Texas A&M University, College Station, USA

    Sohrab MirzaAbedini

  2. Department of Mechanical Engineering, University of North Texas, Denton, USA

    Haifeng Zhang

Authors
  1. Sohrab MirzaAbedini
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  2. Haifeng Zhang
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Correspondence to Sohrab MirzaAbedini.

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Appendices

Appendix

In these appendices, explicit form of the large equations is presented. These equations were abbreviated from the main context to make it easier to read and reach the achievements of this study. However, it can be of significant use for other researchers endeavoring mathematical calculations as below. These equations are all referenced in the main context.

Appendix A: Explicit forms of momentum equations:

$$\begin{aligned} & M_{211} = \mathop \int \limits_{S}^{{}} x_{2} T_{1} dA \\ & \quad = bE_{b} \mathop \int \limits_{{ - x_{n} - \frac{{t_{b} }}{2}}}^{{\frac{{t_{b} }}{2} - x_{n} }} x_{2} \left( { - u_{2,11} x_{2} - u_{3,11} x_{3} } \right)dx_{2} \\ & \quad \quad + b\mathop \int \limits_{{\frac{{{\text{t}}_{{\text{b}}} }}{2} - x_{{\text{n}}} }}^{{\frac{{t_{b} }}{2} + t_{p} - x_{n} }} x_{2} (s_{11}^{ - 1} \left( { - u_{2,11} x_{2} - u_{3,11} x_{3} - E_{3} d_{31}^{*} \cos^{ - 1} (\theta )} \right)dx_{2} \\ & = D_{211} u_{2,11} + G_{211} (V) \\ & M_{311} = \mathop \int \limits_{S}^{{}} x_{3} T_{1} dA = \left( {t_{b} + t_{p} } \right).E_{b} \mathop \int \limits_{{ - \frac{b}{2}}}^{\frac{b}{2}} - x_{3} \left( {u_{2,11} x_{2} + u_{3,11} x_{3} } \right)dx_{3} \\ & \quad + (t_{b} + t_{p} )\mathop \int \limits_{{ - {\text{b}}/2}}^{b/2} x_{3} (s_{11}^{ - 1} \left( { - u_{2,11} x_{2} - u_{3,11} x_{3} - d_{31}^{*} \cos^{ - 1} (\theta )E_{l} } \right)dx_{3} = D_{311} u_{3,11} + G_{311} \left( V \right) \\ \end{aligned}$$
(21)

Explicit form of momentum equations for where \({l}_{1}<{x}_{1}<{l}_{2}:\)

$$\begin{aligned} & M_{221} = - bE_{b} \mathop \int \limits_{{ - \frac{{t_{b} }}{2}}}^{{\frac{{t_{b} }}{2}}} x_{2} \left( {u_{2,11} x_{2} + u_{3,11} x_{3} } \right)dx_{2} = D_{221} u_{2,11} + G_{221} V \\ & M_{321} = - t_{b} E_{b} \mathop \int \limits_{ - b/2}^{b/2} x_{3} \left( {u_{2,11} x_{2} + u_{3,11} x_{3} } \right)dx_{3} = D_{321} u_{3,11} + G_{321} V \\ \end{aligned}$$
(22)

For the 2nd beam, the following momentum equations exist:

$$\begin{aligned} & M_{312} = \mathop \int \limits_{S}^{{}} x_{3} T_{1} dA \\ & \quad = cE_{b} \mathop \int \limits_{{ - x_{n} - \frac{{t_{b} }}{2}}}^{{\frac{{t_{b} }}{2} - x_{n} }} x_{3} \left( { - u_{2,11} x_{2} - u_{3,11} x_{3} } \right)dx_{3} \\ & \quad \quad + c\mathop \int \limits_{{\frac{{{\text{t}}_{{\text{b}}} }}{2} - x_{{\text{n}}} }}^{{\frac{{t_{b} }}{2} + t_{p} - x_{n} }} x_{3} (s_{11}^{ - 1} (\left( { - u_{2,11} x_{2} - u_{3,11} x_{3} - E_{3} d_{31}^{*} \cos^{ - 1} (\theta )} \right)dx_{3} \\ & \quad = D_{312} u_{3,11} + G_{312} (V) \\ & M_{212} = \mathop \int \limits_{S}^{{}} x_{2} T_{1} dA \\ & \quad = \left( {t_{b} + t_{p} } \right).\mathop \int \limits_{{ - \frac{c}{2}}}^{\frac{c}{2}} x_{2} \left( { - u_{2,11} x_{2} - u_{3,11} x_{3} } \right)dx_{2} \\ & \quad \quad + \mathop \int \limits_{{ - \frac{{\text{c}}}{2}}}^{\frac{c}{2}} x_{2} (s_{11}^{ - 1} (\left( { - u_{2,11} x_{2} - u_{3,11} x_{3} - d_{31}^{*} \cos^{ - 1} (\theta )E_{3} } \right)dx_{2} \\ & \quad = D_{212} u_{3,11} + G_{212} (V) \\ & M_{322} = - cE_{b} \mathop \int \limits_{{ - \frac{{t_{b} }}{2}}}^{{\frac{{t_{b} }}{2}}} x_{3} \left( {u_{2,11} x_{2} + u_{3,11} x_{3} } \right)dx_{3} = D_{322} u_{3,11} + G_{322} V \\ & M_{222} = - cE_{b} \mathop \int \limits_{{ - \frac{c}{2}}}^{\frac{c}{2}} x_{2} \left( {u_{2,11} x_{2} + u_{3,11} x_{3} } \right)dx_{2} = D_{222} u_{3,11} + G_{222} V \\ \end{aligned}$$
(23)

Appendix B: Equations for the boundary conditions are:

Displacement B.C. at x1 = 0

\({u}_{211}(0,t)=Asin\theta {\text{exp}}(i\omega t)\)

(24)

 

\({u}_{311}(0,t)=Acos\theta {\text{exp}}(i\omega t)\)

 

\({u}_{212}\left(0,t\right)=Acos\theta {\text{exp}}(i\omega t)\)

 

\({u}_{312}\left(0,t\right)=Asin\theta {\text{exp}}\left(i\omega t\right)\)

Velocity B.C. at x1 = 0

\({u}_{\mathrm{211,1}}\left(0,t\right)=0\)

 

\({u}_{\mathrm{311,1}}\left(0,t\right)=0\)

 

\({u}_{212}\left(0,t\right)=0\)

 

\({u}_{312}\left(0,t\right)=0\)

Displacement B.C. at \({L}_{1}\)

\({u}_{211}\left({L}_{1},t\right)={u}_{221}({L}_{1},t)\)

 

\({u}_{311}\left({L}_{1},t\right)={u}_{321}({L}_{1},t)\)

 

\({u}_{212}\left({L}_{1},t\right)={u}_{222}({L}_{1},t)\)

 

\({u}_{312}\left({L}_{1},t\right)={u}_{322}({L}_{1},t)\)

Velocity B.C. at \({L}_{1}\)

\({u}_{\mathrm{211,1}}\left({L}_{1},t\right)={u}_{\mathrm{221,1}}({L}_{1},t)\)

 

\({u}_{\mathrm{311,1}}\left({L}_{1},t\right)={u}_{\mathrm{321,1}}({L}_{1},t)\)

 

\({u}_{\mathrm{212,1}}\left({L}_{1},t\right)={u}_{\mathrm{222,1}}({L}_{1},t)\)

 

\({u}_{\mathrm{312,1}}\left({L}_{1},t\right)={u}_{\mathrm{322,1}}({L}_{1},t)\)

Momentum B.C at \({L}_{1}\)

\({M}_{211}\left({L}_{1},t\right)={M}_{221}({L}_{1},t)\)

Which equals:\(-{D}_{211}.{u}_{\mathrm{211,11}}=-{D}_{221}.{u}_{\mathrm{221,11}}\)

 

\({M}_{311}\left({L}_{1},t\right)={M}_{321}({L}_{1},t)\)

 

\({M}_{212}\left({L}_{1},t\right)={M}_{222}({L}_{1},t)\)

 

\({M}_{312}\left({L}_{1},t\right)={M}_{322}({L}_{1},t)\)

Shear B.C. at \({L}_{1}\)

\({N}_{211}\left({L}_{1},t\right)={N}_{221}({L}_{1},t)\)

Which equals:\(-{D}_{211}.{u}_{\mathrm{211,111}}=-{D}_{221}.{u}_{\mathrm{221,111}}\)

 

\({N}_{311}\left({L}_{1},t\right)={N}_{321}({L}_{1},t)\)

 

\({N}_{212}\left({L}_{1},t\right)={N}_{222}({L}_{1},t)\)

 

\({N}_{312}\left({L}_{1},t\right)={N}_{322}({L}_{1},t)\)

Momentum B.C. at \({L}_{1}+{L}_{2}\)

\({M}_{221}\left({L}_{1}+{L}_{2},t\right)=0\)

 

\({M}_{321}({L}_{1}+{L}_{2},t)=0\)

 

\({M}_{222}\left({L}_{1}+{L}_{2},t\right)=0\)

 

\({M}_{322}({L}_{1}+{L}_{2},t)=0\)

Shear at \({L}_{2}\)

\(-{N}_{221}\left({L}_{2},t\right)={m}_{0}{\ddot{u}}_{221}\left({L}_{2},t\right)\)

 

\(-{N}_{321}\left({L}_{2},t\right)={m}_{0}{\ddot{u}}_{321}\left({L}_{2},t\right)\)

 

\(-{N}_{222}\left({L}_{2},t\right)={m}_{0}{\ddot{u}}_{222}\left({L}_{2},t\right)\)

 

\(-{N}_{322}\left({L}_{2},t\right)={m}_{0}{\ddot{u}}_{322}\left({L}_{2},t\right)\)

Complex notation for time-harmonic motion:

$$\left\{ {u_{211} ,u_{311} ,u_{221} ,u_{321} ,u_{212} ,u_{312} ,u_{222} ,u_{322} ,V,Q,I} \right\} = Re\{ \left\{ {f_{1} ,f_{2} ,f_{3} ,f_{4} ,f_{5} ,f_{5} ,f_{6} ,f_{7} ,f_{8} ,\overline{V},\overline{Q},\overline{I}} \right\}\exp \left( {i\omega t} \right)\}$$
(25)

The general solutions to (15) for the 1st and 2nd beams:

$$\begin{gathered} f1 = B_{1} \sin \left( {\alpha x_{1} } \right) + B_{2} \cos \left( {\alpha x_{1} } \right) + B_{3} \sinh \left( {\alpha x_{1} } \right) + B_{4} \cosh (\alpha x_{1} ) \hfill \\ f2 : = B_{5} \sin (\beta x_{1} ) + B_{6} \cos (\beta x_{1} ) + B_{7} \sinh (\beta x_{1} ) + B_{8} \cosh (\beta x_{1} ) \hfill \\ f3 : = B_{9} \sin (\gamma x_{1} ) + B_{10} \cos (\gamma x_{1} ) + B_{11} \sinh (\gamma x_{1} ) + B_{12} \cosh (\gamma x_{1} ) \hfill \\ f4 : = B_{13} \sin (\delta x_{1} ) + B_{14} \cos (\delta x_{1} ) + B_{15} \sinh (\delta x_{1} ) + B_{16} \cosh (\delta x_{1} ) \hfill \\ f5 : = B_{17} \sin (\smallint x_{1} ) + B_{18} \cos (\smallint x_{1} ) + B_{19} \sinh (\smallint x_{1} ) + B_{20} \cosh (\smallint x_{1} ) \hfill \\ f6 : = B_{21} \sin (\zeta x_{1} ) + B_{22} \cos (\zeta x\_1) + B_{23} \sinh (\zeta x_{1} ) + B_{24} \cosh (\zeta x_{1} ) \hfill \\ f7 : = B_{25} \sin (\eta x_{1} ) + B_{26} \cos (\eta x_{1} ) + B_{27} \sinh (\eta x_{1} ) + B_{28} \cosh (\eta x_{1} ) \hfill \\ f8 : = B_{29} \sin (\kappa x_{1} ) + B_{30} \cos (\kappa x_{1} ) + B_{31} \sinh (\kappa x_{1} ) + B_{32} \cosh (\kappa x_{1} ) \hfill \\ \end{gathered}$$
(26)

In (26):

$$\begin{gathered} \alpha = \left( {\frac{{m_{1} \omega^{2} }}{{D_{1} }}} \right)^{\frac{1}{4}} ,\beta = \left( {\frac{{m_{2} \omega^{2} }}{{D_{2} }}} \right)^{\frac{1}{4}} ,\gamma = \left( {\frac{{m_{3} \omega^{2} }}{{D_{3} }}} \right)^{\frac{1}{4}} , \delta = \left( {\frac{{m_{4} \omega^{2} }}{{D_{4} }}} \right)^{\frac{1}{4}} , \in = \left( {\frac{{m_{5} \omega^{2} }}{{D_{5} }}} \right)^{\frac{1}{4}} , \hfill \\ \zeta = \left( {\frac{{m_{6} \omega^{2} }}{{D_{6} }}} \right)^{\frac{1}{4}} ,\eta = \left( {\frac{{m_{7} \omega^{2} }}{{D_{7} }}} \right)^{\frac{1}{4}} ,\kappa = \left( {\frac{{m_{8} \omega^{2} }}{{D_{8} }}} \right)^{\frac{1}{4}} \hfill \\ \end{gathered}$$
(27)

Solutions for \({B}_{1}-{B}_{32}\):

$$\begin{aligned} & B_{2} + B_{4} = Asin(\theta ) \\ & B_{6} + B_{8} = Acos(\theta ) \\ & B_{1} \alpha + B_{3} \alpha = 0 \\ & B_{5} \beta + B_{7} \beta = 0 \\ \end{aligned}$$
(28)
$$\begin{aligned} B_{1} sin(\alpha L_{1} ) + B_{2} cos(\alpha L_{1} ) + B_{3} sinh(\alpha L_{1} ) + B_{4} cosh(\alpha L_{1} ) = B_{9} sin(\gamma L_{1} ) + B_{10} cos(\gamma L_{1} ) + B_{11} sinh(\gamma L_{1} ) + B_{12} cosh(\gamma L_{1} )\end{aligned}$$
$$\begin{aligned} B_{5} sin\left( {\beta L1} \right) + B6*cos\left( {\beta L_{1} } \right) + B_{7} sinh\left( {\beta L_{1} } \right) + B_{8} cosh\left( {\beta L_{1} } \right) = B_{13} sin\left( {\delta L_{1} } \right) + B_{14} cos\left( {\delta L_{1} } \right) + B_{15} sinh\left( {\delta L_{1} } \right) + B_{16} cosh(\delta L_{1} ) \end{aligned}$$
$$\begin{aligned} B_{1} cos\left( {\alpha L_{1} } \right)\alpha - B_{2} sin\left( {\alpha L_{1} } \right)\alpha + B_{3} cosh\left( {\alpha L_{1} } \right)\alpha + B_{4} sinh\left( {\alpha L_{1} } \right)\alpha = B_{9} cos\left( {\gamma L_{1} } \right)\gamma - B_{10} sin\left( {\gamma L_{1} } \right)\gamma + B_{11} cosh\left( {\gamma L_{1} } \right)\gamma + B_{12} sinh\left( {\gamma L_{1} } \right)\gamma \end{aligned}$$
$$\begin{aligned} B_{5} cos\left( {\beta L_{1} } \right)\beta - B_{6} sin\left( {\beta L_{1} } \right)\beta + B_{7} cosh\left( {\beta L_{1} } \right)\beta + B_{8} sinh\left( {\beta L_{1} } \right)\beta = B_{13} cos\left( {\delta L_{1} } \right)\delta - B_{14} sin\left( {\delta L_{1} } \right)\delta + B_{15} cosh\left( {\delta L_{1} } \right)\delta + B_{16} sinh\left( {\delta L_{1} } \right)\delta \end{aligned}$$
$$\begin{aligned} & - D_{{211}} \left( { - B_{1} ~\sin \left( {\alpha L_{1} } \right)\alpha ^{2} - B_{2} \cos \left( {\alpha L_{1} } \right)\alpha ^{2} ~ + B_{3} \sinh \left( {\alpha L_{1} } \right)\alpha ^{2} + B_{4} \cosh \left( {\alpha L_{1} } \right)\alpha ^{2} } \right) + D_{{31}} E_{l} G_{{211}} /s_{{11}} ~ \\ & = - D_{{1221}} \left( { - B_{9} \sin \left( {\gamma L_{1} } \right)\gamma ^{2} - B_{{10}} \cos \left( {\gamma L_{1} } \right)\gamma ^{2} + B_{{11}} \sinh \left( {\gamma L_{1} } \right)\gamma ^{2} + B_{{12}} \cosh \left( {\gamma L_{1} } \right)\gamma ^{2} } \right) \\ \end{aligned}$$
$$\begin{aligned} & - D_{{311}} ( - B_{5} \sin \left( {\beta L_{1} } \right)\beta ^{2} - B_{6} \cos \left( {\beta L_{1} } \right)\beta ^{2} + B_{7} \sinh \left( {\beta L_{1} } \right)\beta ^{2} + B_{8} \cosh \left( {\beta L_{1} } \right)\beta ^{2} ) + D_{{31}} E_{l} G_{{311}} /s_{{11}} ~ \\ & = ~ - D_{{321}} ( - B_{{13}} \sin \left( {\delta L_{1} } \right)\delta ^{2} - B_{{14}} \cos \left( {\delta L_{1} } \right)\delta ^{2} + B_{{15}} \sinh \left( {\delta L_{1} } \right)\delta ^{2} + B_{{16}} \cosh \left( {\delta L_{1} } \right)\delta ^{2} ) \\ \end{aligned}$$
$$\begin{aligned} & - D_{{211}} ( - B_{1} \cos \left( {\alpha L_{1} } \right)\alpha ^{3} ~ + B_{2} \sin \left( {\alpha L_{1} } \right)\alpha ^{3} + B_{3} \cosh \left( {\alpha L_{1} } \right)\alpha ^{3} + B_{4} \sinh \left( {\alpha L_{1} } \right)\alpha ^{3} ) \\ & = ~ - D_{{221}} ( - B_{9} \cos \left( {\gamma L_{1} } \right)\gamma ^{3} + B_{{10}} \sin \left( {\gamma L_{1} } \right)\gamma ^{3} + ~B_{{11}} \cosh \left( {\gamma L_{1} } \right)\gamma ^{3} + B_{{12}} \sinh \left( {\gamma L_{1} } \right)\gamma ^{3} ) \\ \end{aligned}$$
$$\begin{aligned} & - D_{{311}} ( - B_{5} \cos \left( {\beta L_{1} } \right)\beta ^{3} + B_{6} \sin \left( {\beta L_{1} } \right)\beta ^{3} + B_{7} \cosh \left( {\beta L_{1} } \right)\beta ^{3} + B_{8} \sinh \left( {\beta L_{1} } \right)\beta ^{3} ) \\ & = ~ - D_{{321}} ( - B_{{13}} \cos \left( {\delta L_{1} } \right)\delta ^{3} + B_{{14}} \sin \left( {\delta L_{1} } \right)\delta ^{3} + B_{{15}} \cosh \left( {\delta L_{1} } \right)\delta ^{3} + B_{{16}} \sinh \left( {\delta L_{1} } \right)\delta ^{3} ) \\ \end{aligned}$$
$$\begin{aligned} - D_{221} ( - B_{9} sin\left( {\gamma \left( {L_{1} + L_{2} } \right)} \right)\gamma^{2} - B_{10} cos\left( {\gamma \left( {L_{1} + L_{2} } \right)} \right)\gamma^{2} + B_{11} sinh\left( {\gamma \left( {L_{1} + L_{2} } \right)} \right)\gamma^{2} + B_{12} cosh\left( {\gamma \left( {L_{1} + L_{2} } \right)} \right)\gamma^{2} ) = 0 \end{aligned}$$
$$\begin{aligned} - D_{321} ( - B_{13} sin\left( {\delta \left( {L_{1} + L_{2} } \right)} \right)\delta^{2} - B_{14} cos\left( {\delta \left( {L_{1} + L_{2} } \right)} \right)\delta^{2} + B_{15} sinh\left( {\delta \left( {L_{1} + L_{2} } \right)} \right)\delta^{2} + B_{16} cosh\left( {\delta \left( {L_{1} + L_{2} } \right)} \right)\delta^{2} ) = 0 \end{aligned}$$
$$\begin{aligned} - D_{221} \left( { - B_{9} cos\left( {\gamma L_{2} } \right)\gamma^{3} + B_{10} sin\left( {\gamma L_{2} } \right)\gamma^{3} + B_{11} cosh\left( {\gamma L_{2} } \right)\gamma^{3} + B_{12} sinh\left( {\gamma L_{2} } \right)\gamma^{3} } \right) = m_{0} \omega^{2} (B_{9} sin(\gamma L_{2} ) + B_{10} cos(\gamma L_{2} ) + B_{11} sinh(\gamma L_{2} ) + B_{12} cosh(\gamma L_{2} )) \end{aligned}$$
$$\begin{aligned} - D_{321} ( - B_{13} cos\left( {\delta L_{2} } \right)\delta^{3} + B_{14} sin\left( {\delta L_{2} } \right)\delta^{3} + B_{15} cosh\left( {\delta L_{2} } \right)\delta^{3} + B_{16} sinh\left( {\delta L_{2} } \right)\delta^{3} ) = m_{0} \omega^{2} (B_{13} sin(\delta L_{2} ) + B_{14} cos(\delta L_{2} ) + B_{15} sinh(\delta L_{2} ) + B_{16} cosh(\delta L_{2} )) \end{aligned}$$
$$B_{18} + B_{20} = A.cos(\theta )$$
$$B_{22} + B_{24} = A.sin(\theta )$$
$$B_{17} \in + B_{19} \in = 0$$
$$B_{21} \zeta + B_{23} \zeta = 0$$
$$\begin{aligned} & B_{{17}} \sin ( \in L_{1} ) + B_{{18}} \cos ( \in L_{1} ) + B_{{19}} \sinh ( \in L_{1} ) + B_{{20}} \cosh ( \in L_{1} )~ \\ & = B_{{25}} \sin (\eta L_{1} ) + B_{{26}} \cos \left( {\eta L_{1} } \right) + B_{{27}} \sinh (\eta L_{1} ) + B_{{28}} \cosh \left( {\eta L_{1} } \right) \\ \end{aligned}$$
$$\begin{aligned} & B_{{21}} \sin (\zeta L_{1} ) + B_{{22}} \cos (\zeta L_{1} ) + B_{{23}} \sinh (\zeta L_{1} ) + B_{{24}} \cosh (\zeta L_{1} )~ \\ & = B_{{29}} \sin (\kappa L_{1} ) + B_{{30}} \cos (\kappa L_{1} ) + B_{{31}} \sinh (\kappa L_{1} ) + B_{{32}} \cosh (\kappa L_{1} ) \\ \end{aligned}$$
$$\begin{aligned} & B_{{17}} \cos \left( { \in L_{1} } \right) \in - B_{{18}} \sin \left( { \in L_{1} } \right) \in + B_{{19}} \cosh \left( { \in L_{1} } \right) \in + B_{{20}} \sinh \left( { \in L_{1} } \right) \in ~ \\ & = B_{{25}} \cos \left( {\eta L_{1} } \right)\eta - B_{{26}} \sin \left( {\eta L_{1} } \right)\eta + B_{{27}} \cosh \left( {\eta L_{1} } \right)\eta + B_{{28}} \sinh \left( {\eta L_{1} } \right)\eta \\ \end{aligned}$$
$$\begin{aligned} & B_{{21}} \cos \left( {\zeta L_{1} } \right)\zeta - B_{{22}} \sin \left( {\zeta L_{1} } \right)\zeta + B_{{23}} \cosh \left( {\zeta L_{1} } \right)\zeta + B_{{24}} \sinh \left( {\zeta L_{1} } \right)\zeta ~ \\ & = B_{{29}} \cos \left( {\kappa L_{1} } \right)\kappa - B_{{30}} \sin \left( {\kappa L_{1} } \right)\kappa + B_{{31}} \cosh \left( {\kappa L_{1} } \right)\kappa + B_{{32}} \sinh \left( {\kappa L_{1} } \right)\kappa \\ \end{aligned}$$
$$\begin{gathered} - D_{{312}} \left( { - B_{{17}} \sin \left( { \in L_{1} } \right) \in ^{2} - B_{{18}} \cos \left( { \in L_{1} } \right) \in ^{2} + B_{{19}} \sinh \left( { \in L_{1} } \right) \in ^{2} + B_{{20}} \cosh \left( { \in L_{1} } \right) \in ^{2} } \right) + d_{{31}} E_{l} G_{{312}} /s_{{11}} ~ \hfill \\ = - D_{{322}} \left( { - B_{{25}} \sin \left( {\eta L_{1} } \right)\eta ^{2} - B_{{26}} \cos \left( {\eta L_{1} } \right)\eta ^{2} + B_{{27}} \sinh \left( {\eta L_{1} } \right)\eta ^{2} + B_{{28}} \cosh \left( {\eta L_{1} } \right)\eta ^{2} } \right) \hfill \\ \end{gathered}$$
$$\begin{gathered} - D_{{212}} \left( { - B_{{21}} \sin \left( {\zeta L_{1} } \right)\zeta ^{2} - B_{{22}} \cos \left( {\zeta L_{1} } \right)\zeta ^{2} + B_{{23}} \sinh \left( {\zeta L_{1} } \right)\zeta ^{2} + B_{{24}} \cosh \left( {\zeta L_{1} } \right)\zeta ^{2} } \right) + d_{{31}} E_{l} G_{{212}} /s_{{11}} ~ \hfill \\ = - D_{{222}} \left( { - B_{{29}} \sin \left( {\kappa L_{1} } \right)\kappa ^{2} - B_{{30}} \cos \left( {\kappa L_{1} } \right)\kappa ^{2} + B_{{31}} \sinh \left( {\kappa L_{1} } \right)\kappa ^{2} + B_{{32}} \cosh \left( {\kappa L_{1} } \right)\kappa ^{2} } \right) \hfill \\ \end{gathered}$$
$$\begin{aligned} & - D312\left( { - B_{{17}} \cos \left( { \in L_{1} } \right) \in ^{3} + B_{{18}} \sin \left( {\smallint L_{1} } \right) \in ^{3} + B_{{19}} \cosh \left( { \in L_{1} } \right) \in ^{3} + B_{{20}} \sinh \left( { \in L_{1} } \right) \in ^{3} } \right)~ \\ & = - D_{{322}} \left( { - B_{{25}} \cos \left( {\eta L_{1} } \right)\eta ^{3} + B_{{26}} \sin \left( {\eta L_{1} } \right)\eta ^{3} + B_{{27}} \cosh \left( {\eta L_{1} } \right)\eta ^{3} + B_{{28}} \sinh \left( {\eta L_{1} } \right)\eta ^{3} } \right) \\ \end{aligned}$$
$$\begin{aligned} & - D_{{212}} \left( { - B_{{21}} \cos \left( {\zeta L_{1} } \right)\zeta ^{3} + B_{{22}} \sin \left( {\zeta L_{1} } \right)\zeta ^{3} + B_{{23}} \cosh \left( {\zeta L_{1} } \right)\zeta ^{3} + B_{{24}} \sinh \left( {\zeta L_{1} } \right)\zeta ^{3} } \right)~ \\ & = - D_{{222}} \left( { - B_{{29}} \cos \left( {\kappa L_{1} } \right)\kappa ^{3} + B_{{30}} \sin \left( {\kappa L_{1} } \right)\kappa ^{3} + B_{{31}} \cosh \left( {\kappa L_{1} } \right)\kappa ^{3} + B_{{32}} \sinh \left( {\kappa L_{1} } \right)\kappa ^{3} } \right) \\ \end{aligned}$$
$$- D_{322} ( - B_{25} \sin \left( {\eta \left( {L_{1} + L_{2} } \right)} \right)\eta^{2} - B_{26} \cos \left( {\eta \left( {L_{1} + L_{2} } \right)} \right)\eta^{2} + B_{27} \sinh \left( {\eta \left( {L_{1} + L_{2} } \right)} \right)\eta^{2} + B_{28} \cosh \left( {\eta \left( {L_{1} + L_{2} } \right)} \right)\eta^{2} ) = 0$$
$$\begin{aligned} - D_{222} ( - B_{29} \sin \left( {\kappa \left( {L_{1} + L_{2} } \right)} \right)\kappa^{2} - B_{30} \cos \left( {\kappa \left( {L_{1} + L_{2} } \right)} \right)\kappa^{2} + B_{31} \sinh \left( {\kappa \left( {L_{1} + L_{2} } \right)} \right)\kappa^{2} + B_{32} \cosh \left( {\kappa \left( {L_{1} + L_{2} } \right)} \right)\kappa^{2} ) = 0 \end{aligned}$$
$$\begin{aligned} & - D_{{322}} \left( { - B_{{25}} \cos \left( {\eta L_{2} } \right)\eta ^{3} + B_{{26}} \sin \left( {\eta L_{2} } \right)\eta ^{3} + B_{{27}} \cosh \left( {\eta L_{2} } \right)\eta ^{3} + B_{{28}} \sinh \left( {\eta L_{2} } \right)\eta ^{3} } \right) \\ & = m_{0} \omega ^{2} \left( {B_{{25}} \sin (\eta L_{2} ) + B_{{26}} \cos (\eta L_{2} ) + B_{{27}} \sinh (\eta L_{2} ) + B_{{28}} \cosh (\eta L_{2} )} \right) \\ \end{aligned}$$
$$\begin{aligned} & - D_{{222}} \left( { - B_{{29}} \cos \left( {\kappa L_{2} } \right)\kappa^{3} + B_{{30}} \sin \left( {\kappa L_{2} } \right)\kappa^{3} + B_{{31}} \cosh \left( {\kappa L_{2} } \right)\kappa^{3} + B_{{32}} \sinh \left( {\kappa L_{2} } \right)\kappa^{3} } \right) \\ & = ~m_{0} \omega^{2} \left( {B_{{29}} \sin (\kappa L_{2} ) + B_{{30}} \cos (\kappa L_{2} ) + B_{{31}} \sinh (\kappa L_{2} ) + B_{{32}} \cosh (\kappa L_{2} )} \right) \\ \end{aligned}$$

Voltage equations of the first and second beam, individually:

$$\begin{gathered} - i\omega \left\{ {\frac{{bd_{31} }}{{s_{11} }}\left( {t_{b} + t_{p} } \right)\left[ {B_{1} \alpha \cos \alpha L_{1} - B_{2} \alpha \sin \alpha L_{1} + B_{3} \alpha \cosh \alpha L_{1} + B_{4} \alpha \sinh \alpha L_{1} - \alpha B_{1} - \alpha B_{3} } \right] + \overline{{\varepsilon^{*} }}_{33} \sin^{ - 1} (\theta )\frac{{\overline{V}}}{{t_{p} }}bL_{1} } \right\} \hfill \\ - i\omega \left\{ {\frac{{bd_{31}^{*} \cos^{ - 1} (\theta )}}{{s_{11} }}\left( {t_{b} + t_{p} } \right)\left[ {B_{5} \beta \cos \beta L_{1} - B_{6} \beta \sin \beta L_{1} + B_{8} \beta \cosh \beta L_{1} + B_{9} \beta \sinh \beta L_{1} - \beta B_{5} - \beta B_{7} } \right] + \overline{{\varepsilon^{*} }}_{33} \sin^{ - 1} (\theta )\frac{{\overline{V}}}{{t_{p} }}\left( {t_{b} + t_{p} } \right)L_{1} } \right\} \hfill \\ = \frac{{\overline{V}}}{{Z_{L} }} \hfill \\ \end{gathered}$$
(29)
$$\begin{gathered} - i\omega \left\{ {\frac{{cd_{31} }}{{s_{11} }}\left( {t_{b} + t_{p} } \right)\left[ {B_{17} \in \cos \in L_{1} - B_{18} \in \sin \in L_{1} + B_{19} \in \cosh \in L_{1} + B_{20} \in \sinh \in L_{1} - \in B_{17} - \in B_{19} } \right] + \overline{{\varepsilon^{*} }}_{33} \sin^{ - 1} (\theta )\frac{{\overline{V}}}{{t_{p} }}cL_{1} } \right\} \hfill \\ - i\omega \left\{ {\frac{{cd_{31}^{*} \cos^{ - 1} (\theta )}}{{s_{11} }}\left( {t_{b} + t_{p} } \right)\left[ {B_{21} \zeta \cos \zeta L_{1} - B_{22} \zeta \sin \zeta L_{1} + B_{23} \zeta \cosh \zeta L_{1} + B_{24} \zeta \sinh \zeta L_{1} - \zeta B_{21} - \zeta B_{23} } \right] + \overline{{\varepsilon^{*} }}_{33} \sin^{ - 1} (\theta )\frac{{\overline{V}}}{{t_{p} }}\left( {t_{b} + t_{p} } \right)L_{1} } \right\} \hfill \\ = \frac{{\overline{V}}}{{Z_{L} }} \hfill \\ \end{gathered}$$
(30)

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MirzaAbedini, S., Zhang, H. Use of Angularity on Piezoelectric Crystal to Create Frequency Phase Shift for a Wide-Band Energy Harvester. J. Vib. Eng. Technol. (2024). https://doi.org/10.1007/s42417-024-01355-7

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