Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study

Abstract

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

Appendix A

$$\begin{aligned} {h_{11}}= & {} \int _0^1 {{\textstyle {{\rho (x)} \over {{\rho _b}{A_b}}}}\psi _n^2 \text {d}x} ,\\ {h_{12}}= & {} \int _0^1 {\delta (x - \beta )\psi _n^2 \text {d}x} ,\\ {h_{13}}= & {} \int _0^1 {\delta (x - \beta )\psi _{nx}^2 \text {d}x} ,\\ {h_{21}}= & {} {h_{11}} ,\\ {h_{31}}= & {} \int _0^1 {E{I^ * }(x)\psi _n^2 \text {d}x} ,\\ {h_{32}}= & {} \int _0^1 {{\textstyle {{{\rho _b}{A_b}E{I^ * }(x)} \over {\rho (x)}}}\psi _n^2 \text {d}x},\\ {h_{33}}= & {} \int _0^1 {{\textstyle {\rho \over {{\rho _b}{A_b}}}}(1 - x)\psi _{nx}^2 \text {d}x},\\ {h_{34}}= & {} \int _0^1 {\int _x^1 {\delta (\xi - \beta )d\xi \psi _n^2 \text {d}x} } ,\\ {h_{41}}= & {} \frac{1}{2}\int _0^1 {E{I^ * }(x){\psi _k}{\psi _{lx}}{\psi _{mx}}{\psi _n} \text {d}x},\\ {h_{42}}= & {} \frac{1}{2}\int _0^1 {\delta (x - \beta )E{I^ * }(x){\psi _k}{\psi _{lx}}{\psi _{mx}}{\psi _n} \text {d}x} ,\\ {h_{43}}= & {} 3\int _0^1 {E{I^ * }(x){\psi _{kx}}{\psi _{lxx}}{\psi _{mxxx}}{\psi _n} \text {d}x} \\&+ \int _0^1 {E{I^ * }(x){\psi _{kxx}}{\psi _{lxx}}{\psi _{mxx}}{\psi _n} \text {d}x} ,\\ {h_{51}}= & {} \int _0^1 {\left\{ {\int _x^1 {\left( {\int _0^\xi {{\psi _{l\eta }}{\psi _{m\eta }}d\eta } } \right) d\xi } } \right\} {\textstyle {{\rho (x)} \over {{\rho _b}{A_b}}}}{\psi _{kx}}{\psi _{nx}}\text {d}x} ,\\ {h_{52}}= & {} \left( {\int _0^\beta {{\psi _{lx}}{\psi _{mx}}\text {d}x} } \right) \left( {\int _0^\beta {{\psi _{kx}}{\psi _{nx}} \text {d}x} } \right) ,\\ {h_{53}}= & {} {\left\{ {{\psi _{kx}}{\psi _{lx}}{\psi _{mx}}{\psi _{nx}}} \right\} _{x = \beta }},\,\,{h_{61}} = {h_{51}}\\ {h_{62}}= & {} \int _0^1 {{\textstyle {1 \over 2}}{\textstyle {{\rho (x)} \over {{\rho _b}{A_b}}}}{\psi _{kx}}{\psi _{lx}}{\psi _{mx}}{\psi _n}\text {d}x}\\&- \int _0^1 {{\textstyle {{\rho (x)} \over {{\rho _b}{A_b}}}}{\psi _{kx}}{\psi _{lxx}}\left( {\int _x^1 {{\psi _m}d\xi } } \right) \text {d}x} \\ {h_{63}}= & {} {h_{52}},\,\,{h_{65}} = 0.5{h_{53}},\\ {h_{64}}= & {} {\textstyle {1 \over 2}}{\left\{ {{\psi _{kx}}{\psi _{lx}}{\psi _m}{\psi _{nx}}} \right\} _{x = \beta }} \\&- {\psi _m}(\beta )\int _0^\beta {{\psi _{kx}}{\psi _{lxx}}{\psi _n}\text {d}x} \\ \zeta _n^*= & {} \frac{{c{h_{21}}}}{{2\epsilon {\hbar _n}{{\rho _b}{A_b}} \theta _1 }};{\hbar _n} = {h_{11}} + \mu {h_{12}} + J{\lambda ^2}{h_{13}},\\ \theta _n^{2}= & {} \frac{{\kappa _n^4({h_{31}} + \mu {h_{32}})}}{{{{\rho _b}{A_b}} {L^4}{\hbar _n}}} - \frac{{g({h_{33}} + \mu {h_{34}})}}{{L{\hbar _n}}},\\ F_{nm}= & {} F_{nm}^* \frac{\varGamma }{\epsilon } = \frac{{{\Omega ^2}{\varGamma }{z_r}}}{{\epsilon \theta _1^{2}{\hbar _n}L}}({h_{33}} + \mu {h_{34}}),\\ \alpha _{klm}^n= & {} \frac{{{\lambda ^2}}}{{\epsilon {{\rho _b}{A_b}} {L^4}{\hbar _n}\theta _1^{2}}}\left\{ {\kappa _k^4\left( {{h_{41}} + \mu {h_{42}}} \right) + {h_{43}}} \right\} ,\\ \beta _{klm}^n= & {} \frac{{{\lambda ^2}}}{{\epsilon {\hbar _n}}}\left\{ {{h_{51}} + \mu {h_{52}} + J{\lambda ^2}{h_{53}}} \right\} ,\\ \gamma _{klm}^n= & {} \frac{{{\lambda ^2}}}{{\epsilon {\hbar _n}}}\left\{ {{h_{61}} - {h_{62}} + \mu \left( {{h_{63}} - {h_{64}}} \right) + J{\lambda ^2}{h_{65}}} \right\} \\ {\alpha _{enj}}= & {} {\alpha _{nj}} + {\beta _{en}} + {\gamma _{en}},\\ {\alpha _{nj}}= & {} \left\{ {\begin{array}{*{20}{c}} {3\alpha _{nnn}^n}&{}{{\mathrm{for}} j = n}\\ {2(\alpha _{njj}^n + \alpha _{jjn}^n + \alpha _{jnj}^n)}&{}{{\mathrm{for}} j \ne n} \end{array}} \right\} \\ {\beta _{nj}}= & {} \left\{ {\begin{array}{*{20}{c}} {\omega _n^2\beta _{nnn}^n}&{}{{\mathrm{for}} j = n}\\ {2\omega _j^2\beta _{njj}^n}&{}{{\mathrm{for}} j \ne n} \end{array}} \right\} \\ {\gamma _{nj}}= & {} \left\{ {\begin{array}{*{20}{c}} { - 3\omega _n^2\gamma _{nnn}^n}&{}{;j = n}\\ { - 2\left\{ {\omega _j^2\left( {\gamma _{jnj}^n + \gamma _{njj}^n} \right) + \omega _n^2\gamma _{jjn}^n} \right\} }&{}{;j \ne n} \end{array}} \right\} \\ Q_1^*= & {} \alpha _{121}^1 + \alpha _{211}^1 + \alpha _{112}^1 + {\omega _1}{\omega _2}\left( {\beta _{121}^1 + \beta _{112}^1} \right) \\&- \omega _1^2\beta _{211}^1 - \left\{ {\omega _1^2\left( {\gamma _{211}^1 + \gamma _{121}^1} \right) + \omega _2^2\gamma _{112}^1} \right\} ,\\ Q_2^*= & {} \alpha _{111}^1 - \omega _1^2\left( {\beta _{111}^1 + \gamma _{111}^1} \right) . \end{aligned}$$

Other parameters are defined as follows:

$$\begin{aligned} {z_r}= & {} 1\,{\mathrm{mm}},\,\,{c = 1\,{\mathrm{Ns - m}}{{\mathrm{m}}^{ - 2}}},\,\,{\mu = 3.6},\\ J= & {} 0.0366,\,\,{\beta = 0.27},\,\,{\lambda = 0.1}\\ {\kappa _1}= & {} {\mathrm{1}}{\mathrm{.791}},\,\,{{\kappa _2} = {\mathrm{3}}{\mathrm{.244}}},\,\,{\epsilon = 0.001},\,\,{{\omega _1} = 1},\\ {\omega _2}= & {} 3.013 \\ {\alpha _{{\mathrm{e11}}}}= & {} - {\mathrm{9}}{\mathrm{.306}},\,\,{{\alpha _{{\mathrm{e12}}}}{\mathrm{= 9}}{\mathrm{.312}}},\,\,{{\alpha _{{\mathrm{e21}}}}{\mathrm{= 60}}{\mathrm{.12}}},\\&{{\alpha _{{\mathrm{e22}}}} - {\mathrm{255}}{\mathrm{.7}}}\\ {f_{{\mathrm{11}}}}= & {} 0 {\mathrm{.033}},\,\,{{f_{{\mathrm{12}}}}{\mathrm{= 0}}{\mathrm{.0124}}},\,\,{{f_{21}}{\mathrm{= 0}}{\mathrm{.0358}}},\\ {f_{{\mathrm{22}}}}= & {} {\mathrm{0}}{\mathrm{.0852}}\\ Q_1^ *= & {} - {\mathrm{10}}{\mathrm{.12}},\,\,{Q_2^ * = - 37.4},\,\,{{\zeta _{\mathrm{1}}}{\mathrm{= 3}}{\mathrm{.126}}{\textstyle {\nu \over \epsilon }}},\\ {\zeta _{\mathrm{2}}}= & {} {\mathrm{1}}{\mathrm{.267}}{\textstyle {\nu \over \epsilon }}. \end{aligned}$$