Frequency-Domain Analysis of Shock-Excited Magneto-Electro-Elastic Energy Harvesters with Different Unimorph and Bimorph Configurations

Abstract

Bearing in mind that the environmental kinetic energy may be available in the form of shock pulse acceleration instead of the common harmonic type, this study investigates the response of unimorph and bimorph cantilever-based functionally graded magneto-electro-elastic energy harvesters under mechanical shock. Using Ritz's method together with the Euler–Bernoulli beam theory, the reduced equations of motion are obtained. Then, neglecting the influence of the terms related to the magneto-electro-elastic layer and leading the gradient index to infinity, the present findings are compared and verified by those available in the literature. Afterward, given the fact that the frequency domain analysis provides a better physical understanding from a dynamical system, using the mass-spring shock spectrum diagram, the influence of shock duration on the values of the harvested power is investigated. Doing so, the optimal duration is determined. The results reveal that although employing the shock spectrum diagram associated with a mass-spring system can predict the optimal duration with relatively good accuracy, to have more accurate results, one should hire the shock spectrum diagram associated with the harvested power. Therefore, introducing the power spectrum diagram, a detailed parametric study is then performed to investigate the effects of the through-thickness material gradation index, the external coil and the piezoelectric circuit resistances as well as the harvester configurations. Finally, comparing the maximum harvested powers from functionally graded piezoelectric and magneto-electro-elastic harvesters, the advantages of using the present system are addressed. It is worth mentioning that the present work is the first attempt on providing a frequency-domain analysis for FGMEEM-based hybrid energy harvesters undergoing mechanical shock.

Appendix

The dimensionless coefficients associated with the reduced equations of motion given in (26) for each configuration are given in this section. The coefficients $d_{i} = \frac{{c_{i + 1} }}{{c_{1} }}$$\left( {i = 1,2,... \, 6} \right)$. Also, the coefficients $c_{i} \left( {i = 1 - 7} \right)$, $s_{i} \left( {i = 1 - 4} \right)$, $k_{i} \left( {i = 1 - 4} \right)$ and $h_{i} \left( {i = 1 - 4} \right)$ for the unimorph layout are given in Eqs. (30). These quantities for the bimorph configuration are also given in Eqs. (31).

$$\begin{gathered} c_{1} = - \frac{{\tilde{I}_{2} L^{2} }}{{\tilde{D}_{S} T^{2} }}\int_{0}^{1} {\psi^{{\prime}{2}} } dx - \frac{{\tilde{I}_{0} L^{4} }}{{\tilde{D}_{S} T^{2} }}\int_{0}^{1} {\psi^{2} dx} - \left[ {\frac{{M^{ * } M_{{{\text{tip}}}} L^{2} }}{{\tilde{D}_{S} T^{2} }}\left( {\psi^{2} } \right)} \right]_{x = 1} , \, c_{2} = - \frac{{c_{t} c^{ * } L^{2} }}{{\tilde{D}_{S} T}}\int_{0}^{1} {\psi^{2} dx} , \hfill \\ \, \hfill \\ c_{3} = \frac{{\left( {\tilde{B}_{M} + \tilde{B}_{S} } \right)\tilde{B}_{M} }}{{\tilde{D}_{S} \left( {\tilde{A}_{S} + \tilde{A}_{M} } \right)}}\int_{0}^{1} {\left( {\psi^{\prime\prime}} \right)^{2} dx} - \frac{{\left( {\tilde{D}_{M} + \tilde{D}_{S} } \right)}}{{\tilde{D}_{S} }}\int_{0}^{1} {\left( {\psi^{\prime\prime}} \right)^{2} dx} + \frac{{\left( {\tilde{B}_{M} + \tilde{B}_{S} } \right)\tilde{B}_{M} }}{{\tilde{D}_{S} \left( {\tilde{A}_{S} + \tilde{A}_{M} } \right)}}\left( {\int_{0}^{1} {\psi^{\prime\prime}dx} } \right)^{2} ,\, \hfill \\ c_{4} = \frac{{\left( {\tilde{B}_{M} + \tilde{B}_{S} } \right)L}}{{\tilde{D}_{S} }}\int_{0}^{1} {\psi^{\prime\prime}dx} , \, c_{5} = \frac{{\tilde{B}_{e} V_{p}^{ * } L}}{{t_{M} \tilde{D}_{S} }}\int_{0}^{1} {\psi^{\prime\prime}dx} ,\,\,\,c_{6} = - \frac{{\tilde{B}_{m} NV_{m}^{ * } L}}{{R_{m} t_{M} \tilde{D}_{S} }}\int_{0}^{1} {\psi^{\prime\prime}dx} , \hfill \\ c_{7} = {\mathbb{Z}}_{0} \left( { - \frac{{\tilde{I}_{0} L^{4} }}{{\tilde{D}_{S} T^{2} }}\int_{0}^{1} {\psi dx} - \left[ {\frac{{M^{ * } M_{{{\text{tip}}}} L^{2} }}{{\tilde{D}_{S} T^{2} }}\left( \psi \right)} \right]_{x = 1} } \right), \hfill \\ \end{gathered}$$

(30a)

$$s_{1} = \frac{{T^{2} \left( {\tilde{A}_{S} + \tilde{A}_{M} } \right)}}{{M^{ * } M_{{{\text{tip}}}} }},\,\,s_{2} = \frac{{T^{2} \tilde{B}_{M} \int_{0}^{1} {\psi^{\prime\prime}dx} }}{{M^{ * } LM_{{{\text{tip}}}} }},\,\,s_{3} = \frac{{T^{2} \tilde{A}_{e} V_{p}^{ * } }}{{M^{ * } t_{M} M_{{{\text{tip}}}} }},s_{4} = \frac{{T^{2} \tilde{A}_{m} NV_{m}^{ * } }}{{M^{ * } R_{m} t_{M} M_{{{\text{tip}}}} }},$$

(30b)

$$k_{1} = \frac{{t_{M} \tilde{A}_{e} }}{{\tilde{A}_{{h_{33} }} V_{p}^{ * } }},\,\,k_{2} = \frac{{t_{M} \tilde{B}_{e} \int_{0}^{1} {\psi^{\prime\prime}dx} }}{{\tilde{A}_{{h_{33} }} LV_{p}^{ * } }},\,\,k_{3} = \frac{{ - \tilde{A}_{{g_{33} }} NV_{m}^{ * } }}{{R_{m} \tilde{A}_{{h_{33} }} V_{p}^{ * } }},\,\,k_{4} = \frac{{\left( {t_{M} } \right)^{2} T}}{{R_{e} \tilde{A}_{{h_{33} }} L}},$$

(30c)

$$h_{1} = \frac{{R_{m} t_{M} \tilde{A}_{m} }}{{N\tilde{A}_{{\mu_{33} }} V_{m}^{ * } }},\,h_{2} = \frac{{ - R_{m} t_{M} \tilde{B}_{m} \int_{0}^{1} {\psi^{\prime\prime}} dx}}{{N\tilde{A}_{{\mu_{33} }} LV_{m}^{ * } }},\,\,h_{3} = \frac{{ - R_{m} \tilde{A}_{{g_{33} }} V_{p}^{ * } }}{{N\tilde{A}_{{\mu_{33} }} V_{m}^{ * } }},\,\,h_{4} = \frac{{R_{m} \left( {t_{M} } \right)^{2} T}}{{N^{2} \tilde{A}_{{\mu_{33} }} L^{2} }}.$$

(30d)

$$\begin{gathered} c_{1} = - \frac{{\tilde{I}_{2} L^{2} }}{{\tilde{D}_{S} T^{2} }}\int_{0}^{1} {\psi^{{\prime}{2}} } dx - \frac{{\tilde{I}_{0} L^{4} }}{{\tilde{D}_{S} T^{2} }}\int_{0}^{1} {\psi^{2} dx} - \left[ {\frac{{M^{ * } M_{{{\text{tip}}}} L^{2} }}{{\tilde{D}_{S} T^{2} }}\left( {\psi^{2} } \right)} \right]_{x = 1} , \, c_{2} = - \frac{{c_{t} c^{ * } L^{2} }}{{\tilde{D}_{S} T}}\int_{0}^{1} {\psi^{2} dx} , \hfill \\ c_{3} = - \frac{{\left( {\tilde{D}_{M} + \tilde{D}_{S} } \right)}}{{\tilde{D}_{S} }}\int_{0}^{1} {\left( {\psi^{\prime\prime}} \right)^{2} dx} ,\, \, c_{4} = 0, \, c_{5} = \frac{{\tilde{B}_{e} V_{p}^{ * } L}}{{2t_{M} \tilde{D}_{S} }}\int_{0}^{1} {\psi^{\prime\prime}dx} , \hfill \\ c_{6} = - \frac{{\tilde{B}_{m} NV_{m}^{ * } L}}{{R_{m} t_{M} \tilde{D}_{S} }}\int_{0}^{1} {\psi^{\prime\prime}dx} , \, c_{7} = {\mathbb{Z}}_{0} \left( { - \frac{{\tilde{I}_{0} L^{4} }}{{\tilde{D}_{S} T^{2} }}\int_{0}^{1} {\psi dx} - \left[ {\frac{{M^{ * } M_{{{\text{tip}}}} L^{2} }}{{\tilde{D}_{S} T^{2} }}\left( \psi \right)} \right]_{x = 1} } \right), \hfill \\ \end{gathered}$$

(31a)

$$s_{1} = \frac{{T^{2} \left( {\tilde{A}_{S} + \tilde{A}_{M} } \right)}}{{M^{ * } M_{{{\text{tip}}}} }},\,\,s_{2} = 0,\,\,s_{3} = 0,s_{4} = 0,$$

(31b)

$$k_{1} = 0,\,\,k_{2} = \frac{{2t_{M} \tilde{B}_{e} \int_{0}^{1} {\psi^{\prime\prime}dx} }}{{\tilde{A}_{{h_{33} }} LV_{p}^{ * } }},\,\,k_{3} = \frac{{ - 2\tilde{A}_{{g_{33} }} NV_{m}^{ * } }}{{R_{m} \tilde{A}_{{h_{33} }} V_{p}^{ * } }},\,\,k_{4} = \frac{{4\left( {t_{M} } \right)^{2} T}}{{R_{e} \tilde{A}_{{h_{33} }} L}},$$

(31c)

$$h_{1} = 0,\, \, h_{2} = \frac{{ - R_{m} t_{M} \tilde{B}_{m} \int_{0}^{1} {\psi^{\prime\prime}} dx}}{{N\tilde{A}_{{\mu_{33} }} LV_{m}^{ * } }},\, \, h_{3} = \frac{{ - R_{m} \tilde{A}_{{g_{33} }} V_{p}^{ * } }}{{2N\tilde{A}_{{\mu_{33} }} V_{m}^{ * } }},\,\,h_{4} = \frac{{2R_{m} \left( {t_{M} } \right)^{2} T}}{{N^{2} \tilde{A}_{{\mu_{33} }} L^{2} }}.$$

(31d)

where $\tilde{I}_{0}$, $\tilde{I}_{1}$ and $\tilde{I}_{2}$ for the unimorph configuration are given in Eqs. (32). These quantities for the case of a system with a bimorph layout are also given in Eqs. (33).

$$\left[ {\tilde{I}_{0} ,\tilde{I}_{1} ,\tilde{I}_{2} } \right] = b\left( {\int_{{\frac{{ - \left( {t_{S} + t_{M} } \right)}}{2}}}^{{\frac{{\left( {t_{S} - t_{M} } \right)}}{2}}} {\rho_{S} \left\{ {\left. {1,\tilde{z},\tilde{z}^{2} } \right\}} \right.d\tilde{z}} + \int_{{\frac{{\left( {t_{S} - t_{M} } \right)}}{2}}}^{{\frac{{\left( {t_{S} + t_{M} } \right)}}{2}}} {\rho_{M} \left\{ {\left. {1,\tilde{z},\tilde{z}^{2} } \right\}} \right.d\tilde{z}} } \right),$$

(32)

$$\left[ {\tilde{I}_{0} ,\tilde{I}_{1} ,\tilde{I}_{2} } \right] = b\left( {\int_{{ - t_{M} - \frac{{t_{S} }}{2}}}^{{ - \frac{{t_{S} }}{2}}} {\rho_{M} \left\{ {\left. {1,\tilde{z},\tilde{z}^{2} } \right\}} \right.d\tilde{z}} + \int_{{\frac{{ - t_{S} }}{2}}}^{{\frac{{t_{S} }}{2}}} {\rho_{S} \left\{ {\left. {1,\tilde{z},\tilde{z}^{2} } \right\}} \right.d\tilde{z}} + \int_{{\frac{{t_{S} }}{2}}}^{{t_{M} + \frac{{t_{S} }}{2}}} {\rho_{M} \left\{ {\left. {1,\tilde{z},\tilde{z}^{2} } \right\}} \right.d\tilde{z}} } \right).$$

(33)