Investigation of output voltage, vibrations and dynamic characteristic of 2DOF nonlinear functionally graded piezoelectric energy harvester

Abstract

With the aim of providing the larger frequency band and progress the energy harvesting proficiency of the bistable piezoelectric energy harvester (BPEH), a 2-DOF nonlinear distributed parameter model of BPEH with an elastic magnifier is considered in this paper. This study deals with bimorph beam with functionally graded piezoelectric (FGP) layers carrying concentrated mass at tip while taking into account the geometrical nonlinear terms and the nonlinear magnetic force to enhance energy harvesting performance. By presenting the magnifier, oscillations of the tip are inspired into high-energy orbits. Mathematical model of BPEH with a dynamic magnifier has been developed, then the nonlinear coupled equation of motion derived using Hamilton principle. Initially, frequency response of harvester derived according to harmonic balance method and the adjusted model has been confirmed by parametric studies of earlier lumped parameter model. Analytical results show that the harvesting voltage can be improved with correct selection of design parameters of the magnifier. The proposed 2-DOF harvester system can provide higher output over a wider frequency band and may cause a considerable change in harvester performance. A parametric study is consummated to expose the effects of power law index of FGP on the output voltage and dynamic characteristic.

Appendix

$$\begin{aligned}&M_{0}={\int _{V_{s}}}^0 {\rho _{s}\phi ^{2}(x)\mathrm{d}x} +\sum \limits _{i=1}^2 {\int _{{V_pi}}}^0 {\rho _{pi}(z)\phi ^{2}\left( x \right) \mathrm{d}x+M_{t}\phi ^{2}\left( L \right) +I_{t}\phi ^{'2}\left( L \right) } \\&K_{0}={\int _{V_{s}}}^0 {C_{s}{[-z\phi }^{''}(x)]^{2}{\mathrm{d}V}_{S}} +\sum \limits _{i=1}^2 {\int _{{V_pi}}}^0 {C_{11}^{E}(z)[{-z\phi }^{''}(x)]^{2}{\mathrm{d}V}_{pi}} \\&B_{2}={\int _{V_{s}}}^0 {\rho _{s}\phi (x)\mathrm{d}V_{s}} +\sum \limits _{i=1}^2 {\int _{{V_pi}}}^0 {\rho _{pi}(z)\phi \left( x \right) {\mathrm{d}V}_{pi}+M_{t}\phi \left( L \right) } \\&\alpha =\sum \limits _{i=1}^2 {\int _{{V_pi}}}^0 {C_{s}(z){[-z\phi }^{''}\left( x \right) e_{31}(z)[-\mathrm {\nabla }\psi \left( z \right) ]{\mathrm{d}V}_{pi}} \\&C_{p}=\sum \limits _{i=1}^2 {\int _{{V_pi}}}^0 {\varepsilon _{33}^{s}(z)[-\nabla \psi _{v}\left( z \right) ]^{2}{\mathrm{d}V}_{pi}} \\&k_{1}=\frac{\mu _{0}M_{A}V_{A}M_{B}V_{B}}{\mathrm {2\pi }}\left[ 6\phi ^{2}\left( L \right) +d^{2}\phi ^{'2}\left( L \right) +3d\phi \left( L \right) \phi ^{'}\left( L \right) \right] k/d^{5} \\&k_{3}=\left( \int _0^L {\varphi \left( x \right) \varphi ^{''}\left( x \right) \mathrm{d}x} \right) \left( \int _0^L {{\varphi ^{'}\left( x \right) }^{2}\mathrm{d}x} \right) \nonumber \\&\qquad \quad +\,\frac{{3\mu }_{0}M_{A}V_{A}M_{B}V_{B}}{\mathrm {8\pi }} \frac{\left[ 30\phi ^{4}\left( L \right) +13d^{2}\phi ^{2}\left( L \right) \phi ^{'2}\left( L \right) +20d\phi ^{3}\left( L \right) \phi ^{'}\left( L \right) \right] }{d^{7}} \\ \end{aligned}$$