Dynamic Analysis of Parametrically Excited Piezoelectric Bimorph Beam for Energy Harvesting

Abstract

In this work, the dynamics of an energy harvester which is modeled as a base excited cantilever beam with tip mass having piezoelectric patches on both top and bottom surfaces has been investigated. The governing equation of motion of the system is developed using extended Hamilton’s principle and it is solved by using method of multiple scales. Closed form solution has been developed to find out the voltage generated from this nonlinear energy harvester. While in most of the previously considered energy harvesters the system is considered without tip mass and axial load, here putting the additional tip mass and periodic axial load, different resonance conditions have been explored. A parametric study has been carried out to investigate the variation of generated voltage with different system parameters. The study will find application in the generation of voltage for a wide range of frequencies available in ambient vibration.

Appendix

$\begin{aligned} \bar{\omega }_{n} & = K_{n} - P_{n} \\ M & = \int\limits_{0}^{{L_{b} }} {m_{b} \phi_{n}^{2} ds} \\ K_{n} & = (E_{s} I_{s} + 2E_{p} I_{p} )\int\limits_{0}^{{L_{b} }} {\phi_{n} (\phi^{\prime\prime}_{n} } )^{\prime\prime}ds \\ P_{n} & = p_{0} \int\limits_{0}^{{L_{b} }} {\phi_{n} } \phi^{\prime\prime}_{n} ds \\ \bar{\Lambda}_{n} & = (E_{s} I_{s} + 2E_{p} I_{p} )\int\limits_{0}^{{L_{b} }} {\phi_{n} [\phi^{\prime}_{n} (\phi^{\prime}_{n} \phi^{\prime\prime}_{n} )^{\prime}]}^{\prime } ds \\ \bar{F}_{2} & = p\cos ({\bar{\Omega }}_{2}\,\bar{t}) \\ \end{aligned}$

$\begin{aligned} \theta_{an} & = - \int\limits_{0}^{{L_{b} }} {\phi_{n} K^{\prime\prime}_{a} } ds \\ \theta_{bn} & = - \int\limits_{0}^{{L_{b} }} {[\frac{1}{2}\phi_{n} (K_{a} \phi^{{{\prime }2}}_{n} )^{\prime\prime} - \phi_{n} (K_{a} \phi^{\prime}_{n} \phi^{\prime\prime}_{n} )^{\prime}]} ds \\ C_{p} & = \frac{{b_{p} \hat{\varepsilon }L_{b} }}{{2t_{p} }} \\ \theta_{cn} & = \frac{3}{2}\int\limits_{0}^{{L_{b} }} {K_{a} \phi^{{{\prime }2}}_{n} } \phi_{n} ds \\ \bar{F}_{1} & = m_{b}{g}\int\limits_{0}^{{L_{b} }} {\phi_{n} ds} \\ z &= \left( {\omega}_{n}^{2} + i {\omega} _{n} r_{e}\right) / \left( \omega_{n}^{2} + r_{e}^{2}\right) \end{aligned}$

$$ \phi_{i} (s) = - \frac{{\sin \beta_{n} L_{b} + \sinh \beta_{n} L_{b} }}{{\cos \beta_{n} L_{b} + \cosh \beta_{n} L_{b} }}(\cos \beta_{n} s - \cosh \beta_{n} s) + (\sin \beta_{n} s - \sinh \beta_{n} s) $$

$\beta_{n}$ is obtained by solving the equation

$$ 1 + \cosh (\beta_{n} L_{b} )\cos (\beta_{n} L_{b} ) + (\frac{{m_{t} }}{{m_{b} }})(\cos \beta_{n} L_{b} \sinh \beta_{n} L_{b} - \sin \beta_{n} L_{b} \cosh \beta_{n} L_{b} ) = 0 $$