Performance analysis of piezoelectric energy harvesters with a tip mass and nonlinearities of geometry and damping under parametric and external excitations

Abstract

Based on the Hamilton’s principle, a nonlinear mathematical model of the cantilever-type piezoelectric energy harvester with a tip mass is systematically derived under parametric and external excitations. The proposed model accounts for geometric and electro-mechanical coupling nonlinearities, damping nonlinearity and the inextensibility condition of beam. Using the Galerkin approach, the proposed model is converted into the electro-mechanical coupling Mathieu–Duffing equations. Analytical solutions of the frequency–response curves are presented by the multiple scales method. Nonlinear characteristics of the energy harvesters are explored under parametric excitation and hybrid parametric and external excitations. Analytical results provided new insights into the effects of tip mass and nonlinear damping on the performance of the energy harvester. The results show that with the tip mass increasing, the frequency–response curves of the energy harvester change from the nonlinear hardening type to the nonlinear softening type and the operating bandwidth and the output voltages of the energy harvester enlarge. For parametrical excitation, variation of the quadratic damping does not alter the initial threshold of the harvesters and the position of two transcritical bifurcation points of the frequency–response curves. The initiation threshold decreases with the tip mass increasing. Hybrid parametric and external excitations enhance the bandwidth and output voltage of the energy harvester, which will probably be used as an ideal way to improve the performance of the energy harvesting system.

Appendix A

The coefficients of Eqs. (22) and (23) are defined as:

$$\begin{aligned} a_{1n} \left( {\bar{{s}}} \right)= & {} \int _{\bar{{s}}}^1 {\int _0^{\bar{{\xi }}} {{X'}^{2}_{n} \left( {\bar{{\eta }}} \right) \mathrm{d}\bar{{\eta }}\mathrm{d}\bar{{\xi }}} } ,a_{2n} \left( {\bar{{s}}} \right) =\int _0^{\bar{{s}}} {{X'}^{2}_{n} \left( {\bar{{\eta }}} \right) \mathrm{d}\bar{{\eta }}} ,b_{1nn} =\int _0^1 {X_{n} X_{n} \mathrm{d}\bar{{s}}} \nonumber \\ b_{2nn}= & {} \int _0^1 {{X}''_{n} X_{n} \mathrm{d}\bar{{s}}} -\int _0^1 {\bar{{s}}{X}''_{n} X_{n} \mathrm{d}\bar{{s}}} -\int _0^1 {{X}'_{n} X_{n} \mathrm{d}\bar{{s}}} -\bar{{M}}{X}'_{n} (1)X_{n} (1) \nonumber \\ b_{3nn}= & {} \int _0^1 {\left( {\bar{{\beta }}_{n}^{4} {X'}^{2}_{n} X_{n} +4{X}'_{n} {X}''_{n} {X}'''_{n} +{X''}^{3}_{n}} \right) X_{n} \mathrm{d}\bar{{s}}} \nonumber \\ b_{4nn}= & {} \int _0^1 {a_{2n} \left( {\bar{{s}}} \right) {X}'_{n} X_{n} \mathrm{d}\bar{{s}}} -\int _0^1 {a_{1n} \left( {\bar{{s}}} \right) {X}''_{n} X_{n}\mathrm{d}\bar{{s}}} ,b_{5nn} =\bar{{\alpha }}{X}'_{n} \left( 1 \right) \nonumber \\ b_{6nn}= & {} \bar{{\alpha }}{X'}^{3}_{n} \left( 1 \right) ,b_{7nn} =\int _0^1 {X_{n} \mathrm{d}\bar{{s}}} +\bar{{M}}X_{n} (1),b_{8nn} =\bar{{\alpha }}\int _0^1{{X}''_{n} \mathrm{d}\bar{{s}}} \nonumber \\ b_{9nn}= & {} \bar{{\alpha }}\int _0^1 {{X'}^{2}_{n} {X}''_{n} \mathrm{d}\bar{{s}}} ,b_{10nn} =\int _0^1 {X_{n}^{2} } \left| {X_{n} } \right| \mathrm{d}\bar{{s}}\nonumber \\ \tilde{{c}}_{a}= & {} \frac{1}{2}\bar{{c}}_{a} ,\quad \tilde{{c}}_{q} =\bar{{c}}_{q} b_{10nn} , \bar{{\omega }}_{n}^{2} =\bar{{\beta }}_{n}^{4} , \bar{{\sigma }}_{n} =\frac{1}{2}\frac{b_{2nn} }{b_{1nn} }, \beta _{n} =\frac{b_{3nn} }{b_{1nn} }, \kappa _{n} =\frac{b_{4nn} }{b_{1nn} }\nonumber \\ \zeta _{n}= & {} \frac{b_{5nn} }{b_{1nn} }, \gamma _{n} =\frac{b_{6nn} }{2b_{1nn} }, \bar{{\lambda }}_{n} =\frac{b_{7nn} }{b_{1nn} }, \eta _{n} =b_{8nn} , \chi _{n} =\frac{3}{2}b_{9nn} \end{aligned}$$

(A1)