Comprehensive theoretical and experimental investigation of the rotational impact energy harvester with the centrifugal softening effect

Abstract

Rotation-based energy harvesting has attracted considerable interest in recent years. This paper presents a comprehensive theoretical model to analyze a rotational impact energy harvester using the centrifugal softening effect. The harvester is composed of a centrifugal-softening driving beam that impacts two rigid piezoelectric beams to generate electrical energy through the gravity excitation. The theoretical model is derived based on Hamilton’s principle and Hertzian contact theory. An impact force model is used to overcome the limitation of the previous piecewise linear model, which cannot reflect the influence of the deformations of the driving and generating beams on the impact force and the energy output. Furthermore, an analytical impact force model is originally proposed for such a harvester based on Lee’s method to understand the impact mechanism. The proposed analytical model is validated through comparison with Runge–Kutta method. Both numerical and experimental results show that the centrifugal softening effect can amplify the relative motion between the driving and generating beams and increase the impact force, thus improving output power at low rotational frequencies. The maximum output power is increased by 135.5% at 11.5 Hz for the impact gap of 0.75 mm. In addition, with the large impact stiffness, the impact force can successfully prevent the inverted driving beam from continuously deflecting and suffering the static divergence. Based on the validated theoretical model, parametric studies are conducted to further investigate the effects of the impact stiffness and the centrifugal softening coefficient.

Appendix A: Matrices and vectors in the electromechanical model

$$\begin{aligned} {{{\mathbf {M}}_{\mathrm{d}}}}= & {} {m_{\mathrm{d}}}\int _0^L {{\mathbf {\Phi }_{\mathrm{d}}^{2}(x_{\mathrm{d}})}} \mathrm{d}x_{\mathrm{d}} + {M_0}{{\mathbf {\Phi }_{\mathrm{d}}^{2}(L)}} \nonumber \\&+2{S_0}{{\mathbf {\Phi }_{\mathrm{d}}(L)}}{{\mathbf {\Phi }_{\mathrm{d}}^{'}(L)}} + {I_{\mathrm{t}}}{{\mathbf {\Phi }_{\mathrm{d}}^{'2}(L)}} \end{aligned}$$

(47)

$$\begin{aligned} {{{\mathbf {M}}_{\mathrm{u}}}}= & {} {m_{\mathrm{u}}}\int _0^h {{\mathbf {\Phi }_{\mathrm{u}}^{2}(x_{\mathrm{u}})}} \mathrm{d}x_{\mathrm{u}}; {{{\mathbf {M}}_{\mathrm{l}}}} = {m_{\mathrm{l}}}\int _0^h {{\mathbf {\Phi }_{\mathrm{l}}^{2}(x_{\mathrm{l}})}} \mathrm{d}x_\mathrm{l}\nonumber \\ \end{aligned}$$

(48)

$$\begin{aligned} {{{\mathbf {C}}_\mathrm{{d}}}}= & {} {c_{\mathrm{d}}}\int _0^L {{\mathbf {\Phi }_{\mathrm{d}}(x_{\mathrm{d}})}} \mathrm{d}x_{\mathrm{d}};{{{\mathbf {C}}_{\mathrm{u}}}} = {c_{\mathrm{u}} \int _0^h {{\mathbf {\Phi }_{\mathrm{u}}}(x_{\mathrm{u}})}} \mathrm{d}x_{\mathrm{u}};{{{\mathbf {C}}_{\mathrm{l}}}} \nonumber \\= & {} {c_{\mathrm{l}}}\int _0^h {{\mathbf {\Phi }_{\mathrm{l}}(x_{\mathrm{l}})}} \mathrm{d}x_{\mathrm{l}} \end{aligned}$$

(49)

$$\begin{aligned} {{{\mathbf {K}}_{\mathrm{d}}}}= & {} {{{\mathbf {M}}_{\mathrm{d}}}}\omega _{\mathrm{d}}^2; {{{\mathbf {K}}_{{\mathrm{u}}}}} ={{{\mathbf {M}}_{\mathrm{u}}}}\omega _{\mathrm{u}}^2; {{{\mathbf {K}}_{{\mathrm{l}}}}} ={{{\mathbf {M}}_{\mathrm{l}}}}\omega _{\mathrm{l}}^2 \end{aligned}$$

(50)

$$\begin{aligned} {\mathbf {\Lambda }_{\mathrm{d}}}= & {} {M_0}{{\mathbf {\Phi }_{\mathrm{d}}(L)}}+{m_{\mathrm{d}}}\int _0^L {{\mathbf {\Phi }_{\mathrm{d}}(x_{\mathrm{d}})}} \mathrm{d}x_{\mathrm{d}}+{S_0}{{\mathbf {\Phi }_{\mathrm{d}}^{'}(L)}}\nonumber \\ \end{aligned}$$

(51)

$$\begin{aligned} {\mathbf {\Lambda }_{\mathrm{u}}}= & {} {m_{\mathrm{u}}}\int _0^h {{\mathbf {\Phi }_{\mathrm{u}}(x_{\mathrm{u}})}} \mathrm{d}x_{\mathrm{u}};{\mathbf {\Lambda }_{\mathrm{l}}}={m_{\mathrm{l}}}\int _0^h {{\mathbf {\Phi }_{\mathrm{l}}(x_{\mathrm{l}})}} \mathrm{d}x_{\mathrm{l}} \end{aligned}$$

(52)

$$\begin{aligned} {\mathbf {\Theta }_{\mathrm{u}}}= & {} \frac{{w_{\mathrm{g}}}{(t_g+t_{\mathrm{p}})}{e_{31}}\int _0^{l_{\mathrm{p}}}{{\mathbf {\Phi }_{\mathrm{u}}^{''}(x_{\mathrm{u}})}}\mathrm{d}x_{\mathrm{u}}}{2}\end{aligned}$$

(53)

$$\begin{aligned} {\mathbf {\Theta }_{\mathrm{l}}}= & {} \frac{{w_{\mathrm{g}}}{(t_g+t_{\mathrm{p}})}{e_{31}}\int _0^{l_{\mathrm{p}}}{{\mathbf {\Phi }_{\mathrm{l}}^{''}(x_l)}}\mathrm{d}x_\mathrm{l}}{2} \end{aligned}$$

(54)