Nonlinear characteristic of a circular composite plate energy harvester: experiments and simulations

Abstract

A piezoelectric energy harvester is investigated with the focus on its nonlinear behavior. The harvester consists of a circular composite plate with the clamped boundary, a proof mass and two steel rings. A reduced-order model of the harvester is established, and the parameters are identified from the experimental data. A technique is proposed to identify electrical parameters with the outcomes agreeing with the theoretical values, and a fifth-order polynomial is employed to approximate the nonlinear restoring force. Both the experimental and the numerical results demonstrate that the harvester changes its characteristic from linearity to a softening nonlinearity and finally to a combined softening and hardening nonlinearity as the excitation increases from low to high.

Appendix: Electricity parameters derived from the elasticity theory

The parameters of piezoelectric ceramics are listed in Table 5. The thin composite plate has an axially symmetric structure. So the polar coordinate of the plate is established in Fig. 12. Because of the proof mass attached at the center of the plate, the structure is consider as a circular plate with a corresponding concentrated force and the clamped boundary condition.

The dynamic deflection can be derived from the elasticity solution [14] as

$$\begin{aligned} \begin{aligned} w\left( {r,t} \right) =&\,z\left( t \right) \phi \left( r \right) \\ \hbox {where},\,\, \phi \left( r \right) =&\,\frac{r^{2}\left( {2\ln r-2\ln R_\mathrm{s} -1} \right) +R_\mathrm{s}^2 }{R_\mathrm{s}^{2}} \end{aligned} \end{aligned}$$

(A1)

In Eq. (A1), w(r, t) is the deflection of the plate in the z-direction at position r and time t, ris the radius of deflection point. The static deflection curve $\phi (r)$ is used to approximate fundamental vibrating mode of the circular composite plate.

Fig. 12

The structure of circular composite plate

Unlike beam models, there are two stress terms in the plate. As shown in Fig. 12, one is the radial direction stress $\sigma _{r}$ and the other is the angular direction stress $\sigma _{\theta }$. For the linear distribution of the bending strain, the average strain for the PZT is employed in the calculation. The distance from the center of the PZT layer to the center of the composite plate is ($h_{\mathrm{p}}+h_{\mathrm{b}})/2$. Thus the two average strains terms, $\varepsilon _{r}$ and $\varepsilon _{\theta }$ in the PZT layer can be expressed as

$$\begin{aligned} \varepsilon _r= & {} -\frac{h_\mathrm{p} +h_\mathrm{b} }{2}\frac{\partial ^{2}w\left( {r,t} \right) }{\partial r^{2}} \end{aligned}$$

(A2a)

$$\begin{aligned} \varepsilon _\theta= & {} -\frac{h_\mathrm{p} +h_\mathrm{b} }{2r}\frac{\partial w\left( {r,t} \right) }{\partial r} \end{aligned}$$

(A2b)

The average strains terms are calculated by the classical elasticity solution, and the piezoelectric constitutive equations can be written for the circular plate

$$\begin{aligned}&\varepsilon _r =s_{rr}^E \left( {\sigma _r -\mu \sigma _\theta } \right) -d_{zr} E_z \end{aligned}$$

(A3a)

$$\begin{aligned}&\varepsilon _\theta =s_{rr}^E \left( {\sigma _\theta -\mu \sigma _r } \right) -d_{zr} E_z \end{aligned}$$

(A3b)

$$\begin{aligned}&D_z =-d_{zr} \left( {\sigma _r +\sigma _\theta } \right) +\varepsilon _{zz}^{T} E_z \end{aligned}$$

(A3c)

where $\mu$ is Poisson’ s ratio of the PZT layer, $\varepsilon _{zz}^{T}$ is the permittivity of PZT, $d_{zr}$ is the piezoelectric constant, $D_{z}$ is the electric displacement, $E_{z}$ is the electric field, and $s_{rr}^{E}$ is the elastic compliance constant for PZT. By Using Eqs. (A3a) and (A3b), one can describe the stress in terms of strain and electric field

$$\begin{aligned} \sigma _r= & {} \frac{1}{s_{rr}^E \left( {1-\mu ^{2}} \right) }\left[ {\varepsilon _r +\mu \varepsilon _\theta +\left( {1+\mu } \right) d_{zr} E_z } \right] \end{aligned}$$

(A4a)

$$\begin{aligned} \sigma _\theta= & {} \frac{1}{s_{rr}^E \left( {1-\mu ^{2}} \right) }\left[ {\mu \varepsilon _r +\varepsilon _\theta +\left( {1+\mu } \right) d_{zr} E_z } \right] \end{aligned}$$

(A4b)

Inserting Eq. (A4) into (A3c) leads to the relationship between the electric displacement and the strain. Because of $E_{3}=-u(t)/h_{\mathrm{p}}$, it can be expressed as

$$\begin{aligned} D_z= & {} -\frac{d_{zr} }{s_{rr}^E \left( {1-\mu } \right) }\left( {\varepsilon _r +\varepsilon _\theta } \right) \nonumber \\&-\frac{\varepsilon _{zz}^T s_{rr}^E \left( {1-\mu } \right) -2d_{zr}^2 }{s_{rr}^E \left( {1-\mu } \right) h_\mathrm{p} }u\left( t \right) \end{aligned}$$

(A5)

Charge Q(t) can be obtained by integrating Eqs. (A2) and (A5) on the electrode surface,

$$\begin{aligned} Q\left( t \right)= & {} \frac{n_\mathrm{p} \pi d_{zr} \left( {h_\mathrm{p} +h_\mathrm{b} } \right) }{2s_{rr}^E \left( {1-\mu } \right) }\int \limits _0^{R_\mathrm{p} } \left( r\frac{\partial ^{2}w\left( {r,t} \right) }{\partial r^{2}}\right. \nonumber \\&\left. +\frac{\partial w\left( {r,t} \right) }{\partial r} \right) \mathrm{d}r\nonumber \\&-\,n_\mathrm{p} \pi R_\mathrm{p}^2 \frac{\varepsilon _{zz}^T s_{rr}^E \left( {1-\mu } \right) -2d_{zr}^2 }{s_{rr}^E \left( {1-\mu } \right) h_\mathrm{p} }\frac{u\left( t \right) }{n_\mathrm{s} } \end{aligned}$$

(A6)

where the number of the PZT plate in parallel connection is $n_{\mathrm{p}}$ and in series connection is $n_{\mathrm{s}}$.

The output current can be calculated by taking the derivative of Eq. (A6) with respect to the time,

$$\begin{aligned} i\left( t \right)= & {} \frac{\mathrm{d}Q\left( t \right) }{\mathrm{d}t}\nonumber \\= & {} \frac{n_\mathrm{p} \pi d_{zr} \left( {h_\mathrm{p} +h_\mathrm{b} } \right) }{2s_{rr}^E \left( {1-\mu } \right) }\int \limits _0^{R_\mathrm{p} } \left( r\frac{\partial ^{2}w\left( {r,t} \right) }{\partial r^{2}\partial t}\right. \nonumber \\&\left. +\frac{\partial w\left( {r,t} \right) }{\partial r\partial t} \right) \mathrm{d}r-C_\mathrm{p} \dot{u}\left( t \right) \end{aligned}$$

(A7)

The equivalent capacitance $C_{\mathrm{p}}$ is expressed as

$$\begin{aligned} C_\mathrm{p} =\pi R_\mathrm{p}^2 n_\mathrm{p} \frac{\varepsilon _{zz}^T s_{rr}^E \left( {1-\mu } \right) -2d_{zr}^2 }{s_{rr}^E \left( {1-\mu } \right) h_\mathrm{p} n_\mathrm{s} } \end{aligned}$$

(A8)

Based on Eqs. (A1) and (A7) with $u(t)=i(t)R_{\mathrm{L}}$, the electromechanical equation is expressed as.

$$\begin{aligned} -\eta \dot{z}\left( t \right) +C_\mathrm{p} \dot{u}\left( t \right) +u\left( t \right) /R_\mathrm{L} =0 \end{aligned}$$

(A9)

and the electromechanical coupling coefficient $\eta$ is expressed as

$$\begin{aligned} \eta =\frac{4n_\mathrm{p} \pi d_{zr} \left( {h_\mathrm{p} +h_\mathrm{b} } \right) \gamma ^{2}\ln \gamma }{s_{rr}^E \left( {1-\mu } \right) } \end{aligned}$$

(A10)

where $\gamma$ is the radius ratio $\gamma ={R}_{\mathrm{p}}/R_{\mathrm{s}}$, and $R_{\mathrm{L}}$ is the load resistance.