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Open Access
Research article

Dynamic Characteristic Analysis of Tri-Stable Piezoelectric Energy Harvester with Double Elastic Amplifiers

dawei man1*,
yingying bai2,
qingnan hu2,
huaiming xu2,
gaozheng xu2,
liping tang1
1
School of Civil Engineering, Anhui Jianzhu University, 230601 Hefei, China BIM Engineering Center of Anhui Province, 230601 Hefei, China
2
School of Civil Engineering, Anhui Jianzhu University, 230601 Hefei, China
Journal of Intelligent Systems and Control
|
Volume 2, Issue 2, 2023
|
Pages 54-69
Received: 01-13-2023,
Revised: 02-15-2023,
Accepted: 02-28-2023,
Available online: 04-02-2023
View Full Article|Download PDF

Abstract:

In order to further improve the vibration energy harvesting efficiency of piezoelectric energy harvester under low frequency environmental excitation, this paper, based on the traditional magnetic tri-stable piezoelectric energy collector model, proposes a tri-stable piezoelectric energy harvester (TPEH+DEM) model with two elastic amplifiers which are installed between the U-shaped frame and the base and between the fixed end of the piezoelectric cantilever beam and the U-shaped frame respectively. Based on Hamilton principle, the motion equation of electromechanical coupling of TPEH+DEM system is established, and the analytical solutions of displacement, output voltage and power of the system are obtained by harmonic balance method. The effects of the mass of elastic amplifier, spring stiffness, magnet spacing and load resistance on the dynamic characteristics of energy harvesting of TPEH+DEM system are analyzed. The result shows that there are two peaks in the response output power of TPEH+DEM system in the operating frequency range. By adjusting the mass and stiffness of the elastic amplifier reasonably, the system can move into the inter-well motion under low external excitation intensity, and produce high output power. Compared with the traditional model which only has an elastic amplifier on the base of piezoelectric energy harvester, TPEH+DEM model has better energy harvesting performance under low frequency and low intensity external excitation.

Keywords: Piezoelectric energy harvester, Electromechanical coupling, Elastic amplifier, Harmonic balance method, Inter-well motion

1. Introduction

Piezoelectric energy harvester can convert mechanical energy in the environment into electrical energy by using piezoelectric effect to supply power to microelectronic devices. The early linear piezoelectric energy harvester has a narrow working frequency band, so it is difficult to effectively match with the broadband vibration frequency in the environment, which leads to low energy harvesting efficiency [1], [2], [3], [4], [5], [6], [7], [8]. To solve this problem, scholars at home and abroad begin to study the multi-stable piezoelectric energy harvester model to improve the energy harvesting performance of the system [9], [10], [11], [12], [13], [14], [15], [16]. Chen et al. [17], based on the bi-stable piezoelectric energy harvesting model, analyzed the effects of material parameters, frequency and amplitude of external excitation on the system performance. Kim and Seok [18] studied a magnetic tri-stable piezoelectric cantilever beam energy harvester, and described in detail the advantages of tri-stable piezoelectric energy harvester for broadband vibration energy harvesting under low intensity excitation. Zhou et al. [19], based on the magnetic tri-stable piezoelectric energy harvester model, used numerical simulation and experimental method to demonstrate that the tri-stable piezoelectric energy harvester has better energy harvesting effect than bi-stable energy harvester under low external excitation level. Zhou and Zuo [20] obtained the analytical expression of steady-state response of asymmetric tri-stable piezoelectric cantilever beam by harmonic balance method, and analyzed the influence of potential well depth on energy harvesting effect under different excitation conditions.

At present, most of the studied multi-stable piezoelectric energy harvesters are fixed on rigid bases. Once the positions of cantilever beams and magnets are fixed, it will be difficult to adjust the frequency bandwidth of piezoelectric harvesters to match the external excitation frequency by changing the relative positions between magnets, so as to realize large-scale inter-well motion [21], [22]. Considering the above shortcomings of rigid base and to further improve the power output of multi-stable piezoelectric energy harvester under weak excitation intensity, researchers introduce elastic amplifier base into the structure of multi-stable piezoelectric energy harvester [23]. Liu et al. [24] proposed an elastic amplifier piezoelectric energy harvester with added mass, and demonstrated through experiments that the energy harvester can enhance vibration harvesting and reduce resonance frequency while broadening its operating frequency range. Wang et al. [25] demonstrated through numerical simulation and experimental method that the bi-stable piezoelectric energy harvester (BPEH+EM) system with elastic amplifier is easier to jump out of the barrier and move into large-scale inter-well motion at lower excitation level, and produces higher output power. Wang et al. [26] designed a piezoelectric cantilever beam energy harvester model with elastic amplifier, which provides enough kinetic energy to overcome the tri-stable potential well barrier, so that the system moves into large-scale well motion.

Based on the magnetic tri-stable piezoelectric cantilever beam energy harvester model with elastic amplifier base, a new type of amplifier composed of mass block Mf and spring kf is added to the fixed end of the piezoelectric cantilever beam to form a TPEH+DEM model with double elastic amplifiers. Considering the eccentric distance and moment of inertia of the free end magnet of the beam, the motion control equation of electromechanical coupling of TPEH+DEM system is established based on Hamilton principle, and the analytical solution of the equation is obtained by using harmonic balance method. Focus is put on the effects of the relative position of magnet spacing, the mass and stiffness of elastic amplifier on the energy harvesting performance of TPEH+DEM system, and the calculation result of TPEH+DEM model is compared with that of traditional TPEH+EM model.

2. Modeling of Piezoelectric Energy Harvester

Figure 1 is a structural model of the TPEH+DEM with double elastic amplifiers established in this paper. TPEH consists of a cantilever piezoelectric beam of length l and width b and a beam-end magnet (expressed by A) and two external magnets (expressed by B and C) fixed to the right side of the U-shaped frame. The horizontal distance between the piezoelectric beam cantilever end magnet A and the U-shaped frame fixed magnet is dh, and the vertical distance from the free end magnet of the beam to the external magnet B (external magnet C) is dv. The cantilever beam consists of a metal substrate with a thickness of hs and two piezoelectric patches with a thickness of tp covering the upper and lower surfaces of the cantilever beam. The two piezoelectric patches have opposite polarization in the thickness direction and are connected in series with an external load resistor R. DEM consists of two elastic amplifiers (denoted EM1 and EM2), of which EM1 comprises a U-shaped frame and a spring kb between TPEH and the substrate; EM2 consists of a spring kf connecting the beam end mass Mf and the bottom of the U-shaped frame.

In Figure 1, z(s, t), zm(t) and zb(t) represent the relative vertical displacement between a certain point on the section of piezoelectric cantilever beam and the fixed end of the beam, the vibration displacement of the U-shaped frame and the vertical vibration displacement of the base, respectively. The eccentric distance of the beam-end magnet is represented by e. Assuming that the metal substrate and the piezoelectric layer are completely tightly bonded, the constitutive relation is expressed as follows:

$\left.\begin{array}{l}T_1^{\mathrm{s}}=Y_{\mathrm{s}} S_1^{\mathrm{s}} \\ T_1^{\mathrm{p}}=Y_{\mathrm{p}}\left(S_1^{\mathrm{p}}-d_{31} E_3\right) \\ D_3=d_{31} T_1+\varepsilon_{33}^{\mathrm{T}} E_3\end{array}\right\}$
(1)

where, parameters related to the substrate are denoted by superscript s, and parameters related to the piezoelectric layer are denoted by subscript p; x and y directions are represented by 1 and 3 in subscripts, respectively; T, S, Y, D3 and d31 denote the stress, strain, Young modulus, electric displacement, and piezoelectric constant of the piezoelectric cantilever beam, respectively; $\varepsilon_{33}^T$ is the dielectric constant of the piezoelectric layer when the stress is constant; E3=-V(t)/(2tP), E represents electric field strength and V(t) is voltage; the expression of the relationship between displacement and strain is $S_1^s=S_1^p=-y z^{\prime \prime}$, where y is the distance between the neutral axis of the piezoelectric cantilever beam and any point on its cross section, z is the curvature of the piezoelectric cantilever beam.

The Lagrange equation of the system is as follows:

$\int_{t_1}^{t_2}\left[\delta\left(T_K+W_P-U_e-U_m-U_d\right)+\delta W\right] d t=0$
(2)
Figure 1. Structural model of TPEH+DEM

where, Tk is kinetic energy, Ue is strain energy, Wp is electric potential energy of electric field, Um is magnetic potential energy between magnets, Ud is elastic potential energy, and W is external work. Their expressions are as follows.

$\begin{aligned} & T_k=\frac{1}{2} \int_0^l m\left(\dot{z}+\dot{z}_m(t)\right)^2 d s+\frac{1}{2} M_t\left(\dot{z}(l, t)+e \dot{z}^{\prime}(l, t)+\dot{z}_m(t)\right)^2 \\ & +\frac{1}{2} J \dot{z}^{\prime}(l, t)^2+\frac{1}{2} M_f\left(\dot{z}(0, t)+\dot{z}_m(t)\right)^2+\frac{1}{2} M_m \dot{z}_m(t)^2\end{aligned}$
(3)

where, z(l, t) is the displacement of the piezoelectric cantilever beam at s=l; m=2ρptpb+ρshsb is the equivalent mass on the unit beam, where ρp is the density of the piezoelectric layer and ρs is the density of the substrate; the mass, eccentric distance and moment of inertia of free end magnet A of the piezoelectric cantilever beam are represented by Mt, e and J, respectively; (˙) denotes the derivative of time t.

$U_e=\frac{1}{2} \int_0^l\left[Y I z^{\prime \prime 2}-Y_P b d_{31}\left(h+\frac{t_P}{2}\right) Z(t) z^{\prime \prime}\right] d s$
(4)

where,

$h=h_{\mathrm{s}} / 2, Y I=Y_{\mathrm{s}} I_{\mathrm{s}}+Y_{\mathrm{p}} I_{\mathrm{p}} Y I=\frac{2}{3}\left[Y_s b h^3+Y_P b\left(3 h^2 t_P+3 h t_P{ }^2+t_P{ }^3\right)\right]$
(5)
$W_P=\frac{1}{2} Y_P b d_{31}\left(\mathrm{~h}+\frac{t_P}{2}\right) V(t) \int_0^l z^{\prime \prime} d s+b l \varepsilon_{33}^S \frac{V(t)^2}{4 t_P}$
(6)
$U_d=\frac{1}{2} k_f z(0, t)^2+\frac{1}{2} k_b z_m{ }^2$
(7)
$\begin{aligned} & \delta W=\delta z_m \ddot{z}_b\left(M_m+M_t+m l+M_f\right) \\ & +\delta X(t) \ddot{z}_b\left(M_t \varphi_1(l)+m \int_0^l \varphi_1(s) d s+M_t e \varphi_1^{\prime}(l)+M_f \varphi_1(0)\right)\end{aligned}$
(8)

where, $\varepsilon_{33}^S=\varepsilon_{33}^T-d_{31}^2 Y_p$, $\varepsilon_{33}^S$ is the dielectric constant of the piezoelectric layer when the stress is constant.

According to the magnetic dipole model [27, 28] after considering eccentric distance at the free end magnet of the beam, it can be obtained that:

$\begin{aligned} U_{\mathrm{m}}= & \frac{\mu_0}{4 \pi}\left[\frac{\mathbf{m}_{\mathrm{B}}}{\left\|\mathbf{r}_{\mathrm{BA}}\right\|_2^3}+\frac{\mathbf{m}_{\mathrm{C}}}{\left\|\mathbf{r}_{\mathrm{CA}}\right\|_2^3}-\frac{\left(\mathbf{m}_{\mathrm{B}} \cdot \mathbf{r}_{\mathrm{BA}}\right) 3 \mathbf{r}_{\mathrm{BA}}}{\left\|\mathbf{r}_{\mathrm{BA}}\right\|_2^5}\right. \\ & \left.-\frac{\left(\mathbf{m}_{\mathrm{C}} \cdot \mathbf{r}_{\mathrm{CA}}\right) 3 \mathbf{r}_{\mathrm{CA}}}{\left\|\mathbf{r}_{\mathrm{CA}}\right\|_2^5}\right] \cdot \mathbf{m}_{\mathrm{A}}\end{aligned}$
(9)

where, μ0=4π×10-7Hm-1 indicates vacuum magnetic conductivity,

$\mathbf{r}_{\mathrm{BA}}=\left[\begin{array}{ll}-d_{\mathrm{h}}-\Delta x & v(l, t)+e \sin \beta-d_{\mathrm{v} 1}\end{array}\right]$ ,

$\mathbf{r}_{\mathrm{CA}}=\left[\begin{array}{ll}-d_{\mathrm{h}}-\Delta x & v(l, t)+e \sin \beta+d_{\mathrm{v} 2}\end{array}\right]$ ,

The magnetic dipole moments are mA=[MAVAcosβ MAVAsinβ], mB=[-MBVB 0] and mC=[-MCVC 0]. In Δxe(1-cosβ), β=arctanv'(l), MA, MB and MC are the magnetization intensity of magnets A, B and C, respectively, and VA, VB and VC are the volumes of magnets A, B and C, respectively.

Using Galerkin method, the displacement z(s, t) is expressed as:

$z(s, t)=\phi_r(s) X_r(t)$
(10)

where, Xr(t) is the r-th modal coordinate of the piezoelectric cantilever beam, ϕr(s) represents the r-th modal function of the beam, and its normalized condition expression is:

$\begin{aligned} & \int_0^l \varphi_s(s) m \varphi_r(s) d s+\varphi_s(l) M_t \varphi_r(l)+\varphi_s(l) M_t e \varphi_r^{\prime}(l) \\ & +\varphi_s(0) M_f \varphi_r(0)+\varphi_s^{\prime}(l)\left(J+M_t e^2\right) \varphi_r^{\prime}(l) \\ & +\varphi_s^{\prime}(l) M_t e \varphi_r(l)=\delta_{r s}\end{aligned}$
(11)
$\int_0^l \frac{d^2 \varphi_s(s)}{d s^2} Y I \frac{d^2 \varphi_r(s)}{d s^2} d s+\varphi_s(0) k_f \varphi_r(0)=\omega_r^2 \delta_{r s}$
(12)

where, δrs is the Kronecker function. When s=r, $\delta_{r s}$ is 1; when $s \neq r$, $\delta_{r s}$ is 0. The expression of inherent frequency of the piezoelectric cantilever beam under undamped vibration is $\omega_r=\lambda_r^2 \sqrt{Y I /\left(m l^4\right)}$, where λr is its eigen value. The calculation method of λr and vibration mode function can be seen in reference [18].

Considering only the first-order mode and substituting formula (10) into formula (9), this paper expands at X1(t)=0 by using Taylor’s formula and can obtain:

$\begin{aligned} & U_m=k_0-\frac{1}{2} k_1 X_1^2+\frac{1}{4} k_2 X_1^4+\frac{1}{6} k_3 X_1^6 +o\left(X_1^7\right)\end{aligned}$
(13)

where,

$k_0=\frac{2 k q_1}{\left(d_h^2+d_v^2\right)^2}$,

$\begin{aligned} & k_1=\frac{4 k}{\left(d_h{ }^2+d_v{ }^2\right)^{5 / 2}}\left[q_1\left(2.5 q_2-17.5 q_3{ }^2\right)\right. \\ & \left.-5 q_3 q_4\left(d_h{ }^2+d_v{ }^2\right)-q_5\right]\end{aligned}$

$\begin{aligned} & k_2=\frac{8 k}{\left(d_h^2+d_v^2\right)^{5 / 2}}\left[q_1\left(-2.5 q_6-17.5 q_3 q_7\right.\right. \\ & \left.+4.38 q_2{ }^2-78.75 q_2 q_3{ }^2+144.375 q_3{ }^4\right) \\ & +q_4\left(-2.5 q_7-17.5 q_2 q_3+52.5 q_3{ }^3\left(d_h{ }^2+d_v{ }^2\right)\right. \\ & \left.q_5\left(-2.5 q_2+17.5 q_3{ }^2\right)+5 q_3 q_8+q_9\right]\end{aligned}$

$\begin{aligned} & k_3=\frac{12 k}{\left(d_h{ }^2+d_v{ }^2\right)^{5 / 2}}\left[q_1\left(-2.5 q_{10}+13.125 q_3 q_{11}+8.75 q_2 q_6\right.\right. \\ & +4.375 q_7{ }^2-78.78 q_3{ }^2 q_6+78.25 q_2 q_3 q_7-288.75 q_3{ }^3 q_7 \\ & \left.+216.55 q_2{ }^2 q_3{ }^2-6.56 q_2{ }^3-938.44 q_2 q_3{ }^4+938.44 q_3{ }^6\right) \\ & +q_4\left(1.875 q_{11}-17.5 q_3 q_6+8.75 q_2 q_7-78.75 q_3{ }^2 q_7+\right. \\ & \left.39.38 q_2{ }^2 q_3-288.75 q_2 q_3{ }^2+375.375 q_3{ }^5\right)+q_5\left(-2.5 q_6\right. \\ & \left.-17.5 q_3 q_7+4.375 q_2{ }^2-78.75 q_2 q_3{ }^2+144.375 q_3{ }^4\right) \\ & +q_8\left(-2.5 q_7-17.5 q_2 q_3+52.5 q_3{ }^2\right)+q_9\left(-2.5 q_2+17.5 q_3{ }^2\right) \\ & \left.+5 q_3 q_{12}+q_{13}\right]\end{aligned}$

Coefficient κ, the expression of qi=1…26 is shown in the previous reference [18].

Considering only the first-order mode, formula (2) is substituted into the Lagrange variational equation shown in formula (14):

$\left\{\begin{array}{l}\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{z}_m}\right)-\frac{\partial L}{\partial z_m}+\frac{\partial W}{\partial z_m}=0 \\ \frac{d}{d t}\left(\frac{\partial L}{\partial \dot{X}}\right)-\frac{\partial L}{\partial X}+\frac{\partial W}{\partial X}=F(t) \\ \frac{d}{d t}\left(\frac{\partial L}{\partial \dot{V}}\right)-\frac{\partial L}{\partial V}+\frac{\partial W}{\partial V}=Q(t)\end{array}\right.$
(14)

where, $F(t)=-2 \xi_1 \omega_1 \dot{\eta}_1(t)$ is the generalized dissipative force of TPEH+DEM system, ω1 is the first-order natural frequency of the system, ξ1 is its damping ratio and Q(t) is generalized output charge. $\dot{Q}(t)=-V(t) / R$, where R represents the load resistance of external circuit. Through formula (14), the differential equation of motion of TPEH+DEM system can be obtained as follows:

$\left\{\begin{array}{c}M_0 \ddot{X}_1(t)+M_1 \ddot{z}_m(t)+k_b z_m=-M_1 \ddot{z}_b(t) \\ \ddot{X}_1(t)+2 \xi_1 \omega_1 \dot{X}_1(t)+\omega_1^2 X_1(t)-k_1 X_1(t) \\ +k_2 X_1(t)^3+k_3 X_1(t)^5-\theta_1 V(t) \\ +M_0 \ddot{z}_m(t)=-M_0 \ddot{z}_b(t) \\ C_P \dot{V}(t)+\frac{V(t)}{R}+\theta_1 \dot{X}_1(t)=0\end{array}\right.$
(15)

where,

$\begin{gathered}M_0=m \int_o^l \phi_1(s) d s+M_t \phi_1(l)+M_t e \phi_1^{\prime}(l) \\ +M_{\mathrm{f}} \phi_1(0),\, M_1=m l+M_t+M_f+M_m, \\ \omega_1^2=Y I \int_0^l \phi_1^{\prime \prime}(s)^2 d s+k_f \phi_1(0)^2, \,\theta_1=Y_P b d_{31}\left(h+\frac{t_P}{2}\right) \int_0^l \phi_1^{\prime \prime}(s) d s, \,C_P=\frac{b l \varepsilon_{33}^s}{2 t_P} .\end{gathered}$

where,

$\begin{gathered}\omega_1^2=Y I \int_0^l \varphi_1^{\prime \prime} d s,\, g_0=m g \int_0^l \varphi_1(s) d s \\ +M_t g \varphi_1(l), \,\Gamma_1=m \int_0^l \varphi_1(s) d s+M_{\mathrm{t}}\left(\varphi_1(l)+e \varphi_1^{\prime}(l)\right) \\ \theta_1=Y_P b d_{31}\left(h+\frac{t_P}{2}\right) \int_0^l \varphi_1(s) d s, C_P=\frac{b l \varepsilon_{33}^S}{2 t_P} .\end{gathered}$

External excitation is set to $\ddot{v}_b(t)=\bar{v}_b \cos \left(\omega_e t\right)$, where $\bar{v}_b$ is the amplitude of external excitation, $\omega_{\mathrm{e}}$ is the circular frequency of excitation. Substituting dimensionless transformation $x=\eta_1 / l$, $\bar{V}=V C_p /\left(l \theta_1\right)$ and $\tau=\omega_1 t$ into formula (15), we can get

$\left\{\begin{array}{c}M_0 \ddot{x}+M_1 \ddot{V}_{\mathrm{m}}+K_{\mathrm{b}} V_{\mathrm{m}}=-M_1 \ddot{V}_{\mathrm{b}} \\\\ \ddot{x}+2 \xi_1 \dot{x}+\left(1-K_1\right) x+K_2 x^3+K_3 x^5 \\ -\Theta \bar{V}+M_0 \ddot{V}_{\mathrm{m}}=-M_0 \ddot{V}_{\mathrm{b}} \\\\ \dot{\bar{V}}+\alpha \bar{V}+\dot{x}=0\end{array}\right.$
(16)

where, $K_b=\frac{k_b}{\omega_1^2}, K_1=\frac{k_1}{\omega_1^2}, K_2=\frac{k_2 l^2}{\omega_1^2}, \quad \Theta=\frac{\theta_1^2}{C_P \omega_1^2}, \alpha=\frac{1}{C_P R_L \omega_1}, F=-\frac{M_0 \bar{z}_b}{\omega_1^2 l}$.

The first and second equations of formula (16) are used to eliminate the variable Vm, and formula (16) is simplified to

$\left\{\begin{array}{c}\frac{M_1-M_0^2}{K_b} x^{(4)}+\frac{2 M_1 \xi_1}{K_b} x^{(3)}+\frac{M_1\left(1-K_1\right)+K_b}{K_b} \ddot{x} \\ +2 \xi_1 \dot{x}+\left(1-K_1\right) x+K_2 x^3+K_3 x^5+\frac{M_1 K_2}{K_b}\left(6 x \dot{x}^2+3 x^2 \ddot{x}\right) \\ +\frac{M_1 K_3}{K_b}\left(20 x^3 \dot{x}^2+5 x^4 \ddot{x}\right)-\frac{M_1 \Theta}{K_b} \ddot{\bar{V}}-\Theta \bar{V}=F \cos (\omega t) \\ \dot{\bar{V}}+\alpha \bar{V}+\dot{x}=0\end{array}\right.$
(17)

3. Solution by Harmonic Balance Method

It is assumed that the solution of formula (17) can be expressed as:

$\left\{\begin{array}{l}x=A(\tau) \sin (\omega \tau)+B(\tau) \cos (\omega \tau) \\ \bar{V}=C(\tau) \sin (\omega \tau)+D(\tau) \cos (\omega \tau)\end{array}\right.$
(18)

where A, B, C and D are uncertain coefficients, the displacement amplitude of the piezoelectric cantilever beam can be expressed as $a=\sqrt{A^2+B^2}$, and the output voltage amplitude can be expressed as $u=\sqrt{C^2+D^2}$. We substitute formula (18) into formula (17), take the constant terms on both sides, and the coefficients of cos(ωτ) and sin(ωτ) are equal, and ignore the higher-order harmonic term, then the following equation is obtained:

$\Pi_1(\ddot{B}-2 \omega \dot{A})+2 \xi_1 \dot{B}+\Pi_3 B-\Pi_2 A+\Pi_4 D-F=0$
(19)
$\Pi_1(\ddot{A}-2 \omega \dot{B})+2 \xi_1 \dot{A}+\Pi_2 B+\Pi_3 A+\Pi_4 C=0$
(20)
$\dot{C}-\omega D+\alpha C+\dot{A}-\omega B=0$
(21)
$\dot{D}+\omega C+\alpha D+\dot{B}+\omega A=0$
(22)

where,

$\begin{gathered}\Pi_1=\frac{K_1 M_1+K_b}{K_b} \\ \Pi_2=\frac{2 \xi_1 M_1}{K_b} \omega^3-2 \xi_1 \omega, \\ \Pi_3=\frac{M_1-M_0^2}{K_b} \omega^4-\frac{\left(1-K_1\right) M_1+K_b}{K_b} \omega^2+1-K_1 \\ +\frac{3}{4} K_2 a^2+\frac{5}{8} K_3 a^4-\frac{3}{4} \frac{K_2 M_1}{K_b} \omega^2 a^2-\frac{5}{8} \frac{K_3 M_1}{K_b} \omega^4 a^4 \\ \Pi_4=\frac{\Theta M_1}{K_b} \omega^2-\Theta\end{gathered}$

In the steady state, all derivatives of time are 0. Therefore, the analytical expressions of displacement amplitude and voltage amplitude of the system can be obtained by using formulas (19-22) as follows:

$a^2\left[\left(\Pi_2+\Pi_4 \frac{\alpha \omega}{\omega^2+\alpha^2}\right)^2+\left(\Pi_3-\Pi_4 \frac{\omega^2}{\omega^2+\alpha^2}\right)^2\right]=F^2$
(23)

The steady-state displacement amplitude a of TPEH+DEM system can be calculated by formula (23). The expressions of steady-state output voltage and power amplitude are:

$u=\left(\frac{\omega}{\sqrt{\omega^2+\alpha^2}}\right) a$
(24)
$P=\frac{l^2 \theta_1^2 u^2}{C_P{ }^2 R}$
(25)

4. Analysis of Calculation Result

In this chapter, focus will be put on the effects of relative position of magnets, load resistance, mass of elastic amplifier and spring stiffness ratio on the performance of TPEH+DEM (Table 1). The main physical and geometric parameters of the system are as follows [1].

Table 1. The geometric and material properties of the TPEH+DEM

Parameter

Value

Length l

75mm

Width b

20mm

Thickness hs

0.2mm

Piezoelectric modulus Ys

70Gpa

Piezoelectric modulus Yp

60.98Gpa

Density $\rho_s$

2700kg/mm3

Density $\rho_p$

7750kg/mm3

Piezoelectric constant d31

-1.7×10-10C/N

Permittivity of free space $\varepsilon_{33}^s$

1.33×10-8F/m

Damping ratio $\xi_1$

0.01

Magnetization of magnets (A,B,C)

1×10-6m3

Volume of magnet (A,B,C)

1.22×10-6m3

Eccentricity e

5mm

Mass of tip magnet Mt

14.9g

Figure 2. Variation curves of displacement amplitude of the system with excitation frequency under different models
Figure 3. Variation curves of output power amplitude of the system with excitation frequency under different models
Figure 4. Variation curves of displacement amplitude of the system with excitation frequency in different values of $M_{\mathrm{f}}$
Figure 5. Variation curves of output power amplitude of the system with excitation frequency in different values of $M_{\mathrm{f}}$
Figure 6. Variation curves of left peak power of the system with load resistance in different values of $M_{\mathrm{f}}$
Figure 7. Variation curves of right peak power of the system with load resistance in different values of $M_{\mathrm{f}}$

We take Mm=60 g, dh=21 mm, dv=8 mm and kb=8,000 N/mm. The variation curves of displacement and output power amplitudes of the two models of piezoelectric energy harvester models (TPEH+DEM, TPEH+EM) with excitation frequency are shown in Figure 2 and Figure 3, respectively. TPEH+EM is equipped with only an elastic amplifier, i.e. a spring kb located between TPEH and the substrate.

We take Mm=60 g, dh=21 mm, dv=8 mm, kf=50,000 N/mm and kb=8,000 N/mm. Figure 4 and Figure 5 show the variation curves of displacement amplitude a and power amplitude P of TPEH+DEM system with excitation frequency in different values of Mf s, respectively. As can be seen from Figure 4 and Figure 5, with the increase of Mf, left displacement and power peak of the system increase continuously, while right displacement and power peak change little, but the frequency band range moves to the low frequency band.

Figure 6 and Figure 7 show the variation curves of left and right peak power of TPEH+DEM system with load resistance R in different values of Mf when Mm=60 g, dv=8 mm, dh=21 mm, kf=50,000 N/mm and kb=8,000 N/mm. It can be seen from Figure 6 that with the increase of R, left peak power of the system increases rapidly first, and the first local maximum Pmax1 appears, corresponding to the load resistance Ropt1, and then continues to increase after greatly decreasing (when the load resistance is Ropt2, it reaches the second local maximum Pmax2 of the curve, and finally decreases gradually). In addition, it can be seen from Figure 6 that under the same Mf, the two local maxima are very close, and increasing Mf can obviously improve Pmax1 and Pmax2. As shown in Figure 7, when Mf = 0.02g, left peak power of the system is 0.1130 W; when Mf increases to 0.06 g, right peak power of the system is 0.1115 W, that is, with the change of Mf, the peak power changes little.

Figure 8. Variation curves of left peak power of the system with load resistance in different values of $d_{\mathrm{v}}$ and $d_{\mathrm{h}}$
Figure 9. Variation curves of right peak power of the system with load resistance in different values of $d_{\mathrm{v}}$ and $d_{\mathrm{h}}$

We take Mf=60 g, Mm=60 g, kf=50,000 N/mm and kb=8,000 N/mm. Figure 8 and Figure 9 show the variation curves of left and right peak power of TPEH+DEM system corresponding to different relative positions between magnets with load resistance R. As can be seen from Figure 8, the variation trend of left peak power of the system with load resistance R is the same as that of Figure 6, and the peak power of the system corresponding to different dv and dh is relatively close. Figure 9 shows that the peak power of the system can be significantly increased by increasing dh while keeping dv constant. However, when dh is kept constant, the peak power decreases slightly with the increase of dv.

We take kf=50,000 N/mm, kb=8,000 N/mm, dv=8 mm and dh=21 mm. Figure 10 and Figure 11 show the variation curves of steady-state voltage amplitude u with excitation acceleration when w is 0.8 and 1.2 respectively, and the mass of elastic amplifier is taken for different values. As can be seen from Figure 10 and Figure 11, each working condition corresponds to an excitation acceleration threshold, which enables the system to move into large-scale inter-well motion and generate high output voltage. When w is the same, the threshold of excitation acceleration required for the system to move into inter-well motion can be reduced by increasing the mass Mf and Mm of two elastic amplifiers, but the change of output voltage is not obvious. When the mass of elastic amplifiers is constant, the threshold of excitation acceleration increases with the increase of excitation frequency.

Figure 10. Variation curves of output voltage with excitation amplitude in different values of $M_{\mathrm{f}}$ and $M_{\mathrm{m}}$ when $\omega$=0.8
Figure 11. Variation curves of output voltage with excitation amplitude in different values of $M_{\mathrm{f}}$ and $M_{\mathrm{m}}$ when $\omega$=1.2

Taking the load resistance R=300 kΩ, as shown in Figure 12 and Figure 13, the steady-state voltage amplitude u of TPEH+DEM system varies with the excitation amplitude when the excitation acceleration w is 0.6 and 1.2, respectively and kf/kb is in different values. As shown in Figure 12 and Figure 13, when w=0.6, with the increase of kf/kb, the excitation threshold of the system decreases, while the steady-state voltage amplitude u changes little. When w is increased to 1.2, the excitation threshold obviously decreases with the increase of kf/kb, and the output voltage amplitude increases with the increase of kf/kb after moving into inter-well motion. It can be seen that when the excitation frequency is high, increasing the kf/kb value can make the system move into inter-well motion and produce higher output voltage at lower excitation intensity.

Figure 14 and Figure 15 show the variation curves of power and displacement amplitudes of the system when Mm is different in values and when R=300 kΩ, Mf=60 g, dh=21, dv=8, kf=50,000 N/mm and kb=50,000 N/mm. The result shows that, with the increase of excitation frequency, the power and displacement amplitudes of the system increase significantly at first, then decrease sharply after reaching the maximum value rapidly, then increase again, and then decrease after reaching the second local peak value when Mm keeps constant. In addition, with the increase of Mm, the peak power and displacement of the system increase significantly, and can enter the secondary ascending section at lower excitation frequency, resulting in higher local peak power.

Figure 12. Variation curves of output voltage with excitation amplitude in different values of kf/kb when $\omega$=0.6
Figure 13. Variation curves of output voltage with excitation amplitude in different values of kf/kb when $\omega$=1.2
Figure 14. Curves of power amplitude changing with frequency in different values of $M_{\mathrm{m}}$
Figure 15. Curves of displacement amplitude changing with excitation frequency in different values of $M_{\mathrm{m}}$

5. Conclusion

(1) Based on Hamilton variational principle, the differential equation of motion of a tri-stable piezoelectric energy harvesting system with double elastic amplifiers is derived, and the analytical solution of electromechanical coupling equation describing the energy output characteristics of the system is obtained by using the harmonic balance method.

(2) With the increase of excitation frequency, the output power amplitude of TPEH+DEM system can produce two high and low peaks. Compared with TPEH+EM system, TPEH+DEM system has better energy harvesting performance under low frequency external excitation.

(3) At the same excitation frequency, the excitation acceleration threshold required for TPEH+DEM system to move into inter-well motion can be reduced by increasing the mass of elastic amplifiers; when the excitation frequency is high, increasing the stiffness ratio kf/kb of the two elastic amplifiers can make TPEH+DEM system move into inter-well motion at lower excitation intensity and produce higher output voltage.

(4) Increasing Mm can significantly improve the maximum peak power of TPEH+DEM system, and can make the system enter the secondary ascending section at lower excitation frequency, resulting in higher local peak power.

Funding
This research was funded by the Anhui Provincial University Provincial Natural Science Research Project-Key Project (Grant No.: 2022AH050240); Doctoral Startup Foundation of Anhui Jianzhu University (Grant No.: 2020QDZ07); Scientific research Project of Anhui Education Department- Key Project (Grant No.: KJ2021JD01); Anhui Provincial Natural Science Foundation (Grant No.: 2108085MA28).
Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Man, D. W., Bai, Y. Y., Hu, Q. N., Xu, H. M., Xu, G. Z., & Tang, L. P. (2023). Dynamic Characteristic Analysis of Tri-Stable Piezoelectric Energy Harvester with Double Elastic Amplifiers. J. Intell Syst. Control, 2(2), 54-69. https://doi.org/10.56578/jisc020201
D. W. Man, Y. Y. Bai, Q. N. Hu, H. M. Xu, G. Z. Xu, and L. P. Tang, "Dynamic Characteristic Analysis of Tri-Stable Piezoelectric Energy Harvester with Double Elastic Amplifiers," J. Intell Syst. Control, vol. 2, no. 2, pp. 54-69, 2023. https://doi.org/10.56578/jisc020201
@research-article{Man2023DynamicCA,
title={Dynamic Characteristic Analysis of Tri-Stable Piezoelectric Energy Harvester with Double Elastic Amplifiers},
author={Dawei Man and Yingying Bai and Qingnan Hu and Huaiming Xu and Gaozheng Xu and Liping Tang},
journal={Journal of Intelligent Systems and Control},
year={2023},
page={54-69},
doi={https://doi.org/10.56578/jisc020201}
}
Dawei Man, et al. "Dynamic Characteristic Analysis of Tri-Stable Piezoelectric Energy Harvester with Double Elastic Amplifiers." Journal of Intelligent Systems and Control, v 2, pp 54-69. doi: https://doi.org/10.56578/jisc020201
Dawei Man, Yingying Bai, Qingnan Hu, Huaiming Xu, Gaozheng Xu and Liping Tang. "Dynamic Characteristic Analysis of Tri-Stable Piezoelectric Energy Harvester with Double Elastic Amplifiers." Journal of Intelligent Systems and Control, 2, (2023): 54-69. doi: https://doi.org/10.56578/jisc020201
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