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

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 M_f and spring k_f 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.

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 d_h, and the vertical distance from the free end magnet of the beam to the external magnet B (external magnet C) is d_v. The cantilever beam consists of a metal substrate with a thickness of h_s and two piezoelectric patches with a thickness of t_p 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 k_b between TPEH and the substrate; EM₂ consists of a spring k_f connecting the beam end mass M_f and the bottom of the U-shaped frame.

In Figure 1, z(s, t), z_m(t) and z_b(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:

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)/(2t_P), 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:

where, T_k is kinetic energy, U_e is strain energy, W_p is electric potential energy of electric field, U_m is magnetic potential energy between magnets, U_d is elastic potential energy, and W is external work. Their expressions are as follows.

where, z(l, t) is the displacement of the piezoelectric cantilever beam at s=l; m=2ρ_pt_pb+ρ_sh_sb is the equivalent mass on the unit beam, where ρ_p is the density of the piezoelectric layer and ρ_s 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 M_t, e and J, respectively; (˙) denotes the derivative of time t.

where,

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:

where, μ₀=4π×10^-7H⋅m^-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 m_A=[M_AV_Acosβ M_AV_Asinβ], m_B=[-M_BV_B 0] and m_C=[-M_CV_C 0]. In Δx≈e(1-cosβ), β=arctanv^'(l), M_A, M_B and M_C are the magnetization intensity of magnets A, B and C, respectively, and V_A, V_B and V_C are the volumes of magnets A, B and C, respectively.

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

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

where, δ_rs 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 λ_r is its eigen value. The calculation method of λ_r 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:

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 q_i=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):

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:

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

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 V_m, and formula (16) is simplified to

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

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:

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:

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:

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].

We take M_m=60 g, d_h=21 mm, d_v=8 mm and k_b=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 k_b located between TPEH and the substrate.

We take M_m=60 g, d_h=21 mm, d_v=8 mm, k_f=50,000 N/mm and k_b=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 M_f s, respectively. As can be seen from Figure 4 and Figure 5, with the increase of M_f, 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 M_f when M_m=60 g, d_v=8 mm, d_h=21 mm, k_f=50,000 N/mm and k_b=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 P_max1 appears, corresponding to the load resistance R_opt1, and then continues to increase after greatly decreasing (when the load resistance is R_opt2, it reaches the second local maximum P_max2 of the curve, and finally decreases gradually). In addition, it can be seen from Figure 6 that under the same M_f, the two local maxima are very close, and increasing M_f can obviously improve P_max1 and P_max2. As shown in Figure 7, when M_f = 0.02g, left peak power of the system is 0.1130 W; when M_f increases to 0.06 g, right peak power of the system is 0.1115 W, that is, with the change of M_f, the peak power changes little.

We take M_f=60 g, M_m=60 g, k_f=50,000 N/mm and k_b=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 d_v and d_h is relatively close. Figure 9 shows that the peak power of the system can be significantly increased by increasing d_h while keeping dv constant. However, when d_h is kept constant, the peak power decreases slightly with the increase of d_v.

We take k_f=50,000 N/mm, k_b=8,000 N/mm, d_v=8 mm and d_h=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 M_f and M_m 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.

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 k_f/k_b is in different values. As shown in Figure 12 and Figure 13, when w=0.6, with the increase of k_f/k_b, 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 k_f/k_b, and the output voltage amplitude increases with the increase of k_f/k_b after moving into inter-well motion. It can be seen that when the excitation frequency is high, increasing the k_f/k_b 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 M_m is different in values and when R=300 kΩ, M_f=60 g, d_h=21, d_v=8, k_f=50,000 N/mm and k_b=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 M_m keeps constant. In addition, with the increase of M_m, 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.

(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 k_f/k_b 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 M_m 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.