Comparison of modeling methods and parametric study for a piezoelectric wind energy harvester

In the preceding study, a simple SDOF model was established to simulate the electroaeroelastic behavior of a GPEH [36]. In that model, the harvester was considered to oscillate close to the fundamental frequency, which was confirmed with visual observation during the experiment. The governing equations are given as

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl {M}_{\mathrm{eff}}\ddot{w}(L,t)+{C}_{\mathrm{eff}}\dot {w}(L,t)+{K}_{\mathrm{eff}}w(L,t)+\Theta V(t)\\ \displaystyle \tqs ={F}_{z}(t)=\frac{1}{2}{\rho }_{\mathrm{a}}h{l}_{\mathrm{tip}}{U}^{2}{C}_{\mathrm{ F}z}\\ \displaystyle \fl \frac{V(t)}{{R}_{L}}+{C}_{\mathrm{P}}\dot {V}(t)-\Theta \dot {w}(L,t)=0 \end{array} \end{eqnarray} \tag{ 7 }$$

where w(L,t) is the displacement of the bluff body in the direction normal to the wind flow; C_eff and K_eff are the effective damping and stiffness of the harvester; V(t) is the generated voltage across R_L; C_P is the total capacitance of two piezoelectric sheets with parallel connection; Θ is the electromechanical coupling term; h and l_tip are the frontal dimension and length of the bluff body, the product of which equals S_tip; and M_eff is the effective mass approximated as M_eff = 33/140M_b + M_tip, where M_b and M_tip are the mass of the cantilever and the bluff body. Θ is determined easily though experiments by

$$\begin{eqnarray} &&\Theta =\sqrt{({\omega }_{\mathrm{noc}}^{2}-{\omega }_{\mathrm{nsc}}^{2}){M}_{\mathrm{eff}}{C}_{\mathrm{P}}} \end{eqnarray} \tag{ 8 }$$

where ω_noc and ω_nsc are the open circuit and short circuit resonant frequencies of the harvester. C_Fz is expressed as in equation (3), and the attack angle is modified to $\alpha =\dot {w}(L,t)/U+{w}^{\prime}(L,t)$, where w'(L,t) is the rotation angle at the free end due to the deflection of the beam, approximated as w'(L,t) = 3w(L,t)/2L.

By defining a state vector X,

$$\begin{eqnarray} &&\boldsymbol{X}=\left \{ \begin{array}{@{}c@{}} \displaystyle {X}_{1}\\ \displaystyle {X}_{2}\\ \displaystyle {X}_{3} \end{array}\right \} =\left \{ \begin{array}{@{}c@{}} \displaystyle w(L,t)\\ \displaystyle \dot {w}(L,t)\\ \displaystyle V(t) \end{array}\right \} \end{eqnarray} \tag{ 9 }$$

the governing equations can be written in the state space form as

$$\begin{eqnarray} \dot {\boldsymbol{X}}&=&\left \{ \begin{array}{@{}c@{}} \displaystyle \dot {w}(L,t)\\ \displaystyle \ddot{w}(L,t)\\ \displaystyle \dot {V}(t) \end{array}\right \} \nonumber\\ &=&\left \{ \begin{array}{@{}c@{}} \displaystyle {X}_{2}\\ \displaystyle -2 \zeta {\omega }_{n}{X}_{2}-{\omega }_{n}^{2}{X}_{1}-\frac{\Theta }{{M}_{\mathrm{eff}}}{X}_{3} \\\qquad \displaystyle +~\frac{{\rho }_{\mathrm{a}}h{l}_{\mathrm{tip}}{U}^{2}}{2{M}_{\mathrm{eff}}}\left [\mathop{\sum }\limits _{r=1,2,\ldots }{A}_{r}\left (\frac{{X}_{2}}{U}+\frac{3{X}_{1}}{2 L}\right )^{r}\right ] \\ \displaystyle \frac{\Theta }{{C}_{\mathrm{P}}}{X}_{2}-\frac{{X}_{3}}{{R}_{L}{C}_{\mathrm{P}}} \end{array}\right \} \nonumber\\ \end{eqnarray} \tag{ 10 }$$

where 2ζω_n = C_eff/M_eff and ω_n = ω_nsc. Equation (10) can then be numerically solved in MATLAB using a solver like ode45 to determine the vibration response of the beam, cut-in wind speed, and generated voltage across R_L. The average power P_ave is related to the root mean square (RMS) voltage V_RMS as ${P}_{\mathrm{ave}}={V}_{\mathrm{RMS}}^{2}/{R}_{\mathrm{L}}$.

Distributed parameter electromechanical models based on Euler–Bernoulli beam theory have been studied extensively for vibration piezoelectric energy harvesters with base-excitations [5, 46, 47]. For a GPEH, the electromechanical equations are developed based on similar assumptions: (a) the Euler–Bernoulli beam assumptions are applied to the composite beam; (b) the perfectly conductive electrodes fully cover the top and bottom surfaces of the piezoelectric sheet inducing uniform electric field through the thickness; (c) only the z-direction vibration (transverse vibration) is considered; (d) the damping mechanisms including the internal strain rate damping and external air damping satisfy the proportional damping criterion. The schematics of the GPEH showing the coordinate directions and the cross-section of the composite beam are presented in figure 3. We consider the electrically parallel connection for the two piezoelectric sheets. The coupled governing equations for the GPEH can be written as [5, 39]

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl \frac{{\partial }^{2}}{\partial {x}^{2}}\left [Y I(x)\frac{{\partial }^{2}w(x,t)}{\partial {x}^{2}}\right ]+{c}_{s}I(x)\frac{{\partial }^{5}w(x,t)}{\partial {x}^{4}\partial t}+{c}_{\mathrm{a}}\frac{\partial w(x,t)}{\partial t} \\ \displaystyle +~m(x)\frac{{\partial }^{2}w(x,t)}{\partial {t}^{2}}+\theta V(t) \\ \displaystyle \times ~\left [\frac{\mathrm{d}\delta (x-{x}_{1})}{\mathrm{d}x}-\frac{\mathrm{d}\delta (x-{x}_{2})}{\mathrm{d}x}\right ]={F}_{z}\delta (x-L) \\ \displaystyle \fl \frac{V(t)}{{R}_{L}}+{C}_{\mathrm{p}}\frac{\mathrm{d}V(t)}{\mathrm{d}t}-\theta \int \nolimits _{{x}_{1}}^{{x}_{2}}\frac{{\partial }^{3}w(x,t)}{\partial {x}^{2}\partial t}\mathrm{d}x=0 \end{array} \end{eqnarray} \tag{ 11 }$$

where w(x,t) is the transverse deflection of the beam in z direction; δ(x) is the Dirac delta function; c_s and c_a are the coefficients of strain rate damping and air damping; m(x) is the distributed mass of the beam, being expressed as m(x) = ρ_sh_sb_s for 0 < x < x₁ and x₂ < x < L, and m(x) = ρ_sh_sb_s + 2ρ_ph_pb_p for x₁ < x < x₂, where ρ_s,h_s and b_s are the mass density, thickness and width of the substrate, respectively, ρ_p,h_p and b_p are the corresponding terms of the piezoelectric sheet, and x₁ and x₂ are respectively the starting and ending positions of the piezoelectric sheets along the beam; YI(x) is the bending stiffness of the composite beam expressed as $Y I(x)={Y}_{\mathrm{s}}{b}_{\mathrm{s}}{h}_{\mathrm{s}}^{3}/1 2$ for 0 < x < x₁ and x₂ < x < L, and $Y I(x)={Y}_{\mathrm{s}}{b}_{\mathrm{s}}{h}_{\mathrm{s}}^{3}/1 2+2{Y}_{\mathrm{p}}{b}_{\mathrm{p}}[({h}_{\mathrm{p}}+{h}_{\mathrm{s}}/2)^{3}-{h}_{\mathrm{s}}^{3}/8]/3$ for x₁ < x < x₂, where Y_s and Y_p are the Young's modulus of the substrate and piezoelectric material, respectively; θ is the piezoelectric coupling term given by θ =− Y_pd₃₁b_p(h_p + h_s), where d₃₁ is the piezoelectric constant; and C_p is the total capacitance as in the SDOF model, being given by ${C}_{\mathrm{p}}=2{\varepsilon }_{3 3}^{S}{b}_{\mathrm{p}}({x}_{2}-{x}_{1})/{h}_{\mathrm{p}}$ ((x₂ − x₁) equals the length of the piezoelectric sheets L_p), where ${\varepsilon }_{3 3}^{S}$ is the permittivity at constant strain. As in the SDOF model, F_z is the aerodynamic force due to galloping.

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The transverse deflection w(x,t) can be represented as

$$\begin{eqnarray} &&w(x,t)=\sum _{r=1}^{\infty }{\phi }_{r}(x){\eta }_{r}(t) \end{eqnarray} \tag{ 12 }$$

where ϕ_r(x) is the mass normalized mode shape for the rth mode of the corresponding undamped free vibration problem, and η_r(t) is the modal coordinate. Erturk and Inman [5, 46, 47] presented the procedure for the exact analytical solution of ϕ_r(x) for a cantilever fully and uniformly covered with piezoelectric materials. As for a cantilever beam which is partially covered by piezoelectric sheets (the case for our prototype in the experiment), Abdelkefi et al [39] developed the Euler–Bernoulli distributed parameter model by deriving the exact analytical mode shape function to analyze the electroaeroelastic behavior of a GPEH. Yet such an analytical solution for the segmented mode shapes is quite cumbersome. Single mode (fundamental mode only) was considered in their analysis. In this paper, we obtain ϕ_r(x) and ${\phi }_{r}^{\prime}(x)$ easily by the finite element method in Matlab. Hermite cubic functions are employed as the interpolation functions for the beam element.

F_z is still in the form as in the SDOF model (equation (7)), but the attack angle is here expressed in modal coordinates,

$$\begin{eqnarray} &&\alpha =\frac{\sum _{r=1}^{\infty }{\phi }_{r}(L){\eta }_{r}(t)}{U}+\sum _{r=1}^{\infty }{\phi }_{r}^{\prime}(L){\eta }_{r}(t). \end{eqnarray} \tag{ 13 }$$

Subsequently, introducing equations (12) and (13) into equation (11) yields the coupled governing equations in modal coordinates,

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl {\ddot{\eta }}_{r}(t)+2{\zeta }_{r}{\omega }_{r}{\dot {\eta }}_{r}(t)+{\omega }_{r}^{2}{\eta }_{r}(t)+{\chi }_{r}V(t)\\ \displaystyle ={\phi }_{r}(L)\times \frac{1}{2}{\rho }_{\mathrm{a}}h{l}_{\mathrm{tip}}{U}^{2}\\ \displaystyle \times ~\mathop{\sum }\limits _{i=1,2,\ldots }{A}_{i}\left [\frac{\sum _{r=1}^{\infty }{\phi }_{r}(L){\dot {\eta }}_{r}(t)}{U}+\sum _{r=1}^{\infty }\phi _{r}^{\prime}(L){\eta }_{r}(t)\right ]^{i}\\ \displaystyle \fl \frac{V(t)}{{R}_{L}}+{C}_{\mathrm{p}}\dot {V}(t)-\sum _{r=1}^{\infty }{\chi }_{r}{\dot {\eta }}_{r}(t)=0 \end{array} \end{eqnarray} \tag{ 14 }$$

where ζ_r is the modal mechanical damping; ω_r is the undamped natural frequency of the rth vibration mode; and χ_r is the modal electromechanical coupling term written as ${\chi }_{r}=\theta [{\phi }_{r}^{\prime}({x}_{2})-{\phi }_{r}^{\prime}({x}_{1})]$. Before proceeding to solve equation (14), a short explanation of how to evaluate ζ_r is presented here. As mentioned in assumption (d), the mechanical damping which includes the internal strain rate damping and external air damping is treated with the proportional damping (Rayleigh damping) assumption. For a uniform piezoelectric beam of which the flexural rigidity YI and mass per unit length m are both uniformly distributed [5, 46, 47], ζ_r can be simply expressed as ζ_r = c_sIω_r/2YI + c_a/2mω_r. Once the damping ratios for two separate vibration modes are known, the two unknown constant damping coefficients c_s and c_a can be mathematically calculated. Yet for a non-uniform piezoelectric beam which is partially covered by piezoelectric sheets as in our case, c_s and c_a are no longer constant along the beam length. Whereas we can still assume that c_sI(x) and c_a(x) are stiffness proportional and mass proportional, respectively, and obtain two constant values of (c_sI/YI)_ave and (c_a/m)_ave. However, to avoid the calculation of these coefficients as well as the inaccuracy caused by the proportional damping assumption, we can obtain ζ_r experimentally using the frequency response function (FRF) obtained in base-vibration condition [48] and proceed to numerical solution of equation (14) for the mechanical and electric responses of the GPEH.

Assume the first nth modes are considered. Again, we introduce the state vector,

$$\begin{eqnarray} &&\fl \boldsymbol{X}=\left \{ \begin{array}{@{}c@{}} \displaystyle {X}_{1}\\ \displaystyle {X}_{2}\\ \displaystyle \vdots \\ \displaystyle {X}_{2 r-1}\\ \displaystyle {X}_{2 r}\\ \displaystyle \vdots \\ \displaystyle {X}_{2 n+1} \end{array}\right \} =\left \{ \begin{array}{@{}c@{}} \displaystyle {\eta }_{1}(t)\\ \displaystyle {\dot {\eta }}_{1}(t)\\ \displaystyle \vdots \\ \displaystyle {\eta }_{r}(t)\\ \displaystyle {\dot {\eta }}_{r}(t)\\ \displaystyle \vdots \\ \displaystyle V(t) \end{array}\right \} \tqs (r=1,2,\ldots ,n). \end{eqnarray} \tag{ 15 }$$

The governing equations can then be written in the state space form as

$$\begin{eqnarray} &&\fl \dot {\boldsymbol{X}}=\left \{ \begin{array}{@{}c@{}} \displaystyle {\dot {\eta }}_{1}(t)\\ \displaystyle {\ddot{\eta }}_{1}(t)\\ \displaystyle \vdots \\ \displaystyle {\dot {\eta }}_{r}(t)\\ \displaystyle {\ddot{\eta }}_{r}(t)\\ \displaystyle \vdots \\ \displaystyle \dot {V}(t) \end{array}\right \} \nonumber\\ &&\mbox { $ \fl =\left \{ \begin{array}{@{}c@{}} \displaystyle {X}_{2}\\ \displaystyle -2{\zeta }_{1}{\omega }_{1}{X}_{2}-{\omega }_{1}^{2}{X}_{1}-{\chi }_{1}{X}_{2 n+1}+{\phi }_{1}(L) \\\qquad \displaystyle \times \ \frac{{\rho }_{\mathrm{a}}h{l}_{\mathrm{tip}}{U}^{2}}{2}\left [\mathop{\sum }\limits _{i=1,2,\ldots }{A}_{i}\left (\frac{{\phi }_{1}(L){X}_{2}}{U}+\phi _{1}^{\prime}(L){X}_{1}\right )^{i}\right ]\\ \displaystyle \vdots \\ \displaystyle {X}_{2 r}\\ \displaystyle -2{\zeta }_{r}{\omega }_{r}{X}_{2 r}-{\omega }_{r}^{2}{X}_{2 r-1}-{\chi }_{r}{X}_{2 n+1}+{\phi }_{r}(L) \\\qquad \displaystyle \times \ \frac{{\rho }_{\mathrm{a}}h{l}_{\mathrm{tip}}{U}^{2}}{2}\left [\mathop{\sum }\limits _{i=1,2,\ldots }{A}_{i}\left (\frac{{\phi }_{r}(L){X}_{2 r}}{U}\phi _{r}^{\prime}(L){X}_{2 r-1}\right )^{i}\right ]\\ \displaystyle \vdots \\ \displaystyle \frac{\sum _{r=1}^{n}{\chi }_{r}{X}_{2 r}}{{C}_{\mathrm{P}}}-\frac{{X}_{2 n+1}}{{R}_{L}{C}_{\mathrm{P}}} \end{array}\right \} .$ } \nonumber\\ \end{eqnarray} \tag{ 16 }$$

Like the SDOF model, equation (16) can be solved in MATLAB using ode45. Two cases with different numbers of modes employed will be considered in section 4. Firstly, we only consider the fundamental mode by letting n equal 1 in equation (16), since a GPEH is oscillating near its fundamental frequency [37, 38]. Secondly, in order to get a more accurate expression for the galloping aerodynamic force, we take into account the first three modes (n = 3) to investigate how the higher modes influence the overall responses of a GPEH.

A prototype device is fabricated and tested in the wind tunnel following the same experimental setup procedure as the preceding study [36], and the measured results are presented to validate the aforementioned SDOF and distributed parameter models. The effectiveness of the models is compared based on the agreement between their predictions and experiments. The cantilever beam is the same as the one in [36], which is composed of an aluminum substrate with two piezoelectric sheets (DuraAct P-876.A12 from Physik Instrumente) bonded to each side of the root, being connected in parallel with the total capacitance C_p = 180 nF. A metal support is employed to fix the root of the cantilever in the wind tunnel. A bluff body is attached to the free end. According to the preceding experimental study on the cross-section geometry of the bluff body [36], a square section is employed, for its better performance than other geometries in the laminar flow, with the lowest cut-in wind speed and the largest output power. Three Configurations are tested in the wind tunnel with different bluff body properties for adequate model comparisons. The properties of the beam and piezoelectric sheets are listed in table 1, while those of the bluff body are shown in table 2. The prototype and the installation in the wind tunnel are shown in figure 4(a). Prior to the wind tunnel test, some parameters such as ζ,ω_nsc and ω_noc need to be measured under base-excitation. The damping ratio ζ for the SDOF model is the same as ζ₁ in the distributed parameter model. For Configuration 1, ζ₁ = 0.005 [36], ζ₂ = 0.015 and ζ₃ = 0.034; for Configuration 2, ζ₁ = 0.008,ζ₂ = 0.019 and ζ₃ = 0.041; for Configuration 3, ζ₁ = 0.004,ζ₂ = 0.012 and ζ₃ = 0.031. During the wind tunnel test, the wind speed is measured by a Pitot tube anemometer, and the voltage across the load resistance is measured by the NI 9229 data acquisition module. The overall experimental setup is presented in figure 4(b).

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The measured and predicted values of average power P_ave versus load resistance R_L are first compared in figure 5(a). The data presented are obtained for Configuration 1 at three different wind speeds. As shown in the picture, for U = 4, 5 and 6 m s⁻¹, P_ave initially increases with R_L until it reaches an optimum value, then decreases when R_L continues growing. The optimum R_L barely changes for the three wind speeds, which are all around 105–120 kΩ. All three models can capture this trend well, yet with some discrepancies. Also, both the single mode distributed parameter model (shortened to Distri. (mode 1) here and hereafter) and multiple mode distributed parameter model (shortened to Distri.(mode 1–3) here and hereafter) predict a shallow valley around the optimum R_L for U = 4 m s⁻¹, which is an important aspect for the influence of R_L on P_ave and will be addressed in section 5. Moreover, no obvious difference is observed between the predictions of Distri. (mode 1) and Distri. (mode 1–3). Figure 5(b) shows the measured and predicted P_ave versus wind speed U for the three different GPEH configurations at R_L = 105 kΩ. It is noted that P_ave increases monotonically with U. Again, all three models can predict consistent results with the experiments. As for the cut-in wind speed U_cr, the SDOF model predicts better results for all three configurations, while as for the harvested power level, no obvious superiority is observed for one model over the others since small discrepancies exist for all three models. The main source of the error is probably the unavoidable small turbulence in the wind tunnel which disturbs the bluff body, to start oscillation before the wind speed reaches the real cut-in speed. In general, the three models can successfully predict the electroaeroelastic behavior with good agreement with the experiments. The SDOF model is advantageous for its simplicity and ease of obtaining the coupling coefficient for a fabricated prototype, while the distributed models have their merits in the better representation of the aerodynamic force and ease of parametric study. In the following study about the effects of different parameters on the output power, Distri. (mode 1) is employed to obtain the simulation results.

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Figure 6(a) shows the variation of the average output power P_ave with the wind speed U for different load resistances R_L. Here, Configuration 1 is considered and its properties are shown in tables 1 and 2. It can be seen that the growth rate of P_ave is greatly affected by R_L. The growth rate of P_ave firstly increases with R_L until 105 kΩ, then decreases when R_L is further increasing. An enlarged view is provided to show this trend more clearly around the threshold of galloping. It is noted that the value of R_L which owns the largest growth rate of P_ave also gives the largest cut-in wind speed U_cr. The variation of U_cr with R_L is presented in figure 6(b). As can be seen from this curve, U_cr increases with R_L up to the maximum value then decreases with R_L. As for the harvested power level, the largest power is obtained with R_L around 105 kΩ for U > 4 m s⁻¹. Features of the optimum R_L can be seen from figure 6(c) which displays the variation of P_ave with R_L at different wind speeds. For U = 2 m s⁻¹, P_ave is zero between 20 and 800 kΩ since this speed is lower than the respective U_cr for these R_L values (figure 6(b)). When U is slightly larger than U_cr (i.e., U = 3 and 4 m s⁻¹), a shallow valley exists around 105 kΩ. When U continues growing (i.e., U = 5–8 m s⁻¹), there is a peak for P_ave between 100 and 120 kΩ, where lies the optimum R_L = 105 kΩ. It is worth mentioning that R_L for the valley and the peak of P_ave overlap with each other, corresponding to that with the largest U_cr. From here onwards, we regard R_L as the optimum one if the largest U_cr and power growth rate are achieved in the response of P_ave versus U (i.e., the curve for R_L = 105 kΩ in figure 6(a)). At the optimum R_L, either a valley or a peak appears in the response of P_ave versus R_L at a specific U.

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Figure 7 shows the effects of the wind exposure area of the bluff body S_tip (h × l_tip) on the electroaeroelastic behavior of the GPEH. The harvester properties are the same as those listed in tables 1 and 2 (Configuration 1) except the length of the bluff body l_tip. Figure 7(a) is obtained with U = 6 m s⁻¹. It is determined that the optimum R_L for the various S_tip are equal to each other, which is reasonable since the natural frequencies are the same with equal M_tip. The variation of P_ave with U at the optimum R_L for different S_tip is shown in figure 7(b). With increasing S_tip, the cut-in wind speed decreases while the harvested power increases, both beneficial for harvesting the wind power. This can be expected since the increasing wind exposure area results in an increase in the aerodynamic force. Moreover, the growth rate of power is not much affected by S_tip.

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Figure 8 shows the effects of the mass of the bluff body M_tip on the electroaeroelastic behavior of the GPEH. The harvester properties are the same as those listed in tables 1 and 2 (Configuration 1) except M_tip. The most obvious change due to varying M_tip is in the fundamental frequency of the harvester as indicated in the legend. Figure 8(a) shows the variation of P_ave with R_L for U = 6 m s⁻¹, from which the respective optimum R_L can be obtained for each M_tip. It is noted that the optimum R_L varies with different M_tip (i.e., different fundamental frequency ω₁). With the respective optimum R_L, the variation of P_ave with U is displayed in figure 8(b). As can be seen from this figure, the cut-in wind speed increases and the harvested power decreases with the increasing M_tip (or the decreasing of ω₁). Moreover, the growth rate of the harvested power also decreases with the increasing M_tip. It can summarized that for the GPEH with a specific piezoelectric cantilever, reducing M_tip can achieve a better performance in extracting power from the wind.

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Figure 9 shows the effects of the lengths of the piezoelectric sheets L_p on the electroaeroelastic behavior of the GPEH. The harvester properties are the same as those listed in tables 1 and 2 (Configuration 1) except L_p and C_p, since the latter one varies linearly with the former one when other parameters are kept unchanged. The respective optimum R_L is determined from figure 9(a), which varies with L_p because of the change in the total stiffness of the composite beam. Figure 9(b) shows the variation of P_ave with U at the respective optimum R_L. The cut-in wind speed as well as the growth rate of the harvested power increases with the increasing L_p. The largest L_p can extract the highest power at relatively high wind speeds (>12 m s⁻¹). A useful factor to evaluate the performance of the GPEH is the power density which is calculated as power per piezoelectric volume in this paper. Figure 9(c) shows the power density versus the wind speed for different L_p. It is noted that the power density is not monotonically increasing with L_p. At high wind speeds the highest power density is obtained by a medium value of L_p, which is 100 mm among the considered discrete values. A maximum power density of 12.96 mW cm⁻³ is achieved at 14 m s⁻¹ with L_p = 100 mm. A comparison of the power density with other piezoelectric wind energy harvesters is shown in figure 10. As can be seen from this figure, the present GPEH shows good performance in power density, especially at relatively high wind speeds. The present GPEH is less advantageous at lower wind speeds when compared to the harvesters in [15, 37]. However, the present GPEH has the largest overall range of wind speed for effective power generation with considerable power density. Note that the data for the present GPEH in the figure are not optimized in the aspects of S_tip,M_tip, etc.

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