Temperature impact on the performance of galloping-based piezoaeroelastic energy harvesters

The energy harvester shown in figure 1 consists of a piezoelectric bimorph cantilever beam with a prismatic structure attached to its free end. This structure undergoes galloping in the transverse direction when subjected to an incoming flow. The cantilever beam is composed of aluminum and two piezoelectric layers (PZT-5H). The piezoelectric sheets are bonded by two in-plane electrodes of negligible thickness connected in parallel with opposite polarity to a load resistance R. The geometric and material properties of the harvester are presented in table 1.

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The Euler–Bernoulli beam assumptions are used to model the multilayered cantilever beam. Based on these assumptions, the transverse vibration, v(x;t), of the cantilever beam is described by

$$\begin{eqnarray} &&\fl \frac{{\partial }^{2}M(x,t)}{\partial {x}^{2}}+{c}_{\mathrm{a}}\frac{\partial v(x,t)}{\partial t}+m\frac{{\partial }^{2}v(x,t)}{\partial {t}^{2}}\nonumber\\ &&=~{F}_{\mathrm{tip}}\delta (x-L)-{M}_{\mathrm{tip}}\frac{\mathrm{d}\delta (x-L)}{\mathrm{d}x} \end{eqnarray} \tag{ 1 }$$

where δ(x) is the Dirac delta function, F_tip and M_tip are, respectively, the galloping aerodynamic force and moment at the tip of the beam that are caused by the oscillation of the structure, L is the length of the beam, c_a is the viscous air damping coefficient, m is the mass of the beam per unit length, and M(x;t) is the internal moment. This moment is composed of three components. The first is the resistance to bending and is given by $E I\frac{{\partial }^{2}v(x,t)}{\partial {x}^{2}}$. The second is due to the strain rate damping effect and is represented by ${c}_{\mathrm{s}}I\frac{{\partial }^{3}v(x,t)}{\partial {x}^{2}\partial t}$. The third component is the contribution of the piezoelectric sheets which are connected in parallel. This contribution is represented by (H(x) − H(x − L))ϑ_pV(t) where H(x) is the Heaviside step function, V(t) is the generated voltage, and ϑ_p is the piezoelectric coupling term [26]. This term is given by

$$\begin{eqnarray} &&{\vartheta }_{\mathrm{p}}=-{e}_{3 1}{b}_{2}({h}_{\mathrm{p}}+{h}_{\mathrm{s}}) \end{eqnarray} \tag{ 2 }$$

where e₃₁ is the piezoelectric stress coefficient, b₂ is the width of the piezoelectric layer and h_s and h_p are the thicknesses of the aluminum and piezoelectric layers, respectively.

Substituting for the moment M(x;t) its three components in equation (1), the equation of motion of the electromechanical system is rewritten as

$$\begin{eqnarray} &&\fl Y I\frac{{\partial }^{4}v(x,t)}{\partial {x}^{4}}+{c}_{\mathrm{s}}I\frac{{\partial }^{5}v(x,t)}{\partial {x}^{4}\partial t}+{c}_{\mathrm{a}}\frac{\partial v(x,t)}{\partial t}\nonumber\\ &&+~m\frac{{\partial }^{2}v(x,t)}{\partial {t}^{2}}+\left (\frac{\mathrm{d}\delta (x)}{\mathrm{d}x}-\frac{\mathrm{d}\delta (x-L)}{\mathrm{d}x}\right ){\vartheta }_{\mathrm{p}}V(t)\nonumber\\ &&=~{F}_{\mathrm{tip}}\delta (x-L)-{M}_{\mathrm{tip}}\frac{\mathrm{d}\delta (x-L)}{\mathrm{d}x}. \end{eqnarray} \tag{ 3 }$$

In this equation, the stiffness EI and mass of the beam per unit length m are given by $E I=\frac{1}{1 2}{b}_{1}{E}_{\mathrm{s}}{h}_{\mathrm{s}}^{3}+\frac{2}{3}{b}_{2}{E}_{\mathrm{p}}[({h}_{\mathrm{p}}+\frac{{h}_{\mathrm{s}}}{2})^{3}-\frac{{h}_{\mathrm{s}}^{3}}{8}]$ and m = b₁ρ_sh_s + 2b₂ρ_ph_p where E_s and E_p are the Young's moduli of the aluminum and piezoelectric layers, respectively, and ρ_s and ρ_p are the respective densities of these layers.

To complete the problem formulation, we relate the mechanical and electrical variables by using the Gauss law [27]

$$\begin{eqnarray} &&\frac{\mathrm{d}}{\mathrm{d}t}\int \mathbf{D}\boldsymbol{ \cdot }\mathbf{n}\hspace{0.167em} \mathrm{d}A=\frac{\mathrm{d}}{\mathrm{d}t}\int {D}_{2}\hspace{0.167em} \mathrm{d}A=\frac{V}{R} \end{eqnarray} \tag{ 4 }$$

where D is the electric displacement vector and n is the normal vector to the plane of the beam. The electric displacement component D₂ is given by the following relation [26]:

$$\begin{eqnarray} &&{D}_{2}={e}_{3 1}{\varepsilon }_{1 1}(x,t)+{\varepsilon }_{3 3}^{s}{E}_{2} \end{eqnarray} \tag{ 5 }$$

where ε₁₁ is the axial strain component in the aluminum and piezoelectric layers and is given by ${\varepsilon }_{1 1}(x,y,t)=-y\frac{{\partial }^{2}v(x,t)}{{\partial }^{2}{x}^{2}},{\varepsilon }_{3 3}^{s}$ is the permittivity component at constant strain. Substituting (5) into (4), we obtain the equation governing the strain–voltage relation,

$$\begin{eqnarray} &&\fl \!-{e}_{3 1}({h}_{\mathrm{p}}+{h}_{\mathrm{s}}){b}_{2}\!\int \nolimits _{0}^{L}\!\frac{{\partial }^{3}v(x,t)}{\partial t \partial {x}^{2}}\hspace{0.167em} \mathrm{d}x-\frac{2{\varepsilon }_{3 3}^{s}{b}_{2}L}{{h}_{\mathrm{p}}}\frac{\mathrm{d}V(t)}{\mathrm{d}t}=\frac{V(t)}{R}.\nonumber\\ \end{eqnarray} \tag{ 6 }$$

To perform the linear and nonlinear analyses, we first discretize the system using the Galerkin procedure which requires the exact mode shapes of the structure. To determine the mode shapes of the considered system, we drop the damping, forcing, and polarization from equation (3), and let v(x,t) = ϕ(x)e^iωt. The resulting eigenvalue problem is given by

$$\begin{eqnarray} &&E I{\phi }^{i v}-m{\omega }^{2}\phi =0 \end{eqnarray} \tag{ 7 }$$

with the following boundary conditions:

$$\begin{eqnarray} &&\phi (0)=0\tqs {\phi }^{\prime}(0)=0 \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} &&E I{\phi }^{\prime\prime}(L)-{\omega }^{2}{M}_{\mathrm{ t}}{L}_{c}\phi (L)-{\omega }^{2}{I}_{\mathrm{ t}}{\phi }^{\prime}(L)=0 \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} &&E I{\phi }^{\prime\prime\prime}(L)+{\omega }^{2}{M}_{\mathrm{ t}}{L}_{c}{\phi }^{\prime}(L)+{\omega }^{2}{M}_{\mathrm{ t}}\phi (L)=0 \end{eqnarray} \tag{ 10 }$$

where I_t is the rotary inertia of the tip structure M_t and L_c is half of the length of the tip mass. The general expression of the mode shapes is written as

$$\begin{eqnarray} &&\phi (x)=A\sin \beta x+B\cos \beta x+C\sinh \beta x+D\cosh \beta x \end{eqnarray} \tag{ 11 }$$

where β and ω are related by $\omega ={\beta }^{2}\sqrt{\frac{E I}{m}}$.

To obtain the relation between the different coefficients in (11), we normalize the eigenfunctions and use the following orthogonality conditions:

$$\begin{eqnarray} &&\fl \int \nolimits _{0}^{L}{\phi }_{s}(x)m{\phi }_{r}(x)\hspace{0.167em} \mathrm{d}x+{\phi }_{s}(L){M}_{\mathrm{t}}{\phi }_{r}(L)+{\phi }_{{s}^{\prime}}(L){I}_{\mathrm{t}}{\phi }_{r}^{\prime}(L)\nonumber\\ &&+~{\phi }_{s}(L){M}_{\mathrm{t}}{I}_{\mathrm{t}}{\phi }_{r}^{\prime}(L)+{\phi }_{r}^{\prime}(L){I}_{\mathrm{ t}}{\phi }_{s}^{\prime}(L)={\delta }_{r s} \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} &&\fl \int \nolimits _{0}^{L}{\phi }_{s}^{\prime\prime}(x)E I{\phi }_{r}^{\prime\prime}(x)\hspace{0.167em} \mathrm{d}x={\delta }_{r s}{\omega }_{r}^{2} \end{eqnarray} \tag{ 13 }$$

where s and r are used to represent the vibration modes and δ_rs is the Kronecker delta, defined as unity when s is equal to r and zero otherwise.

Using the Galerkin discretization, we express the transverse displacement, v(x,t), in the following form:

$$\begin{eqnarray} &&v(x,t)=\sum _{i=1}^{\infty }{\phi }_{i}(x){q}_{i}(t) \end{eqnarray} \tag{ 14 }$$

where q_i(t) are the modal coordinates and ϕ_i(x) are the mode shapes of the cantilever beam with the prismatic tip structure. Substituting equation (14) into equations (3) and (6) and considering the first mode only, we obtain the following coupled equations of motions:

$$\begin{eqnarray} &&\ddot{q}(t)+2 \xi \omega \dot {q}(t)+{\omega }^{2}q(t)+{\theta }_{\mathrm{ p}}V(t)=f(t) \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} &&\frac{V(t)}{R}+{C}_{\mathrm{p}}\dot {V}(t)-{\theta }_{\mathrm{p}}\dot {q}(t)=0 \end{eqnarray} \tag{ 16 }$$

where ξ is the mechanical damping coefficient, f(t) is the galloping force of the first mode which is expressed as f(t) = ϕ(L)F_tip + ϕ'(L)M_tip,ω is the fundamental natural frequency of the structure, and the coefficients θ_p and C_p are the piezoelectric coupling term and the capacitance of the harvester which are given by θ_p = ϕ'(L)ϑ_p and ${C}_{\mathrm{p}}=\frac{2{\varepsilon }_{3 3}^{s}{b}_{2}L}{{h}_{\mathrm{p}}}$.

Inspecting the governing equations of the harvester (equations (3) and (6)), we note that these equations depend on the properties of the piezoelectric material (E_p,e₃₁, and ${\varepsilon }_{3 3}^{s}$). It has been demonstrated in the literature [10] that the temperature can significantly affect the properties of the piezoelectric material. The plotted curves in figure 2 show the effects of the temperature on the piezoelectric and dielectric constants and Young's modulus of the PZT-5H. The red points represent the values measured by Hooker [10] and the solid lines are fitted polynomials to be considered in our subsequent analysis. Figure 2(a) shows that the temperature affects the piezoelectric constant d₃₁ = e₃₁/E_p significantly. In fact, an increase of about 200% in the piezoelectric constant is obtained when the temperature is varied from −70 to 120 °C. For the dielectric constant, ${\varepsilon }_{3 3}^{T}/{\varepsilon }_{0}$, where ε₀ is the permittivity of free space and ${\varepsilon }_{3 3}^{T}={\varepsilon }_{3 3}^{s}+{d}_{3 1}^{2}{E}_{\mathrm{p}}$ is the permittivity component at constant stress, an increase in the operating temperature from −70 to 120 °C causes a significant increase of about 300%, as shown in figure 2(b). The variation of the PZT-5H Young's modulus as a function of the temperature is plotted in figure 2(c). A decrease of about 30% is obtained when the temperature is increased from −20 to 50 °C. Clearly, the properties of the PZT-5H are strong functions of the operating temperature.

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Noting that the structural natural frequency of the harvester, ω, depends on the Young's modulus through the relation $\omega ={\beta }^{2}\sqrt{\frac{E I}{m}}$, where $E I=\frac{1}{1 2}{b}_{1}{E}_{\mathrm{s}}{h}_{\mathrm{s}}^{3}+\frac{2}{3}{b}_{2}{E}_{\mathrm{p}}[({h}_{\mathrm{p}}+\frac{{h}_{\mathrm{s}}}{2})^{3}-\frac{{h}_{\mathrm{s}}^{3}}{8}]$, and that varying the temperatures causes variations in the value of E_p, it is clear that the structural natural frequency depends on the temperature. Figure 3(a) shows the variation of the structural natural frequency as a function of the temperature. The plots show that increasing the temperature from −20 to 50 °C causes the natural frequency to decrease by about 15%. This is expected because the Young's modulus, E_p, is reduced when the temperature is increased, as shown in figure 2(c). The variations of the piezoelectric constant and dielectric constant, as shown in figures 2(a) and (b), result in variations in the capacitance, C_p, and piezoelectric coupling, θ_p. In fact, the capacitance is directly related to ${\varepsilon }_{3 3}^{s}$ by the relation ${C}_{\mathrm{p}}=\frac{2{\varepsilon }_{3 3}^{s}{b}_{2}L}{{h}_{\mathrm{p}}}$ and the piezoelectric coupling coefficient is expressed as a function of e₃₁ by θ_p =− e₃₁b₂(h_p + h_s)ϕ'(L). The variations of the capacitance and piezoelectric coupling coefficients as functions of the temperature are plotted in figures 3(b) and (c), respectively. The plot in figure 3(b) shows that increasing the temperature from −20 to 50 °C causes the capacitance of the piezoelectric material to increase by about 78%. This is due to the increase in the permittivity at constant strain ${\varepsilon }_{3 3}^{s}$. On the other hand, there is an optimum value of the temperature at which the piezoelectric coupling is maximized, as shown in figure 3(c). This coupling increases when the temperature is increased from −20 °C to almost 0 °C and then decreases as the temperature is raised from 0 to 50 °C. Because of the significant impact of the temperature on the piezoelectric coupling, the capacitance of the piezoelectric material, and the natural frequency of the harvester, one would expect the harvester's response to be strongly dependent on the operating temperature.

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The effects of the temperature and electrical load resistance on the onset speed of galloping are determined from a linear analysis. The matrix B includes all parameters that affect the linear part of the system. In this matrix, three variables are dependent on the temperature. These are the structural natural frequency, ω, the capacitance of the piezoelectric material, C_p, and the piezoelectric coupling, θ_p. B(U) has a set of three eigenvalues λ_i, i = 1,2,3, which vary with the temperature. The first two eigenvalues are similar to those of a pure galloping problem in the absence of the piezoelectricity effect. The third eigenvalue, λ₃, is a result of the electromechanical coupling and is always real and negative, as in the case of piezoelectric systems subjected to base or aeroelastic excitations [33–35, 17]. The first two eigenvalues are complex conjugates (${\lambda }_{2}=\overline{{\lambda }_{1}}$). The real part of these eigenvalues represents the electromechanical damping coefficient and the positive imaginary part corresponds to the global frequency of the coupled system. Because λ₃ is always real and negative, the stability of the trivial solution depends only on the real part of the first two eigenvalues. The solution of the linear part is asymptotically stable if the real part of λ₁ is negative. On the other hand, if the real part of λ₁ is positive, the solution of the linearized system is unstable. The speed, U_g, for which the real part of λ₁ is zero corresponds to the onset of instability or galloping. Because Real (dλ₁/dU) is nonzero at U_g, the instability or bifurcation is a Hopf bifurcation.

Figures 4(a) and (b) show the variation of the onset speed of galloping with the electrical load resistance for the two considered cross-section geometries and for different values of the temperature when using the physical and geometric properties described in table 1. The plots show that the load resistance affects the onset speed of galloping significantly for both considered configurations and for different temperature values. In the lower range (R < 10³ Ω) and for different values of the temperature, the rate of variation is negligible. This rate increases significantly over a middle range of resistance values and yields a peak for the onset speed when the electrical load resistance is between 10⁴ and 10⁵ Ω for all considered temperature values. At higher load resistances (R > 10⁷ Ω), the onset speed of galloping decreases again to values that are close to those obtained in the low-resistance range for all considered temperatures. The minimum values for the onset speed of galloping in the low- and high-resistance ranges correspond to the minimum values of the coupled electromechanical damping over the same range of load resistances. Furthermore, the plots show that the temperature affects the onset speed of galloping strongly in the middle range of the resistance values (i.e. between 10⁴ and 10⁵ Ω). As the temperature is increased, the onset speed of galloping is decreased. This is expected because this value is dependent on the structural natural frequency, the capacitance of the piezoelectric material, and the piezoelectric coupling; these vary with the temperature and their variations affect the coupled electromechanical damping. The plots also show that the speed at the onset of instability strongly depends on the geometry of the cross-section. Indeed, the isosceles triangle with δ = 30° has a lower speed for any load resistance when compared to the isosceles triangle with the δ = 53° cross-section. On inspecting the relation between the onset speed of galloping and the linear system parameters, it is noted that the larger the value of the linear coefficient a₁ is, the smaller the onset speed is.

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To further investigate the effects of the temperature on the onset speed of galloping, we plot in figures 5(a) and (b) the variation of the onset speed of galloping as a function of the temperature for different values of the electrical load resistance and for the two cross-section geometries. For R = 10³ and 10⁷ Ω, the impact of the temperature on the onset speed of galloping is very negligible. For R = 10⁴ and 10⁶ Ω, varying the temperature affects the onset speed of galloping significantly. In fact, at higher temperatures, the onset speed of galloping decreases when the temperature is increased. For R = 10⁵ Ω, the onset speed of galloping changes significantly when the temperature is varied. At low temperature values, the onset speed is very high, with a maximum value near T ≈− 15 °C. At higher temperature values, it decreases drastically. This significant variation in the onset speed of galloping when varying the temperature is due to the dependence of the piezoelectric properties on the temperature and their effects on the coupled electromechanical damping. As we note the same tendency for the two cross-section geometries considered, we will consider only the isosceles triangle with δ = 30° in the rest of this work.

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To investigate the impact of the temperature, electrical load resistance, and wind speed on the performance of the harvester, we perform a full nonlinear analysis of the governing equations. The bifurcation diagrams of the tip displacement of the bluff body for different temperatures and when the electrical load resistance is set equal to 10³ Ω,10⁴ Ω,10⁵ Ω, and 10⁶ Ω are shown in figures 6(a)–(d), respectively. On inspecting these curves, we find that the load resistance affects the onset speed of galloping strongly, as demonstrated by the linear analysis performed in section 4.1. The onset speed of galloping is always lower at higher temperature values for all considered load resistances. The impact of the temperature on the onset speed is very negligible for R = 10³ Ω, as shown in figure 6(a). For R = 10⁵ Ω, the onset speed changes significantly when the temperature value is varied, as shown in figure 6(c). The displacement values are zero for T =− 20 and 0 °C for wind speeds less than 20 m s⁻¹. This is expected because the onset speeds of galloping for these two temperatures are larger than 20 m s⁻¹ for a load resistance of 10⁵ Ω, as shown in figure 5(a). Additionally, the maximum displacement values are observed when T = 40 °C for all considered load resistances. This is explained by the low values of the electromechanical damping at this temperature.

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Figure 7 shows the impact of the temperature on the voltage output for different values of the load resistance when varying the wind speed. For R = 10³ Ω, a higher value of the generated voltage is obtained when the temperature is 0 °C, as shown in figure 7(a). Furthermore, there is an intersection point between the different curves of the generated voltage when varying the temperature for R = 10³,10⁴, and 10⁶ Ω, as shown in figures 7(a), (b), and (d). The wind speed corresponding to this intersection point is closer to the onset speed of galloping when the impact of the temperature on the onset speed is less pronounced. For R = 10³ Ω, the intersection point is very close to the onset of instability. For R = 10⁴ Ω, the intersection point is significantly higher than the onset speed. Beyond this intersection point, the generated voltage, for the cases where the onset speed of galloping is relatively larger, becomes more important. This result is very clear in figures 7(a) and (d).

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The plotted curves in figure 8 show the bifurcation diagrams of the harvested power for different temperatures and load resistances. The lowest values of the harvested power are obtained at T = 40°C for R = 10³ and 10⁶ Ω, as shown in figures 8(a) and (d), respectively. It is important to note that these minimum values in the harvested power are associated with maximum values of the tip displacement, as shown in figures 6(a) and (d), The highest values of harvested power are obtained in cases where the onset speeds of galloping are higher. These maximum values of the harvested power are associated with minimum values of the tip displacement, as shown in figures 6(a) and (d), for R = 10⁴ and 10⁵ Ω. All of these results show the significant impact of the temperature on the response of the harvester.

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To further elucidate the effects of the temperature on the system's outputs, we plot in figures 9 and 10 the variation of the tip displacement, generated voltage, and harvested power as functions of the temperature for different load resistances and for two distinct values of the wind speed, namely 5 and 15 m s⁻¹. For U = 5 m s⁻¹, figures 9(a)–(c) show that no power can be generated for electrical load resistance values of 10⁴ or 10⁵ Ω. For R = 10³ Ω and U = 5 m s⁻¹, increase of the temperature is accompanied by a slight increase in the tip displacement, as shown in figure 9(a), and a slight decrease in the harvested power, as shown in figure 9(c). For R = 10⁶ Ω and U = 5 m s⁻¹, the tip displacement increases as the temperature is increased. This is explained by the low electromechanical damping terms at these temperatures. Figure 9(c) shows that increasing the temperature also causes an increase in the harvested power level.

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At U = 15 m s⁻¹, figure 10(a) shows that increase of the temperature is accompanied by an increase in the displacement for the considered load resistances. This is due to the low electromechanical damping at the high temperature values. On the other hand, the generated voltage and harvested power show different tendencies when using different load resistances. Because the intersection wind speed values when the load resistances are set equal to 10³ and 10⁶ Ω are smaller than this wind speed (15 m s⁻¹), the variations of the generated voltage and harvested power with the temperature, as shown in figures 10(b) and (c), have similar tendencies to the variation of the onset speed of galloping with the temperature. These results show that maximum levels of harvested power are accompanied by minimum values of tip displacement when varying the temperature, and vice versa. For a load resistance of 10⁴ Ω, there is a minimum value in the tip displacement when the temperature is between −5 and 0 °C, as shown in figure 10(a). This range of temperature corresponds to a higher onset speed of galloping, as shown in figure 5(a). Because the intersection wind speed value is larger than the considered speed, the curves of the tip displacement, generated voltage, and harvested power have the same tendency when varying the temperature, as shown in figures 10(a)–(c). For R = 10⁵ Ω, the harvester does not produce any power at temperatures lower than 26 °C. This is expected because the onset speed of galloping is larger than 15 m s⁻¹ in this range of temperatures, as shown in figure 5(a). In addition, increase of the temperature is associated with an increase in all of the system's outputs.

To further investigate the effects of the electrical load resistance on the system outputs for different values of the temperature, we present in figures 11, 12 and 13 the variation of the tip displacement, generated voltage, and harvested power, respectively, with the electrical load resistance for four values of the temperature and two distinct wind speeds, namely U = 5 m s⁻¹ and U = 15 m s⁻¹. On inspecting figures 11(a), 12(a), and 13(a), it is noted that for U = 5 m s⁻¹ there is a range of load resistance where no oscillations (zero-tip displacement) are observed. This range of load resistance depends on the temperature. This is expected because the onset speed of galloping in this region of load resistance is larger than U = 5 m s⁻¹. These zero-displacement values can be explained by the maximum value of the electromechanical damping in this region of electrical load resistance. When the wind speed is increased to U = 15 m s⁻¹, as shown in figures 11(b), 12(b), and 13(b), we find that the range of zero-tip displacement is decreased progressively for all considered temperature values. For all considered temperature values, for low (R < 10³ Ω) and high (R > 10⁷ Ω) load resistance values, the variation of the displacement with the electrical load resistance is negligible for both wind speeds, as shown in figures 11(a) and (b).

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Figures 12(a) and (b) show the variation of the generated voltage with the load resistance for different temperature values. It is noted that, for low and high values of the load resistance, increase of the electrical load resistance is associated with an increase in the generated voltage for all considered temperature values and for both considered wind speeds. Moreover, the generated voltage stabilizes for a specific value of the electrical load resistance, which is around 10⁷ Ω when U = 5 m s⁻¹ and 10⁶ Ω when U = 15 m s⁻¹. As mentioned above, for a specific range of load resistance, only at high temperature (T = 40 °C) can the system harvest energy. Furthermore, the level of generated voltage is higher at lower temperatures.

The plots in figure 13 show the variation of the harvested power with the load resistance for different values of the temperature and for two distinct wind speeds. The plots show that there are optimum values of the load resistance for which more energy can be harvested and these optimum values depend on the temperature and the wind speed considered. Similarly to the variation of the tip displacement and generated voltage with the load resistance, the harvested power is almost zero over a specific range of load resistance that is temperature-dependent. Moreover, the region of load resistance over which the harvested power is maximum matches the region of load resistance over which the displacement is minimum, and vice versa, for all considered temperature values and for both wind speeds. These results show that at higher temperature values, energy harvesting can be performed at lower wind speeds. In the low-temperature range, the level of harvested power can be maximized but this will be achieved only at higher wind speeds.