Dynamic Behavior of Galloping Micro Energy Harvester with the Elliptical Bluff Body Using CFD Simulation

Abstract

Energy extraction from flow-induced oscillations based on piezoelectric structures has recently been tackled by several researchers. This paper presents a study of the dynamic behavior analysis and parametric characteristics of a galloping piezoelectric micro energy harvester (GPEH) applied to self-powered micro-electro-mechanical systems (MEMS). The mechanical performance of a piezoelectric micro energy harvester cantilever beam with two layers of elastic silicon and piezoelectric (PZT-5A) attached to a tip elliptical cylinder is numerically simulated. Using size-dependent beam formulation on the basis of the modified couple stress theory and Gauss’ law, the coupled electro-mechanical non-linear governing equations of the energy harvester are obtained. The mode summation and Galerkin methods are used to derive the extracted power from the system. The study also models the flow field effect on the beam oscillations via CFD simulation. The effect of elliptical cylinder mass, damping ratio, beam thickness, and load resistance on the dynamic behavior and harvested power of the system is studied. Findings reveal that increasing the normalized tip mass from 0 to 0.5 and 1 increases the output power density from 0.12 to 0.2 and 0.22, respectively, and the corresponding electrical load resistance of maximum power increases from 175 to 280 kΩ and 375 kΩ, respectively. An approximately linear relation between the elliptical cylinder mass and the load resistance is observed. By increasing/decreasing the cylinder mass, the required electrical load resistance for maximum output power proportionally changes. The damping analysis shows that a higher damping ratio increases the onset velocity of galloping and decreases the extracted power.

1. Introduction

Many microscale devices, like health monitoring sensors [1], pacemakers [2], flow sensors [3], etc., require electrical power. In recent years, common power supply devices, such as micro-fuel cells [4] and micro batteries [5] with limited lifetime and high maintenance requirements, are being replaced by electrical power extracted from the ambient and mechanical vibrations [6,7]. There are many different conversion mechanisms that can be applied to convert these vibration sources into electrical energy, such as electromagnetic induction [8], the piezoelectric effect [9], the electrostatic effect [10], and the triboelectric effect [11]. Among these sources, piezoelectric energy harvesters received the most attention because of their small volume, simple configuration, and high efficiency and power density [12].

Piezoelectric energy harvesters are categorized into vortex-induced vibration (VIV) energy harvesters [13], flutter-based energy harvesters [14], and galloping energy harvesters [15]. VIV energy harvesters extract power through the vibration of deformable structures caused by generated vortices behind the bluff body [16]. When the frequency of vortex shedding approaches the natural frequency, the lock-in phenomenon occurs, and the maximum output power is achieved [17]. In flutter-based energy harvesters, an airfoil is attached to a beam, and coupling occurs between the aerodynamics and the structure. Since the aeroelasticity becomes larger than the damping force, the vibration amplitude increases, and fluttering effect occurs [14]. Galloping energy harvesters consist of a low-frequency and large-amplitude motion, which mainly occur when the environmental wind speed is higher than the cut-in wind speed. The structure oscillates with a large amplitude but instability does not occur [18]. In comparison to the VIV energy harvesters, the galloping-based system obtains energy over a wider fluid speed range with higher power density; therefore, many studies have been conducted regarding these systems at both the micro- and macro-scale [19,20].

Analysis of the bluff body cross section regarding the galloping energy harvester efficiency was performed by Yang et al. [21]. They considered squares, rectangles with distinct aspect ratios, equilateral triangles, and D-section cross sections and show that the square cross section gives the maximum output power. Tan and Yan [22] provide a theoretical design and optimization procedure for galloping-based piezoelectric energy harvesters. They derived the analytical solutions for non-linear distributed parameter models. The electromechanical motion of a galloping wind energy harvester was studied by Sobhanirad and Afsharfard [23]. They used a harmonic balance method to derive the periodic response of the harvester and verified the findings via experiments. Shi et al. [24] proposed a prismatic triangular bluff body connected to a cantilever beam as a velocity sensor based on the galloping effect. Their system works without requiring an additional power supply and is designed for wind velocities ranging from 4.45 to 10 m/s. A novel galloping piezoelectric wind energy harvester with a cuboid bluff body was analyzed by Lai et al. [25]. To transform mechanical vibrational energy into electric power, an integrated vibro-impact dielectric elastomer generator was used in their design. Zhao et al. [26] presented a funnel-shaped galloping energy harvester with a wide operational wind-speed range and high harvesting power. They showed that the funnel-shaped bluff body was utilized to avoid vortex reattachment, which is an enhancement in structural non-stream fluid flow, and allow the pressure direction to be determined along with the lift force. A galloping triboelectric nanogenerator (GTENG) based on contact electrification between two flexible beams at low speed flows was introduced by Zhang et al. [27]. They found that the impact behavior between the main and auxiliary beam occurred when the main beam was subjected to cross flows beyond the critical value. Optimization and mechanical behavior prediction of galloping energy harvesters via base excitation was implemented by Huang et al. [28]. Chen et al. [29] designed a two-degree-of-freedom (2DOF) piezoelectric energy harvester (PEH), which employs VIV and wake galloping mechanisms to improve the wind energy-harvesting performance in a wide wind-speed range. They verified their results via wind tunnel experiments and showed that the proposed 2DOF energy harvester overcame the limitations of conventional one-degree-of-freedom VIVs and wake galloping energy harvesters. They also proposed a 2DOF quasi-zero stiffness non-linear galloping energy harvester for ultra-low wind speeds [30]. Application of a galloping energy harvester in a heat exchanger was investigated by Wang et al. [18]. This work focused on investigating the effects of different ash deposition types on energy-harvesting performance.

Mechanical elements used in microsystems, such as beams and plates, have small thicknesses in the order of microns. Experiments conducted on microscale structures proved that the classical continuum theory is unable to precisely predict the static and dynamic behavior of microscale systems [31,32]. Non-classical theories, such as strain gradient theory [33], couple stress theory [34], and modified couple stress theory [35], have been developed to cover the incapability of the classical continuum theory. Comparing the outputs of the modified couple stress theory to experimental findings indicates that this theory can successfully take the size-dependent behavior of microscale systems into account and remove the gap between theoretical and experimental results [36,37]. Due to accurate results, the modified couple stress theory is widely used to investigate the mechanical behavior of microscale devices [38,39,40].

Based on the literature review conducted earlier, there appears to be a research gap regarding the dynamic behavior of GPEH with regard to an elliptical tip cylinder. To address this gap, the present study proposes a size-dependent non-linear governing equation of the cantilever beam based on the modified couple stress theory. The coupled electro-mechanical equations of the GPEH using galloping force and Gauss’ law are devised and, by considering the mechanical properties of silicon and PZT-5A as cantilever beam layers, numerically solved. This study presents original work by providing new insights into the effect of elliptical cylinder mass, damping ratio, beam thickness, and load resistance on the GPEHs’ mechanical and electrical behaviors.

2. Mathematical Modeling

Piezoelectric energy extraction flow-induced oscillation is a method of generating electricity via the natural motion of fluid using piezoelectric materials. These materials can generate an electric charge when subjected to mechanical stress or pressure. The process of piezoelectric energy extraction flow-induced oscillation involves installing a piezoelectric device in a flowing fluid. As the fluid flows over the device, it causes the piezoelectric device to oscillate and deform piezoelectric materials, which generate an electric charge. This electric charge can be collected and stored. In this paper, we used a bi-layered micro energy harvester cantilever beam with two isotropic elastic solid layers (made of silicon) and a piezoelectric layer (PZT-5A) and elliptic tip cylinder, and the device is schematically illustrated in Figure 1.

According to Figure 1, the neutral axis position ($\overline{z}$) could be written as follows:

$$\begin{array}{ccc}& \overline{z}=\frac{\left(h_p+h_s\right)E_p{h}_s}{2(E_s{h}_s+E_p{h}_p)}+\frac{h_s}{2},& \\ z_0=-\overline{z},& z_1=h_s-\overline{z},& z_2=h_s+h_p-\overline{z},\end{array}$$

(1)

where $E_S$ and $E_p$ are the Young’s modulus of the elastic solid and piezoelectric layers at a constant electric field, and $h_s$ and $h_p$ are the elastic and piezoelectric beam thickness, respectively.

2.1. Coupled Electromechanical Governing Equations

Based on the modified couple stress theory [35] and using the bi-layered micro cantilever beam formulation [13,41], the equation governing the motion of galloping of the micro energy harvester with solid and piezoelectric layers and the corresponding boundary conditions could be written in the following form:

$$\begin{array}{ll}\left({\left(EI\right)}_{nc}+{\left(EI\right)}_c\right)& \frac{{\partial }^{4}w}{\partial x^4}-\frac{1}{2}{e}_{31}b\left(z_1+z_2\right)V\left(t\right)\frac{d}{dx}\left(\delta \left(x\right)-\delta \left(x-L\right)\right)+c\frac{\partial w}{\partial t}\\ & +\left({\rho }_s{A}_s+{\rho }_p{A}_p+M\delta \left(x-L\right)\right)\frac{{\partial }^{2}w}{\partial t^2}=F_{gallop}\left(\delta \left(x-L\right)\right)\end{array}$$

(2)

B.c:

$$\begin{array}{ll}w\left(0,t\right)=0,& \frac{\partial w\left(0,t\right)}{\partial x}=0,\\ \frac{{\partial }^{2}w\left(L,t\right)}{\partial x^2}=0,& \frac{{\partial }^{3}w\left(L,t\right)}{\partial x^3}=0,\end{array}$$

(3)

where $w$, $L$, and $b$ are the transverse beam deflection, length, and width, respectively; $c$ is damping coefficient; $M$ is tip mass, and $V\left(t\right)$ is the voltage between the piezoelectric electrodes. $A_s$, ${\rho }_s$ and $A_p$, ${\rho }_p$ are the solid and piezoelectric layers’ cross sections and densities, respectively. $F_{gallop}$ is the galloping force that acts on the device and is explained in Equation (6). The system’s flexural rigidity can be divided into two parts: ${\left(EI\right)}_c$ represents the classical component based on classical beam theory (Equation (4)), and ${\left(EI\right)}_{nc}$ represents the non-classical component based on the modified couple stress beam theory (Equation (5)).

$${\left(EI\right)}_c=\frac{1}{3}b\left[E_s\left(z_1^3-z_0^3\right)+E_p\left(z_2^3-z_1^3\right)\right]$$

(4)

$${\left(EI\right)}_{nc}=b{h}_s{\mu }_s{l}_s^2$$

(5)

In which ${\mu }_s$ and $l_s$ are the shear modulus of the elasticity and material length scale parameter of base layer, respectively. The galloping force that acted on the energy harvester could be written in the following form [19]:

$$F_{gallop}=\frac{1}{2}\rho U^{2}A\left[a_1\left(\frac{1}{U}\frac{\partial w\left(L,t\right)}{\partial t}\right)+a_3{\left(\frac{1}{U}\frac{\partial w\left(L,t\right)}{\partial t}\right)}^3\right]$$

(6)

In Equation (6), $\rho$ and $U$ are the fluid density and flow velocity, respectively; $A$ is the area of the bluff body facing the flow stream; and $a_1$ and $a_3$ are empirical coefficients, which were obtained through the applied polynomial fit of the aerodynamic coefficient $C_y$ versus $\left(\frac{1}{U}\frac{\partial w\left(L,t\right)}{\partial t}\right)$. Coefficients $a_1$ and $a_3$ are dependent on the cross-section geometry and aspect ratio of the bluff body, as well as the characteristics of the flow, and are independent of the Reynolds number. For piezoelectric beams, the electro-mechanical relationship [42] was expressed via Equation (7).

$${D_e=e_{31}{ϵ}_{xx}+E_{33}^{s}E}_2,$$

(7)

In which $D_e$ is vector of electrical displacement, and $E_{33}^s$ and $e_{31}$ are the permittivity at constant strain and piezoelectric stress coefficient, respectively. $E_2\left(t\right)=-\frac{V\left(t\right)}{h_p}$ represents the electric field in the poling direction, while $V\left(t\right)$ is the voltage between the piezoelectric electrodes. Differentiating Gauss’ law ($q(t)={\int }_A^{}D.dA$) with respect to time and using the above equation, the coupled electro-mechanical governing equation was obtained [42].

$$-e_{31}b{h}_p{\int }_0^L\frac{{\partial }^{3}w}{\partial x^2\partial t}dx{-E}_{33}^s\frac{bL}{h_p}\frac{dV}{dt}=\frac{V}{R}$$

(8)

Using the defined dimensionless parameters

$$\begin{array}{ll}\widehat{x}=\frac{x}{L}& \widehat{w}=\frac{w}{L}\\ t=k\widehat{t}={\left(\frac{\left({\rho }_s{A}_s+{\rho }_p{A}_p\right)L^4}{{\left(EI\right)}_c}\right)}^{\frac{1}{2}}\widehat{t}& \widehat{M}=\frac{M}{L\left({\rho }_s{A}_s+{\rho }_p{A}_p\right)}\\ \widehat{V}=\frac{1}{2}{e}_{31}b\left(z_1+z_2\right)\frac{L}{{\left(EI\right)}_c}V& \widehat{c}=\frac{c{L}^4}{{\left(EI\right)}_c\mathrm{k}}\\ \widehat{F}=\frac{F_{gallo p}{L}^2}{{\left(EI\right)}_c}=\frac{1}{2}\frac{\rho U^{2}A{L}^2}{{\left(EI\right)}_c}\left[a_1\beta \left(\frac{\partial \widehat{w}\left(L,t\right)}{\partial \widehat{t}}\right)+a_3{\beta }^3{\left(\frac{\partial \widehat{w}\left(L,t\right)}{\partial \widehat{t}}\right)}^3\right]\\ \widehat{{\left(EI\right)}_{nc}}=\frac{{\mu }_s}{{\left(EI\right)}_c}{l}_s^{2}b{h}_s& \mathsf{\beta }=\frac{L}{\mathrm{U}\mathrm{k}}\end{array}$$

(9)

the following dimensionless coupled governing equations could be obtained.

$$\begin{array}{ll}\left(\widehat{{\left(EI\right)}_{nc}}+1\right)& \frac{{\partial }^4\widehat{w}}{\partial {\widehat{x}}^4}-\widehat{V}\left({\delta }^{\prime \left(\widehat{x}\right)}-{\delta }^{\prime \left(\widehat{x}-1\right)}\right)+\widehat{c}\frac{\partial \widehat{w}}{\partial \widehat{t}}+\left(1+\widehat{M}\delta \left(\widehat{x}-1\right)\right)\frac{{\partial }^2\widehat{w}}{\partial {\widehat{t}}^2}\\ & ={\widehat{F}}_{gallop}\left(\delta \left(\widehat{x}-1\right)\right)\end{array}$$

(10)

$$\gamma \widehat{V}+\alpha {\int }_0^1\frac{{\partial }^3\widehat{w}}{\partial {\widehat{x}}^2\partial \widehat{t}}d\widehat{x}+\frac{d\widehat{V}}{d\widehat{t}}=0$$

(11)

where

$$\begin{array}{ll}\alpha =-\frac{e_{31}b{h}_p\left(z_1+z_2\right)}{2{\left(EI\right)}_c{E}_{33}^s}& \gamma =-\frac{k{h}_p}{E_{33}^{s}bLR}\end{array}$$

(12)

Utilizing the vibrational mode shapes of the cantilever beam ($\widehat{w}\left(\widehat{x},\widehat{t}\right)=\sum _{i=1}^n{\varphi }_i\left(\widehat{x}\right){\eta }_i\left(\widehat{t}\right)$) and the Galerkin approach, the solution to the governing equations can be obtained. On the basis of the first mode shape of the cantilever beam [43] (see Equation (13)), we simplified Equations (10) and (11) to identify Equations (14) and (15).

$${\varphi }_1\left(x\right)=\mathit{cos}\left({\beta }_{1}x\right)-\mathit{cosh}\left({\beta }_{1}x\right)-\frac{\mathit{cos}\left({\beta }_1\right)+\mathit{cosh}\left({\beta }_1\right)}{\mathit{sin}\left({\beta }_1\right)+\mathit{sinh}\left({\beta }_1\right)}\left(\mathit{sin}\left({\beta }_{1}x\right)-\mathit{sinh}\left({\beta }_{1}x\right)\right), {\beta }_1=1.8751$$

(13)

$$\begin{array}{ll}\left(\widehat{{\left(EI\right)}_{nc}}+1\right)& {\eta }_1\left({\int }_0^1{{\varphi }_1}^{\left(iv\right)}\varphi dx\right)-\widehat{V}\left({\int }_0^1{\varphi }_1\left({\delta }^{\prime }\left(x\right)-{\delta }^{\prime }\left(x-1\right)\right)dx\right)\\ & +\widehat{c}\frac{d{\eta }_1}{d\widehat{t}}\left({\int }_0^1{{\varphi }_1}^{2}dx\right)+\frac{d^2{\eta }_1}{d{\widehat{t}}^2}\left({\int }_0^1{{\varphi }_1}^2\left(1+\widehat{M}\delta \left(x-1\right)\right)dx\right)\\ & ={\widehat{F}}_{gallop}\left({\int }_0^1{\varphi }_1\delta \left(x-1\right)dx\right)\end{array}$$

(14)

$$\gamma \widehat{V}+\alpha \frac{d{\mathsf{\eta }}_1}{d\widehat{t}}\left({\int }_0^1{\varphi }_1″dx\right)+\frac{d\widehat{V}}{d\widehat{t}}=0$$

(15)

The coupled time-dependent elastic solid electrical governing equations were numerically solved using MATLAB software version 9.3 R2017b, and the mechanical and electrical responses of the system are presented in Section 3.

2.2. Flow Field Simulation

To determine the actuating force component in the direction normal to the flow direction, the transverse galloping phenomena was utilized. The galloping force ($F_{gallop}$) was related to the fluid dynamic pressure, and the instantaneous fluid force coefficient in the transverse direction was related to the flow stream ($C_y$). As the time scale of the body oscillations was considerably higher than the flow field characteristic time scale, it was completely reasonable to apply the quasi-steady assumption. As the fluid force coefficient could be related to the lift and drag coefficient at each angle of attack, as it was expressed in Equation (6), the following formulation could be applied to estimate the force coefficient as a function of the angle of attack ($\tan \left(\alpha \right)=\dot{y}/U$).

$$F_{gallop}=\frac{1}{2}\rho U^{2}A[a_1\frac{\dot{y}}{U}+a_3{\left(\frac{\dot{y}}{U}\right)}^3]$$

(16)

We noted that as the bluff body cross section is symmetric around the line passing through the center in the flow direction, only the odd coefficients remained at non-zero values. The coefficients of $a_1$ and $a_3$ were calculated using the polynomial fit of the galloping force versus the angle of attack. As the galloping force was a direct function of lift and drag coefficients, these two parameters were determined using the steady CFD simulations at various angles of attack.

The flow field around the oscillating elliptical bluff body was determined via 3D simulation using the CFD module of COMSOL Multiphysics software version 5.4. The flow was assumed to be incompressible and laminar. Therefore, the governing equations of continuity and momentum could be expressed as follows:

$$\frac{\partial u_i}{\partial x_i}=0$$

(17)

$$u_j\frac{\partial u_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial \mathrm{p}}{\partial x_j}+\frac{\mu }{\rho }\frac{{\partial }^2{u}_i}{\partial x_i\partial x_j}$$

(18)

In the above equation, $\mu$, $\rho$, $p$, and $u_i$ denote the fluid viscosity and density, the pressure field, and the average component of the velocity field, respectively.

2.3. CFD Problem Definition and Numerical Considerations

The schematic of the CFD simulation accompanied by the assigned boundary conditions is represented in Figure 2. To ensure the validity of the results, the computational domain was sufficiently extended around the cylinder to increase the accuracy of the assigned boundary conditions.

The quality of the computational grid cells, including the proper sizing and distribution, was an important factor required to obtain accurate results. To do so, the combination of tetrahedral unstructured mesh with body conformed boundary layer mesh was implemented. The density of the grids near the elliptical body was higher and gradually decreased toward the surrounding boundaries at a maximum rate of 1.15. Five layers of boundary layers with minimum heights of 1.2 × 10⁻⁵ m were applied around the body. The schematics of the final grid are illustrated in Figure 3 at the sectional view (part a) and near the elliptical body (part b).

To ensure the sufficiency of the cells’ quantity, four similar grids with cell numbers of 987,124 (coarse), 1,203,654 (medium), 1,422,924 (fine), and 1,763,290 (very fine) were utilized. Based on the calculated values of lift coefficient and the computational time, it is concluded that the fine mesh has the advantages of accurate results and a reasonable computational time. Thus, the aforementioned grid is implemented for all of the calculations made in the present investigation. To ensure the convergence of the solution, the convergence criteria was set at 10⁻⁵ for all of the governing equations.

3. Results

To investigate the dynamic behavior of the energy harvester, the base beam and piezoelectric layer are assumed to be made of silicon and lead zirconate titanate (PZT-5A), respectively.

Table 1. The utilized magnitudes of the implemented parameters.

Dimensions and Properties	Base Beam (Silicon) [44]	Piezoelectric Layer (PZT-5A) [45]	Other Parameters
Length $\left(\mathsf{\mu }\mathrm{m}\right)$	500	500	---
Width $\left(\mathsf{\mu }\mathrm{m}\right)$	50	50	---
Thickness $\left(\mathsf{\mu }\mathrm{m}\right)$	5	1	---
Density $(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)$	2332	7800	---
Young modulus $\left(\mathrm{G}\mathrm{P}\mathrm{a}\right)$	150	66	---
Poisson’s ratio	0.23	0.35	---
Length scale parameter $\left(\mathsf{\mu }\mathrm{m}\right)$	0.27	---	---
Permittivity at constant strain $E_{33}^s\left(\mathrm{n}\mathrm{F}{\mathrm{m}}^{-1}\right)$	---	13.28	---
Piezoelectric stress coefficient $e_{31}\left(\mathrm{C}{\mathrm{m}}^{-2}\right)$	---	−12.54	---
Fluid density $(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)$	---	---	1.225
Fluid viscosity (Pa.s)			0.00001812
Elliptical cylinder big, small diameters $\left(\mathsf{\mu }\mathrm{m}\right)$	---	---	500,100
Elliptical cylinder length $\left(\mathsf{\mu }\mathrm{m}\right)$	---	---	500

provides a list of the beam layers’ material properties, along with benchmark values for other parameters [44,45].

3.1. CFD Results

Regarding the validation of the numerical results obtained using available data, it should be pointed out that the experimental measurements of the electrical power output for the present configuration and its specific details are not available in the literature. Thus, validation was performed for a simplified case to ensure the accuracy of the applied CFD simulation. To do so, the mean and maximum deviation from the mean lift coefficient for a rigid ellipse versus the angle of attack is plotted for 2D [46] and 3D (present study) models (Figure 4). According to the results, it can be concluded that the results obtained for the magnitudes of C_L and C_L,max-C_L are within a maximum 4% margin of error compared to the previous 2D study.

The streamlines at the midsection plane are illustrated in Figure 5, Figure 6 and Figure 7 for various angles of attack (i.e., 0, 20, and 40°) as well as wind speeds (i.e., 0.5, 2, and 10 m/s). For the examined airstream speed, the Reynolds number (Re) based on the thickness of the elliptical harvester varies from 3.38 to 67.6 for air velocities of 0.5 to 10 m/s. The streamline coloring scheme is selected to match the velocity magnitude. For the lowest examined wind speed (0.5 m/s), due to the low flow inertia, the flow stream has a high tendency to follow the body’s surface, and no dominant wake is observable in the downstream region (Figure 5).

As the wind speed increases toward 2 m/s (see Figure 6), no significant disturbances are observed at angles of attack of up to 20°. However, at a steep orientation with α = 40°, a dominant vortex is formed downstream of the elliptical object. The mentioned vortex has direct impacts on the lift and drag coefficients, which will be discussed in the following subsections of this paper.

The flow field for the highest examined wind speed (10 m/s) is depicted in Figure 7. As is obvious, the vortex regions behind the object are intensified at all angles of attack. Even for parallel flow to the body midline, two small vortices appeared behind the body. The wake regions and vorticity in the downstream significantly increase at higher angles of attack and, consequently, affect the aerodynamic characteristics.

It should be mentioned that the flow field is 3D, and to better visualize the flow pattern, the 3D streamlines around the elliptical body are presented as U_∞ = 2 m/s. In addition to the flow behavior reported in Figure 5, Figure 6 and Figure 7, the cross stream velocity component is observable, especially near the elliptical object edges, as shown in Figure 8.

The variations in lift and drag coefficients are illustrated in Figure 9 for several wind speeds and angles of attack. The wind speed varies between 0.5 and 10 m/s, while the elliptical body is oriented based on angles of attack ranging from 0 to 50° to represent a wide range of oscillatory performances. Regarding the behavior of the drag coefficient, it is obvious that the drag force is non-linearly increased with the angle of attack (Figure 9a). At lower angle of attack, the drag force growth is lower than the same parameter at steeper orientations. Moreover, at each angle of attack the drag coefficient is increased toward lower velocity magnitudes. In other words, at a specific angle of attack of 20°, the drag coefficient is increased by 353% due to wind speed reduction from 10 to 0.5 m/s. It should also be added that the drag coefficient is more sensitive to the angle of attack at higher wind speeds. As the angle of attack varies from 0 to 50, C_d is increased by 56.6% and 268.1% for wind speeds of 0.5 m/s and 10 m/s, respectively. Focusing on the C_l chart, it is observed that at lower angle of attacks, a monotonic increasing behavior is achieved, while beyond an angle of attack of 20°, the increase rate of C_l versus α is reduced toward a maximum magnitude (Figure 9b). The lift coefficient decreases beyond the maximum value at a 30 to 40° angle of attack. C_l is independent of the wind speed for very low angles of attack (i.e., 0 to 10°), while at higher wind speeds, a considerable difference is observed for various wind speeds. Decreasing the wind speed from 10 to 0.5 m/s enhances the C_l by 61.2% at a 50° orientation of the elliptical body. The reported trend for drag and lift coefficient is close to those of the available literature for this body in wind streams, as the cross-sectional aspect ratio of the examined body represents a nearly thin body. The component of fluid force in the transverse direction (C_y) is illustrated in Figure 9c. This graph is provided as a function of tan(α), which is to be used for the calculation of a₁ and a₃ parameters in Equation (1). At large angles of attack, a monotonic increasing trend is observed as a function of tan(α), while at initial stages of elliptical body rotation, a more complicated behavior is occurred. For the minimum simulated wind speed, C_y remains unchanged near a value of zero up to tan(α) = 0.4, while a rapid increase occurs beyond that value. However, for higher values of wind speed, a slight reduction, followed by rapid increase, is observed as a function of tan(α). To be more specific, except for U_∞ = 0.5 m/s, a minimum value of C_y is distinguishable between tan(α) values of 0.4 and 0.6.

To explain the effect of the fluid dynamics and the flow field on the electrical power produced, it can be stated that the main source of actuating force on the micro energy harvester is the variations in the aerodynamic force components on the elliptical body’s surface. As the harvester oscillates, its orientation (angle of attack) and velocity alter during each time period, which results in the acceleration and deceleration of the harvester movement to form a galloping effect. As the flow field and its coupled effect on the harvester motion is complicated, the explained force is applied to the system dynamics via calculation of the transverse force component based on the lift and drag forces.

3.2. Galloping Effect

This section discusses the observed galloping effect of the proposed energy harvester using the material properties listed in Table 1 and the CFD results obtained. The tip deflection of the cantilever beam for different damping ratio values is illustrated in Figure 10. The galloping effect occurs at different wind speeds, and our findings reveal that increasing the damping ratio from 0.1 to 0.2 and 0.3 increases the stiffness and natural frequency of the system, thereby increasing the onset velocity of galloping from 3.2 to 4.7 and 6.5 m/s, respectively. Higher damping ratios give stiffer energy harvesters and decrease the beam deflection, and at a specific wind speed (e.g., U = 9 m/s), the maximum deflection obtained for c = 0.1 is 14% and 33% higher than the maximum deflection of c = 0.2 and c = 0.3, respectively.

Figure 11 illustrates the extracted power for the different damping ratio values of the piezoelectric energy harvester. The galloping effect is observed, and, for example, at $c=0.1$, the extracted power is zero for wind speeds lower than 3.2 m/s and sharply increases as the wind velocity increases. Similar to Figure 10, a stiffer beam generates lower power. For instance, at $U=9 \mathrm{m}/\mathrm{s}$, the maximum power density of $c=0.1$ is 33% and 60% greater than $c=0.2$ and $c=0.3$, respectively. These findings support the non-linear relationship between the output power and the beam deflection of the proposed energy harvesters.

3.3. Load Resistance

The extracted power versus electrical load resistance of the system for different wind velocities is plotted in Figure 12. It is observed that the maximum output power is obtained at load resistances between 15 and 150 kΩ (with the maximum value occurring at 50 kΩ) for all velocities. For other load resistances, the obtained power significantly decreases. This figure shows that when wind velocity increases from 8 to 9 and 10 m/s, the power density of the proposed PEH increases from 0.016 to 0.022 (38%) and 0.0335 (110%), respectively.

3.4. Elliptical Cylinder Mass Effect

Changes in the tip mass (mass of the elliptical cylinder) affects the natural frequency of the energy harvester and has a significant effect on its dynamic behavior and extracted power. It should be noted that to consider a change in the mass of elliptical cylinder without altering its external shape, a different material with a different density or a hollow cylinder should be utilized. Therefore, the following figures demonstrate the effects of normalized tip mass (tip mass/solid and piezoelectric beam mass). Figure 13 shows the extracted power of the device with respect to electrical load resistance for three normalized mass values ($\widehat{M}$= 0, 0.5, 1). As seen in the Figure, increasing the normalized tip mass from 0 to 0.5 and 1 increases the output power density from 0.12 to 0.2 and 0.22, respectively. Additionally, the corresponding electrical load resistance at maximum power increases from 175 to 280 kΩ and 375 kΩ, respectively.

These findings suggest that the relationship between the normalized tip mass and the load resistance value should be considered when designing energy harvesters. This relationship is illustrated in Figure 14 for different wind speeds. This figure shows that the relationship between tip mass and load resistance is approximately linear, while changing the tip mass value proportionally changes the electrical load resistance required to achieve maximum output power. For instance, for $\widehat{M }$= 0.6, 1.2, and 1.8, the maximum output power is extracted at load resistances of R = 32, 40, and 50 kΩ, respectively.

The effect of wind speed on output power at different tip masses is presented in Figure 15. As expected, increasing the wind speed and normalized mass increases the output power, though for higher normalized mass, the power increment is decreased. Additionally, Figure 14 reveals that the wind speed has no effect on the tip mass–resistance relationship.

Figure 16 presents the time response of the extracted power of the system for different values of normalized mass at U = 9 m/s. In all figures, the power starts to increase at t = 80, and tip mass has no effect on the increasing time. However, the average and maximum values of the power are significantly influenced by the magnitude of tip mass. In the transverse galloping phenomenon, the body oscillations are usually much larger than the flow characteristic timescale [47]. This Figure shows that increasing the tip mass decreases the natural frequency of the energy harvester, and the difference between the body’s natural frequencies and galloping force frequency (flow frequency) is reduced. As a result, the beating phenomenon becomes more apparent in the time response of the system.

4. Conclusions

In this paper, the mechanical behavior analysis of a galloping piezoelectric micro energy harvester (GPEH) is performed. Based on the modified couple stress theory and Gauss’ law, the coupled electro-mechanical non-linear governing equation of the energy harvester is obtained. The proposed energy harvester is made of a cantilever beam with two layers of elastic silicon and piezoelectric (PZT-5A) attached to a elliptical cylinder at the tip. Using the Galerkin method and cantilever beam mode shapes, the dynamic behavior of the system and the effect of the elliptical cylinder’s mass, damping ratio, and load resistance on the harvested power is investigated. The results indicate that there is an approximately linear relationship between the elliptical cylinder mass and the load resistance, and the electrical load resistance required to obtain maximum output power is proportional to the cylinder mass. Increasing the normalized mass from 0 to 0.5 and 1 increases the output power from 0.12 to 0.2 and 0.22, respectively, and the corresponding electrical load resistance at maximum power increases from 175 to 280 kΩ and 375 kΩ, respectively. Additionally, increasing the damping ratio from 0.1 to 0.3 increases the onset velocity of galloping from 3.2 to 6.5 m/s. The time response analysis shows that for any tip mass value, the power starts to increase at dimensionless time = 80, and tip mass has no effect on it. However, the rms and maximum value of the power are significantly affected by the magnitude of the tip mass. Increasing the tip mass decreases the natural frequency of the energy harvester, and the difference between the body’s natural frequencies and galloping force frequency (flow frequency) is reduced, resulting in the clear observation of the beating phenomenon in the time response of the system.