Optimal Design Guidelines for Efficient Energy Harvesting in Piezoelectric Bladeless Wind Turbines

Abstract

This study presents an optimal design methodology for a piezoelectric-based bladeless wind turbine (BWT) that efficiently converts wind-induced vibration of a cantilever-mounted cylinder into electrical energy. A lumped-parameter model integrating structural dynamics, fluid-structure interaction, and piezoelectric energy conversion is introduced and simplified to derive key dimensionless design parameters and optimal conditions for maximizing power output. The optimal design criteria are as follows: tuning the resonance between the structural natural frequency and vortex shedding frequency; setting the dimensionless load resistance R* to unity; and minimizing ω_nR_LC_eq to a value smaller than unity. Numerical simulations and wind tunnel experiments validate the model, showing good agreement with less than 7% error in power prediction under resonance conditions and successfully predicting the coupled behavior of fluid, structure, and piezoelectric components. The proposed optimal design methodology facilitates the development of compact and efficient piezoelectric-based bladeless wind energy harvesting systems suitable for urban and space-constrained environments.

1. Introduction

1.1. Research Backgrounds

Wind power generation is a technology that converts the kinetic energy of wind into electrical energy and is attracting attention as a representative renewable energy source that can reduce the use of fossil fuels in response to the climate crisis [1,2]. Conventional wind turbines produce electricity with rotating turbine blades driven by the wind, and they have relatively high power-generation [2]. However, this method of wind power generation has several disadvantages due to the large rotating parts: high installation and maintenance costs; it can only be used in large open spaces; noise and vibration caused by rotation; and it can pose a threat to wildlife such as birds [3,4,5,6].

To overcome these problems, a new wind power generation method called a bladeless wind turbine (BWT), also known as a bladeless wind energy harvester (BWEH), which utilizes vortex-induced vibration (VIV) induced as wind passes around the structure, has been proposed [7,8,9,10,11,12]. Compared to conventional rotating wind turbines, the BWT system offers several advantages: since there are no large rotating blades, installation and maintenance costs are reduced and durability is improved [13]; it requires less installation space and can be installed near urban areas or residential buildings [9,13]; there is no vibration caused by rotating blades [13]; and it does not threaten the lives of birds or other wild animals.

BWT energy harvesting mechanisms are classified into electromagnetic [14,15,16,17,18] and piezoelectric types [19,20,21,22,23,24]. The former is effective for large-scale systems but less suitable for small devices due to the size of magnets and coils, whereas the latter enables lightweight and simple configurations, ideal for compact BWT applications [17,18].

In an effort to model and apply the fluid-induced vibration and power generation characteristics of piezoelectric bladeless wind turbines (BWTs) to practical design, it is essential to analyze the fluid vortex behavior, the electrical response of the piezoelectric material, and their coupling with structural vibration. Barrero-Gil et al. [25] modeled a vibrating cylinder as a single-degree-of-freedom (1-DOF) spring–mass–damper system and analyzed the effects of key parameters using experimentally measured vortex data. Dai et al. [23] experimentally compared the power generation performance of four different BWT configurations and subsequently proposed a distributed-parameter model [26] based on the vortex-lift model of Facchinetti et al. [27]. Jia et al. [24] adopted the model proposed by Dai et al. [26] to develop a numerical framework simulating the dynamic behavior and energy harvesting performance of a vertically oriented BWT, and validated the results through experiments. In addition, several researchers [28] modeled BWTs as equivalent analog circuits to predict power output using commercial circuit analysis software. However, most previous studies focused on the “analysis perspective” to predict complex system behaviors, which requires significant time and computational resources due to repetitive simulations for parameter tuning. To overcome this limitation, a shift to a “design perspective” is necessary. Therefore, this study aims to derive “closed-form, dimensionless design rules” that allow engineers to determine optimal parameters immediately without extensive numerical iterations.

1.2. Research Objectives

This study integrates and simplifies the theoretical models established by Facchinetti et al. [27] and Jia et al. [24] into a unified lumped-parameter framework to develop an optimal design methodology for piezoelectric-based BWTs. In particular, the following research objectives are set:

(a)

To integrate the mechanical and electrical modeling of the BWT structure and piezoelectric elements and to identify key design parameters;

(b)

To develop a design methodology that achieves optimal power generation at a given rated wind speed;

(c)

To verify the proposed optimum design criteria using both simulations based on existing models and experimental results.

To achieve these goals, Section 2 reviews an existing mathematical model [24] of BWT systems. Section 3 presents the optimal design methodology. Section 4 validates the numerical model and the proposed design methodology by comparing simulation results with experimental data.

Through this study, a more systematic approach to the optimal design of BWTs is proposed improving the practical feasibility of piezoelectric-based wind energy harvesting systems.

2. Mathematical Model for BWT Energy Harvester

2.1. System Concept and Assumptions for BWT

To perform the optimal design of a bladeless wind turbine (BWT) system, this section introduces and analyzes a mathematical model of a piezoelectric-based BWT and analyzes the key design variables. First, the basic structure and operating principle of the BWT system are explained, followed by the main assumptions used in the analysis.

The bladeless wind turbine (BWT) system converts wind energy into electrical energy by utilizing vortex-induced vibrations (VIV). Figure 1 illustrates the configuration of the BWT system investigated in this study. A lightweight foam cylinder is attached around the upper section of a thin Al 1060 cantilever beam, while piezoelectric (PZT) layers are bonded to both sides of its lower section. The outer surfaces of the two PZT layers are connected to a common electrode serving as the output terminal, whereas the metal cantilever functions as the ground electrode. The base of the cantilever is firmly fixed using a vise clamp. When the cylinder oscillates laterally under wind excitation, it induces corresponding vibrations in the cantilever. These vibrations cause cyclic stretching and compression of the piezoelectric layers, thereby generating electrical power across the load resistor.

2.1.1. Main Assumptions for Analysis

To effectively analyze the system behavior and perform the optimal design, the following assumptions are made:

(a)

The cantilever is assumed to vibrate only in its first bending mode, which dominates the system response.

(b)

The cantilever beam is thin relative to its length and is modeled as an Euler–Bernoulli beam; Axial deformations are neglected.

(c)

Deformation of the foam cylinder, and the cantilever section where the cylinder is attached is neglected.

(d)

Additional damping effects due to clamping, cylinder bending, or other factors are not modeled. Instead, the damping ratio for the resonance mode is determined experimentally.

(e)

The incoming wind is assumed to be laminar.

2.1.2. Basic Working Principles of BWT System

The power generation process of the BWT system consists of the following three stages:

(a)

Vortex-Induced Vibration (VIV):

When wind flows past the cylinder, a Kármán vortex street forms, causing the cylinder to vibrate laterally, perpendicular to the wind direction. The angular frequency, ${\omega }_f$ of the vortex shedding is determined by the Strouhal number ($S_t$) as follows:

$${\omega }_f={\displaystyle \frac{2\pi S_{t}U}{D_c}}$$

(1)

where $U$ is the wind speed, $D_c$ is the diameter of the cylinder. In this study, the Reynolds number is approximately 1.3 × 10⁴, falling within the sub-critical regime ($300<Re<3 \times {10}^5$). In this regime, the Strouhal number for a smooth circular cylinder is well-established to be approximately constant at 0.21 [29].

(b)

Coupled Vibration of Cylinder and Cantilever:

The lateral vibration of the cylinder forces the cantilever to vibrate. The cantilever primarily vibrates in its first bending mode, which dominates the system’s dynamic response.

(c)

Piezoelectric Energy Conversion:

The vibration of the cantilever causes the piezoelectric layers to experience alternating tension and compression. This mechanical deformation alters the internal electric field of the piezoelectric material, thereby generating electrical energy. The generated electrical energy is either stored as electric charge in the capacitance of the piezoelectric layers or dissipated through the load resistance.

2.2. Mathematical Model

A dynamic model [24] comprising a cylinder, a cantilever, and piezoelectric layers is introduced and simplified to derive optimal design guidelines for the BWT system. The operation of the system is governed by three key coupled equations that represent the structural vibration, electrical behavior, and vortex dynamics.

2.2.1. Vibrational Behavior of Cantilever Beam

The cantilever beam’s vibration is modeled using Euler–Bernoulli beam theory, assuming that the beam is thin and flexible. The equation of motion for the cantilever vibration is given by:

$$\rho A{\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}+EI{\displaystyle \frac{{\partial }^{4}w}{\partial x^4}}=F\left(x,t\right)$$

(2)

where the terms $\rho$, $A$, $E$, $I$ on the left-hand-side mean the material density, cross-sectional area, Young’s modulus, and area moment of inertia, while $F$ on the right-hand-side means external force per unit length. The transverse displacement of beam, $w\left(x,t\right)$ is separated into spatial and temporal components as:

$$w\left(x,t\right)=\varphi \left(x\right)r\left(t\right)$$

(3)

Once the modal analysis of a vibrating cantilever structure is performed to determine the natural frequency ${\omega }_n$ and mode shape $\varphi \left(x\right)$, the lumped equation of motion of $r\left(t\right)$ can be simplified to the following second-order differential equation for the modal response:

$$\ddot{r}\left(t\right)+\left(2\zeta {\omega }_n+\eta \right)\dot{r}\left(t\right)+{\omega }_n^{2}r\left(t\right)=-\theta V\left(t\right)+Kq\left(t\right)$$

(4)

where the terms $\zeta$ and $\eta$ on the left-hand side mean the structural damping ratio and fluid damping, respectively, while $\theta$ and $K$ on the right-hand side are coupling coefficients for piezoelectric and fluid equations, respectively. The aerodynamic coupling term $K$ in Equation (4) is expressed as:

$$K={\displaystyle \frac{C_{L0}{\rho }_f{U}^2{D}_c}{4}}\left[L_c\varphi \left(L_s\right)+{\displaystyle \frac{L_c^2}{2}}{\varphi }^{\prime }\left(L_s\right)\right]$$

(5)

here $K$ is a constant determined by the mode shape and flow velocity and remains unchanged once the structural mode and flow conditions are fixed. The definition of $\theta$ is introduced in the next section.

2.2.2. Electrical Behavior of Piezoelectric Energy Harvester

The electrical behavior of the piezoelectric energy harvester is described by the following governing equation:

$$C_{eq}\dot{V}\left(t\right)+{\displaystyle \frac{V\left(t\right)}{R_L}}=\theta \dot{r}\left(t\right)$$

(6)

here $C_{eq}$ is the effective parallel capacitance of the two parallel piezoelectric sheets:

$$C_{eq}=2{\displaystyle \frac{{\epsilon }_{33}{b}_p{L}_p}{h_p}}$$

(7)

where ${\epsilon }_{33}$ is electric permittivity, and $b_p$, $L_p$, $h_p$ are width, length, and thickness, respectively.

The voltage $V$ in Equation (6) is the output voltage, $R_L$ is the load resistance, and $\theta$ on the right-hand side is defined as:

$$\theta =-2e_{31}{b}_p\overline{y}\varphi \left(L_p\right)$$

(8)

where $e_{31}$ is the electromechanical coupling coefficient, and $\overline{y}$ is the distance from the beam’s neutral axis to the center of the piezoelectric layer.

2.2.3. Vibrational Behavior of Karman Vortex

The normalized lift force $q$ acting transversely on the cylinder by the flow around a cylinder of diameter $D_c$ is modeled by the van der Pol equation as follows [24,27]:

$$\ddot{q}\left(t\right)+{\epsilon }_f{\omega }_f\left(q^2-1\right)\dot{q}\left(t\right)+{\omega }_f^{2}q\left(t\right)={\displaystyle \frac{A_f}{D_c}}\left[\varphi \left(L_s\right)+{\displaystyle \frac{L_c}{2}}\varphi \left(L_s\right)\right]{\ddot{r}}_i\left(t\right)$$

(9)

where ${\epsilon }_f$ and $A_f$ are set to 0.3 and 12, respectively for cylindrical body [27].

3. Optimal Design Method

3.1. Derivation of Optimal Design Rule

The three governing equations, Equations (4), (6) and (9), introduced in Section 2, are time-dependent differential equations that are mutually coupled, with Equation (9) being inherently nonlinear. Therefore, obtaining an analytical solution is highly challenging. To simplify the model for optimal design, the following key assumptions are made:

(a)

Optimal energy harvesting is achieved when the natural frequency of the structure matches the vortex shedding frequency.

(b)

For the direct resistive load configuration, the generated current primarily flows through the load resistance under optimal conditions.

(c)

The solution $q\left(t\right)$ of the van der Pol equation is assumed to be a sinusoidal wave with constant angular frequency ${\omega }_f$ and constant amplitude $q_0$ under the optimal operating condition, with a default amplitude value of $q_0=2$.

The van der Pol oscillator is well known for its nonlinear behavior and does not admit a closed-form analytic solution. For the purpose of deriving a simplified design rule, it is necessary to adopt a representative limit-cycle amplitude. In this study, we assume the fundamental amplitude of the van der Pol oscillation as $q_0=2$, which corresponds to the typical limit-cycle amplitude in the absence of strong fluid–structure coupling. The validity and implications of this assumption are examined in a later section.

According to assumption (a), the structural natural frequency and the vortex shedding frequency must be equal at the optimal resonance condition:

$${\omega }_n={\omega }_f$$

(10)

Since the structure vibrates at the resonance frequency ${\omega }_n$, the first and third terms on the left-hand side of Equation (4) cancel out in the phase domain, yielding the simplified form:

$$j{\omega }_n\left(2\zeta {\omega }_n+\eta \right)r=-\theta V+Kq$$

(11)

In Equation (6), the first and second terms on the left-hand side represent the current flowing through the equivalent capacitance and the load resistor, respectively, while the right-hand side represents the current generated by the piezoelectric strain. To maximize power output, the current dissipated through the load resistance must be much larger than the current flowing into the capacitance (assumption (b)). Therefore, assuming $\omega C_{eq}\ll 1/R_L$, Equation (6) is simplified as:

$$V=j{\omega }_n{R}_L\theta r$$

(12)

Substituting Equation (12) into Equation (11), the two equations are combined into:

$$\left({\displaystyle \frac{2\zeta {\omega }_n+\eta }{R_L\theta }}+\theta \right)V=Kq$$

(13)

By applying assumption (c), the term $q$ becomes a constant $q_0$. Taking the time-averaged power $P=V^2/\left(2R_L\right)$, the following simplified expression is obtained:

$$P^{\ast }={\displaystyle \frac{1}{2}}{\displaystyle \frac{R^{\ast }}{{\left(1+R^{\ast }\right)}^2+{\left(\omega R_L{C}_{eq}\right)}^2}}$$

(14)

where the dimensionless power $P^{\ast }$ is defined as:

$$P^{\ast }={\displaystyle \frac{\left(2\zeta {\omega }_n+\eta \right)}{{\left(K{q}_0\right)}^2}}P$$

(15)

and the non-dimensional load resistance $R^{\ast }$ is defined as:

$$R^{\ast }={\displaystyle \frac{R_L{\theta }^2}{\left(2\zeta {\omega }_n+\eta \right)}}$$

(16)

Figure 2 illustrates how the non-dimensional power $P^{\ast }$ varies with $R^{\ast }$ and ${\omega }_n{R}_L{C}_{eq}$. The solid black line represents the case when ${\omega }_n{R}_L{C}_{eq}=0$. As $R^{\ast }$ approaches either zero or infinity, $P^{\ast }$ becomes approximately proportional to $R^{\ast }$ and $1/R^{\ast }$, respectively, and the maximum $P^{\ast }$ value of 0.125 is achieved when $R^{\ast }=1$.

Physically, the non-dimensional load resistance $R^{\ast }$ represents the balance between the electrical energy harvested through the load resistor and the total energy dissipation caused by structural damping and fluid resistance. Maximum power is obtained when these two rates become equal, which occurs at $R^{\ast }=1$.

Additionally, the graph shows how the maximum $P^{\ast }$ decreases as ${\omega }_n{R}_L{C}_{eq}$ increases, indicating that increasing the capacitance reduces power output. However, when ${\omega }_n{R}_L{C}_{eq}$ becomes smaller than 0.1, the difference from the ideal ${\omega }_n{R}_L{C}_{eq}=0$ case becomes negligible. This result confirms the validity of Assumption (b), demonstrating that this approximation is specifically applicable to the direct resistive coupling configuration used in this study, where complex impedance matching components are absent.

The above result can be interpreted as follows: (a) Maximum power is obtained when the energy harvested through the load resistance (numerator of Equation (16)) equals the energy loss due to structural damping and fluid resistance (denominator), which occurs when $R^{\ast }=1$; (b) Additionally, maximum power is achieved when the system’s resonance frequency ${\omega }_n$ is much lower than cutoff frequency $1/\left(R_L{C}_{eq}\right)$ of the RC low-pass filter formed by the intrinsic capacitance of the PZT layer, satisfying the condition ${\omega }_n{R}_L{C}_{eq}<<1$.

Based on the above analysis, the derived optimal design criteria can be summarized as follows:

(1)

${\omega }_n={\omega }_f$

(2)

$R^{\ast }=1$

(3)

${\omega }_n{R}_L{C}_{eq}<<1$

3.2. Optimal Design Execution

In this section, a small-scale BWT system is designed based on the optimal design criteria derived in Section 3.1. The designed parameters are later verified through experiments and numerical analysis presented in Section 4. The nominal wind speed for the design is set to 4 m/s, considering that the system is intended for small-scale wind energy harvesting. The diameter and length of the foam cylinder are fixed at 50 mm and 160 mm, respectively.

3.2.1. Cantilever Design

The vortex shedding frequency at the rated wind speed is calculated using Equation (1). According to the first design rule, the natural frequency of the BWT structure is designed to match the vortex shedding frequency by adjusting the material, length, and thickness of the cantilever. Additionally, the structural damping loss should be much smaller than the loss due to fluid resistance to maximize the power. In this study, aluminum (Al 1060), known for its low damping coefficient, is selected. The structural damping loss is estimated to be less than 10% of the fluid-resistance loss.

3.2.2. PZT Design

The width $b_p$ and length $L_p$ of the piezoelectric sheets are primarily constrained by the geometry of the cylinder and the cantilever. Within this geometric limitation, the piezoelectric material and thickness were carefully selected to satisfy the second and third optimal design rules described in Section 3.1. Several commercially available piezoelectric materials (including ceramic and polymer-based sheets) were evaluated, and the chosen PZT material provided the highest predicted electromechanical coupling and strain energy harvesting performance under the intended operating conditions.

The length of the PZT sheet was determined to maximize the strain energy harvested along the cantilever while avoiding unnecessary extension into regions with negligible curvature. Although increasing $L_p$ generally improves energy harvesting, the gain becomes marginal when the PZT covers a sufficiently large portion of the high-strain region of the substrate. The thickness of the PZT layer was selected by considering both the target resonance frequency and the RC filtering condition ${\omega }_n{R}_L{C}_{eq}<<1$. A thicker PZT layer increases the effective stiffness of the cantilever, raising the resonance frequency and reducing the generated voltage due to weaker electromechanical coupling. Conversely, excessively thin PZT sheets may compromise manufacturability and mechanical durability. The selected thickness of 0.2 mm represents the optimal balance among these considerations and is the thinnest option available within our fabrication constraints.

With the chosen dimensions and material properties, the resulting value of ${\omega }_n{R}_L{C}_{eq}$ remains sufficiently below unity, ensuring compliance with the third optimal design rule and enabling effective power extraction through the load resistance. The final PZT-sheet configuration, along with other structural parameters, is summarized in

Table 1. Design specifications of the BWT system.

Part	Parameter	Description	Values
Foam cylinder	$L_c$	Length	160 mm
	$D_c$	Diameter	50 mm
	${\rho }_c$	Density	55 kg/m3
Cantilever substrate	$L_s$	Length	55 mm
	$b_s$	Width	42 mm
	$h_s$	Thickness	0.6 mm
	${\rho }_s$	Density	2700 kg/m3
	$E_s$	Young’s modulus	69 GPa
Piezo layers	$L_p$	Length	51 mm
	$b_p$	Width	38 mm
	$h_p$	Thickness	0.2 mm
	${\rho }_p$	Density	7650 kg/m3
	$E_p$	Young’s modulus	87 GPa
	${\epsilon }_{33}$	Dielectric constant	1050
	$e_{31}$	Electromechanical coupling coefficient	9.85 C/m2

.

The calculation results show that the vortex shedding frequency at 4 m/s wind speed is 105.56 rad/s (16.80 Hz), and the structural natural frequency is 101.47 rad/s (16.15 Hz), which is tuned close to the vortex frequency. The optimal load resistance $R_L$ is calculated to be 12 kΩ, and the dimensionless parameter $\omega R_L{C}_{eq}$ is 0.22, which is sufficiently smaller than 1.

Based on the design specifications listed in Table 1, the key lumped parameters for the numerical simulation were calculated as follows: $\theta =-0.014$, $K=0.0635$, $\eta =1.369$. The value of $\zeta$ was taken from experiment as $\zeta =0.0049$, without mathematical derivation. The detailed mathematical derivations of these coefficients are adapted from the formulations provided in Jia et al. [24].

4. Results and Discussion

4.1. Free Vibration of the Structure

First, a free vibration test of the structure without aerodynamic force was conducted. The fabricated BWT prototype was excited by an impulse applied to the top of the device, and the output voltage from the PZT under open-circuit conditions was measured.

This test aims to verify whether the natural frequency predicted by the numerical model matches the experimental results, thereby confirming the model’s validity. As shown in Figure 3, the cantilever was firmly fixed using a vise clamp, and the top of the foam cylinder was struck with a hammer to apply an impulse. The vibration of the cantilever and the corresponding voltage output from the piezoelectric sheets were measured over time.

Figure 4 shows the measured PZT voltage response. The initial peak voltage of 3.4 V gradually attenuates, reaching 2 V after 1 s. The experimental data was fitted to extract the natural frequency and damping ratio.

The results showed that the natural frequency was 15.93 Hz, and the damping ratio $\zeta$ was measured as 0.0049.

Although the Al 1060 material used in the experiment typically exhibits a very low intrinsic damping ratio in the range of 0.0001–0.001, a relatively larger value was observed in this test. This increase is attributed to additional damping sources introduced by the experimental setup, including bending deformation of the foam cylinder, imperfect clamping at the bottom vise, and possible shear deformation at the PZT bonding interface. Similar observations have been reported in recent aeroelastic studies of slender wind turbine structures, where the combined effects of structural flexibility, mounting conditions, and fluid–structure interaction significantly amplify the effective damping and modify the dynamic response characteristics [30]. Such findings support the interpretation that the elevated damping ratio measured in our experiment is not unusual for lightweight, high–aspect-ratio structures subjected to wind-induced excitation.

The simulated natural frequency was 16.15 Hz, showing a very small error of about 1.4%, which confirms the model accurately predicts the free vibration of the BWT structure. Since the damping ratio in the simulation was set to the experimental value, the simulation results match the experimental data. The results are summarized in

Table 2. Results of impact response test.

Value	Experiment	Simulation	Error
Natural frequency	15.93 Hz	16.15 Hz	1.4%
Damping ratio	0.0049	0.0049	-

.

4.2. PZT Output Voltage at 4 m/s Wind Speed

The fabricated BWT device was installed in a wind tunnel to measure its energy harvesting performance under various wind speeds. As shown in Figure 5, the device was subjected to a laminar airflow at 4 m/s. A foam cylinder with a diameter of 50 mm and a length of 160 mm was attached to the upper section of the cantilever to receive lateral aerodynamic force from the wind.

Figure 6a shows the time history of the output voltage generated by the PZT at a wind speed of 4 m/s. The system vibrates stably at a frequency of 15.9 Hz. Figure 6b compares this experimental result with the simulation under a load resistance of 9.1 kΩ. Both the amplitude and the period of the simulated and experimental PZT voltages are in close agreement. The simulation produces a clean sinusoidal voltage representing the first mode response only. In contrast, the experimental result exhibits slight distortions, primarily manifested as small-magnitude high-frequency components. These components are attributed to the higher-order structural modes excited by the inherent nonlinear characteristics of the Kármán vortex shedding.

4.3. Power Output vs. Wind Speed

To evaluate the performance of the fabricated BWT, power output was measured in a wind tunnel under wind speeds ranging from 2.8 m/s to 5.5 m/s. The results in Figure 7 show that power generation is negligible below 3.6 m/s wind speed. Significant energy harvesting occurs in the finite wind speed range of approximately 3.7–4.7 m/s, indicating a lock-in bandwidth of about 1.0 m/s. This synchronization characteristic is qualitatively consistent with the experimental observations by Jia et al. [24], who also reported a comparable lock-in bandwidth of approximately 1.1 m/s (ranging from 3.7 m/s to 4.8 m/s). Both studies confirm the typical VIV behavior where the structural vibration remains synchronized with vortex shedding over a specific, limited range of flow velocities.

At the designed resonance-wind-speed of 4 m/s, the measured output power is 1.39 mW, which is about 3% higher than the predicted value of 1.35 mW. This difference can be explained by the contribution of the higher mode vibrations, discussed in Section 4.1.

Interestingly, the maximum output was observed at 4.2 m/s rather than at the exact resonance condition of 4 m/s. This peak shift is observed both from the experimental and simulation results. The measured output at 4.2 m/s was 1.66 mW, while the predicted value was 1.55 mW, resulting in an error of 7.1%.

This shift in the optimal point can be explained by the balance between frequency matching (lock-in phenomena) and the increase in available wind energy. Although the frequency mismatch reduces the power efficiency, the aerodynamic power increases with the cube of the wind speed, resulting in a region where the power continues to rise slightly beyond the ideal frequency-matching point. The time-dependent numerical simulation clearly captures this shift in the optimum wind speed and the corresponding increase in peak power. This behavior is also consistent with the experimental results reported by Jia et al. [24].

The simulation results accurately predict the lock-in phenomenon observed in the experiment, as well as the wind speed (4.2 m/s) and power output (within 7.1% error) at which the maximum power is generated. This confirms that the simulation sufficiently captures the complex coupled behavior of the fluid, structure, and piezoelectric elements in the BWT system. However, a detailed investigation of the nonlinear lock-in phenomenon and its precise bandwidth is a subject reserved for our future research.

4.4. Power Output vs. Load Resistance

Figure 8 shows the measured power output as the load resistance $R_L$ was varied from 1 kΩ to 100 kΩ. The output power increased with increasing $R_L$, reaching a maximum of 1.42 mW at 9.1 kΩ. Beyond this point, the output decreased as $R_L$ increased further.

The numerical simulation showed a similar trend, predicting a power output of 1.3 mW at 9.1 kΩ and a maximum power of 1.35 mW at 12 kΩ. The difference between the simulation and experimental results at the peak was within 9.2%, indicating that the numerical model accurately predicts the trend of electrical behavior of the piezoelectric energy harvester.

This result confirms that the simulation effectively captures not only the fluid–structure coupling but also the coupled behavior of the piezoelectric output with a high level of quantitative accuracy.

Under the given wind speed condition, the optimum-design rule ($R^{\ast }=1$) yields an optimal load resistance of approximately 12 kΩ which is confirmed by the simulation, whereas the experimental result indicates a value of 9.1 kΩ, corresponding to a difference of about 23%. This discrepancy arises from assuming a fixed van der Pol oscillation amplitude ($q_0=2$) in Assumption (c) of Section 3.1. Nevertheless, this assumption does not significantly affect the determination of the optimal load resistance, because once the system is tuned to resonance, variations in the oscillation amplitude near the optimum point remain relatively small even when only the load resistance ($R_L$) is varied. The experimental results confirm this behavior, demonstrating that the proposed design rule remains valid despite the simplified amplitude assumption.

5. Conclusions

In this study, an optimal design methodology for a piezoelectric-based bladeless wind turbine (BWT) was proposed and validated through both experiments and numerical simulations. A simplified lumped-parameter model was developed not only to reduce complexity but to facilitate a fundamental shift from an analysis-based approach to a design-based one. Unlike conventional numerical studies, this work established explicit optimal design rules (e.g., $R^{\ast }=1$ and $\omega R_L{C}_{eq}\ll 1$) that provide immediate guidelines for practical BWT implementation without relying on computationally expensive simulations.

The key optimal design rules are summarized as follows:

(a)

Tuning the structural natural frequency to match the vortex shedding frequency for resonance;

(b)

Designing the load resistance such that the non-dimensional load resistance $R^{\ast }$ equals to 1;

(c)

Selecting piezoelectric materials and load resistance that ensure the value of ${\omega }_n{R}_L{C}_{eq}$ remains sufficiently smaller than unity.

Wind tunnel experiments validated the numerical model, demonstrating good quantitative accuracy with 7.1% error in power prediction under resonance conditions. Moreover, the model showed consistent qualitative agreement over a wide range of wind speeds and load resistances.

The verified simulation model confirmed that the three proposed design rules (structural natural frequency tuning, $R^{\ast }=1$ and ${\omega }_n{R}_L{C}_{eq}\ll 1$) lead to optimal energy harvesting performance.

The results of this study are expected to serve as useful guidelines for the design and practical implementation of small-scale wind energy harvesting systems. In particular, this research can contribute to the development of compact and efficient piezoelectric bladeless wind power systems suitable for urban environments and space-constrained applications.