Vibration Characteristics and Experimental Research of Bistable Composite-Beam Wind Energy Harvester

Abstract

Vibration energy technology holds promise for self-powered miniature wireless sensor devices in underground coal mines. This study proposes a magnetically coupled bistable composite-beam wind energy harvester (BCBWEH) to enhance the response threshold for harvesting weak ambient wind energy. A nonlinear magnetic model based on magnetic dipoles is established, and the system’s dynamic equations are formulated using the lumped parameter method. Numerical simulations analyze the effect of magnetic spacing on static bifurcation, and discuss the influences of initial static position, wind excitation, and magnetic moment on the system’s dynamic behavior; experimental results validate the accuracy of the numerical predictions. By adjusting the magnetic spacing under the same level of excitation, the system’s motion can transition from single-well oscillation to efficient inter-well vibration. When the initial position lies closer to the shallower potential well, relatively small wind excitation can trigger large-amplitude inter-well vibrations, thereby increasing output power. This study offers guidance for optimizing structural configurations and tuning design parameters of piezoelectric energy harvesters based on composite-beam architectures.

1. Introduction

Coal occupies a pivotal strategic position in national energy security and acts as an indispensable guarantee for steady energy provision [1,2]. Driven by the ongoing intelligent upgrading of underground coal mining, massive monitoring sensors have been arranged in tunneling, ventilation and safety auxiliary subsystems. Unstable power supply has emerged as a core bottleneck hindering the intelligent construction of underground coal mines [3]. At present, conventional chemical batteries are the primary power source for underground wireless sensors; nevertheless, their inherent drawbacks including environmental contamination and excessive maintenance costs render them incompatible with severe underground operating environments [4]. Accordingly, self-powered energy harvesting technology is urgently demanded to satisfy the power consumption requirements of intelligent coal mining equipment.

Wind-induced vibration energy harvesting can capture airflow energy in coal mine tunnels to enable self-powering of sensors, providing an effective solution to underground power supply challenges [5]. Common wind energy conversion mechanisms include magnetoelectric transduction [6], piezoelectric transduction [7], and triboelectric transduction [8]. Among them, piezoelectric transduction offers advantages such as high sensitivity, compact structure, and ease of integration—making it well suited to the harsh working environment of coal mines. However, traditional piezoelectric energy harvesters have a high starting wind speed, low power output at low wind speeds, and operate effectively only within a narrow wind speed range, which limits their practical applications.

To improve energy harvesting performance, many scholars have introduced nonlinear magnetic coupling to optimize vibration structures and broaden the effective operating range of piezoelectric harvesters. Erturk and Inman [9] established a linear theoretical model of piezoelectric cantilever beams and analyzed structural dynamic responses, laying a foundational theoretical framework for piezoelectric energy harvesting research. Ma et al. [10] developed a nonlinear dynamical model for an asymmetric tristable energy harvesting system; using the Melnikov criterion, they determined the onset conditions for chaotic behavior and demonstrated that the asymmetry in potential well geometry facilitates transitions between stable equilibria and significantly boosts voltage output under weak excitations. Zhou et al. [11] established the kinetic formulation of an asymmetric tristable harvester and explored its coexisting solution characteristics and energy enhancement mechanism, confirming that asymmetric structures readily achieve high-energy inter-well motion under weak excitation. Li et al. [12] introduced a hybrid bistable energy harvester that integrates piezoelectric and electromagnetic transduction mechanisms via magnetic coupling; comprehensive multiphysics simulations and experimental validation confirmed its ability to lower the galloping initiation wind speed and extend the effective wind speed interval for sustained power generation. Zhang et al. [13] optimized the dynamic performance of nonlinear harvesters via magnetic parameter tuning and clarified the optimal parameters and low-speed energy harvesting mechanism. Man et al. [14] designed a hybrid tri-stable piezoelectric energy harvester with asymmetric potential wells to enhance energy extraction from rotational vibration. Li et al. [15] developed a piezoelectric–electromagnetic flutter energy harvester leveraging magnetic coupling; experimental and analytical results confirmed its effectiveness in reducing the critical vibration amplitude required for onset of flutter and improving robustness across varying ambient conditions. Li et al. [16] proposed a broadband vortex-induced vibration power harvester based on nonlinear magnetic force, broadening the operational frequency band and enhancing energy collection capacity in complex flow fields. Su et al. [17] designed a nonlinear bidirectional VIV-based vibration harvester capable of adapting to multidirectional incoming flow and improving harvesting stability. Zhou et al. [18] applied the 0–1 test method to identify the motion states of nonlinear VIV harvesters, revealing their dynamic evolution mechanism. Chen et al. [19] proposed a tunable curved-beam bistable piezoelectric harvester and clarified how structural and excitation parameters govern vibration and power generation performance. Zhang et al. [20] optimized a bistable piezoelectric structure and verified its ability to reduce excitation thresholds and improve energy efficiency in weak-vibration environments. Zhang et al. [21] established a gravity-dependent composite-beam tristable energy harvesting model and revealed that gravity can modulate steady-state responses and broaden the operational frequency band. Ma et al. [22] investigated the nonlinear kinetic features of multistable wake-galloping harvesters and revealed the governing law of multistable parameters on vibration responses and energy output. Li et al. [23] proposed a bistable harvester with nonlinear elastic constraints and systematically analyzed its dynamic response and energy output. Chen et al. [24] established the kinetic model of a magnet-connected straight–curved composite-beam harvester and explored its structural vibration characteristics.

The forces exerted by fluids on structures are multidirectional. In recent years, arched and multi-directional structures have been widely studied to improve environmental adaptability. Lin et al. [25] proposed a pipeline-adaptive downstream piezoelectric wind energy harvester to improve energy harvesting efficiency under directional airflow. Sun et al. [26] designed a three-dimensional tunable multi-directional vibration energy harvester that enables multi-mode energy harvesting and enhances performance under random excitation. Xia et al. [27] developed an arched-beam vortex-induced vibration energy harvester that overcomes the insufficient airflow-angle adaptability and limited lock-in bandwidth of traditional devices. Xiao et al. [28] fabricated a flexible arched triboelectric–piezoelectric hybrid energy harvester that enables dual-mechanism synergistic power generation and improves energy harvesting performance under complex operating conditions. Zhang et al. [29] developed a linear-arch composite beam for multidirectional vibration energy harvesting.

Existing studies have fully verified that introducing structural nonlinearity can substantially improve the overall performance of vibration energy harvesting systems. For wind energy harvesting applications, nonlinear dynamic characteristics can effectively reduce the critical startup airflow velocity, expand the usable airflow velocity bandwidth, and significantly enhance the output performance and environmental adaptability of harvesters under low-wind conditions. Most existing studies have employed straight beams combined with permanent magnets to achieve only single magnetic coupling bistability with nonlinear behavior, and the composite nonlinear mechanism involving the intrinsic geometric nonlinearity of a linear-arch combined beam coupled with magnetic bistability has not yet been investigated. To address the unique environmental conditions in underground coal mines—such as low wind speeds and multidirectional coupled airflow loads—this paper proposes a magnetic-coupled bistable composite beam wind energy harvester (BCBWEH). The design incorporates a novel combination of geometric nonlinearity from the linear-arch beam structure and magnetic coupling nonlinearity, which represents an unexplored aspect in current research. The remainder of this paper is organized as follows. Section 2 details the geometric layout of the BCBWEH, develops analytical models for magnetic force, structural restoring force, and galloping aerodynamic force, and establishes the system’s dynamic governing equation via the lumped-parameter method. Section 3 investigates the static bifurcation behavior and potential energy evolution as functions of magnet spacing, and numerically explores the influences of wind excitation intensity, initial equilibrium position, and magnet separation distance on the system’s dynamic responses. Section 4 elaborates the experimental test platform and the fabricated sample to validate the preceding numerical predictions. Finally, major conclusions are drawn in Section 5.

2. Configuration Design and Modeling of BCBWEH

2.1. Configuration Design of the BCBWEH

Figure 1 presents a schematic illustration of the bistable composite-beam wind energy harvester (BCBWEH), comprising four principal components: a polyvinylidene fluoride (PVDF) piezoelectric film, a straight–curved composite beam, a bluff body, and a pair of coaxially aligned permanent magnets configured in a repulsive orientation. The composite beam integrates a rigid straight segment with a compliant arch-shaped segment; the arch is clamped at one end, while the straight segment extends freely, with the bluff body rigidly affixed to its distal tip. The PVDF film serves as the primary electromechanical transduction element. It is bonded to the top surface of the beam using epoxy resin. The load resistance R is infinite, and the system is in an open-circuit state. Magnet A is positioned coaxially at the beam’s free end, while Magnet B is secured on a fixed base directly facing Magnet A. As indicated in the figure, L denotes the total lateral length of the composite beam, and $d_x$ specifies the initial center-to-center horizontal separation between the two magnets along the x-axis. Under aerodynamic excitation, the bluff body drives oscillatory bending deformation of the beam, inducing strain in the PVDF layer and thereby facilitating electromechanical conversion of vibrational mechanical energy into electrical power.

2.2. Mathematical Model of the BCBWEH

2.2.1. Formulation of Nonlinear Magnetic Interaction Force

Accurate modeling of the energy harvester’s dynamic response necessitates a quantitative characterization of the nonlinear magnetic coupling between Magnet A and Magnet B. In this study, an analytical formulation for the system’s magnetic interaction is derived using the magnetic dipole approximation. The relative layout of Magnet A and Magnet B is illustrated in Figure 2.

The magnetization intensities of Magnet A and Magnet B are defined as $M_A$ and $M_B$, respectively. $S_A$ and $S_B$ represent the surface areas of the two permanent magnets, and e denotes the magnet width. Within the framework of the magnetic dipole approximation, the analytical expression for the magnetic interaction between permanent magnets A and B is derived as follows:

$$F_{AB}={\displaystyle \frac{K_{AB}}{4\pi }}\left({\displaystyle \frac{\Delta y_{31}}{r_{31}^3}}-{\displaystyle \frac{\Delta y_{32}}{r_{32}^3}}-{\displaystyle \frac{\Delta y_{41}}{r_{41}^3}}+{\displaystyle \frac{\Delta y_{42}}{r_{42}^3}}\right)$$

(1)

where $K_{AB}={\mu }_0{M}_A{S}_A{M}_B{S}_B$, $\Delta y_{31}=u$, ${\displaystyle r_{31}=\sqrt{{\left(e+\mathrm{d}x\right)}^2+u^2}}$, $\Delta y_{32}=u+e\sin \theta$, ${\displaystyle r_{32}=\sqrt{{\left(\mathrm{d}x\right)}^2+{\left(u+e\sin \theta \right)}^2}}$, $\Delta y_{41}=u$, ${\displaystyle r_{41}=\sqrt{{\left(2e+\mathrm{d}x\right)}^2+u^2}}$, $\Delta y_{42}=u+e\sin \theta$, ${\displaystyle r_{42}=\sqrt{{\left(e+\mathrm{d}x\right)}^2+{\left(u+e\sin \theta \right)}^2}}$. Here, ${\mu }_0$ denotes the permeability of free space.

Accordingly, the total potential energy associated with the interaction between the two magnets is expressed as follows:

$$U_{AB}=\int F_{AB}\mathrm{d}u$$

(2)

2.2.2. Parameter Fitting of the Nonlinear Restoring Force

Unlike traditional straight beams, the proposed composite beam includes an arched segment that exhibits significant geometric nonlinearity. To accurately characterize its nonlinear restoring behavior, quasi-static loading tests were conducted using a YLK-10 high-precision force sensor. Multiple independent force–displacement datasets were averaged to suppress measurement noise, followed by fitting using a cubic polynomial regression based on the least squares method. The empirical relationship between the restoring force $F_R$ and the beam end displacement u is given by:

$$F_R=k_{1}u+k_2{u}^2+k_3{u}^3$$

(3)

where $k_1$, $k_2$, and $k_3$ represent the fitted stiffness parameters.

Figure 3 shows the experimentally measured force–displacement response of the composite beam’s nonlinear elastic restoring force, along with the best-fit cubic polynomial curve. Regression analysis yielded stiffness parameters $k_1=30.50\mathrm{N}/\mathrm{m}$, $k_2=54.13\mathrm{N}/\mathrm{m}$, and $k_3=-7241.28\mathrm{N}/\mathrm{m}$, all rounded to two decimal places. Statistical evaluation indicates excellent goodness of fit, with $R^2=0.99984$, demonstrating that the cubic polynomial model robustly captures the geometric nonlinear behavior characterizing the beam’s restoring properties.

Therefore, the potential energy expression of the structural restoring force can be derived as follows:

$$U_R=\int F_R\mathrm{d}u$$

(4)

2.2.3. Galloping Aerodynamic Force Model

Galloping is a typical flow-induced vibration of bluff bodies under unsteady aerodynamic forces. The square-section bluff body adopted in this study effectively excites galloping responses while suppressing vortex-induced oscillations [30]. Since the natural period of galloping oscillation is approximately $2\pi /{\omega }_s$ far exceeding the time scale of ambient airflow passing over the bluff body—the quasi-steady assumption is employed to model the aerodynamic forces. The in-line fluid forces acting on the spoiler are described by the Morison equation, with the specific expression given as follows [24]:

$$F_g={\displaystyle \frac{1}{2}}\rho l{v}^2{C}_L\sin \left({\omega }_{s}t+\varphi \right)$$

(5)

$${\omega }_s={\displaystyle \frac{2\pi {\mathit{S}}_{\mathit{t}}v}{D}}$$

(6)

where $F_g$ is the lift force induced by galloping, $\rho$ denotes the fluid density, l represents the characteristic length of the bluff body, v is the incoming flow velocity, $C_L$ is the lift coefficient, ${\omega }_s$ is the vortex shedding frequency, $S_t$ is the Strouhal number, and $\varphi$ denotes the phase angle between the fluid-induced force and the structural displacement.

2.2.4. Motion Equation of the Energy Harvester

The $\mathit{M}$-$\mathit{C}$-$\mathit{K}$ model is a classic lumped-parameter model widely used in vibration dynamics research. Based on this, the proposed bistable composite-beam wind energy harvester is simplified into a single-degree-of-freedom system within the theoretical framework of the $\mathit{M}$-$\mathit{C}$-$\mathit{K}$ model. By incorporating the aforementioned nonlinear magnetic force, structural restoring force, and flutter aerodynamic forces, the governing equation of this energy harvester can be expressed as follows:

$$M\ddot{u}+C\dot{u}+F_{AB}+F_R-\vartheta v_t=F_g$$

(7)

$$\vartheta {\dot{u}}_t+C_P{\dot{v}}_t+{\displaystyle \frac{v_t}{R}}=0$$

(8)

where M is the equivalent mass, C represents the equivalent damping coefficient obtained via the damping decay method, $\vartheta$ denotes the electromechanical coupling coefficient, $C_P$ is the equivalent capacitance, $v_t$ is the output voltage, and R is the external load resistance.

The specific expression of the equivalent mass is given as follows [29]:

$$M={\displaystyle \frac{33}{144}}{M}_b+M_t$$

(9)

where $M_b$ is the mass of the composite beam and $M_t$ is the mass of the terminal attachment.

The calculation formula of the electromechanical coupling coefficient is presented as follows:

$$\vartheta ={\displaystyle \frac{1}{2}}{e}_{31}b\left(h_P+h_S\right){\varphi }^{\prime }\left(L\right)$$

(10)

where $e_{31}$ is the piezoelectric constant, b is the width of the composite beam, $h_P$ is the thickness of the piezoelectric layer, $h_S$ is the thickness of the substrate layer, and $\varphi \left(x\right)$ denotes the mode shape function. For the single-degree-of-freedom system, the mode shape function is defined as $\varphi \left(x\right)=1-\cos \left({\displaystyle \frac{\pi x}{2L}}\right)$.

The system damping coefficient can be obtained using the damping decay method. As shown in Figure 4, this is the decay curve of the BCBWEH system, and the damping coefficient expression is as follows:

$$C=2\zeta \sqrt{MK}$$

(11)

In the equation, $\zeta$ is the damping ratio, M is the equivalent mass, and K is the equivalent stiffness.

The damping ratio calculation formula is as follows:

$$\zeta ={\displaystyle \frac{\delta }{\sqrt{4{\pi }^2+{\delta }^2}}}$$

(12)

In the equation, $\delta$ is the logarithmic reduction rate.

To reduce measurement error, the logarithmic decrement $\delta$ is calculated using multiple amplitude peaks on the same side, with the calculation formula as follows:

$$\delta =\ln \left({\displaystyle \frac{A_i+A_{i+1}+A_{i+2}+A_{i+3}+\cdots +A_{i+n-1}}{A_{i+1}+A_{i+2}+A_{i+3}+A_{i+4}+\cdots +A_{i+n}}}\right)$$

(13)

3. Simulation Analysis of the Energy Harvester

3.1. Static Bifurcation Analysisl

Table 1. Geometric dimensions and material properties of the BCBWEH energy harvester.

	Parameter	Value
Substrate layer	Length × Width × Height	100 × 20 × 0.2 mm3
	Material density	8500 kg/m3
	Elastic modulus	128 × 109 Pa
Piezoelectric layer	Length × Width × Height	100 × 200 × 0.11 mm3
	Material density	1750 kg/m3
	Elastic modulus	3 × 109 Pa
Magnet (NdFeB, Grade N30)	Length × Width × Height	10 × 10 × 5 mm3
	Material density	7500 kg/m3
	Magnetization	0.955 T
Bluff body	Length × Width ×	40 × 40 × 100 mm3
	Material density	50 kg/m3

systematically lists the geometric dimensions and constitutive material properties of the substrate layer, PVDF piezoelectric film, bluff body, and permanent magnets used in the BCBWEH prototype.

To obtain the static equilibrium solutions of the system, all time-dependent dynamic terms and aerodynamic excitation terms in Equation (7) are set to zero, i.e., $\dot{u}=\ddot{u}=\dot{v}=\ddot{v}=F_g=0$. The static equilibrium solution of the system is derived as follows:

$$F_{AB}+F_R=0$$

(14)

Based on Equation (14), the static solutions of the system in terms of magnet spacing $dx$ are illustrated in Figure 5. The setup exhibits obvious bifurcation behavior with changes in the permanent magnet dipole moment.

Figure 5 reveals the typical bistable dynamic evolution characteristics of the nonlinear system. The magnet spacing $dx\approx 16\mathrm{mm}$ represents the critical threshold for static bifurcation; as the magnet spacing varies, the system undergoes a dynamic transition between monostable and bistable states. When $dx>16\mathrm{mm}$, the system has only one stable equilibrium point at zero displacement ($u=0$), exhibiting purely monostable behavior with no bistable response. When $dx<16\mathrm{mm}$, the equilibrium points bifurcate, forming three equilibrium positions. In this case, the zero-displacement region OB becomes an unstable equilibrium zone, while regions AB and BC become stable equilibrium zones, corresponding to the typical potential barrier and potential well features in a bistable structure, thereby enabling the system to exhibit stable bistable behavior. Under this configuration, appropriate external excitation can induce large-amplitude inter-well oscillations, thereby meeting the ideal operating conditions required for vibration energy harvesting.

3.2. Potential Energy Distribution Characteristics of the System

The total potential energy of the system consists of two parts: the magnetic potential energy stored in the structure and the elastic strain energy, as expressed by the equation below.

$$U=U_{AB}+U_R$$

(15)

As illustrated in Figure 6, the magnetic force $F_m$ increases with the growth of displacement u. It is worth noting that the positive and negative signs only represent the force direction rather than the magnitude. Two extreme values of $F_m$ exist in the positive and negative displacement intervals, which form two independent potential energy traps during the vibration of the BCBWEH.

Figure 7 clearly demonstrates that the system possesses two asymmetric potential wells. Such asymmetric characteristics are mainly attributed to the inherent nonlinear asymmetry of the structural restoring force. As illustrated in Figure 7b, with the magnet separation fixed at $d_x=8\mathrm{mm}$, the left and right potential wells display markedly unequal depths, specifically fulfilling $\Delta U_2>\Delta U_3$.

The magnet spacing $d_x$ is a key parameter determining the system’s magnetic potential energy and total potential energy, directly influencing the system’s steady-state behavior and potential energy landscape. As $d_x$ increases, the system gradually shifts from bistability toward monostability. Figure 7b compares the potential energy curves under three typical operating conditions (corresponding to magnet spacings of $5\mathrm{mm}$, $8\mathrm{mm}$, and $16\mathrm{mm}$). When $d_x=5\mathrm{mm}$ and $8\mathrm{mm}$, the potential energy curve displays two potential wells and a single barrier, corresponding to two stable equilibrium points and one unstable equilibrium point. In contrast, when $d_x=16\mathrm{mm}$, the system is in a monostable state, with only one equilibrium position. In a bistable configuration, the system achieves the maximum potential barrier height and potential well depth at $dx=5\mathrm{mm}$. When $dx$ increases to 8 mm, the potential barrier height decreases significantly, which satisfies the relationship $\Delta U_2<\Delta U_1$. This potential energy characteristic indicates that compared with the condition of $dx=8\mathrm{mm}$, higher excitation energy is required for the system to generate large-amplitude inter-well vibration at $dx=5\mathrm{mm}$.

3.3. Dynamic Response Analysis

3.3.1. Effects of Incoming Wind Velocity on the System’s Dynamic Behaviors

Wind field excitation significantly influences the transient response characteristics of the system. In this study, the bluff body with a square cross-section is employed. This configuration effectively induces galloping vibration while suppressing vortex-induced vibration. In the numerical simulation, the small rotational angle of the bluff body during motion is omitted from the model, and aerodynamic force simulations are conducted under zero-angle-of-attack conditions across various incoming airflow velocities. The lift coefficient data are processed using the Fast Fourier Transform (FFT) to extract the dominant excitation frequency f and the Strouhal number $St$ of the wind load over a range of wind speeds. This paper focuses on low-velocity conditions in underground coal mines, studying only wind speeds up to $4\mathrm{m}/\mathrm{s}$. Figure 8 displays the relevant outcomes.

Based on the above analysis, the left stable equilibrium position is taken as the system’s initial state to investigate how varying incoming wind speeds affect its transient response characteristics. The system’s dynamic governing equations are numerically integrated using the ODE45 solver. To accurately characterize the system’s dynamic response, a time interval of 20–22 s following stabilization of the transient response is selected for subsequent data analysis. With the magnet spacing fixed at $d_x=8\mathrm{mm}$, three representative incoming wind speeds—1 m/s, 3 m/s, and 4 m/s—are chosen for dynamic analysis. The corresponding system dynamic responses and time-domain voltage outputs are shown in Figure 9. The system’s dynamic behavior and output characteristics evolve distinctly with incoming wind speed. At the low wind speed of 1 m/s (Figure 9a), magnetic forces dominate the response, confining the vibration trajectory entirely within the left potential well; the system exhibits only small-amplitude intra-well oscillation, yielding a weak output voltage. When the wind speed increases to 3 m/s (Figure 9b), external excitation is insufficient to overcome the barrier; the system continues to oscillate within the left potential well, but both the amplitude and voltage output are higher compared to the 1 m/s condition. As the wind speed further rises to 4 m/s (Figure 9c), sufficient external excitation enables the system to surmount the potential barrier and sustain stable, large-amplitude cross-well oscillation. Under these operating conditions, the system performs optimally, with voltage output significantly better than under low wind speed conditions. This indicates that, under the same configuration, the greater the external driving force, the easier it is for the system to achieve inter-well oscillation, thereby enabling high-energy output.

3.3.2. Effect of Initial Position on System Dynamic Performance

Based on previous analysis of magnetic force and system potential energy characteristics, this bistable system exhibits a double-well potential energy distribution, with the two potential wells located precisely at the positions where the magnetic force amplitude is maximized. According to bifurcation characteristics, the system’s stable equilibrium positions are identified as the minima of the potential energy function—meaning that the lowest potential energy state corresponds to the stable equilibrium condition of the energy harvester. This study focuses on the static and dynamic response characteristics of the system when the initial velocity is zero; thus, dynamic responses under external wind field excitation are investigated by setting the system’s initial state at different stable equilibrium positions. In the numerical simulations, the wind speed is fixed at 3 m/s, and calculations are performed using initial displacements corresponding to the left and right stable equilibrium positions.

Figure 10 shows phase-plane trajectories and time-domain voltage output profiles for motions initiated at the left and right stable equilibrium positions. At $d_x=5\mathrm{mm}$, the configuration features deep potential wells and a high potential barrier. Under this configuration, the wind excitation provides insufficient energy to overcome the potential barrier, preventing the device from sustaining large-amplitude cross-well oscillation. As illustrated in Figure 10a,b, with initial conditions set at the two stable equilibrium positions, the device exhibits only small-amplitude intra-well oscillation and no cross-well hopping, thereby producing only a small peak output voltage. Moreover, phase-plane trajectories originating from the two stable equilibrium positions exhibit pronounced asymmetry. This asymmetry arises from the unequal distribution of elastic restoring force along the composite beam on either side of the unstable equilibrium position, inducing asymmetric vibrational behavior in the entire system.

Compared with a $d_x=5\mathrm{mm}$ operating condition, the system at $d_x=8\mathrm{mm}$ exhibits clear differences in potential energy characteristics and dynamic response behavior. Figure 11 shows phase-plane trajectories and time-domain voltage output waveforms for motions initiated at the left and right stable equilibrium positions. When the system starts from the left stable equilibrium position (Figure 11a), it remains confined within the left potential well and undergoes large-amplitude intra-well oscillation under wind excitation, generating an output voltage significantly higher than that obtained under the $d_x=5\mathrm{mm}$ condition. When the system is initiated from the right stable equilibrium position (Figure 11b), the input energy from the incoming airflow is sufficient to overcome the potential barrier, driving the system into sustained, large-amplitude cross-well oscillation. Under these conditions, the cantilever beam experiences significant structural strain, enabling the energy harvester to achieve optimal power generation performance, with an output voltage amplitude higher than when the system starts from the stable equilibrium position on the left side. This indicates that, under the same conditions, starting from a shallower potential well helps generate larger amplitude oscillations, thereby improving power generation performance.

At $d_x=16\mathrm{mm}$, the large magnet spacing nearly eliminates the magnetic force effect, and the system completely degenerates from a bistable structure to a monostable one. In this state, the system has only one unique stable equilibrium position—namely, the zero point of the structural restoring force. Under external wind field excitation, the system exhibits regular periodic vibration centered about $u=0\mathrm{mm}$. The results in Figure 12 indicate that the output voltage amplitude under this monostable condition is generally lower than that under the aforementioned bistable conditions. Moreover, the voltage response characteristics are independent of the system’s initial startup position.

Figure 13 presents the bifurcation diagram of system displacement versus excitation frequency under fixed magnet spacing $d_x=8\mathrm{mm}$ and a dimensionless excitation amplitude of $0.32$, clearly illustrating the evolution of the system’s dynamic behavior with frequency. When the dominant excitation frequency increases from 0 to $6\mathrm{Hz}$, the system displacement stabilizes near $-7\mathrm{mm}$, and the stroboscopic sampling points converge onto a horizontal line, indicating stable single-period oscillation. In the range of 6–8 Hz, the sampling points become scattered, showing a gradual transition from single-period to multi-period oscillations, accompanied by a slight tendency toward bistable jumps. As the excitation frequency further increases to 8–12 Hz, the displacement samples clearly disperse into multiple distinct clusters, confirming that the system enters a coupled oscillation state—resulting from combined effects of inter-well jumps in bistability and multi-period vibrations. Between 12 and $17\mathrm{Hz}$, the bifurcation diagram exhibits a pronounced upward trend: system displacement steadily rises from $-25\mathrm{mm}$ to $-5\mathrm{mm}$, primarily characterized by regular large-amplitude periodic motion. When the frequency reaches 17–20 Hz, the displacement samples are densely distributed across the entire range from $-25\mathrm{mm}$ to $+15\mathrm{mm}$ without evident periodicity, indicating significant chaotic oscillations. The system displays irregular and random transitions between the two potential wells. In this study, an inflow velocity of $3\mathrm{m}/\mathrm{s}$ corresponds to a dimensionless excitation amplitude of $0.32$ and a primary frequency component of $11.2\mathrm{Hz}$. Consistent with the bifurcation evolution, the system operates under typical bistable oscillation mode under these conditions, strongly supporting and effectively validating the dynamic response results shown in Figure 11b.

3.3.3. Effect of Magnetic Moment on System Dynamic Performance

Tuning the distance between the magnets modifies the height of the potential barrier, thereby significantly influencing the system’s transient response and energy harvesting performance. To further elucidate the underlying mechanisms through which magnetic interactions govern system dynamics, comparative simulations were conducted for three representative magnet separations: 5 mm, 8 mm, and 16 mm. During numerical simulations, the incoming wind speed was held constant at 2 m/s, and all runs were initialized from the right-side stable equilibrium position. The resulting dynamic evolution trajectories and time-domain voltage outputs under these distinct operating conditions are presented in Figure 14.

As shown in Figure 14, when the magnet spacing is $5\mathrm{mm}$ (Figure 14a), the system exhibits a deep potential well. Under a wind speed excitation of $2\mathrm{m}/\mathrm{s}$, the provided energy is insufficient to trigger large-amplitude inter-well oscillations; thus, the device only undergoes small-amplitude intra-well oscillations, resulting in a low peak output voltage. When the magnet spacing increases to $8\mathrm{mm}$ (Figure 14b), the potential barrier height decreases significantly. Under the same incoming wind excitation, the device can overcome the barrier and achieve stable large-amplitude inter-well oscillations, leading to a substantial increase in output voltage compared to the case with $d_x=5\mathrm{mm}$, and a significant enhancement in energy harvesting capability. When the magnet spacing further increases to $16\mathrm{mm}$ (Figure 14c), the magnetic interaction becomes negligible, and under light wind excitation, the device generates only a small output voltage. Meanwhile, due to the structure’s nonlinear elastic restoring characteristics, the phase plane trajectory exhibits an asymmetric shape.

For the wind speed of $2\mathrm{m}/\mathrm{s}$, the corresponding excitation amplitude and dominant excitation frequency are $0.086$ and $8.16\mathrm{Hz}$, respectively. Under this constant excitation condition, the dynamic behavior of the system exhibits significant piecewise evolution characteristics with the variation in magnet spacing $d_x$, and the corresponding stroboscopic displacement bifurcation diagram is illustrated in Figure 15. When $d_x$ ranges from $5\mathrm{mm}$ to $9\mathrm{mm}$, the system displacement remains stable within the range of $-9\mathrm{mm}$ to $-7\mathrm{mm}$, and the stroboscopic sampling points lie on a smooth oblique line. The system maintains stable periodic vibration within a single potential well without inter-well transition behavior. As $d_x$ increases to 9–$11\mathrm{mm}$, the system displacement response range expands to $-10\mathrm{mm}$ to $+5\mathrm{mm}$, and bidirectional positive and negative displacements are simultaneously observed. The vertically discrete points in the bifurcation diagram correspond to multi-period motion characteristics. This indicates that the potential barrier height matches well with the external excitation energy in this interval, enabling the system to achieve alternating transitions between the two potential wells and effectively activate the bistable effect. When $d_x$ further increases to 11–$20\mathrm{mm}$, the displacement sampling points rapidly converge to the vicinity of $0\mathrm{mm}$ and form a horizontal steady-state line. The system degenerates into small-amplitude single-period vibration with monostable characteristics. With the continuous attenuation of the magnetic force, the inherent bistable nonlinearity completely disappears.

4. Experimental Validation

To assess the fidelity of the preceding numerical simulations, an experimental setup tailored for the BCBWEH system was implemented, and a physical prototype was fabricated. The layout of this experimental configuration is depicted in Figure 16. The measurement system consists of a computer, a frequency controller, an HG-C1200 miniature laser displacement sensor ((Panasonic Corporation, Kadoma City, Japan) Measurement range: $-80$ mm to $+80$ mm; displacement repeatability accuracy: 200 μm), and a DSOX3024T oscilloscope ((Keysight Technologies, Santa Rosa, CA, USA) Sampling frequency: 20 kHz; voltage measurement range: $\pm 50$ V); airflow excitation is provided by a wind tunnel. This platform enables regulation of airflow excitation, measurement of structural displacement, and acquisition of voltage signals. Figure 17 illustrates the detailed component layout of the BCBWEH prototype, including the fixed frame, base plate, polyvinylidene fluoride (PVDF) piezoelectric film, composite cantilever beam, bluff body, and permanent magnets. The fixed end of the composite beam is securely anchored to the base plate, while the permanent magnets are mounted at the tip of the bluff body and arranged symmetrically. A thin PVDF piezoelectric film is evenly adhered to the beam’s surface to transform mechanical strain into measurable voltage output. All structural dimensions and testing parameters of the experimental setup are highly consistent with those used in the numerical simulation, ensuring the validity of the experimental verification.

During testing, the fabricated energy harvester was securely mounted inside the test section of the wind tunnel. The fan speed was adjusted via a frequency controller to vary the incoming airflow velocity. A HG-C1200 miniature laser displacement sensor was used to collect real-time dynamic displacement data of the composite beam; subsequently, numerical differentiation of the sampled displacement data yielded the structural vibration velocity. Meanwhile, a DSOX3024T oscilloscope simultaneously recorded the piezoelectric output voltage from the prototype. This setup enabled synchronized measurement of both the structural dynamic characteristics and the electrical output signals.

It is well known that the load has a crucial impact on the system’s output. To further investigate this effect, we selected a wind speed of 3 m/s and a magnetic moment of 8 mm, initiating oscillation from the static equilibrium position on the right side. The system’s output voltage and power as functions of resistance are shown in Figure 18. From Figure 18a, it can be observed that the output voltage initially increases rapidly with increasing resistance, then gradually stabilizes. As seen in Figure 18b, the output power first rises sharply and then slowly decreases with increasing resistance. According to circuit theory, maximum power output occurs when the external load matches or closely approximates the internal resistance. Therefore, the resistance value corresponding to the peak power represents the optimal load for the system, which is 200 kΩ.

Figure 19 shows the experimental phase trajectory and time-domain voltage curve of the BCBWEH system under a magnet spacing $d_x=8\mathrm{mm}$ and an incoming wind speed of $4\mathrm{m}/\mathrm{s}$. In this experiment, oscillations were consistently initiated from the stable equilibrium position on the left side of the system. Under this wind speed condition, sufficient airflow excitation enables the system to overcome the potential barrier effect, allowing the energy harvesting device to stably switch between two potential wells and achieve continuous large-amplitude vibration-based power generation. Numerical simulation results (Figure 9c) indicate that, under the same conditions, the system can sustain stable inter-well oscillations. Due to the asymmetric restoring force, the phase trajectory exhibits significant asymmetry, with larger vibration displacement on the left side than on the right; the total vibration displacement reaches up to 110 mm, and the steady-state output voltage can reach as high as 40 V. The corresponding experimental results shown in Figure 19 also display clear asymmetric dynamic characteristics consistent with the simulation trends. Under experimental conditions, the effective vibration displacement is approximately 40 mm, and the steady-state output voltage is about 18 V. Although there are certain discrepancies in numerical amplitudes between simulation and experiment, the evolution patterns of vibration, the asymmetric features of phase trajectories, and the trends in voltage output are highly consistent, effectively validating the accuracy and effectiveness of the numerical simulation model presented in this study.

Figure 20 shows the measured phase plane trajectory and time-domain output voltage curve of the BCBWEH system under an inflow wind speed of $3\mathrm{m}/\mathrm{s}$ and a magnet spacing of $d_x=8\mathrm{mm}$. To investigate the influence of initial equilibrium states on the system’s oscillation characteristics and power generation performance, excitation experiments were conducted starting from both the left and right stable equilibrium positions. As shown in Figure 20a,b, the initial excitation position significantly affects the dynamic response of the system. When the system is initiated from the left stable equilibrium point, structural motion remains confined within a deeper single-side potential well, resulting in only small-amplitude intra-well vibrations with a displacement of approximately $4\mathrm{mm}$ and a steady-state output voltage of merely $0.8\mathrm{V}$. In contrast, when the system starts from the right stable equilibrium point, airflow excitation effectively triggers large-amplitude inter-well oscillations, significantly enhancing energy harvesting performance, with vibration displacements reaching about $35\mathrm{mm}$ and a steady-state output voltage increasing to approximately $11\mathrm{V}$. The corresponding numerical simulation results are shown in Figure 11. In the simulations, when initialized from the left equilibrium position, the system remains restricted to oscillations within the left potential well, with displacement distributed solely along the negative axis, an effective vibration amplitude of about $10\mathrm{mm}$, and a steady-state output voltage of approximately $4.3\mathrm{V}$. However, when initiated from the right equilibrium position, the system exhibits pronounced asymmetric inter-well oscillations, with a vibration amplitude of around $30\mathrm{mm}$ and a steady-state output voltage reaching approximately $11\mathrm{V}$. Both experimental and simulation results indicate that the power generation performance from the right initial equilibrium position is significantly superior to that from the left, and the two cases show high consistency in vibration patterns and voltage output trends, effectively validating the accuracy of the numerical model in predicting dynamics dependent on initial conditions.

Figure 21 shows the measured phase plane trajectory and periodic output voltage curve of the system under conditions where the magnet spacing $d_x=8\mathrm{mm}$, the initial oscillation position is at the right stable equilibrium point, and the incoming wind speed is constant at $2\mathrm{m}/\mathrm{s}$. Experimental results indicate that under these conditions, the BCBWEH system achieves stable large-amplitude oscillations across potential wells, with an effective vibration displacement of approximately $30\mathrm{mm}$ and a peak output voltage reaching up to $10\mathrm{V}$. The numerical simulation results for the same conditions are shown in Figure 14b, where the simulated system also exhibits stable inter-well vibrational response, with an effective vibration displacement of about $25\mathrm{mm}$ and a steady-state output voltage consistent with experimental data, approximately $10\mathrm{V}$. The trends in experimental and simulation data are highly consistent, effectively verifying the magnetic coupling structure’s ability to regulate the system’s nonlinear dynamic characteristics and power generation performance, further confirming the reliability of the device’s working mechanism and the accuracy of the numerical model.

The numerical simulation and experimental results show good agreement in overall trends, but there are some minor numerical discrepancies. The main reasons are as follows: (1) Although the wind tunnel is equipped with filtration devices to optimize airflow quality, the actual wind field cannot fully replicate the ideal, uniform flow field assumed in the numerical simulation; (2) During continuous vibration, the bending section of the composite beam undergoes tensile deformation, which introduces additional deformation and system errors, affecting the displacement and velocity data measured by the laser sensor, thereby reducing measurement accuracy.

The displacement sensor has an effective measurement range of 160 mm, with a full-scale linearity deviation of $\pm 0.2\%$ FS, corresponding to a maximum linearity limit error of $0.32$ mm, and a measurement repeatability limit error of 200 μm. By combining these two limit error components using the root sum square method, the total system limit error is approximately $0.377$ mm. The voltage acquisition module uses an oscilloscope to capture signals, with the vertical channel’s full-scale allowable error being $\pm 2\%$ FS. For this experiment, the voltage acquisition range is set to 0–$50\mathrm{V}$, corresponding to a maximum measurement limit error of 1 V within this range. This paper relies on the original equipment manufacturer’s manual to perform quantitative error calculations, fully demonstrating the validity and reliability of the experimental data.These inherent differences between idealized numerical assumptions and actual experimental conditions are the primary cause of the minor numerical deviations, yet they remain within an acceptable range for engineering research.

5. Conclusions

This work establishes the nonlinear magnetic governing equations and dynamic model for the proposed BCBWEH harvester. Through a combination of numerical simulation and experimental measurement, the effects of magnetic moment configuration and incoming airflow excitation on the system’s static and dynamic behaviors, as well as its power generation capability, are systematically investigated. Key findings are summarized below.

(1) The asymmetric nonlinear restoring force characteristics exhibited by the composite beam result in an asymmetric distribution of potential wells on either side of the system. This is the fundamental reason why significant differences in dynamic response and output performance occur when the system is initiated from different initial equilibrium positions.

(2) The initial equilibrium position strongly governs the system’s dynamic response and energy harvesting performance. When the system starts from the shallower potential well, the potential barrier height is lower, enabling the system to readily overcome the barrier under the same external excitation. In this case, large-amplitude inter-well oscillations are readily triggered, accompanied by larger oscillation amplitude and higher output voltage. In contrast, when the system is initiated from the deeper potential well, the higher potential barrier impedes inter-well motion, and the structure predominantly exhibits small-amplitude intra-well oscillations, thereby degrading both dynamic response and energy harvesting performance.

(3) At the optimal magnet spacing of $dx=8\mathrm{mm}$, the dynamic behavior of the system evolves regularly with increasing wind excitation intensity. As the wind speed rises, the system successively undergoes small-amplitude intra-well vibration, large-amplitude intra-well vibration, and eventually stable large-amplitude inter-well vibration.

(4) The magnet spacing parameter is the core factor governing the steady-state characteristics and vibration modes of the system. With increasing magnet spacing, the system’s dynamic behavior undergoes a sequential evolution from single-side intra-well oscillation to bilateral inter-well oscillation and finally to monostable intra-well oscillation, and the system gradually transitions from a bistable state to a monostable state. Within the bistable working range, reasonable regulation of magnet spacing can optimize the potential well depth and barrier height, reduce the energy threshold for inter-well transition, and facilitate large-amplitude inter-well vibration under constant external excitation. Accordingly, the piezoelectric energy harvesting efficiency and output performance can be effectively improved.