# Enhancing Wind Energy Harvesting Using Passive Turbulence Control Devices
## 2 Physical Mathematical Model  
### Physical and Mathematical Model  
Galloping vibration can be a kind of bending vibration of the cantilever-based structure caused by the self-excitation of airflows [55; 56; 57; 58; 59]. For the VIV vibration transforming the galloping vibration, [PERSON] et al. [31; 32] studied the influence of attached rods with cross-sections and equipping angles on the transformation from vortex-induced vibration to galloping vibration, and they explored the influence of attack angle on transverse force coefficients. Inspired by [PERSON] et al. [31; 32], [PERSON] et al. [33; 34], and our previous study [13], the physical model in our paper uses quite a different structure called a Passive Turbulence Control (PTC) device, which is made of sanding belts with very coarse surface. In addition, this paper focuses on the influence of different electrical circuits components, including electromechanical coupling strengths, wind speeds, and load resistances on the performance of the harvester. Considering a curved section structure mounted on a flexible support, the surface along the axial direction of 60\({}^{\circ}\) is symmetrical with two belts (to induce turbulence conditions for PTC). A previous study has proved that 60\({}^{\circ}\) Y-shaped attachments can arouse galloping phenomenon [13] and VIV is caused by the vortex shedding effect, while resonance occurs when the vortex shedding frequency is close to the natural frequency of the vibratory system. In other words, VIV is mainly caused by the vortex shedding effect. Equipping a Passive Turbulence Control (PTC) device on the bluff body can affect the formation of the vortex street behind the bluff body, and different flow-induced motions (FIM) can be observed depending primarily on the circumferential location of the two strips [60; 61]. Furthermore, galloping can be aroused when a negative nonlinear aerodynamic damping appears for the equipment of the PTC in this work. In this study, two compact belts are designed to assist the passive turbulence control to improve the performance of the wind energy harvester, as shown in Figure 1. As shown in Figure 1a, wind speed \(U\) is perpendicular to the cylindrical section. Based on the Euler-Bernoulli beam theory, Kirchhoff's law, piezoelectric effect, and galloping self-induced vibration, the aero-electromechanical coupling model of the energy harvester is able to be obtained in a distributed parameter model, which can be included into a lumped parameter model form [13; 62]. The structure shown in Figure 1a is simplified to an equivalent Single-degree-of-freedom (SDOF) mass-damping-spring (\(M\)-\(C\)-\(K\)) model in Figure 1b. The cantilever beam is equivalent to a spring oscillator, and \(\alpha\) is the angle of attack. Therefore, the governing equations can be expressed as:  
\[M_{eff}\ddot{y}(t)+C_{eff}\dot{y}(t)+K_{eff}y(t)+\theta V(t)=F_{y}(t) \tag{1}\]  
\[\frac{V(t)}{R}+C_{p}\dot{V}(t)-\theta\dot{y}(t)=0 \tag{2}\]  
where \(M_{eff}\) is the equivalent mass. \(M_{eff}=(33/140)m_{1}+m_{2}\), (\(m_{1}\) and \(m_{2}\) are the mass of the piezoelectric beam and the bluff body, respectively. \(C_{eff}=2\zeta\omega_{n}M_{eff}\) is the equivalent mechanical damping, where \(\zeta\) is the damping ratio. \(K_{eff}=\omega^{2}{}_{n}M_{eff}\) is the equivalent stiffness, where \(\omega_{n}\) is the natural frequency. \(F_{y}(t)\) is the aerodynamic force. \(y(t)\) is the displacement in the \(y\) direction. \(V(t)\) represents output voltage across the load resistance \(R\). \(\theta\) is the electromechanical coupling coefficient, which can be obtained by:  
\[\theta=\sqrt{(\omega_{oc}{}^{2}-\omega_{sc}{}^{2})M_{eff}C_{p}} \tag{3}\]  
where \(\omega_{oc}\) and \(\omega_{sc}\) are open circuit and short circuit frequencies, respectively. \(C_{p}\) is the piezoelectric capacitance. In order to analyze the performance of the harvester under the condition of strong and weak couplings, the dimensionless quantity \(k\) should be defined (\(k\) represents the electromechanical coupling strength).