## 3 Design of a Resonance-Tunable BWT  
### BWT Design and System Modeling  
The proposed BWT system consists of a cylinder mast, a rigid rod, and a variable-stiffness elastomer, as shown in Figure 1. For this research, the variable-stiffness elastomer is made of magnetorheological elastomers (MREs), a class of smart materials that are widely used for resonance-tuning applications.  
When the lift force \(F_{L}\) is exerted by the vortex shedding, the mast oscillates about the pin joint in a direction perpendicular to the force. Although the lift force \(F_{L}(t)\) is distributed along the mast surface, this study assumes that the lift force is exerted as a single-point force at the center point of the mast cylinder, which is denoted as \(l_{1}\) from the joint. Additionally,  
Figure 2: Conceptual change in the lock-in range and vibration-level gain due to structural resonance tuning in the BWT prototype.  
the shift moment by the center of gravity can be ignored if the mass centroid is designed to be very close to the rotation axis. The reaction forces to the rotation energy by the lifting force are provided by the stiffness (\(k_{mre}\)) and damping (\(c_{mre}\)) of the MREs. Note that the MREs provide the equivalent stiffness and damping values of the BWT.  
By assuming that the mast and rod are rigidly connected, the structure can be modeled as a first-order mass-spring-damper system that oscillates about the pin joint. Thus, the system dynamics due to the lift force of vortex shedding and the MRE reaction forces are expressed as  
\[J\ddot{\theta}(t)+c_{mre}l_{2}\dot{\theta}(t)+k_{mre}l_{2}\theta(t)=l_{1}F_{L} (t) \tag{6}\]  
where \(J\) represents the rotational moment of inertia of the mast, \(l_{1}\) is the point where the lift force \(f_{L}(t)\) is applied, and \(l_{2}\) is the length of the rod from the pin joint. Additionally, the MRE stiffness (\(k_{mre}\)) is a function of the nominal stiffness and a variable increment by an external magnetic field (\(B\)).  
\[k_{mre}=k_{mre,0}+\Delta k_{mre}(B) \tag{7}\]  
Assuming that the mast rotation \(\theta\) is small, the shear displacement of the MRE \(x\) due to the rod's oscillation can be approximated as \(x(t)\approx l_{2}\theta(t)\). Then, the BWT dynamics in terms of the displacement \(x\) can be expressed using Equation (6) as  
\[\frac{J}{l_{2}}\ddot{x}(t)+c_{mre}\dot{x}(t)+k_{mre}x(t)=\frac{l_{1}}{l_{2}}F_ {L}(t) \tag{8}\]  
The structural resonant frequency of the proposed BWT is calculated using Equations (5) and (8) as a function of the magnetic field.  
\[\begin{array}{l}f_{BWT}(B)=\frac{1}{2\pi}\sqrt{\frac{l_{2}^{2}}{J}k_{mre}(B) -\left(\frac{b^{2}}{2J}c_{mre}\right)^{2}}\\ =\frac{1}{2\pi}\sqrt{\frac{l_{2}^{2}}{J}\Delta k_{mre}(B)+C_{0}}\end{array} \tag{9}\]  
where \(C_{0}\) is a constant variable of \(C_{0}=\frac{l_{2}^{2}k_{mre,0}}{J}-\left(\frac{l_{2}^{2}}{2J}c_{mre}\right)^ {2}\). This equation shows that the structural resonant frequency can be tuned from the nominal state by increasing the MRE stiffness with the amount of \(\Delta k_{mre}(B)\).  
Due to the vortex-induced vibration, a periodic lift force that oscillates the BWT is generated. The BWT's vibration motion due to the periodic lift force is converted into the frequency domain to analyze the vibrations for various oscillation periods. Thus, the vibration displacement caused by the lift force in the frequency domain can be derived from Equation (8) as the transfer function of  
\[\frac{X(j\omega)}{F_{L}(j\omega)}=\frac{l_{1}l_{2}}{k_{mre}{l_{2}}^{2}-J\omega ^{2}+j\omega c_{mre}{l_{2}}^{2}}. \tag{10}\]