# Experimental Investigation on Flow-Induced Rotation of Two Mechanically Tandem-Coupled Cylinders
## 2 Theoretical Basis
### Dynamic Model of Flow-Induced Rotation (FIR) System  
#### 2.2.1 Pendulum System  
The pendulum system is the rotation part of the FIR system, including double circular cylinders in tandem arrangement and two rectangular diversion plates, as shown in Figure 1. The function of this diversion plate is to mechanically couple the cylinders and facilitate changing the spacing between the two cylinders. In the process of motion, the force of the pendulum system is shown in Figure 2, including the fluid force, \(F_{fluid,up}\), on the upstream cylinder, the fluid force, \(F_{fluid,down}\), on the downstream cylinder, the total torque, \(2M_{1}\), provided by the left and right transmission belts, the torques generated by viscous traction acting on each cylinder surface, \(M_{VT,up}\) and \(M_{VT,down}\), and the mechanical damping force (this force is not shown in Figure 2). According to this, the kinetic equation of the pendulum system is as follows:  
\[ \begin{split} J_{1}\ddot{\theta}_{1}+c_{1}\dot{\theta}_{1}=F_{fluid,up}\cdot\frac{L}{2}+F_{fluid,down}\cdot\frac{L}{2}-2M_{1}+M_{VT,up}+M_{VT, down}\end{split}  \tag{1}\]  
where \(\dot{\theta}_{1}\), \(\dot{\theta}_{1}\) and \(\theta_{1}\) are the angular acceleration, angular velocity and angular displacement of the pendulum system, respectively, and \(c_{1}\) is the damping coefficient of the pendulum system. \(J_{1}\) is the moment of inertia of the pendulum system to the rotary shaft \(O\). \(L\) is the center distance of the upstream and downstream cylinders. \(M_{1}\) is the torque provided by a single transmission belt. Figure 1: Schematic diagram of the flow-induced rotation. According to previous research, the \(F_{fluid}\) acting on the structure includes two parts: the inertial force, \(F_{inviscid}\), and the viscous force, \(F_{viscous}\)[50]. Obviously, the \(F_{inviscid}\) on the upstream and downstream cylinders is the same and the torques of \(F_{inviscid}\) to shaft \(O\) is calculated as follows:  
\[M_{o}(F_{inviscid})=-J_{a}\ddot{\theta}_{1} \tag{2}\]  
where \(J_{a}\) is the additional moment of inertia of the pendulum system to the rotary shaft \(O\). However, the torques of \(F_{viscous}\) on the upstream and downstream cylinders to shaft \(O\) are calculated as follows:  
\[M_{O}(F_{viscous,up})=F_{viscous,up}\cdot\frac{L}{2}=\frac{1}{2}c_{ tangent,up}\rho(Ucos\theta_{1})^{2}DH\cdot\frac{L}{2}=\frac{1}{4}c_{ tangent,up}\rho U^{2}DHLcos^{2}\theta_{1} \tag{3}\]  
\[M_{O}(F_{viscous,down})=F_{viscous,down}\cdot\frac{L}{2}=\frac{1}{2}c_{ tangent,down}\rho(Ucos\theta_{1})^{2}DH\cdot\frac{L}{2}=\frac{1}{4}c_{ tangent,down}\rho U^{2}DHLcos^{2}\theta_{1} \tag{4}\]  
where \(c_{ tangent,up}\) and \(c_{ tangent,down}\) is the instantaneous lift coefficient of the upstream and downstream cylinders, respectively, and \(P\) is the fluid density. \(U\) is the incoming flow velocity. \(D\) is the diameter of the circular cylinder. \(H\) is the length of the cylinder. Additionally, it is clear that \(M_{VT,up}\) and \(M_{VT,down}\), acting on the upstream and downstream cylinders, are equal, namely, \(M_{VT,up}=M_{VT,down}=M_{VT}\). The shear stress acting on the surface of each cylinder is named \(\tau_{0}\), and the expression of \(\tau_{0}\) is:  
\[\tau_{0}=\frac{1}{2}C_{f}\rho\bigg{(}\frac{D}{2}\dot{\theta}_{1}\bigg{)}^{2}= \frac{1}{8}C_{f}\rho D^{2}\bigg{(}\dot{\theta}_{1}\bigg{)}^{2} \tag{5}\]  
where \(C_{f}\) is the friction resistance coefficient of the boundary layer on the cylinder surface, \(C_{f}=f(Re)\).