# Experimental Investigation on Flow-Induced Rotation of Two Mechanically Tandem-Coupled Cylinders
## 3 Experimental Methods
### Free Decay Tests  
Free decay tests, with different spacing ratios, were conducted to obtain the damping ratio (\(\zeta_{total}\)) and natural frequency (\(f_{n,air}\)) of the FIR system. The specific free decay test procedure was as follows:  
First, the pendulum system was artificially deflected to a certain rotational angle under the condition of no water in the test water tunnel, and then released. Meanwhile, the test data was recorded with the help of a data acquisition system, shown in Figure 8c.  
Four tests were performed at each \(L/D\), and the average value of the four tests (\(\zeta_{total,ave}\) and \(f_{n,air,ave}\)) was used as the result. Figure 9a-d corresponded to the free decay curves with \(L/D\) = 4.0, 6.0, 7.0 and 9.0, respectively. The free decay test results of all cases were shown in Tables 4 and 5. Here, the damping ratio \(\zeta_{total}\) expression was defined as [18]:  
\[\zeta_{total}=\frac{\ln\eta}{2\pi}=\frac{1}{2\pi}\ln(\frac{A_{i}}{A_{i+1}}) \tag{24}\]  
where \(A_{i}\) is the \(i\)th peak of the free decay curve, and \(A_{i+1}\) is the \(i\)+1 th peak of the free decay curve. Additionally, the relative error of natural frequency between various cases was defined as:  
\[\delta=\frac{Case_{i}-Case_{i+1}}{Case_{i+1}}\times 100\%(i=1,2,\cdot\cdot \cdot\cdot\cdot,7) \tag{25}\]  
Figure 8: Investigation process of flow-induced rotation: (**a**) The FIR system; (**b**) Angular displacement transducer; (**c**) Data acquisition and processing system; (**d**) Experimental data.  
where \(Case_{i}\) is the natural frequency of the \(i\)th case, and \(Case_{i+1}\) is the natural frequency of the \(i\)+1 th case. Moreover, the relative error of the damping ratio between various cases is defined as:  
\[\varepsilon=\frac{Condition_{j}-Condition_{j+1}}{Condition_{j+1}}\times 100\%(j=1,2, \cdot\cdot\cdot\cdot,7) \tag{26}\]  
where \(Condition_{j}\) is the damping ratio of the \(i\)th working condition, and \(Condition_{j+1}\) is the damping ratio of the \(j\)+1 th working condition.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**L/D**} & **Test 1** & **Test 2** & **Test 3** & **Test 4** & **Average** & **Relative Error** \\  & \(f_{n,air,1}\) **(Hz)** & \(f_{n,air,2}\) **(Hz)** & \(f_{n,air,3}\) **(Hz)** & \(f_{n,air,4}\) **(Hz)** & \(f_{n,air,ave}\) **(Hz)** & \(\delta\) \\ \hline
4.0 & 1.146 & 1.147 & 1.147 & 1.146 & 1.1465 & — \\
4.5 & 1.102 & 1.092 & 1.090 & 1.085 & 1.0923 & 0.0496 \\
5.0 & 1.048 & 1.049 & 1.000 & 1.048 & 1.0363 & 0.0540 \\
5.5 & 1.020 & 1.013 & 1.011 & 1.013 & 1.0142 & 0.0218 \\
6.0 & 0.953 & 0.953 & 0.953 & 0.953 & 0.9530 & 0.0642 \\
7.0 & 0.903 & 0.855 & 0.856 & 0.902 & 0.8790 & 0.0842 \\
8.0 & 0.806 & 0.807 & 0.805 & 0.806 & 0.8060 & 0.0906 \\
9.0 & 0.758 & 0.756 & 0.756 & 0.758 & 0.7570 & 0.0647 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Free decay test results of \(f_{n,air}\) at various spacing ratios.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**L/D**} & **Test 1** & **Test 2** & **Test 3** & **Test 4** & **Average** & **Relative Error** \\  & \(\xi_{total,1}\) & \(\xi_{total,2}\) & \(\xi_{total,3}\) & \(\xi_{total,4}\) & \(\xi_{total,ave}\) & \(\varepsilon\) \\ \hline
4.0 & 0.060 & 0.055 & 0.052 & 0.052 & 0.0548 & — \\
4.5 & 0.054 & 0.051 & 0.050 & 0.048 & 0.0508 & 0.0788 \\
5.0 & 0.056 &As can be seen from Table 4, the \(f_{n,air}\) would slightly decrease with an increase of \(L/D\), but the \(f_{n,air}\) of all cases was in the range of 0.757-1.147 Hz. Furthermore, the \(\zeta_{total}\) of all work conditions was controlled within the interval of 0.039 to 0.054, as seen in Table 5.