# Experimental Investigation on Flow-Induced Rotation of Two Mechanically Tandem-Coupled Cylinders
## 4 Results and Discussion
### L/D = 7.0, 8.0 and 9.0  
#### 4.3.1 Rotation Response  
This section analyzes and summarizes the rotation characteristics when \(L/D=7.0\), 8.0 and 9.0. Figure 18 presents the rotational vibration responses of the FIR system at these three spacing ratios. The specific analysis is as follows:  
Figure 17: Rotational oscillation time history and spectral characteristics at three typical velocities when \(L/D=6.0\): (**a**) \(U=0.302\) m/s (\(U_{t}=3.164\)); (**b**) \(U=0.556\) m/s (\(U_{t}=5.83\)); (**c**) \(U=0.810\) m/s (\(U_{t}=8.496\)).  
Figure 18: Complete rotation response of FIR system at \(L/D=7.0\), 8.0 and 9.0: (**a**) Arc length response; (**b**) Frequency response.  
(1) On the whole, within the test flow rate range, the rotational responses of these three cases still showed the typical VIV law. With the increase of \(U_{r}\), the \(A^{*}\) first increased and then decreased, while the \(f^{*}\) presented continuous growth. Additionally, the maximum \(A^{*}\) was 0.53-0.72 at these three spacing ratios, which was lower than the value when \(L/D=5.0\), 5.5 and 6.0, indicating that, when the distance between two cylinders exceeded 0.6 m (corresponding to \(L/D=6.0\)), the oscillation level of the FIR system would decrease instead. By analyzing the definition of arc length ratio, \(A^{*}\), it was found that when \(L/D>6.0\), the \(\theta\) would be significantly reduced, resulting in the \(A^{*}\) being unable to reach the previous level, which indicated that the distance between the two cylinders should be controlled within a reasonable range to ensure the rotational oscillation reached an ideal level. Additionally, the range of synchronization interval at \(L/D=7.0\), 8.0 and 9.0 was significantly larger than those in the other cases. This meant that the FIR system with \(L/D=7.0\), 8.0 and 9.0 had stronger adaptability to the current velocity and was more favorable to FIR power generation.  
(2) Comparing the \(A^{*}\)-\(U_{r}\) curves of the three cases, it could be seen that in the initial and lower branches of VIV, the \(A^{*}\) increased with the increase of \(L/D\), while in the upper branch of VIV, the \(A^{*}\) decreased with the increase of \(L/D\), indicating that the relationship between the size of the synchronization interval in the three cases was as follows: \(L/D=9.0\) was the largest, followed by \(L/D=8.0\), and \(L/D=7.0\), which was the smallest. Further analyzing the frequency responsed of these three working conditions, it was found that when \(U_{r}>8.5\), the \(f^{*}\) value of \(L/D=8.0\) and 9.0 exceeded 1.0 (\(f^{*}>1.0\)), and \(f^{*}\) continued to increase with increase of \(U_{r}\). The reason might have been that, when \(L/D=8.0\) and 9.0, the rotational oscillation would also enter the nonlinear resonance zone, but the characteristics of nonlinear resonance zone of these two cases were different from those when \(L/D\leq 7.0\). The specific performance was such that the oscillation frequency was no longer locked, but continued to increase with the increase of \(U_{r}\) until \(f^{*}>1.0\). It should be noted that at this time, the locked interval of arc length response still existed and becames significantly wider. This phenomenon was highly similar to the research results of [PERSON] [53].  
In conclusion, when the spacing ratios of two tandem-coupled cylinders were 7.0, 8.0 and 9.0, the rotation responses still presenteds the typical VIV law. Compared with other cases, the rotational oscillation in these three cases were at a medium level, and the maximum \(A^{*}\) was between 0.53 and 0.72. Another notable characteristic of the rotation response was that the range of synchronization interval was larger, and this range tended to increase with the increase of \(L/D\). Furthermore, it should be noted that the longer arc length response could only be generated when the distance between the two cylinders of the FIR system was in a rational range.