# 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
#### 4.3.2 Oscillation Time History and Spectral Characteristics  
In order to further understand the rotational oscillation characteristics of the FIR system when \(L/D=7.0\), 8.0 and 9.0, herein, four typical velocities (\(U=0.265\) m/s, 0.483 m/s, 0.628 m/s and 0.810 m/s) were selected for each \(L/D\) to analyze the time history and spectral characteristics of \(\theta\), as shown in Figures 19-21, respectively. The specific analysis is as follows:  
(1) \(U=0.265\) m/s: It can be seen from the time-history diagram that the \(\theta\) of the three cases was small at this velocity. The cases of \(L/D=7.0\) and 8.0 maintained about 4.6\({}^{\circ}\), while the case of \(L/D=9.0\) could reach 5.8\({}^{\circ}\). Furthermore, it can be seen from the spectrogram that no frequency doubling occurred, but the frequency band was wide, which indicated that the vibration of the three cases was in an unstable state and the energy was not concentrated. Figure 19: Rotational oscillation time history and spectral characteristics at \(L/D\) = 7.0: (a) \(U\) = 0.265 m/s (\(U_{r}\) = 2.940); (b) \(U\) = 0.483 m/s (\(U_{r}\) = 5.355); (c) \(U\) = 0.628 m/s (\(U_{r}\) = 6.965); (d) \(U\) = 0.810 m/s (\(U_{r}\) = 8.977). (2) \(U=0.483\) m/s: Firstly, compared with the time history diagram when \(U=0.265\) m/s, it could be found that the \(\theta\) of \(L/D=7.0\) and \(8.0\) were greatly improved, increasing to about \(11.8^{\circ}\) and \(8.9^{\circ}\), respectively, while the \(\theta\) of \(L/D=9.0\) only increased slightly to \(6.7^{\circ}\). Additionally, the waveforms of the three cases at this flow rate were uniform. Then, the analysis of the spectral diagram shows that the \(f_{\rm osc}\) of the three cases at this flow velocity were also increased, compared with that at \(U=0.265\) m/s, the frequency band narrowed, and there was no frequency doubling phenomenon. This showed that the rotational vibration becames stable and the vibrational energy was large. (3) \(U=0.628\) m/s: When \(U\) increased to \(0.628\) m/s, it can be seen from the time history diagram that the waveforms of \(L/D=7.0\) and \(L/D=8.0\) became disordered, while the waveform of \(L/D=9.0\) was relatively stable. Meanwhile, the \(\theta\) of \(L/D=7.0\) decreased sharply to about \(6.1^{\circ}\), while the \(\theta\) of \(L/D=8.0\) and \(9.0\) decreased slightly to about \(5.5^{\circ}\) and \(5.9^{\circ}\), respectively. By analyzing the spectral diagram, it could be found that the \(f_{\rm osc}\) of the three cases continued to increase and the frequency band became wider than that of \(U=0.483\) m/s. Comprehensive analysis of the time history and frequency spectral characteristics of the FIR system with \(L/D=7.0\), \(8.0\) and \(9.0\) at the above three typical flow rates, indicated that, with the increase of the test flow rate, the \(\theta\) of the three cases first increased and then decreased, and the \(f_{\rm osc}\) kept increasing, which meant that the three cases all showed the response law of VIV. After entering the lower branch of VIV, the suppression effect of \(L/D=7.0\) was most significant, followed by \(L/D=8.0\), and was the smallest when \(L/D=9.0\), which led to the relationship between the range of synchronous intervals of the three cases such that \(L/D=9.0\) was the largest, \(L/D=8.0\) was second and \(L/D=7.0\) was the smallest. (4) \(U=0.810\) m/s: When \(U\) increased to \(0.810\) m/s, the \(\theta\) of \(L/D=7.0\), \(8.0\) and \(9.0\) all decreased to about \(3.5^{\circ}\) and the waveforms of the time history were all out of order. Meanwhile, the frequency bands became wider and there were the characteristics of frequency doubling, indicating that, after the vibration response of these three cases entered the lower branch of VIV, there was no \"re-growth region\", and the vibration was still very unstable. In conclusion, when the spacing ratios of the FIR system were 7.0, 8.0 and 9.0, the rotational oscillation responses presented the typical VIV law, and the range of the synchronization interval increased with increase of \(L/D\).