# Experimental Investigation on Flow-Induced Rotation of Two Mechanically Tandem-Coupled Cylinders
## 4 Results and Discussion
### L/D = 4.0 and 4.5
#### 4.1.2 Oscillation Time History and Spectral Characteristics  
In order to further study the oscillation characteristics of the FIR system at \(L/D=4.0\) and \(4.5\), Figures 12 and 13 show the time history and frequency spectra of the rotation angle (\(\theta\)) at six typical velocities (\(U=0.302\) m/s, \(0.483\) m/s, \(0.519\) m/s, \(0.592\) m/s, \(0.665\) m/s and \(0.882\) m/s), respectively. The ordinate in the spectrum was normalized and the dominant oscillation frequency highlighted. (1) \(U=0.302\)m/s: It can be seen from the time history diagram that the \(\theta\) of \(L/D=4.0\) and 4.5 was small at this flow rate, reaching only about 3.3\({}^{\circ}\) and 4.6\({}^{\circ}\), respectively, and the waveform was relatively uniform. According to the spectrum analysis, the frequency bands of the two cases were narrow, the \(f_{osc}\) values were 0.566 Hz and 0.554 Hz, respectively, and there were no frequency doubling characteristics. The results indicated that the rotational vibration was relatively stable and the oscillation energy was low at \(U=0.302\) m/s. (2) \(U=0.483\) m/s: The waveforms of time history became more uniform and the \(\theta\) increased remarkably in both cases, increasing to 10.5\({}^{\circ}\) and 12.6\({}^{\circ}\), respectively. Additionally, it can be seen from the spectrum diagram that the values of the \(f_{osc}\) of the two cases were higher than those of \(U=0.302\) m/s, and the frequency bands became narrower, indicating that the rotational oscillation of the two cases was stable and the vibration energy was concentrated at this velocity. (3) \(U=0.519\) m/s: For \(L/D=4.0\), the \(\theta\) began to decrease and the waveform of time history was disordered, the \(f_{osc}\) increased to 0.915 Hz but the frequency band became wider, indicating that the rotary oscillation was suppressed and in an unstable state at  
Figure 13: Rotational oscillation time history and spectral characteristics at six typical velocities when \(L/D=4.5\): (a) \(U=0.302\) m/s (\(U_{r}=2.764\)); (b) \(U=0.483\)m/s (\(U_{r}=4.428\)); (c) \(U=0.519\) m/s (\(U_{r}=4.76\)); (d) \(U=0.592\) m/s (\(U_{r}=5.426\)); (e) \(U=0.665\) m/s (\(U_{r}=6.091\)); (f) \(U=0.882\) m/s (\(U_{r}=8.088\)). \(U=0.519\) m/s. For \(L/D=4.5\), the \(\theta\) was still growing, reaching about 14.2\({}^{\circ}\). The \(f_{osc}\) increased to 0.752 Hz and the frequency band was narrow, which suggested that the rotational vibration was not inhibited at this rate and the vibration was still fine. Further analysis of the time history and spectrum diagram with \(L/D=4.0\) found that the arc length response first increased and then decreased, and the frequency response continued to increase, indicating that the flow-induced rotation characteristics with \(L/D=4.0\) conformed to the VIV in the range of \(U=0.302\)-0.519 m/s. (4) \(U=0.592\) m/s: For \(L/D=4.0\), the waveform was unstable, but the \(\theta\) rose slightly compared with that of \(U=0.519\) m/s. The frequency band was still wide. However, for \(L/D=4.5\), the waveform of time history became disorganized and the \(\theta\) was significantly reduced. Additionally, the frequency band became wider and the \(f_{osc}\) increased to 0.840 Hz. The results suggested that the rotational vibration with \(L/D=4.5\) was suppressed at \(U=0.592\) m/s. Meanwhile, it could also be concluded that for \(L/D=4.5\), the arc length response also first increased and then decreased when \(U=0.302\)-0.592 m/s, and the dominant frequency always increased, which was also in line with the response law of VIV. (5) \(U=0.665\) m/s: For \(L/D=4.0\), the \(\theta\) continued to increase, reaching about 7.3\({}^{\circ}\), but did not exceed the value reached at \(U=0.483\) m/s. Furthermore, compared with \(U=0.592\) m/s, the frequency band became narrower and the \(f_{osc}\) increased to 0.911 Hz. For \(L/D=4.5\), the \(\theta\) also increased, but the growth range was smaller than that of \(L/D=4.0\). The frequency band was narrow and the \(f_{osc}\) increased to 0.879 Hz.