# Experimental Study on Streamwise Vortex-Induced Vibration of a Flexible, Slender Cylinder
## 4 Results and Discussion
It can be seen that the transverse amplitude was much smaller than its streamwise counterpart in the first excitation region. The peak values, \(x_{Max}/D\approx 0.12\) (streamwise) and \(y_{Max}/D\approx 0.03\) (transverse), were simultaneously obtained at \(V_{r}\approx 2.3\). The non-dimensional vibration frequencies, normalized by the first order natural frequencies of the cylinder model under different axial tensions, were roughly unified in the first excitation region (i.e., \(f_{x}\approx f_{y}\approx f_{1}\), where \(f_{x}\) and \(f_{y}\) were the dominant response frequencies in the streamwise and transverse directions, respectively). This agrees with the results of [PERSON] and [PERSON] [34], in which a 2-DOF spring-supported rigid cylinder was tested. Figure 5: Dimensionless streamwise root-mean-square (RMS) response amplitudes versus the reduced velocity (grey circles represent five typical streamwise VIV response regions, which will be discussed later, including the beginning of the first excitation region, the maximum streamwise amplitude in the first excitation region, the transition region between the first and second excitation region, the maximum streamwise amplitude in the second excitation region, and the end of the second excitation region). Figure 6: Dimensionless maximum amplitudes: (**a**) in the streamwise and transverse directions, together with the frequency ratios, and (**b**), versus the reduced velocity. Figure 7 shows the \(x\)-\(y\) trajectories of the cylinder at \(V_{r}\approx 2.3\), together with the distribution of \(x_{RMS}/D\) along the axis of the cylinder model. In the first excitation region, the cylinder followed a slightly inclined linear orbit, which agrees with the results in the 2D tests (elastically-supported rigid cylinder) of [PERSON] and [PERSON] [9] and [PERSON] [34], but is different from the flat figure-of-eight trajectory observed in the 3D tests (pivoted cylinder) of [PERSON] and [PERSON] [7]. At the transition point of \(V_{r}\approx 2.6\), the ratio between the streamwise and transverse frequencies was roughly 2, i.e., \(f_{x}\approx 2f_{y}\approx f_{1}\), due to the alternate vortex shedding, which indicates the onset of the \"lock-in\" phenomenon and the beginning of the monotonic increase of the transverse amplitude. In the second excitation, the frequency ratio was kept at \(f_{x}\approx 2f_{y}\) and the orbit of the cylinder showed a figure-of-eight pattern, as shown in Figure 8. These observations are consistent with those in [PERSON] and [PERSON] [9], [PERSON] and [PERSON] [34], and [PERSON] and [PERSON] [7], except that the figure-of-eight was slightly skewed at the top side. At the end of the second excitation region, \(V_{r}\approx 3.8\), a significant drop could be found in \(f_{x}/f_{1}\) and the vibration frequencies in the streamwise and transverse directions became equal again. The streamwise vibration frequency obviously fell outside the resonance region, which resulted in the large reduction in the streamwise maximum amplitude. The transverse maximum amplitude shows a mild increment at this reduced velocity, which agrees with the results of [PERSON] and [PERSON] [9], but disagrees with those in [PERSON] and [PERSON] [7], where the transverse maximum amplitude plummeted. The typical streamwise VIV behaviors of a flexible cylinder were further investigated in the following part. Five representative cases, as indicated by the markers with grey shadow in Figure 5, were selected. These representative cases corresponded to the initial zone, middle of the first excitation region, the transition region, and the middle and end of the second excitation region, respectively. Figures 9-13 give the streamwise displacement response and vibration frequency spectra at five measuring points. The first column graph shows the time history of streamwise displacement for the last 30 s of a complete sampling period at different measuring positions along the cylinder model.