# Experimental Study on Streamwise Vortex-Induced Vibration of a Flexible, Slender Cylinder
## 4 Results and Discussion
In the second excitation region, the streamwise displacement showed evident beatings in its time history, as shown in Figure 5:3 The time history of the displacement response was regular and periodic, and attained the maximum amplitude of 0.16. Two large peaks were found in the spectral plot. One occurred at \(f_{x}/f_{1}=1.0\), and the other occurred at one-half of this value, which was apparently the effects of the transverse vibrations. [PERSON] and [PERSON] [34] performed the laboratory tests on the vortex-induced vibration of an elastically supported cylinder, which could vibrate in streamwise and transverse directions, and could also be constrained to move only in a streamwise direction. In the first branch, they found that the transverse response could be negligible, while the freedom to move in the transverse directions had a significant effect on streamwise VIV in the second branch. The frequency of the transverse oscillations was equal to half the frequency of the streamwise vibrations, and the cylinder followed a figure-of-eight trajectory in the second branch [7]. This is the reason why two frequency peaks can be seen in the plot of spectra of the displacement in Figure 5:4  
At the end of the second excitation region, irregularities were checked one more time in the time history of the streamwise displacement, as shown in Figure 5:5 However, different from the results shown in Figures 9 and 11, a predominant peak which obviously deviated from \(f_{x}/f_{1}\approx 1.0\) was observed in the spectral plot, indicating the end of the resonance. Previous experiments carried out during the past several decades indicate that the maximum amplitude of a transverse VIV is highly dependant on a combined mass-damping parameter \(m^{*}\zeta\). The so-called 'Griffin plot' is presented by compiling a large variety of investigations on VIV [3; 5]. Although there are conflicting opinions regarding its validity, the Griffin plot is widely used in industry and academic fields. Similarly, it was also proved that the maximum amplitude of the streamwise VIV followed closely with the reduced mass-damping parameter \(C_{n}\)[20; 24; 34; 35]. A series of experiments on streamwise VIV of a circular cylinder was carried out by [PERSON] and colleagues [20; 33; 36; 37] to investigate the sensitivity of the amplitude response to \(C_{n}\). In this study, an extensive compilation of a large variety of published data, together with the present results, is depicted in two log-log plots, as shown in Figure 5:6 Two least-square fitted curves are superimposed on the plots and their expressions are defined as:  
Figure 5:7  
in the first excitation region (see Figure 14a); and  
Figure 5:8  
in the second excitation region (see Figure 14b). It is shown that the two curves follow closely with the data, especially when \(C_{n}<1.0\), which covers the majority of the streamwise VIV of marine cylinders. A small deviation can be observed at a larger \(C_{n}\). These two formulae could be used to predict the response amplitudes according to the mass-damping parameter \(C_{n}\).