# Experimental Study on Streamwise Vortex-Induced Vibration of a Flexible, Slender Cylinder
## 3 Modal Analysis Approach  
In order to eliminate the noises from the measured strain signals, a band-pass filtering, ranging from 0.5 to 30 Hz, was applied. Next, the filtered data was used in the modal analysis to obtain the displacement responses of the cylinder model. Assuming that the time-varying displacement could be expressed as a sum of the product of the mode shapes and the modal weights:  
\begin{table}
\begin{tabular}{c c} \hline
**Items** & **Values** \\ \hline Cylinder model length, \(L\) & 3.91 (m) \\ Outer diameter, \(D\) & 0.020 (m) \\ Aspect ratio, \(L/D\) & 195.5 \\ Bending stiffness, \(EI\) & 324.94 (N\(\cdot\)m\({}^{2}\)) \\ Axial tension, \(T\) & 0, 50, 100, 150 (N) \\
1 st order natural frequency in still water, \(f_{1}\) & 1.96, 2.18, 2.38, 2.56 (Hz) \\ Mass per unit length (with trapped water), \(m_{s}\) & 0.573 (kg/m) \\ Mass ratio, \(m^{\dagger}\) & 1.39 \\ Structural damping ratio in still water, \(\zeta\) & 0.0598 \\ \hline \end{tabular}
\end{table}
Table 1: Key parameters of the cylinder model.  
Figure 3: Locations of strain gauges along the cylinder model.  
Figure 2: Flexible cylinder model.  
\[x(z,t)=\sum_{n=1}^{\infty}w_{n}(t)\varphi_{n}(z) \tag{1}\]  
where \(x\) (\(z\), \(t\)) is the time varying displacement in the streamwise direction, \(z\) is the coordinate along the axis of the cylinder model, \(t\) is time, \(w_{n}\) (\(t\)) is the modal weight, \(\varphi_{n}(z)\) is the mode shape, and \(n=1\), \(2\), \(3\), \(\dots\) is the order of vibration mode.  
For the pined-pined boundary conditions applied in the present experiment, the nth mode shape could be written as:  
\[\varphi_{n}(z)=\sin\frac{n\pi z}{L} \tag{2}\]  
where \(L\) is the length of the cylinder model. Taking into account the relationship between curvature and strain (i.e., \(k(z,t)=\varepsilon(z,t)/R\)), Equation (2) resulted in:  
\[\frac{\varepsilon(z,t)}{R}=k(z,t)=x^{\prime\prime}(z,t)=-\sum_{n=1}^{\infty} \left(\frac{n\pi}{L}\right)^{2}w_{n}(t)\sin\frac{n\pi z}{L} \tag{3}\]  
where \(R\) denotes the radius of the cylinder model. In Equation (3), the unknown time-varying modal weight \(w_{n}\) (t) could be calculated using the filtered strain signals. Readers are referred to [PERSON] et al. [29], [PERSON] et al. [30], [PERSON] and [PERSON] [31], and [PERSON] et al. [32] for more details about the modal analysis approach.  
Figure 4 shows an example of the modal analysis. The left column plots gave the time histories of the strain signals measured at five measuring points during the last 30 s. The corresponding modal weights evaluated using Equation (3) are depicted in the middle column plots. It can be seen that the streamwise displacement response was dominated by the first mode in this case. The time histories of the displacement at the five measuring points were calculated using Equation (1), and are presented in the right column plots. In the following section, the displacement responses were obtained by the modal analysis approach, which was introduced in this section.  
Figure 4: An example of the modal analysis approach with the axial tension (\(T=150\) N) and towing velocity (\(u=0.15\) m/s). (**a**) Streamwise strain signals at five measuring points; (**b**) modal weights of streamwise VIV; and (**c**), time histories of the displacements at five measuring points.