## 3 Results and Discussion
### Grey Relational Analysis of Multiple Parameters  
The parameters considered in this paper that affect the amplitude of the VIV for the risers are modulus, flow velocity, top tension force, constraint number and density. In this paper, grey relational analysis (Equations (1)-(8)) is utilized to determine the grey relational grade of the 5 parameters on the amplitude of VIV for the riser. \[ \begin{array}{l}X_{0}=\{X_{0}(1),\,X_{0}(2),\,\ldots\,\ldots,X_{0}(36)\}\\ X_{1}=\{X_{1}(1),X_{1}(2),\,\ldots\,\ldots,X_{1}(36)\}\\ X_{2}=\{X_{2}(1),X_{2}(2),\,\ldots\,\ldots,X_{2}(36)\}\\ X_{3}=\{X_{3}(1),X_{3}(2),\,\ldots\,\ldots,X_{3}(36)\}\\ X_{4}=\{X_{4}(1),X_{4}(2),\,\ldots\,\ldots,X_{4}(36)\}\\ X_{5}=\{X_{5}(1),X_{5}(2),\,\ldots\,\ldots,X_{5}(36)\}\end{array}  \tag{1}\]  
where, \(X_{0}\) is original (reference) data and \(X_{1}\) to \(X_{5}\) are the comparative series. More specifically, \(X_{0}\) is the amplitude of VIV; \(X_{1}\) is the modulus of material; \(X_{2}\) is the flow velocity; \(X_{3}\) is the top tension force; \(X_{4}\) is the number of constraints; \(X_{5}\) is the density of material; and numbers 1-36 in brackets are the experiment case number. More specifically, the values of \(X_{0}\) are obtained from Figures 4-7; the values of \(X_{1}\) and \(X_{5}\) are obtained from Table 1; Table 3 gives the values \(X_{2}\) and \(X_{3}\); in terms of the constraint number \(X_{4}\), 5 is for S-S (U\({}_{x}\), U\({}_{y}\), U\({}_{x}\) at bottom and U\({}_{x}\), U\({}_{y}\) at top of the riser) and 8 is for S-F (U\({}_{x}\), U\({}_{y}\), U\({}_{x}\), \(\theta_{x}\), \(\theta_{y}\), \(\theta_{x}\) at bottom and U\({}_{x}\), U\({}_{y}\) at top of the riser). The detailed values for \(X_{0}\) to \(X_{5}\) are presented below:  
Figure 10: Effects of top tension force on the amplitude of the VIV for UPVC riser. Here we have to note that, for \(X_{0}\) to \(X_{5}\), their measurement units are different, and their values have a very large range, which might cause inaccurate results in the analysis. Therefore, these sequences are normalized to comparability sequences using a process called grey relational generation in GRA. More specifically, the data from Equation (1) is normalized using \(Y_{i}=\frac{X_{i}(n)}{\sum\limits_{n=1}^{36}X_{i}(n)}\) to obtain Equation (2). Where, \(\sum\limits_{n=1}^{36}X_{i}(n)\) is the mean value of \(X_{i}(n)\), \(n\in\) (1,36). \[\left\{\begin{array}{l}Y_{0}=\{Y_{0}(1),Y_{0}(2),\ldots\ldots,Y_{0}(36)\}= \left\{ \begin{array}{l}\frac{X_{0}(1)}{\sum_{n=1}^{36}X_{0}(n)},\frac{X_{0}(2 )}{\sum_{n=1}^{36}X_{0}(n)},\ldots\ldots,\frac{X_{0}(36)}{\sum_{n=1}^{36}X_{0} (n)}\right\}\\ Y_{1}=\{Y_{1}(1),Y_{1}(2),\ldots\ldots,Y_{1}(36)\}=\left\{\begin{array}{l} \frac{X_{1}(1)}{\sum_{n=1}^{36}X_{1}(n)},\frac{X_{1}(2)}{\sum_{n=1}^{36}X_{1} (n)},\ldots\ldots,\frac{X_{1}(36)}{\sum_{n=1}^{36}X_{1}(n)}\end{array}\right\} \\ Y_{2}=\{Y_{2}(1),Y_{2}(2),\ldots\ldots,Y_{2}(36)\}=\left\{\begin{array}{l} \frac{X_{2}(1)}{\sum_{n=1}^{36}X_{2}(n)},\frac{X_{2}(2)}{\sum_{n=1}^{36}X_{2} (n)},\ldots\ldots,\frac{X_{2}(36)}{\sum_{n=1}^{36}X_{2}(n)}\end{array}\right\} \\ Y_{3}=\{Y_{3}(1),Y_{3}(2),\ldots\ldots,Y_{3}(36)\}=\left\{\begin{array}{l} \frac{X_{3}(1)}{\sum_{n=1}^{36}X_{3}(n)},\frac{X_{3}(2)}{\sum_{n=1}^{36}X_{3} (n)},\ldots\ldots,\frac{X_{3}(36)}{\sum_{n=1}^{36}X_{3}(n)}\end{array}\right\} \\ Y_{4}=\{Y_{4}(1),Y_{4}(2),\ldots\ldots,Y_{4}(36)\}=\left\{\begin{array}{l} \frac{X_{4}(1)}{\sum_{n=1}^{36}X_{4}(n)},\frac{X_{4}(2)}{\sum_{n=1}^{36}X_{4} (n)},\ldots\ldots,\frac{X_{4}(36)}{\sum_{n=1}^{36}X_{4}(n)}\end{array}\right\} \\ Y_{5}=\{Y_{5}(1),Y_{5}(2),\ldots\ldots,Y_{5}(36)\}=\left\{\begin{array}{l} \frac{X_{5}(1)}{\sum_{n=1}^{36}X_{5}(n)},\frac{X_{5}(2)}{\sum_{n=1}^{36}X_{5} (n)},\ldots\frac{X_{5}(36)}{\sum_{n=1}^{36}X_{5}(n)}\end{array}\right\} \end{array}