## 5 Two Different Tuning Modes of d-MTFDI
### Distributed Tuning Mode  
For the d-MTFDI with distributed tuning mode, it is assumed that (1) the tuning frequency interval is unified and (2) the frequency range is symmetric about \(\
u=1\). The frequency distribution is shown in Figure 8, where the normalized frequency range is  
Figure 7: (a) ABS model; (b) wind azimuths; (c) PSD of \(p(t)\).  
denoted as \(\Delta\), and the uniform normalized frequency interval is presented as \(\delta\). Thus, the tuning frequency ratio of the \(i\)th TFDI is as follows:  
\[\
u_{di}=1-\frac{R-2i+1}{2(R-1)}\Delta \tag{12}\]  
Apparently, the normalized frequency range \(\Delta\) and unified damping ratio \(\xi_{d}\) directly determine the WIV control performance of d-MTFDI with distributed tuning frequency. Based on the index of \(J\), the performance will be evaluated with the variance of \(\Delta\) and \(\xi_{d}\). Considering the constraint of engineering application, \(\Delta\in[0,\,0.4]\), \(\xi_{d}\in[0.001,\,0.2]\). And for the arrangement of d-MTFDI, nine different [\(R,\,p,\,\beta\)] combinations are selected to reflect the influence of each parameter, namely [2, 2, 0.2], [6, 2, 0.2], [10, 2, 0.2], [6, 2, 0.4], [6, 4, 0.4], [6, 10, 0.4], [10, 10, 0.2], [10, 0.4], [10, 10, 0.4], [10, 1]. Figure 9 plots the variation of WIV control efficiency \(J\) with the normalized frequency range \(\Delta\) and damping ratio \(\xi_{d}\) with nine different [\(R,\,p,\,\beta\)] combinations. The values of optimal [\(\Delta,\,\xi_{d}\)] corresponding to the maximum \(J\) of every [\(R,\,p,\,\beta\)] combination are also presented in a [\(\Delta,\,\xi_{d},\,J\)] format in every figure. Several phenomena can be seen in the figure, as follows:  
1. For all [\(R,\,p,\,\beta\)] combinations, there always exists an optimal [\(\Delta,\,\xi_{d}\)] combination;
2. Figure 9a-c plot the variance of \(R\) with \(p=2\) and \(\beta=0.2\). The increase of \(R\) will increase optimal \(\Delta\) but decrease optimal \(\xi_{d}\), but the effect on \(J\) is not clear. Such tendency can also be proven by the comparison of Figure 9f,h.
3. Figure 9d-f plot the variance of \(p\) with \(R=6\) and \(\beta=0.4\). The increase of \(p\) will both increase optimal \(\Delta\) and \(\xi_{d}\), and \(J\) is also increased. Such tendency can be also proven by the comparison of Figure 9b,g.
4. Figure 9g-i plot the variance of \(\beta\) with \(R=10\) and \(p=10\). The increase of \(\beta\) will increase optimal \(\Delta\), and \(J\) is also increased, but the effect on \(\xi_{d}\) is not clear. Such tendency can also be proven by the comparison of Figure 9b,d.
5. The robustness toward the change of \(\Delta\) and \(\xi_{d}\) will be increased with the increase of either \(p\) or \(\beta\), but the change of \(R\) exerts almost no influence. More importantly, when \(\xi_{d}\) is smaller than the optimal value, \(J\) always takes a steep drop, but when \(\xi_{d}\) is larger than the optimal value and \(p\) or \(\beta\) are large enough, especially \(p\) is large enough, the robustness toward \(\xi_{d}\) will be greatly enhanced. Such a phenomenon indicates that the designed \(\xi_{d}\) should be larger than the optimal value to make sure \(J\) will not be located in the steep drop part due to the installation error or material property randomness.  
Since the working mechanism of distributed tuning mode is more complex than that of the unified tuning mode, the optimal combination of [\(\Delta,\,\xi_{d}\)] should be determined using numerical optimization technology and will be presented later.  
Figure 8: Tuning frequency distribution of d-MTFDI.