## 5 Two Different Tuning Modes of d-MTFDI
### Comparison with Two Tuning Modes  
This section will show the differences of d-MTFDI with two tuning modes with respect to the WIV control performance and optimal damping ratio. Considering the constraint of engineering application, the variation ranges are as follows: \(R\in[2,\,10]\), \(p\in[2,\,10]\), and \(\beta\in[0,\,1]\).  
Figure 10 plots the variation of \(J^{opt}\) with respect to \(\beta\) and \(p\) for two tuning modes, utilizing the optimal tuning parameter values obtained in Sections 5.1 and 5.2. The \(J^{opt}\) results for the d-MTFD (i.e., no inerter) using the same \(u\) and optimal tuning parameters are also shown in Figure 10 (i.e., horizontal dash line). It can be seen in Figure 10 that regardless of the tuning mode, \(J^{opt}\) increases with either an increase in \(p\) or \(\beta\). However, when \(p\) is small (e.g., \(p=2\)), there is a value range of \(\beta\) for which the \(J^{opt}\) of d-MTFD is lower than that of d-MTFD. This indicates a negative contribution from the inerter to the  
Figure 9: Plot of \(J\) with the variation of \(\Delta\) and \(p\) for \([R,\,p,\,\beta]\) combinations of (**a**) [2, 2, 0.2]; (**b**) [6, 2, 0.2]; (**c**) [10, 2, 0.2]; (**d**) [6, 2, 0.4]; (**e**) [6, 4, 0.4]; (**f**) [6, 10, 0.4]; (**g**) [10, 10, 0.2]; (**h**) [10, 10, 0.4]; and (**i**) [10, 10, 1] (The red hollow dots are the maximum points and the blue ones are the corresponding projection on the X-Y plane).  
vibration control, attributed to the negative stiffness effect of the inerter [38]. Therefore, such a value range should be avoided when designing a d-MTFDI.  
Furthermore, a comparison of the two tuning modes reveals that \(f^{opt}\) with distributed tuning mode is consistently greater than that with unified tuning mode. While this increase does not exhibit a clear correlation with \(\beta\) and \(p\), it does show a slight upward trend with an increase in \(R\).  
Apart from the WIV control performance, comparing the optimal damping ratios is crucial, as excessively high damping may be challenging to implement in practice. Figure 11 plots the variation of \(\xi_{d}^{opt}\) with \(\beta\) and \(p\) across the two tuning modes. It is evident that, regardless of the tuning mode, \(\xi_{d}^{opt}\) increases with both \(\beta\) and \(p\), while it decreases with an increase in \(R\). Notably, the decreasing trend for the distributed tuning mode is significantly more pronounced than for the unified tuning mode. This indicates that the \(\xi_{d}^{opt}\) for the distributed tuning mode are more attainable in engineering applications, thereby enhancing the feasibility of the d-MTFDI.  
Figure 11: Comparison of \(\xi_{d}^{opt}\) with two tuning modes with the variation of \(\beta\) and p based on optimal tuning (**a**) \(R=2\); (**b**) \(R=6\); (**c**) \(R=10\).  
Figure 10: Comparison of \(f^{opt}\) with two tuning modes with the variation of \(\beta\) and \(p\) based on optimal tuning: (**a**) \(R=2\); (**b**) \(R=6\); (**c**) \(R=10\).  
Taking combination [\(R\), \(p\), \(\beta\)] = [10, 8, 0.25] as an example, the optimal parameters of d-MTFDI with two tuning modes are shown in Table 1. As can be seen in the table, the \(\phi_{d}^{opt}\) of distributed tuning mode is only about 25% of that with unified frequency, but the \(\int^{opt}\) is higher by 1.9%. The corresponding two NTFs are plotted in Figure 12. The NTF of unified tuning mode exhibits two peaks, while that of distributed tuning mode is flattened, resulting in a broader frequency range being controlled and an improved WIV control performance.  
Based on the analysis, the design process is presented in Figure 13. As can be seen in Figure 13, the preliminary design phase is to determine the MOS and inerters preset parameters that match the demand of WIV control performance, and for the optimal parameters design phase, two tuning modes offer two methods