## 7 Conclusions  
This study proposed a new inerter-enhanced damping system, namely d-MTFDI, by using part of the outer skins of DSF and combining inerters. The configuration and working mechanism of the d-MTFDI-controlled system are presented in detail. Based on a 76-floor benchmark building with wind tunnel test data, the WIV control effectiveness is verified, and the performance with two tuning modes is studied and compared. The corresponding design framework is proposed. The advantages of d-MTFDI over d-MTMD are presented. The main conclusions are as follows:  
1. By utilizing the DSF system, the d-MTFDI not only addresses the challenge of space limitations for multi-layer inerter installations in high-rise buildings but also significantly reduces the additional mass required for the tuned damping system.
2. For the unified tuning mode, the formulas of optimal tuning frequency ratio and damping ratio are derived. The WIV mitigation ratio \(J\) typically increases with \(p\) and \(\beta\), but in cases of small \(\beta\), introducing inerters to MOS may negatively impact the WIV control effect.
3. For the distributed tuning mode, there is always an optimal frequency range \(\Delta\) and damping ratio \(\xi_{d}\) combination, with both the corresponding values and robustness toward the change of \(\Delta\) and \(\xi_{d}\) increasing with \(p\) and \(\beta\). However, in practical engineering applications, it is recommended to select a damping ratio \(\xi_{d}\) slightly larger than the optimal value to ensure that performance \(J\) does not decrease significantly due to potential errors or randomness.
4. The d-MTFDI with distributed tuning mode consistently outperforms the unified tuning mode across varying values of \(R\), \(p\), and \(\beta\), primarily due to the resulting  
Figure 14: Comparison of acceleration PSD with three types of dampers (\(R=10\), \(p=10\), \(\beta=0.25\)).  
flattened transfer function. The optimal damping ratio in the distributed tuning mode is significantly lower than that in the unified mode, and an increase in \(R\) further reduces the demand for \(\xi_{d}\), enhancing the practical application of d-MTFDI.
5. Compared with the traditional d-MTMD with the same mass ratio and installation positions, the d-MTFDI demonstrates a notable improvement in WIV control performance, exceeding a 20% increase, which indicates the significant advantage provided by the incorporation of inerters. The limitations of this article that should be further addressed in the future: 1. Experimental validation of d-MTFDI through an aeroelastic wind tunnel test; 2. Effect of floor action excited by wind and seismic actions on the outer skins of DSF, especially for the MOS.  
All authors contributed to the study conception and design. Material preparation was performed by [PERSON], and data collection and analysis were performed by [PERSON] and [PERSON]. The first draft of the manuscript was written by [PERSON] and revised by [PERSON] All authors commented on previous versions of the manuscript. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 51508502, 51978614).  
Not applicable.  
The data presented in this study are available in [32].  
The authors declare no conflicts of interest.