**5. Conclusions**

The response deformation of a structure is relatively small during the early stage of the seismic response, which leads to poor performance of velocity-dependent damping compared with the displacement-dependent damping with the same equivalent damper ratio [40,42]. In this study, the ability of the displacement-dependent damper to generate a larger control force during the early stage of the excitation is exploited and further enhanced by a supplemental inerter-spring system tuned to the primary structure, realizing an effective DDIS. A summary of the study and the main conclusions drawn from this investigation are reported below:


**Author Contributions:** Conceptualization, R.Z.; methodology, R.Z. and Z.Z.; software, Z.Z. and Y.J.; validation, Z.Z., R.Z., D.D.D. and Y.J.; formal analysis, Z.Z. and Y.J.; investigation, Z.Z., Y.J. and C.P.; data curation, Z.Z.; writing—original draft preparation, Z.Z., R.Z., Y.J., D.D.D. and C.P.; writing—review and editing, Z.Z., R.Z., C.P. and D.D.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was supported by the National Natural Science Foundation of China under Grant (grant no. 51978525 and 51778489), the Natural Science Foundation of Shandong Province (No. ZR2018BEE033); and the Fundamental Research Funds for the Central Universities (grant no. 22120180064).

**Acknowledgments:** The authors wish to acknowledge Kohju Ikago of Tohoku University for his help in interpreting the significance of the results of this study.

**Conflicts of Interest:** The authors declare no conflict of interest.
