# Displacement-Dependent Damping Inerter System for Seismic Response Control
## 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:  
1. The DDIS is observed to suppress the structural responses in a timely manner as soon as a peak response occurs during the early stage of the excitation. The proposed equivalent linearization method is effective to conduct the stochastic dynamic analysis of the DDIS-equipped structure. The dynamic response of the DDIS controlled systems are further evaluated in the time domain considering the non-linearity and validating the accuracy of the stochastic analysis.
2. The interaction between the inerter, spring and the DDE constitutes the damping enhancement mechanism of the DDIS. Compared with an identical DDE, the proposed mechanism amplifies the deformation of the DDE in the DDIS by over 60%; thus, the DDIS is characterized by a higher energy dissipation capability.
3. The proposed DDIS considerably reduces the structural displacement and acceleration, which is a result of its damping and mass-enhancement mechanism. For the DDIS with medium value of stiffness ratio and inertance-mass ratio, the displacement and acceleration responses of original uncontrolled structure are reduced by 60% and 40%, respectively.
4. The benefits and characteristics of the proposed DDIS are obtained from an extensive parametric analysis and not limited to any specific optimal design procedures. Further studies should be conducted on the parameter design methodology and practical design procedure of the DDIS. The equivalent linearization method proposed in this paper would help reduce the computational cost required to develop a design method.  
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.  
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. ZR2018 BEE033); and the Fundamental Research Funds for the Central Universities (grant no. 22120180064).  
Figure 16: Hysteresis curves of the DDEs of the DDIS and displacement-dependent damper under (**a**) El Centro and (**b**) SS excitation.  
**Acknowledgments:** The authors wish to acknowledge [PERSON] 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.