## 5 Conclusions  
This paper investigates the dynamic characteristics of a rotary axial ECD and its seismic reduction performance when applied in a linear elastic SDOF system. The nonlinear force-velocity relationship of the ECD is expressed using the [PERSON]'s model which are validated by Finite element simulation and experimental testing. The energy dissipation capacity of the ECD is obtained analytically and its optimal parameters for maximum energy dissipation are also derived. The control performance of ECDs is investigated and evaluated under harmonic excitations and real earthquake ground motions. The most important findings are summarized below:  
1. The force-velocity constitutive behavior of the ECD can be well depicted by the [PERSON]'s model. The eddy current damping force is linearly proportional to the velocity for low speed region, gradually increasing with decreasing slope when the velocity become higher, reaching a maximum at the critical speed, and then decreasing for much higher speeds. These unique characteristics can protect the damper and structure from damage when an over-load is exerted on the damper.
2. When the ratio of the critical velocity of the ECD to the maximum velocity is 0.786, the energy dissipation capacity of ECD reaches its maximum under harmonic motions. It always can find a better design of ECD when the velocity exponent \(\alpha\) of FVD is larger than 0.2 such that the energy dissipation capacity of ECD is larger than that of FVD under the same harmonic motion.
3. In the resonance frequency band, both the displacement and acceleration responses of SDOF-ECDs are smaller than those of SDOF-FVDs under the same harmonic excitation. As the velocity exponent \(\alpha\) of the FVD increases, the control performance of the corresponding ECD gets better and better compared with that of FVD.
4. The displacement and acceleration reduction ratios of ECDs are larger than those of FVDs in most of the cases under real earthquake excitations, and the displacement reduction ratios are generally greater than the acceleration reduction ratios for both ECDs and FVDs. The energy dissipation capacity of ECDs outperforms that of FVDs in most of the cases under real earthquake excitations.  
Currently, this paper is limited to the numerical evaluation of ECDs for elastic SDOF systems. The conclusions summarized above are based on the limited data presented in this paper. To further extrapolate the findings, the next step is to experimentally test the performance of ECDs with structures. Meanwhile, the further studies on applying ECDs in MDOF systems will also be developed in the future.  
All authors made scientific contributions: [PERSON] derived the formulas and made the calculations, [PERSON] conceived the idea of this research and wrote the manuscript, and [PERSON] revised the manuscript.  
This work was supported in part by the National Natural Science Foundation of China under grant No. 51708203, the Fundamental Research Funds for the Central Universities under grant No. 531118010047, the Open Research Fund Program of Guangdong Key Laboratory of Earthquake Engineering and Application Technology under grant No. 2014B0303010075-02, the Open Research Fund Program for innovation platforms of universities in the Hunan province from the Education Department of Hunan Province under grant No. 17K022, and the independent research project funds from the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body in Hunan University under grant No. 71860006.  
The authors declare no conflict of interest.