# Comparative Analysis of Viscous Damping Model and Hysteretic Damping Model
## 5 Conclusions  
This paper performed a comparative analysis of the viscous damping and hysteresis damping models. The dynamic response consists of transient and steady responses. So, the transient and steady responses of the two damping models were compared and analyzed, respectively. Some conclusions are as follows. 1. The attenuation coefficient and damped natural frequency are important parameters of the transient response. When the relative error of the attenuation coefficients for the two damping models was less than 10%, the damping ratio was less than 0.5041. When the relative error of damped natural frequencies for the two damping models was less than 10%, the damping ratio is less than 0.3308. 2. Earthquake waves can be decomposed into a series of harmonic waves. The vibration amplitude of the steady response is an important parameter. When the relative error of the maximum and minimum values of the vibration amplitude ratio for the two damping models was less than 10%, the damping ratio was less than 0.1736. 3. The numerical examples show that with the increase in damping ratio, the difference between the dynamic responses calculated by the two damping models gradually increases. Based on the comparative analysis of the transient and steady responses, the threshold of the damping ratio was selected as 0.1736, conservatively. When the damping ratio was less than 0.1736, the dynamic responses of the two damping models were approximately equal. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Category**} & \multicolumn{2}{c}{**EI Centro Wave**} & \multicolumn{2}{c}{**Tianjin Wave**} & \multicolumn{2}{c}{**Chichi Wave**} \\ \cline{2-7}  & **VT** & **HT** & **VT** & **HT** & **VT** & **HT** \\ \hline Peek displacement (mm) & 1.4355 & 1.3101 & 0.8906 & 0.7958 & 6.8338 & 5.9648 \\ Relative error (\%) & — & 8.74 & — & 10.64 & — & 12.72 \\ Peek velocity (mm/s) & 16.8375 & 15.6483 & 11.2117 & 10.6375 & 75.9327 & 72.7688 \\ Relative error (\%) & — & 7.06 & — & 5.12 & — & 4.17 \\ Peek acceleration (mm/s\({}^{2}\)) & 246.3872 & 261.6063 & 143.7406 & 150.9666 & 1399.0967 & 1483.3127 \\ Relative error (\%) & — & 6.17 & — & 5.03 & — & 6.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparative analysis of Model C. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Category**} & \multicolumn{2}{c}{**EI Centro Wave**} & \multicolumn{2}{c}{**Tianjin Wave**} & \multicolumn{2}{c}{**Chichi Wave**} \\ \cline{2-7}  & **VT** & **HT** & **VT** & **HT** & **VT** & **HT** \\ \hline Peek displacement (mm) & 1.7662 & 1.6725 & 1.0725 & 1.0148 & 8.0569 & 7.7329 \\ Relative error (\%) & — & 5.31 & — & 5.38 & — & 4.02 \\ Peek velocity (mm/s) & 20.9349 & 19.9960 & 14.4211 & 13.9641 & 93.1646 & 94.2103 \\ Relative error (\%) & — & 4.48 & — & 3.17 & — & 1.12 \\ Peek acceleration (mm/s\({}^{2}\)) & 313.6426 & 331.0309 & 184.3038 & 180.2010 & 1622.9458 & 1606.6127 \\ Relative error (\%) & — & 5.54 & — & 2.23 & — & 1.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparative analysis of Model B. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Category**} & \multicolumn{2}{c}{**EI Centro wave**} & \multicolumn{2}{c}{**Tianjin wave**} & \multicolumn{2}{c}{**Chichi wave**} \\ \cline{2-7}  & **VT** & **HT** & **VT** & **HT** & **VT** & **HT** \\ \hline Peek displacement (mm) & 1.2325 & 1.0669 & 0.8261 & 0.6449 & 6.0843 & 4.7873 \\ Relative error (\%) & — & 13.44 & — & 21.93 & — & 21.32 \\ Peak velocity (mm/s) & 13.9592 & 12.8003 & 8.9614 & 7.6885 & 64.2697 & 61.1640 \\ Relative error (\%) & — & 8.30 & — & 14.20 & — & 4.83 \\ Peek acceleration (mm/s\({}^{2}\)) & 206.7430 & 220.9314 & 120.8636 & 131.8000 & 1292.1100 & 1425.8649 \\ Relative error (\%) & — & 6.86 & — & 9.05 & — & 10.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparative analysis of Model D.  
**Author Contributions:** Conceptualization, Q.L. and P.S. ; methodology, Q.L. and P.S. ; software, Y.W. ; data curation, Y.W. ; formal analysis, P.S. ; writing--original draft, Q.L.