# Comparative Analysis of Viscous Damping Model and Hysteretic Damping Model
## 4 Numerical Examples  
The numerical examples for free vibration responses were analyzed. Then natural frequency of the numerical model is 14 rad/s. The loss factors of some viscoelastic materials are greater than 0.10 [5], and the equivalent loss factor of some isolation dampers can reach 0.40 [25]. So, the material damping ratios were selected as 0.10, 020, 0.30 and 0.40, respectively. These numerical models are defined as Model A, Model B, Model C and Model D. El Centro wave, Tianjin wave and Chi-Chi wave are used to calculate the dynamic responses. The dominant frequencies and peak ground accelerations of the three earthquake waves are different. The corresponding information is shown in Table 1. The ratios of dominant frequency and natural frequency are 1.06, 0.50 and 0.30, respectively. Different ratios can represent different frequency bands of earthquake waves. The law of the calculation results is general. The time-domain displacements of Model A, Model B, Model C and Model D can be calculated by the numerical calculation methods of viscous damping model (VT) and hysteretic damping model (HT), respectively. The corresponding calculation results are shown in Figures 6-8. In order to ensure the calculation accuracy and convergence, the time-domain methods based on the viscous damping model and hysteretic damping model are precise calculation methods [22]. Earthquake waves can be decomposed into a series of harmonic waves in the methods. Figures 6-8 show that with the increase in damping ratio, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Earthquake Name** & **Time (s)** & **Peak Ground Acceleration (g)** & **Dominant Frequency (Rad/s)** \\ \hline El Centro & 22 & 0.0051 & 14.8796 \\ Tianjin & 20 & 0.0102 & 7.0563 \\ Chi-Chi & 20 & 0.0204 & 4.1417 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Information of earthquake waves. Figure 5: The change curve of vibration amplitude ratios with different damping ratios. The comparative analyses of peak displacements, peak velocities and peak accelerations based on the viscous damping model and hysteretic damping model are shown in Tables 2-5. With the increase in damping ratio, the difference among the peak displacements, peak velocities and peak accelerations calculated based on viscous damping model and hysteretic damping model gradually increases. In the case of small damping, the dynamic responses calculated by the two damping models are approximately equal. In the case of large damping, the dynamic responses calculated by the two damping models are different and the difference cannot be ignored. Specially, when the damping ratio is 0.30, the relative error of peak displacements based on viscous damping model and hysteretic damping model is greater than 10%. When the damping ratio is 0.40, the relative error of peak velocities and peak accelerations based on viscous damping model and hysteretic damping model is greater than 10%. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Category**} & \multicolumn{2}{c}{**El 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) & 2.3749 & 2.3368 & 1.3369 & 1.3124 & 10.6377 & 10.7779 \\ Relative error (\%) & — & 1.60 & — & 1.83 & — & 1.32 \\ Peek velocity (mm/s) & 28.1130 & 27.3384 & 19.1281 & 19.1565 & 133.0985 & 133.3709 \\ Relative error (\%) & — & 2.76 & — & 0.15 & — & 0.20 \\ Peek acceleration & 457.7155 & 449.7389 & 261.4425 & 261.8755 & 2197.0835 & 2144.0104 \\ (mm/s\({}^{2}\)) & & & & & & \\ Relative error (\%) & — & 1.74 & — & 0.17 & — & 2.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparative analysis of Model A.  
Figure 8: The time-domain displacements of different models due to Chi-Chi wave: (**a**) Model A; (**b**) Model B; (**c**) Model C; (**d**) Model D.  
Based on the comparative analysis of the transient and steady responses in Sections 2 and 3.