## 5 Comparisons of the Control Performances
### The Comparison with Other DVAs under Random Excitation
The results show that, under the same initial parameters, the presented Maxwell DVA has the minimum mean square response, which shows that it has good vibration absorption performance under random excitation. Moreover, the proposed viscoelastic Maxwell DVA with two positive grounded stiffness shows better performance. To simulate engineering reality, a random force excitation with zero mean value and unit variance is constructed in the 50 s, and its time history is shown in \[Y_{V}= v^{4}(1+\mu)^{2}+v^{2}\big{[}-2-\mu+4\xi^{2}(1+\mu)\big{]}+1\]8 The mass and stiffness of the primary system is selected as \(m_{1}=1\,\mathrm{kg},k_{1}=100\,\mathrm{N/m}\), and the mass of DVA is \(m_{2}=0.1\,\mathrm{kg}\). Then, the corresponding time history of different DVAs attached to primary system are shown in \[Y_{V}= v^{4}(1+\mu)^{2}+v^{2}\big{[}-2-\mu+4\xi^{2}(1+\mu)\big{]}+1\]9 Table 4 summarizes the variances and decrease ratios of the the displacements of the primary system. From Figure 15 and Table 4, it can be seen that, when the primary system is subjected to random excitation, the proposed DVA, under two sets of optimal parameters, has lower amplitude than classical DVAs. Compared with the IN-Maxwell-type DVA, it is found that under the same initial conditions \(\mu=0.1\), \(\beta=0.3\), the proposed model for case 2c has better control performance under random excitation, which indicates that the lever element plays a role in vibration absorption. Compared with the L-grounded-type DVA, the model proposed in this paper for case 1c has a better control performance under the same initial conditions \(\mu=0.1,L=3.5\), which indicates that the inerter plays a role in vibration absorption. Furthermore, when \(\mu L^{2}\) is in the range of 1 to 4, the proposed DVA has satisfactory reduction vibration effect, and the DVA has two positive grounded stiffness. Although the performance of the DVA, with negative grounded stiffness in this paper, may be worse than that of the DVA with positive grounded stiffness, sometimes it may need to be selected according to the actual engineering situation. The model and two groups of optimization formulas proposed in this paper can be alternately implemented in building protection and bridge seismic resistance. \begin{table}
\begin{tabular}{c c c} \hline
**Model of DVA** & **Variances** & **Decrease Ratios (\%)** \\ \hline Without DVA & \(2.8464\times 10^{-4}\) & \\ Voigt-type DVA & \(3.7631\times 10^{-5}\) & 86.78 \\ Grounded-type DVA & \(3.0478\times 10^{-5}\) & 89.29 \\ IN-Maxwell-type DVA & \(2.7244\times 10^{-5}\) & 90.43 \\ L-grounded-type DVA & \(2.2196\times 10^{-6}\) & 99.22 \\ DVA of case 1c & \(7.5018\times 10^{-7}\) & 99.74 \\ DVA of case 2c & \(1.0018\times 10^{-5}\) & 96.48 \\ \hline \end{tabular}
\end{table} 
Table 4: The variances and decrease ratios of the displacements of the primary system. Figure 14: Random force excitation. Figure 15: Comparison of the time histories of the primary system with different DVAs: (**a**) Comparison of classical DVAs with our results. (**b**) Comparison of L-grounded-type DVA with case 1c. (**c**) Comparison of IN-Maxwell-type DVA with case 2c. (**d**) Comparison of case 1c with case 2c.