# Dynamics Analysis of a Variable Stiffness Tuned Mass Damper Enhanced by an Inerter
## 4 Results and Discussion
### Parameter Analysis  
The RMS of \(u_{1}\) for different values of the negative stiffness ratio \(\beta_{4}\) are shown in Figure 4a. The displacement excitation amplitude \(f_{\rm e}\) = 0.3 mm. The response amplitude decreases when the negative stiffness ratio \(\beta_{4}\) increases from 0.0465 to 0.1862. When the negative stiffness ratio \(\beta_{4}\) = 0.4655, the primary structure exhibits an unstable branch. As the negative stiffness ratio \(\beta_{4}\) increases, the dynamic behavior of the primary structure becomes more complex, and the range of unstable band widens. It can be seen that the response amplitude significantly decreases with the increasing VS-TMD-I negative stiffness ratio \(\beta_{4}\). Figure 4b shows the influence of the VS-TMD-I for the RMS of \(u_{2}\) with the different negative stiffness ratio \(\beta_{4}\). When the negative stiffness ratio \(\beta_{4}\) = 0.4655, the primary structure exhibits an unstable branch that bends to the upper left. The negative stiffness ratio \(\beta_{4}\) increases to 2.3274, the range of unstable band widens, and the unstable branch bends to the upper right.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Excitation**} & \multirow{2}{*}{**Parameters**} & \multicolumn{2}{c}{**Values**} \\
**Amplitudes** & & & & \\ \hline \multirow{6}{*}{\(f_{\rm e}\) = 0.2 mm} & & VS-TMD & VS-TMD-I & VS-TMD \\  & Mass ratio & \(\lambda_{\rm M}\) = 0.001 & \(\lambda_{\rm I}\) = 0.001 & \(\lambda_{\rm M}\) = 0.1 \\  & Inertance ratio & \(\lambda_{\rm m}\) = 0.001 & \(\lambda_{\rm b}\) = 0.1 & \(\lambda_{\rm m}\) = 0.1 \\ \hline \multirow{2}{*}{\(f_{\rm e}\) = 0.2 mm} & RMS of \(u_{1}\) & 1.51 & 0.28 & 0.283 \\  & Decreased (\%) & \(-\)0.7 & 81.33 & 81.13 \\ \multirow{2}{*}{\(f_{\rm e}\) = 0.65 mm} & RMS of \(u_{1}\) & 4.91 & 2.15 & 2.58 \\  & Decreased (\%) & \(-\)0.2 & 56.22 & 47.34 \\ \multirow{2}{*}{\(f_{\rm e}\) = 1 mm} & RMS of \(u_{1}\) & 7.55 & 4.15 & 5.07 \\  & Decreased (\%) & \(-\)0.7 & 44.67 & 32.4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Dimensionless parameters of the primary structure and the VS-TMD-I.  
Figure 3: Vibration reduction performance with different mass ratios and inertance ratios under different displacement excitations: (**a**)\(f_{\rm e}\) = 0.2 mm; (**b**)\(f_{\rm e}\) = 0.65 mm; (**c**)\(f_{\rm e}\) = 1 mm.  
Figure 5a illustrates the effect of the negative stiffness ratio \(\beta_{5}\) of the VS-TMD-I for the primary structure. It can be shown that the response amplitude significantly decreases with the increasing VS-TMD-I negative stiffness ratio \(\beta_{5}\). The response amplitude decreases and the primary structure is not unstable when the negative stiffness ratio \(\beta_{5}\) increases from 0.448 to 0.896. It can be observed that when the negative stiffness ratio \(\beta_{5}=2.241\), the primary structure exhibits an unstable branch that bends to the lower right. The range of the unstable band of the primary structure widens as the negative stiffness ratio \(\beta_{5}\) increases. Figure 5b shows the influence of the VS-TMD-I for the primary structure with the different inertance ratio \(\lambda_{3}\). The response amplitude of \(u_{1}\) decreases and the primary structure is not unstable when the inertance ratio \(\lambda_{3}\) decreases from 25.21 to 20.35. When the inertance ratio \(\lambda_{3}=16.67\), the primary structure exhibits an unstable branch. The primary structure exhibits an unstable branch that bends to the upper left when the inertance ratio \(\lambda_{3}\) decreases to 4.55. It can be seen that the resonant frequency of the primary structure is shifted to the left with the decreasing inertance ratio \(\lambda_{3}\). The response amplitude decreases with the decreasing VS-TMD-I inertance ratio \(\lambda_{3}\).