# Displacement-Dependent Damping Inerter System for Seismic Response Control
## 3 Characteristics of DDIS
### Parametric Analysis of DDIS
#### 3.2.2 Influence of DDIS Parameters  
In this section, several cases are considered to demonstrate the vibration mitigation effect of the DDIS, with \(\kappa\) varying in the range of \(10^{-2}\)-1.00, and \(\mu\) varying in the range of \(10^{-3}\)-3.00. \(\lambda\) is set to 0.20, 1.00, and 1000.00 to simulate low, medium, and high stiffness of the support (i.e., the spring), respectively. A broad-band stochastic excitation is adopted, with \(\omega_{g}=15\), and \(\xi_{g}=0.60\)[57; 63] to simulate the commonly used firm soil condition. The analysis results of \(\gamma_{Dis}\gamma_{Acc}\), and \(\gamma_{E}\) are summarized in Figures 6-8, respectively. As expected, the displacement performance of the structure-DDIS system, \(\gamma_{Dis}\), gradually decreases with the increase in hysteretic stiffness ratio \(\kappa\), especially for medium and rigid stiffness of the support. Similar to a conventional displacement-dependent damper, an increase in \(\kappa\) implies greater stiffness of the primary structure to resist vibration, and hence, decreased structural displacement through increased non-linear damping. As can be observed from the surf plots in Figure 6, a decrease in \(\mu\) is accompanied by an increase in \(\gamma_{Dis}\). Considering the specific case in which \(\mu\) decreases to zero, the DDIS degenerates into a conventional displacement-dependent damper with a stiff support. The proposed DDIS thus provides much greater displacement mitigation through the tuning effect of the spring and the energy storage effect of the inerter. Regarding the acceleration mitigation ratio \(\gamma_{Acc}\), Figure 7 shows that, for the DDIS with a tunable spring (Figure 7a,b), a medium inertance (\(\mu\approx 0.50\)) is beneficial to the mitigation of the acceleration  
Figure 8: Variation of \(\gamma_{E}\) for: (**a**) \(\lambda=0.2\), (**b**) \(\lambda=1.00\), and (**c**) \(\lambda=1000\). response, while a small \(\kappa\) (\(<0.05\)) and large inertance (\(\mu>1.0\)) have a detrimental effect. This can be attributed to a weak DDE being incapable of timely dissipation of the large amount of energy stored by a large inerter, with the energy filtering into the primary structure. Figure 7a shows that when the DDE and inerter are set in parallel with a low-stiffness spring, the supplemental inerter substantially decreases the structural acceleration effectively. For the DDIS without a tunable spring (Figure 7c), the significant reduction of structural acceleration (small \(\gamma_{Acc}\)) is reached at a high price of large inertance (\(\mu>1.0\)). With regard to \(\gamma_{E}\), Figure 8 shows that an increase in \(\kappa\) (\(\kappa\in[0.01,\ 1.0]\)) causes a decrease in \(\gamma_{E}\), implying greater dissipation of vibrational energy by a large DDE (\(\kappa>0.50\)), with less energy filtering into the primary structure. Regarding the effect of the added inerter, an increase in inertance decreases \(\gamma_{E}\), especially when the DDIS springs are tunable stiffness (\(\lambda<1.0\)). From the perspective of energy dissipation, the application of the inerter enhances the energy dissipation and vibration mitigation effect of the DDIS compared with only a stiff-supported DDE identical to that used in the system. For the DDIS without a tunable spring (Figure 8c), this benefit of the inerter is not evident. The damping enhancement effect of the proposed DDIS may be attributed to its combination mechanism in which the spring is used to tune the frequency of the inerter to improve energy absorption. This fundamental mechanism of the DDIS provides an alternative explanation of its advantage over a DDE identical to that utilized in the system. An additional parametric analysis is conducted to investigate the effect of the variation of the mechanical parameters of the DDIS on its performance (evaluated with respect to \(\kappa\) and \(\lambda\)). The corresponding surfs plots are shown in Figures 9 and 10, which consider wide ranges of \(\kappa\) and \(\lambda\) values.