# Displacement-Dependent Damping Inerter System for Seismic Response Control
## 4 Non-Linear Time-History Analysis and Performance of DDIS  
### Displacement Mitigation Effect in Early Stage of Seismic Response  
As mentioned above, the classic inerter system utilizes a VDE for energy dissipation. The damping reaction force of the system is proportional to the relative velocity of the damping element; furthermore, there is an inevitable time difference between the times when maximum damping force and maximum displacement are generated. The DDE of the proposed DDIS avoids this problem, as the maximum displacement and maximum damping force occur simultaneously. When an earthquake occurs, the deformation of the DDE induced by the structural vibration directly causes the DDIS to produce a timely reaction force for effective vibration control. This results in significant reduction of the dynamic responses of the structure during the early stage of the seismic response.  
To characterize the displacement mitigation effect of the DDIS, its results are compared with those of a conventional VDIS when both were applied to the single-span, single-floor frame structure described in Section 3.2. The viscous damping ratio is considered to be equal to the equivalent damping ratio of the DDE [42; 47] to achieve the maximum hysteresis loop area of VDE under the dynamic excitation being equal to that of DDE. The stiffness of the support (\(\lambda\)) is set to infinity to ignore its tuning effect and only compare the VDIS and DDIS in terms of the structural displacement mitigation difference induced by the difference in the damping element. The design parameters of the DDIS \(\mu\) and \(\kappa\) are both 0.50, a value that is chosen arbitrarily from the parametric analysis results. The characteristics of the DDIS and the benefits produced by the inerter also hold true for the DDIS with other parameter values. Regarding the hysteresis of the DDIS, \(\alpha\) and \(u_{dy}\) were assumed to be 0.02 and 0.001, respectively. The main conclusions drawn from the test results, as discussed below, are true for other parameter combinations (\(\mu\), \(\kappa\), and \(\lambda\)). The Kanai-Tajimi spectrum is adopted as the input power spectrum, with the predominant frequency of the dynamic excitation, \(\omega_{g}\), set to 0.5\(\omega_{s0}\), \(\omega_{s0}\), and 2.0\(\omega_{s0}\) (\(\omega_{s0}\) denoting the circular frequency of the primary structure) to simulate low-frequency seismic (LS) excitation, the severest seismic (SS) excitation (resonance condition), and high-frequency seismic (HS) excitation, respectively, and artificial records are generated accordingly. The well-studied ground motion record of the 1940 Imperial Valley Earthquake recorded at El Centro (N-S component) is also used as a representative natural excitation. Under the four excitations, the equivalent damping ratio \(\xi_{VDIS}=c_{d}/2\sqrt{mk}\) (\(c_{d}\) is the damping coefficient of the VDIS) is designed as 0.025, 0.032, 0.039, and 0.025. The displacement responses of the structures with the DDIS and VDIS under different excitations are shown in Figure 13. The figure shows that for all the excitations, the roof displacement is reduced more effectively by the DDIS compared with the VDIS despite the same equivalent damping ratio. The roof displacements of the structures with the DDIS and VDIS are substantially suppressed after \(t_{DDIS}\) and \(t_{VDIS}\), respectively, while negligible reduction is achieved before \(t_{DDIS}\) and \(t_{VDIS}\). Furthermore, \(t_{DDIS}\) is definitely lower than \(t_{VDIS}\), demonstrating the timely control advantage of the DDIS over the VDIS in the early stage of the seismic response. As shown in the hysteretic curves of the damping element in DDIS and VDIS (Figure 14), at the beginning of the excitation, the energy dissipated by DDIS (dotted line: 1-2-3-4-5-6) is definitely larger than that in the case of VDIS (solid line: 1'-2'-3'-4'). The larger damping force produced by DDIS in the early stage is beneficial for the roof displacement reduction in a timely and more effective manner.