# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 4 Damping of the Building Model
### Frequency Response Analysis  
The present resonant tuning of the viscous inerter damper hinges on the creation of a bifurcation point in the complex root diagram; see Figures 8-10 in the previous section. Thus, the initial validation in Section 4.4 investigates the root loci for the target mode 1 for the pure viscous inerter damper (\(k_{0}=0\)) and for sufficiently small values of the restoring spring's stiffness \(k_{0}\). However, in structural dynamics, the tuning of resonant dampers is most commonly assessed by the reduction in frequency amplitudes, i.e., by the inspection of the frequency response curves (or FRFs). The ideal frequency response curves for the present resonant damping device are obtained from the sdof FRF in (21), in which the optimal damper ratio \(\beta=\beta_{opt}\) is contained in \(\eta=i\rho\beta\). These sdof FRF curves are shown in Figure 11. From the frequency displacement curve in Figure 11a it is seen that the equal damping calibration used in Section 3.3 to determine the optimal mass ratio will not provide a flat plateau in the FRF for the displacement variable. Instead, the curve is slightly inclined, with a larger low-frequency peak. However, the equal damping calibration yields a flat plateau for the structure velocity amplitude in Figure 11a. Although not shown, the acceleration amplitude will then have an opposite inclination to that for the displacement in (a), whereby the present tuning (with a flat velocity amplitude curve) might be a decent compromise between minimizing displacement to optimize load-carrying capacity and minimizing acceleration to optimize comfort. Figure 12 shows the frequency response curves for the full 20-story building model with the viscous inerter damper placed between the ground and first floor, as shown in Figure 1. The curves are obtained by solving the frequency equation of motion in (45) with evenly distributed forcing by  
\[\mathbf{f}_{ext}=f_{ext}[\,1,\,1,\ldots,\,1\,]^{T}\]  
where the scalar \(f_{ext}\) denotes the loading intensity. Since the response curves in Figure 12 are normalized by the corresponding static deflection \(\mathbf{q}_{0}=\mathbf{K}^{-1}\mathbf{f}_{ext}\), the actual value of \(f_{ext}\) is irrelevant in the present linear analysis. Figure 12a shows the top floor displacement amplitude, while (b) shows the corresponding velocity amplitude. The two bottom subfigures (c) and (d) show the displacement and velocity amplitudes for the damper stroke \(v=u-y\) across the damper dashpot. The blue curves are for vanishing damper stiffness (\(k_{0}=0\)), while the red curves represent  
Figure 11: Frequency response functions for idealized sdof model from (21): (**a**) modal displacement amplitude and (**b**) modal velocity amplitude. Figure 12: Frequency response functions for mode 1: (**a,b**) Top-floor motion \(q_{top}\), (**c,d**) damper motion \(v\), (**a,c**) displacement amplitude, and (**b,d**) velocity amplitude. \(k_{0}=0.1\,\omega_{1}^{2}m\) (red), as in the previous root locus diagrams. The solid-line curves represent the optimal damper parameter \(\beta=\beta_{opt}\) from (42), while the dashed curves are obtained for the bifurcation-point damping at \(\beta=\beta_{opt}/\sqrt{2}\). The response amplification associated with the dashed curves (for maximum damping) explains why a slightly increased damper ratio \(\beta\) (compared to \(\beta_{bif}\)) is typically chosen in a damper tuning that aims at minimizing response amplitudes. It is seen by comparing the blue-solid top-row curves in Figure 12a,b with the ideal sdorf-curves in Figure 11 that the curves for the full 20-story model are practically identical to the idealized curves, with a slight inclination for the displacement amplitude and an almost perfectly flat curve around resonance for the velocity amplitude. This comparison shows that the present tuning procedure, relying on the iterative scheme in Table 1, is very accurate for the present case with the inerter damper placed between ground and the first floor of the 20-story building model in Figure 1.