# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 4 Damping of the Building Model
### Tuning Parameters  
The first overall task in the damper tuning is to determine the front-end damper inertance \(m\) using the iterative procedure in Table 1. In the present case, a first guess is chosen as \(m=0.05m_{1}\), i.e., 5% mode 1 modal mass. The iterative procedure is run for 10 iterations, with a relative change of less than \(10^{-12}\) for \(\tilde{\mu}_{1}=0.1051\) in the final iteration. The final inertance is then obtained as \(m=919.4\times 10^{6}\,\mathrm{kg}\), which results in the governing tuning parameters in Table 3.  
For the optimal front-end inertance \(m\), the clamped natural frequencies \(\hat{\omega}_{j}\) are obtained by solving the generalized eigenvalue problem (9). These frequencies are presented as \(\hat{\rho}_{j}=\hat{\omega}_{j}/\omega_{1}\) for target mode \(s=1\) in the bottom half of Table 2. It is seen that \(\hat{\rho}_{1}\hat{\rho}_{2}=1\)  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(j\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \(\hat{\rho}_{j}\) & 1.00 & 2.99 & 4.97 & 6.92 & 8.82 & 10.68 & 12.47 & 14.19 & 15.83 & 17.37 \\ \(m_{j}/m_{1}\) & 1.000 & 0.113 & 0.042 & 0.022 & 0.015 & 0.011 & 0.008 & 0.007 & 0.006 & 0.006 \\ \hline \(\hat{\rho}_{j}\) & 0.853 & 1.17 & 3.16 & 5.23 & 7.27 & 9.26 & 11.2 & 13.1 & 14.8 & 16.5 \\ \(\hat{m}_{j}/m_{1}\) & 0.185 & 0.259 & 9.37 & 29.6 & 61.6 & 106.87 & 168.35 & 250.1 & 358.3 & 502.1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Normalized natural frequencies and modal masses for free (top row) and clamped (bottom row) conditions. Frequencies are normalized by the free mode 1 natural frequency \(\omega_{1}=2\pi\times 0.14\,\mathrm{Hz}=0.88\,\mathrm{rad/s}\), and modal masses are normalized by the free mode 1 modal mass \(m_{1}=8746\times 10^{6}\,\mathrm{kg}\). The front-end inertance \(m=919.4\times 10^{6}\,\mathrm{kg}\) is obtained by the design procedure in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mu\) & \(\tilde{\mu}_{1}\) & \(\mu_{1}^{\prime}\) & \(\kappa_{1}^{\prime}\) & \(\beta_{opt}\) \\ \hline
0.1051 & 0.1025 & 4.1440 & 0.1025 & 0.2264 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Output from tuning procedure in Table 1.  
as required by the inverse point relation in (37a) and enforced by step (4) of the iterative tuning procedure in Table 1.  
It is seen in Table 3 that the effective mass ratio \(\bar{\mu}_{1}=0.1025\) is approximately equal to the actual mass ratio \(\mu=0.1050\), which indicates that the correction by \(\mu_{1}^{\prime}=4.1440\) in (23) is seemingly small. Thus, when using the common mass ratio in the damper tuning, i.e., \(\bar{\mu}_{1}=\mu\), only moderate detuning is expected. In Table 3, the correction stiffness ratio \(\kappa_{1}^{\prime}=\bar{\mu}_{1}\), as required by (37), while the optimal damper ratio \(\beta=\beta_{opt}\) from (42) secures a flat plateau in the frequency response curves. It is noted that the attainable damping is half of the damper ratio, which for the present case yields \(\zeta_{*}=\frac{1}{2}\beta=0.113\) for damping of the first vibration mode (\(s=1\)).