# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 3 Viscous Inerter Damper
### Optimal Damping  
When the optimal front-end inertance is determined by the iterative procedure in Table 1, the optimal viscous coefficient \(c\) is chosen to optimize the attainable damping. In theory, the maximum damping is obtained exactly at the bifurcation point. However, as demonstrated in [9; 18; 43], the bifurcation point damper tuning results in interaction between the two modes associated with the first vibration form, which yields undesirable  
\begin{table}
\begin{tabular}{l l} \hline (1) & Choose \(m\) and determine \(\mu=m/m_{1}\) \\ (2) & Solve (9) and determined \(\hat{\rho}_{1}^{2}\) and \(\hat{\rho}_{2}^{2}\) from (31) \\ (3) & Determine \(\kappa_{1}^{\prime}\) from (34) and \(\mu_{1}^{\prime}\) from (35) \\ (4) & Update \(\mu\) from (38) and \(m=\mu m_{1}\) \\ (5) & Repeat (2) to (4) until \(\mu\) has converged \\ \hline \end{tabular}
\end{table}
Table 1: Tuning procedure for optimal front-end inertance \(m\).  
response amplification in FRF curves. Thus, for the present damper, the ratio \(\beta_{bif}\) at the bifurcation point is multiplied by \(\sqrt{2}\) to achieve a flat plateau in the frequency response curves. The FRF curves for optimal and bifurcation point damper ratios are compared in the numerical example in Section 4.5.  
For the optimal mass ratio tuning \(\beta_{1}/\kappa_{1}^{\prime}=1\) in (37), the characteristic Equation (30) is readily reduced to  
\[\rho^{4}-\rho^{2}(2+\bar{\mu}_{s})+1+i\rho\,\frac{\bar{\mu}_{s}}{\beta}\,(- \rho^{2}+1)=0 \tag{39}\]  
This characteristic equation secures that the two roots are inverse points to unity when \(\beta\to\infty\), i.e., at the clamped damper limit. When reducing \(\beta\), the complex poles move along semi-circular paths, still as inverse points to the unit circle. At the particular value \(\beta=\beta_{bif}\), the two roots meet at a bifurcation point, after which they split and (approximately) trace the unit circle, with the pole approaching the real axis being the less damped and thus the pole governing the structural dynamics. When locating the poles with sufficient distance to the bifurcation point to avoid the dynamic interference between the two modes, the desired characteristic equation can be written as  
\[\rho^{4}-\rho^{2}\Big{(}2+8\xi_{*}^{2}\Big{)}+1+i\rho(4\xi_{*})(-\rho^{2}+1)=0 \tag{40}\]  
as derived in [9; 18; 43]. In this equation, \(\zeta_{*}\) represents the attainable damping ratio introduced by the inerter damper to the target mode, which in this case is the fundamental mode \(s=1\).  
The optimal \(\beta=\beta_{opt}\) and the corresponding attainable damping ratio \(\zeta_{*}\) are determined by one-to-one comparison between the actual characteristic Equation (39) for the present viscous inerter damper and the desired characteristic Equation (40). It follows directly from a comparison of the quadratic terms that the attainable damping ratio is  
\[\zeta_{*}=\sqrt{\tfrac{1}{8}\bar{\mu}_{1}} \tag{41}\]  
which indicates that the damping ratio scales with the square root of the combined mass ratio \(\bar{\mu}_{1}\), in practice evaluated by (33). Subsequently, the optimal damper ratio \(\beta_{opt}\) is derived by comparison of the factors to the \((-\rho^{2}+1)\) parenthesis. This gives the optimal damper ratio  
\[\beta_{opt}=\sqrt{\tfrac{1}{2}\bar{\mu}_{1}}=2\zeta_{*} \tag{42}\]  
as twice the attainable damping ratio. The tuning expressions (41) and (42) canonize the combined mass ratio \(\bar{\mu}_{1}\) as the overall governing damper parameter that determines both tuning and performance.